1 \input texinfo @c -*-texinfo-*-
2 @comment %**start of header (This is for running Texinfo on a region.)
4 @setfilename ../../info/calc
6 @settitle GNU Emacs Calc Manual
8 @comment %**end of header (This is for running Texinfo on a region.)
10 @c The following macros are used for conditional output for single lines.
12 @c `foo' will appear only in TeX output
14 @c `foo' will appear only in non-TeX output
16 @c @expr{expr} will typeset an expression;
17 @c $x$ in TeX, @samp{x} otherwise.
22 @alias infoline=comment
36 @alias texline=comment
37 @macro infoline{stuff}
56 % Suggested by Karl Berry <karl@@freefriends.org>
57 \gdef\!{\mskip-\thinmuskip}
60 @c Fix some other things specifically for this manual.
63 @mathcode`@:=`@: @c Make Calc fractions come out right in math mode
65 \gdef\coloneq{\mathrel{\mathord:\mathord=}}
67 \gdef\beforedisplay{\vskip-10pt}
68 \gdef\afterdisplay{\vskip-5pt}
69 \gdef\beforedisplayh{\vskip-25pt}
70 \gdef\afterdisplayh{\vskip-10pt}
72 @newdimen@kyvpos @kyvpos=0pt
73 @newdimen@kyhpos @kyhpos=0pt
74 @newcount@calcclubpenalty @calcclubpenalty=1000
77 @newtoks@calcoldeverypar @calcoldeverypar=@everypar
78 @everypar={@calceverypar@the@calcoldeverypar}
79 @ifx@turnoffactive@undefinedzzz@def@turnoffactive{}@fi
80 @ifx@ninett@undefinedzzz@font@ninett=cmtt9@fi
81 @catcode`@\=0 \catcode`\@=11
83 \catcode`\@=0 @catcode`@\=@active
89 This file documents Calc, the GNU Emacs calculator.
92 This file documents Calc, the GNU Emacs calculator, included with GNU Emacs 23.1.
95 Copyright @copyright{} 1990, 1991, 2001, 2002, 2003, 2004,
96 2005, 2006, 2007, 2008, 2009 Free Software Foundation, Inc.
99 Permission is granted to copy, distribute and/or modify this document
100 under the terms of the GNU Free Documentation License, Version 1.3 or
101 any later version published by the Free Software Foundation; with the
102 Invariant Sections being just ``GNU GENERAL PUBLIC LICENSE'', with the
103 Front-Cover texts being ``A GNU Manual,'' and with the Back-Cover
104 Texts as in (a) below. A copy of the license is included in the section
105 entitled ``GNU Free Documentation License.''
107 (a) The FSF's Back-Cover Text is: ``You have the freedom to copy and
108 modify this GNU manual. Buying copies from the FSF supports it in
109 developing GNU and promoting software freedom.''
115 * Calc: (calc). Advanced desk calculator and mathematical tool.
120 @center @titlefont{Calc Manual}
122 @center GNU Emacs Calc
125 @center Dave Gillespie
126 @center daveg@@synaptics.com
129 @vskip 0pt plus 1filll
130 Copyright @copyright{} 1990, 1991, 2001, 2002, 2003, 2004,
131 2005, 2006, 2007, 2008, 2009 Free Software Foundation, Inc.
144 @node Top, Getting Started, (dir), (dir)
145 @chapter The GNU Emacs Calculator
148 @dfn{Calc} is an advanced desk calculator and mathematical tool
149 written by Dave Gillespie that runs as part of the GNU Emacs environment.
151 This manual, also written (mostly) by Dave Gillespie, is divided into
152 three major parts: ``Getting Started,'' the ``Calc Tutorial,'' and the
153 ``Calc Reference.'' The Tutorial introduces all the major aspects of
154 Calculator use in an easy, hands-on way. The remainder of the manual is
155 a complete reference to the features of the Calculator.
159 For help in the Emacs Info system (which you are using to read this
160 file), type @kbd{?}. (You can also type @kbd{h} to run through a
161 longer Info tutorial.)
165 * Getting Started:: General description and overview.
167 * Interactive Tutorial::
169 * Tutorial:: A step-by-step introduction for beginners.
171 * Introduction:: Introduction to the Calc reference manual.
172 * Data Types:: Types of objects manipulated by Calc.
173 * Stack and Trail:: Manipulating the stack and trail buffers.
174 * Mode Settings:: Adjusting display format and other modes.
175 * Arithmetic:: Basic arithmetic functions.
176 * Scientific Functions:: Transcendentals and other scientific functions.
177 * Matrix Functions:: Operations on vectors and matrices.
178 * Algebra:: Manipulating expressions algebraically.
179 * Units:: Operations on numbers with units.
180 * Store and Recall:: Storing and recalling variables.
181 * Graphics:: Commands for making graphs of data.
182 * Kill and Yank:: Moving data into and out of Calc.
183 * Keypad Mode:: Operating Calc from a keypad.
184 * Embedded Mode:: Working with formulas embedded in a file.
185 * Programming:: Calc as a programmable calculator.
187 * Copying:: How you can copy and share Calc.
188 * GNU Free Documentation License:: The license for this documentation.
189 * Customizing Calc:: Customizing Calc.
190 * Reporting Bugs:: How to report bugs and make suggestions.
192 * Summary:: Summary of Calc commands and functions.
194 * Key Index:: The standard Calc key sequences.
195 * Command Index:: The interactive Calc commands.
196 * Function Index:: Functions (in algebraic formulas).
197 * Concept Index:: General concepts.
198 * Variable Index:: Variables used by Calc (both user and internal).
199 * Lisp Function Index:: Internal Lisp math functions.
203 @node Getting Started, Interactive Tutorial, Top, Top
206 @node Getting Started, Tutorial, Top, Top
208 @chapter Getting Started
210 This chapter provides a general overview of Calc, the GNU Emacs
211 Calculator: What it is, how to start it and how to exit from it,
212 and what are the various ways that it can be used.
216 * About This Manual::
217 * Notations Used in This Manual::
218 * Demonstration of Calc::
220 * History and Acknowledgements::
223 @node What is Calc, About This Manual, Getting Started, Getting Started
224 @section What is Calc?
227 @dfn{Calc} is an advanced calculator and mathematical tool that runs as
228 part of the GNU Emacs environment. Very roughly based on the HP-28/48
229 series of calculators, its many features include:
233 Choice of algebraic or RPN (stack-based) entry of calculations.
236 Arbitrary precision integers and floating-point numbers.
239 Arithmetic on rational numbers, complex numbers (rectangular and polar),
240 error forms with standard deviations, open and closed intervals, vectors
241 and matrices, dates and times, infinities, sets, quantities with units,
242 and algebraic formulas.
245 Mathematical operations such as logarithms and trigonometric functions.
248 Programmer's features (bitwise operations, non-decimal numbers).
251 Financial functions such as future value and internal rate of return.
254 Number theoretical features such as prime factorization and arithmetic
255 modulo @var{m} for any @var{m}.
258 Algebraic manipulation features, including symbolic calculus.
261 Moving data to and from regular editing buffers.
264 Embedded mode for manipulating Calc formulas and data directly
265 inside any editing buffer.
268 Graphics using GNUPLOT, a versatile (and free) plotting program.
271 Easy programming using keyboard macros, algebraic formulas,
272 algebraic rewrite rules, or extended Emacs Lisp.
275 Calc tries to include a little something for everyone; as a result it is
276 large and might be intimidating to the first-time user. If you plan to
277 use Calc only as a traditional desk calculator, all you really need to
278 read is the ``Getting Started'' chapter of this manual and possibly the
279 first few sections of the tutorial. As you become more comfortable with
280 the program you can learn its additional features. Calc does not
281 have the scope and depth of a fully-functional symbolic math package,
282 but Calc has the advantages of convenience, portability, and freedom.
284 @node About This Manual, Notations Used in This Manual, What is Calc, Getting Started
285 @section About This Manual
288 This document serves as a complete description of the GNU Emacs
289 Calculator. It works both as an introduction for novices and as
290 a reference for experienced users. While it helps to have some
291 experience with GNU Emacs in order to get the most out of Calc,
292 this manual ought to be readable even if you don't know or use Emacs
295 This manual is divided into three major parts:@: the ``Getting
296 Started'' chapter you are reading now, the Calc tutorial (chapter 2),
297 and the Calc reference manual (the remaining chapters and appendices).
299 @c This manual has been printed in two volumes, the @dfn{Tutorial} and the
300 @c @dfn{Reference}. Both volumes include a copy of the ``Getting Started''
303 If you are in a hurry to use Calc, there is a brief ``demonstration''
304 below which illustrates the major features of Calc in just a couple of
305 pages. If you don't have time to go through the full tutorial, this
306 will show you everything you need to know to begin.
307 @xref{Demonstration of Calc}.
309 The tutorial chapter walks you through the various parts of Calc
310 with lots of hands-on examples and explanations. If you are new
311 to Calc and you have some time, try going through at least the
312 beginning of the tutorial. The tutorial includes about 70 exercises
313 with answers. These exercises give you some guided practice with
314 Calc, as well as pointing out some interesting and unusual ways
317 The reference section discusses Calc in complete depth. You can read
318 the reference from start to finish if you want to learn every aspect
319 of Calc. Or, you can look in the table of contents or the Concept
320 Index to find the parts of the manual that discuss the things you
323 @c @cindex Marginal notes
324 Every Calc keyboard command is listed in the Calc Summary, and also
325 in the Key Index. Algebraic functions, @kbd{M-x} commands, and
326 variables also have their own indices.
328 @c @infoline In the printed manual, each
329 @c paragraph that is referenced in the Key or Function Index is marked
330 @c in the margin with its index entry.
332 @c [fix-ref Help Commands]
333 You can access this manual on-line at any time within Calc by pressing
334 the @kbd{h i} key sequence. Outside of the Calc window, you can press
335 @kbd{C-x * i} to read the manual on-line. From within Calc the command
336 @kbd{h t} will jump directly to the Tutorial; from outside of Calc the
337 command @kbd{C-x * t} will jump to the Tutorial and start Calc if
338 necessary. Pressing @kbd{h s} or @kbd{C-x * s} will take you directly
339 to the Calc Summary. Within Calc, you can also go to the part of the
340 manual describing any Calc key, function, or variable using
341 @w{@kbd{h k}}, @kbd{h f}, or @kbd{h v}, respectively. @xref{Help Commands}.
344 The Calc manual can be printed, but because the manual is so large, you
345 should only make a printed copy if you really need it. To print the
346 manual, you will need the @TeX{} typesetting program (this is a free
347 program by Donald Knuth at Stanford University) as well as the
348 @file{texindex} program and @file{texinfo.tex} file, both of which can
349 be obtained from the FSF as part of the @code{texinfo} package.
350 To print the Calc manual in one huge tome, you will need the
351 source code to this manual, @file{calc.texi}, available as part of the
352 Emacs source. Once you have this file, type @kbd{texi2dvi calc.texi}.
353 Alternatively, change to the @file{man} subdirectory of the Emacs
354 source distribution, and type @kbd{make calc.dvi}. (Don't worry if you
355 get some ``overfull box'' warnings while @TeX{} runs.)
356 The result will be a device-independent output file called
357 @file{calc.dvi}, which you must print in whatever way is right
358 for your system. On many systems, the command is
371 @c Printed copies of this manual are also available from the Free Software
374 @node Notations Used in This Manual, Demonstration of Calc, About This Manual, Getting Started
375 @section Notations Used in This Manual
378 This section describes the various notations that are used
379 throughout the Calc manual.
381 In keystroke sequences, uppercase letters mean you must hold down
382 the shift key while typing the letter. Keys pressed with Control
383 held down are shown as @kbd{C-x}. Keys pressed with Meta held down
384 are shown as @kbd{M-x}. Other notations are @key{RET} for the
385 Return key, @key{SPC} for the space bar, @key{TAB} for the Tab key,
386 @key{DEL} for the Delete key, and @key{LFD} for the Line-Feed key.
387 The @key{DEL} key is called Backspace on some keyboards, it is
388 whatever key you would use to correct a simple typing error when
389 regularly using Emacs.
391 (If you don't have the @key{LFD} or @key{TAB} keys on your keyboard,
392 the @kbd{C-j} and @kbd{C-i} keys are equivalent to them, respectively.
393 If you don't have a Meta key, look for Alt or Extend Char. You can
394 also press @key{ESC} or @kbd{C-[} first to get the same effect, so
395 that @kbd{M-x}, @kbd{@key{ESC} x}, and @kbd{C-[ x} are all equivalent.)
397 Sometimes the @key{RET} key is not shown when it is ``obvious''
398 that you must press @key{RET} to proceed. For example, the @key{RET}
399 is usually omitted in key sequences like @kbd{M-x calc-keypad @key{RET}}.
401 Commands are generally shown like this: @kbd{p} (@code{calc-precision})
402 or @kbd{C-x * k} (@code{calc-keypad}). This means that the command is
403 normally used by pressing the @kbd{p} key or @kbd{C-x * k} key sequence,
404 but it also has the full-name equivalent shown, e.g., @kbd{M-x calc-precision}.
406 Commands that correspond to functions in algebraic notation
407 are written: @kbd{C} (@code{calc-cos}) [@code{cos}]. This means
408 the @kbd{C} key is equivalent to @kbd{M-x calc-cos}, and that
409 the corresponding function in an algebraic-style formula would
410 be @samp{cos(@var{x})}.
412 A few commands don't have key equivalents: @code{calc-sincos}
415 @node Demonstration of Calc, Using Calc, Notations Used in This Manual, Getting Started
416 @section A Demonstration of Calc
419 @cindex Demonstration of Calc
420 This section will show some typical small problems being solved with
421 Calc. The focus is more on demonstration than explanation, but
422 everything you see here will be covered more thoroughly in the
425 To begin, start Emacs if necessary (usually the command @code{emacs}
426 does this), and type @kbd{C-x * c} to start the
427 Calculator. (You can also use @kbd{M-x calc} if this doesn't work.
428 @xref{Starting Calc}, for various ways of starting the Calculator.)
430 Be sure to type all the sample input exactly, especially noting the
431 difference between lower-case and upper-case letters. Remember,
432 @key{RET}, @key{TAB}, @key{DEL}, and @key{SPC} are the Return, Tab,
433 Delete, and Space keys.
435 @strong{RPN calculation.} In RPN, you type the input number(s) first,
436 then the command to operate on the numbers.
439 Type @kbd{2 @key{RET} 3 + Q} to compute
440 @texline @math{\sqrt{2+3} = 2.2360679775}.
441 @infoline the square root of 2+3, which is 2.2360679775.
444 Type @kbd{P 2 ^} to compute
445 @texline @math{\pi^2 = 9.86960440109}.
446 @infoline the value of `pi' squared, 9.86960440109.
449 Type @key{TAB} to exchange the order of these two results.
452 Type @kbd{- I H S} to subtract these results and compute the Inverse
453 Hyperbolic sine of the difference, 2.72996136574.
456 Type @key{DEL} to erase this result.
458 @strong{Algebraic calculation.} You can also enter calculations using
459 conventional ``algebraic'' notation. To enter an algebraic formula,
460 use the apostrophe key.
463 Type @kbd{' sqrt(2+3) @key{RET}} to compute
464 @texline @math{\sqrt{2+3}}.
465 @infoline the square root of 2+3.
468 Type @kbd{' pi^2 @key{RET}} to enter
469 @texline @math{\pi^2}.
470 @infoline `pi' squared.
471 To evaluate this symbolic formula as a number, type @kbd{=}.
474 Type @kbd{' arcsinh($ - $$) @key{RET}} to subtract the second-most-recent
475 result from the most-recent and compute the Inverse Hyperbolic sine.
477 @strong{Keypad mode.} If you are using the X window system, press
478 @w{@kbd{C-x * k}} to get Keypad mode. (If you don't use X, skip to
482 Click on the @key{2}, @key{ENTER}, @key{3}, @key{+}, and @key{SQRT}
483 ``buttons'' using your left mouse button.
486 Click on @key{PI}, @key{2}, and @tfn{y^x}.
489 Click on @key{INV}, then @key{ENTER} to swap the two results.
492 Click on @key{-}, @key{INV}, @key{HYP}, and @key{SIN}.
495 Click on @key{<-} to erase the result, then click @key{OFF} to turn
496 the Keypad Calculator off.
498 @strong{Grabbing data.} Type @kbd{C-x * x} if necessary to exit Calc.
499 Now select the following numbers as an Emacs region: ``Mark'' the
500 front of the list by typing @kbd{C-@key{SPC}} or @kbd{C-@@} there,
501 then move to the other end of the list. (Either get this list from
502 the on-line copy of this manual, accessed by @w{@kbd{C-x * i}}, or just
503 type these numbers into a scratch file.) Now type @kbd{C-x * g} to
504 ``grab'' these numbers into Calc.
515 The result @samp{[1.23, 1.97, 1.6, 2, 1.19, 1.08]} is a Calc ``vector.''
516 Type @w{@kbd{V R +}} to compute the sum of these numbers.
519 Type @kbd{U} to Undo this command, then type @kbd{V R *} to compute
520 the product of the numbers.
523 You can also grab data as a rectangular matrix. Place the cursor on
524 the upper-leftmost @samp{1} and set the mark, then move to just after
525 the lower-right @samp{8} and press @kbd{C-x * r}.
528 Type @kbd{v t} to transpose this
529 @texline @math{3\times2}
532 @texline @math{2\times3}
534 matrix. Type @w{@kbd{v u}} to unpack the rows into two separate
535 vectors. Now type @w{@kbd{V R + @key{TAB} V R +}} to compute the sums
536 of the two original columns. (There is also a special
537 grab-and-sum-columns command, @kbd{C-x * :}.)
539 @strong{Units conversion.} Units are entered algebraically.
540 Type @w{@kbd{' 43 mi/hr @key{RET}}} to enter the quantity 43 miles-per-hour.
541 Type @w{@kbd{u c km/hr @key{RET}}}. Type @w{@kbd{u c m/s @key{RET}}}.
543 @strong{Date arithmetic.} Type @kbd{t N} to get the current date and
544 time. Type @kbd{90 +} to find the date 90 days from now. Type
545 @kbd{' <25 dec 87> @key{RET}} to enter a date, then @kbd{- 7 /} to see how
546 many weeks have passed since then.
548 @strong{Algebra.} Algebraic entries can also include formulas
549 or equations involving variables. Type @kbd{@w{' [x + y} = a, x y = 1] @key{RET}}
550 to enter a pair of equations involving three variables.
551 (Note the leading apostrophe in this example; also, note that the space
552 in @samp{x y} is required.) Type @w{@kbd{a S x,y @key{RET}}} to solve
553 these equations for the variables @expr{x} and @expr{y}.
556 Type @kbd{d B} to view the solutions in more readable notation.
557 Type @w{@kbd{d C}} to view them in C language notation, @kbd{d T}
558 to view them in the notation for the @TeX{} typesetting system,
559 and @kbd{d L} to view them in the notation for the La@TeX{} typesetting
560 system. Type @kbd{d N} to return to normal notation.
563 Type @kbd{7.5}, then @kbd{s l a @key{RET}} to let @expr{a = 7.5} in these formulas.
564 (That's the letter @kbd{l}, not the numeral @kbd{1}.)
567 @strong{Help functions.} You can read about any command in the on-line
568 manual. Type @kbd{C-x * c} to return to Calc after each of these
569 commands: @kbd{h k t N} to read about the @kbd{t N} command,
570 @kbd{h f sqrt @key{RET}} to read about the @code{sqrt} function, and
571 @kbd{h s} to read the Calc summary.
574 @strong{Help functions.} You can read about any command in the on-line
575 manual. Remember to type the letter @kbd{l}, then @kbd{C-x * c}, to
576 return here after each of these commands: @w{@kbd{h k t N}} to read
577 about the @w{@kbd{t N}} command, @kbd{h f sqrt @key{RET}} to read about the
578 @code{sqrt} function, and @kbd{h s} to read the Calc summary.
581 Press @key{DEL} repeatedly to remove any leftover results from the stack.
582 To exit from Calc, press @kbd{q} or @kbd{C-x * c} again.
584 @node Using Calc, History and Acknowledgements, Demonstration of Calc, Getting Started
588 Calc has several user interfaces that are specialized for
589 different kinds of tasks. As well as Calc's standard interface,
590 there are Quick mode, Keypad mode, and Embedded mode.
594 * The Standard Interface::
595 * Quick Mode Overview::
596 * Keypad Mode Overview::
597 * Standalone Operation::
598 * Embedded Mode Overview::
599 * Other C-x * Commands::
602 @node Starting Calc, The Standard Interface, Using Calc, Using Calc
603 @subsection Starting Calc
606 On most systems, you can type @kbd{C-x *} to start the Calculator.
607 The key sequence @kbd{C-x *} is bound to the command @code{calc-dispatch},
608 which can be rebound if convenient (@pxref{Customizing Calc}).
610 When you press @kbd{C-x *}, Emacs waits for you to press a second key to
611 complete the command. In this case, you will follow @kbd{C-x *} with a
612 letter (upper- or lower-case, it doesn't matter for @kbd{C-x *}) that says
613 which Calc interface you want to use.
615 To get Calc's standard interface, type @kbd{C-x * c}. To get
616 Keypad mode, type @kbd{C-x * k}. Type @kbd{C-x * ?} to get a brief
617 list of the available options, and type a second @kbd{?} to get
620 To ease typing, @kbd{C-x * *} also works to start Calc. It starts the
621 same interface (either @kbd{C-x * c} or @w{@kbd{C-x * k}}) that you last
622 used, selecting the @kbd{C-x * c} interface by default.
624 If @kbd{C-x *} doesn't work for you, you can always type explicit
625 commands like @kbd{M-x calc} (for the standard user interface) or
626 @w{@kbd{M-x calc-keypad}} (for Keypad mode). First type @kbd{M-x}
627 (that's Meta with the letter @kbd{x}), then, at the prompt,
628 type the full command (like @kbd{calc-keypad}) and press Return.
630 The same commands (like @kbd{C-x * c} or @kbd{C-x * *}) that start
631 the Calculator also turn it off if it is already on.
633 @node The Standard Interface, Quick Mode Overview, Starting Calc, Using Calc
634 @subsection The Standard Calc Interface
637 @cindex Standard user interface
638 Calc's standard interface acts like a traditional RPN calculator,
639 operated by the normal Emacs keyboard. When you type @kbd{C-x * c}
640 to start the Calculator, the Emacs screen splits into two windows
641 with the file you were editing on top and Calc on the bottom.
647 --**-Emacs: myfile (Fundamental)----All----------------------
648 --- Emacs Calculator Mode --- |Emacs Calculator Trail
656 --%*-Calc: 12 Deg (Calculator)----All----- --%*- *Calc Trail*
660 In this figure, the mode-line for @file{myfile} has moved up and the
661 ``Calculator'' window has appeared below it. As you can see, Calc
662 actually makes two windows side-by-side. The lefthand one is
663 called the @dfn{stack window} and the righthand one is called the
664 @dfn{trail window.} The stack holds the numbers involved in the
665 calculation you are currently performing. The trail holds a complete
666 record of all calculations you have done. In a desk calculator with
667 a printer, the trail corresponds to the paper tape that records what
670 In this case, the trail shows that four numbers (17.3, 3, 2, and 4)
671 were first entered into the Calculator, then the 2 and 4 were
672 multiplied to get 8, then the 3 and 8 were subtracted to get @mathit{-5}.
673 (The @samp{>} symbol shows that this was the most recent calculation.)
674 The net result is the two numbers 17.3 and @mathit{-5} sitting on the stack.
676 Most Calculator commands deal explicitly with the stack only, but
677 there is a set of commands that allow you to search back through
678 the trail and retrieve any previous result.
680 Calc commands use the digits, letters, and punctuation keys.
681 Shifted (i.e., upper-case) letters are different from lowercase
682 letters. Some letters are @dfn{prefix} keys that begin two-letter
683 commands. For example, @kbd{e} means ``enter exponent'' and shifted
684 @kbd{E} means @expr{e^x}. With the @kbd{d} (``display modes'') prefix
685 the letter ``e'' takes on very different meanings: @kbd{d e} means
686 ``engineering notation'' and @kbd{d E} means ``@dfn{eqn} language mode.''
688 There is nothing stopping you from switching out of the Calc
689 window and back into your editing window, say by using the Emacs
690 @w{@kbd{C-x o}} (@code{other-window}) command. When the cursor is
691 inside a regular window, Emacs acts just like normal. When the
692 cursor is in the Calc stack or trail windows, keys are interpreted
695 When you quit by pressing @kbd{C-x * c} a second time, the Calculator
696 windows go away but the actual Stack and Trail are not gone, just
697 hidden. When you press @kbd{C-x * c} once again you will get the
698 same stack and trail contents you had when you last used the
701 The Calculator does not remember its state between Emacs sessions.
702 Thus if you quit Emacs and start it again, @kbd{C-x * c} will give you
703 a fresh stack and trail. There is a command (@kbd{m m}) that lets
704 you save your favorite mode settings between sessions, though.
705 One of the things it saves is which user interface (standard or
706 Keypad) you last used; otherwise, a freshly started Emacs will
707 always treat @kbd{C-x * *} the same as @kbd{C-x * c}.
709 The @kbd{q} key is another equivalent way to turn the Calculator off.
711 If you type @kbd{C-x * b} first and then @kbd{C-x * c}, you get a
712 full-screen version of Calc (@code{full-calc}) in which the stack and
713 trail windows are still side-by-side but are now as tall as the whole
714 Emacs screen. When you press @kbd{q} or @kbd{C-x * c} again to quit,
715 the file you were editing before reappears. The @kbd{C-x * b} key
716 switches back and forth between ``big'' full-screen mode and the
717 normal partial-screen mode.
719 Finally, @kbd{C-x * o} (@code{calc-other-window}) is like @kbd{C-x * c}
720 except that the Calc window is not selected. The buffer you were
721 editing before remains selected instead. If you are in a Calc window,
722 then @kbd{C-x * o} will switch you out of it, being careful not to
723 switch you to the Calc Trail window. So @kbd{C-x * o} is a handy
724 way to switch out of Calc momentarily to edit your file; you can then
725 type @kbd{C-x * c} to switch back into Calc when you are done.
727 @node Quick Mode Overview, Keypad Mode Overview, The Standard Interface, Using Calc
728 @subsection Quick Mode (Overview)
731 @dfn{Quick mode} is a quick way to use Calc when you don't need the
732 full complexity of the stack and trail. To use it, type @kbd{C-x * q}
733 (@code{quick-calc}) in any regular editing buffer.
735 Quick mode is very simple: It prompts you to type any formula in
736 standard algebraic notation (like @samp{4 - 2/3}) and then displays
737 the result at the bottom of the Emacs screen (@mathit{3.33333333333}
738 in this case). You are then back in the same editing buffer you
739 were in before, ready to continue editing or to type @kbd{C-x * q}
740 again to do another quick calculation. The result of the calculation
741 will also be in the Emacs ``kill ring'' so that a @kbd{C-y} command
742 at this point will yank the result into your editing buffer.
744 Calc mode settings affect Quick mode, too, though you will have to
745 go into regular Calc (with @kbd{C-x * c}) to change the mode settings.
747 @c [fix-ref Quick Calculator mode]
748 @xref{Quick Calculator}, for further information.
750 @node Keypad Mode Overview, Standalone Operation, Quick Mode Overview, Using Calc
751 @subsection Keypad Mode (Overview)
754 @dfn{Keypad mode} is a mouse-based interface to the Calculator.
755 It is designed for use with terminals that support a mouse. If you
756 don't have a mouse, you will have to operate Keypad mode with your
757 arrow keys (which is probably more trouble than it's worth).
759 Type @kbd{C-x * k} to turn Keypad mode on or off. Once again you
760 get two new windows, this time on the righthand side of the screen
761 instead of at the bottom. The upper window is the familiar Calc
762 Stack; the lower window is a picture of a typical calculator keypad.
766 \advance \dimen0 by 24\baselineskip%
767 \ifdim \dimen0>\pagegoal \vfill\eject \fi%
772 |--- Emacs Calculator Mode ---
776 |--%*-Calc: 12 Deg (Calcul
777 |----+----+--Calc---+----+----1
778 |FLR |CEIL|RND |TRNC|CLN2|FLT |
779 |----+----+----+----+----+----|
780 | LN |EXP | |ABS |IDIV|MOD |
781 |----+----+----+----+----+----|
782 |SIN |COS |TAN |SQRT|y^x |1/x |
783 |----+----+----+----+----+----|
784 | ENTER |+/- |EEX |UNDO| <- |
785 |-----+---+-+--+--+-+---++----|
786 | INV | 7 | 8 | 9 | / |
787 |-----+-----+-----+-----+-----|
788 | HYP | 4 | 5 | 6 | * |
789 |-----+-----+-----+-----+-----|
790 |EXEC | 1 | 2 | 3 | - |
791 |-----+-----+-----+-----+-----|
792 | OFF | 0 | . | PI | + |
793 |-----+-----+-----+-----+-----+
797 Keypad mode is much easier for beginners to learn, because there
798 is no need to memorize lots of obscure key sequences. But not all
799 commands in regular Calc are available on the Keypad. You can
800 always switch the cursor into the Calc stack window to use
801 standard Calc commands if you need. Serious Calc users, though,
802 often find they prefer the standard interface over Keypad mode.
804 To operate the Calculator, just click on the ``buttons'' of the
805 keypad using your left mouse button. To enter the two numbers
806 shown here you would click @w{@kbd{1 7 .@: 3 ENTER 5 +/- ENTER}}; to
807 add them together you would then click @kbd{+} (to get 12.3 on
810 If you click the right mouse button, the top three rows of the
811 keypad change to show other sets of commands, such as advanced
812 math functions, vector operations, and operations on binary
815 Because Keypad mode doesn't use the regular keyboard, Calc leaves
816 the cursor in your original editing buffer. You can type in
817 this buffer in the usual way while also clicking on the Calculator
818 keypad. One advantage of Keypad mode is that you don't need an
819 explicit command to switch between editing and calculating.
821 If you press @kbd{C-x * b} first, you get a full-screen Keypad mode
822 (@code{full-calc-keypad}) with three windows: The keypad in the lower
823 left, the stack in the lower right, and the trail on top.
825 @c [fix-ref Keypad Mode]
826 @xref{Keypad Mode}, for further information.
828 @node Standalone Operation, Embedded Mode Overview, Keypad Mode Overview, Using Calc
829 @subsection Standalone Operation
832 @cindex Standalone Operation
833 If you are not in Emacs at the moment but you wish to use Calc,
834 you must start Emacs first. If all you want is to run Calc, you
835 can give the commands:
845 emacs -f full-calc-keypad
849 which run a full-screen Calculator (as if by @kbd{C-x * b C-x * c}) or
850 a full-screen X-based Calculator (as if by @kbd{C-x * b C-x * k}).
851 In standalone operation, quitting the Calculator (by pressing
852 @kbd{q} or clicking on the keypad @key{EXIT} button) quits Emacs
855 @node Embedded Mode Overview, Other C-x * Commands, Standalone Operation, Using Calc
856 @subsection Embedded Mode (Overview)
859 @dfn{Embedded mode} is a way to use Calc directly from inside an
860 editing buffer. Suppose you have a formula written as part of a
874 and you wish to have Calc compute and format the derivative for
875 you and store this derivative in the buffer automatically. To
876 do this with Embedded mode, first copy the formula down to where
877 you want the result to be, leaving a blank line before and after the
892 Now, move the cursor onto this new formula and press @kbd{C-x * e}.
893 Calc will read the formula (using the surrounding blank lines to tell
894 how much text to read), then push this formula (invisibly) onto the Calc
895 stack. The cursor will stay on the formula in the editing buffer, but
896 the line with the formula will now appear as it would on the Calc stack
897 (in this case, it will be left-aligned) and the buffer's mode line will
898 change to look like the Calc mode line (with mode indicators like
899 @samp{12 Deg} and so on). Even though you are still in your editing
900 buffer, the keyboard now acts like the Calc keyboard, and any new result
901 you get is copied from the stack back into the buffer. To take the
902 derivative, you would type @kbd{a d x @key{RET}}.
916 (Note that by default division had lower precedence than multiplication
917 in Calc, so that @samp{1 / ln(x) x} is equivalent to @samp{1 / (ln(x) x)}.)
919 To make this look nicer, you might want to press @kbd{d =} to center
920 the formula, and even @kbd{d B} to use Big display mode.
929 % [calc-mode: justify: center]
930 % [calc-mode: language: big]
938 Calc has added annotations to the file to help it remember the modes
939 that were used for this formula. They are formatted like comments
940 in the @TeX{} typesetting language, just in case you are using @TeX{} or
941 La@TeX{}. (In this example @TeX{} is not being used, so you might want
942 to move these comments up to the top of the file or otherwise put them
945 As an extra flourish, we can add an equation number using a
946 righthand label: Type @kbd{d @} (1) @key{RET}}.
950 % [calc-mode: justify: center]
951 % [calc-mode: language: big]
952 % [calc-mode: right-label: " (1)"]
960 To leave Embedded mode, type @kbd{C-x * e} again. The mode line
961 and keyboard will revert to the way they were before.
963 The related command @kbd{C-x * w} operates on a single word, which
964 generally means a single number, inside text. It searches for an
965 expression which ``looks'' like a number containing the point.
966 Here's an example of its use:
969 A slope of one-third corresponds to an angle of 1 degrees.
972 Place the cursor on the @samp{1}, then type @kbd{C-x * w} to enable
973 Embedded mode on that number. Now type @kbd{3 /} (to get one-third),
974 and @kbd{I T} (the Inverse Tangent converts a slope into an angle),
975 then @w{@kbd{C-x * w}} again to exit Embedded mode.
978 A slope of one-third corresponds to an angle of 18.4349488229 degrees.
981 @c [fix-ref Embedded Mode]
982 @xref{Embedded Mode}, for full details.
984 @node Other C-x * Commands, , Embedded Mode Overview, Using Calc
985 @subsection Other @kbd{C-x *} Commands
988 Two more Calc-related commands are @kbd{C-x * g} and @kbd{C-x * r},
989 which ``grab'' data from a selected region of a buffer into the
990 Calculator. The region is defined in the usual Emacs way, by
991 a ``mark'' placed at one end of the region, and the Emacs
992 cursor or ``point'' placed at the other.
994 The @kbd{C-x * g} command reads the region in the usual left-to-right,
995 top-to-bottom order. The result is packaged into a Calc vector
996 of numbers and placed on the stack. Calc (in its standard
997 user interface) is then started. Type @kbd{v u} if you want
998 to unpack this vector into separate numbers on the stack. Also,
999 @kbd{C-u C-x * g} interprets the region as a single number or
1002 The @kbd{C-x * r} command reads a rectangle, with the point and
1003 mark defining opposite corners of the rectangle. The result
1004 is a matrix of numbers on the Calculator stack.
1006 Complementary to these is @kbd{C-x * y}, which ``yanks'' the
1007 value at the top of the Calc stack back into an editing buffer.
1008 If you type @w{@kbd{C-x * y}} while in such a buffer, the value is
1009 yanked at the current position. If you type @kbd{C-x * y} while
1010 in the Calc buffer, Calc makes an educated guess as to which
1011 editing buffer you want to use. The Calc window does not have
1012 to be visible in order to use this command, as long as there
1013 is something on the Calc stack.
1015 Here, for reference, is the complete list of @kbd{C-x *} commands.
1016 The shift, control, and meta keys are ignored for the keystroke
1017 following @kbd{C-x *}.
1020 Commands for turning Calc on and off:
1024 Turn Calc on or off, employing the same user interface as last time.
1026 @item =, +, -, /, \, &, #
1027 Alternatives for @kbd{*}.
1030 Turn Calc on or off using its standard bottom-of-the-screen
1031 interface. If Calc is already turned on but the cursor is not
1032 in the Calc window, move the cursor into the window.
1035 Same as @kbd{C}, but don't select the new Calc window. If
1036 Calc is already turned on and the cursor is in the Calc window,
1037 move it out of that window.
1040 Control whether @kbd{C-x * c} and @kbd{C-x * k} use the full screen.
1043 Use Quick mode for a single short calculation.
1046 Turn Calc Keypad mode on or off.
1049 Turn Calc Embedded mode on or off at the current formula.
1052 Turn Calc Embedded mode on or off, select the interesting part.
1055 Turn Calc Embedded mode on or off at the current word (number).
1058 Turn Calc on in a user-defined way, as defined by a @kbd{Z I} command.
1061 Quit Calc; turn off standard, Keypad, or Embedded mode if on.
1062 (This is like @kbd{q} or @key{OFF} inside of Calc.)
1069 Commands for moving data into and out of the Calculator:
1073 Grab the region into the Calculator as a vector.
1076 Grab the rectangular region into the Calculator as a matrix.
1079 Grab the rectangular region and compute the sums of its columns.
1082 Grab the rectangular region and compute the sums of its rows.
1085 Yank a value from the Calculator into the current editing buffer.
1092 Commands for use with Embedded mode:
1096 ``Activate'' the current buffer. Locate all formulas that
1097 contain @samp{:=} or @samp{=>} symbols and record their locations
1098 so that they can be updated automatically as variables are changed.
1101 Duplicate the current formula immediately below and select
1105 Insert a new formula at the current point.
1108 Move the cursor to the next active formula in the buffer.
1111 Move the cursor to the previous active formula in the buffer.
1114 Update (i.e., as if by the @kbd{=} key) the formula at the current point.
1117 Edit (as if by @code{calc-edit}) the formula at the current point.
1124 Miscellaneous commands:
1128 Run the Emacs Info system to read the Calc manual.
1129 (This is the same as @kbd{h i} inside of Calc.)
1132 Run the Emacs Info system to read the Calc Tutorial.
1135 Run the Emacs Info system to read the Calc Summary.
1138 Load Calc entirely into memory. (Normally the various parts
1139 are loaded only as they are needed.)
1142 Read a region of written keystroke names (like @kbd{C-n a b c @key{RET}})
1143 and record them as the current keyboard macro.
1146 (This is the ``zero'' digit key.) Reset the Calculator to
1147 its initial state: Empty stack, and initial mode settings.
1150 @node History and Acknowledgements, , Using Calc, Getting Started
1151 @section History and Acknowledgements
1154 Calc was originally started as a two-week project to occupy a lull
1155 in the author's schedule. Basically, a friend asked if I remembered
1157 @texline @math{2^{32}}.
1158 @infoline @expr{2^32}.
1159 I didn't offhand, but I said, ``that's easy, just call up an
1160 @code{xcalc}.'' @code{Xcalc} duly reported that the answer to our
1161 question was @samp{4.294967e+09}---with no way to see the full ten
1162 digits even though we knew they were there in the program's memory! I
1163 was so annoyed, I vowed to write a calculator of my own, once and for
1166 I chose Emacs Lisp, a) because I had always been curious about it
1167 and b) because, being only a text editor extension language after
1168 all, Emacs Lisp would surely reach its limits long before the project
1169 got too far out of hand.
1171 To make a long story short, Emacs Lisp turned out to be a distressingly
1172 solid implementation of Lisp, and the humble task of calculating
1173 turned out to be more open-ended than one might have expected.
1175 Emacs Lisp didn't have built-in floating point math (now it does), so
1176 this had to be simulated in software. In fact, Emacs integers would
1177 only comfortably fit six decimal digits or so---not enough for a decent
1178 calculator. So I had to write my own high-precision integer code as
1179 well, and once I had this I figured that arbitrary-size integers were
1180 just as easy as large integers. Arbitrary floating-point precision was
1181 the logical next step. Also, since the large integer arithmetic was
1182 there anyway it seemed only fair to give the user direct access to it,
1183 which in turn made it practical to support fractions as well as floats.
1184 All these features inspired me to look around for other data types that
1185 might be worth having.
1187 Around this time, my friend Rick Koshi showed me his nifty new HP-28
1188 calculator. It allowed the user to manipulate formulas as well as
1189 numerical quantities, and it could also operate on matrices. I
1190 decided that these would be good for Calc to have, too. And once
1191 things had gone this far, I figured I might as well take a look at
1192 serious algebra systems for further ideas. Since these systems did
1193 far more than I could ever hope to implement, I decided to focus on
1194 rewrite rules and other programming features so that users could
1195 implement what they needed for themselves.
1197 Rick complained that matrices were hard to read, so I put in code to
1198 format them in a 2D style. Once these routines were in place, Big mode
1199 was obligatory. Gee, what other language modes would be useful?
1201 Scott Hemphill and Allen Knutson, two friends with a strong mathematical
1202 bent, contributed ideas and algorithms for a number of Calc features
1203 including modulo forms, primality testing, and float-to-fraction conversion.
1205 Units were added at the eager insistence of Mass Sivilotti. Later,
1206 Ulrich Mueller at CERN and Przemek Klosowski at NIST provided invaluable
1207 expert assistance with the units table. As far as I can remember, the
1208 idea of using algebraic formulas and variables to represent units dates
1209 back to an ancient article in Byte magazine about muMath, an early
1210 algebra system for microcomputers.
1212 Many people have contributed to Calc by reporting bugs and suggesting
1213 features, large and small. A few deserve special mention: Tim Peters,
1214 who helped develop the ideas that led to the selection commands, rewrite
1215 rules, and many other algebra features;
1216 @texline Fran\c{c}ois
1218 Pinard, who contributed an early prototype of the Calc Summary appendix
1219 as well as providing valuable suggestions in many other areas of Calc;
1220 Carl Witty, whose eagle eyes discovered many typographical and factual
1221 errors in the Calc manual; Tim Kay, who drove the development of
1222 Embedded mode; Ove Ewerlid, who made many suggestions relating to the
1223 algebra commands and contributed some code for polynomial operations;
1224 Randal Schwartz, who suggested the @code{calc-eval} function; Juha
1225 Sarlin, who first worked out how to split Calc into quickly-loading
1226 parts; Bob Weiner, who helped immensely with the Lucid Emacs port; and
1227 Robert J. Chassell, who suggested the Calc Tutorial and exercises as
1228 well as many other things.
1230 @cindex Bibliography
1231 @cindex Knuth, Art of Computer Programming
1232 @cindex Numerical Recipes
1233 @c Should these be expanded into more complete references?
1234 Among the books used in the development of Calc were Knuth's @emph{Art
1235 of Computer Programming} (especially volume II, @emph{Seminumerical
1236 Algorithms}); @emph{Numerical Recipes} by Press, Flannery, Teukolsky,
1237 and Vetterling; Bevington's @emph{Data Reduction and Error Analysis
1238 for the Physical Sciences}; @emph{Concrete Mathematics} by Graham,
1239 Knuth, and Patashnik; Steele's @emph{Common Lisp, the Language}; the
1240 @emph{CRC Standard Math Tables} (William H. Beyer, ed.); and
1241 Abramowitz and Stegun's venerable @emph{Handbook of Mathematical
1242 Functions}. Also, of course, Calc could not have been written without
1243 the excellent @emph{GNU Emacs Lisp Reference Manual}, by Bil Lewis and
1246 Final thanks go to Richard Stallman, without whose fine implementations
1247 of the Emacs editor, language, and environment, Calc would have been
1248 finished in two weeks.
1253 @c This node is accessed by the `C-x * t' command.
1254 @node Interactive Tutorial, Tutorial, Getting Started, Top
1258 Some brief instructions on using the Emacs Info system for this tutorial:
1260 Press the space bar and Delete keys to go forward and backward in a
1261 section by screenfuls (or use the regular Emacs scrolling commands
1264 Press @kbd{n} or @kbd{p} to go to the Next or Previous section.
1265 If the section has a @dfn{menu}, press a digit key like @kbd{1}
1266 or @kbd{2} to go to a sub-section from the menu. Press @kbd{u} to
1267 go back up from a sub-section to the menu it is part of.
1269 Exercises in the tutorial all have cross-references to the
1270 appropriate page of the ``answers'' section. Press @kbd{f}, then
1271 the exercise number, to see the answer to an exercise. After
1272 you have followed a cross-reference, you can press the letter
1273 @kbd{l} to return to where you were before.
1275 You can press @kbd{?} at any time for a brief summary of Info commands.
1277 Press the number @kbd{1} now to enter the first section of the Tutorial.
1283 @node Tutorial, Introduction, Interactive Tutorial, Top
1286 @node Tutorial, Introduction, Getting Started, Top
1291 This chapter explains how to use Calc and its many features, in
1292 a step-by-step, tutorial way. You are encouraged to run Calc and
1293 work along with the examples as you read (@pxref{Starting Calc}).
1294 If you are already familiar with advanced calculators, you may wish
1296 to skip on to the rest of this manual.
1298 @c to skip on to volume II of this manual, the @dfn{Calc Reference}.
1300 @c [fix-ref Embedded Mode]
1301 This tutorial describes the standard user interface of Calc only.
1302 The Quick mode and Keypad mode interfaces are fairly
1303 self-explanatory. @xref{Embedded Mode}, for a description of
1304 the Embedded mode interface.
1306 The easiest way to read this tutorial on-line is to have two windows on
1307 your Emacs screen, one with Calc and one with the Info system. Press
1308 @kbd{C-x * t} to set this up; the on-line tutorial will be opened in the
1309 current window and Calc will be started in another window. From the
1310 Info window, the command @kbd{C-x * c} can be used to switch to the Calc
1311 window and @kbd{C-x * o} can be used to switch back to the Info window.
1312 (If you have a printed copy of the manual you can use that instead; in
1313 that case you only need to press @kbd{C-x * c} to start Calc.)
1315 This tutorial is designed to be done in sequence. But the rest of this
1316 manual does not assume you have gone through the tutorial. The tutorial
1317 does not cover everything in the Calculator, but it touches on most
1321 You may wish to print out a copy of the Calc Summary and keep notes on
1322 it as you learn Calc. @xref{About This Manual}, to see how to make a
1323 printed summary. @xref{Summary}.
1326 The Calc Summary at the end of the reference manual includes some blank
1327 space for your own use. You may wish to keep notes there as you learn
1333 * Arithmetic Tutorial::
1334 * Vector/Matrix Tutorial::
1336 * Algebra Tutorial::
1337 * Programming Tutorial::
1339 * Answers to Exercises::
1342 @node Basic Tutorial, Arithmetic Tutorial, Tutorial, Tutorial
1343 @section Basic Tutorial
1346 In this section, we learn how RPN and algebraic-style calculations
1347 work, how to undo and redo an operation done by mistake, and how
1348 to control various modes of the Calculator.
1351 * RPN Tutorial:: Basic operations with the stack.
1352 * Algebraic Tutorial:: Algebraic entry; variables.
1353 * Undo Tutorial:: If you make a mistake: Undo and the trail.
1354 * Modes Tutorial:: Common mode-setting commands.
1357 @node RPN Tutorial, Algebraic Tutorial, Basic Tutorial, Basic Tutorial
1358 @subsection RPN Calculations and the Stack
1360 @cindex RPN notation
1363 Calc normally uses RPN notation. You may be familiar with the RPN
1364 system from Hewlett-Packard calculators, FORTH, or PostScript.
1365 (Reverse Polish Notation, RPN, is named after the Polish mathematician
1370 Calc normally uses RPN notation. You may be familiar with the RPN
1371 system from Hewlett-Packard calculators, FORTH, or PostScript.
1372 (Reverse Polish Notation, RPN, is named after the Polish mathematician
1376 The central component of an RPN calculator is the @dfn{stack}. A
1377 calculator stack is like a stack of dishes. New dishes (numbers) are
1378 added at the top of the stack, and numbers are normally only removed
1379 from the top of the stack.
1383 In an operation like @expr{2+3}, the 2 and 3 are called the @dfn{operands}
1384 and the @expr{+} is the @dfn{operator}. In an RPN calculator you always
1385 enter the operands first, then the operator. Each time you type a
1386 number, Calc adds or @dfn{pushes} it onto the top of the Stack.
1387 When you press an operator key like @kbd{+}, Calc @dfn{pops} the appropriate
1388 number of operands from the stack and pushes back the result.
1390 Thus we could add the numbers 2 and 3 in an RPN calculator by typing:
1391 @kbd{2 @key{RET} 3 @key{RET} +}. (The @key{RET} key, Return, corresponds to
1392 the @key{ENTER} key on traditional RPN calculators.) Try this now if
1393 you wish; type @kbd{C-x * c} to switch into the Calc window (you can type
1394 @kbd{C-x * c} again or @kbd{C-x * o} to switch back to the Tutorial window).
1395 The first four keystrokes ``push'' the numbers 2 and 3 onto the stack.
1396 The @kbd{+} key ``pops'' the top two numbers from the stack, adds them,
1397 and pushes the result (5) back onto the stack. Here's how the stack
1398 will look at various points throughout the calculation:
1406 C-x * c 2 @key{RET} 3 @key{RET} + @key{DEL}
1410 The @samp{.} symbol is a marker that represents the top of the stack.
1411 Note that the ``top'' of the stack is really shown at the bottom of
1412 the Stack window. This may seem backwards, but it turns out to be
1413 less distracting in regular use.
1415 @cindex Stack levels
1416 @cindex Levels of stack
1417 The numbers @samp{1:} and @samp{2:} on the left are @dfn{stack level
1418 numbers}. Old RPN calculators always had four stack levels called
1419 @expr{x}, @expr{y}, @expr{z}, and @expr{t}. Calc's stack can grow
1420 as large as you like, so it uses numbers instead of letters. Some
1421 stack-manipulation commands accept a numeric argument that says
1422 which stack level to work on. Normal commands like @kbd{+} always
1423 work on the top few levels of the stack.
1425 @c [fix-ref Truncating the Stack]
1426 The Stack buffer is just an Emacs buffer, and you can move around in
1427 it using the regular Emacs motion commands. But no matter where the
1428 cursor is, even if you have scrolled the @samp{.} marker out of
1429 view, most Calc commands always move the cursor back down to level 1
1430 before doing anything. It is possible to move the @samp{.} marker
1431 upwards through the stack, temporarily ``hiding'' some numbers from
1432 commands like @kbd{+}. This is called @dfn{stack truncation} and
1433 we will not cover it in this tutorial; @pxref{Truncating the Stack},
1434 if you are interested.
1436 You don't really need the second @key{RET} in @kbd{2 @key{RET} 3
1437 @key{RET} +}. That's because if you type any operator name or
1438 other non-numeric key when you are entering a number, the Calculator
1439 automatically enters that number and then does the requested command.
1440 Thus @kbd{2 @key{RET} 3 +} will work just as well.
1442 Examples in this tutorial will often omit @key{RET} even when the
1443 stack displays shown would only happen if you did press @key{RET}:
1456 Here, after pressing @kbd{3} the stack would really show @samp{1: 2}
1457 with @samp{Calc:@: 3} in the minibuffer. In these situations, you can
1458 press the optional @key{RET} to see the stack as the figure shows.
1460 (@bullet{}) @strong{Exercise 1.} (This tutorial will include exercises
1461 at various points. Try them if you wish. Answers to all the exercises
1462 are located at the end of the Tutorial chapter. Each exercise will
1463 include a cross-reference to its particular answer. If you are
1464 reading with the Emacs Info system, press @kbd{f} and the
1465 exercise number to go to the answer, then the letter @kbd{l} to
1466 return to where you were.)
1469 Here's the first exercise: What will the keystrokes @kbd{1 @key{RET} 2
1470 @key{RET} 3 @key{RET} 4 + * -} compute? (@samp{*} is the symbol for
1471 multiplication.) Figure it out by hand, then try it with Calc to see
1472 if you're right. @xref{RPN Answer 1, 1}. (@bullet{})
1474 (@bullet{}) @strong{Exercise 2.} Compute
1475 @texline @math{(2\times4) + (7\times9.4) + {5\over4}}
1476 @infoline @expr{2*4 + 7*9.5 + 5/4}
1477 using the stack. @xref{RPN Answer 2, 2}. (@bullet{})
1479 The @key{DEL} key is called Backspace on some keyboards. It is
1480 whatever key you would use to correct a simple typing error when
1481 regularly using Emacs. The @key{DEL} key pops and throws away the
1482 top value on the stack. (You can still get that value back from
1483 the Trail if you should need it later on.) There are many places
1484 in this tutorial where we assume you have used @key{DEL} to erase the
1485 results of the previous example at the beginning of a new example.
1486 In the few places where it is really important to use @key{DEL} to
1487 clear away old results, the text will remind you to do so.
1489 (It won't hurt to let things accumulate on the stack, except that
1490 whenever you give a display-mode-changing command Calc will have to
1491 spend a long time reformatting such a large stack.)
1493 Since the @kbd{-} key is also an operator (it subtracts the top two
1494 stack elements), how does one enter a negative number? Calc uses
1495 the @kbd{_} (underscore) key to act like the minus sign in a number.
1496 So, typing @kbd{-5 @key{RET}} won't work because the @kbd{-} key
1497 will try to do a subtraction, but @kbd{_5 @key{RET}} works just fine.
1499 You can also press @kbd{n}, which means ``change sign.'' It changes
1500 the number at the top of the stack (or the number being entered)
1501 from positive to negative or vice-versa: @kbd{5 n @key{RET}}.
1503 @cindex Duplicating a stack entry
1504 If you press @key{RET} when you're not entering a number, the effect
1505 is to duplicate the top number on the stack. Consider this calculation:
1509 1: 3 2: 3 1: 9 2: 9 1: 81
1513 3 @key{RET} @key{RET} * @key{RET} *
1518 (Of course, an easier way to do this would be @kbd{3 @key{RET} 4 ^},
1519 to raise 3 to the fourth power.)
1521 The space-bar key (denoted @key{SPC} here) performs the same function
1522 as @key{RET}; you could replace all three occurrences of @key{RET} in
1523 the above example with @key{SPC} and the effect would be the same.
1525 @cindex Exchanging stack entries
1526 Another stack manipulation key is @key{TAB}. This exchanges the top
1527 two stack entries. Suppose you have computed @kbd{2 @key{RET} 3 +}
1528 to get 5, and then you realize what you really wanted to compute
1529 was @expr{20 / (2+3)}.
1533 1: 5 2: 5 2: 20 1: 4
1537 2 @key{RET} 3 + 20 @key{TAB} /
1542 Planning ahead, the calculation would have gone like this:
1546 1: 20 2: 20 3: 20 2: 20 1: 4
1551 20 @key{RET} 2 @key{RET} 3 + /
1555 A related stack command is @kbd{M-@key{TAB}} (hold @key{META} and type
1556 @key{TAB}). It rotates the top three elements of the stack upward,
1557 bringing the object in level 3 to the top.
1561 1: 10 2: 10 3: 10 3: 20 3: 30
1562 . 1: 20 2: 20 2: 30 2: 10
1566 10 @key{RET} 20 @key{RET} 30 @key{RET} M-@key{TAB} M-@key{TAB}
1570 (@bullet{}) @strong{Exercise 3.} Suppose the numbers 10, 20, and 30 are
1571 on the stack. Figure out how to add one to the number in level 2
1572 without affecting the rest of the stack. Also figure out how to add
1573 one to the number in level 3. @xref{RPN Answer 3, 3}. (@bullet{})
1575 Operations like @kbd{+}, @kbd{-}, @kbd{*}, @kbd{/}, and @kbd{^} pop two
1576 arguments from the stack and push a result. Operations like @kbd{n} and
1577 @kbd{Q} (square root) pop a single number and push the result. You can
1578 think of them as simply operating on the top element of the stack.
1582 1: 3 1: 9 2: 9 1: 25 1: 5
1586 3 @key{RET} @key{RET} * 4 @key{RET} @key{RET} * + Q
1591 (Note that capital @kbd{Q} means to hold down the Shift key while
1592 typing @kbd{q}. Remember, plain unshifted @kbd{q} is the Quit command.)
1594 @cindex Pythagorean Theorem
1595 Here we've used the Pythagorean Theorem to determine the hypotenuse of a
1596 right triangle. Calc actually has a built-in command for that called
1597 @kbd{f h}, but let's suppose we can't remember the necessary keystrokes.
1598 We can still enter it by its full name using @kbd{M-x} notation:
1606 3 @key{RET} 4 @key{RET} M-x calc-hypot
1610 All Calculator commands begin with the word @samp{calc-}. Since it
1611 gets tiring to type this, Calc provides an @kbd{x} key which is just
1612 like the regular Emacs @kbd{M-x} key except that it types the @samp{calc-}
1621 3 @key{RET} 4 @key{RET} x hypot
1625 What happens if you take the square root of a negative number?
1629 1: 4 1: -4 1: (0, 2)
1637 The notation @expr{(a, b)} represents a complex number.
1638 Complex numbers are more traditionally written @expr{a + b i};
1639 Calc can display in this format, too, but for now we'll stick to the
1640 @expr{(a, b)} notation.
1642 If you don't know how complex numbers work, you can safely ignore this
1643 feature. Complex numbers only arise from operations that would be
1644 errors in a calculator that didn't have complex numbers. (For example,
1645 taking the square root or logarithm of a negative number produces a
1648 Complex numbers are entered in the notation shown. The @kbd{(} and
1649 @kbd{,} and @kbd{)} keys manipulate ``incomplete complex numbers.''
1653 1: ( ... 2: ( ... 1: (2, ... 1: (2, ... 1: (2, 3)
1661 You can perform calculations while entering parts of incomplete objects.
1662 However, an incomplete object cannot actually participate in a calculation:
1666 1: ( ... 2: ( ... 3: ( ... 1: ( ... 1: ( ...
1676 Adding 5 to an incomplete object makes no sense, so the last command
1677 produces an error message and leaves the stack the same.
1679 Incomplete objects can't participate in arithmetic, but they can be
1680 moved around by the regular stack commands.
1684 2: 2 3: 2 3: 3 1: ( ... 1: (2, 3)
1685 1: 3 2: 3 2: ( ... 2 .
1689 2 @key{RET} 3 @key{RET} ( M-@key{TAB} M-@key{TAB} )
1694 Note that the @kbd{,} (comma) key did not have to be used here.
1695 When you press @kbd{)} all the stack entries between the incomplete
1696 entry and the top are collected, so there's never really a reason
1697 to use the comma. It's up to you.
1699 (@bullet{}) @strong{Exercise 4.} To enter the complex number @expr{(2, 3)},
1700 your friend Joe typed @kbd{( 2 , @key{SPC} 3 )}. What happened?
1701 (Joe thought of a clever way to correct his mistake in only two
1702 keystrokes, but it didn't quite work. Try it to find out why.)
1703 @xref{RPN Answer 4, 4}. (@bullet{})
1705 Vectors are entered the same way as complex numbers, but with square
1706 brackets in place of parentheses. We'll meet vectors again later in
1709 Any Emacs command can be given a @dfn{numeric prefix argument} by
1710 typing a series of @key{META}-digits beforehand. If @key{META} is
1711 awkward for you, you can instead type @kbd{C-u} followed by the
1712 necessary digits. Numeric prefix arguments can be negative, as in
1713 @kbd{M-- M-3 M-5} or @w{@kbd{C-u - 3 5}}. Calc commands use numeric
1714 prefix arguments in a variety of ways. For example, a numeric prefix
1715 on the @kbd{+} operator adds any number of stack entries at once:
1719 1: 10 2: 10 3: 10 3: 10 1: 60
1720 . 1: 20 2: 20 2: 20 .
1724 10 @key{RET} 20 @key{RET} 30 @key{RET} C-u 3 +
1728 For stack manipulation commands like @key{RET}, a positive numeric
1729 prefix argument operates on the top @var{n} stack entries at once. A
1730 negative argument operates on the entry in level @var{n} only. An
1731 argument of zero operates on the entire stack. In this example, we copy
1732 the second-to-top element of the stack:
1736 1: 10 2: 10 3: 10 3: 10 4: 10
1737 . 1: 20 2: 20 2: 20 3: 20
1742 10 @key{RET} 20 @key{RET} 30 @key{RET} C-u -2 @key{RET}
1746 @cindex Clearing the stack
1747 @cindex Emptying the stack
1748 Another common idiom is @kbd{M-0 @key{DEL}}, which clears the stack.
1749 (The @kbd{M-0} numeric prefix tells @key{DEL} to operate on the
1752 @node Algebraic Tutorial, Undo Tutorial, RPN Tutorial, Basic Tutorial
1753 @subsection Algebraic-Style Calculations
1756 If you are not used to RPN notation, you may prefer to operate the
1757 Calculator in Algebraic mode, which is closer to the way
1758 non-RPN calculators work. In Algebraic mode, you enter formulas
1759 in traditional @expr{2+3} notation.
1761 @strong{Warning:} Note that @samp{/} has lower precedence than
1762 @samp{*}, so that @samp{a/b*c} is interpreted as @samp{a/(b*c)}. See
1765 You don't really need any special ``mode'' to enter algebraic formulas.
1766 You can enter a formula at any time by pressing the apostrophe (@kbd{'})
1767 key. Answer the prompt with the desired formula, then press @key{RET}.
1768 The formula is evaluated and the result is pushed onto the RPN stack.
1769 If you don't want to think in RPN at all, you can enter your whole
1770 computation as a formula, read the result from the stack, then press
1771 @key{DEL} to delete it from the stack.
1773 Try pressing the apostrophe key, then @kbd{2+3+4}, then @key{RET}.
1774 The result should be the number 9.
1776 Algebraic formulas use the operators @samp{+}, @samp{-}, @samp{*},
1777 @samp{/}, and @samp{^}. You can use parentheses to make the order
1778 of evaluation clear. In the absence of parentheses, @samp{^} is
1779 evaluated first, then @samp{*}, then @samp{/}, then finally
1780 @samp{+} and @samp{-}. For example, the expression
1783 2 + 3*4*5 / 6*7^8 - 9
1790 2 + ((3*4*5) / (6*(7^8)) - 9
1794 or, in large mathematical notation,
1809 $$ 2 + { 3 \times 4 \times 5 \over 6 \times 7^8 } - 9 $$
1814 The result of this expression will be the number @mathit{-6.99999826533}.
1816 Calc's order of evaluation is the same as for most computer languages,
1817 except that @samp{*} binds more strongly than @samp{/}, as the above
1818 example shows. As in normal mathematical notation, the @samp{*} symbol
1819 can often be omitted: @samp{2 a} is the same as @samp{2*a}.
1821 Operators at the same level are evaluated from left to right, except
1822 that @samp{^} is evaluated from right to left. Thus, @samp{2-3-4} is
1823 equivalent to @samp{(2-3)-4} or @mathit{-5}, whereas @samp{2^3^4} is equivalent
1824 to @samp{2^(3^4)} (a very large integer; try it!).
1826 If you tire of typing the apostrophe all the time, there is
1827 Algebraic mode, where Calc automatically senses
1828 when you are about to type an algebraic expression. To enter this
1829 mode, press the two letters @w{@kbd{m a}}. (An @samp{Alg} indicator
1830 should appear in the Calc window's mode line.)
1832 Press @kbd{m a}, then @kbd{2+3+4} with no apostrophe, then @key{RET}.
1834 In Algebraic mode, when you press any key that would normally begin
1835 entering a number (such as a digit, a decimal point, or the @kbd{_}
1836 key), or if you press @kbd{(} or @kbd{[}, Calc automatically begins
1839 Functions which do not have operator symbols like @samp{+} and @samp{*}
1840 must be entered in formulas using function-call notation. For example,
1841 the function name corresponding to the square-root key @kbd{Q} is
1842 @code{sqrt}. To compute a square root in a formula, you would use
1843 the notation @samp{sqrt(@var{x})}.
1845 Press the apostrophe, then type @kbd{sqrt(5*2) - 3}. The result should
1846 be @expr{0.16227766017}.
1848 Note that if the formula begins with a function name, you need to use
1849 the apostrophe even if you are in Algebraic mode. If you type @kbd{arcsin}
1850 out of the blue, the @kbd{a r} will be taken as an Algebraic Rewrite
1851 command, and the @kbd{csin} will be taken as the name of the rewrite
1854 Some people prefer to enter complex numbers and vectors in algebraic
1855 form because they find RPN entry with incomplete objects to be too
1856 distracting, even though they otherwise use Calc as an RPN calculator.
1858 Still in Algebraic mode, type:
1862 1: (2, 3) 2: (2, 3) 1: (8, -1) 2: (8, -1) 1: (9, -1)
1863 . 1: (1, -2) . 1: 1 .
1866 (2,3) @key{RET} (1,-2) @key{RET} * 1 @key{RET} +
1870 Algebraic mode allows us to enter complex numbers without pressing
1871 an apostrophe first, but it also means we need to press @key{RET}
1872 after every entry, even for a simple number like @expr{1}.
1874 (You can type @kbd{C-u m a} to enable a special Incomplete Algebraic
1875 mode in which the @kbd{(} and @kbd{[} keys use algebraic entry even
1876 though regular numeric keys still use RPN numeric entry. There is also
1877 Total Algebraic mode, started by typing @kbd{m t}, in which all
1878 normal keys begin algebraic entry. You must then use the @key{META} key
1879 to type Calc commands: @kbd{M-m t} to get back out of Total Algebraic
1880 mode, @kbd{M-q} to quit, etc.)
1882 If you're still in Algebraic mode, press @kbd{m a} again to turn it off.
1884 Actual non-RPN calculators use a mixture of algebraic and RPN styles.
1885 In general, operators of two numbers (like @kbd{+} and @kbd{*})
1886 use algebraic form, but operators of one number (like @kbd{n} and @kbd{Q})
1887 use RPN form. Also, a non-RPN calculator allows you to see the
1888 intermediate results of a calculation as you go along. You can
1889 accomplish this in Calc by performing your calculation as a series
1890 of algebraic entries, using the @kbd{$} sign to tie them together.
1891 In an algebraic formula, @kbd{$} represents the number on the top
1892 of the stack. Here, we perform the calculation
1893 @texline @math{\sqrt{2\times4+1}},
1894 @infoline @expr{sqrt(2*4+1)},
1895 which on a traditional calculator would be done by pressing
1896 @kbd{2 * 4 + 1 =} and then the square-root key.
1903 ' 2*4 @key{RET} $+1 @key{RET} Q
1908 Notice that we didn't need to press an apostrophe for the @kbd{$+1},
1909 because the dollar sign always begins an algebraic entry.
1911 (@bullet{}) @strong{Exercise 1.} How could you get the same effect as
1912 pressing @kbd{Q} but using an algebraic entry instead? How about
1913 if the @kbd{Q} key on your keyboard were broken?
1914 @xref{Algebraic Answer 1, 1}. (@bullet{})
1916 The notations @kbd{$$}, @kbd{$$$}, and so on stand for higher stack
1917 entries. For example, @kbd{' $$+$ @key{RET}} is just like typing @kbd{+}.
1919 Algebraic formulas can include @dfn{variables}. To store in a
1920 variable, press @kbd{s s}, then type the variable name, then press
1921 @key{RET}. (There are actually two flavors of store command:
1922 @kbd{s s} stores a number in a variable but also leaves the number
1923 on the stack, while @w{@kbd{s t}} removes a number from the stack and
1924 stores it in the variable.) A variable name should consist of one
1925 or more letters or digits, beginning with a letter.
1929 1: 17 . 1: a + a^2 1: 306
1932 17 s t a @key{RET} ' a+a^2 @key{RET} =
1937 The @kbd{=} key @dfn{evaluates} a formula by replacing all its
1938 variables by the values that were stored in them.
1940 For RPN calculations, you can recall a variable's value on the
1941 stack either by entering its name as a formula and pressing @kbd{=},
1942 or by using the @kbd{s r} command.
1946 1: 17 2: 17 3: 17 2: 17 1: 306
1947 . 1: 17 2: 17 1: 289 .
1951 s r a @key{RET} ' a @key{RET} = 2 ^ +
1955 If you press a single digit for a variable name (as in @kbd{s t 3}, you
1956 get one of ten @dfn{quick variables} @code{q0} through @code{q9}.
1957 They are ``quick'' simply because you don't have to type the letter
1958 @code{q} or the @key{RET} after their names. In fact, you can type
1959 simply @kbd{s 3} as a shorthand for @kbd{s s 3}, and likewise for
1960 @kbd{t 3} and @w{@kbd{r 3}}.
1962 Any variables in an algebraic formula for which you have not stored
1963 values are left alone, even when you evaluate the formula.
1967 1: 2 a + 2 b 1: 34 + 2 b
1974 Calls to function names which are undefined in Calc are also left
1975 alone, as are calls for which the value is undefined.
1979 1: 2 + log10(0) + log10(x) + log10(5, 6) + foo(3)
1982 ' log10(100) + log10(0) + log10(x) + log10(5,6) + foo(3) @key{RET}
1987 In this example, the first call to @code{log10} works, but the other
1988 calls are not evaluated. In the second call, the logarithm is
1989 undefined for that value of the argument; in the third, the argument
1990 is symbolic, and in the fourth, there are too many arguments. In the
1991 fifth case, there is no function called @code{foo}. You will see a
1992 ``Wrong number of arguments'' message referring to @samp{log10(5,6)}.
1993 Press the @kbd{w} (``why'') key to see any other messages that may
1994 have arisen from the last calculation. In this case you will get
1995 ``logarithm of zero,'' then ``number expected: @code{x}''. Calc
1996 automatically displays the first message only if the message is
1997 sufficiently important; for example, Calc considers ``wrong number
1998 of arguments'' and ``logarithm of zero'' to be important enough to
1999 report automatically, while a message like ``number expected: @code{x}''
2000 will only show up if you explicitly press the @kbd{w} key.
2002 (@bullet{}) @strong{Exercise 2.} Joe entered the formula @samp{2 x y},
2003 stored 5 in @code{x}, pressed @kbd{=}, and got the expected result,
2004 @samp{10 y}. He then tried the same for the formula @samp{2 x (1+y)},
2005 expecting @samp{10 (1+y)}, but it didn't work. Why not?
2006 @xref{Algebraic Answer 2, 2}. (@bullet{})
2008 (@bullet{}) @strong{Exercise 3.} What result would you expect
2009 @kbd{1 @key{RET} 0 /} to give? What if you then type @kbd{0 *}?
2010 @xref{Algebraic Answer 3, 3}. (@bullet{})
2012 One interesting way to work with variables is to use the
2013 @dfn{evaluates-to} (@samp{=>}) operator. It works like this:
2014 Enter a formula algebraically in the usual way, but follow
2015 the formula with an @samp{=>} symbol. (There is also an @kbd{s =}
2016 command which builds an @samp{=>} formula using the stack.) On
2017 the stack, you will see two copies of the formula with an @samp{=>}
2018 between them. The lefthand formula is exactly like you typed it;
2019 the righthand formula has been evaluated as if by typing @kbd{=}.
2023 2: 2 + 3 => 5 2: 2 + 3 => 5
2024 1: 2 a + 2 b => 34 + 2 b 1: 2 a + 2 b => 20 + 2 b
2027 ' 2+3 => @key{RET} ' 2a+2b @key{RET} s = 10 s t a @key{RET}
2032 Notice that the instant we stored a new value in @code{a}, all
2033 @samp{=>} operators already on the stack that referred to @expr{a}
2034 were updated to use the new value. With @samp{=>}, you can push a
2035 set of formulas on the stack, then change the variables experimentally
2036 to see the effects on the formulas' values.
2038 You can also ``unstore'' a variable when you are through with it:
2043 1: 2 a + 2 b => 2 a + 2 b
2050 We will encounter formulas involving variables and functions again
2051 when we discuss the algebra and calculus features of the Calculator.
2053 @node Undo Tutorial, Modes Tutorial, Algebraic Tutorial, Basic Tutorial
2054 @subsection Undo and Redo
2057 If you make a mistake, you can usually correct it by pressing shift-@kbd{U},
2058 the ``undo'' command. First, clear the stack (@kbd{M-0 @key{DEL}}) and exit
2059 and restart Calc (@kbd{C-x * * C-x * *}) to make sure things start off
2060 with a clean slate. Now:
2064 1: 2 2: 2 1: 8 2: 2 1: 6
2072 You can undo any number of times. Calc keeps a complete record of
2073 all you have done since you last opened the Calc window. After the
2074 above example, you could type:
2086 You can also type @kbd{D} to ``redo'' a command that you have undone
2091 . 1: 2 2: 2 1: 6 1: 6
2100 It was not possible to redo past the @expr{6}, since that was placed there
2101 by something other than an undo command.
2104 You can think of undo and redo as a sort of ``time machine.'' Press
2105 @kbd{U} to go backward in time, @kbd{D} to go forward. If you go
2106 backward and do something (like @kbd{*}) then, as any science fiction
2107 reader knows, you have changed your future and you cannot go forward
2108 again. Thus, the inability to redo past the @expr{6} even though there
2109 was an earlier undo command.
2111 You can always recall an earlier result using the Trail. We've ignored
2112 the trail so far, but it has been faithfully recording everything we
2113 did since we loaded the Calculator. If the Trail is not displayed,
2114 press @kbd{t d} now to turn it on.
2116 Let's try grabbing an earlier result. The @expr{8} we computed was
2117 undone by a @kbd{U} command, and was lost even to Redo when we pressed
2118 @kbd{*}, but it's still there in the trail. There should be a little
2119 @samp{>} arrow (the @dfn{trail pointer}) resting on the last trail
2120 entry. If there isn't, press @kbd{t ]} to reset the trail pointer.
2121 Now, press @w{@kbd{t p}} to move the arrow onto the line containing
2122 @expr{8}, and press @w{@kbd{t y}} to ``yank'' that number back onto the
2125 If you press @kbd{t ]} again, you will see that even our Yank command
2126 went into the trail.
2128 Let's go further back in time. Earlier in the tutorial we computed
2129 a huge integer using the formula @samp{2^3^4}. We don't remember
2130 what it was, but the first digits were ``241''. Press @kbd{t r}
2131 (which stands for trail-search-reverse), then type @kbd{241}.
2132 The trail cursor will jump back to the next previous occurrence of
2133 the string ``241'' in the trail. This is just a regular Emacs
2134 incremental search; you can now press @kbd{C-s} or @kbd{C-r} to
2135 continue the search forwards or backwards as you like.
2137 To finish the search, press @key{RET}. This halts the incremental
2138 search and leaves the trail pointer at the thing we found. Now we
2139 can type @kbd{t y} to yank that number onto the stack. If we hadn't
2140 remembered the ``241'', we could simply have searched for @kbd{2^3^4},
2141 then pressed @kbd{@key{RET} t n} to halt and then move to the next item.
2143 You may have noticed that all the trail-related commands begin with
2144 the letter @kbd{t}. (The store-and-recall commands, on the other hand,
2145 all began with @kbd{s}.) Calc has so many commands that there aren't
2146 enough keys for all of them, so various commands are grouped into
2147 two-letter sequences where the first letter is called the @dfn{prefix}
2148 key. If you type a prefix key by accident, you can press @kbd{C-g}
2149 to cancel it. (In fact, you can press @kbd{C-g} to cancel almost
2150 anything in Emacs.) To get help on a prefix key, press that key
2151 followed by @kbd{?}. Some prefixes have several lines of help,
2152 so you need to press @kbd{?} repeatedly to see them all.
2153 You can also type @kbd{h h} to see all the help at once.
2155 Try pressing @kbd{t ?} now. You will see a line of the form,
2158 trail/time: Display; Fwd, Back; Next, Prev, Here, [, ]; Yank: [MORE] t-
2162 The word ``trail'' indicates that the @kbd{t} prefix key contains
2163 trail-related commands. Each entry on the line shows one command,
2164 with a single capital letter showing which letter you press to get
2165 that command. We have used @kbd{t n}, @kbd{t p}, @kbd{t ]}, and
2166 @kbd{t y} so far. The @samp{[MORE]} means you can press @kbd{?}
2167 again to see more @kbd{t}-prefix commands. Notice that the commands
2168 are roughly divided (by semicolons) into related groups.
2170 When you are in the help display for a prefix key, the prefix is
2171 still active. If you press another key, like @kbd{y} for example,
2172 it will be interpreted as a @kbd{t y} command. If all you wanted
2173 was to look at the help messages, press @kbd{C-g} afterwards to cancel
2176 One more way to correct an error is by editing the stack entries.
2177 The actual Stack buffer is marked read-only and must not be edited
2178 directly, but you can press @kbd{`} (the backquote or accent grave)
2179 to edit a stack entry.
2181 Try entering @samp{3.141439} now. If this is supposed to represent
2182 @cpi{}, it's got several errors. Press @kbd{`} to edit this number.
2183 Now use the normal Emacs cursor motion and editing keys to change
2184 the second 4 to a 5, and to transpose the 3 and the 9. When you
2185 press @key{RET}, the number on the stack will be replaced by your
2186 new number. This works for formulas, vectors, and all other types
2187 of values you can put on the stack. The @kbd{`} key also works
2188 during entry of a number or algebraic formula.
2190 @node Modes Tutorial, , Undo Tutorial, Basic Tutorial
2191 @subsection Mode-Setting Commands
2194 Calc has many types of @dfn{modes} that affect the way it interprets
2195 your commands or the way it displays data. We have already seen one
2196 mode, namely Algebraic mode. There are many others, too; we'll
2197 try some of the most common ones here.
2199 Perhaps the most fundamental mode in Calc is the current @dfn{precision}.
2200 Notice the @samp{12} on the Calc window's mode line:
2203 --%*-Calc: 12 Deg (Calculator)----All------
2207 Most of the symbols there are Emacs things you don't need to worry
2208 about, but the @samp{12} and the @samp{Deg} are mode indicators.
2209 The @samp{12} means that calculations should always be carried to
2210 12 significant figures. That is why, when we type @kbd{1 @key{RET} 7 /},
2211 we get @expr{0.142857142857} with exactly 12 digits, not counting
2212 leading and trailing zeros.
2214 You can set the precision to anything you like by pressing @kbd{p},
2215 then entering a suitable number. Try pressing @kbd{p 30 @key{RET}},
2216 then doing @kbd{1 @key{RET} 7 /} again:
2221 2: 0.142857142857142857142857142857
2226 Although the precision can be set arbitrarily high, Calc always
2227 has to have @emph{some} value for the current precision. After
2228 all, the true value @expr{1/7} is an infinitely repeating decimal;
2229 Calc has to stop somewhere.
2231 Of course, calculations are slower the more digits you request.
2232 Press @w{@kbd{p 12}} now to set the precision back down to the default.
2234 Calculations always use the current precision. For example, even
2235 though we have a 30-digit value for @expr{1/7} on the stack, if
2236 we use it in a calculation in 12-digit mode it will be rounded
2237 down to 12 digits before it is used. Try it; press @key{RET} to
2238 duplicate the number, then @w{@kbd{1 +}}. Notice that the @key{RET}
2239 key didn't round the number, because it doesn't do any calculation.
2240 But the instant we pressed @kbd{+}, the number was rounded down.
2245 2: 0.142857142857142857142857142857
2252 In fact, since we added a digit on the left, we had to lose one
2253 digit on the right from even the 12-digit value of @expr{1/7}.
2255 How did we get more than 12 digits when we computed @samp{2^3^4}? The
2256 answer is that Calc makes a distinction between @dfn{integers} and
2257 @dfn{floating-point} numbers, or @dfn{floats}. An integer is a number
2258 that does not contain a decimal point. There is no such thing as an
2259 ``infinitely repeating fraction integer,'' so Calc doesn't have to limit
2260 itself. If you asked for @samp{2^10000} (don't try this!), you would
2261 have to wait a long time but you would eventually get an exact answer.
2262 If you ask for @samp{2.^10000}, you will quickly get an answer which is
2263 correct only to 12 places. The decimal point tells Calc that it should
2264 use floating-point arithmetic to get the answer, not exact integer
2267 You can use the @kbd{F} (@code{calc-floor}) command to convert a
2268 floating-point value to an integer, and @kbd{c f} (@code{calc-float})
2269 to convert an integer to floating-point form.
2271 Let's try entering that last calculation:
2275 1: 2. 2: 2. 1: 1.99506311689e3010
2279 2.0 @key{RET} 10000 @key{RET} ^
2284 @cindex Scientific notation, entry of
2285 Notice the letter @samp{e} in there. It represents ``times ten to the
2286 power of,'' and is used by Calc automatically whenever writing the
2287 number out fully would introduce more extra zeros than you probably
2288 want to see. You can enter numbers in this notation, too.
2292 1: 2. 2: 2. 1: 1.99506311678e3010
2296 2.0 @key{RET} 1e4 @key{RET} ^
2300 @cindex Round-off errors
2302 Hey, the answer is different! Look closely at the middle columns
2303 of the two examples. In the first, the stack contained the
2304 exact integer @expr{10000}, but in the second it contained
2305 a floating-point value with a decimal point. When you raise a
2306 number to an integer power, Calc uses repeated squaring and
2307 multiplication to get the answer. When you use a floating-point
2308 power, Calc uses logarithms and exponentials. As you can see,
2309 a slight error crept in during one of these methods. Which
2310 one should we trust? Let's raise the precision a bit and find
2315 . 1: 2. 2: 2. 1: 1.995063116880828e3010
2319 p 16 @key{RET} 2. @key{RET} 1e4 ^ p 12 @key{RET}
2324 @cindex Guard digits
2325 Presumably, it doesn't matter whether we do this higher-precision
2326 calculation using an integer or floating-point power, since we
2327 have added enough ``guard digits'' to trust the first 12 digits
2328 no matter what. And the verdict is@dots{} Integer powers were more
2329 accurate; in fact, the result was only off by one unit in the
2332 @cindex Guard digits
2333 Calc does many of its internal calculations to a slightly higher
2334 precision, but it doesn't always bump the precision up enough.
2335 In each case, Calc added about two digits of precision during
2336 its calculation and then rounded back down to 12 digits
2337 afterward. In one case, it was enough; in the other, it
2338 wasn't. If you really need @var{x} digits of precision, it
2339 never hurts to do the calculation with a few extra guard digits.
2341 What if we want guard digits but don't want to look at them?
2342 We can set the @dfn{float format}. Calc supports four major
2343 formats for floating-point numbers, called @dfn{normal},
2344 @dfn{fixed-point}, @dfn{scientific notation}, and @dfn{engineering
2345 notation}. You get them by pressing @w{@kbd{d n}}, @kbd{d f},
2346 @kbd{d s}, and @kbd{d e}, respectively. In each case, you can
2347 supply a numeric prefix argument which says how many digits
2348 should be displayed. As an example, let's put a few numbers
2349 onto the stack and try some different display modes. First,
2350 use @kbd{M-0 @key{DEL}} to clear the stack, then enter the four
2355 4: 12345 4: 12345 4: 12345 4: 12345 4: 12345
2356 3: 12345. 3: 12300. 3: 1.2345e4 3: 1.23e4 3: 12345.000
2357 2: 123.45 2: 123. 2: 1.2345e2 2: 1.23e2 2: 123.450
2358 1: 12.345 1: 12.3 1: 1.2345e1 1: 1.23e1 1: 12.345
2361 d n M-3 d n d s M-3 d s M-3 d f
2366 Notice that when we typed @kbd{M-3 d n}, the numbers were rounded down
2367 to three significant digits, but then when we typed @kbd{d s} all
2368 five significant figures reappeared. The float format does not
2369 affect how numbers are stored, it only affects how they are
2370 displayed. Only the current precision governs the actual rounding
2371 of numbers in the Calculator's memory.
2373 Engineering notation, not shown here, is like scientific notation
2374 except the exponent (the power-of-ten part) is always adjusted to be
2375 a multiple of three (as in ``kilo,'' ``micro,'' etc.). As a result
2376 there will be one, two, or three digits before the decimal point.
2378 Whenever you change a display-related mode, Calc redraws everything
2379 in the stack. This may be slow if there are many things on the stack,
2380 so Calc allows you to type shift-@kbd{H} before any mode command to
2381 prevent it from updating the stack. Anything Calc displays after the
2382 mode-changing command will appear in the new format.
2386 4: 12345 4: 12345 4: 12345 4: 12345 4: 12345
2387 3: 12345.000 3: 12345.000 3: 12345.000 3: 1.2345e4 3: 12345.
2388 2: 123.450 2: 123.450 2: 1.2345e1 2: 1.2345e1 2: 123.45
2389 1: 12.345 1: 1.2345e1 1: 1.2345e2 1: 1.2345e2 1: 12.345
2392 H d s @key{DEL} U @key{TAB} d @key{SPC} d n
2397 Here the @kbd{H d s} command changes to scientific notation but without
2398 updating the screen. Deleting the top stack entry and undoing it back
2399 causes it to show up in the new format; swapping the top two stack
2400 entries reformats both entries. The @kbd{d @key{SPC}} command refreshes the
2401 whole stack. The @kbd{d n} command changes back to the normal float
2402 format; since it doesn't have an @kbd{H} prefix, it also updates all
2403 the stack entries to be in @kbd{d n} format.
2405 Notice that the integer @expr{12345} was not affected by any
2406 of the float formats. Integers are integers, and are always
2409 @cindex Large numbers, readability
2410 Large integers have their own problems. Let's look back at
2411 the result of @kbd{2^3^4}.
2414 2417851639229258349412352
2418 Quick---how many digits does this have? Try typing @kbd{d g}:
2421 2,417,851,639,229,258,349,412,352
2425 Now how many digits does this have? It's much easier to tell!
2426 We can actually group digits into clumps of any size. Some
2427 people prefer @kbd{M-5 d g}:
2430 24178,51639,22925,83494,12352
2433 Let's see what happens to floating-point numbers when they are grouped.
2434 First, type @kbd{p 25 @key{RET}} to make sure we have enough precision
2435 to get ourselves into trouble. Now, type @kbd{1e13 /}:
2438 24,17851,63922.9258349412352
2442 The integer part is grouped but the fractional part isn't. Now try
2443 @kbd{M-- M-5 d g} (that's meta-minus-sign, meta-five):
2446 24,17851,63922.92583,49412,352
2449 If you find it hard to tell the decimal point from the commas, try
2450 changing the grouping character to a space with @kbd{d , @key{SPC}}:
2453 24 17851 63922.92583 49412 352
2456 Type @kbd{d , ,} to restore the normal grouping character, then
2457 @kbd{d g} again to turn grouping off. Also, press @kbd{p 12} to
2458 restore the default precision.
2460 Press @kbd{U} enough times to get the original big integer back.
2461 (Notice that @kbd{U} does not undo each mode-setting command; if
2462 you want to undo a mode-setting command, you have to do it yourself.)
2463 Now, type @kbd{d r 16 @key{RET}}:
2466 16#200000000000000000000
2470 The number is now displayed in @dfn{hexadecimal}, or ``base-16'' form.
2471 Suddenly it looks pretty simple; this should be no surprise, since we
2472 got this number by computing a power of two, and 16 is a power of 2.
2473 In fact, we can use @w{@kbd{d r 2 @key{RET}}} to see it in actual binary
2477 2#1000000000000000000000000000000000000000000000000000000 @dots{}
2481 We don't have enough space here to show all the zeros! They won't
2482 fit on a typical screen, either, so you will have to use horizontal
2483 scrolling to see them all. Press @kbd{<} and @kbd{>} to scroll the
2484 stack window left and right by half its width. Another way to view
2485 something large is to press @kbd{`} (back-quote) to edit the top of
2486 stack in a separate window. (Press @kbd{C-c C-c} when you are done.)
2488 You can enter non-decimal numbers using the @kbd{#} symbol, too.
2489 Let's see what the hexadecimal number @samp{5FE} looks like in
2490 binary. Type @kbd{16#5FE} (the letters can be typed in upper or
2491 lower case; they will always appear in upper case). It will also
2492 help to turn grouping on with @kbd{d g}:
2498 Notice that @kbd{d g} groups by fours by default if the display radix
2499 is binary or hexadecimal, but by threes if it is decimal, octal, or any
2502 Now let's see that number in decimal; type @kbd{d r 10}:
2508 Numbers are not @emph{stored} with any particular radix attached. They're
2509 just numbers; they can be entered in any radix, and are always displayed
2510 in whatever radix you've chosen with @kbd{d r}. The current radix applies
2511 to integers, fractions, and floats.
2513 @cindex Roundoff errors, in non-decimal numbers
2514 (@bullet{}) @strong{Exercise 1.} Your friend Joe tried to enter one-third
2515 as @samp{3#0.1} in @kbd{d r 3} mode with a precision of 12. He got
2516 @samp{3#0.0222222...} (with 25 2's) in the display. When he multiplied
2517 that by three, he got @samp{3#0.222222...} instead of the expected
2518 @samp{3#1}. Next, Joe entered @samp{3#0.2} and, to his great relief,
2519 saw @samp{3#0.2} on the screen. But when he typed @kbd{2 /}, he got
2520 @samp{3#0.10000001} (some zeros omitted). What's going on here?
2521 @xref{Modes Answer 1, 1}. (@bullet{})
2523 @cindex Scientific notation, in non-decimal numbers
2524 (@bullet{}) @strong{Exercise 2.} Scientific notation works in non-decimal
2525 modes in the natural way (the exponent is a power of the radix instead of
2526 a power of ten, although the exponent itself is always written in decimal).
2527 Thus @samp{8#1.23e3 = 8#1230.0}. Suppose we have the hexadecimal number
2528 @samp{f.e8f} times 16 to the 15th power: We write @samp{16#f.e8fe15}.
2529 What is wrong with this picture? What could we write instead that would
2530 work better? @xref{Modes Answer 2, 2}. (@bullet{})
2532 The @kbd{m} prefix key has another set of modes, relating to the way
2533 Calc interprets your inputs and does computations. Whereas @kbd{d}-prefix
2534 modes generally affect the way things look, @kbd{m}-prefix modes affect
2535 the way they are actually computed.
2537 The most popular @kbd{m}-prefix mode is the @dfn{angular mode}. Notice
2538 the @samp{Deg} indicator in the mode line. This means that if you use
2539 a command that interprets a number as an angle, it will assume the
2540 angle is measured in degrees. For example,
2544 1: 45 1: 0.707106781187 1: 0.500000000001 1: 0.5
2552 The shift-@kbd{S} command computes the sine of an angle. The sine
2554 @texline @math{\sqrt{2}/2};
2555 @infoline @expr{sqrt(2)/2};
2556 squaring this yields @expr{2/4 = 0.5}. However, there has been a slight
2557 roundoff error because the representation of
2558 @texline @math{\sqrt{2}/2}
2559 @infoline @expr{sqrt(2)/2}
2560 wasn't exact. The @kbd{c 1} command is a handy way to clean up numbers
2561 in this case; it temporarily reduces the precision by one digit while it
2562 re-rounds the number on the top of the stack.
2564 @cindex Roundoff errors, examples
2565 (@bullet{}) @strong{Exercise 3.} Your friend Joe computed the sine
2566 of 45 degrees as shown above, then, hoping to avoid an inexact
2567 result, he increased the precision to 16 digits before squaring.
2568 What happened? @xref{Modes Answer 3, 3}. (@bullet{})
2570 To do this calculation in radians, we would type @kbd{m r} first.
2571 (The indicator changes to @samp{Rad}.) 45 degrees corresponds to
2572 @cpiover{4} radians. To get @cpi{}, press the @kbd{P} key. (Once
2573 again, this is a shifted capital @kbd{P}. Remember, unshifted
2574 @kbd{p} sets the precision.)
2578 1: 3.14159265359 1: 0.785398163398 1: 0.707106781187
2585 Likewise, inverse trigonometric functions generate results in
2586 either radians or degrees, depending on the current angular mode.
2590 1: 0.707106781187 1: 0.785398163398 1: 45.
2593 .5 Q m r I S m d U I S
2598 Here we compute the Inverse Sine of
2599 @texline @math{\sqrt{0.5}},
2600 @infoline @expr{sqrt(0.5)},
2601 first in radians, then in degrees.
2603 Use @kbd{c d} and @kbd{c r} to convert a number from radians to degrees
2608 1: 45 1: 0.785398163397 1: 45.
2615 Another interesting mode is @dfn{Fraction mode}. Normally,
2616 dividing two integers produces a floating-point result if the
2617 quotient can't be expressed as an exact integer. Fraction mode
2618 causes integer division to produce a fraction, i.e., a rational
2623 2: 12 1: 1.33333333333 1: 4:3
2627 12 @key{RET} 9 / m f U / m f
2632 In the first case, we get an approximate floating-point result.
2633 In the second case, we get an exact fractional result (four-thirds).
2635 You can enter a fraction at any time using @kbd{:} notation.
2636 (Calc uses @kbd{:} instead of @kbd{/} as the fraction separator
2637 because @kbd{/} is already used to divide the top two stack
2638 elements.) Calculations involving fractions will always
2639 produce exact fractional results; Fraction mode only says
2640 what to do when dividing two integers.
2642 @cindex Fractions vs. floats
2643 @cindex Floats vs. fractions
2644 (@bullet{}) @strong{Exercise 4.} If fractional arithmetic is exact,
2645 why would you ever use floating-point numbers instead?
2646 @xref{Modes Answer 4, 4}. (@bullet{})
2648 Typing @kbd{m f} doesn't change any existing values in the stack.
2649 In the above example, we had to Undo the division and do it over
2650 again when we changed to Fraction mode. But if you use the
2651 evaluates-to operator you can get commands like @kbd{m f} to
2656 1: 12 / 9 => 1.33333333333 1: 12 / 9 => 1.333 1: 12 / 9 => 4:3
2659 ' 12/9 => @key{RET} p 4 @key{RET} m f
2664 In this example, the righthand side of the @samp{=>} operator
2665 on the stack is recomputed when we change the precision, then
2666 again when we change to Fraction mode. All @samp{=>} expressions
2667 on the stack are recomputed every time you change any mode that
2668 might affect their values.
2670 @node Arithmetic Tutorial, Vector/Matrix Tutorial, Basic Tutorial, Tutorial
2671 @section Arithmetic Tutorial
2674 In this section, we explore the arithmetic and scientific functions
2675 available in the Calculator.
2677 The standard arithmetic commands are @kbd{+}, @kbd{-}, @kbd{*}, @kbd{/},
2678 and @kbd{^}. Each normally takes two numbers from the top of the stack
2679 and pushes back a result. The @kbd{n} and @kbd{&} keys perform
2680 change-sign and reciprocal operations, respectively.
2684 1: 5 1: 0.2 1: 5. 1: -5. 1: 5.
2691 @cindex Binary operators
2692 You can apply a ``binary operator'' like @kbd{+} across any number of
2693 stack entries by giving it a numeric prefix. You can also apply it
2694 pairwise to several stack elements along with the top one if you use
2699 3: 2 1: 9 3: 2 4: 2 3: 12
2700 2: 3 . 2: 3 3: 3 2: 13
2701 1: 4 1: 4 2: 4 1: 14
2705 2 @key{RET} 3 @key{RET} 4 M-3 + U 10 M-- M-3 +
2709 @cindex Unary operators
2710 You can apply a ``unary operator'' like @kbd{&} to the top @var{n}
2711 stack entries with a numeric prefix, too.
2716 2: 3 2: 0.333333333333 2: 3.
2720 2 @key{RET} 3 @key{RET} 4 M-3 & M-2 &
2724 Notice that the results here are left in floating-point form.
2725 We can convert them back to integers by pressing @kbd{F}, the
2726 ``floor'' function. This function rounds down to the next lower
2727 integer. There is also @kbd{R}, which rounds to the nearest
2745 Since dividing-and-flooring (i.e., ``integer quotient'') is such a
2746 common operation, Calc provides a special command for that purpose, the
2747 backslash @kbd{\}. Another common arithmetic operator is @kbd{%}, which
2748 computes the remainder that would arise from a @kbd{\} operation, i.e.,
2749 the ``modulo'' of two numbers. For example,
2753 2: 1234 1: 12 2: 1234 1: 34
2757 1234 @key{RET} 100 \ U %
2761 These commands actually work for any real numbers, not just integers.
2765 2: 3.1415 1: 3 2: 3.1415 1: 0.1415
2769 3.1415 @key{RET} 1 \ U %
2773 (@bullet{}) @strong{Exercise 1.} The @kbd{\} command would appear to be a
2774 frill, since you could always do the same thing with @kbd{/ F}. Think
2775 of a situation where this is not true---@kbd{/ F} would be inadequate.
2776 Now think of a way you could get around the problem if Calc didn't
2777 provide a @kbd{\} command. @xref{Arithmetic Answer 1, 1}. (@bullet{})
2779 We've already seen the @kbd{Q} (square root) and @kbd{S} (sine)
2780 commands. Other commands along those lines are @kbd{C} (cosine),
2781 @kbd{T} (tangent), @kbd{E} (@expr{e^x}) and @kbd{L} (natural
2782 logarithm). These can be modified by the @kbd{I} (inverse) and
2783 @kbd{H} (hyperbolic) prefix keys.
2785 Let's compute the sine and cosine of an angle, and verify the
2787 @texline @math{\sin^2x + \cos^2x = 1}.
2788 @infoline @expr{sin(x)^2 + cos(x)^2 = 1}.
2789 We'll arbitrarily pick @mathit{-64} degrees as a good value for @expr{x}.
2790 With the angular mode set to degrees (type @w{@kbd{m d}}), do:
2794 2: -64 2: -64 2: -0.89879 2: -0.89879 1: 1.
2795 1: -64 1: -0.89879 1: -64 1: 0.43837 .
2798 64 n @key{RET} @key{RET} S @key{TAB} C f h
2803 (For brevity, we're showing only five digits of the results here.
2804 You can of course do these calculations to any precision you like.)
2806 Remember, @kbd{f h} is the @code{calc-hypot}, or square-root of sum
2807 of squares, command.
2810 @texline @math{\displaystyle\tan x = {\sin x \over \cos x}}.
2811 @infoline @expr{tan(x) = sin(x) / cos(x)}.
2815 2: -0.89879 1: -2.0503 1: -64.
2823 A physical interpretation of this calculation is that if you move
2824 @expr{0.89879} units downward and @expr{0.43837} units to the right,
2825 your direction of motion is @mathit{-64} degrees from horizontal. Suppose
2826 we move in the opposite direction, up and to the left:
2830 2: -0.89879 2: 0.89879 1: -2.0503 1: -64.
2831 1: 0.43837 1: -0.43837 . .
2839 How can the angle be the same? The answer is that the @kbd{/} operation
2840 loses information about the signs of its inputs. Because the quotient
2841 is negative, we know exactly one of the inputs was negative, but we
2842 can't tell which one. There is an @kbd{f T} [@code{arctan2}] function which
2843 computes the inverse tangent of the quotient of a pair of numbers.
2844 Since you feed it the two original numbers, it has enough information
2845 to give you a full 360-degree answer.
2849 2: 0.89879 1: 116. 3: 116. 2: 116. 1: 180.
2850 1: -0.43837 . 2: -0.89879 1: -64. .
2854 U U f T M-@key{RET} M-2 n f T -
2859 The resulting angles differ by 180 degrees; in other words, they
2860 point in opposite directions, just as we would expect.
2862 The @key{META}-@key{RET} we used in the third step is the
2863 ``last-arguments'' command. It is sort of like Undo, except that it
2864 restores the arguments of the last command to the stack without removing
2865 the command's result. It is useful in situations like this one,
2866 where we need to do several operations on the same inputs. We could
2867 have accomplished the same thing by using @kbd{M-2 @key{RET}} to duplicate
2868 the top two stack elements right after the @kbd{U U}, then a pair of
2869 @kbd{M-@key{TAB}} commands to cycle the 116 up around the duplicates.
2871 A similar identity is supposed to hold for hyperbolic sines and cosines,
2872 except that it is the @emph{difference}
2873 @texline @math{\cosh^2x - \sinh^2x}
2874 @infoline @expr{cosh(x)^2 - sinh(x)^2}
2875 that always equals one. Let's try to verify this identity.
2879 2: -64 2: -64 2: -64 2: 9.7192e54 2: 9.7192e54
2880 1: -64 1: -3.1175e27 1: 9.7192e54 1: -64 1: 9.7192e54
2883 64 n @key{RET} @key{RET} H C 2 ^ @key{TAB} H S 2 ^
2888 @cindex Roundoff errors, examples
2889 Something's obviously wrong, because when we subtract these numbers
2890 the answer will clearly be zero! But if you think about it, if these
2891 numbers @emph{did} differ by one, it would be in the 55th decimal
2892 place. The difference we seek has been lost entirely to roundoff
2895 We could verify this hypothesis by doing the actual calculation with,
2896 say, 60 decimal places of precision. This will be slow, but not
2897 enormously so. Try it if you wish; sure enough, the answer is
2898 0.99999, reasonably close to 1.
2900 Of course, a more reasonable way to verify the identity is to use
2901 a more reasonable value for @expr{x}!
2903 @cindex Common logarithm
2904 Some Calculator commands use the Hyperbolic prefix for other purposes.
2905 The logarithm and exponential functions, for example, work to the base
2906 @expr{e} normally but use base-10 instead if you use the Hyperbolic
2911 1: 1000 1: 6.9077 1: 1000 1: 3
2919 First, we mistakenly compute a natural logarithm. Then we undo
2920 and compute a common logarithm instead.
2922 The @kbd{B} key computes a general base-@var{b} logarithm for any
2927 2: 1000 1: 3 1: 1000. 2: 1000. 1: 6.9077
2928 1: 10 . . 1: 2.71828 .
2931 1000 @key{RET} 10 B H E H P B
2936 Here we first use @kbd{B} to compute the base-10 logarithm, then use
2937 the ``hyperbolic'' exponential as a cheap hack to recover the number
2938 1000, then use @kbd{B} again to compute the natural logarithm. Note
2939 that @kbd{P} with the hyperbolic prefix pushes the constant @expr{e}
2942 You may have noticed that both times we took the base-10 logarithm
2943 of 1000, we got an exact integer result. Calc always tries to give
2944 an exact rational result for calculations involving rational numbers
2945 where possible. But when we used @kbd{H E}, the result was a
2946 floating-point number for no apparent reason. In fact, if we had
2947 computed @kbd{10 @key{RET} 3 ^} we @emph{would} have gotten an
2948 exact integer 1000. But the @kbd{H E} command is rigged to generate
2949 a floating-point result all of the time so that @kbd{1000 H E} will
2950 not waste time computing a thousand-digit integer when all you
2951 probably wanted was @samp{1e1000}.
2953 (@bullet{}) @strong{Exercise 2.} Find a pair of integer inputs to
2954 the @kbd{B} command for which Calc could find an exact rational
2955 result but doesn't. @xref{Arithmetic Answer 2, 2}. (@bullet{})
2957 The Calculator also has a set of functions relating to combinatorics
2958 and statistics. You may be familiar with the @dfn{factorial} function,
2959 which computes the product of all the integers up to a given number.
2963 1: 100 1: 93326215443... 1: 100. 1: 9.3326e157
2971 Recall, the @kbd{c f} command converts the integer or fraction at the
2972 top of the stack to floating-point format. If you take the factorial
2973 of a floating-point number, you get a floating-point result
2974 accurate to the current precision. But if you give @kbd{!} an
2975 exact integer, you get an exact integer result (158 digits long
2978 If you take the factorial of a non-integer, Calc uses a generalized
2979 factorial function defined in terms of Euler's Gamma function
2980 @texline @math{\Gamma(n)}
2981 @infoline @expr{gamma(n)}
2982 (which is itself available as the @kbd{f g} command).
2986 3: 4. 3: 24. 1: 5.5 1: 52.342777847
2987 2: 4.5 2: 52.3427777847 . .
2991 M-3 ! M-0 @key{DEL} 5.5 f g
2996 Here we verify the identity
2997 @texline @math{n! = \Gamma(n+1)}.
2998 @infoline @expr{@var{n}!@: = gamma(@var{n}+1)}.
3000 The binomial coefficient @var{n}-choose-@var{m}
3001 @texline or @math{\displaystyle {n \choose m}}
3003 @texline @math{\displaystyle {n! \over m! \, (n-m)!}}
3004 @infoline @expr{n!@: / m!@: (n-m)!}
3005 for all reals @expr{n} and @expr{m}. The intermediate results in this
3006 formula can become quite large even if the final result is small; the
3007 @kbd{k c} command computes a binomial coefficient in a way that avoids
3008 large intermediate values.
3010 The @kbd{k} prefix key defines several common functions out of
3011 combinatorics and number theory. Here we compute the binomial
3012 coefficient 30-choose-20, then determine its prime factorization.
3016 2: 30 1: 30045015 1: [3, 3, 5, 7, 11, 13, 23, 29]
3020 30 @key{RET} 20 k c k f
3025 You can verify these prime factors by using @kbd{V R *} to multiply
3026 together the elements of this vector. The result is the original
3030 Suppose a program you are writing needs a hash table with at least
3031 10000 entries. It's best to use a prime number as the actual size
3032 of a hash table. Calc can compute the next prime number after 10000:
3036 1: 10000 1: 10007 1: 9973
3044 Just for kicks we've also computed the next prime @emph{less} than
3047 @c [fix-ref Financial Functions]
3048 @xref{Financial Functions}, for a description of the Calculator
3049 commands that deal with business and financial calculations (functions
3050 like @code{pv}, @code{rate}, and @code{sln}).
3052 @c [fix-ref Binary Number Functions]
3053 @xref{Binary Functions}, to read about the commands for operating
3054 on binary numbers (like @code{and}, @code{xor}, and @code{lsh}).
3056 @node Vector/Matrix Tutorial, Types Tutorial, Arithmetic Tutorial, Tutorial
3057 @section Vector/Matrix Tutorial
3060 A @dfn{vector} is a list of numbers or other Calc data objects.
3061 Calc provides a large set of commands that operate on vectors. Some
3062 are familiar operations from vector analysis. Others simply treat
3063 a vector as a list of objects.
3066 * Vector Analysis Tutorial::
3071 @node Vector Analysis Tutorial, Matrix Tutorial, Vector/Matrix Tutorial, Vector/Matrix Tutorial
3072 @subsection Vector Analysis
3075 If you add two vectors, the result is a vector of the sums of the
3076 elements, taken pairwise.
3080 1: [1, 2, 3] 2: [1, 2, 3] 1: [8, 8, 3]
3084 [1,2,3] s 1 [7 6 0] s 2 +
3089 Note that we can separate the vector elements with either commas or
3090 spaces. This is true whether we are using incomplete vectors or
3091 algebraic entry. The @kbd{s 1} and @kbd{s 2} commands save these
3092 vectors so we can easily reuse them later.
3094 If you multiply two vectors, the result is the sum of the products
3095 of the elements taken pairwise. This is called the @dfn{dot product}
3109 The dot product of two vectors is equal to the product of their
3110 lengths times the cosine of the angle between them. (Here the vector
3111 is interpreted as a line from the origin @expr{(0,0,0)} to the
3112 specified point in three-dimensional space.) The @kbd{A}
3113 (absolute value) command can be used to compute the length of a
3118 3: 19 3: 19 1: 0.550782 1: 56.579
3119 2: [1, 2, 3] 2: 3.741657 . .
3120 1: [7, 6, 0] 1: 9.219544
3123 M-@key{RET} M-2 A * / I C
3128 First we recall the arguments to the dot product command, then
3129 we compute the absolute values of the top two stack entries to
3130 obtain the lengths of the vectors, then we divide the dot product
3131 by the product of the lengths to get the cosine of the angle.
3132 The inverse cosine finds that the angle between the vectors
3133 is about 56 degrees.
3135 @cindex Cross product
3136 @cindex Perpendicular vectors
3137 The @dfn{cross product} of two vectors is a vector whose length
3138 is the product of the lengths of the inputs times the sine of the
3139 angle between them, and whose direction is perpendicular to both
3140 input vectors. Unlike the dot product, the cross product is
3141 defined only for three-dimensional vectors. Let's double-check
3142 our computation of the angle using the cross product.
3146 2: [1, 2, 3] 3: [-18, 21, -8] 1: [-0.52, 0.61, -0.23] 1: 56.579
3147 1: [7, 6, 0] 2: [1, 2, 3] . .
3151 r 1 r 2 V C s 3 M-@key{RET} M-2 A * / A I S
3156 First we recall the original vectors and compute their cross product,
3157 which we also store for later reference. Now we divide the vector
3158 by the product of the lengths of the original vectors. The length of
3159 this vector should be the sine of the angle; sure enough, it is!
3161 @c [fix-ref General Mode Commands]
3162 Vector-related commands generally begin with the @kbd{v} prefix key.
3163 Some are uppercase letters and some are lowercase. To make it easier
3164 to type these commands, the shift-@kbd{V} prefix key acts the same as
3165 the @kbd{v} key. (@xref{General Mode Commands}, for a way to make all
3166 prefix keys have this property.)
3168 If we take the dot product of two perpendicular vectors we expect
3169 to get zero, since the cosine of 90 degrees is zero. Let's check
3170 that the cross product is indeed perpendicular to both inputs:
3174 2: [1, 2, 3] 1: 0 2: [7, 6, 0] 1: 0
3175 1: [-18, 21, -8] . 1: [-18, 21, -8] .
3178 r 1 r 3 * @key{DEL} r 2 r 3 *
3182 @cindex Normalizing a vector
3183 @cindex Unit vectors
3184 (@bullet{}) @strong{Exercise 1.} Given a vector on the top of the
3185 stack, what keystrokes would you use to @dfn{normalize} the
3186 vector, i.e., to reduce its length to one without changing its
3187 direction? @xref{Vector Answer 1, 1}. (@bullet{})
3189 (@bullet{}) @strong{Exercise 2.} Suppose a certain particle can be
3190 at any of several positions along a ruler. You have a list of
3191 those positions in the form of a vector, and another list of the
3192 probabilities for the particle to be at the corresponding positions.
3193 Find the average position of the particle.
3194 @xref{Vector Answer 2, 2}. (@bullet{})
3196 @node Matrix Tutorial, List Tutorial, Vector Analysis Tutorial, Vector/Matrix Tutorial
3197 @subsection Matrices
3200 A @dfn{matrix} is just a vector of vectors, all the same length.
3201 This means you can enter a matrix using nested brackets. You can
3202 also use the semicolon character to enter a matrix. We'll show
3207 1: [ [ 1, 2, 3 ] 1: [ [ 1, 2, 3 ]
3208 [ 4, 5, 6 ] ] [ 4, 5, 6 ] ]
3211 [[1 2 3] [4 5 6]] ' [1 2 3; 4 5 6] @key{RET}
3216 We'll be using this matrix again, so type @kbd{s 4} to save it now.
3218 Note that semicolons work with incomplete vectors, but they work
3219 better in algebraic entry. That's why we use the apostrophe in
3222 When two matrices are multiplied, the lefthand matrix must have
3223 the same number of columns as the righthand matrix has rows.
3224 Row @expr{i}, column @expr{j} of the result is effectively the
3225 dot product of row @expr{i} of the left matrix by column @expr{j}
3226 of the right matrix.
3228 If we try to duplicate this matrix and multiply it by itself,
3229 the dimensions are wrong and the multiplication cannot take place:
3233 1: [ [ 1, 2, 3 ] * [ [ 1, 2, 3 ]
3234 [ 4, 5, 6 ] ] [ 4, 5, 6 ] ]
3242 Though rather hard to read, this is a formula which shows the product
3243 of two matrices. The @samp{*} function, having invalid arguments, has
3244 been left in symbolic form.
3246 We can multiply the matrices if we @dfn{transpose} one of them first.
3250 2: [ [ 1, 2, 3 ] 1: [ [ 14, 32 ] 1: [ [ 17, 22, 27 ]
3251 [ 4, 5, 6 ] ] [ 32, 77 ] ] [ 22, 29, 36 ]
3252 1: [ [ 1, 4 ] . [ 27, 36, 45 ] ]
3257 U v t * U @key{TAB} *
3261 Matrix multiplication is not commutative; indeed, switching the
3262 order of the operands can even change the dimensions of the result
3263 matrix, as happened here!
3265 If you multiply a plain vector by a matrix, it is treated as a
3266 single row or column depending on which side of the matrix it is
3267 on. The result is a plain vector which should also be interpreted
3268 as a row or column as appropriate.
3272 2: [ [ 1, 2, 3 ] 1: [14, 32]
3281 Multiplying in the other order wouldn't work because the number of
3282 rows in the matrix is different from the number of elements in the
3285 (@bullet{}) @strong{Exercise 1.} Use @samp{*} to sum along the rows
3287 @texline @math{2\times3}
3289 matrix to get @expr{[6, 15]}. Now use @samp{*} to sum along the columns
3290 to get @expr{[5, 7, 9]}.
3291 @xref{Matrix Answer 1, 1}. (@bullet{})
3293 @cindex Identity matrix
3294 An @dfn{identity matrix} is a square matrix with ones along the
3295 diagonal and zeros elsewhere. It has the property that multiplication
3296 by an identity matrix, on the left or on the right, always produces
3297 the original matrix.
3301 1: [ [ 1, 2, 3 ] 2: [ [ 1, 2, 3 ] 1: [ [ 1, 2, 3 ]
3302 [ 4, 5, 6 ] ] [ 4, 5, 6 ] ] [ 4, 5, 6 ] ]
3303 . 1: [ [ 1, 0, 0 ] .
3308 r 4 v i 3 @key{RET} *
3312 If a matrix is square, it is often possible to find its @dfn{inverse},
3313 that is, a matrix which, when multiplied by the original matrix, yields
3314 an identity matrix. The @kbd{&} (reciprocal) key also computes the
3315 inverse of a matrix.
3319 1: [ [ 1, 2, 3 ] 1: [ [ -2.4, 1.2, -0.2 ]
3320 [ 4, 5, 6 ] [ 2.8, -1.4, 0.4 ]
3321 [ 7, 6, 0 ] ] [ -0.73333, 0.53333, -0.2 ] ]
3329 The vertical bar @kbd{|} @dfn{concatenates} numbers, vectors, and
3330 matrices together. Here we have used it to add a new row onto
3331 our matrix to make it square.
3333 We can multiply these two matrices in either order to get an identity.
3337 1: [ [ 1., 0., 0. ] 1: [ [ 1., 0., 0. ]
3338 [ 0., 1., 0. ] [ 0., 1., 0. ]
3339 [ 0., 0., 1. ] ] [ 0., 0., 1. ] ]
3342 M-@key{RET} * U @key{TAB} *
3346 @cindex Systems of linear equations
3347 @cindex Linear equations, systems of
3348 Matrix inverses are related to systems of linear equations in algebra.
3349 Suppose we had the following set of equations:
3363 $$ \openup1\jot \tabskip=0pt plus1fil
3364 \halign to\displaywidth{\tabskip=0pt
3365 $\hfil#$&$\hfil{}#{}$&
3366 $\hfil#$&$\hfil{}#{}$&
3367 $\hfil#$&${}#\hfil$\tabskip=0pt plus1fil\cr
3376 This can be cast into the matrix equation,
3381 [ [ 1, 2, 3 ] [ [ a ] [ [ 6 ]
3382 [ 4, 5, 6 ] * [ b ] = [ 2 ]
3383 [ 7, 6, 0 ] ] [ c ] ] [ 3 ] ]
3390 $$ \pmatrix{ 1 & 2 & 3 \cr 4 & 5 & 6 \cr 7 & 6 & 0 }
3392 \pmatrix{ a \cr b \cr c } = \pmatrix{ 6 \cr 2 \cr 3 }
3397 We can solve this system of equations by multiplying both sides by the
3398 inverse of the matrix. Calc can do this all in one step:
3402 2: [6, 2, 3] 1: [-12.6, 15.2, -3.93333]
3413 The result is the @expr{[a, b, c]} vector that solves the equations.
3414 (Dividing by a square matrix is equivalent to multiplying by its
3417 Let's verify this solution:
3421 2: [ [ 1, 2, 3 ] 1: [6., 2., 3.]
3424 1: [-12.6, 15.2, -3.93333]
3432 Note that we had to be careful about the order in which we multiplied
3433 the matrix and vector. If we multiplied in the other order, Calc would
3434 assume the vector was a row vector in order to make the dimensions
3435 come out right, and the answer would be incorrect. If you
3436 don't feel safe letting Calc take either interpretation of your
3437 vectors, use explicit
3438 @texline @math{N\times1}
3441 @texline @math{1\times N}
3443 matrices instead. In this case, you would enter the original column
3444 vector as @samp{[[6], [2], [3]]} or @samp{[6; 2; 3]}.
3446 (@bullet{}) @strong{Exercise 2.} Algebraic entry allows you to make
3447 vectors and matrices that include variables. Solve the following
3448 system of equations to get expressions for @expr{x} and @expr{y}
3449 in terms of @expr{a} and @expr{b}.
3462 $$ \eqalign{ x &+ a y = 6 \cr
3469 @xref{Matrix Answer 2, 2}. (@bullet{})
3471 @cindex Least-squares for over-determined systems
3472 @cindex Over-determined systems of equations
3473 (@bullet{}) @strong{Exercise 3.} A system of equations is ``over-determined''
3474 if it has more equations than variables. It is often the case that
3475 there are no values for the variables that will satisfy all the
3476 equations at once, but it is still useful to find a set of values
3477 which ``nearly'' satisfy all the equations. In terms of matrix equations,
3478 you can't solve @expr{A X = B} directly because the matrix @expr{A}
3479 is not square for an over-determined system. Matrix inversion works
3480 only for square matrices. One common trick is to multiply both sides
3481 on the left by the transpose of @expr{A}:
3483 @samp{trn(A)*A*X = trn(A)*B}.
3487 $A^T A \, X = A^T B$, where $A^T$ is the transpose \samp{trn(A)}.
3490 @texline @math{A^T A}
3491 @infoline @expr{trn(A)*A}
3492 is a square matrix so a solution is possible. It turns out that the
3493 @expr{X} vector you compute in this way will be a ``least-squares''
3494 solution, which can be regarded as the ``closest'' solution to the set
3495 of equations. Use Calc to solve the following over-determined
3511 $$ \openup1\jot \tabskip=0pt plus1fil
3512 \halign to\displaywidth{\tabskip=0pt
3513 $\hfil#$&$\hfil{}#{}$&
3514 $\hfil#$&$\hfil{}#{}$&
3515 $\hfil#$&${}#\hfil$\tabskip=0pt plus1fil\cr
3519 2a&+&4b&+&6c&=11 \cr}
3525 @xref{Matrix Answer 3, 3}. (@bullet{})
3527 @node List Tutorial, , Matrix Tutorial, Vector/Matrix Tutorial
3528 @subsection Vectors as Lists
3532 Although Calc has a number of features for manipulating vectors and
3533 matrices as mathematical objects, you can also treat vectors as
3534 simple lists of values. For example, we saw that the @kbd{k f}
3535 command returns a vector which is a list of the prime factors of a
3538 You can pack and unpack stack entries into vectors:
3542 3: 10 1: [10, 20, 30] 3: 10
3551 You can also build vectors out of consecutive integers, or out
3552 of many copies of a given value:
3556 1: [1, 2, 3, 4] 2: [1, 2, 3, 4] 2: [1, 2, 3, 4]
3557 . 1: 17 1: [17, 17, 17, 17]
3560 v x 4 @key{RET} 17 v b 4 @key{RET}
3564 You can apply an operator to every element of a vector using the
3569 1: [17, 34, 51, 68] 1: [289, 1156, 2601, 4624] 1: [17, 34, 51, 68]
3577 In the first step, we multiply the vector of integers by the vector
3578 of 17's elementwise. In the second step, we raise each element to
3579 the power two. (The general rule is that both operands must be
3580 vectors of the same length, or else one must be a vector and the
3581 other a plain number.) In the final step, we take the square root
3584 (@bullet{}) @strong{Exercise 1.} Compute a vector of powers of two
3586 @texline @math{2^{-4}}
3587 @infoline @expr{2^-4}
3588 to @expr{2^4}. @xref{List Answer 1, 1}. (@bullet{})
3590 You can also @dfn{reduce} a binary operator across a vector.
3591 For example, reducing @samp{*} computes the product of all the
3592 elements in the vector:
3596 1: 123123 1: [3, 7, 11, 13, 41] 1: 123123
3604 In this example, we decompose 123123 into its prime factors, then
3605 multiply those factors together again to yield the original number.
3607 We could compute a dot product ``by hand'' using mapping and
3612 2: [1, 2, 3] 1: [7, 12, 0] 1: 19
3621 Recalling two vectors from the previous section, we compute the
3622 sum of pairwise products of the elements to get the same answer
3623 for the dot product as before.
3625 A slight variant of vector reduction is the @dfn{accumulate} operation,
3626 @kbd{V U}. This produces a vector of the intermediate results from
3627 a corresponding reduction. Here we compute a table of factorials:
3631 1: [1, 2, 3, 4, 5, 6] 1: [1, 2, 6, 24, 120, 720]
3634 v x 6 @key{RET} V U *
3638 Calc allows vectors to grow as large as you like, although it gets
3639 rather slow if vectors have more than about a hundred elements.
3640 Actually, most of the time is spent formatting these large vectors
3641 for display, not calculating on them. Try the following experiment
3642 (if your computer is very fast you may need to substitute a larger
3647 1: [1, 2, 3, 4, ... 1: [2, 3, 4, 5, ...
3650 v x 500 @key{RET} 1 V M +
3654 Now press @kbd{v .} (the letter @kbd{v}, then a period) and try the
3655 experiment again. In @kbd{v .} mode, long vectors are displayed
3656 ``abbreviated'' like this:
3660 1: [1, 2, 3, ..., 500] 1: [2, 3, 4, ..., 501]
3663 v x 500 @key{RET} 1 V M +
3668 (where now the @samp{...} is actually part of the Calc display).
3669 You will find both operations are now much faster. But notice that
3670 even in @w{@kbd{v .}} mode, the full vectors are still shown in the Trail.
3671 Type @w{@kbd{t .}} to cause the trail to abbreviate as well, and try the
3672 experiment one more time. Operations on long vectors are now quite
3673 fast! (But of course if you use @kbd{t .} you will lose the ability
3674 to get old vectors back using the @kbd{t y} command.)
3676 An easy way to view a full vector when @kbd{v .} mode is active is
3677 to press @kbd{`} (back-quote) to edit the vector; editing always works
3678 with the full, unabbreviated value.
3680 @cindex Least-squares for fitting a straight line
3681 @cindex Fitting data to a line
3682 @cindex Line, fitting data to
3683 @cindex Data, extracting from buffers
3684 @cindex Columns of data, extracting
3685 As a larger example, let's try to fit a straight line to some data,
3686 using the method of least squares. (Calc has a built-in command for
3687 least-squares curve fitting, but we'll do it by hand here just to
3688 practice working with vectors.) Suppose we have the following list
3689 of values in a file we have loaded into Emacs:
3716 If you are reading this tutorial in printed form, you will find it
3717 easiest to press @kbd{C-x * i} to enter the on-line Info version of
3718 the manual and find this table there. (Press @kbd{g}, then type
3719 @kbd{List Tutorial}, to jump straight to this section.)
3721 Position the cursor at the upper-left corner of this table, just
3722 to the left of the @expr{1.34}. Press @kbd{C-@@} to set the mark.
3723 (On your system this may be @kbd{C-2}, @kbd{C-@key{SPC}}, or @kbd{NUL}.)
3724 Now position the cursor to the lower-right, just after the @expr{1.354}.
3725 You have now defined this region as an Emacs ``rectangle.'' Still
3726 in the Info buffer, type @kbd{C-x * r}. This command
3727 (@code{calc-grab-rectangle}) will pop you back into the Calculator, with
3728 the contents of the rectangle you specified in the form of a matrix.
3732 1: [ [ 1.34, 0.234 ]
3739 (You may wish to use @kbd{v .} mode to abbreviate the display of this
3742 We want to treat this as a pair of lists. The first step is to
3743 transpose this matrix into a pair of rows. Remember, a matrix is
3744 just a vector of vectors. So we can unpack the matrix into a pair
3745 of row vectors on the stack.
3749 1: [ [ 1.34, 1.41, 1.49, ... ] 2: [1.34, 1.41, 1.49, ... ]
3750 [ 0.234, 0.298, 0.402, ... ] ] 1: [0.234, 0.298, 0.402, ... ]
3758 Let's store these in quick variables 1 and 2, respectively.
3762 1: [1.34, 1.41, 1.49, ... ] .
3770 (Recall that @kbd{t 2} is a variant of @kbd{s 2} that removes the
3771 stored value from the stack.)
3773 In a least squares fit, the slope @expr{m} is given by the formula
3777 m = (N sum(x y) - sum(x) sum(y)) / (N sum(x^2) - sum(x)^2)
3783 $$ m = {N \sum x y - \sum x \sum y \over
3784 N \sum x^2 - \left( \sum x \right)^2} $$
3790 @texline @math{\sum x}
3791 @infoline @expr{sum(x)}
3792 represents the sum of all the values of @expr{x}. While there is an
3793 actual @code{sum} function in Calc, it's easier to sum a vector using a
3794 simple reduction. First, let's compute the four different sums that
3802 r 1 V R + t 3 r 1 2 V M ^ V R + t 4
3809 1: 13.613 1: 33.36554
3812 r 2 V R + t 5 r 1 r 2 V M * V R + t 6
3818 These are @samp{sum(x)}, @samp{sum(x^2)}, @samp{sum(y)}, and @samp{sum(x y)},
3819 respectively. (We could have used @kbd{*} to compute @samp{sum(x^2)} and
3824 These are $\sum x$, $\sum x^2$, $\sum y$, and $\sum x y$,
3825 respectively. (We could have used \kbd{*} to compute $\sum x^2$ and
3829 Finally, we also need @expr{N}, the number of data points. This is just
3830 the length of either of our lists.
3842 (That's @kbd{v} followed by a lower-case @kbd{l}.)
3844 Now we grind through the formula:
3848 1: 633.94526 2: 633.94526 1: 67.23607
3852 r 7 r 6 * r 3 r 5 * -
3859 2: 67.23607 3: 67.23607 2: 67.23607 1: 0.52141679
3860 1: 1862.0057 2: 1862.0057 1: 128.9488 .
3864 r 7 r 4 * r 3 2 ^ - / t 8
3868 That gives us the slope @expr{m}. The y-intercept @expr{b} can now
3869 be found with the simple formula,
3873 b = (sum(y) - m sum(x)) / N
3879 $$ b = {\sum y - m \sum x \over N} $$
3886 1: 13.613 2: 13.613 1: -8.09358 1: -0.425978
3890 r 5 r 8 r 3 * - r 7 / t 9
3894 Let's ``plot'' this straight line approximation,
3895 @texline @math{y \approx m x + b},
3896 @infoline @expr{m x + b},
3897 and compare it with the original data.
3901 1: [0.699, 0.735, ... ] 1: [0.273, 0.309, ... ]
3909 Notice that multiplying a vector by a constant, and adding a constant
3910 to a vector, can be done without mapping commands since these are
3911 common operations from vector algebra. As far as Calc is concerned,
3912 we've just been doing geometry in 19-dimensional space!
3914 We can subtract this vector from our original @expr{y} vector to get
3915 a feel for the error of our fit. Let's find the maximum error:
3919 1: [0.0387, 0.0112, ... ] 1: [0.0387, 0.0112, ... ] 1: 0.0897
3927 First we compute a vector of differences, then we take the absolute
3928 values of these differences, then we reduce the @code{max} function
3929 across the vector. (The @code{max} function is on the two-key sequence
3930 @kbd{f x}; because it is so common to use @code{max} in a vector
3931 operation, the letters @kbd{X} and @kbd{N} are also accepted for
3932 @code{max} and @code{min} in this context. In general, you answer
3933 the @kbd{V M} or @kbd{V R} prompt with the actual key sequence that
3934 invokes the function you want. You could have typed @kbd{V R f x} or
3935 even @kbd{V R x max @key{RET}} if you had preferred.)
3937 If your system has the GNUPLOT program, you can see graphs of your
3938 data and your straight line to see how well they match. (If you have
3939 GNUPLOT 3.0 or higher, the following instructions will work regardless
3940 of the kind of display you have. Some GNUPLOT 2.0, non-X-windows systems
3941 may require additional steps to view the graphs.)
3943 Let's start by plotting the original data. Recall the ``@var{x}'' and ``@var{y}''
3944 vectors onto the stack and press @kbd{g f}. This ``fast'' graphing
3945 command does everything you need to do for simple, straightforward
3950 2: [1.34, 1.41, 1.49, ... ]
3951 1: [0.234, 0.298, 0.402, ... ]
3958 If all goes well, you will shortly get a new window containing a graph
3959 of the data. (If not, contact your GNUPLOT or Calc installer to find
3960 out what went wrong.) In the X window system, this will be a separate
3961 graphics window. For other kinds of displays, the default is to
3962 display the graph in Emacs itself using rough character graphics.
3963 Press @kbd{q} when you are done viewing the character graphics.
3965 Next, let's add the line we got from our least-squares fit.
3967 (If you are reading this tutorial on-line while running Calc, typing
3968 @kbd{g a} may cause the tutorial to disappear from its window and be
3969 replaced by a buffer named @samp{*Gnuplot Commands*}. The tutorial
3970 will reappear when you terminate GNUPLOT by typing @kbd{g q}.)
3975 2: [1.34, 1.41, 1.49, ... ]
3976 1: [0.273, 0.309, 0.351, ... ]
3979 @key{DEL} r 0 g a g p
3983 It's not very useful to get symbols to mark the data points on this
3984 second curve; you can type @kbd{g S g p} to remove them. Type @kbd{g q}
3985 when you are done to remove the X graphics window and terminate GNUPLOT.
3987 (@bullet{}) @strong{Exercise 2.} An earlier exercise showed how to do
3988 least squares fitting to a general system of equations. Our 19 data
3989 points are really 19 equations of the form @expr{y_i = m x_i + b} for
3990 different pairs of @expr{(x_i,y_i)}. Use the matrix-transpose method
3991 to solve for @expr{m} and @expr{b}, duplicating the above result.
3992 @xref{List Answer 2, 2}. (@bullet{})
3994 @cindex Geometric mean
3995 (@bullet{}) @strong{Exercise 3.} If the input data do not form a
3996 rectangle, you can use @w{@kbd{C-x * g}} (@code{calc-grab-region})
3997 to grab the data the way Emacs normally works with regions---it reads
3998 left-to-right, top-to-bottom, treating line breaks the same as spaces.
3999 Use this command to find the geometric mean of the following numbers.
4000 (The geometric mean is the @var{n}th root of the product of @var{n} numbers.)
4009 The @kbd{C-x * g} command accepts numbers separated by spaces or commas,
4010 with or without surrounding vector brackets.
4011 @xref{List Answer 3, 3}. (@bullet{})
4014 As another example, a theorem about binomial coefficients tells
4015 us that the alternating sum of binomial coefficients
4016 @var{n}-choose-0 minus @var{n}-choose-1 plus @var{n}-choose-2, and so
4017 on up to @var{n}-choose-@var{n},
4018 always comes out to zero. Let's verify this
4022 As another example, a theorem about binomial coefficients tells
4023 us that the alternating sum of binomial coefficients
4024 ${n \choose 0} - {n \choose 1} + {n \choose 2} - \cdots \pm {n \choose n}$
4025 always comes out to zero. Let's verify this
4031 1: [1, 2, 3, 4, 5, 6, 7] 1: [0, 1, 2, 3, 4, 5, 6]
4041 1: [1, -6, 15, -20, 15, -6, 1] 1: 0
4044 V M ' (-1)^$ choose(6,$) @key{RET} V R +
4048 The @kbd{V M '} command prompts you to enter any algebraic expression
4049 to define the function to map over the vector. The symbol @samp{$}
4050 inside this expression represents the argument to the function.
4051 The Calculator applies this formula to each element of the vector,
4052 substituting each element's value for the @samp{$} sign(s) in turn.
4054 To define a two-argument function, use @samp{$$} for the first
4055 argument and @samp{$} for the second: @kbd{V M ' $$-$ @key{RET}} is
4056 equivalent to @kbd{V M -}. This is analogous to regular algebraic
4057 entry, where @samp{$$} would refer to the next-to-top stack entry
4058 and @samp{$} would refer to the top stack entry, and @kbd{' $$-$ @key{RET}}
4059 would act exactly like @kbd{-}.
4061 Notice that the @kbd{V M '} command has recorded two things in the
4062 trail: The result, as usual, and also a funny-looking thing marked
4063 @samp{oper} that represents the operator function you typed in.
4064 The function is enclosed in @samp{< >} brackets, and the argument is
4065 denoted by a @samp{#} sign. If there were several arguments, they
4066 would be shown as @samp{#1}, @samp{#2}, and so on. (For example,
4067 @kbd{V M ' $$-$} will put the function @samp{<#1 - #2>} on the
4068 trail.) This object is a ``nameless function''; you can use nameless
4069 @w{@samp{< >}} notation to answer the @kbd{V M '} prompt if you like.
4070 Nameless function notation has the interesting, occasionally useful
4071 property that a nameless function is not actually evaluated until
4072 it is used. For example, @kbd{V M ' $+random(2.0)} evaluates
4073 @samp{random(2.0)} once and adds that random number to all elements
4074 of the vector, but @kbd{V M ' <#+random(2.0)>} evaluates the
4075 @samp{random(2.0)} separately for each vector element.
4077 Another group of operators that are often useful with @kbd{V M} are
4078 the relational operators: @kbd{a =}, for example, compares two numbers
4079 and gives the result 1 if they are equal, or 0 if not. Similarly,
4080 @w{@kbd{a <}} checks for one number being less than another.
4082 Other useful vector operations include @kbd{v v}, to reverse a
4083 vector end-for-end; @kbd{V S}, to sort the elements of a vector
4084 into increasing order; and @kbd{v r} and @w{@kbd{v c}}, to extract
4085 one row or column of a matrix, or (in both cases) to extract one
4086 element of a plain vector. With a negative argument, @kbd{v r}
4087 and @kbd{v c} instead delete one row, column, or vector element.
4089 @cindex Divisor functions
4090 (@bullet{}) @strong{Exercise 4.} The @expr{k}th @dfn{divisor function}
4094 is the sum of the @expr{k}th powers of all the divisors of an
4095 integer @expr{n}. Figure out a method for computing the divisor
4096 function for reasonably small values of @expr{n}. As a test,
4097 the 0th and 1st divisor functions of 30 are 8 and 72, respectively.
4098 @xref{List Answer 4, 4}. (@bullet{})
4100 @cindex Square-free numbers
4101 @cindex Duplicate values in a list
4102 (@bullet{}) @strong{Exercise 5.} The @kbd{k f} command produces a
4103 list of prime factors for a number. Sometimes it is important to
4104 know that a number is @dfn{square-free}, i.e., that no prime occurs
4105 more than once in its list of prime factors. Find a sequence of
4106 keystrokes to tell if a number is square-free; your method should
4107 leave 1 on the stack if it is, or 0 if it isn't.
4108 @xref{List Answer 5, 5}. (@bullet{})
4110 @cindex Triangular lists
4111 (@bullet{}) @strong{Exercise 6.} Build a list of lists that looks
4112 like the following diagram. (You may wish to use the @kbd{v /}
4113 command to enable multi-line display of vectors.)
4122 [1, 2, 3, 4, 5, 6] ]
4127 @xref{List Answer 6, 6}. (@bullet{})
4129 (@bullet{}) @strong{Exercise 7.} Build the following list of lists.
4137 [10, 11, 12, 13, 14],
4138 [15, 16, 17, 18, 19, 20] ]
4143 @xref{List Answer 7, 7}. (@bullet{})
4145 @cindex Maximizing a function over a list of values
4146 @c [fix-ref Numerical Solutions]
4147 (@bullet{}) @strong{Exercise 8.} Compute a list of values of Bessel's
4148 @texline @math{J_1(x)}
4150 function @samp{besJ(1,x)} for @expr{x} from 0 to 5 in steps of 0.25.
4151 Find the value of @expr{x} (from among the above set of values) for
4152 which @samp{besJ(1,x)} is a maximum. Use an ``automatic'' method,
4153 i.e., just reading along the list by hand to find the largest value
4154 is not allowed! (There is an @kbd{a X} command which does this kind
4155 of thing automatically; @pxref{Numerical Solutions}.)
4156 @xref{List Answer 8, 8}. (@bullet{})
4158 @cindex Digits, vectors of
4159 (@bullet{}) @strong{Exercise 9.} You are given an integer in the range
4160 @texline @math{0 \le N < 10^m}
4161 @infoline @expr{0 <= N < 10^m}
4162 for @expr{m=12} (i.e., an integer of less than
4163 twelve digits). Convert this integer into a vector of @expr{m}
4164 digits, each in the range from 0 to 9. In vector-of-digits notation,
4165 add one to this integer to produce a vector of @expr{m+1} digits
4166 (since there could be a carry out of the most significant digit).
4167 Convert this vector back into a regular integer. A good integer
4168 to try is 25129925999. @xref{List Answer 9, 9}. (@bullet{})
4170 (@bullet{}) @strong{Exercise 10.} Your friend Joe tried to use
4171 @kbd{V R a =} to test if all numbers in a list were equal. What
4172 happened? How would you do this test? @xref{List Answer 10, 10}. (@bullet{})
4174 (@bullet{}) @strong{Exercise 11.} The area of a circle of radius one
4175 is @cpi{}. The area of the
4176 @texline @math{2\times2}
4178 square that encloses that circle is 4. So if we throw @var{n} darts at
4179 random points in the square, about @cpiover{4} of them will land inside
4180 the circle. This gives us an entertaining way to estimate the value of
4181 @cpi{}. The @w{@kbd{k r}}
4182 command picks a random number between zero and the value on the stack.
4183 We could get a random floating-point number between @mathit{-1} and 1 by typing
4184 @w{@kbd{2.0 k r 1 -}}. Build a vector of 100 random @expr{(x,y)} points in
4185 this square, then use vector mapping and reduction to count how many
4186 points lie inside the unit circle. Hint: Use the @kbd{v b} command.
4187 @xref{List Answer 11, 11}. (@bullet{})
4189 @cindex Matchstick problem
4190 (@bullet{}) @strong{Exercise 12.} The @dfn{matchstick problem} provides
4191 another way to calculate @cpi{}. Say you have an infinite field
4192 of vertical lines with a spacing of one inch. Toss a one-inch matchstick
4193 onto the field. The probability that the matchstick will land crossing
4194 a line turns out to be
4195 @texline @math{2/\pi}.
4196 @infoline @expr{2/pi}.
4197 Toss 100 matchsticks to estimate @cpi{}. (If you want still more fun,
4198 the probability that the GCD (@w{@kbd{k g}}) of two large integers is
4200 @texline @math{6/\pi^2}.
4201 @infoline @expr{6/pi^2}.
4202 That provides yet another way to estimate @cpi{}.)
4203 @xref{List Answer 12, 12}. (@bullet{})
4205 (@bullet{}) @strong{Exercise 13.} An algebraic entry of a string in
4206 double-quote marks, @samp{"hello"}, creates a vector of the numerical
4207 (ASCII) codes of the characters (here, @expr{[104, 101, 108, 108, 111]}).
4208 Sometimes it is convenient to compute a @dfn{hash code} of a string,
4209 which is just an integer that represents the value of that string.
4210 Two equal strings have the same hash code; two different strings
4211 @dfn{probably} have different hash codes. (For example, Calc has
4212 over 400 function names, but Emacs can quickly find the definition for
4213 any given name because it has sorted the functions into ``buckets'' by
4214 their hash codes. Sometimes a few names will hash into the same bucket,
4215 but it is easier to search among a few names than among all the names.)
4216 One popular hash function is computed as follows: First set @expr{h = 0}.
4217 Then, for each character from the string in turn, set @expr{h = 3h + c_i}
4218 where @expr{c_i} is the character's ASCII code. If we have 511 buckets,
4219 we then take the hash code modulo 511 to get the bucket number. Develop a
4220 simple command or commands for converting string vectors into hash codes.
4221 The hash code for @samp{"Testing, 1, 2, 3"} is 1960915098, which modulo
4222 511 is 121. @xref{List Answer 13, 13}. (@bullet{})
4224 (@bullet{}) @strong{Exercise 14.} The @kbd{H V R} and @kbd{H V U}
4225 commands do nested function evaluations. @kbd{H V U} takes a starting
4226 value and a number of steps @var{n} from the stack; it then applies the
4227 function you give to the starting value 0, 1, 2, up to @var{n} times
4228 and returns a vector of the results. Use this command to create a
4229 ``random walk'' of 50 steps. Start with the two-dimensional point
4230 @expr{(0,0)}; then take one step a random distance between @mathit{-1} and 1
4231 in both @expr{x} and @expr{y}; then take another step, and so on. Use the
4232 @kbd{g f} command to display this random walk. Now modify your random
4233 walk to walk a unit distance, but in a random direction, at each step.
4234 (Hint: The @code{sincos} function returns a vector of the cosine and
4235 sine of an angle.) @xref{List Answer 14, 14}. (@bullet{})
4237 @node Types Tutorial, Algebra Tutorial, Vector/Matrix Tutorial, Tutorial
4238 @section Types Tutorial
4241 Calc understands a variety of data types as well as simple numbers.
4242 In this section, we'll experiment with each of these types in turn.
4244 The numbers we've been using so far have mainly been either @dfn{integers}
4245 or @dfn{floats}. We saw that floats are usually a good approximation to
4246 the mathematical concept of real numbers, but they are only approximations
4247 and are susceptible to roundoff error. Calc also supports @dfn{fractions},
4248 which can exactly represent any rational number.
4252 1: 3628800 2: 3628800 1: 518400:7 1: 518414:7 1: 7:518414
4256 10 ! 49 @key{RET} : 2 + &
4261 The @kbd{:} command divides two integers to get a fraction; @kbd{/}
4262 would normally divide integers to get a floating-point result.
4263 Notice we had to type @key{RET} between the @kbd{49} and the @kbd{:}
4264 since the @kbd{:} would otherwise be interpreted as part of a
4265 fraction beginning with 49.
4267 You can convert between floating-point and fractional format using
4268 @kbd{c f} and @kbd{c F}:
4272 1: 1.35027217629e-5 1: 7:518414
4279 The @kbd{c F} command replaces a floating-point number with the
4280 ``simplest'' fraction whose floating-point representation is the
4281 same, to within the current precision.
4285 1: 3.14159265359 1: 1146408:364913 1: 3.1416 1: 355:113
4288 P c F @key{DEL} p 5 @key{RET} P c F
4292 (@bullet{}) @strong{Exercise 1.} A calculation has produced the
4293 result 1.26508260337. You suspect it is the square root of the
4294 product of @cpi{} and some rational number. Is it? (Be sure
4295 to allow for roundoff error!) @xref{Types Answer 1, 1}. (@bullet{})
4297 @dfn{Complex numbers} can be stored in both rectangular and polar form.
4301 1: -9 1: (0, 3) 1: (3; 90.) 1: (6; 90.) 1: (2.4495; 45.)
4309 The square root of @mathit{-9} is by default rendered in rectangular form
4310 (@w{@expr{0 + 3i}}), but we can convert it to polar form (3 with a
4311 phase angle of 90 degrees). All the usual arithmetic and scientific
4312 operations are defined on both types of complex numbers.
4314 Another generalized kind of number is @dfn{infinity}. Infinity
4315 isn't really a number, but it can sometimes be treated like one.
4316 Calc uses the symbol @code{inf} to represent positive infinity,
4317 i.e., a value greater than any real number. Naturally, you can
4318 also write @samp{-inf} for minus infinity, a value less than any
4319 real number. The word @code{inf} can only be input using
4324 2: inf 2: -inf 2: -inf 2: -inf 1: nan
4325 1: -17 1: -inf 1: -inf 1: inf .
4328 ' inf @key{RET} 17 n * @key{RET} 72 + A +
4333 Since infinity is infinitely large, multiplying it by any finite
4334 number (like @mathit{-17}) has no effect, except that since @mathit{-17}
4335 is negative, it changes a plus infinity to a minus infinity.
4336 (``A huge positive number, multiplied by @mathit{-17}, yields a huge
4337 negative number.'') Adding any finite number to infinity also
4338 leaves it unchanged. Taking an absolute value gives us plus
4339 infinity again. Finally, we add this plus infinity to the minus
4340 infinity we had earlier. If you work it out, you might expect
4341 the answer to be @mathit{-72} for this. But the 72 has been completely
4342 lost next to the infinities; by the time we compute @w{@samp{inf - inf}}
4343 the finite difference between them, if any, is undetectable.
4344 So we say the result is @dfn{indeterminate}, which Calc writes
4345 with the symbol @code{nan} (for Not A Number).
4347 Dividing by zero is normally treated as an error, but you can get
4348 Calc to write an answer in terms of infinity by pressing @kbd{m i}
4349 to turn on Infinite mode.
4353 3: nan 2: nan 2: nan 2: nan 1: nan
4354 2: 1 1: 1 / 0 1: uinf 1: uinf .
4358 1 @key{RET} 0 / m i U / 17 n * +
4363 Dividing by zero normally is left unevaluated, but after @kbd{m i}
4364 it instead gives an infinite result. The answer is actually
4365 @code{uinf}, ``undirected infinity.'' If you look at a graph of
4366 @expr{1 / x} around @w{@expr{x = 0}}, you'll see that it goes toward
4367 plus infinity as you approach zero from above, but toward minus
4368 infinity as you approach from below. Since we said only @expr{1 / 0},
4369 Calc knows that the answer is infinite but not in which direction.
4370 That's what @code{uinf} means. Notice that multiplying @code{uinf}
4371 by a negative number still leaves plain @code{uinf}; there's no
4372 point in saying @samp{-uinf} because the sign of @code{uinf} is
4373 unknown anyway. Finally, we add @code{uinf} to our @code{nan},
4374 yielding @code{nan} again. It's easy to see that, because
4375 @code{nan} means ``totally unknown'' while @code{uinf} means
4376 ``unknown sign but known to be infinite,'' the more mysterious
4377 @code{nan} wins out when it is combined with @code{uinf}, or, for
4378 that matter, with anything else.
4380 (@bullet{}) @strong{Exercise 2.} Predict what Calc will answer
4381 for each of these formulas: @samp{inf / inf}, @samp{exp(inf)},
4382 @samp{exp(-inf)}, @samp{sqrt(-inf)}, @samp{sqrt(uinf)},
4383 @samp{abs(uinf)}, @samp{ln(0)}.
4384 @xref{Types Answer 2, 2}. (@bullet{})
4386 (@bullet{}) @strong{Exercise 3.} We saw that @samp{inf - inf = nan},
4387 which stands for an unknown value. Can @code{nan} stand for
4388 a complex number? Can it stand for infinity?
4389 @xref{Types Answer 3, 3}. (@bullet{})
4391 @dfn{HMS forms} represent a value in terms of hours, minutes, and
4396 1: 2@@ 30' 0" 1: 3@@ 30' 0" 2: 3@@ 30' 0" 1: 2.
4397 . . 1: 1@@ 45' 0." .
4400 2@@ 30' @key{RET} 1 + @key{RET} 2 / /
4404 HMS forms can also be used to hold angles in degrees, minutes, and
4409 1: 0.5 1: 26.56505 1: 26@@ 33' 54.18" 1: 0.44721
4417 First we convert the inverse tangent of 0.5 to degrees-minutes-seconds
4418 form, then we take the sine of that angle. Note that the trigonometric
4419 functions will accept HMS forms directly as input.
4422 (@bullet{}) @strong{Exercise 4.} The Beatles' @emph{Abbey Road} is
4423 47 minutes and 26 seconds long, and contains 17 songs. What is the
4424 average length of a song on @emph{Abbey Road}? If the Extended Disco
4425 Version of @emph{Abbey Road} added 20 seconds to the length of each
4426 song, how long would the album be? @xref{Types Answer 4, 4}. (@bullet{})
4428 A @dfn{date form} represents a date, or a date and time. Dates must
4429 be entered using algebraic entry. Date forms are surrounded by
4430 @samp{< >} symbols; most standard formats for dates are recognized.
4434 2: <Sun Jan 13, 1991> 1: 2.25
4435 1: <6:00pm Thu Jan 10, 1991> .
4438 ' <13 Jan 1991>, <1/10/91, 6pm> @key{RET} -
4443 In this example, we enter two dates, then subtract to find the
4444 number of days between them. It is also possible to add an
4445 HMS form or a number (of days) to a date form to get another
4450 1: <4:45:59pm Mon Jan 14, 1991> 1: <2:50:59am Thu Jan 17, 1991>
4457 @c [fix-ref Date Arithmetic]
4459 The @kbd{t N} (``now'') command pushes the current date and time on the
4460 stack; then we add two days, ten hours and five minutes to the date and
4461 time. Other date-and-time related commands include @kbd{t J}, which
4462 does Julian day conversions, @kbd{t W}, which finds the beginning of
4463 the week in which a date form lies, and @kbd{t I}, which increments a
4464 date by one or several months. @xref{Date Arithmetic}, for more.
4466 (@bullet{}) @strong{Exercise 5.} How many days until the next
4467 Friday the 13th? @xref{Types Answer 5, 5}. (@bullet{})
4469 (@bullet{}) @strong{Exercise 6.} How many leap years will there be
4470 between now and the year 10001 A.D.? @xref{Types Answer 6, 6}. (@bullet{})
4472 @cindex Slope and angle of a line
4473 @cindex Angle and slope of a line
4474 An @dfn{error form} represents a mean value with an attached standard
4475 deviation, or error estimate. Suppose our measurements indicate that
4476 a certain telephone pole is about 30 meters away, with an estimated
4477 error of 1 meter, and 8 meters tall, with an estimated error of 0.2
4478 meters. What is the slope of a line from here to the top of the
4479 pole, and what is the equivalent angle in degrees?
4483 1: 8 +/- 0.2 2: 8 +/- 0.2 1: 0.266 +/- 0.011 1: 14.93 +/- 0.594
4487 8 p .2 @key{RET} 30 p 1 / I T
4492 This means that the angle is about 15 degrees, and, assuming our
4493 original error estimates were valid standard deviations, there is about
4494 a 60% chance that the result is correct within 0.59 degrees.
4496 @cindex Torus, volume of
4497 (@bullet{}) @strong{Exercise 7.} The volume of a torus (a donut shape) is
4498 @texline @math{2 \pi^2 R r^2}
4499 @infoline @w{@expr{2 pi^2 R r^2}}
4500 where @expr{R} is the radius of the circle that
4501 defines the center of the tube and @expr{r} is the radius of the tube
4502 itself. Suppose @expr{R} is 20 cm and @expr{r} is 4 cm, each known to
4503 within 5 percent. What is the volume and the relative uncertainty of
4504 the volume? @xref{Types Answer 7, 7}. (@bullet{})
4506 An @dfn{interval form} represents a range of values. While an
4507 error form is best for making statistical estimates, intervals give
4508 you exact bounds on an answer. Suppose we additionally know that
4509 our telephone pole is definitely between 28 and 31 meters away,
4510 and that it is between 7.7 and 8.1 meters tall.
4514 1: [7.7 .. 8.1] 2: [7.7 .. 8.1] 1: [0.24 .. 0.28] 1: [13.9 .. 16.1]
4518 [ 7.7 .. 8.1 ] [ 28 .. 31 ] / I T
4523 If our bounds were correct, then the angle to the top of the pole
4524 is sure to lie in the range shown.
4526 The square brackets around these intervals indicate that the endpoints
4527 themselves are allowable values. In other words, the distance to the
4528 telephone pole is between 28 and 31, @emph{inclusive}. You can also
4529 make an interval that is exclusive of its endpoints by writing
4530 parentheses instead of square brackets. You can even make an interval
4531 which is inclusive (``closed'') on one end and exclusive (``open'') on
4536 1: [1 .. 10) 1: (0.1 .. 1] 2: (0.1 .. 1] 1: (0.2 .. 3)
4540 [ 1 .. 10 ) & [ 2 .. 3 ) *
4545 The Calculator automatically keeps track of which end values should
4546 be open and which should be closed. You can also make infinite or
4547 semi-infinite intervals by using @samp{-inf} or @samp{inf} for one
4550 (@bullet{}) @strong{Exercise 8.} What answer would you expect from
4551 @samp{@w{1 /} @w{(0 .. 10)}}? What about @samp{@w{1 /} @w{(-10 .. 0)}}? What
4552 about @samp{@w{1 /} @w{[0 .. 10]}} (where the interval actually includes
4553 zero)? What about @samp{@w{1 /} @w{(-10 .. 10)}}?
4554 @xref{Types Answer 8, 8}. (@bullet{})
4556 (@bullet{}) @strong{Exercise 9.} Two easy ways of squaring a number
4557 are @kbd{@key{RET} *} and @w{@kbd{2 ^}}. Normally these produce the same
4558 answer. Would you expect this still to hold true for interval forms?
4559 If not, which of these will result in a larger interval?
4560 @xref{Types Answer 9, 9}. (@bullet{})
4562 A @dfn{modulo form} is used for performing arithmetic modulo @var{m}.
4563 For example, arithmetic involving time is generally done modulo 12
4568 1: 17 mod 24 1: 3 mod 24 1: 21 mod 24 1: 9 mod 24
4571 17 M 24 @key{RET} 10 + n 5 /
4576 In this last step, Calc has divided by 5 modulo 24; i.e., it has found a
4577 new number which, when multiplied by 5 modulo 24, produces the original
4578 number, 21. If @var{m} is prime and the divisor is not a multiple of
4579 @var{m}, it is always possible to find such a number. For non-prime
4580 @var{m} like 24, it is only sometimes possible.
4584 1: 10 mod 24 1: 16 mod 24 1: 1000000... 1: 16
4587 10 M 24 @key{RET} 100 ^ 10 @key{RET} 100 ^ 24 %
4592 These two calculations get the same answer, but the first one is
4593 much more efficient because it avoids the huge intermediate value
4594 that arises in the second one.
4596 @cindex Fermat, primality test of
4597 (@bullet{}) @strong{Exercise 10.} A theorem of Pierre de Fermat
4599 @texline @w{@math{x^{n-1} \bmod n = 1}}
4600 @infoline @expr{x^(n-1) mod n = 1}
4601 if @expr{n} is a prime number and @expr{x} is an integer less than
4602 @expr{n}. If @expr{n} is @emph{not} a prime number, this will
4603 @emph{not} be true for most values of @expr{x}. Thus we can test
4604 informally if a number is prime by trying this formula for several
4605 values of @expr{x}. Use this test to tell whether the following numbers
4606 are prime: 811749613, 15485863. @xref{Types Answer 10, 10}. (@bullet{})
4608 It is possible to use HMS forms as parts of error forms, intervals,
4609 modulo forms, or as the phase part of a polar complex number.
4610 For example, the @code{calc-time} command pushes the current time
4611 of day on the stack as an HMS/modulo form.
4615 1: 17@@ 34' 45" mod 24@@ 0' 0" 1: 6@@ 22' 15" mod 24@@ 0' 0"
4623 This calculation tells me it is six hours and 22 minutes until midnight.
4625 (@bullet{}) @strong{Exercise 11.} A rule of thumb is that one year
4627 @texline @math{\pi \times 10^7}
4628 @infoline @w{@expr{pi * 10^7}}
4629 seconds. What time will it be that many seconds from right now?
4630 @xref{Types Answer 11, 11}. (@bullet{})
4632 (@bullet{}) @strong{Exercise 12.} You are preparing to order packaging
4633 for the CD release of the Extended Disco Version of @emph{Abbey Road}.
4634 You are told that the songs will actually be anywhere from 20 to 60
4635 seconds longer than the originals. One CD can hold about 75 minutes
4636 of music. Should you order single or double packages?
4637 @xref{Types Answer 12, 12}. (@bullet{})
4639 Another kind of data the Calculator can manipulate is numbers with
4640 @dfn{units}. This isn't strictly a new data type; it's simply an
4641 application of algebraic expressions, where we use variables with
4642 suggestive names like @samp{cm} and @samp{in} to represent units
4643 like centimeters and inches.
4647 1: 2 in 1: 5.08 cm 1: 0.027778 fath 1: 0.0508 m
4650 ' 2in @key{RET} u c cm @key{RET} u c fath @key{RET} u b
4655 We enter the quantity ``2 inches'' (actually an algebraic expression
4656 which means two times the variable @samp{in}), then we convert it
4657 first to centimeters, then to fathoms, then finally to ``base'' units,
4658 which in this case means meters.
4662 1: 9 acre 1: 3 sqrt(acre) 1: 190.84 m 1: 190.84 m + 30 cm
4665 ' 9 acre @key{RET} Q u s ' $+30 cm @key{RET}
4672 1: 191.14 m 1: 36536.3046 m^2 1: 365363046 cm^2
4680 Since units expressions are really just formulas, taking the square
4681 root of @samp{acre} is undefined. After all, @code{acre} might be an
4682 algebraic variable that you will someday assign a value. We use the
4683 ``units-simplify'' command to simplify the expression with variables
4684 being interpreted as unit names.
4686 In the final step, we have converted not to a particular unit, but to a
4687 units system. The ``cgs'' system uses centimeters instead of meters
4688 as its standard unit of length.
4690 There is a wide variety of units defined in the Calculator.
4694 1: 55 mph 1: 88.5139 kph 1: 88.5139 km / hr 1: 8.201407e-8 c
4697 ' 55 mph @key{RET} u c kph @key{RET} u c km/hr @key{RET} u c c @key{RET}
4702 We express a speed first in miles per hour, then in kilometers per
4703 hour, then again using a slightly more explicit notation, then
4704 finally in terms of fractions of the speed of light.
4706 Temperature conversions are a bit more tricky. There are two ways to
4707 interpret ``20 degrees Fahrenheit''---it could mean an actual
4708 temperature, or it could mean a change in temperature. For normal
4709 units there is no difference, but temperature units have an offset
4710 as well as a scale factor and so there must be two explicit commands
4715 1: 20 degF 1: 11.1111 degC 1: -20:3 degC 1: -6.666 degC
4718 ' 20 degF @key{RET} u c degC @key{RET} U u t degC @key{RET} c f
4723 First we convert a change of 20 degrees Fahrenheit into an equivalent
4724 change in degrees Celsius (or Centigrade). Then, we convert the
4725 absolute temperature 20 degrees Fahrenheit into Celsius. Since
4726 this comes out as an exact fraction, we then convert to floating-point
4727 for easier comparison with the other result.
4729 For simple unit conversions, you can put a plain number on the stack.
4730 Then @kbd{u c} and @kbd{u t} will prompt for both old and new units.
4731 When you use this method, you're responsible for remembering which
4732 numbers are in which units:
4736 1: 55 1: 88.5139 1: 8.201407e-8
4739 55 u c mph @key{RET} kph @key{RET} u c km/hr @key{RET} c @key{RET}
4743 To see a complete list of built-in units, type @kbd{u v}. Press
4744 @w{@kbd{C-x * c}} again to re-enter the Calculator when you're done looking
4747 (@bullet{}) @strong{Exercise 13.} How many seconds are there really
4748 in a year? @xref{Types Answer 13, 13}. (@bullet{})
4750 @cindex Speed of light
4751 (@bullet{}) @strong{Exercise 14.} Supercomputer designs are limited by
4752 the speed of light (and of electricity, which is nearly as fast).
4753 Suppose a computer has a 4.1 ns (nanosecond) clock cycle, and its
4754 cabinet is one meter across. Is speed of light going to be a
4755 significant factor in its design? @xref{Types Answer 14, 14}. (@bullet{})
4757 (@bullet{}) @strong{Exercise 15.} Sam the Slug normally travels about
4758 five yards in an hour. He has obtained a supply of Power Pills; each
4759 Power Pill he eats doubles his speed. How many Power Pills can he
4760 swallow and still travel legally on most US highways?
4761 @xref{Types Answer 15, 15}. (@bullet{})
4763 @node Algebra Tutorial, Programming Tutorial, Types Tutorial, Tutorial
4764 @section Algebra and Calculus Tutorial
4767 This section shows how to use Calc's algebra facilities to solve
4768 equations, do simple calculus problems, and manipulate algebraic
4772 * Basic Algebra Tutorial::
4773 * Rewrites Tutorial::
4776 @node Basic Algebra Tutorial, Rewrites Tutorial, Algebra Tutorial, Algebra Tutorial
4777 @subsection Basic Algebra
4780 If you enter a formula in Algebraic mode that refers to variables,
4781 the formula itself is pushed onto the stack. You can manipulate
4782 formulas as regular data objects.
4786 1: 2 x^2 - 6 1: 6 - 2 x^2 1: (6 - 2 x^2) (3 x^2 + y)
4789 ' 2x^2-6 @key{RET} n ' 3x^2+y @key{RET} *
4793 (@bullet{}) @strong{Exercise 1.} Do @kbd{' x @key{RET} Q 2 ^} and
4794 @kbd{' x @key{RET} 2 ^ Q} both wind up with the same result (@samp{x})?
4795 Why or why not? @xref{Algebra Answer 1, 1}. (@bullet{})
4797 There are also commands for doing common algebraic operations on
4798 formulas. Continuing with the formula from the last example,
4802 1: 18 x^2 + 6 y - 6 x^4 - 2 x^2 y 1: (18 - 2 y) x^2 - 6 x^4 + 6 y
4810 First we ``expand'' using the distributive law, then we ``collect''
4811 terms involving like powers of @expr{x}.
4813 Let's find the value of this expression when @expr{x} is 2 and @expr{y}
4818 1: 17 x^2 - 6 x^4 + 3 1: -25
4821 1:2 s l y @key{RET} 2 s l x @key{RET}
4826 The @kbd{s l} command means ``let''; it takes a number from the top of
4827 the stack and temporarily assigns it as the value of the variable
4828 you specify. It then evaluates (as if by the @kbd{=} key) the
4829 next expression on the stack. After this command, the variable goes
4830 back to its original value, if any.
4832 (An earlier exercise in this tutorial involved storing a value in the
4833 variable @code{x}; if this value is still there, you will have to
4834 unstore it with @kbd{s u x @key{RET}} before the above example will work
4837 @cindex Maximum of a function using Calculus
4838 Let's find the maximum value of our original expression when @expr{y}
4839 is one-half and @expr{x} ranges over all possible values. We can
4840 do this by taking the derivative with respect to @expr{x} and examining
4841 values of @expr{x} for which the derivative is zero. If the second
4842 derivative of the function at that value of @expr{x} is negative,
4843 the function has a local maximum there.
4847 1: 17 x^2 - 6 x^4 + 3 1: 34 x - 24 x^3
4850 U @key{DEL} s 1 a d x @key{RET} s 2
4855 Well, the derivative is clearly zero when @expr{x} is zero. To find
4856 the other root(s), let's divide through by @expr{x} and then solve:
4860 1: (34 x - 24 x^3) / x 1: 34 x / x - 24 x^3 / x 1: 34 - 24 x^2
4863 ' x @key{RET} / a x a s
4870 1: 34 - 24 x^2 = 0 1: x = 1.19023
4873 0 a = s 3 a S x @key{RET}
4878 Notice the use of @kbd{a s} to ``simplify'' the formula. When the
4879 default algebraic simplifications don't do enough, you can use
4880 @kbd{a s} to tell Calc to spend more time on the job.
4882 Now we compute the second derivative and plug in our values of @expr{x}:
4886 1: 1.19023 2: 1.19023 2: 1.19023
4887 . 1: 34 x - 24 x^3 1: 34 - 72 x^2
4890 a . r 2 a d x @key{RET} s 4
4895 (The @kbd{a .} command extracts just the righthand side of an equation.
4896 Another method would have been to use @kbd{v u} to unpack the equation
4897 @w{@samp{x = 1.19}} to @samp{x} and @samp{1.19}, then use @kbd{M-- M-2 @key{DEL}}
4898 to delete the @samp{x}.)
4902 2: 34 - 72 x^2 1: -68. 2: 34 - 72 x^2 1: 34
4906 @key{TAB} s l x @key{RET} U @key{DEL} 0 s l x @key{RET}
4911 The first of these second derivatives is negative, so we know the function
4912 has a maximum value at @expr{x = 1.19023}. (The function also has a
4913 local @emph{minimum} at @expr{x = 0}.)
4915 When we solved for @expr{x}, we got only one value even though
4916 @expr{34 - 24 x^2 = 0} is a quadratic equation that ought to have
4917 two solutions. The reason is that @w{@kbd{a S}} normally returns a
4918 single ``principal'' solution. If it needs to come up with an
4919 arbitrary sign (as occurs in the quadratic formula) it picks @expr{+}.
4920 If it needs an arbitrary integer, it picks zero. We can get a full
4921 solution by pressing @kbd{H} (the Hyperbolic flag) before @kbd{a S}.
4925 1: 34 - 24 x^2 = 0 1: x = 1.19023 s1 1: x = -1.19023
4928 r 3 H a S x @key{RET} s 5 1 n s l s1 @key{RET}
4933 Calc has invented the variable @samp{s1} to represent an unknown sign;
4934 it is supposed to be either @mathit{+1} or @mathit{-1}. Here we have used
4935 the ``let'' command to evaluate the expression when the sign is negative.
4936 If we plugged this into our second derivative we would get the same,
4937 negative, answer, so @expr{x = -1.19023} is also a maximum.
4939 To find the actual maximum value, we must plug our two values of @expr{x}
4940 into the original formula.
4944 2: 17 x^2 - 6 x^4 + 3 1: 24.08333 s1^2 - 12.04166 s1^4 + 3
4948 r 1 r 5 s l @key{RET}
4953 (Here we see another way to use @kbd{s l}; if its input is an equation
4954 with a variable on the lefthand side, then @kbd{s l} treats the equation
4955 like an assignment to that variable if you don't give a variable name.)
4957 It's clear that this will have the same value for either sign of
4958 @code{s1}, but let's work it out anyway, just for the exercise:
4962 2: [-1, 1] 1: [15.04166, 15.04166]
4963 1: 24.08333 s1^2 ... .
4966 [ 1 n , 1 ] @key{TAB} V M $ @key{RET}
4971 Here we have used a vector mapping operation to evaluate the function
4972 at several values of @samp{s1} at once. @kbd{V M $} is like @kbd{V M '}
4973 except that it takes the formula from the top of the stack. The
4974 formula is interpreted as a function to apply across the vector at the
4975 next-to-top stack level. Since a formula on the stack can't contain
4976 @samp{$} signs, Calc assumes the variables in the formula stand for
4977 different arguments. It prompts you for an @dfn{argument list}, giving
4978 the list of all variables in the formula in alphabetical order as the
4979 default list. In this case the default is @samp{(s1)}, which is just
4980 what we want so we simply press @key{RET} at the prompt.
4982 If there had been several different values, we could have used
4983 @w{@kbd{V R X}} to find the global maximum.
4985 Calc has a built-in @kbd{a P} command that solves an equation using
4986 @w{@kbd{H a S}} and returns a vector of all the solutions. It simply
4987 automates the job we just did by hand. Applied to our original
4988 cubic polynomial, it would produce the vector of solutions
4989 @expr{[1.19023, -1.19023, 0]}. (There is also an @kbd{a X} command
4990 which finds a local maximum of a function. It uses a numerical search
4991 method rather than examining the derivatives, and thus requires you
4992 to provide some kind of initial guess to show it where to look.)
4994 (@bullet{}) @strong{Exercise 2.} Given a vector of the roots of a
4995 polynomial (such as the output of an @kbd{a P} command), what
4996 sequence of commands would you use to reconstruct the original
4997 polynomial? (The answer will be unique to within a constant
4998 multiple; choose the solution where the leading coefficient is one.)
4999 @xref{Algebra Answer 2, 2}. (@bullet{})
5001 The @kbd{m s} command enables Symbolic mode, in which formulas
5002 like @samp{sqrt(5)} that can't be evaluated exactly are left in
5003 symbolic form rather than giving a floating-point approximate answer.
5004 Fraction mode (@kbd{m f}) is also useful when doing algebra.
5008 2: 34 x - 24 x^3 2: 34 x - 24 x^3
5009 1: 34 x - 24 x^3 1: [sqrt(51) / 6, sqrt(51) / -6, 0]
5012 r 2 @key{RET} m s m f a P x @key{RET}
5016 One more mode that makes reading formulas easier is Big mode.
5025 1: [-----, -----, 0]
5034 Here things like powers, square roots, and quotients and fractions
5035 are displayed in a two-dimensional pictorial form. Calc has other
5036 language modes as well, such as C mode, FORTRAN mode, @TeX{} mode
5041 2: 34*x - 24*pow(x, 3) 2: 34*x - 24*x**3
5042 1: @{sqrt(51) / 6, sqrt(51) / -6, 0@} 1: /sqrt(51) / 6, sqrt(51) / -6, 0/
5053 2: [@{\sqrt@{51@} \over 6@}, @{\sqrt@{51@} \over -6@}, 0]
5054 1: @{2 \over 3@} \sqrt@{5@}
5057 d T ' 2 \sqrt@{5@} \over 3 @key{RET}
5062 As you can see, language modes affect both entry and display of
5063 formulas. They affect such things as the names used for built-in
5064 functions, the set of arithmetic operators and their precedences,
5065 and notations for vectors and matrices.
5067 Notice that @samp{sqrt(51)} may cause problems with older
5068 implementations of C and FORTRAN, which would require something more
5069 like @samp{sqrt(51.0)}. It is always wise to check over the formulas
5070 produced by the various language modes to make sure they are fully
5073 Type @kbd{m s}, @kbd{m f}, and @kbd{d N} to reset these modes. (You
5074 may prefer to remain in Big mode, but all the examples in the tutorial
5075 are shown in normal mode.)
5077 @cindex Area under a curve
5078 What is the area under the portion of this curve from @expr{x = 1} to @expr{2}?
5079 This is simply the integral of the function:
5083 1: 17 x^2 - 6 x^4 + 3 1: 5.6666 x^3 - 1.2 x^5 + 3 x
5091 We want to evaluate this at our two values for @expr{x} and subtract.
5092 One way to do it is again with vector mapping and reduction:
5096 2: [2, 1] 1: [12.93333, 7.46666] 1: 5.46666
5097 1: 5.6666 x^3 ... . .
5099 [ 2 , 1 ] @key{TAB} V M $ @key{RET} V R -
5103 (@bullet{}) @strong{Exercise 3.} Find the integral from 1 to @expr{y}
5105 @texline @math{x \sin \pi x}
5106 @infoline @w{@expr{x sin(pi x)}}
5107 (where the sine is calculated in radians). Find the values of the
5108 integral for integers @expr{y} from 1 to 5. @xref{Algebra Answer 3,
5111 Calc's integrator can do many simple integrals symbolically, but many
5112 others are beyond its capabilities. Suppose we wish to find the area
5114 @texline @math{\sin x \ln x}
5115 @infoline @expr{sin(x) ln(x)}
5116 over the same range of @expr{x}. If you entered this formula and typed
5117 @kbd{a i x @key{RET}} (don't bother to try this), Calc would work for a
5118 long time but would be unable to find a solution. In fact, there is no
5119 closed-form solution to this integral. Now what do we do?
5121 @cindex Integration, numerical
5122 @cindex Numerical integration
5123 One approach would be to do the integral numerically. It is not hard
5124 to do this by hand using vector mapping and reduction. It is rather
5125 slow, though, since the sine and logarithm functions take a long time.
5126 We can save some time by reducing the working precision.
5130 3: 10 1: [1, 1.1, 1.2, ... , 1.8, 1.9]
5135 10 @key{RET} 1 @key{RET} .1 @key{RET} C-u v x
5140 (Note that we have used the extended version of @kbd{v x}; we could
5141 also have used plain @kbd{v x} as follows: @kbd{v x 10 @key{RET} 9 + .1 *}.)
5145 2: [1, 1.1, ... ] 1: [0., 0.084941, 0.16993, ... ]
5149 ' sin(x) ln(x) @key{RET} s 1 m r p 5 @key{RET} V M $ @key{RET}
5164 (If you got wildly different results, did you remember to switch
5167 Here we have divided the curve into ten segments of equal width;
5168 approximating these segments as rectangular boxes (i.e., assuming
5169 the curve is nearly flat at that resolution), we compute the areas
5170 of the boxes (height times width), then sum the areas. (It is
5171 faster to sum first, then multiply by the width, since the width
5172 is the same for every box.)
5174 The true value of this integral turns out to be about 0.374, so
5175 we're not doing too well. Let's try another approach.
5179 1: sin(x) ln(x) 1: 0.84147 x - 0.84147 + 0.11957 (x - 1)^2 - ...
5182 r 1 a t x=1 @key{RET} 4 @key{RET}
5187 Here we have computed the Taylor series expansion of the function
5188 about the point @expr{x=1}. We can now integrate this polynomial
5189 approximation, since polynomials are easy to integrate.
5193 1: 0.42074 x^2 + ... 1: [-0.0446, -0.42073] 1: 0.3761
5196 a i x @key{RET} [ 2 , 1 ] @key{TAB} V M $ @key{RET} V R -
5201 Better! By increasing the precision and/or asking for more terms
5202 in the Taylor series, we can get a result as accurate as we like.
5203 (Taylor series converge better away from singularities in the
5204 function such as the one at @code{ln(0)}, so it would also help to
5205 expand the series about the points @expr{x=2} or @expr{x=1.5} instead
5208 @cindex Simpson's rule
5209 @cindex Integration by Simpson's rule
5210 (@bullet{}) @strong{Exercise 4.} Our first method approximated the
5211 curve by stairsteps of width 0.1; the total area was then the sum
5212 of the areas of the rectangles under these stairsteps. Our second
5213 method approximated the function by a polynomial, which turned out
5214 to be a better approximation than stairsteps. A third method is
5215 @dfn{Simpson's rule}, which is like the stairstep method except
5216 that the steps are not required to be flat. Simpson's rule boils
5217 down to the formula,
5221 (h/3) * (f(a) + 4 f(a+h) + 2 f(a+2h) + 4 f(a+3h) + ...
5222 + 2 f(a+(n-2)*h) + 4 f(a+(n-1)*h) + f(a+n*h))
5229 \qquad {h \over 3} (f(a) + 4 f(a+h) + 2 f(a+2h) + 4 f(a+3h) + \cdots
5230 \hfill \cr \hfill {} + 2 f(a+(n-2)h) + 4 f(a+(n-1)h) + f(a+n h)) \qquad
5236 where @expr{n} (which must be even) is the number of slices and @expr{h}
5237 is the width of each slice. These are 10 and 0.1 in our example.
5238 For reference, here is the corresponding formula for the stairstep
5243 h * (f(a) + f(a+h) + f(a+2h) + f(a+3h) + ...
5244 + f(a+(n-2)*h) + f(a+(n-1)*h))
5250 $$ h (f(a) + f(a+h) + f(a+2h) + f(a+3h) + \cdots
5251 + f(a+(n-2)h) + f(a+(n-1)h)) $$
5255 Compute the integral from 1 to 2 of
5256 @texline @math{\sin x \ln x}
5257 @infoline @expr{sin(x) ln(x)}
5258 using Simpson's rule with 10 slices.
5259 @xref{Algebra Answer 4, 4}. (@bullet{})
5261 Calc has a built-in @kbd{a I} command for doing numerical integration.
5262 It uses @dfn{Romberg's method}, which is a more sophisticated cousin
5263 of Simpson's rule. In particular, it knows how to keep refining the
5264 result until the current precision is satisfied.
5266 @c [fix-ref Selecting Sub-Formulas]
5267 Aside from the commands we've seen so far, Calc also provides a
5268 large set of commands for operating on parts of formulas. You
5269 indicate the desired sub-formula by placing the cursor on any part
5270 of the formula before giving a @dfn{selection} command. Selections won't
5271 be covered in the tutorial; @pxref{Selecting Subformulas}, for
5272 details and examples.
5274 @c hard exercise: simplify (2^(n r) - 2^(r*(n - 1))) / (2^r - 1) 2^(n - 1)
5275 @c to 2^((n-1)*(r-1)).
5277 @node Rewrites Tutorial, , Basic Algebra Tutorial, Algebra Tutorial
5278 @subsection Rewrite Rules
5281 No matter how many built-in commands Calc provided for doing algebra,
5282 there would always be something you wanted to do that Calc didn't have
5283 in its repertoire. So Calc also provides a @dfn{rewrite rule} system
5284 that you can use to define your own algebraic manipulations.
5286 Suppose we want to simplify this trigonometric formula:
5290 1: 1 / cos(x) - sin(x) tan(x)
5293 ' 1/cos(x) - sin(x) tan(x) @key{RET} s 1
5298 If we were simplifying this by hand, we'd probably replace the
5299 @samp{tan} with a @samp{sin/cos} first, then combine over a common
5300 denominator. There is no Calc command to do the former; the @kbd{a n}
5301 algebra command will do the latter but we'll do both with rewrite
5302 rules just for practice.
5304 Rewrite rules are written with the @samp{:=} symbol.
5308 1: 1 / cos(x) - sin(x)^2 / cos(x)
5311 a r tan(a) := sin(a)/cos(a) @key{RET}
5316 (The ``assignment operator'' @samp{:=} has several uses in Calc. All
5317 by itself the formula @samp{tan(a) := sin(a)/cos(a)} doesn't do anything,
5318 but when it is given to the @kbd{a r} command, that command interprets
5319 it as a rewrite rule.)
5321 The lefthand side, @samp{tan(a)}, is called the @dfn{pattern} of the
5322 rewrite rule. Calc searches the formula on the stack for parts that
5323 match the pattern. Variables in a rewrite pattern are called
5324 @dfn{meta-variables}, and when matching the pattern each meta-variable
5325 can match any sub-formula. Here, the meta-variable @samp{a} matched
5326 the actual variable @samp{x}.
5328 When the pattern part of a rewrite rule matches a part of the formula,
5329 that part is replaced by the righthand side with all the meta-variables
5330 substituted with the things they matched. So the result is
5331 @samp{sin(x) / cos(x)}. Calc's normal algebraic simplifications then
5332 mix this in with the rest of the original formula.
5334 To merge over a common denominator, we can use another simple rule:
5338 1: (1 - sin(x)^2) / cos(x)
5341 a r a/x + b/x := (a+b)/x @key{RET}
5345 This rule points out several interesting features of rewrite patterns.
5346 First, if a meta-variable appears several times in a pattern, it must
5347 match the same thing everywhere. This rule detects common denominators
5348 because the same meta-variable @samp{x} is used in both of the
5351 Second, meta-variable names are independent from variables in the
5352 target formula. Notice that the meta-variable @samp{x} here matches
5353 the subformula @samp{cos(x)}; Calc never confuses the two meanings of
5356 And third, rewrite patterns know a little bit about the algebraic
5357 properties of formulas. The pattern called for a sum of two quotients;
5358 Calc was able to match a difference of two quotients by matching
5359 @samp{a = 1}, @samp{b = -sin(x)^2}, and @samp{x = cos(x)}.
5361 @c [fix-ref Algebraic Properties of Rewrite Rules]
5362 We could just as easily have written @samp{a/x - b/x := (a-b)/x} for
5363 the rule. It would have worked just the same in all cases. (If we
5364 really wanted the rule to apply only to @samp{+} or only to @samp{-},
5365 we could have used the @code{plain} symbol. @xref{Algebraic Properties
5366 of Rewrite Rules}, for some examples of this.)
5368 One more rewrite will complete the job. We want to use the identity
5369 @samp{sin(x)^2 + cos(x)^2 = 1}, but of course we must first rearrange
5370 the identity in a way that matches our formula. The obvious rule
5371 would be @samp{@w{1 - sin(x)^2} := cos(x)^2}, but a little thought shows
5372 that the rule @samp{sin(x)^2 := 1 - cos(x)^2} will also work. The
5373 latter rule has a more general pattern so it will work in many other
5378 1: (1 + cos(x)^2 - 1) / cos(x) 1: cos(x)
5381 a r sin(x)^2 := 1 - cos(x)^2 @key{RET} a s
5385 You may ask, what's the point of using the most general rule if you
5386 have to type it in every time anyway? The answer is that Calc allows
5387 you to store a rewrite rule in a variable, then give the variable
5388 name in the @kbd{a r} command. In fact, this is the preferred way to
5389 use rewrites. For one, if you need a rule once you'll most likely
5390 need it again later. Also, if the rule doesn't work quite right you
5391 can simply Undo, edit the variable, and run the rule again without
5392 having to retype it.
5396 ' tan(x) := sin(x)/cos(x) @key{RET} s t tsc @key{RET}
5397 ' a/x + b/x := (a+b)/x @key{RET} s t merge @key{RET}
5398 ' sin(x)^2 := 1 - cos(x)^2 @key{RET} s t sinsqr @key{RET}
5400 1: 1 / cos(x) - sin(x) tan(x) 1: cos(x)
5403 r 1 a r tsc @key{RET} a r merge @key{RET} a r sinsqr @key{RET} a s
5407 To edit a variable, type @kbd{s e} and the variable name, use regular
5408 Emacs editing commands as necessary, then type @kbd{C-c C-c} to store
5409 the edited value back into the variable.
5410 You can also use @w{@kbd{s e}} to create a new variable if you wish.
5412 Notice that the first time you use each rule, Calc puts up a ``compiling''
5413 message briefly. The pattern matcher converts rules into a special
5414 optimized pattern-matching language rather than using them directly.
5415 This allows @kbd{a r} to apply even rather complicated rules very
5416 efficiently. If the rule is stored in a variable, Calc compiles it
5417 only once and stores the compiled form along with the variable. That's
5418 another good reason to store your rules in variables rather than
5419 entering them on the fly.
5421 (@bullet{}) @strong{Exercise 1.} Type @kbd{m s} to get Symbolic
5422 mode, then enter the formula @samp{@w{(2 + sqrt(2))} / @w{(1 + sqrt(2))}}.
5423 Using a rewrite rule, simplify this formula by multiplying the top and
5424 bottom by the conjugate @w{@samp{1 - sqrt(2)}}. The result will have
5425 to be expanded by the distributive law; do this with another
5426 rewrite. @xref{Rewrites Answer 1, 1}. (@bullet{})
5428 The @kbd{a r} command can also accept a vector of rewrite rules, or
5429 a variable containing a vector of rules.
5433 1: [tsc, merge, sinsqr] 1: [tan(x) := sin(x) / cos(x), ... ]
5436 ' [tsc,merge,sinsqr] @key{RET} =
5443 1: 1 / cos(x) - sin(x) tan(x) 1: cos(x)
5446 s t trig @key{RET} r 1 a r trig @key{RET} a s
5450 @c [fix-ref Nested Formulas with Rewrite Rules]
5451 Calc tries all the rules you give against all parts of the formula,
5452 repeating until no further change is possible. (The exact order in
5453 which things are tried is rather complex, but for simple rules like
5454 the ones we've used here the order doesn't really matter.
5455 @xref{Nested Formulas with Rewrite Rules}.)
5457 Calc actually repeats only up to 100 times, just in case your rule set
5458 has gotten into an infinite loop. You can give a numeric prefix argument
5459 to @kbd{a r} to specify any limit. In particular, @kbd{M-1 a r} does
5460 only one rewrite at a time.
5464 1: 1 / cos(x) - sin(x)^2 / cos(x) 1: (1 - sin(x)^2) / cos(x)
5467 r 1 M-1 a r trig @key{RET} M-1 a r trig @key{RET}
5471 You can type @kbd{M-0 a r} if you want no limit at all on the number
5472 of rewrites that occur.
5474 Rewrite rules can also be @dfn{conditional}. Simply follow the rule
5475 with a @samp{::} symbol and the desired condition. For example,
5479 1: exp(2 pi i) + exp(3 pi i) + exp(4 pi i)
5482 ' exp(2 pi i) + exp(3 pi i) + exp(4 pi i) @key{RET}
5489 1: 1 + exp(3 pi i) + 1
5492 a r exp(k pi i) := 1 :: k % 2 = 0 @key{RET}
5497 (Recall, @samp{k % 2} is the remainder from dividing @samp{k} by 2,
5498 which will be zero only when @samp{k} is an even integer.)
5500 An interesting point is that the variables @samp{pi} and @samp{i}
5501 were matched literally rather than acting as meta-variables.
5502 This is because they are special-constant variables. The special
5503 constants @samp{e}, @samp{phi}, and so on also match literally.
5504 A common error with rewrite
5505 rules is to write, say, @samp{f(a,b,c,d,e) := g(a+b+c+d+e)}, expecting
5506 to match any @samp{f} with five arguments but in fact matching
5507 only when the fifth argument is literally @samp{e}!
5509 @cindex Fibonacci numbers
5514 Rewrite rules provide an interesting way to define your own functions.
5515 Suppose we want to define @samp{fib(n)} to produce the @var{n}th
5516 Fibonacci number. The first two Fibonacci numbers are each 1;
5517 later numbers are formed by summing the two preceding numbers in
5518 the sequence. This is easy to express in a set of three rules:
5522 ' [fib(1) := 1, fib(2) := 1, fib(n) := fib(n-1) + fib(n-2)] @key{RET} s t fib
5527 ' fib(7) @key{RET} a r fib @key{RET}
5531 One thing that is guaranteed about the order that rewrites are tried
5532 is that, for any given subformula, earlier rules in the rule set will
5533 be tried for that subformula before later ones. So even though the
5534 first and third rules both match @samp{fib(1)}, we know the first will
5535 be used preferentially.
5537 This rule set has one dangerous bug: Suppose we apply it to the
5538 formula @samp{fib(x)}? (Don't actually try this.) The third rule
5539 will match @samp{fib(x)} and replace it with @w{@samp{fib(x-1) + fib(x-2)}}.
5540 Each of these will then be replaced to get @samp{fib(x-2) + 2 fib(x-3) +
5541 fib(x-4)}, and so on, expanding forever. What we really want is to apply
5542 the third rule only when @samp{n} is an integer greater than two. Type
5543 @w{@kbd{s e fib @key{RET}}}, then edit the third rule to:
5546 fib(n) := fib(n-1) + fib(n-2) :: integer(n) :: n > 2
5554 1: fib(6) + fib(x) + fib(0) 1: 8 + fib(x) + fib(0)
5557 ' fib(6)+fib(x)+fib(0) @key{RET} a r fib @key{RET}
5562 We've created a new function, @code{fib}, and a new command,
5563 @w{@kbd{a r fib @key{RET}}}, which means ``evaluate all @code{fib} calls in
5564 this formula.'' To make things easier still, we can tell Calc to
5565 apply these rules automatically by storing them in the special
5566 variable @code{EvalRules}.
5570 1: [fib(1) := ...] . 1: [8, 13]
5573 s r fib @key{RET} s t EvalRules @key{RET} ' [fib(6), fib(7)] @key{RET}
5577 It turns out that this rule set has the problem that it does far
5578 more work than it needs to when @samp{n} is large. Consider the
5579 first few steps of the computation of @samp{fib(6)}:
5585 fib(4) + fib(3) + fib(3) + fib(2) =
5586 fib(3) + fib(2) + fib(2) + fib(1) + fib(2) + fib(1) + 1 = ...
5591 Note that @samp{fib(3)} appears three times here. Unless Calc's
5592 algebraic simplifier notices the multiple @samp{fib(3)}s and combines
5593 them (and, as it happens, it doesn't), this rule set does lots of
5594 needless recomputation. To cure the problem, type @code{s e EvalRules}
5595 to edit the rules (or just @kbd{s E}, a shorthand command for editing
5596 @code{EvalRules}) and add another condition:
5599 fib(n) := fib(n-1) + fib(n-2) :: integer(n) :: n > 2 :: remember
5603 If a @samp{:: remember} condition appears anywhere in a rule, then if
5604 that rule succeeds Calc will add another rule that describes that match
5605 to the front of the rule set. (Remembering works in any rule set, but
5606 for technical reasons it is most effective in @code{EvalRules}.) For
5607 example, if the rule rewrites @samp{fib(7)} to something that evaluates
5608 to 13, then the rule @samp{fib(7) := 13} will be added to the rule set.
5610 Type @kbd{' fib(8) @key{RET}} to compute the eighth Fibonacci number, then
5611 type @kbd{s E} again to see what has happened to the rule set.
5613 With the @code{remember} feature, our rule set can now compute
5614 @samp{fib(@var{n})} in just @var{n} steps. In the process it builds
5615 up a table of all Fibonacci numbers up to @var{n}. After we have
5616 computed the result for a particular @var{n}, we can get it back
5617 (and the results for all smaller @var{n}) later in just one step.
5619 All Calc operations will run somewhat slower whenever @code{EvalRules}
5620 contains any rules. You should type @kbd{s u EvalRules @key{RET}} now to
5621 un-store the variable.
5623 (@bullet{}) @strong{Exercise 2.} Sometimes it is possible to reformulate
5624 a problem to reduce the amount of recursion necessary to solve it.
5625 Create a rule that, in about @var{n} simple steps and without recourse
5626 to the @code{remember} option, replaces @samp{fib(@var{n}, 1, 1)} with
5627 @samp{fib(1, @var{x}, @var{y})} where @var{x} and @var{y} are the
5628 @var{n}th and @var{n+1}st Fibonacci numbers, respectively. This rule is
5629 rather clunky to use, so add a couple more rules to make the ``user
5630 interface'' the same as for our first version: enter @samp{fib(@var{n})},
5631 get back a plain number. @xref{Rewrites Answer 2, 2}. (@bullet{})
5633 There are many more things that rewrites can do. For example, there
5634 are @samp{&&&} and @samp{|||} pattern operators that create ``and''
5635 and ``or'' combinations of rules. As one really simple example, we
5636 could combine our first two Fibonacci rules thusly:
5639 [fib(1 ||| 2) := 1, fib(n) := ... ]
5643 That means ``@code{fib} of something matching either 1 or 2 rewrites
5646 You can also make meta-variables optional by enclosing them in @code{opt}.
5647 For example, the pattern @samp{a + b x} matches @samp{2 + 3 x} but not
5648 @samp{2 + x} or @samp{3 x} or @samp{x}. The pattern @samp{opt(a) + opt(b) x}
5649 matches all of these forms, filling in a default of zero for @samp{a}
5650 and one for @samp{b}.
5652 (@bullet{}) @strong{Exercise 3.} Your friend Joe had @samp{2 + 3 x}
5653 on the stack and tried to use the rule
5654 @samp{opt(a) + opt(b) x := f(a, b, x)}. What happened?
5655 @xref{Rewrites Answer 3, 3}. (@bullet{})
5657 (@bullet{}) @strong{Exercise 4.} Starting with a positive integer @expr{a},
5658 divide @expr{a} by two if it is even, otherwise compute @expr{3 a + 1}.
5659 Now repeat this step over and over. A famous unproved conjecture
5660 is that for any starting @expr{a}, the sequence always eventually
5661 reaches 1. Given the formula @samp{seq(@var{a}, 0)}, write a set of
5662 rules that convert this into @samp{seq(1, @var{n})} where @var{n}
5663 is the number of steps it took the sequence to reach the value 1.
5664 Now enhance the rules to accept @samp{seq(@var{a})} as a starting
5665 configuration, and to stop with just the number @var{n} by itself.
5666 Now make the result be a vector of values in the sequence, from @var{a}
5667 to 1. (The formula @samp{@var{x}|@var{y}} appends the vectors @var{x}
5668 and @var{y}.) For example, rewriting @samp{seq(6)} should yield the
5669 vector @expr{[6, 3, 10, 5, 16, 8, 4, 2, 1]}.
5670 @xref{Rewrites Answer 4, 4}. (@bullet{})
5672 (@bullet{}) @strong{Exercise 5.} Define, using rewrite rules, a function
5673 @samp{nterms(@var{x})} that returns the number of terms in the sum
5674 @var{x}, or 1 if @var{x} is not a sum. (A @dfn{sum} for our purposes
5675 is one or more non-sum terms separated by @samp{+} or @samp{-} signs,
5676 so that @expr{2 - 3 (x + y) + x y} is a sum of three terms.)
5677 @xref{Rewrites Answer 5, 5}. (@bullet{})
5679 (@bullet{}) @strong{Exercise 6.} A Taylor series for a function is an
5680 infinite series that exactly equals the value of that function at
5681 values of @expr{x} near zero.
5685 cos(x) = 1 - x^2 / 2! + x^4 / 4! - x^6 / 6! + ...
5691 $$ \cos x = 1 - {x^2 \over 2!} + {x^4 \over 4!} - {x^6 \over 6!} + \cdots $$
5695 The @kbd{a t} command produces a @dfn{truncated Taylor series} which
5696 is obtained by dropping all the terms higher than, say, @expr{x^2}.
5697 Calc represents the truncated Taylor series as a polynomial in @expr{x}.
5698 Mathematicians often write a truncated series using a ``big-O'' notation
5699 that records what was the lowest term that was truncated.
5703 cos(x) = 1 - x^2 / 2! + O(x^3)
5709 $$ \cos x = 1 - {x^2 \over 2!} + O(x^3) $$
5714 The meaning of @expr{O(x^3)} is ``a quantity which is negligibly small
5715 if @expr{x^3} is considered negligibly small as @expr{x} goes to zero.''
5717 The exercise is to create rewrite rules that simplify sums and products of
5718 power series represented as @samp{@var{polynomial} + O(@var{var}^@var{n})}.
5719 For example, given @samp{1 - x^2 / 2 + O(x^3)} and @samp{x - x^3 / 6 + O(x^4)}
5720 on the stack, we want to be able to type @kbd{*} and get the result
5721 @samp{x - 2:3 x^3 + O(x^4)}. Don't worry if the terms of the sum are
5722 rearranged or if @kbd{a s} needs to be typed after rewriting. (This one
5723 is rather tricky; the solution at the end of this chapter uses 6 rewrite
5724 rules. Hint: The @samp{constant(x)} condition tests whether @samp{x} is
5725 a number.) @xref{Rewrites Answer 6, 6}. (@bullet{})
5727 Just for kicks, try adding the rule @code{2+3 := 6} to @code{EvalRules}.
5728 What happens? (Be sure to remove this rule afterward, or you might get
5729 a nasty surprise when you use Calc to balance your checkbook!)
5731 @xref{Rewrite Rules}, for the whole story on rewrite rules.
5733 @node Programming Tutorial, Answers to Exercises, Algebra Tutorial, Tutorial
5734 @section Programming Tutorial
5737 The Calculator is written entirely in Emacs Lisp, a highly extensible
5738 language. If you know Lisp, you can program the Calculator to do
5739 anything you like. Rewrite rules also work as a powerful programming
5740 system. But Lisp and rewrite rules take a while to master, and often
5741 all you want to do is define a new function or repeat a command a few
5742 times. Calc has features that allow you to do these things easily.
5744 One very limited form of programming is defining your own functions.
5745 Calc's @kbd{Z F} command allows you to define a function name and
5746 key sequence to correspond to any formula. Programming commands use
5747 the shift-@kbd{Z} prefix; the user commands they create use the lower
5748 case @kbd{z} prefix.
5752 1: 1 + x + x^2 / 2 + x^3 / 6 1: 1 + x + x^2 / 2 + x^3 / 6
5755 ' 1 + x + x^2/2! + x^3/3! @key{RET} Z F e myexp @key{RET} @key{RET} @key{RET} y
5759 This polynomial is a Taylor series approximation to @samp{exp(x)}.
5760 The @kbd{Z F} command asks a number of questions. The above answers
5761 say that the key sequence for our function should be @kbd{z e}; the
5762 @kbd{M-x} equivalent should be @code{calc-myexp}; the name of the
5763 function in algebraic formulas should also be @code{myexp}; the
5764 default argument list @samp{(x)} is acceptable; and finally @kbd{y}
5765 answers the question ``leave it in symbolic form for non-constant
5770 1: 1.3495 2: 1.3495 3: 1.3495
5771 . 1: 1.34986 2: 1.34986
5775 .3 z e .3 E ' a+1 @key{RET} z e
5780 First we call our new @code{exp} approximation with 0.3 as an
5781 argument, and compare it with the true @code{exp} function. Then
5782 we note that, as requested, if we try to give @kbd{z e} an
5783 argument that isn't a plain number, it leaves the @code{myexp}
5784 function call in symbolic form. If we had answered @kbd{n} to the
5785 final question, @samp{myexp(a + 1)} would have evaluated by plugging
5786 in @samp{a + 1} for @samp{x} in the defining formula.
5788 @cindex Sine integral Si(x)
5793 (@bullet{}) @strong{Exercise 1.} The ``sine integral'' function
5794 @texline @math{{\rm Si}(x)}
5795 @infoline @expr{Si(x)}
5796 is defined as the integral of @samp{sin(t)/t} for
5797 @expr{t = 0} to @expr{x} in radians. (It was invented because this
5798 integral has no solution in terms of basic functions; if you give it
5799 to Calc's @kbd{a i} command, it will ponder it for a long time and then
5800 give up.) We can use the numerical integration command, however,
5801 which in algebraic notation is written like @samp{ninteg(f(t), t, 0, x)}
5802 with any integrand @samp{f(t)}. Define a @kbd{z s} command and
5803 @code{Si} function that implement this. You will need to edit the
5804 default argument list a bit. As a test, @samp{Si(1)} should return
5805 0.946083. (If you don't get this answer, you might want to check that
5806 Calc is in Radians mode. Also, @code{ninteg} will run a lot faster if
5807 you reduce the precision to, say, six digits beforehand.)
5808 @xref{Programming Answer 1, 1}. (@bullet{})
5810 The simplest way to do real ``programming'' of Emacs is to define a
5811 @dfn{keyboard macro}. A keyboard macro is simply a sequence of
5812 keystrokes which Emacs has stored away and can play back on demand.
5813 For example, if you find yourself typing @kbd{H a S x @key{RET}} often,
5814 you may wish to program a keyboard macro to type this for you.
5818 1: y = sqrt(x) 1: x = y^2
5821 ' y=sqrt(x) @key{RET} C-x ( H a S x @key{RET} C-x )
5823 1: y = cos(x) 1: x = s1 arccos(y) + 2 pi n1
5826 ' y=cos(x) @key{RET} X
5831 When you type @kbd{C-x (}, Emacs begins recording. But it is also
5832 still ready to execute your keystrokes, so you're really ``training''
5833 Emacs by walking it through the procedure once. When you type
5834 @w{@kbd{C-x )}}, the macro is recorded. You can now type @kbd{X} to
5835 re-execute the same keystrokes.
5837 You can give a name to your macro by typing @kbd{Z K}.
5841 1: . 1: y = x^4 1: x = s2 sqrt(s1 sqrt(y))
5844 Z K x @key{RET} ' y=x^4 @key{RET} z x
5849 Notice that we use shift-@kbd{Z} to define the command, and lower-case
5850 @kbd{z} to call it up.
5852 Keyboard macros can call other macros.
5856 1: abs(x) 1: x = s1 y 1: 2 / x 1: x = 2 / y
5859 ' abs(x) @key{RET} C-x ( ' y @key{RET} a = z x C-x ) ' 2/x @key{RET} X
5863 (@bullet{}) @strong{Exercise 2.} Define a keyboard macro to negate
5864 the item in level 3 of the stack, without disturbing the rest of
5865 the stack. @xref{Programming Answer 2, 2}. (@bullet{})
5867 (@bullet{}) @strong{Exercise 3.} Define keyboard macros to compute
5868 the following functions:
5873 @texline @math{\displaystyle{\sin x \over x}},
5874 @infoline @expr{sin(x) / x},
5875 where @expr{x} is the number on the top of the stack.
5878 Compute the base-@expr{b} logarithm, just like the @kbd{B} key except
5879 the arguments are taken in the opposite order.
5882 Produce a vector of integers from 1 to the integer on the top of
5886 @xref{Programming Answer 3, 3}. (@bullet{})
5888 (@bullet{}) @strong{Exercise 4.} Define a keyboard macro to compute
5889 the average (mean) value of a list of numbers.
5890 @xref{Programming Answer 4, 4}. (@bullet{})
5892 In many programs, some of the steps must execute several times.
5893 Calc has @dfn{looping} commands that allow this. Loops are useful
5894 inside keyboard macros, but actually work at any time.
5898 1: x^6 2: x^6 1: 360 x^2
5902 ' x^6 @key{RET} 4 Z < a d x @key{RET} Z >
5907 Here we have computed the fourth derivative of @expr{x^6} by
5908 enclosing a derivative command in a ``repeat loop'' structure.
5909 This structure pops a repeat count from the stack, then
5910 executes the body of the loop that many times.
5912 If you make a mistake while entering the body of the loop,
5913 type @w{@kbd{Z C-g}} to cancel the loop command.
5915 @cindex Fibonacci numbers
5916 Here's another example:
5925 1 @key{RET} @key{RET} 20 Z < @key{TAB} C-j + Z >
5930 The numbers in levels 2 and 1 should be the 21st and 22nd Fibonacci
5931 numbers, respectively. (To see what's going on, try a few repetitions
5932 of the loop body by hand; @kbd{C-j}, also on the Line-Feed or @key{LFD}
5933 key if you have one, makes a copy of the number in level 2.)
5935 @cindex Golden ratio
5936 @cindex Phi, golden ratio
5937 A fascinating property of the Fibonacci numbers is that the @expr{n}th
5938 Fibonacci number can be found directly by computing
5939 @texline @math{\phi^n / \sqrt{5}}
5940 @infoline @expr{phi^n / sqrt(5)}
5941 and then rounding to the nearest integer, where
5942 @texline @math{\phi} (``phi''),
5943 @infoline @expr{phi},
5944 the ``golden ratio,'' is
5945 @texline @math{(1 + \sqrt{5}) / 2}.
5946 @infoline @expr{(1 + sqrt(5)) / 2}.
5947 (For convenience, this constant is available from the @code{phi}
5948 variable, or the @kbd{I H P} command.)
5952 1: 1.61803 1: 24476.0000409 1: 10945.9999817 1: 10946
5959 @cindex Continued fractions
5960 (@bullet{}) @strong{Exercise 5.} The @dfn{continued fraction}
5962 @texline @math{\phi}
5963 @infoline @expr{phi}
5965 @texline @math{1 + 1/(1 + 1/(1 + 1/( \ldots )))}.
5966 @infoline @expr{1 + 1/(1 + 1/(1 + 1/( ...@: )))}.
5967 We can compute an approximate value by carrying this however far
5968 and then replacing the innermost
5969 @texline @math{1/( \ldots )}
5970 @infoline @expr{1/( ...@: )}
5972 @texline @math{\phi}
5973 @infoline @expr{phi}
5974 using a twenty-term continued fraction.
5975 @xref{Programming Answer 5, 5}. (@bullet{})
5977 (@bullet{}) @strong{Exercise 6.} Linear recurrences like the one for
5978 Fibonacci numbers can be expressed in terms of matrices. Given a
5979 vector @w{@expr{[a, b]}} determine a matrix which, when multiplied by this
5980 vector, produces the vector @expr{[b, c]}, where @expr{a}, @expr{b} and
5981 @expr{c} are three successive Fibonacci numbers. Now write a program
5982 that, given an integer @expr{n}, computes the @expr{n}th Fibonacci number
5983 using matrix arithmetic. @xref{Programming Answer 6, 6}. (@bullet{})
5985 @cindex Harmonic numbers
5986 A more sophisticated kind of loop is the @dfn{for} loop. Suppose
5987 we wish to compute the 20th ``harmonic'' number, which is equal to
5988 the sum of the reciprocals of the integers from 1 to 20.
5997 0 @key{RET} 1 @key{RET} 20 Z ( & + 1 Z )
6002 The ``for'' loop pops two numbers, the lower and upper limits, then
6003 repeats the body of the loop as an internal counter increases from
6004 the lower limit to the upper one. Just before executing the loop
6005 body, it pushes the current loop counter. When the loop body
6006 finishes, it pops the ``step,'' i.e., the amount by which to
6007 increment the loop counter. As you can see, our loop always
6010 This harmonic number function uses the stack to hold the running
6011 total as well as for the various loop housekeeping functions. If
6012 you find this disorienting, you can sum in a variable instead:
6016 1: 0 2: 1 . 1: 3.597739
6020 0 t 7 1 @key{RET} 20 Z ( & s + 7 1 Z ) r 7
6025 The @kbd{s +} command adds the top-of-stack into the value in a
6026 variable (and removes that value from the stack).
6028 It's worth noting that many jobs that call for a ``for'' loop can
6029 also be done more easily by Calc's high-level operations. Two
6030 other ways to compute harmonic numbers are to use vector mapping
6031 and reduction (@kbd{v x 20}, then @w{@kbd{V M &}}, then @kbd{V R +}),
6032 or to use the summation command @kbd{a +}. Both of these are
6033 probably easier than using loops. However, there are some
6034 situations where loops really are the way to go:
6036 (@bullet{}) @strong{Exercise 7.} Use a ``for'' loop to find the first
6037 harmonic number which is greater than 4.0.
6038 @xref{Programming Answer 7, 7}. (@bullet{})
6040 Of course, if we're going to be using variables in our programs,
6041 we have to worry about the programs clobbering values that the
6042 caller was keeping in those same variables. This is easy to
6047 . 1: 0.6667 1: 0.6667 3: 0.6667
6052 Z ` p 4 @key{RET} 2 @key{RET} 3 / s 7 s s a @key{RET} Z ' r 7 s r a @key{RET}
6057 When we type @kbd{Z `} (that's a back-quote character), Calc saves
6058 its mode settings and the contents of the ten ``quick variables''
6059 for later reference. When we type @kbd{Z '} (that's an apostrophe
6060 now), Calc restores those saved values. Thus the @kbd{p 4} and
6061 @kbd{s 7} commands have no effect outside this sequence. Wrapping
6062 this around the body of a keyboard macro ensures that it doesn't
6063 interfere with what the user of the macro was doing. Notice that
6064 the contents of the stack, and the values of named variables,
6065 survive past the @kbd{Z '} command.
6067 @cindex Bernoulli numbers, approximate
6068 The @dfn{Bernoulli numbers} are a sequence with the interesting
6069 property that all of the odd Bernoulli numbers are zero, and the
6070 even ones, while difficult to compute, can be roughly approximated
6072 @texline @math{\displaystyle{2 n! \over (2 \pi)^n}}.
6073 @infoline @expr{2 n!@: / (2 pi)^n}.
6074 Let's write a keyboard macro to compute (approximate) Bernoulli numbers.
6075 (Calc has a command, @kbd{k b}, to compute exact Bernoulli numbers, but
6076 this command is very slow for large @expr{n} since the higher Bernoulli
6077 numbers are very large fractions.)
6084 10 C-x ( @key{RET} 2 % Z [ @key{DEL} 0 Z : ' 2 $! / (2 pi)^$ @key{RET} = Z ] C-x )
6089 You can read @kbd{Z [} as ``then,'' @kbd{Z :} as ``else,'' and
6090 @kbd{Z ]} as ``end-if.'' There is no need for an explicit ``if''
6091 command. For the purposes of @w{@kbd{Z [}}, the condition is ``true''
6092 if the value it pops from the stack is a nonzero number, or ``false''
6093 if it pops zero or something that is not a number (like a formula).
6094 Here we take our integer argument modulo 2; this will be nonzero
6095 if we're asking for an odd Bernoulli number.
6097 The actual tenth Bernoulli number is @expr{5/66}.
6101 3: 0.0756823 1: 0 1: 0.25305 1: 0 1: 1.16659
6106 10 k b @key{RET} c f M-0 @key{DEL} 11 X @key{DEL} 12 X @key{DEL} 13 X @key{DEL} 14 X
6110 Just to exercise loops a bit more, let's compute a table of even
6115 3: [] 1: [0.10132, 0.03079, 0.02340, 0.033197, ...]
6120 [ ] 2 @key{RET} 30 Z ( X | 2 Z )
6125 The vertical-bar @kbd{|} is the vector-concatenation command. When
6126 we execute it, the list we are building will be in stack level 2
6127 (initially this is an empty list), and the next Bernoulli number
6128 will be in level 1. The effect is to append the Bernoulli number
6129 onto the end of the list. (To create a table of exact fractional
6130 Bernoulli numbers, just replace @kbd{X} with @kbd{k b} in the above
6131 sequence of keystrokes.)
6133 With loops and conditionals, you can program essentially anything
6134 in Calc. One other command that makes looping easier is @kbd{Z /},
6135 which takes a condition from the stack and breaks out of the enclosing
6136 loop if the condition is true (non-zero). You can use this to make
6137 ``while'' and ``until'' style loops.
6139 If you make a mistake when entering a keyboard macro, you can edit
6140 it using @kbd{Z E}. First, you must attach it to a key with @kbd{Z K}.
6141 One technique is to enter a throwaway dummy definition for the macro,
6142 then enter the real one in the edit command.
6146 1: 3 1: 3 Calc Macro Edit Mode.
6147 . . Original keys: 1 <return> 2 +
6154 C-x ( 1 @key{RET} 2 + C-x ) Z K h @key{RET} Z E h
6159 A keyboard macro is stored as a pure keystroke sequence. The
6160 @file{edmacro} package (invoked by @kbd{Z E}) scans along the
6161 macro and tries to decode it back into human-readable steps.
6162 Descriptions of the keystrokes are given as comments, which begin with
6163 @samp{;;}, and which are ignored when the edited macro is saved.
6164 Spaces and line breaks are also ignored when the edited macro is saved.
6165 To enter a space into the macro, type @code{SPC}. All the special
6166 characters @code{RET}, @code{LFD}, @code{TAB}, @code{SPC}, @code{DEL},
6167 and @code{NUL} must be written in all uppercase, as must the prefixes
6168 @code{C-} and @code{M-}.
6170 Let's edit in a new definition, for computing harmonic numbers.
6171 First, erase the four lines of the old definition. Then, type
6172 in the new definition (or use Emacs @kbd{M-w} and @kbd{C-y} commands
6173 to copy it from this page of the Info file; you can of course skip
6174 typing the comments, which begin with @samp{;;}).
6177 Z` ;; calc-kbd-push (Save local values)
6178 0 ;; calc digits (Push a zero onto the stack)
6179 st ;; calc-store-into (Store it in the following variable)
6180 1 ;; calc quick variable (Quick variable q1)
6181 1 ;; calc digits (Initial value for the loop)
6182 TAB ;; calc-roll-down (Swap initial and final)
6183 Z( ;; calc-kbd-for (Begin the "for" loop)
6184 & ;; calc-inv (Take the reciprocal)
6185 s+ ;; calc-store-plus (Add to the following variable)
6186 1 ;; calc quick variable (Quick variable q1)
6187 1 ;; calc digits (The loop step is 1)
6188 Z) ;; calc-kbd-end-for (End the "for" loop)
6189 sr ;; calc-recall (Recall the final accumulated value)
6190 1 ;; calc quick variable (Quick variable q1)
6191 Z' ;; calc-kbd-pop (Restore values)
6195 Press @kbd{C-c C-c} to finish editing and return to the Calculator.
6206 The @file{edmacro} package defines a handy @code{read-kbd-macro} command
6207 which reads the current region of the current buffer as a sequence of
6208 keystroke names, and defines that sequence on the @kbd{X}
6209 (and @kbd{C-x e}) key. Because this is so useful, Calc puts this
6210 command on the @kbd{C-x * m} key. Try reading in this macro in the
6211 following form: Press @kbd{C-@@} (or @kbd{C-@key{SPC}}) at
6212 one end of the text below, then type @kbd{C-x * m} at the other.
6224 (@bullet{}) @strong{Exercise 8.} A general algorithm for solving
6225 equations numerically is @dfn{Newton's Method}. Given the equation
6226 @expr{f(x) = 0} for any function @expr{f}, and an initial guess
6227 @expr{x_0} which is reasonably close to the desired solution, apply
6228 this formula over and over:
6232 new_x = x - f(x)/f'(x)
6237 $$ x_{\rm new} = x - {f(x) \over f'(x)} $$
6242 where @expr{f'(x)} is the derivative of @expr{f}. The @expr{x}
6243 values will quickly converge to a solution, i.e., eventually
6244 @texline @math{x_{\rm new}}
6245 @infoline @expr{new_x}
6246 and @expr{x} will be equal to within the limits
6247 of the current precision. Write a program which takes a formula
6248 involving the variable @expr{x}, and an initial guess @expr{x_0},
6249 on the stack, and produces a value of @expr{x} for which the formula
6250 is zero. Use it to find a solution of
6251 @texline @math{\sin(\cos x) = 0.5}
6252 @infoline @expr{sin(cos(x)) = 0.5}
6253 near @expr{x = 4.5}. (Use angles measured in radians.) Note that
6254 the built-in @w{@kbd{a R}} (@code{calc-find-root}) command uses Newton's
6255 method when it is able. @xref{Programming Answer 8, 8}. (@bullet{})
6257 @cindex Digamma function
6258 @cindex Gamma constant, Euler's
6259 @cindex Euler's gamma constant
6260 (@bullet{}) @strong{Exercise 9.} The @dfn{digamma} function
6261 @texline @math{\psi(z) (``psi'')}
6262 @infoline @expr{psi(z)}
6263 is defined as the derivative of
6264 @texline @math{\ln \Gamma(z)}.
6265 @infoline @expr{ln(gamma(z))}.
6266 For large values of @expr{z}, it can be approximated by the infinite sum
6270 psi(z) ~= ln(z) - 1/2z - sum(bern(2 n) / 2 n z^(2 n), n, 1, inf)
6275 $$ \psi(z) \approx \ln z - {1\over2z} -
6276 \sum_{n=1}^\infty {\code{bern}(2 n) \over 2 n z^{2n}}
6283 @texline @math{\sum}
6284 @infoline @expr{sum}
6285 represents the sum over @expr{n} from 1 to infinity
6286 (or to some limit high enough to give the desired accuracy), and
6287 the @code{bern} function produces (exact) Bernoulli numbers.
6288 While this sum is not guaranteed to converge, in practice it is safe.
6289 An interesting mathematical constant is Euler's gamma, which is equal
6290 to about 0.5772. One way to compute it is by the formula,
6291 @texline @math{\gamma = -\psi(1)}.
6292 @infoline @expr{gamma = -psi(1)}.
6293 Unfortunately, 1 isn't a large enough argument
6294 for the above formula to work (5 is a much safer value for @expr{z}).
6295 Fortunately, we can compute
6296 @texline @math{\psi(1)}
6297 @infoline @expr{psi(1)}
6299 @texline @math{\psi(5)}
6300 @infoline @expr{psi(5)}
6301 using the recurrence
6302 @texline @math{\psi(z+1) = \psi(z) + {1 \over z}}.
6303 @infoline @expr{psi(z+1) = psi(z) + 1/z}.
6304 Your task: Develop a program to compute
6305 @texline @math{\psi(z)};
6306 @infoline @expr{psi(z)};
6307 it should ``pump up'' @expr{z}
6308 if necessary to be greater than 5, then use the above summation
6309 formula. Use looping commands to compute the sum. Use your function
6311 @texline @math{\gamma}
6312 @infoline @expr{gamma}
6313 to twelve decimal places. (Calc has a built-in command
6314 for Euler's constant, @kbd{I P}, which you can use to check your answer.)
6315 @xref{Programming Answer 9, 9}. (@bullet{})
6317 @cindex Polynomial, list of coefficients
6318 (@bullet{}) @strong{Exercise 10.} Given a polynomial in @expr{x} and
6319 a number @expr{m} on the stack, where the polynomial is of degree
6320 @expr{m} or less (i.e., does not have any terms higher than @expr{x^m}),
6321 write a program to convert the polynomial into a list-of-coefficients
6322 notation. For example, @expr{5 x^4 + (x + 1)^2} with @expr{m = 6}
6323 should produce the list @expr{[1, 2, 1, 0, 5, 0, 0]}. Also develop
6324 a way to convert from this form back to the standard algebraic form.
6325 @xref{Programming Answer 10, 10}. (@bullet{})
6328 (@bullet{}) @strong{Exercise 11.} The @dfn{Stirling numbers of the
6329 first kind} are defined by the recurrences,
6333 s(n,n) = 1 for n >= 0,
6334 s(n,0) = 0 for n > 0,
6335 s(n+1,m) = s(n,m-1) - n s(n,m) for n >= m >= 1.
6341 $$ \eqalign{ s(n,n) &= 1 \qquad \hbox{for } n \ge 0, \cr
6342 s(n,0) &= 0 \qquad \hbox{for } n > 0, \cr
6343 s(n+1,m) &= s(n,m-1) - n \, s(n,m) \qquad
6344 \hbox{for } n \ge m \ge 1.}
6348 (These numbers are also sometimes written $\displaystyle{n \brack m}$.)
6351 This can be implemented using a @dfn{recursive} program in Calc; the
6352 program must invoke itself in order to calculate the two righthand
6353 terms in the general formula. Since it always invokes itself with
6354 ``simpler'' arguments, it's easy to see that it must eventually finish
6355 the computation. Recursion is a little difficult with Emacs keyboard
6356 macros since the macro is executed before its definition is complete.
6357 So here's the recommended strategy: Create a ``dummy macro'' and assign
6358 it to a key with, e.g., @kbd{Z K s}. Now enter the true definition,
6359 using the @kbd{z s} command to call itself recursively, then assign it
6360 to the same key with @kbd{Z K s}. Now the @kbd{z s} command will run
6361 the complete recursive program. (Another way is to use @w{@kbd{Z E}}
6362 or @kbd{C-x * m} (@code{read-kbd-macro}) to read the whole macro at once,
6363 thus avoiding the ``training'' phase.) The task: Write a program
6364 that computes Stirling numbers of the first kind, given @expr{n} and
6365 @expr{m} on the stack. Test it with @emph{small} inputs like
6366 @expr{s(4,2)}. (There is a built-in command for Stirling numbers,
6367 @kbd{k s}, which you can use to check your answers.)
6368 @xref{Programming Answer 11, 11}. (@bullet{})
6370 The programming commands we've seen in this part of the tutorial
6371 are low-level, general-purpose operations. Often you will find
6372 that a higher-level function, such as vector mapping or rewrite
6373 rules, will do the job much more easily than a detailed, step-by-step
6376 (@bullet{}) @strong{Exercise 12.} Write another program for
6377 computing Stirling numbers of the first kind, this time using
6378 rewrite rules. Once again, @expr{n} and @expr{m} should be taken
6379 from the stack. @xref{Programming Answer 12, 12}. (@bullet{})
6384 This ends the tutorial section of the Calc manual. Now you know enough
6385 about Calc to use it effectively for many kinds of calculations. But
6386 Calc has many features that were not even touched upon in this tutorial.
6388 The rest of this manual tells the whole story.
6390 @c Volume II of this manual, the @dfn{Calc Reference}, tells the whole story.
6393 @node Answers to Exercises, , Programming Tutorial, Tutorial
6394 @section Answers to Exercises
6397 This section includes answers to all the exercises in the Calc tutorial.
6400 * RPN Answer 1:: 1 @key{RET} 2 @key{RET} 3 @key{RET} 4 + * -
6401 * RPN Answer 2:: 2*4 + 7*9.5 + 5/4
6402 * RPN Answer 3:: Operating on levels 2 and 3
6403 * RPN Answer 4:: Joe's complex problems
6404 * Algebraic Answer 1:: Simulating Q command
6405 * Algebraic Answer 2:: Joe's algebraic woes
6406 * Algebraic Answer 3:: 1 / 0
6407 * Modes Answer 1:: 3#0.1 = 3#0.0222222?
6408 * Modes Answer 2:: 16#f.e8fe15
6409 * Modes Answer 3:: Joe's rounding bug
6410 * Modes Answer 4:: Why floating point?
6411 * Arithmetic Answer 1:: Why the \ command?
6412 * Arithmetic Answer 2:: Tripping up the B command
6413 * Vector Answer 1:: Normalizing a vector
6414 * Vector Answer 2:: Average position
6415 * Matrix Answer 1:: Row and column sums
6416 * Matrix Answer 2:: Symbolic system of equations
6417 * Matrix Answer 3:: Over-determined system
6418 * List Answer 1:: Powers of two
6419 * List Answer 2:: Least-squares fit with matrices
6420 * List Answer 3:: Geometric mean
6421 * List Answer 4:: Divisor function
6422 * List Answer 5:: Duplicate factors
6423 * List Answer 6:: Triangular list
6424 * List Answer 7:: Another triangular list
6425 * List Answer 8:: Maximum of Bessel function
6426 * List Answer 9:: Integers the hard way
6427 * List Answer 10:: All elements equal
6428 * List Answer 11:: Estimating pi with darts
6429 * List Answer 12:: Estimating pi with matchsticks
6430 * List Answer 13:: Hash codes
6431 * List Answer 14:: Random walk
6432 * Types Answer 1:: Square root of pi times rational
6433 * Types Answer 2:: Infinities
6434 * Types Answer 3:: What can "nan" be?
6435 * Types Answer 4:: Abbey Road
6436 * Types Answer 5:: Friday the 13th
6437 * Types Answer 6:: Leap years
6438 * Types Answer 7:: Erroneous donut
6439 * Types Answer 8:: Dividing intervals
6440 * Types Answer 9:: Squaring intervals
6441 * Types Answer 10:: Fermat's primality test
6442 * Types Answer 11:: pi * 10^7 seconds
6443 * Types Answer 12:: Abbey Road on CD
6444 * Types Answer 13:: Not quite pi * 10^7 seconds
6445 * Types Answer 14:: Supercomputers and c
6446 * Types Answer 15:: Sam the Slug
6447 * Algebra Answer 1:: Squares and square roots
6448 * Algebra Answer 2:: Building polynomial from roots
6449 * Algebra Answer 3:: Integral of x sin(pi x)
6450 * Algebra Answer 4:: Simpson's rule
6451 * Rewrites Answer 1:: Multiplying by conjugate
6452 * Rewrites Answer 2:: Alternative fib rule
6453 * Rewrites Answer 3:: Rewriting opt(a) + opt(b) x
6454 * Rewrites Answer 4:: Sequence of integers
6455 * Rewrites Answer 5:: Number of terms in sum
6456 * Rewrites Answer 6:: Truncated Taylor series
6457 * Programming Answer 1:: Fresnel's C(x)
6458 * Programming Answer 2:: Negate third stack element
6459 * Programming Answer 3:: Compute sin(x) / x, etc.
6460 * Programming Answer 4:: Average value of a list
6461 * Programming Answer 5:: Continued fraction phi
6462 * Programming Answer 6:: Matrix Fibonacci numbers
6463 * Programming Answer 7:: Harmonic number greater than 4
6464 * Programming Answer 8:: Newton's method
6465 * Programming Answer 9:: Digamma function
6466 * Programming Answer 10:: Unpacking a polynomial
6467 * Programming Answer 11:: Recursive Stirling numbers
6468 * Programming Answer 12:: Stirling numbers with rewrites
6471 @c The following kludgery prevents the individual answers from
6472 @c being entered on the table of contents.
6474 \global\let\oldwrite=\write
6475 \gdef\skipwrite#1#2{\let\write=\oldwrite}
6476 \global\let\oldchapternofonts=\chapternofonts
6477 \gdef\chapternofonts{\let\write=\skipwrite\oldchapternofonts}
6480 @node RPN Answer 1, RPN Answer 2, Answers to Exercises, Answers to Exercises
6481 @subsection RPN Tutorial Exercise 1
6484 @kbd{1 @key{RET} 2 @key{RET} 3 @key{RET} 4 + * -}
6487 @texline @math{1 - (2 \times (3 + 4)) = -13}.
6488 @infoline @expr{1 - (2 * (3 + 4)) = -13}.
6490 @node RPN Answer 2, RPN Answer 3, RPN Answer 1, Answers to Exercises
6491 @subsection RPN Tutorial Exercise 2
6494 @texline @math{2\times4 + 7\times9.5 + {5\over4} = 75.75}
6495 @infoline @expr{2*4 + 7*9.5 + 5/4 = 75.75}
6497 After computing the intermediate term
6498 @texline @math{2\times4 = 8},
6499 @infoline @expr{2*4 = 8},
6500 you can leave that result on the stack while you compute the second
6501 term. With both of these results waiting on the stack you can then
6502 compute the final term, then press @kbd{+ +} to add everything up.
6511 2 @key{RET} 4 * 7 @key{RET} 9.5 *
6518 4: 8 3: 8 2: 8 1: 75.75
6519 3: 66.5 2: 66.5 1: 67.75 .
6528 Alternatively, you could add the first two terms before going on
6529 with the third term.
6533 2: 8 1: 74.5 3: 74.5 2: 74.5 1: 75.75
6534 1: 66.5 . 2: 5 1: 1.25 .
6538 ... + 5 @key{RET} 4 / +
6542 On an old-style RPN calculator this second method would have the
6543 advantage of using only three stack levels. But since Calc's stack
6544 can grow arbitrarily large this isn't really an issue. Which method
6545 you choose is purely a matter of taste.
6547 @node RPN Answer 3, RPN Answer 4, RPN Answer 2, Answers to Exercises
6548 @subsection RPN Tutorial Exercise 3
6551 The @key{TAB} key provides a way to operate on the number in level 2.
6555 3: 10 3: 10 4: 10 3: 10 3: 10
6556 2: 20 2: 30 3: 30 2: 30 2: 21
6557 1: 30 1: 20 2: 20 1: 21 1: 30
6561 @key{TAB} 1 + @key{TAB}
6565 Similarly, @kbd{M-@key{TAB}} gives you access to the number in level 3.
6569 3: 10 3: 21 3: 21 3: 30 3: 11
6570 2: 21 2: 30 2: 30 2: 11 2: 21
6571 1: 30 1: 10 1: 11 1: 21 1: 30
6574 M-@key{TAB} 1 + M-@key{TAB} M-@key{TAB}
6578 @node RPN Answer 4, Algebraic Answer 1, RPN Answer 3, Answers to Exercises
6579 @subsection RPN Tutorial Exercise 4
6582 Either @kbd{( 2 , 3 )} or @kbd{( 2 @key{SPC} 3 )} would have worked,
6583 but using both the comma and the space at once yields:
6587 1: ( ... 2: ( ... 1: (2, ... 2: (2, ... 2: (2, ...
6588 . 1: 2 . 1: (2, ... 1: (2, 3)
6595 Joe probably tried to type @kbd{@key{TAB} @key{DEL}} to swap the
6596 extra incomplete object to the top of the stack and delete it.
6597 But a feature of Calc is that @key{DEL} on an incomplete object
6598 deletes just one component out of that object, so he had to press
6599 @key{DEL} twice to finish the job.
6603 2: (2, ... 2: (2, 3) 2: (2, 3) 1: (2, 3)
6604 1: (2, 3) 1: (2, ... 1: ( ... .
6607 @key{TAB} @key{DEL} @key{DEL}
6611 (As it turns out, deleting the second-to-top stack entry happens often
6612 enough that Calc provides a special key, @kbd{M-@key{DEL}}, to do just that.
6613 @kbd{M-@key{DEL}} is just like @kbd{@key{TAB} @key{DEL}}, except that it doesn't exhibit
6614 the ``feature'' that tripped poor Joe.)
6616 @node Algebraic Answer 1, Algebraic Answer 2, RPN Answer 4, Answers to Exercises
6617 @subsection Algebraic Entry Tutorial Exercise 1
6620 Type @kbd{' sqrt($) @key{RET}}.
6622 If the @kbd{Q} key is broken, you could use @kbd{' $^0.5 @key{RET}}.
6623 Or, RPN style, @kbd{0.5 ^}.
6625 (Actually, @samp{$^1:2}, using the fraction one-half as the power, is
6626 a closer equivalent, since @samp{9^0.5} yields @expr{3.0} whereas
6627 @samp{sqrt(9)} and @samp{9^1:2} yield the exact integer @expr{3}.)
6629 @node Algebraic Answer 2, Algebraic Answer 3, Algebraic Answer 1, Answers to Exercises
6630 @subsection Algebraic Entry Tutorial Exercise 2
6633 In the formula @samp{2 x (1+y)}, @samp{x} was interpreted as a function
6634 name with @samp{1+y} as its argument. Assigning a value to a variable
6635 has no relation to a function by the same name. Joe needed to use an
6636 explicit @samp{*} symbol here: @samp{2 x*(1+y)}.
6638 @node Algebraic Answer 3, Modes Answer 1, Algebraic Answer 2, Answers to Exercises
6639 @subsection Algebraic Entry Tutorial Exercise 3
6642 The result from @kbd{1 @key{RET} 0 /} will be the formula @expr{1 / 0}.
6643 The ``function'' @samp{/} cannot be evaluated when its second argument
6644 is zero, so it is left in symbolic form. When you now type @kbd{0 *},
6645 the result will be zero because Calc uses the general rule that ``zero
6646 times anything is zero.''
6648 @c [fix-ref Infinities]
6649 The @kbd{m i} command enables an @dfn{Infinite mode} in which @expr{1 / 0}
6650 results in a special symbol that represents ``infinity.'' If you
6651 multiply infinity by zero, Calc uses another special new symbol to
6652 show that the answer is ``indeterminate.'' @xref{Infinities}, for
6653 further discussion of infinite and indeterminate values.
6655 @node Modes Answer 1, Modes Answer 2, Algebraic Answer 3, Answers to Exercises
6656 @subsection Modes Tutorial Exercise 1
6659 Calc always stores its numbers in decimal, so even though one-third has
6660 an exact base-3 representation (@samp{3#0.1}), it is still stored as
6661 0.3333333 (chopped off after 12 or however many decimal digits) inside
6662 the calculator's memory. When this inexact number is converted back
6663 to base 3 for display, it may still be slightly inexact. When we
6664 multiply this number by 3, we get 0.999999, also an inexact value.
6666 When Calc displays a number in base 3, it has to decide how many digits
6667 to show. If the current precision is 12 (decimal) digits, that corresponds
6668 to @samp{12 / log10(3) = 25.15} base-3 digits. Because 25.15 is not an
6669 exact integer, Calc shows only 25 digits, with the result that stored
6670 numbers carry a little bit of extra information that may not show up on
6671 the screen. When Joe entered @samp{3#0.2}, the stored number 0.666666
6672 happened to round to a pleasing value when it lost that last 0.15 of a
6673 digit, but it was still inexact in Calc's memory. When he divided by 2,
6674 he still got the dreaded inexact value 0.333333. (Actually, he divided
6675 0.666667 by 2 to get 0.333334, which is why he got something a little
6676 higher than @code{3#0.1} instead of a little lower.)
6678 If Joe didn't want to be bothered with all this, he could have typed
6679 @kbd{M-24 d n} to display with one less digit than the default. (If
6680 you give @kbd{d n} a negative argument, it uses default-minus-that,
6681 so @kbd{M-- d n} would be an easier way to get the same effect.) Those
6682 inexact results would still be lurking there, but they would now be
6683 rounded to nice, natural-looking values for display purposes. (Remember,
6684 @samp{0.022222} in base 3 is like @samp{0.099999} in base 10; rounding
6685 off one digit will round the number up to @samp{0.1}.) Depending on the
6686 nature of your work, this hiding of the inexactness may be a benefit or
6687 a danger. With the @kbd{d n} command, Calc gives you the choice.
6689 Incidentally, another consequence of all this is that if you type
6690 @kbd{M-30 d n} to display more digits than are ``really there,''
6691 you'll see garbage digits at the end of the number. (In decimal
6692 display mode, with decimally-stored numbers, these garbage digits are
6693 always zero so they vanish and you don't notice them.) Because Calc
6694 rounds off that 0.15 digit, there is the danger that two numbers could
6695 be slightly different internally but still look the same. If you feel
6696 uneasy about this, set the @kbd{d n} precision to be a little higher
6697 than normal; you'll get ugly garbage digits, but you'll always be able
6698 to tell two distinct numbers apart.
6700 An interesting side note is that most computers store their
6701 floating-point numbers in binary, and convert to decimal for display.
6702 Thus everyday programs have the same problem: Decimal 0.1 cannot be
6703 represented exactly in binary (try it: @kbd{0.1 d 2}), so @samp{0.1 * 10}
6704 comes out as an inexact approximation to 1 on some machines (though
6705 they generally arrange to hide it from you by rounding off one digit as
6706 we did above). Because Calc works in decimal instead of binary, you can
6707 be sure that numbers that look exact @emph{are} exact as long as you stay
6708 in decimal display mode.
6710 It's not hard to show that any number that can be represented exactly
6711 in binary, octal, or hexadecimal is also exact in decimal, so the kinds
6712 of problems we saw in this exercise are likely to be severe only when
6713 you use a relatively unusual radix like 3.
6715 @node Modes Answer 2, Modes Answer 3, Modes Answer 1, Answers to Exercises
6716 @subsection Modes Tutorial Exercise 2
6718 If the radix is 15 or higher, we can't use the letter @samp{e} to mark
6719 the exponent because @samp{e} is interpreted as a digit. When Calc
6720 needs to display scientific notation in a high radix, it writes
6721 @samp{16#F.E8F*16.^15}. You can enter a number like this as an
6722 algebraic entry. Also, pressing @kbd{e} without any digits before it
6723 normally types @kbd{1e}, but in a high radix it types @kbd{16.^} and
6724 puts you in algebraic entry: @kbd{16#f.e8f @key{RET} e 15 @key{RET} *} is another
6725 way to enter this number.
6727 The reason Calc puts a decimal point in the @samp{16.^} is to prevent
6728 huge integers from being generated if the exponent is large (consider
6729 @samp{16#1.23*16^1000}, where we compute @samp{16^1000} as a giant
6730 exact integer and then throw away most of the digits when we multiply
6731 it by the floating-point @samp{16#1.23}). While this wouldn't normally
6732 matter for display purposes, it could give you a nasty surprise if you
6733 copied that number into a file and later moved it back into Calc.
6735 @node Modes Answer 3, Modes Answer 4, Modes Answer 2, Answers to Exercises
6736 @subsection Modes Tutorial Exercise 3
6739 The answer he got was @expr{0.5000000000006399}.
6741 The problem is not that the square operation is inexact, but that the
6742 sine of 45 that was already on the stack was accurate to only 12 places.
6743 Arbitrary-precision calculations still only give answers as good as
6746 The real problem is that there is no 12-digit number which, when
6747 squared, comes out to 0.5 exactly. The @kbd{f [} and @kbd{f ]}
6748 commands decrease or increase a number by one unit in the last
6749 place (according to the current precision). They are useful for
6750 determining facts like this.
6754 1: 0.707106781187 1: 0.500000000001
6764 1: 0.707106781187 1: 0.707106781186 1: 0.499999999999
6771 A high-precision calculation must be carried out in high precision
6772 all the way. The only number in the original problem which was known
6773 exactly was the quantity 45 degrees, so the precision must be raised
6774 before anything is done after the number 45 has been entered in order
6775 for the higher precision to be meaningful.
6777 @node Modes Answer 4, Arithmetic Answer 1, Modes Answer 3, Answers to Exercises
6778 @subsection Modes Tutorial Exercise 4
6781 Many calculations involve real-world quantities, like the width and
6782 height of a piece of wood or the volume of a jar. Such quantities
6783 can't be measured exactly anyway, and if the data that is input to
6784 a calculation is inexact, doing exact arithmetic on it is a waste
6787 Fractions become unwieldy after too many calculations have been
6788 done with them. For example, the sum of the reciprocals of the
6789 integers from 1 to 10 is 7381:2520. The sum from 1 to 30 is
6790 9304682830147:2329089562800. After a point it will take a long
6791 time to add even one more term to this sum, but a floating-point
6792 calculation of the sum will not have this problem.
6794 Also, rational numbers cannot express the results of all calculations.
6795 There is no fractional form for the square root of two, so if you type
6796 @w{@kbd{2 Q}}, Calc has no choice but to give you a floating-point answer.
6798 @node Arithmetic Answer 1, Arithmetic Answer 2, Modes Answer 4, Answers to Exercises
6799 @subsection Arithmetic Tutorial Exercise 1
6802 Dividing two integers that are larger than the current precision may
6803 give a floating-point result that is inaccurate even when rounded
6804 down to an integer. Consider @expr{123456789 / 2} when the current
6805 precision is 6 digits. The true answer is @expr{61728394.5}, but
6806 with a precision of 6 this will be rounded to
6807 @texline @math{12345700.0/2.0 = 61728500.0}.
6808 @infoline @expr{12345700.@: / 2.@: = 61728500.}.
6809 The result, when converted to an integer, will be off by 106.
6811 Here are two solutions: Raise the precision enough that the
6812 floating-point round-off error is strictly to the right of the
6813 decimal point. Or, convert to Fraction mode so that @expr{123456789 / 2}
6814 produces the exact fraction @expr{123456789:2}, which can be rounded
6815 down by the @kbd{F} command without ever switching to floating-point
6818 @node Arithmetic Answer 2, Vector Answer 1, Arithmetic Answer 1, Answers to Exercises
6819 @subsection Arithmetic Tutorial Exercise 2
6822 @kbd{27 @key{RET} 9 B} could give the exact result @expr{3:2}, but it
6823 does a floating-point calculation instead and produces @expr{1.5}.
6825 Calc will find an exact result for a logarithm if the result is an integer
6826 or (when in Fraction mode) the reciprocal of an integer. But there is
6827 no efficient way to search the space of all possible rational numbers
6828 for an exact answer, so Calc doesn't try.
6830 @node Vector Answer 1, Vector Answer 2, Arithmetic Answer 2, Answers to Exercises
6831 @subsection Vector Tutorial Exercise 1
6834 Duplicate the vector, compute its length, then divide the vector
6835 by its length: @kbd{@key{RET} A /}.
6839 1: [1, 2, 3] 2: [1, 2, 3] 1: [0.27, 0.53, 0.80] 1: 1.
6840 . 1: 3.74165738677 . .
6847 The final @kbd{A} command shows that the normalized vector does
6848 indeed have unit length.
6850 @node Vector Answer 2, Matrix Answer 1, Vector Answer 1, Answers to Exercises
6851 @subsection Vector Tutorial Exercise 2
6854 The average position is equal to the sum of the products of the
6855 positions times their corresponding probabilities. This is the
6856 definition of the dot product operation. So all you need to do
6857 is to put the two vectors on the stack and press @kbd{*}.
6859 @node Matrix Answer 1, Matrix Answer 2, Vector Answer 2, Answers to Exercises
6860 @subsection Matrix Tutorial Exercise 1
6863 The trick is to multiply by a vector of ones. Use @kbd{r 4 [1 1 1] *} to
6864 get the row sum. Similarly, use @kbd{[1 1] r 4 *} to get the column sum.
6866 @node Matrix Answer 2, Matrix Answer 3, Matrix Answer 1, Answers to Exercises
6867 @subsection Matrix Tutorial Exercise 2
6880 $$ \eqalign{ x &+ a y = 6 \cr
6886 Just enter the righthand side vector, then divide by the lefthand side
6891 1: [6, 10] 2: [6, 10] 1: [6 - 4 a / (b - a), 4 / (b - a) ]
6896 ' [6 10] @key{RET} ' [1 a; 1 b] @key{RET} /
6900 This can be made more readable using @kbd{d B} to enable Big display
6906 1: [6 - -----, -----]
6911 Type @kbd{d N} to return to Normal display mode afterwards.
6913 @node Matrix Answer 3, List Answer 1, Matrix Answer 2, Answers to Exercises
6914 @subsection Matrix Tutorial Exercise 3
6918 @texline @math{A^T A \, X = A^T B},
6919 @infoline @expr{trn(A) * A * X = trn(A) * B},
6921 @texline @math{A' = A^T A}
6922 @infoline @expr{A2 = trn(A) * A}
6924 @texline @math{B' = A^T B};
6925 @infoline @expr{B2 = trn(A) * B};
6926 now, we have a system
6927 @texline @math{A' X = B'}
6928 @infoline @expr{A2 * X = B2}
6929 which we can solve using Calc's @samp{/} command.
6944 $$ \openup1\jot \tabskip=0pt plus1fil
6945 \halign to\displaywidth{\tabskip=0pt
6946 $\hfil#$&$\hfil{}#{}$&
6947 $\hfil#$&$\hfil{}#{}$&
6948 $\hfil#$&${}#\hfil$\tabskip=0pt plus1fil\cr
6952 2a&+&4b&+&6c&=11 \cr}
6957 The first step is to enter the coefficient matrix. We'll store it in
6958 quick variable number 7 for later reference. Next, we compute the
6965 1: [ [ 1, 2, 3 ] 2: [ [ 1, 4, 7, 2 ] 1: [57, 84, 96]
6966 [ 4, 5, 6 ] [ 2, 5, 6, 4 ] .
6967 [ 7, 6, 0 ] [ 3, 6, 0, 6 ] ]
6968 [ 2, 4, 6 ] ] 1: [6, 2, 3, 11]
6971 ' [1 2 3; 4 5 6; 7 6 0; 2 4 6] @key{RET} s 7 v t [6 2 3 11] *
6976 Now we compute the matrix
6983 2: [57, 84, 96] 1: [-11.64, 14.08, -3.64]
6984 1: [ [ 70, 72, 39 ] .
6994 (The actual computed answer will be slightly inexact due to
6997 Notice that the answers are similar to those for the
6998 @texline @math{3\times3}
7000 system solved in the text. That's because the fourth equation that was
7001 added to the system is almost identical to the first one multiplied
7002 by two. (If it were identical, we would have gotten the exact same
7004 @texline @math{4\times3}
7006 system would be equivalent to the original
7007 @texline @math{3\times3}
7011 Since the first and fourth equations aren't quite equivalent, they
7012 can't both be satisfied at once. Let's plug our answers back into
7013 the original system of equations to see how well they match.
7017 2: [-11.64, 14.08, -3.64] 1: [5.6, 2., 3., 11.2]
7029 This is reasonably close to our original @expr{B} vector,
7030 @expr{[6, 2, 3, 11]}.
7032 @node List Answer 1, List Answer 2, Matrix Answer 3, Answers to Exercises
7033 @subsection List Tutorial Exercise 1
7036 We can use @kbd{v x} to build a vector of integers. This needs to be
7037 adjusted to get the range of integers we desire. Mapping @samp{-}
7038 across the vector will accomplish this, although it turns out the
7039 plain @samp{-} key will work just as well.
7044 1: [1, 2, 3, 4, 5, 6, 7, 8, 9] 1: [-4, -3, -2, -1, 0, 1, 2, 3, 4]
7047 2 v x 9 @key{RET} 5 V M - or 5 -
7052 Now we use @kbd{V M ^} to map the exponentiation operator across the
7057 1: [0.0625, 0.125, 0.25, 0.5, 1, 2, 4, 8, 16]
7064 @node List Answer 2, List Answer 3, List Answer 1, Answers to Exercises
7065 @subsection List Tutorial Exercise 2
7068 Given @expr{x} and @expr{y} vectors in quick variables 1 and 2 as before,
7069 the first job is to form the matrix that describes the problem.
7079 $$ m \times x + b \times 1 = y $$
7084 @texline @math{19\times2}
7086 matrix with our @expr{x} vector as one column and
7087 ones as the other column. So, first we build the column of ones, then
7088 we combine the two columns to form our @expr{A} matrix.
7092 2: [1.34, 1.41, 1.49, ... ] 1: [ [ 1.34, 1 ]
7093 1: [1, 1, 1, ...] [ 1.41, 1 ]
7097 r 1 1 v b 19 @key{RET} M-2 v p v t s 3
7103 @texline @math{A^T y}
7104 @infoline @expr{trn(A) * y}
7106 @texline @math{A^T A}
7107 @infoline @expr{trn(A) * A}
7112 1: [33.36554, 13.613] 2: [33.36554, 13.613]
7113 . 1: [ [ 98.0003, 41.63 ]
7117 v t r 2 * r 3 v t r 3 *
7122 (Hey, those numbers look familiar!)
7126 1: [0.52141679, -0.425978]
7133 Since we were solving equations of the form
7134 @texline @math{m \times x + b \times 1 = y},
7135 @infoline @expr{m*x + b*1 = y},
7136 these numbers should be @expr{m} and @expr{b}, respectively. Sure
7137 enough, they agree exactly with the result computed using @kbd{V M} and
7140 The moral of this story: @kbd{V M} and @kbd{V R} will probably solve
7141 your problem, but there is often an easier way using the higher-level
7142 arithmetic functions!
7144 @c [fix-ref Curve Fitting]
7145 In fact, there is a built-in @kbd{a F} command that does least-squares
7146 fits. @xref{Curve Fitting}.
7148 @node List Answer 3, List Answer 4, List Answer 2, Answers to Exercises
7149 @subsection List Tutorial Exercise 3
7152 Move to one end of the list and press @kbd{C-@@} (or @kbd{C-@key{SPC}} or
7153 whatever) to set the mark, then move to the other end of the list
7154 and type @w{@kbd{C-x * g}}.
7158 1: [2.3, 6, 22, 15.1, 7, 15, 14, 7.5, 2.5]
7163 To make things interesting, let's assume we don't know at a glance
7164 how many numbers are in this list. Then we could type:
7168 2: [2.3, 6, 22, ... ] 2: [2.3, 6, 22, ... ]
7169 1: [2.3, 6, 22, ... ] 1: 126356422.5
7179 2: 126356422.5 2: 126356422.5 1: 7.94652913734
7180 1: [2.3, 6, 22, ... ] 1: 9 .
7188 (The @kbd{I ^} command computes the @var{n}th root of a number.
7189 You could also type @kbd{& ^} to take the reciprocal of 9 and
7190 then raise the number to that power.)
7192 @node List Answer 4, List Answer 5, List Answer 3, Answers to Exercises
7193 @subsection List Tutorial Exercise 4
7196 A number @expr{j} is a divisor of @expr{n} if
7197 @texline @math{n \mathbin{\hbox{\code{\%}}} j = 0}.
7198 @infoline @samp{n % j = 0}.
7199 The first step is to get a vector that identifies the divisors.
7203 2: 30 2: [0, 0, 0, 2, ...] 1: [1, 1, 1, 0, ...]
7204 1: [1, 2, 3, 4, ...] 1: 0 .
7207 30 @key{RET} v x 30 @key{RET} s 1 V M % 0 V M a = s 2
7212 This vector has 1's marking divisors of 30 and 0's marking non-divisors.
7214 The zeroth divisor function is just the total number of divisors.
7215 The first divisor function is the sum of the divisors.
7220 2: [1, 2, 3, 4, ...] 1: [1, 2, 3, 0, ...] 1: 72
7221 1: [1, 1, 1, 0, ...] . .
7224 V R + r 1 r 2 V M * V R +
7229 Once again, the last two steps just compute a dot product for which
7230 a simple @kbd{*} would have worked equally well.
7232 @node List Answer 5, List Answer 6, List Answer 4, Answers to Exercises
7233 @subsection List Tutorial Exercise 5
7236 The obvious first step is to obtain the list of factors with @kbd{k f}.
7237 This list will always be in sorted order, so if there are duplicates
7238 they will be right next to each other. A suitable method is to compare
7239 the list with a copy of itself shifted over by one.
7243 1: [3, 7, 7, 7, 19] 2: [3, 7, 7, 7, 19] 2: [3, 7, 7, 7, 19, 0]
7244 . 1: [3, 7, 7, 7, 19, 0] 1: [0, 3, 7, 7, 7, 19]
7247 19551 k f @key{RET} 0 | @key{TAB} 0 @key{TAB} |
7254 1: [0, 0, 1, 1, 0, 0] 1: 2 1: 0
7262 Note that we have to arrange for both vectors to have the same length
7263 so that the mapping operation works; no prime factor will ever be
7264 zero, so adding zeros on the left and right is safe. From then on
7265 the job is pretty straightforward.
7267 Incidentally, Calc provides the
7268 @texline @dfn{M@"obius} @math{\mu}
7269 @infoline @dfn{Moebius mu}
7270 function which is zero if and only if its argument is square-free. It
7271 would be a much more convenient way to do the above test in practice.
7273 @node List Answer 6, List Answer 7, List Answer 5, Answers to Exercises
7274 @subsection List Tutorial Exercise 6
7277 First use @kbd{v x 6 @key{RET}} to get a list of integers, then @kbd{V M v x}
7278 to get a list of lists of integers!
7280 @node List Answer 7, List Answer 8, List Answer 6, Answers to Exercises
7281 @subsection List Tutorial Exercise 7
7284 Here's one solution. First, compute the triangular list from the previous
7285 exercise and type @kbd{1 -} to subtract one from all the elements.
7298 The numbers down the lefthand edge of the list we desire are called
7299 the ``triangular numbers'' (now you know why!). The @expr{n}th
7300 triangular number is the sum of the integers from 1 to @expr{n}, and
7301 can be computed directly by the formula
7302 @texline @math{n (n+1) \over 2}.
7303 @infoline @expr{n * (n+1) / 2}.
7307 2: [ [0], [0, 1], ... ] 2: [ [0], [0, 1], ... ]
7308 1: [0, 1, 2, 3, 4, 5] 1: [0, 1, 3, 6, 10, 15]
7311 v x 6 @key{RET} 1 - V M ' $ ($+1)/2 @key{RET}
7316 Adding this list to the above list of lists produces the desired
7325 [10, 11, 12, 13, 14],
7326 [15, 16, 17, 18, 19, 20] ]
7333 If we did not know the formula for triangular numbers, we could have
7334 computed them using a @kbd{V U +} command. We could also have
7335 gotten them the hard way by mapping a reduction across the original
7340 2: [ [0], [0, 1], ... ] 2: [ [0], [0, 1], ... ]
7341 1: [ [0], [0, 1], ... ] 1: [0, 1, 3, 6, 10, 15]
7349 (This means ``map a @kbd{V R +} command across the vector,'' and
7350 since each element of the main vector is itself a small vector,
7351 @kbd{V R +} computes the sum of its elements.)
7353 @node List Answer 8, List Answer 9, List Answer 7, Answers to Exercises
7354 @subsection List Tutorial Exercise 8
7357 The first step is to build a list of values of @expr{x}.
7361 1: [1, 2, 3, ..., 21] 1: [0, 1, 2, ..., 20] 1: [0, 0.25, 0.5, ..., 5]
7364 v x 21 @key{RET} 1 - 4 / s 1
7368 Next, we compute the Bessel function values.
7372 1: [0., 0.124, 0.242, ..., -0.328]
7375 V M ' besJ(1,$) @key{RET}
7380 (Another way to do this would be @kbd{1 @key{TAB} V M f j}.)
7382 A way to isolate the maximum value is to compute the maximum using
7383 @kbd{V R X}, then compare all the Bessel values with that maximum.
7387 2: [0., 0.124, 0.242, ... ] 1: [0, 0, 0, ... ] 2: [0, 0, 0, ... ]
7391 @key{RET} V R X V M a = @key{RET} V R + @key{DEL}
7396 It's a good idea to verify, as in the last step above, that only
7397 one value is equal to the maximum. (After all, a plot of
7398 @texline @math{\sin x}
7399 @infoline @expr{sin(x)}
7400 might have many points all equal to the maximum value, 1.)
7402 The vector we have now has a single 1 in the position that indicates
7403 the maximum value of @expr{x}. Now it is a simple matter to convert
7404 this back into the corresponding value itself.
7408 2: [0, 0, 0, ... ] 1: [0, 0., 0., ... ] 1: 1.75
7409 1: [0, 0.25, 0.5, ... ] . .
7416 If @kbd{a =} had produced more than one @expr{1} value, this method
7417 would have given the sum of all maximum @expr{x} values; not very
7418 useful! In this case we could have used @kbd{v m} (@code{calc-mask-vector})
7419 instead. This command deletes all elements of a ``data'' vector that
7420 correspond to zeros in a ``mask'' vector, leaving us with, in this
7421 example, a vector of maximum @expr{x} values.
7423 The built-in @kbd{a X} command maximizes a function using more
7424 efficient methods. Just for illustration, let's use @kbd{a X}
7425 to maximize @samp{besJ(1,x)} over this same interval.
7429 2: besJ(1, x) 1: [1.84115, 0.581865]
7433 ' besJ(1,x), [0..5] @key{RET} a X x @key{RET}
7438 The output from @kbd{a X} is a vector containing the value of @expr{x}
7439 that maximizes the function, and the function's value at that maximum.
7440 As you can see, our simple search got quite close to the right answer.
7442 @node List Answer 9, List Answer 10, List Answer 8, Answers to Exercises
7443 @subsection List Tutorial Exercise 9
7446 Step one is to convert our integer into vector notation.
7450 1: 25129925999 3: 25129925999
7452 1: [11, 10, 9, ..., 1, 0]
7455 25129925999 @key{RET} 10 @key{RET} 12 @key{RET} v x 12 @key{RET} -
7462 1: 25129925999 1: [0, 2, 25, 251, 2512, ... ]
7463 2: [100000000000, ... ] .
7471 (Recall, the @kbd{\} command computes an integer quotient.)
7475 1: [0, 2, 5, 1, 2, 9, 9, 2, 5, 9, 9, 9]
7482 Next we must increment this number. This involves adding one to
7483 the last digit, plus handling carries. There is a carry to the
7484 left out of a digit if that digit is a nine and all the digits to
7485 the right of it are nines.
7489 1: [0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1] 1: [1, 1, 1, 0, 0, 1, ... ]
7499 1: [1, 1, 1, 0, 0, 0, ... ] 1: [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1]
7507 Accumulating @kbd{*} across a vector of ones and zeros will preserve
7508 only the initial run of ones. These are the carries into all digits
7509 except the rightmost digit. Concatenating a one on the right takes
7510 care of aligning the carries properly, and also adding one to the
7515 2: [0, 0, 0, 0, ... ] 1: [0, 0, 2, 5, 1, 2, 9, 9, 2, 6, 0, 0, 0]
7516 1: [0, 0, 2, 5, ... ] .
7519 0 r 2 | V M + 10 V M %
7524 Here we have concatenated 0 to the @emph{left} of the original number;
7525 this takes care of shifting the carries by one with respect to the
7526 digits that generated them.
7528 Finally, we must convert this list back into an integer.
7532 3: [0, 0, 2, 5, ... ] 2: [0, 0, 2, 5, ... ]
7533 2: 1000000000000 1: [1000000000000, 100000000000, ... ]
7534 1: [100000000000, ... ] .
7537 10 @key{RET} 12 ^ r 1 |
7544 1: [0, 0, 20000000000, 5000000000, ... ] 1: 25129926000
7552 Another way to do this final step would be to reduce the formula
7553 @w{@samp{10 $$ + $}} across the vector of digits.
7557 1: [0, 0, 2, 5, ... ] 1: 25129926000
7560 V R ' 10 $$ + $ @key{RET}
7564 @node List Answer 10, List Answer 11, List Answer 9, Answers to Exercises
7565 @subsection List Tutorial Exercise 10
7568 For the list @expr{[a, b, c, d]}, the result is @expr{((a = b) = c) = d},
7569 which will compare @expr{a} and @expr{b} to produce a 1 or 0, which is
7570 then compared with @expr{c} to produce another 1 or 0, which is then
7571 compared with @expr{d}. This is not at all what Joe wanted.
7573 Here's a more correct method:
7577 1: [7, 7, 7, 8, 7] 2: [7, 7, 7, 8, 7]
7581 ' [7,7,7,8,7] @key{RET} @key{RET} v r 1 @key{RET}
7588 1: [1, 1, 1, 0, 1] 1: 0
7595 @node List Answer 11, List Answer 12, List Answer 10, Answers to Exercises
7596 @subsection List Tutorial Exercise 11
7599 The circle of unit radius consists of those points @expr{(x,y)} for which
7600 @expr{x^2 + y^2 < 1}. We start by generating a vector of @expr{x^2}
7601 and a vector of @expr{y^2}.
7603 We can make this go a bit faster by using the @kbd{v .} and @kbd{t .}
7608 2: [2., 2., ..., 2.] 2: [2., 2., ..., 2.]
7609 1: [2., 2., ..., 2.] 1: [1.16, 1.98, ..., 0.81]
7612 v . t . 2. v b 100 @key{RET} @key{RET} V M k r
7619 2: [2., 2., ..., 2.] 1: [0.026, 0.96, ..., 0.036]
7620 1: [0.026, 0.96, ..., 0.036] 2: [0.53, 0.81, ..., 0.094]
7623 1 - 2 V M ^ @key{TAB} V M k r 1 - 2 V M ^
7627 Now we sum the @expr{x^2} and @expr{y^2} values, compare with 1 to
7628 get a vector of 1/0 truth values, then sum the truth values.
7632 1: [0.56, 1.78, ..., 0.13] 1: [1, 0, ..., 1] 1: 84
7640 The ratio @expr{84/100} should approximate the ratio @cpiover{4}.
7644 1: 0.84 1: 3.36 2: 3.36 1: 1.0695
7652 Our estimate, 3.36, is off by about 7%. We could get a better estimate
7653 by taking more points (say, 1000), but it's clear that this method is
7656 (Naturally, since this example uses random numbers your own answer
7657 will be slightly different from the one shown here!)
7659 If you typed @kbd{v .} and @kbd{t .} before, type them again to
7660 return to full-sized display of vectors.
7662 @node List Answer 12, List Answer 13, List Answer 11, Answers to Exercises
7663 @subsection List Tutorial Exercise 12
7666 This problem can be made a lot easier by taking advantage of some
7667 symmetries. First of all, after some thought it's clear that the
7668 @expr{y} axis can be ignored altogether. Just pick a random @expr{x}
7669 component for one end of the match, pick a random direction
7670 @texline @math{\theta},
7671 @infoline @expr{theta},
7672 and see if @expr{x} and
7673 @texline @math{x + \cos \theta}
7674 @infoline @expr{x + cos(theta)}
7675 (which is the @expr{x} coordinate of the other endpoint) cross a line.
7676 The lines are at integer coordinates, so this happens when the two
7677 numbers surround an integer.
7679 Since the two endpoints are equivalent, we may as well choose the leftmost
7680 of the two endpoints as @expr{x}. Then @expr{theta} is an angle pointing
7681 to the right, in the range -90 to 90 degrees. (We could use radians, but
7682 it would feel like cheating to refer to @cpiover{2} radians while trying
7683 to estimate @cpi{}!)
7685 In fact, since the field of lines is infinite we can choose the
7686 coordinates 0 and 1 for the lines on either side of the leftmost
7687 endpoint. The rightmost endpoint will be between 0 and 1 if the
7688 match does not cross a line, or between 1 and 2 if it does. So:
7689 Pick random @expr{x} and
7690 @texline @math{\theta},
7691 @infoline @expr{theta},
7693 @texline @math{x + \cos \theta},
7694 @infoline @expr{x + cos(theta)},
7695 and count how many of the results are greater than one. Simple!
7697 We can make this go a bit faster by using the @kbd{v .} and @kbd{t .}
7702 1: [0.52, 0.71, ..., 0.72] 2: [0.52, 0.71, ..., 0.72]
7703 . 1: [78.4, 64.5, ..., -42.9]
7706 v . t . 1. v b 100 @key{RET} V M k r 180. v b 100 @key{RET} V M k r 90 -
7711 (The next step may be slow, depending on the speed of your computer.)
7715 2: [0.52, 0.71, ..., 0.72] 1: [0.72, 1.14, ..., 1.45]
7716 1: [0.20, 0.43, ..., 0.73] .
7726 1: [0, 1, ..., 1] 1: 0.64 1: 3.125
7729 1 V M a > V R + 100 / 2 @key{TAB} /
7733 Let's try the third method, too. We'll use random integers up to
7734 one million. The @kbd{k r} command with an integer argument picks
7739 2: [1000000, 1000000, ..., 1000000] 2: [78489, 527587, ..., 814975]
7740 1: [1000000, 1000000, ..., 1000000] 1: [324014, 358783, ..., 955450]
7743 1000000 v b 100 @key{RET} @key{RET} V M k r @key{TAB} V M k r
7750 1: [1, 1, ..., 25] 1: [1, 1, ..., 0] 1: 0.56
7753 V M k g 1 V M a = V R + 100 /
7767 For a proof of this property of the GCD function, see section 4.5.2,
7768 exercise 10, of Knuth's @emph{Art of Computer Programming}, volume II.
7770 If you typed @kbd{v .} and @kbd{t .} before, type them again to
7771 return to full-sized display of vectors.
7773 @node List Answer 13, List Answer 14, List Answer 12, Answers to Exercises
7774 @subsection List Tutorial Exercise 13
7777 First, we put the string on the stack as a vector of ASCII codes.
7781 1: [84, 101, 115, ..., 51]
7784 "Testing, 1, 2, 3 @key{RET}
7789 Note that the @kbd{"} key, like @kbd{$}, initiates algebraic entry so
7790 there was no need to type an apostrophe. Also, Calc didn't mind that
7791 we omitted the closing @kbd{"}. (The same goes for all closing delimiters
7792 like @kbd{)} and @kbd{]} at the end of a formula.
7794 We'll show two different approaches here. In the first, we note that
7795 if the input vector is @expr{[a, b, c, d]}, then the hash code is
7796 @expr{3 (3 (3a + b) + c) + d = 27a + 9b + 3c + d}. In other words,
7797 it's a sum of descending powers of three times the ASCII codes.
7801 2: [84, 101, 115, ..., 51] 2: [84, 101, 115, ..., 51]
7802 1: 16 1: [15, 14, 13, ..., 0]
7805 @key{RET} v l v x 16 @key{RET} -
7812 2: [84, 101, 115, ..., 51] 1: 1960915098 1: 121
7813 1: [14348907, ..., 1] . .
7816 3 @key{TAB} V M ^ * 511 %
7821 Once again, @kbd{*} elegantly summarizes most of the computation.
7822 But there's an even more elegant approach: Reduce the formula
7823 @kbd{3 $$ + $} across the vector. Recall that this represents a
7824 function of two arguments that computes its first argument times three
7825 plus its second argument.
7829 1: [84, 101, 115, ..., 51] 1: 1960915098
7832 "Testing, 1, 2, 3 @key{RET} V R ' 3$$+$ @key{RET}
7837 If you did the decimal arithmetic exercise, this will be familiar.
7838 Basically, we're turning a base-3 vector of digits into an integer,
7839 except that our ``digits'' are much larger than real digits.
7841 Instead of typing @kbd{511 %} again to reduce the result, we can be
7842 cleverer still and notice that rather than computing a huge integer
7843 and taking the modulo at the end, we can take the modulo at each step
7844 without affecting the result. While this means there are more
7845 arithmetic operations, the numbers we operate on remain small so
7846 the operations are faster.
7850 1: [84, 101, 115, ..., 51] 1: 121
7853 "Testing, 1, 2, 3 @key{RET} V R ' (3$$+$)%511 @key{RET}
7857 Why does this work? Think about a two-step computation:
7858 @w{@expr{3 (3a + b) + c}}. Taking a result modulo 511 basically means
7859 subtracting off enough 511's to put the result in the desired range.
7860 So the result when we take the modulo after every step is,
7864 3 (3 a + b - 511 m) + c - 511 n
7870 $$ 3 (3 a + b - 511 m) + c - 511 n $$
7875 for some suitable integers @expr{m} and @expr{n}. Expanding out by
7876 the distributive law yields
7880 9 a + 3 b + c - 511*3 m - 511 n
7886 $$ 9 a + 3 b + c - 511\times3 m - 511 n $$
7891 The @expr{m} term in the latter formula is redundant because any
7892 contribution it makes could just as easily be made by the @expr{n}
7893 term. So we can take it out to get an equivalent formula with
7898 9 a + 3 b + c - 511 n'
7904 $$ 9 a + 3 b + c - 511 n' $$
7909 which is just the formula for taking the modulo only at the end of
7910 the calculation. Therefore the two methods are essentially the same.
7912 Later in the tutorial we will encounter @dfn{modulo forms}, which
7913 basically automate the idea of reducing every intermediate result
7914 modulo some value @var{m}.
7916 @node List Answer 14, Types Answer 1, List Answer 13, Answers to Exercises
7917 @subsection List Tutorial Exercise 14
7919 We want to use @kbd{H V U} to nest a function which adds a random
7920 step to an @expr{(x,y)} coordinate. The function is a bit long, but
7921 otherwise the problem is quite straightforward.
7925 2: [0, 0] 1: [ [ 0, 0 ]
7926 1: 50 [ 0.4288, -0.1695 ]
7927 . [ -0.4787, -0.9027 ]
7930 [0,0] 50 H V U ' <# + [random(2.0)-1, random(2.0)-1]> @key{RET}
7934 Just as the text recommended, we used @samp{< >} nameless function
7935 notation to keep the two @code{random} calls from being evaluated
7936 before nesting even begins.
7938 We now have a vector of @expr{[x, y]} sub-vectors, which by Calc's
7939 rules acts like a matrix. We can transpose this matrix and unpack
7940 to get a pair of vectors, @expr{x} and @expr{y}, suitable for graphing.
7944 2: [ 0, 0.4288, -0.4787, ... ]
7945 1: [ 0, -0.1696, -0.9027, ... ]
7952 Incidentally, because the @expr{x} and @expr{y} are completely
7953 independent in this case, we could have done two separate commands
7954 to create our @expr{x} and @expr{y} vectors of numbers directly.
7956 To make a random walk of unit steps, we note that @code{sincos} of
7957 a random direction exactly gives us an @expr{[x, y]} step of unit
7958 length; in fact, the new nesting function is even briefer, though
7959 we might want to lower the precision a bit for it.
7963 2: [0, 0] 1: [ [ 0, 0 ]
7964 1: 50 [ 0.1318, 0.9912 ]
7965 . [ -0.5965, 0.3061 ]
7968 [0,0] 50 m d p 6 @key{RET} H V U ' <# + sincos(random(360.0))> @key{RET}
7972 Another @kbd{v t v u g f} sequence will graph this new random walk.
7974 An interesting twist on these random walk functions would be to use
7975 complex numbers instead of 2-vectors to represent points on the plane.
7976 In the first example, we'd use something like @samp{random + random*(0,1)},
7977 and in the second we could use polar complex numbers with random phase
7978 angles. (This exercise was first suggested in this form by Randal
7981 @node Types Answer 1, Types Answer 2, List Answer 14, Answers to Exercises
7982 @subsection Types Tutorial Exercise 1
7985 If the number is the square root of @cpi{} times a rational number,
7986 then its square, divided by @cpi{}, should be a rational number.
7990 1: 1.26508260337 1: 0.509433962268 1: 2486645810:4881193627
7998 Technically speaking this is a rational number, but not one that is
7999 likely to have arisen in the original problem. More likely, it just
8000 happens to be the fraction which most closely represents some
8001 irrational number to within 12 digits.
8003 But perhaps our result was not quite exact. Let's reduce the
8004 precision slightly and try again:
8008 1: 0.509433962268 1: 27:53
8011 U p 10 @key{RET} c F
8016 Aha! It's unlikely that an irrational number would equal a fraction
8017 this simple to within ten digits, so our original number was probably
8018 @texline @math{\sqrt{27 \pi / 53}}.
8019 @infoline @expr{sqrt(27 pi / 53)}.
8021 Notice that we didn't need to re-round the number when we reduced the
8022 precision. Remember, arithmetic operations always round their inputs
8023 to the current precision before they begin.
8025 @node Types Answer 2, Types Answer 3, Types Answer 1, Answers to Exercises
8026 @subsection Types Tutorial Exercise 2
8029 @samp{inf / inf = nan}. Perhaps @samp{1} is the ``obvious'' answer.
8030 But if @w{@samp{17 inf = inf}}, then @samp{17 inf / inf = inf / inf = 17}, too.
8032 @samp{exp(inf) = inf}. It's tempting to say that the exponential
8033 of infinity must be ``bigger'' than ``regular'' infinity, but as
8034 far as Calc is concerned all infinities are as just as big.
8035 In other words, as @expr{x} goes to infinity, @expr{e^x} also goes
8036 to infinity, but the fact the @expr{e^x} grows much faster than
8037 @expr{x} is not relevant here.
8039 @samp{exp(-inf) = 0}. Here we have a finite answer even though
8040 the input is infinite.
8042 @samp{sqrt(-inf) = (0, 1) inf}. Remember that @expr{(0, 1)}
8043 represents the imaginary number @expr{i}. Here's a derivation:
8044 @samp{sqrt(-inf) = @w{sqrt((-1) * inf)} = sqrt(-1) * sqrt(inf)}.
8045 The first part is, by definition, @expr{i}; the second is @code{inf}
8046 because, once again, all infinities are the same size.
8048 @samp{sqrt(uinf) = uinf}. In fact, we do know something about the
8049 direction because @code{sqrt} is defined to return a value in the
8050 right half of the complex plane. But Calc has no notation for this,
8051 so it settles for the conservative answer @code{uinf}.
8053 @samp{abs(uinf) = inf}. No matter which direction @expr{x} points,
8054 @samp{abs(x)} always points along the positive real axis.
8056 @samp{ln(0) = -inf}. Here we have an infinite answer to a finite
8057 input. As in the @expr{1 / 0} case, Calc will only use infinities
8058 here if you have turned on Infinite mode. Otherwise, it will
8059 treat @samp{ln(0)} as an error.
8061 @node Types Answer 3, Types Answer 4, Types Answer 2, Answers to Exercises
8062 @subsection Types Tutorial Exercise 3
8065 We can make @samp{inf - inf} be any real number we like, say,
8066 @expr{a}, just by claiming that we added @expr{a} to the first
8067 infinity but not to the second. This is just as true for complex
8068 values of @expr{a}, so @code{nan} can stand for a complex number.
8069 (And, similarly, @code{uinf} can stand for an infinity that points
8070 in any direction in the complex plane, such as @samp{(0, 1) inf}).
8072 In fact, we can multiply the first @code{inf} by two. Surely
8073 @w{@samp{2 inf - inf = inf}}, but also @samp{2 inf - inf = inf - inf = nan}.
8074 So @code{nan} can even stand for infinity. Obviously it's just
8075 as easy to make it stand for minus infinity as for plus infinity.
8077 The moral of this story is that ``infinity'' is a slippery fish
8078 indeed, and Calc tries to handle it by having a very simple model
8079 for infinities (only the direction counts, not the ``size''); but
8080 Calc is careful to write @code{nan} any time this simple model is
8081 unable to tell what the true answer is.
8083 @node Types Answer 4, Types Answer 5, Types Answer 3, Answers to Exercises
8084 @subsection Types Tutorial Exercise 4
8088 2: 0@@ 47' 26" 1: 0@@ 2' 47.411765"
8092 0@@ 47' 26" @key{RET} 17 /
8097 The average song length is two minutes and 47.4 seconds.
8101 2: 0@@ 2' 47.411765" 1: 0@@ 3' 7.411765" 1: 0@@ 53' 6.000005"
8110 The album would be 53 minutes and 6 seconds long.
8112 @node Types Answer 5, Types Answer 6, Types Answer 4, Answers to Exercises
8113 @subsection Types Tutorial Exercise 5
8116 Let's suppose it's January 14, 1991. The easiest thing to do is
8117 to keep trying 13ths of months until Calc reports a Friday.
8118 We can do this by manually entering dates, or by using @kbd{t I}:
8122 1: <Wed Feb 13, 1991> 1: <Wed Mar 13, 1991> 1: <Sat Apr 13, 1991>
8125 ' <2/13> @key{RET} @key{DEL} ' <3/13> @key{RET} t I
8130 (Calc assumes the current year if you don't say otherwise.)
8132 This is getting tedious---we can keep advancing the date by typing
8133 @kbd{t I} over and over again, but let's automate the job by using
8134 vector mapping. The @kbd{t I} command actually takes a second
8135 ``how-many-months'' argument, which defaults to one. This
8136 argument is exactly what we want to map over:
8140 2: <Sat Apr 13, 1991> 1: [<Mon May 13, 1991>, <Thu Jun 13, 1991>,
8141 1: [1, 2, 3, 4, 5, 6] <Sat Jul 13, 1991>, <Tue Aug 13, 1991>,
8142 . <Fri Sep 13, 1991>, <Sun Oct 13, 1991>]
8145 v x 6 @key{RET} V M t I
8150 Et voil@`a, September 13, 1991 is a Friday.
8157 ' <sep 13> - <jan 14> @key{RET}
8162 And the answer to our original question: 242 days to go.
8164 @node Types Answer 6, Types Answer 7, Types Answer 5, Answers to Exercises
8165 @subsection Types Tutorial Exercise 6
8168 The full rule for leap years is that they occur in every year divisible
8169 by four, except that they don't occur in years divisible by 100, except
8170 that they @emph{do} in years divisible by 400. We could work out the
8171 answer by carefully counting the years divisible by four and the
8172 exceptions, but there is a much simpler way that works even if we
8173 don't know the leap year rule.
8175 Let's assume the present year is 1991. Years have 365 days, except
8176 that leap years (whenever they occur) have 366 days. So let's count
8177 the number of days between now and then, and compare that to the
8178 number of years times 365. The number of extra days we find must be
8179 equal to the number of leap years there were.
8183 1: <Mon Jan 1, 10001> 2: <Mon Jan 1, 10001> 1: 2925593
8184 . 1: <Tue Jan 1, 1991> .
8187 ' <jan 1 10001> @key{RET} ' <jan 1 1991> @key{RET} -
8194 3: 2925593 2: 2925593 2: 2925593 1: 1943
8195 2: 10001 1: 8010 1: 2923650 .
8199 10001 @key{RET} 1991 - 365 * -
8203 @c [fix-ref Date Forms]
8205 There will be 1943 leap years before the year 10001. (Assuming,
8206 of course, that the algorithm for computing leap years remains
8207 unchanged for that long. @xref{Date Forms}, for some interesting
8208 background information in that regard.)
8210 @node Types Answer 7, Types Answer 8, Types Answer 6, Answers to Exercises
8211 @subsection Types Tutorial Exercise 7
8214 The relative errors must be converted to absolute errors so that
8215 @samp{+/-} notation may be used.
8223 20 @key{RET} .05 * 4 @key{RET} .05 *
8227 Now we simply chug through the formula.
8231 1: 19.7392088022 1: 394.78 +/- 19.739 1: 6316.5 +/- 706.21
8234 2 P 2 ^ * 20 p 1 * 4 p .2 @key{RET} 2 ^ *
8238 It turns out the @kbd{v u} command will unpack an error form as
8239 well as a vector. This saves us some retyping of numbers.
8243 3: 6316.5 +/- 706.21 2: 6316.5 +/- 706.21
8248 @key{RET} v u @key{TAB} /
8253 Thus the volume is 6316 cubic centimeters, within about 11 percent.
8255 @node Types Answer 8, Types Answer 9, Types Answer 7, Answers to Exercises
8256 @subsection Types Tutorial Exercise 8
8259 The first answer is pretty simple: @samp{1 / (0 .. 10) = (0.1 .. inf)}.
8260 Since a number in the interval @samp{(0 .. 10)} can get arbitrarily
8261 close to zero, its reciprocal can get arbitrarily large, so the answer
8262 is an interval that effectively means, ``any number greater than 0.1''
8263 but with no upper bound.
8265 The second answer, similarly, is @samp{1 / (-10 .. 0) = (-inf .. -0.1)}.
8267 Calc normally treats division by zero as an error, so that the formula
8268 @w{@samp{1 / 0}} is left unsimplified. Our third problem,
8269 @w{@samp{1 / [0 .. 10]}}, also (potentially) divides by zero because zero
8270 is now a member of the interval. So Calc leaves this one unevaluated, too.
8272 If you turn on Infinite mode by pressing @kbd{m i}, you will
8273 instead get the answer @samp{[0.1 .. inf]}, which includes infinity
8274 as a possible value.
8276 The fourth calculation, @samp{1 / (-10 .. 10)}, has the same problem.
8277 Zero is buried inside the interval, but it's still a possible value.
8278 It's not hard to see that the actual result of @samp{1 / (-10 .. 10)}
8279 will be either greater than @mathit{0.1}, or less than @mathit{-0.1}. Thus
8280 the interval goes from minus infinity to plus infinity, with a ``hole''
8281 in it from @mathit{-0.1} to @mathit{0.1}. Calc doesn't have any way to
8282 represent this, so it just reports @samp{[-inf .. inf]} as the answer.
8283 It may be disappointing to hear ``the answer lies somewhere between
8284 minus infinity and plus infinity, inclusive,'' but that's the best
8285 that interval arithmetic can do in this case.
8287 @node Types Answer 9, Types Answer 10, Types Answer 8, Answers to Exercises
8288 @subsection Types Tutorial Exercise 9
8292 1: [-3 .. 3] 2: [-3 .. 3] 2: [0 .. 9]
8293 . 1: [0 .. 9] 1: [-9 .. 9]
8296 [ 3 n .. 3 ] @key{RET} 2 ^ @key{TAB} @key{RET} *
8301 In the first case the result says, ``if a number is between @mathit{-3} and
8302 3, its square is between 0 and 9.'' The second case says, ``the product
8303 of two numbers each between @mathit{-3} and 3 is between @mathit{-9} and 9.''
8305 An interval form is not a number; it is a symbol that can stand for
8306 many different numbers. Two identical-looking interval forms can stand
8307 for different numbers.
8309 The same issue arises when you try to square an error form.
8311 @node Types Answer 10, Types Answer 11, Types Answer 9, Answers to Exercises
8312 @subsection Types Tutorial Exercise 10
8315 Testing the first number, we might arbitrarily choose 17 for @expr{x}.
8319 1: 17 mod 811749613 2: 17 mod 811749613 1: 533694123 mod 811749613
8323 17 M 811749613 @key{RET} 811749612 ^
8328 Since 533694123 is (considerably) different from 1, the number 811749613
8331 It's awkward to type the number in twice as we did above. There are
8332 various ways to avoid this, and algebraic entry is one. In fact, using
8333 a vector mapping operation we can perform several tests at once. Let's
8334 use this method to test the second number.
8338 2: [17, 42, 100000] 1: [1 mod 15485863, 1 mod ... ]
8342 [17 42 100000] 15485863 @key{RET} V M ' ($$ mod $)^($-1) @key{RET}
8347 The result is three ones (modulo @expr{n}), so it's very probable that
8348 15485863 is prime. (In fact, this number is the millionth prime.)
8350 Note that the functions @samp{($$^($-1)) mod $} or @samp{$$^($-1) % $}
8351 would have been hopelessly inefficient, since they would have calculated
8352 the power using full integer arithmetic.
8354 Calc has a @kbd{k p} command that does primality testing. For small
8355 numbers it does an exact test; for large numbers it uses a variant
8356 of the Fermat test we used here. You can use @kbd{k p} repeatedly
8357 to prove that a large integer is prime with any desired probability.
8359 @node Types Answer 11, Types Answer 12, Types Answer 10, Answers to Exercises
8360 @subsection Types Tutorial Exercise 11
8363 There are several ways to insert a calculated number into an HMS form.
8364 One way to convert a number of seconds to an HMS form is simply to
8365 multiply the number by an HMS form representing one second:
8369 1: 31415926.5359 2: 31415926.5359 1: 8726@@ 38' 46.5359"
8380 2: 8726@@ 38' 46.5359" 1: 6@@ 6' 2.5359" mod 24@@ 0' 0"
8381 1: 15@@ 27' 16" mod 24@@ 0' 0" .
8389 It will be just after six in the morning.
8391 The algebraic @code{hms} function can also be used to build an
8396 1: hms(0, 0, 10000000. pi) 1: 8726@@ 38' 46.5359"
8399 ' hms(0, 0, 1e7 pi) @key{RET} =
8404 The @kbd{=} key is necessary to evaluate the symbol @samp{pi} to
8405 the actual number 3.14159...
8407 @node Types Answer 12, Types Answer 13, Types Answer 11, Answers to Exercises
8408 @subsection Types Tutorial Exercise 12
8411 As we recall, there are 17 songs of about 2 minutes and 47 seconds
8416 2: 0@@ 2' 47" 1: [0@@ 3' 7" .. 0@@ 3' 47"]
8417 1: [0@@ 0' 20" .. 0@@ 1' 0"] .
8420 [ 0@@ 20" .. 0@@ 1' ] +
8427 1: [0@@ 52' 59." .. 1@@ 4' 19."]
8435 No matter how long it is, the album will fit nicely on one CD.
8437 @node Types Answer 13, Types Answer 14, Types Answer 12, Answers to Exercises
8438 @subsection Types Tutorial Exercise 13
8441 Type @kbd{' 1 yr @key{RET} u c s @key{RET}}. The answer is 31557600 seconds.
8443 @node Types Answer 14, Types Answer 15, Types Answer 13, Answers to Exercises
8444 @subsection Types Tutorial Exercise 14
8447 How long will it take for a signal to get from one end of the computer
8452 1: m / c 1: 3.3356 ns
8455 ' 1 m / c @key{RET} u c ns @key{RET}
8460 (Recall, @samp{c} is a ``unit'' corresponding to the speed of light.)
8464 1: 3.3356 ns 1: 0.81356 ns / ns 1: 0.81356
8468 ' 4.1 ns @key{RET} / u s
8473 Thus a signal could take up to 81 percent of a clock cycle just to
8474 go from one place to another inside the computer, assuming the signal
8475 could actually attain the full speed of light. Pretty tight!
8477 @node Types Answer 15, Algebra Answer 1, Types Answer 14, Answers to Exercises
8478 @subsection Types Tutorial Exercise 15
8481 The speed limit is 55 miles per hour on most highways. We want to
8482 find the ratio of Sam's speed to the US speed limit.
8486 1: 55 mph 2: 55 mph 3: 11 hr mph / yd
8490 ' 55 mph @key{RET} ' 5 yd/hr @key{RET} /
8494 The @kbd{u s} command cancels out these units to get a plain
8495 number. Now we take the logarithm base two to find the final
8496 answer, assuming that each successive pill doubles his speed.
8500 1: 19360. 2: 19360. 1: 14.24
8509 Thus Sam can take up to 14 pills without a worry.
8511 @node Algebra Answer 1, Algebra Answer 2, Types Answer 15, Answers to Exercises
8512 @subsection Algebra Tutorial Exercise 1
8515 @c [fix-ref Declarations]
8516 The result @samp{sqrt(x)^2} is simplified back to @expr{x} by the
8517 Calculator, but @samp{sqrt(x^2)} is not. (Consider what happens
8518 if @w{@expr{x = -4}}.) If @expr{x} is real, this formula could be
8519 simplified to @samp{abs(x)}, but for general complex arguments even
8520 that is not safe. (@xref{Declarations}, for a way to tell Calc
8521 that @expr{x} is known to be real.)
8523 @node Algebra Answer 2, Algebra Answer 3, Algebra Answer 1, Answers to Exercises
8524 @subsection Algebra Tutorial Exercise 2
8527 Suppose our roots are @expr{[a, b, c]}. We want a polynomial which
8528 is zero when @expr{x} is any of these values. The trivial polynomial
8529 @expr{x-a} is zero when @expr{x=a}, so the product @expr{(x-a)(x-b)(x-c)}
8530 will do the job. We can use @kbd{a c x} to write this in a more
8535 1: 34 x - 24 x^3 1: [1.19023, -1.19023, 0]
8545 1: [x - 1.19023, x + 1.19023, x] 1: (x - 1.19023) (x + 1.19023) x
8548 V M ' x-$ @key{RET} V R *
8555 1: x^3 - 1.41666 x 1: 34 x - 24 x^3
8558 a c x @key{RET} 24 n * a x
8563 Sure enough, our answer (multiplied by a suitable constant) is the
8564 same as the original polynomial.
8566 @node Algebra Answer 3, Algebra Answer 4, Algebra Answer 2, Answers to Exercises
8567 @subsection Algebra Tutorial Exercise 3
8571 1: x sin(pi x) 1: (sin(pi x) - pi x cos(pi x)) / pi^2
8574 ' x sin(pi x) @key{RET} m r a i x @key{RET}
8582 2: (sin(pi x) - pi x cos(pi x)) / pi^2
8585 ' [y,1] @key{RET} @key{TAB}
8592 1: [(sin(pi y) - pi y cos(pi y)) / pi^2, (sin(pi) - pi cos(pi)) / pi^2]
8602 1: (sin(pi y) - pi y cos(pi y)) / pi^2 + (pi cos(pi) - sin(pi)) / pi^2
8612 1: (sin(3.14159 y) - 3.14159 y cos(3.14159 y)) / 9.8696 - 0.3183
8622 1: [0., -0.95493, 0.63662, -1.5915, 1.2732]
8625 v x 5 @key{RET} @key{TAB} V M $ @key{RET}
8629 @node Algebra Answer 4, Rewrites Answer 1, Algebra Answer 3, Answers to Exercises
8630 @subsection Algebra Tutorial Exercise 4
8633 The hard part is that @kbd{V R +} is no longer sufficient to add up all
8634 the contributions from the slices, since the slices have varying
8635 coefficients. So first we must come up with a vector of these
8636 coefficients. Here's one way:
8640 2: -1 2: 3 1: [4, 2, ..., 4]
8641 1: [1, 2, ..., 9] 1: [-1, 1, ..., -1] .
8644 1 n v x 9 @key{RET} V M ^ 3 @key{TAB} -
8651 1: [4, 2, ..., 4, 1] 1: [1, 4, 2, ..., 4, 1]
8659 Now we compute the function values. Note that for this method we need
8660 eleven values, including both endpoints of the desired interval.
8664 2: [1, 4, 2, ..., 4, 1]
8665 1: [1, 1.1, 1.2, ... , 1.8, 1.9, 2.]
8668 11 @key{RET} 1 @key{RET} .1 @key{RET} C-u v x
8675 2: [1, 4, 2, ..., 4, 1]
8676 1: [0., 0.084941, 0.16993, ... ]
8679 ' sin(x) ln(x) @key{RET} m r p 5 @key{RET} V M $ @key{RET}
8684 Once again this calls for @kbd{V M * V R +}; a simple @kbd{*} does the
8689 1: 11.22 1: 1.122 1: 0.374
8697 Wow! That's even better than the result from the Taylor series method.
8699 @node Rewrites Answer 1, Rewrites Answer 2, Algebra Answer 4, Answers to Exercises
8700 @subsection Rewrites Tutorial Exercise 1
8703 We'll use Big mode to make the formulas more readable.
8709 1: (2 + sqrt(2)) / (1 + sqrt(2)) 1: --------
8715 ' (2+sqrt(2)) / (1+sqrt(2)) @key{RET} d B
8720 Multiplying by the conjugate helps because @expr{(a+b) (a-b) = a^2 - b^2}.
8725 1: (2 + V 2 ) (V 2 - 1)
8728 a r a/(b+c) := a*(b-c) / (b^2-c^2) @key{RET}
8736 1: 2 + V 2 - 2 1: V 2
8739 a r a*(b+c) := a*b + a*c a s
8744 (We could have used @kbd{a x} instead of a rewrite rule for the
8747 The multiply-by-conjugate rule turns out to be useful in many
8748 different circumstances, such as when the denominator involves
8749 sines and cosines or the imaginary constant @code{i}.
8751 @node Rewrites Answer 2, Rewrites Answer 3, Rewrites Answer 1, Answers to Exercises
8752 @subsection Rewrites Tutorial Exercise 2
8755 Here is the rule set:
8759 [ fib(n) := fib(n, 1, 1) :: integer(n) :: n >= 1,
8761 fib(n, x, y) := fib(n-1, y, x+y) ]
8766 The first rule turns a one-argument @code{fib} that people like to write
8767 into a three-argument @code{fib} that makes computation easier. The
8768 second rule converts back from three-argument form once the computation
8769 is done. The third rule does the computation itself. It basically
8770 says that if @expr{x} and @expr{y} are two consecutive Fibonacci numbers,
8771 then @expr{y} and @expr{x+y} are the next (overlapping) pair of Fibonacci
8774 Notice that because the number @expr{n} was ``validated'' by the
8775 conditions on the first rule, there is no need to put conditions on
8776 the other rules because the rule set would never get that far unless
8777 the input were valid. That further speeds computation, since no
8778 extra conditions need to be checked at every step.
8780 Actually, a user with a nasty sense of humor could enter a bad
8781 three-argument @code{fib} call directly, say, @samp{fib(0, 1, 1)},
8782 which would get the rules into an infinite loop. One thing that would
8783 help keep this from happening by accident would be to use something like
8784 @samp{ZzFib} instead of @code{fib} as the name of the three-argument
8787 @node Rewrites Answer 3, Rewrites Answer 4, Rewrites Answer 2, Answers to Exercises
8788 @subsection Rewrites Tutorial Exercise 3
8791 He got an infinite loop. First, Calc did as expected and rewrote
8792 @w{@samp{2 + 3 x}} to @samp{f(2, 3, x)}. Then it looked for ways to
8793 apply the rule again, and found that @samp{f(2, 3, x)} looks like
8794 @samp{a + b x} with @w{@samp{a = 0}} and @samp{b = 1}, so it rewrote to
8795 @samp{f(0, 1, f(2, 3, x))}. It then wrapped another @samp{f(0, 1, ...)}
8796 around that, and so on, ad infinitum. Joe should have used @kbd{M-1 a r}
8797 to make sure the rule applied only once.
8799 (Actually, even the first step didn't work as he expected. What Calc
8800 really gives for @kbd{M-1 a r} in this situation is @samp{f(3 x, 1, 2)},
8801 treating 2 as the ``variable,'' and @samp{3 x} as a constant being added
8802 to it. While this may seem odd, it's just as valid a solution as the
8803 ``obvious'' one. One way to fix this would be to add the condition
8804 @samp{:: variable(x)} to the rule, to make sure the thing that matches
8805 @samp{x} is indeed a variable, or to change @samp{x} to @samp{quote(x)}
8806 on the lefthand side, so that the rule matches the actual variable
8807 @samp{x} rather than letting @samp{x} stand for something else.)
8809 @node Rewrites Answer 4, Rewrites Answer 5, Rewrites Answer 3, Answers to Exercises
8810 @subsection Rewrites Tutorial Exercise 4
8817 Here is a suitable set of rules to solve the first part of the problem:
8821 [ seq(n, c) := seq(n/2, c+1) :: n%2 = 0,
8822 seq(n, c) := seq(3n+1, c+1) :: n%2 = 1 :: n > 1 ]
8826 Given the initial formula @samp{seq(6, 0)}, application of these
8827 rules produces the following sequence of formulas:
8841 whereupon neither of the rules match, and rewriting stops.
8843 We can pretty this up a bit with a couple more rules:
8847 [ seq(n) := seq(n, 0),
8854 Now, given @samp{seq(6)} as the starting configuration, we get 8
8857 The change to return a vector is quite simple:
8861 [ seq(n) := seq(n, []) :: integer(n) :: n > 0,
8863 seq(n, v) := seq(n/2, v | n) :: n%2 = 0,
8864 seq(n, v) := seq(3n+1, v | n) :: n%2 = 1 ]
8869 Given @samp{seq(6)}, the result is @samp{[6, 3, 10, 5, 16, 8, 4, 2, 1]}.
8871 Notice that the @expr{n > 1} guard is no longer necessary on the last
8872 rule since the @expr{n = 1} case is now detected by another rule.
8873 But a guard has been added to the initial rule to make sure the
8874 initial value is suitable before the computation begins.
8876 While still a good idea, this guard is not as vitally important as it
8877 was for the @code{fib} function, since calling, say, @samp{seq(x, [])}
8878 will not get into an infinite loop. Calc will not be able to prove
8879 the symbol @samp{x} is either even or odd, so none of the rules will
8880 apply and the rewrites will stop right away.
8882 @node Rewrites Answer 5, Rewrites Answer 6, Rewrites Answer 4, Answers to Exercises
8883 @subsection Rewrites Tutorial Exercise 5
8890 If @expr{x} is the sum @expr{a + b}, then `@tfn{nterms(}@var{x}@tfn{)}' must
8891 be `@tfn{nterms(}@var{a}@tfn{)}' plus `@tfn{nterms(}@var{b}@tfn{)}'. If @expr{x}
8892 is not a sum, then `@tfn{nterms(}@var{x}@tfn{)}' = 1.
8896 [ nterms(a + b) := nterms(a) + nterms(b),
8902 Here we have taken advantage of the fact that earlier rules always
8903 match before later rules; @samp{nterms(x)} will only be tried if we
8904 already know that @samp{x} is not a sum.
8906 @node Rewrites Answer 6, Programming Answer 1, Rewrites Answer 5, Answers to Exercises
8907 @subsection Rewrites Tutorial Exercise 6
8910 Here is a rule set that will do the job:
8914 [ a*(b + c) := a*b + a*c,
8915 opt(a) O(x^n) + opt(b) O(x^m) := O(x^n) :: n <= m
8916 :: constant(a) :: constant(b),
8917 opt(a) O(x^n) + opt(b) x^m := O(x^n) :: n <= m
8918 :: constant(a) :: constant(b),
8919 a O(x^n) := O(x^n) :: constant(a),
8920 x^opt(m) O(x^n) := O(x^(n+m)),
8921 O(x^n) O(x^m) := O(x^(n+m)) ]
8925 If we really want the @kbd{+} and @kbd{*} keys to operate naturally
8926 on power series, we should put these rules in @code{EvalRules}. For
8927 testing purposes, it is better to put them in a different variable,
8928 say, @code{O}, first.
8930 The first rule just expands products of sums so that the rest of the
8931 rules can assume they have an expanded-out polynomial to work with.
8932 Note that this rule does not mention @samp{O} at all, so it will
8933 apply to any product-of-sum it encounters---this rule may surprise
8934 you if you put it into @code{EvalRules}!
8936 In the second rule, the sum of two O's is changed to the smaller O.
8937 The optional constant coefficients are there mostly so that
8938 @samp{O(x^2) - O(x^3)} and @samp{O(x^3) - O(x^2)} are handled
8939 as well as @samp{O(x^2) + O(x^3)}.
8941 The third rule absorbs higher powers of @samp{x} into O's.
8943 The fourth rule says that a constant times a negligible quantity
8944 is still negligible. (This rule will also match @samp{O(x^3) / 4},
8945 with @samp{a = 1/4}.)
8947 The fifth rule rewrites, for example, @samp{x^2 O(x^3)} to @samp{O(x^5)}.
8948 (It is easy to see that if one of these forms is negligible, the other
8949 is, too.) Notice the @samp{x^opt(m)} to pick up terms like
8950 @w{@samp{x O(x^3)}}. Optional powers will match @samp{x} as @samp{x^1}
8951 but not 1 as @samp{x^0}. This turns out to be exactly what we want here.
8953 The sixth rule is the corresponding rule for products of two O's.
8955 Another way to solve this problem would be to create a new ``data type''
8956 that represents truncated power series. We might represent these as
8957 function calls @samp{series(@var{coefs}, @var{x})} where @var{coefs} is
8958 a vector of coefficients for @expr{x^0}, @expr{x^1}, @expr{x^2}, and so
8959 on. Rules would exist for sums and products of such @code{series}
8960 objects, and as an optional convenience could also know how to combine a
8961 @code{series} object with a normal polynomial. (With this, and with a
8962 rule that rewrites @samp{O(x^n)} to the equivalent @code{series} form,
8963 you could still enter power series in exactly the same notation as
8964 before.) Operations on such objects would probably be more efficient,
8965 although the objects would be a bit harder to read.
8967 @c [fix-ref Compositions]
8968 Some other symbolic math programs provide a power series data type
8969 similar to this. Mathematica, for example, has an object that looks
8970 like @samp{PowerSeries[@var{x}, @var{x0}, @var{coefs}, @var{nmin},
8971 @var{nmax}, @var{den}]}, where @var{x0} is the point about which the
8972 power series is taken (we've been assuming this was always zero),
8973 and @var{nmin}, @var{nmax}, and @var{den} allow pseudo-power-series
8974 with fractional or negative powers. Also, the @code{PowerSeries}
8975 objects have a special display format that makes them look like
8976 @samp{2 x^2 + O(x^4)} when they are printed out. (@xref{Compositions},
8977 for a way to do this in Calc, although for something as involved as
8978 this it would probably be better to write the formatting routine
8981 @node Programming Answer 1, Programming Answer 2, Rewrites Answer 6, Answers to Exercises
8982 @subsection Programming Tutorial Exercise 1
8985 Just enter the formula @samp{ninteg(sin(t)/t, t, 0, x)}, type
8986 @kbd{Z F}, and answer the questions. Since this formula contains two
8987 variables, the default argument list will be @samp{(t x)}. We want to
8988 change this to @samp{(x)} since @expr{t} is really a dummy variable
8989 to be used within @code{ninteg}.
8991 The exact keystrokes are @kbd{Z F s Si @key{RET} @key{RET} C-b C-b @key{DEL} @key{DEL} @key{RET} y}.
8992 (The @kbd{C-b C-b @key{DEL} @key{DEL}} are what fix the argument list.)
8994 @node Programming Answer 2, Programming Answer 3, Programming Answer 1, Answers to Exercises
8995 @subsection Programming Tutorial Exercise 2
8998 One way is to move the number to the top of the stack, operate on
8999 it, then move it back: @kbd{C-x ( M-@key{TAB} n M-@key{TAB} M-@key{TAB} C-x )}.
9001 Another way is to negate the top three stack entries, then negate
9002 again the top two stack entries: @kbd{C-x ( M-3 n M-2 n C-x )}.
9004 Finally, it turns out that a negative prefix argument causes a
9005 command like @kbd{n} to operate on the specified stack entry only,
9006 which is just what we want: @kbd{C-x ( M-- 3 n C-x )}.
9008 Just for kicks, let's also do it algebraically:
9009 @w{@kbd{C-x ( ' -$$$, $$, $ @key{RET} C-x )}}.
9011 @node Programming Answer 3, Programming Answer 4, Programming Answer 2, Answers to Exercises
9012 @subsection Programming Tutorial Exercise 3
9015 Each of these functions can be computed using the stack, or using
9016 algebraic entry, whichever way you prefer:
9020 @texline @math{\displaystyle{\sin x \over x}}:
9021 @infoline @expr{sin(x) / x}:
9023 Using the stack: @kbd{C-x ( @key{RET} S @key{TAB} / C-x )}.
9025 Using algebraic entry: @kbd{C-x ( ' sin($)/$ @key{RET} C-x )}.
9028 Computing the logarithm:
9030 Using the stack: @kbd{C-x ( @key{TAB} B C-x )}
9032 Using algebraic entry: @kbd{C-x ( ' log($,$$) @key{RET} C-x )}.
9035 Computing the vector of integers:
9037 Using the stack: @kbd{C-x ( 1 @key{RET} 1 C-u v x C-x )}. (Recall that
9038 @kbd{C-u v x} takes the vector size, starting value, and increment
9041 Alternatively: @kbd{C-x ( ~ v x C-x )}. (The @kbd{~} key pops a
9042 number from the stack and uses it as the prefix argument for the
9045 Using algebraic entry: @kbd{C-x ( ' index($) @key{RET} C-x )}.
9047 @node Programming Answer 4, Programming Answer 5, Programming Answer 3, Answers to Exercises
9048 @subsection Programming Tutorial Exercise 4
9051 Here's one way: @kbd{C-x ( @key{RET} V R + @key{TAB} v l / C-x )}.
9053 @node Programming Answer 5, Programming Answer 6, Programming Answer 4, Answers to Exercises
9054 @subsection Programming Tutorial Exercise 5
9058 2: 1 1: 1.61803398502 2: 1.61803398502
9059 1: 20 . 1: 1.61803398875
9062 1 @key{RET} 20 Z < & 1 + Z > I H P
9067 This answer is quite accurate.
9069 @node Programming Answer 6, Programming Answer 7, Programming Answer 5, Answers to Exercises
9070 @subsection Programming Tutorial Exercise 6
9076 [ [ 0, 1 ] * [a, b] = [b, a + b]
9081 Thus @samp{[0, 1; 1, 1]^n * [1, 1]} computes Fibonacci numbers @expr{n+1}
9082 and @expr{n+2}. Here's one program that does the job:
9085 C-x ( ' [0, 1; 1, 1] ^ ($-1) * [1, 1] @key{RET} v u @key{DEL} C-x )
9089 This program is quite efficient because Calc knows how to raise a
9090 matrix (or other value) to the power @expr{n} in only
9091 @texline @math{\log_2 n}
9092 @infoline @expr{log(n,2)}
9093 steps. For example, this program can compute the 1000th Fibonacci
9094 number (a 209-digit integer!) in about 10 steps; even though the
9095 @kbd{Z < ... Z >} solution had much simpler steps, it would have
9096 required so many steps that it would not have been practical.
9098 @node Programming Answer 7, Programming Answer 8, Programming Answer 6, Answers to Exercises
9099 @subsection Programming Tutorial Exercise 7
9102 The trick here is to compute the harmonic numbers differently, so that
9103 the loop counter itself accumulates the sum of reciprocals. We use
9104 a separate variable to hold the integer counter.
9112 1 t 1 1 @key{RET} 4 Z ( t 2 r 1 1 + s 1 & Z )
9117 The body of the loop goes as follows: First save the harmonic sum
9118 so far in variable 2. Then delete it from the stack; the for loop
9119 itself will take care of remembering it for us. Next, recall the
9120 count from variable 1, add one to it, and feed its reciprocal to
9121 the for loop to use as the step value. The for loop will increase
9122 the ``loop counter'' by that amount and keep going until the
9123 loop counter exceeds 4.
9128 1: 3.99498713092 2: 3.99498713092
9132 r 1 r 2 @key{RET} 31 & +
9136 Thus we find that the 30th harmonic number is 3.99, and the 31st
9137 harmonic number is 4.02.
9139 @node Programming Answer 8, Programming Answer 9, Programming Answer 7, Answers to Exercises
9140 @subsection Programming Tutorial Exercise 8
9143 The first step is to compute the derivative @expr{f'(x)} and thus
9145 @texline @math{\displaystyle{x - {f(x) \over f'(x)}}}.
9146 @infoline @expr{x - f(x)/f'(x)}.
9148 (Because this definition is long, it will be repeated in concise form
9149 below. You can use @w{@kbd{C-x * m}} to load it from there. While you are
9150 entering a @kbd{Z ` Z '} body in a macro, Calc simply collects
9151 keystrokes without executing them. In the following diagrams we'll
9152 pretend Calc actually executed the keystrokes as you typed them,
9153 just for purposes of illustration.)
9157 2: sin(cos(x)) - 0.5 3: 4.5
9158 1: 4.5 2: sin(cos(x)) - 0.5
9159 . 1: -(sin(x) cos(cos(x)))
9162 ' sin(cos(x))-0.5 @key{RET} 4.5 m r C-x ( Z ` @key{TAB} @key{RET} a d x @key{RET}
9170 1: x + (sin(cos(x)) - 0.5) / sin(x) cos(cos(x))
9173 / ' x @key{RET} @key{TAB} - t 1
9177 Now, we enter the loop. We'll use a repeat loop with a 20-repetition
9178 limit just in case the method fails to converge for some reason.
9179 (Normally, the @w{@kbd{Z /}} command will stop the loop before all 20
9180 repetitions are done.)
9184 1: 4.5 3: 4.5 2: 4.5
9185 . 2: x + (sin(cos(x)) ... 1: 5.24196456928
9189 20 Z < @key{RET} r 1 @key{TAB} s l x @key{RET}
9193 This is the new guess for @expr{x}. Now we compare it with the
9194 old one to see if we've converged.
9198 3: 5.24196 2: 5.24196 1: 5.24196 1: 5.26345856348
9203 @key{RET} M-@key{TAB} a = Z / Z > Z ' C-x )
9207 The loop converges in just a few steps to this value. To check
9208 the result, we can simply substitute it back into the equation.
9216 @key{RET} ' sin(cos($)) @key{RET}
9220 Let's test the new definition again:
9228 ' x^2-9 @key{RET} 1 X
9232 Once again, here's the full Newton's Method definition:
9236 C-x ( Z ` @key{TAB} @key{RET} a d x @key{RET} / ' x @key{RET} @key{TAB} - t 1
9237 20 Z < @key{RET} r 1 @key{TAB} s l x @key{RET}
9238 @key{RET} M-@key{TAB} a = Z /
9245 @c [fix-ref Nesting and Fixed Points]
9246 It turns out that Calc has a built-in command for applying a formula
9247 repeatedly until it converges to a number. @xref{Nesting and Fixed Points},
9248 to see how to use it.
9250 @c [fix-ref Root Finding]
9251 Also, of course, @kbd{a R} is a built-in command that uses Newton's
9252 method (among others) to look for numerical solutions to any equation.
9253 @xref{Root Finding}.
9255 @node Programming Answer 9, Programming Answer 10, Programming Answer 8, Answers to Exercises
9256 @subsection Programming Tutorial Exercise 9
9259 The first step is to adjust @expr{z} to be greater than 5. A simple
9260 ``for'' loop will do the job here. If @expr{z} is less than 5, we
9261 reduce the problem using
9262 @texline @math{\psi(z) = \psi(z+1) - 1/z}.
9263 @infoline @expr{psi(z) = psi(z+1) - 1/z}. We go
9265 @texline @math{\psi(z+1)},
9266 @infoline @expr{psi(z+1)},
9267 and remember to add back a factor of @expr{-1/z} when we're done. This
9268 step is repeated until @expr{z > 5}.
9270 (Because this definition is long, it will be repeated in concise form
9271 below. You can use @w{@kbd{C-x * m}} to load it from there. While you are
9272 entering a @kbd{Z ` Z '} body in a macro, Calc simply collects
9273 keystrokes without executing them. In the following diagrams we'll
9274 pretend Calc actually executed the keystrokes as you typed them,
9275 just for purposes of illustration.)
9282 1.0 @key{RET} C-x ( Z ` s 1 0 t 2
9286 Here, variable 1 holds @expr{z} and variable 2 holds the adjustment
9287 factor. If @expr{z < 5}, we use a loop to increase it.
9289 (By the way, we started with @samp{1.0} instead of the integer 1 because
9290 otherwise the calculation below will try to do exact fractional arithmetic,
9291 and will never converge because fractions compare equal only if they
9292 are exactly equal, not just equal to within the current precision.)
9301 @key{RET} 5 a < Z [ 5 Z ( & s + 2 1 s + 1 1 Z ) r 1 Z ]
9305 Now we compute the initial part of the sum:
9306 @texline @math{\ln z - {1 \over 2z}}
9307 @infoline @expr{ln(z) - 1/2z}
9308 minus the adjustment factor.
9312 2: 1.79175946923 2: 1.7084261359 1: -0.57490719743
9313 1: 0.0833333333333 1: 2.28333333333 .
9320 Now we evaluate the series. We'll use another ``for'' loop counting
9321 up the value of @expr{2 n}. (Calc does have a summation command,
9322 @kbd{a +}, but we'll use loops just to get more practice with them.)
9326 3: -0.5749 3: -0.5749 4: -0.5749 2: -0.5749
9327 2: 2 2: 1:6 3: 1:6 1: 2.3148e-3
9332 2 @key{RET} 40 Z ( @key{RET} k b @key{TAB} @key{RET} r 1 @key{TAB} ^ * /
9339 3: -0.5749 3: -0.5772 2: -0.5772 1: -0.577215664892
9340 2: -0.5749 2: -0.5772 1: 0 .
9341 1: 2.3148e-3 1: -0.5749 .
9344 @key{TAB} @key{RET} M-@key{TAB} - @key{RET} M-@key{TAB} a = Z / 2 Z ) Z ' C-x )
9348 This is the value of
9349 @texline @math{-\gamma},
9350 @infoline @expr{- gamma},
9351 with a slight bit of roundoff error. To get a full 12 digits, let's use
9356 2: -0.577215664892 2: -0.577215664892
9357 1: 1. 1: -0.577215664901532
9359 1. @key{RET} p 16 @key{RET} X
9363 Here's the complete sequence of keystrokes:
9368 @key{RET} 5 a < Z [ 5 Z ( & s + 2 1 s + 1 1 Z ) r 1 Z ]
9370 2 @key{RET} 40 Z ( @key{RET} k b @key{TAB} @key{RET} r 1 @key{TAB} ^ * /
9371 @key{TAB} @key{RET} M-@key{TAB} - @key{RET} M-@key{TAB} a = Z /
9378 @node Programming Answer 10, Programming Answer 11, Programming Answer 9, Answers to Exercises
9379 @subsection Programming Tutorial Exercise 10
9382 Taking the derivative of a term of the form @expr{x^n} will produce
9384 @texline @math{n x^{n-1}}.
9385 @infoline @expr{n x^(n-1)}.
9386 Taking the derivative of a constant
9387 produces zero. From this it is easy to see that the @expr{n}th
9388 derivative of a polynomial, evaluated at @expr{x = 0}, will equal the
9389 coefficient on the @expr{x^n} term times @expr{n!}.
9391 (Because this definition is long, it will be repeated in concise form
9392 below. You can use @w{@kbd{C-x * m}} to load it from there. While you are
9393 entering a @kbd{Z ` Z '} body in a macro, Calc simply collects
9394 keystrokes without executing them. In the following diagrams we'll
9395 pretend Calc actually executed the keystrokes as you typed them,
9396 just for purposes of illustration.)
9400 2: 5 x^4 + (x + 1)^2 3: 5 x^4 + (x + 1)^2
9405 ' 5 x^4 + (x+1)^2 @key{RET} 6 C-x ( Z ` [ ] t 1 0 @key{TAB}
9410 Variable 1 will accumulate the vector of coefficients.
9414 2: 0 3: 0 2: 5 x^4 + ...
9415 1: 5 x^4 + ... 2: 5 x^4 + ... 1: 1
9419 Z ( @key{TAB} @key{RET} 0 s l x @key{RET} M-@key{TAB} ! / s | 1
9424 Note that @kbd{s | 1} appends the top-of-stack value to the vector
9425 in a variable; it is completely analogous to @kbd{s + 1}. We could
9426 have written instead, @kbd{r 1 @key{TAB} | t 1}.
9430 1: 20 x^3 + 2 x + 2 1: 0 1: [1, 2, 1, 0, 5, 0, 0]
9433 a d x @key{RET} 1 Z ) @key{DEL} r 1 Z ' C-x )
9437 To convert back, a simple method is just to map the coefficients
9438 against a table of powers of @expr{x}.
9442 2: [1, 2, 1, 0, 5, 0, 0] 2: [1, 2, 1, 0, 5, 0, 0]
9443 1: 6 1: [0, 1, 2, 3, 4, 5, 6]
9446 6 @key{RET} 1 + 0 @key{RET} 1 C-u v x
9453 2: [1, 2, 1, 0, 5, 0, 0] 2: 1 + 2 x + x^2 + 5 x^4
9454 1: [1, x, x^2, x^3, ... ] .
9457 ' x @key{RET} @key{TAB} V M ^ *
9461 Once again, here are the whole polynomial to/from vector programs:
9465 C-x ( Z ` [ ] t 1 0 @key{TAB}
9466 Z ( @key{TAB} @key{RET} 0 s l x @key{RET} M-@key{TAB} ! / s | 1
9472 C-x ( 1 + 0 @key{RET} 1 C-u v x ' x @key{RET} @key{TAB} V M ^ * C-x )
9476 @node Programming Answer 11, Programming Answer 12, Programming Answer 10, Answers to Exercises
9477 @subsection Programming Tutorial Exercise 11
9480 First we define a dummy program to go on the @kbd{z s} key. The true
9481 @w{@kbd{z s}} key is supposed to take two numbers from the stack and
9482 return one number, so @key{DEL} as a dummy definition will make
9483 sure the stack comes out right.
9491 4 @key{RET} 2 C-x ( @key{DEL} C-x ) Z K s @key{RET} 2
9495 The last step replaces the 2 that was eaten during the creation
9496 of the dummy @kbd{z s} command. Now we move on to the real
9497 definition. The recurrence needs to be rewritten slightly,
9498 to the form @expr{s(n,m) = s(n-1,m-1) - (n-1) s(n-1,m)}.
9500 (Because this definition is long, it will be repeated in concise form
9501 below. You can use @kbd{C-x * m} to load it from there.)
9511 C-x ( M-2 @key{RET} a = Z [ @key{DEL} @key{DEL} 1 Z :
9518 4: 4 2: 4 2: 3 4: 3 4: 3 3: 3
9519 3: 2 1: 2 1: 2 3: 2 3: 2 2: 2
9520 2: 2 . . 2: 3 2: 3 1: 3
9524 @key{RET} 0 a = Z [ @key{DEL} @key{DEL} 0 Z : @key{TAB} 1 - @key{TAB} M-2 @key{RET} 1 - z s
9529 (Note that the value 3 that our dummy @kbd{z s} produces is not correct;
9530 it is merely a placeholder that will do just as well for now.)
9534 3: 3 4: 3 3: 3 2: 3 1: -6
9535 2: 3 3: 3 2: 3 1: 9 .
9540 M-@key{TAB} M-@key{TAB} @key{TAB} @key{RET} M-@key{TAB} z s * -
9547 1: -6 2: 4 1: 11 2: 11
9551 Z ] Z ] C-x ) Z K s @key{RET} @key{DEL} 4 @key{RET} 2 z s M-@key{RET} k s
9555 Even though the result that we got during the definition was highly
9556 bogus, once the definition is complete the @kbd{z s} command gets
9559 Here's the full program once again:
9563 C-x ( M-2 @key{RET} a =
9564 Z [ @key{DEL} @key{DEL} 1
9566 Z [ @key{DEL} @key{DEL} 0
9567 Z : @key{TAB} 1 - @key{TAB} M-2 @key{RET} 1 - z s
9568 M-@key{TAB} M-@key{TAB} @key{TAB} @key{RET} M-@key{TAB} z s * -
9575 You can read this definition using @kbd{C-x * m} (@code{read-kbd-macro})
9576 followed by @kbd{Z K s}, without having to make a dummy definition
9577 first, because @code{read-kbd-macro} doesn't need to execute the
9578 definition as it reads it in. For this reason, @code{C-x * m} is often
9579 the easiest way to create recursive programs in Calc.
9581 @node Programming Answer 12, , Programming Answer 11, Answers to Exercises
9582 @subsection Programming Tutorial Exercise 12
9585 This turns out to be a much easier way to solve the problem. Let's
9586 denote Stirling numbers as calls of the function @samp{s}.
9588 First, we store the rewrite rules corresponding to the definition of
9589 Stirling numbers in a convenient variable:
9592 s e StirlingRules @key{RET}
9593 [ s(n,n) := 1 :: n >= 0,
9594 s(n,0) := 0 :: n > 0,
9595 s(n,m) := s(n-1,m-1) - (n-1) s(n-1,m) :: n >= m :: m >= 1 ]
9599 Now, it's just a matter of applying the rules:
9603 2: 4 1: s(4, 2) 1: 11
9607 4 @key{RET} 2 C-x ( ' s($$,$) @key{RET} a r StirlingRules @key{RET} C-x )
9611 As in the case of the @code{fib} rules, it would be useful to put these
9612 rules in @code{EvalRules} and to add a @samp{:: remember} condition to
9615 @c This ends the table-of-contents kludge from above:
9617 \global\let\chapternofonts=\oldchapternofonts
9622 @node Introduction, Data Types, Tutorial, Top
9623 @chapter Introduction
9626 This chapter is the beginning of the Calc reference manual.
9627 It covers basic concepts such as the stack, algebraic and
9628 numeric entry, undo, numeric prefix arguments, etc.
9631 @c (Chapter 2, the Tutorial, has been printed in a separate volume.)
9639 * Quick Calculator::
9640 * Prefix Arguments::
9643 * Multiple Calculators::
9644 * Troubleshooting Commands::
9647 @node Basic Commands, Help Commands, Introduction, Introduction
9648 @section Basic Commands
9653 @cindex Starting the Calculator
9654 @cindex Running the Calculator
9655 To start the Calculator in its standard interface, type @kbd{M-x calc}.
9656 By default this creates a pair of small windows, @samp{*Calculator*}
9657 and @samp{*Calc Trail*}. The former displays the contents of the
9658 Calculator stack and is manipulated exclusively through Calc commands.
9659 It is possible (though not usually necessary) to create several Calc
9660 mode buffers each of which has an independent stack, undo list, and
9661 mode settings. There is exactly one Calc Trail buffer; it records a
9662 list of the results of all calculations that have been done. The
9663 Calc Trail buffer uses a variant of Calc mode, so Calculator commands
9664 still work when the trail buffer's window is selected. It is possible
9665 to turn the trail window off, but the @samp{*Calc Trail*} buffer itself
9666 still exists and is updated silently. @xref{Trail Commands}.
9673 In most installations, the @kbd{C-x * c} key sequence is a more
9674 convenient way to start the Calculator. Also, @kbd{C-x * *}
9675 is a synonym for @kbd{C-x * c} unless you last used Calc
9680 @pindex calc-execute-extended-command
9681 Most Calc commands use one or two keystrokes. Lower- and upper-case
9682 letters are distinct. Commands may also be entered in full @kbd{M-x} form;
9683 for some commands this is the only form. As a convenience, the @kbd{x}
9684 key (@code{calc-execute-extended-command})
9685 is like @kbd{M-x} except that it enters the initial string @samp{calc-}
9686 for you. For example, the following key sequences are equivalent:
9687 @kbd{S}, @kbd{M-x calc-sin @key{RET}}, @kbd{x sin @key{RET}}.
9689 Although Calc is designed to be used from the keyboard, some of
9690 Calc's more common commands are available from a menu. In the menu, the
9691 arguments to the functions are given by referring to their stack level
9694 @cindex Extensions module
9695 @cindex @file{calc-ext} module
9696 The Calculator exists in many parts. When you type @kbd{C-x * c}, the
9697 Emacs ``auto-load'' mechanism will bring in only the first part, which
9698 contains the basic arithmetic functions. The other parts will be
9699 auto-loaded the first time you use the more advanced commands like trig
9700 functions or matrix operations. This is done to improve the response time
9701 of the Calculator in the common case when all you need to do is a
9702 little arithmetic. If for some reason the Calculator fails to load an
9703 extension module automatically, you can force it to load all the
9704 extensions by using the @kbd{C-x * L} (@code{calc-load-everything})
9705 command. @xref{Mode Settings}.
9707 If you type @kbd{M-x calc} or @kbd{C-x * c} with any numeric prefix argument,
9708 the Calculator is loaded if necessary, but it is not actually started.
9709 If the argument is positive, the @file{calc-ext} extensions are also
9710 loaded if necessary. User-written Lisp code that wishes to make use
9711 of Calc's arithmetic routines can use @samp{(calc 0)} or @samp{(calc 1)}
9712 to auto-load the Calculator.
9716 If you type @kbd{C-x * b}, then next time you use @kbd{C-x * c} you
9717 will get a Calculator that uses the full height of the Emacs screen.
9718 When full-screen mode is on, @kbd{C-x * c} runs the @code{full-calc}
9719 command instead of @code{calc}. From the Unix shell you can type
9720 @samp{emacs -f full-calc} to start a new Emacs specifically for use
9721 as a calculator. When Calc is started from the Emacs command line
9722 like this, Calc's normal ``quit'' commands actually quit Emacs itself.
9725 @pindex calc-other-window
9726 The @kbd{C-x * o} command is like @kbd{C-x * c} except that the Calc
9727 window is not actually selected. If you are already in the Calc
9728 window, @kbd{C-x * o} switches you out of it. (The regular Emacs
9729 @kbd{C-x o} command would also work for this, but it has a
9730 tendency to drop you into the Calc Trail window instead, which
9731 @kbd{C-x * o} takes care not to do.)
9736 For one quick calculation, you can type @kbd{C-x * q} (@code{quick-calc})
9737 which prompts you for a formula (like @samp{2+3/4}). The result is
9738 displayed at the bottom of the Emacs screen without ever creating
9739 any special Calculator windows. @xref{Quick Calculator}.
9744 Finally, if you are using the X window system you may want to try
9745 @kbd{C-x * k} (@code{calc-keypad}) which runs Calc with a
9746 ``calculator keypad'' picture as well as a stack display. Click on
9747 the keys with the mouse to operate the calculator. @xref{Keypad Mode}.
9751 @cindex Quitting the Calculator
9752 @cindex Exiting the Calculator
9753 The @kbd{q} key (@code{calc-quit}) exits Calc mode and closes the
9754 Calculator's window(s). It does not delete the Calculator buffers.
9755 If you type @kbd{M-x calc} again, the Calculator will reappear with the
9756 contents of the stack intact. Typing @kbd{C-x * c} or @kbd{C-x * *}
9757 again from inside the Calculator buffer is equivalent to executing
9758 @code{calc-quit}; you can think of @kbd{C-x * *} as toggling the
9759 Calculator on and off.
9762 The @kbd{C-x * x} command also turns the Calculator off, no matter which
9763 user interface (standard, Keypad, or Embedded) is currently active.
9764 It also cancels @code{calc-edit} mode if used from there.
9767 @pindex calc-refresh
9768 @cindex Refreshing a garbled display
9769 @cindex Garbled displays, refreshing
9770 The @kbd{d @key{SPC}} key sequence (@code{calc-refresh}) redraws the contents
9771 of the Calculator buffer from memory. Use this if the contents of the
9772 buffer have been damaged somehow.
9777 The @kbd{o} key (@code{calc-realign}) moves the cursor back to its
9778 ``home'' position at the bottom of the Calculator buffer.
9782 @pindex calc-scroll-left
9783 @pindex calc-scroll-right
9784 @cindex Horizontal scrolling
9786 @cindex Wide text, scrolling
9787 The @kbd{<} and @kbd{>} keys are bound to @code{calc-scroll-left} and
9788 @code{calc-scroll-right}. These are just like the normal horizontal
9789 scrolling commands except that they scroll one half-screen at a time by
9790 default. (Calc formats its output to fit within the bounds of the
9791 window whenever it can.)
9795 @pindex calc-scroll-down
9796 @pindex calc-scroll-up
9797 @cindex Vertical scrolling
9798 The @kbd{@{} and @kbd{@}} keys are bound to @code{calc-scroll-down}
9799 and @code{calc-scroll-up}. They scroll up or down by one-half the
9800 height of the Calc window.
9804 The @kbd{C-x * 0} command (@code{calc-reset}; that's @kbd{C-x *} followed
9805 by a zero) resets the Calculator to its initial state. This clears
9806 the stack, resets all the modes to their initial values (the values
9807 that were saved with @kbd{m m} (@code{calc-save-modes})), clears the
9808 caches (@pxref{Caches}), and so on. (It does @emph{not} erase the
9809 values of any variables.) With an argument of 0, Calc will be reset to
9810 its default state; namely, the modes will be given their default values.
9811 With a positive prefix argument, @kbd{C-x * 0} preserves the contents of
9812 the stack but resets everything else to its initial state; with a
9813 negative prefix argument, @kbd{C-x * 0} preserves the contents of the
9814 stack but resets everything else to its default state.
9816 @node Help Commands, Stack Basics, Basic Commands, Introduction
9817 @section Help Commands
9820 @cindex Help commands
9823 The @kbd{?} key (@code{calc-help}) displays a series of brief help messages.
9824 Some keys (such as @kbd{b} and @kbd{d}) are prefix keys, like Emacs'
9825 @key{ESC} and @kbd{C-x} prefixes. You can type
9826 @kbd{?} after a prefix to see a list of commands beginning with that
9827 prefix. (If the message includes @samp{[MORE]}, press @kbd{?} again
9828 to see additional commands for that prefix.)
9831 @pindex calc-full-help
9832 The @kbd{h h} (@code{calc-full-help}) command displays all the @kbd{?}
9833 responses at once. When printed, this makes a nice, compact (three pages)
9834 summary of Calc keystrokes.
9836 In general, the @kbd{h} key prefix introduces various commands that
9837 provide help within Calc. Many of the @kbd{h} key functions are
9838 Calc-specific analogues to the @kbd{C-h} functions for Emacs help.
9844 The @kbd{h i} (@code{calc-info}) command runs the Emacs Info system
9845 to read this manual on-line. This is basically the same as typing
9846 @kbd{C-h i} (the regular way to run the Info system), then, if Info
9847 is not already in the Calc manual, selecting the beginning of the
9848 manual. The @kbd{C-x * i} command is another way to read the Calc
9849 manual; it is different from @kbd{h i} in that it works any time,
9850 not just inside Calc. The plain @kbd{i} key is also equivalent to
9851 @kbd{h i}, though this key is obsolete and may be replaced with a
9852 different command in a future version of Calc.
9856 @pindex calc-tutorial
9857 The @kbd{h t} (@code{calc-tutorial}) command runs the Info system on
9858 the Tutorial section of the Calc manual. It is like @kbd{h i},
9859 except that it selects the starting node of the tutorial rather
9860 than the beginning of the whole manual. (It actually selects the
9861 node ``Interactive Tutorial'' which tells a few things about
9862 using the Info system before going on to the actual tutorial.)
9863 The @kbd{C-x * t} key is equivalent to @kbd{h t} (but it works at
9868 @pindex calc-info-summary
9869 The @kbd{h s} (@code{calc-info-summary}) command runs the Info system
9870 on the Summary node of the Calc manual. @xref{Summary}. The @kbd{C-x * s}
9871 key is equivalent to @kbd{h s}.
9874 @pindex calc-describe-key
9875 The @kbd{h k} (@code{calc-describe-key}) command looks up a key
9876 sequence in the Calc manual. For example, @kbd{h k H a S} looks
9877 up the documentation on the @kbd{H a S} (@code{calc-solve-for})
9878 command. This works by looking up the textual description of
9879 the key(s) in the Key Index of the manual, then jumping to the
9880 node indicated by the index.
9882 Most Calc commands do not have traditional Emacs documentation
9883 strings, since the @kbd{h k} command is both more convenient and
9884 more instructive. This means the regular Emacs @kbd{C-h k}
9885 (@code{describe-key}) command will not be useful for Calc keystrokes.
9888 @pindex calc-describe-key-briefly
9889 The @kbd{h c} (@code{calc-describe-key-briefly}) command reads a
9890 key sequence and displays a brief one-line description of it at
9891 the bottom of the screen. It looks for the key sequence in the
9892 Summary node of the Calc manual; if it doesn't find the sequence
9893 there, it acts just like its regular Emacs counterpart @kbd{C-h c}
9894 (@code{describe-key-briefly}). For example, @kbd{h c H a S}
9895 gives the description:
9898 H a S runs calc-solve-for: a `H a S' v => fsolve(a,v) (?=notes)
9902 which means the command @kbd{H a S} or @kbd{H M-x calc-solve-for}
9903 takes a value @expr{a} from the stack, prompts for a value @expr{v},
9904 then applies the algebraic function @code{fsolve} to these values.
9905 The @samp{?=notes} message means you can now type @kbd{?} to see
9906 additional notes from the summary that apply to this command.
9909 @pindex calc-describe-function
9910 The @kbd{h f} (@code{calc-describe-function}) command looks up an
9911 algebraic function or a command name in the Calc manual. Enter an
9912 algebraic function name to look up that function in the Function
9913 Index or enter a command name beginning with @samp{calc-} to look it
9914 up in the Command Index. This command will also look up operator
9915 symbols that can appear in algebraic formulas, like @samp{%} and
9919 @pindex calc-describe-variable
9920 The @kbd{h v} (@code{calc-describe-variable}) command looks up a
9921 variable in the Calc manual. Enter a variable name like @code{pi} or
9925 @pindex describe-bindings
9926 The @kbd{h b} (@code{calc-describe-bindings}) command is just like
9927 @kbd{C-h b}, except that only local (Calc-related) key bindings are
9931 The @kbd{h n} or @kbd{h C-n} (@code{calc-view-news}) command displays
9932 the ``news'' or change history of Calc. This is kept in the file
9933 @file{README}, which Calc looks for in the same directory as the Calc
9939 The @kbd{h C-c}, @kbd{h C-d}, and @kbd{h C-w} keys display copying,
9940 distribution, and warranty information about Calc. These work by
9941 pulling up the appropriate parts of the ``Copying'' or ``Reporting
9942 Bugs'' sections of the manual.
9944 @node Stack Basics, Numeric Entry, Help Commands, Introduction
9945 @section Stack Basics
9948 @cindex Stack basics
9949 @c [fix-tut RPN Calculations and the Stack]
9950 Calc uses RPN notation. If you are not familiar with RPN, @pxref{RPN
9953 To add the numbers 1 and 2 in Calc you would type the keys:
9954 @kbd{1 @key{RET} 2 +}.
9955 (@key{RET} corresponds to the @key{ENTER} key on most calculators.)
9956 The first three keystrokes ``push'' the numbers 1 and 2 onto the stack. The
9957 @kbd{+} key always ``pops'' the top two numbers from the stack, adds them,
9958 and pushes the result (3) back onto the stack. This number is ready for
9959 further calculations: @kbd{5 -} pushes 5 onto the stack, then pops the
9960 3 and 5, subtracts them, and pushes the result (@mathit{-2}).
9962 Note that the ``top'' of the stack actually appears at the @emph{bottom}
9963 of the buffer. A line containing a single @samp{.} character signifies
9964 the end of the buffer; Calculator commands operate on the number(s)
9965 directly above this line. The @kbd{d t} (@code{calc-truncate-stack})
9966 command allows you to move the @samp{.} marker up and down in the stack;
9967 @pxref{Truncating the Stack}.
9970 @pindex calc-line-numbering
9971 Stack elements are numbered consecutively, with number 1 being the top of
9972 the stack. These line numbers are ordinarily displayed on the lefthand side
9973 of the window. The @kbd{d l} (@code{calc-line-numbering}) command controls
9974 whether these numbers appear. (Line numbers may be turned off since they
9975 slow the Calculator down a bit and also clutter the display.)
9978 @pindex calc-realign
9979 The unshifted letter @kbd{o} (@code{calc-realign}) command repositions
9980 the cursor to its top-of-stack ``home'' position. It also undoes any
9981 horizontal scrolling in the window. If you give it a numeric prefix
9982 argument, it instead moves the cursor to the specified stack element.
9984 The @key{RET} (or equivalent @key{SPC}) key is only required to separate
9985 two consecutive numbers.
9986 (After all, if you typed @kbd{1 2} by themselves the Calculator
9987 would enter the number 12.) If you press @key{RET} or @key{SPC} @emph{not}
9988 right after typing a number, the key duplicates the number on the top of
9989 the stack. @kbd{@key{RET} *} is thus a handy way to square a number.
9991 The @key{DEL} key pops and throws away the top number on the stack.
9992 The @key{TAB} key swaps the top two objects on the stack.
9993 @xref{Stack and Trail}, for descriptions of these and other stack-related
9996 @node Numeric Entry, Algebraic Entry, Stack Basics, Introduction
9997 @section Numeric Entry
10003 @cindex Numeric entry
10004 @cindex Entering numbers
10005 Pressing a digit or other numeric key begins numeric entry using the
10006 minibuffer. The number is pushed on the stack when you press the @key{RET}
10007 or @key{SPC} keys. If you press any other non-numeric key, the number is
10008 pushed onto the stack and the appropriate operation is performed. If
10009 you press a numeric key which is not valid, the key is ignored.
10011 @cindex Minus signs
10012 @cindex Negative numbers, entering
10014 There are three different concepts corresponding to the word ``minus,''
10015 typified by @expr{a-b} (subtraction), @expr{-x}
10016 (change-sign), and @expr{-5} (negative number). Calc uses three
10017 different keys for these operations, respectively:
10018 @kbd{-}, @kbd{n}, and @kbd{_} (the underscore). The @kbd{-} key subtracts
10019 the two numbers on the top of the stack. The @kbd{n} key changes the sign
10020 of the number on the top of the stack or the number currently being entered.
10021 The @kbd{_} key begins entry of a negative number or changes the sign of
10022 the number currently being entered. The following sequences all enter the
10023 number @mathit{-5} onto the stack: @kbd{0 @key{RET} 5 -}, @kbd{5 n @key{RET}},
10024 @kbd{5 @key{RET} n}, @kbd{_ 5 @key{RET}}, @kbd{5 _ @key{RET}}.
10026 Some other keys are active during numeric entry, such as @kbd{#} for
10027 non-decimal numbers, @kbd{:} for fractions, and @kbd{@@} for HMS forms.
10028 These notations are described later in this manual with the corresponding
10029 data types. @xref{Data Types}.
10031 During numeric entry, the only editing key available is @key{DEL}.
10033 @node Algebraic Entry, Quick Calculator, Numeric Entry, Introduction
10034 @section Algebraic Entry
10038 @pindex calc-algebraic-entry
10039 @cindex Algebraic notation
10040 @cindex Formulas, entering
10041 The @kbd{'} (@code{calc-algebraic-entry}) command can be used to enter
10042 calculations in algebraic form. This is accomplished by typing the
10043 apostrophe key, ', followed by the expression in standard format:
10051 @texline @math{2+(3\times4) = 14}
10052 @infoline @expr{2+(3*4) = 14}
10053 and push it on the stack. If you wish you can
10054 ignore the RPN aspect of Calc altogether and simply enter algebraic
10055 expressions in this way. You may want to use @key{DEL} every so often to
10056 clear previous results off the stack.
10058 You can press the apostrophe key during normal numeric entry to switch
10059 the half-entered number into Algebraic entry mode. One reason to do
10060 this would be to fix a typo, as the full Emacs cursor motion and editing
10061 keys are available during algebraic entry but not during numeric entry.
10063 In the same vein, during either numeric or algebraic entry you can
10064 press @kbd{`} (backquote) to switch to @code{calc-edit} mode, where
10065 you complete your half-finished entry in a separate buffer.
10066 @xref{Editing Stack Entries}.
10069 @pindex calc-algebraic-mode
10070 @cindex Algebraic Mode
10071 If you prefer algebraic entry, you can use the command @kbd{m a}
10072 (@code{calc-algebraic-mode}) to set Algebraic mode. In this mode,
10073 digits and other keys that would normally start numeric entry instead
10074 start full algebraic entry; as long as your formula begins with a digit
10075 you can omit the apostrophe. Open parentheses and square brackets also
10076 begin algebraic entry. You can still do RPN calculations in this mode,
10077 but you will have to press @key{RET} to terminate every number:
10078 @kbd{2 @key{RET} 3 @key{RET} * 4 @key{RET} +} would accomplish the same
10079 thing as @kbd{2*3+4 @key{RET}}.
10081 @cindex Incomplete Algebraic Mode
10082 If you give a numeric prefix argument like @kbd{C-u} to the @kbd{m a}
10083 command, it enables Incomplete Algebraic mode; this is like regular
10084 Algebraic mode except that it applies to the @kbd{(} and @kbd{[} keys
10085 only. Numeric keys still begin a numeric entry in this mode.
10088 @pindex calc-total-algebraic-mode
10089 @cindex Total Algebraic Mode
10090 The @kbd{m t} (@code{calc-total-algebraic-mode}) gives you an even
10091 stronger algebraic-entry mode, in which @emph{all} regular letter and
10092 punctuation keys begin algebraic entry. Use this if you prefer typing
10093 @w{@kbd{sqrt( )}} instead of @kbd{Q}, @w{@kbd{factor( )}} instead of
10094 @kbd{a f}, and so on. To type regular Calc commands when you are in
10095 Total Algebraic mode, hold down the @key{META} key. Thus @kbd{M-q}
10096 is the command to quit Calc, @kbd{M-p} sets the precision, and
10097 @kbd{M-m t} (or @kbd{M-m M-t}, if you prefer) turns Total Algebraic
10098 mode back off again. Meta keys also terminate algebraic entry, so
10099 that @kbd{2+3 M-S} is equivalent to @kbd{2+3 @key{RET} M-S}. The symbol
10100 @samp{Alg*} will appear in the mode line whenever you are in this mode.
10102 Pressing @kbd{'} (the apostrophe) a second time re-enters the previous
10103 algebraic formula. You can then use the normal Emacs editing keys to
10104 modify this formula to your liking before pressing @key{RET}.
10107 @cindex Formulas, referring to stack
10108 Within a formula entered from the keyboard, the symbol @kbd{$}
10109 represents the number on the top of the stack. If an entered formula
10110 contains any @kbd{$} characters, the Calculator replaces the top of
10111 stack with that formula rather than simply pushing the formula onto the
10112 stack. Thus, @kbd{' 1+2 @key{RET}} pushes 3 on the stack, and @kbd{$*2
10113 @key{RET}} replaces it with 6. Note that the @kbd{$} key always
10114 initiates algebraic entry; the @kbd{'} is unnecessary if @kbd{$} is the
10115 first character in the new formula.
10117 Higher stack elements can be accessed from an entered formula with the
10118 symbols @kbd{$$}, @kbd{$$$}, and so on. The number of stack elements
10119 removed (to be replaced by the entered values) equals the number of dollar
10120 signs in the longest such symbol in the formula. For example, @samp{$$+$$$}
10121 adds the second and third stack elements, replacing the top three elements
10122 with the answer. (All information about the top stack element is thus lost
10123 since no single @samp{$} appears in this formula.)
10125 A slightly different way to refer to stack elements is with a dollar
10126 sign followed by a number: @samp{$1}, @samp{$2}, and so on are much
10127 like @samp{$}, @samp{$$}, etc., except that stack entries referred
10128 to numerically are not replaced by the algebraic entry. That is, while
10129 @samp{$+1} replaces 5 on the stack with 6, @samp{$1+1} leaves the 5
10130 on the stack and pushes an additional 6.
10132 If a sequence of formulas are entered separated by commas, each formula
10133 is pushed onto the stack in turn. For example, @samp{1,2,3} pushes
10134 those three numbers onto the stack (leaving the 3 at the top), and
10135 @samp{$+1,$-1} replaces a 5 on the stack with 4 followed by 6. Also,
10136 @samp{$,$$} exchanges the top two elements of the stack, just like the
10139 You can finish an algebraic entry with @kbd{M-=} or @kbd{M-@key{RET}} instead
10140 of @key{RET}. This uses @kbd{=} to evaluate the variables in each
10141 formula that goes onto the stack. (Thus @kbd{' pi @key{RET}} pushes
10142 the variable @samp{pi}, but @kbd{' pi M-@key{RET}} pushes 3.1415.)
10144 If you finish your algebraic entry by pressing @key{LFD} (or @kbd{C-j})
10145 instead of @key{RET}, Calc disables the default simplifications
10146 (as if by @kbd{m O}; @pxref{Simplification Modes}) while the entry
10147 is being pushed on the stack. Thus @kbd{' 1+2 @key{RET}} pushes 3
10148 on the stack, but @kbd{' 1+2 @key{LFD}} pushes the formula @expr{1+2};
10149 you might then press @kbd{=} when it is time to evaluate this formula.
10151 @node Quick Calculator, Prefix Arguments, Algebraic Entry, Introduction
10152 @section ``Quick Calculator'' Mode
10157 @cindex Quick Calculator
10158 There is another way to invoke the Calculator if all you need to do
10159 is make one or two quick calculations. Type @kbd{C-x * q} (or
10160 @kbd{M-x quick-calc}), then type any formula as an algebraic entry.
10161 The Calculator will compute the result and display it in the echo
10162 area, without ever actually putting up a Calc window.
10164 You can use the @kbd{$} character in a Quick Calculator formula to
10165 refer to the previous Quick Calculator result. Older results are
10166 not retained; the Quick Calculator has no effect on the full
10167 Calculator's stack or trail. If you compute a result and then
10168 forget what it was, just run @code{C-x * q} again and enter
10169 @samp{$} as the formula.
10171 If this is the first time you have used the Calculator in this Emacs
10172 session, the @kbd{C-x * q} command will create the @code{*Calculator*}
10173 buffer and perform all the usual initializations; it simply will
10174 refrain from putting that buffer up in a new window. The Quick
10175 Calculator refers to the @code{*Calculator*} buffer for all mode
10176 settings. Thus, for example, to set the precision that the Quick
10177 Calculator uses, simply run the full Calculator momentarily and use
10178 the regular @kbd{p} command.
10180 If you use @code{C-x * q} from inside the Calculator buffer, the
10181 effect is the same as pressing the apostrophe key (algebraic entry).
10183 The result of a Quick calculation is placed in the Emacs ``kill ring''
10184 as well as being displayed. A subsequent @kbd{C-y} command will
10185 yank the result into the editing buffer. You can also use this
10186 to yank the result into the next @kbd{C-x * q} input line as a more
10187 explicit alternative to @kbd{$} notation, or to yank the result
10188 into the Calculator stack after typing @kbd{C-x * c}.
10190 If you finish your formula by typing @key{LFD} (or @kbd{C-j}) instead
10191 of @key{RET}, the result is inserted immediately into the current
10192 buffer rather than going into the kill ring.
10194 Quick Calculator results are actually evaluated as if by the @kbd{=}
10195 key (which replaces variable names by their stored values, if any).
10196 If the formula you enter is an assignment to a variable using the
10197 @samp{:=} operator, say, @samp{foo := 2 + 3} or @samp{foo := foo + 1},
10198 then the result of the evaluation is stored in that Calc variable.
10199 @xref{Store and Recall}.
10201 If the result is an integer and the current display radix is decimal,
10202 the number will also be displayed in hex, octal and binary formats. If
10203 the integer is in the range from 1 to 126, it will also be displayed as
10204 an ASCII character.
10206 For example, the quoted character @samp{"x"} produces the vector
10207 result @samp{[120]} (because 120 is the ASCII code of the lower-case
10208 `x'; @pxref{Strings}). Since this is a vector, not an integer, it
10209 is displayed only according to the current mode settings. But
10210 running Quick Calc again and entering @samp{120} will produce the
10211 result @samp{120 (16#78, 8#170, x)} which shows the number in its
10212 decimal, hexadecimal, octal, and ASCII forms.
10214 Please note that the Quick Calculator is not any faster at loading
10215 or computing the answer than the full Calculator; the name ``quick''
10216 merely refers to the fact that it's much less hassle to use for
10217 small calculations.
10219 @node Prefix Arguments, Undo, Quick Calculator, Introduction
10220 @section Numeric Prefix Arguments
10223 Many Calculator commands use numeric prefix arguments. Some, such as
10224 @kbd{d s} (@code{calc-sci-notation}), set a parameter to the value of
10225 the prefix argument or use a default if you don't use a prefix.
10226 Others (like @kbd{d f} (@code{calc-fix-notation})) require an argument
10227 and prompt for a number if you don't give one as a prefix.
10229 As a rule, stack-manipulation commands accept a numeric prefix argument
10230 which is interpreted as an index into the stack. A positive argument
10231 operates on the top @var{n} stack entries; a negative argument operates
10232 on the @var{n}th stack entry in isolation; and a zero argument operates
10233 on the entire stack.
10235 Most commands that perform computations (such as the arithmetic and
10236 scientific functions) accept a numeric prefix argument that allows the
10237 operation to be applied across many stack elements. For unary operations
10238 (that is, functions of one argument like absolute value or complex
10239 conjugate), a positive prefix argument applies that function to the top
10240 @var{n} stack entries simultaneously, and a negative argument applies it
10241 to the @var{n}th stack entry only. For binary operations (functions of
10242 two arguments like addition, GCD, and vector concatenation), a positive
10243 prefix argument ``reduces'' the function across the top @var{n}
10244 stack elements (for example, @kbd{C-u 5 +} sums the top 5 stack entries;
10245 @pxref{Reducing and Mapping}), and a negative argument maps the next-to-top
10246 @var{n} stack elements with the top stack element as a second argument
10247 (for example, @kbd{7 c-u -5 +} adds 7 to the top 5 stack elements).
10248 This feature is not available for operations which use the numeric prefix
10249 argument for some other purpose.
10251 Numeric prefixes are specified the same way as always in Emacs: Press
10252 a sequence of @key{META}-digits, or press @key{ESC} followed by digits,
10253 or press @kbd{C-u} followed by digits. Some commands treat plain
10254 @kbd{C-u} (without any actual digits) specially.
10257 @pindex calc-num-prefix
10258 You can type @kbd{~} (@code{calc-num-prefix}) to pop an integer from the
10259 top of the stack and enter it as the numeric prefix for the next command.
10260 For example, @kbd{C-u 16 p} sets the precision to 16 digits; an alternate
10261 (silly) way to do this would be @kbd{2 @key{RET} 4 ^ ~ p}, i.e., compute 2
10262 to the fourth power and set the precision to that value.
10264 Conversely, if you have typed a numeric prefix argument the @kbd{~} key
10265 pushes it onto the stack in the form of an integer.
10267 @node Undo, Error Messages, Prefix Arguments, Introduction
10268 @section Undoing Mistakes
10274 @cindex Mistakes, undoing
10275 @cindex Undoing mistakes
10276 @cindex Errors, undoing
10277 The shift-@kbd{U} key (@code{calc-undo}) undoes the most recent operation.
10278 If that operation added or dropped objects from the stack, those objects
10279 are removed or restored. If it was a ``store'' operation, you are
10280 queried whether or not to restore the variable to its original value.
10281 The @kbd{U} key may be pressed any number of times to undo successively
10282 farther back in time; with a numeric prefix argument it undoes a
10283 specified number of operations. The undo history is cleared only by the
10284 @kbd{q} (@code{calc-quit}) command. (Recall that @kbd{C-x * c} is
10285 synonymous with @code{calc-quit} while inside the Calculator; this
10286 also clears the undo history.)
10288 Currently the mode-setting commands (like @code{calc-precision}) are not
10289 undoable. You can undo past a point where you changed a mode, but you
10290 will need to reset the mode yourself.
10294 @cindex Redoing after an Undo
10295 The shift-@kbd{D} key (@code{calc-redo}) redoes an operation that was
10296 mistakenly undone. Pressing @kbd{U} with a negative prefix argument is
10297 equivalent to executing @code{calc-redo}. You can redo any number of
10298 times, up to the number of recent consecutive undo commands. Redo
10299 information is cleared whenever you give any command that adds new undo
10300 information, i.e., if you undo, then enter a number on the stack or make
10301 any other change, then it will be too late to redo.
10303 @kindex M-@key{RET}
10304 @pindex calc-last-args
10305 @cindex Last-arguments feature
10306 @cindex Arguments, restoring
10307 The @kbd{M-@key{RET}} key (@code{calc-last-args}) is like undo in that
10308 it restores the arguments of the most recent command onto the stack;
10309 however, it does not remove the result of that command. Given a numeric
10310 prefix argument, this command applies to the @expr{n}th most recent
10311 command which removed items from the stack; it pushes those items back
10314 The @kbd{K} (@code{calc-keep-args}) command provides a related function
10315 to @kbd{M-@key{RET}}. @xref{Stack and Trail}.
10317 It is also possible to recall previous results or inputs using the trail.
10318 @xref{Trail Commands}.
10320 The standard Emacs @kbd{C-_} undo key is recognized as a synonym for @kbd{U}.
10322 @node Error Messages, Multiple Calculators, Undo, Introduction
10323 @section Error Messages
10328 @cindex Errors, messages
10329 @cindex Why did an error occur?
10330 Many situations that would produce an error message in other calculators
10331 simply create unsimplified formulas in the Emacs Calculator. For example,
10332 @kbd{1 @key{RET} 0 /} pushes the formula @expr{1 / 0}; @w{@kbd{0 L}} pushes
10333 the formula @samp{ln(0)}. Floating-point overflow and underflow are also
10334 reasons for this to happen.
10336 When a function call must be left in symbolic form, Calc usually
10337 produces a message explaining why. Messages that are probably
10338 surprising or indicative of user errors are displayed automatically.
10339 Other messages are simply kept in Calc's memory and are displayed only
10340 if you type @kbd{w} (@code{calc-why}). You can also press @kbd{w} if
10341 the same computation results in several messages. (The first message
10342 will end with @samp{[w=more]} in this case.)
10345 @pindex calc-auto-why
10346 The @kbd{d w} (@code{calc-auto-why}) command controls when error messages
10347 are displayed automatically. (Calc effectively presses @kbd{w} for you
10348 after your computation finishes.) By default, this occurs only for
10349 ``important'' messages. The other possible modes are to report
10350 @emph{all} messages automatically, or to report none automatically (so
10351 that you must always press @kbd{w} yourself to see the messages).
10353 @node Multiple Calculators, Troubleshooting Commands, Error Messages, Introduction
10354 @section Multiple Calculators
10357 @pindex another-calc
10358 It is possible to have any number of Calc mode buffers at once.
10359 Usually this is done by executing @kbd{M-x another-calc}, which
10360 is similar to @kbd{C-x * c} except that if a @samp{*Calculator*}
10361 buffer already exists, a new, independent one with a name of the
10362 form @samp{*Calculator*<@var{n}>} is created. You can also use the
10363 command @code{calc-mode} to put any buffer into Calculator mode, but
10364 this would ordinarily never be done.
10366 The @kbd{q} (@code{calc-quit}) command does not destroy a Calculator buffer;
10367 it only closes its window. Use @kbd{M-x kill-buffer} to destroy a
10370 Each Calculator buffer keeps its own stack, undo list, and mode settings
10371 such as precision, angular mode, and display formats. In Emacs terms,
10372 variables such as @code{calc-stack} are buffer-local variables. The
10373 global default values of these variables are used only when a new
10374 Calculator buffer is created. The @code{calc-quit} command saves
10375 the stack and mode settings of the buffer being quit as the new defaults.
10377 There is only one trail buffer, @samp{*Calc Trail*}, used by all
10378 Calculator buffers.
10380 @node Troubleshooting Commands, , Multiple Calculators, Introduction
10381 @section Troubleshooting Commands
10384 This section describes commands you can use in case a computation
10385 incorrectly fails or gives the wrong answer.
10387 @xref{Reporting Bugs}, if you find a problem that appears to be due
10388 to a bug or deficiency in Calc.
10391 * Autoloading Problems::
10392 * Recursion Depth::
10397 @node Autoloading Problems, Recursion Depth, Troubleshooting Commands, Troubleshooting Commands
10398 @subsection Autoloading Problems
10401 The Calc program is split into many component files; components are
10402 loaded automatically as you use various commands that require them.
10403 Occasionally Calc may lose track of when a certain component is
10404 necessary; typically this means you will type a command and it won't
10405 work because some function you've never heard of was undefined.
10408 @pindex calc-load-everything
10409 If this happens, the easiest workaround is to type @kbd{C-x * L}
10410 (@code{calc-load-everything}) to force all the parts of Calc to be
10411 loaded right away. This will cause Emacs to take up a lot more
10412 memory than it would otherwise, but it's guaranteed to fix the problem.
10414 @node Recursion Depth, Caches, Autoloading Problems, Troubleshooting Commands
10415 @subsection Recursion Depth
10420 @pindex calc-more-recursion-depth
10421 @pindex calc-less-recursion-depth
10422 @cindex Recursion depth
10423 @cindex ``Computation got stuck'' message
10424 @cindex @code{max-lisp-eval-depth}
10425 @cindex @code{max-specpdl-size}
10426 Calc uses recursion in many of its calculations. Emacs Lisp keeps a
10427 variable @code{max-lisp-eval-depth} which limits the amount of recursion
10428 possible in an attempt to recover from program bugs. If a calculation
10429 ever halts incorrectly with the message ``Computation got stuck or
10430 ran too long,'' use the @kbd{M} command (@code{calc-more-recursion-depth})
10431 to increase this limit. (Of course, this will not help if the
10432 calculation really did get stuck due to some problem inside Calc.)
10434 The limit is always increased (multiplied) by a factor of two. There
10435 is also an @kbd{I M} (@code{calc-less-recursion-depth}) command which
10436 decreases this limit by a factor of two, down to a minimum value of 200.
10437 The default value is 1000.
10439 These commands also double or halve @code{max-specpdl-size}, another
10440 internal Lisp recursion limit. The minimum value for this limit is 600.
10442 @node Caches, Debugging Calc, Recursion Depth, Troubleshooting Commands
10447 @cindex Flushing caches
10448 Calc saves certain values after they have been computed once. For
10449 example, the @kbd{P} (@code{calc-pi}) command initially ``knows'' the
10450 constant @cpi{} to about 20 decimal places; if the current precision
10451 is greater than this, it will recompute @cpi{} using a series
10452 approximation. This value will not need to be recomputed ever again
10453 unless you raise the precision still further. Many operations such as
10454 logarithms and sines make use of similarly cached values such as
10456 @texline @math{\ln 2}.
10457 @infoline @expr{ln(2)}.
10458 The visible effect of caching is that
10459 high-precision computations may seem to do extra work the first time.
10460 Other things cached include powers of two (for the binary arithmetic
10461 functions), matrix inverses and determinants, symbolic integrals, and
10462 data points computed by the graphing commands.
10464 @pindex calc-flush-caches
10465 If you suspect a Calculator cache has become corrupt, you can use the
10466 @code{calc-flush-caches} command to reset all caches to the empty state.
10467 (This should only be necessary in the event of bugs in the Calculator.)
10468 The @kbd{C-x * 0} (with the zero key) command also resets caches along
10469 with all other aspects of the Calculator's state.
10471 @node Debugging Calc, , Caches, Troubleshooting Commands
10472 @subsection Debugging Calc
10475 A few commands exist to help in the debugging of Calc commands.
10476 @xref{Programming}, to see the various ways that you can write
10477 your own Calc commands.
10480 @pindex calc-timing
10481 The @kbd{Z T} (@code{calc-timing}) command turns on and off a mode
10482 in which the timing of slow commands is reported in the Trail.
10483 Any Calc command that takes two seconds or longer writes a line
10484 to the Trail showing how many seconds it took. This value is
10485 accurate only to within one second.
10487 All steps of executing a command are included; in particular, time
10488 taken to format the result for display in the stack and trail is
10489 counted. Some prompts also count time taken waiting for them to
10490 be answered, while others do not; this depends on the exact
10491 implementation of the command. For best results, if you are timing
10492 a sequence that includes prompts or multiple commands, define a
10493 keyboard macro to run the whole sequence at once. Calc's @kbd{X}
10494 command (@pxref{Keyboard Macros}) will then report the time taken
10495 to execute the whole macro.
10497 Another advantage of the @kbd{X} command is that while it is
10498 executing, the stack and trail are not updated from step to step.
10499 So if you expect the output of your test sequence to leave a result
10500 that may take a long time to format and you don't wish to count
10501 this formatting time, end your sequence with a @key{DEL} keystroke
10502 to clear the result from the stack. When you run the sequence with
10503 @kbd{X}, Calc will never bother to format the large result.
10505 Another thing @kbd{Z T} does is to increase the Emacs variable
10506 @code{gc-cons-threshold} to a much higher value (two million; the
10507 usual default in Calc is 250,000) for the duration of each command.
10508 This generally prevents garbage collection during the timing of
10509 the command, though it may cause your Emacs process to grow
10510 abnormally large. (Garbage collection time is a major unpredictable
10511 factor in the timing of Emacs operations.)
10513 Another command that is useful when debugging your own Lisp
10514 extensions to Calc is @kbd{M-x calc-pass-errors}, which disables
10515 the error handler that changes the ``@code{max-lisp-eval-depth}
10516 exceeded'' message to the much more friendly ``Computation got
10517 stuck or ran too long.'' This handler interferes with the Emacs
10518 Lisp debugger's @code{debug-on-error} mode. Errors are reported
10519 in the handler itself rather than at the true location of the
10520 error. After you have executed @code{calc-pass-errors}, Lisp
10521 errors will be reported correctly but the user-friendly message
10524 @node Data Types, Stack and Trail, Introduction, Top
10525 @chapter Data Types
10528 This chapter discusses the various types of objects that can be placed
10529 on the Calculator stack, how they are displayed, and how they are
10530 entered. (@xref{Data Type Formats}, for information on how these data
10531 types are represented as underlying Lisp objects.)
10533 Integers, fractions, and floats are various ways of describing real
10534 numbers. HMS forms also for many purposes act as real numbers. These
10535 types can be combined to form complex numbers, modulo forms, error forms,
10536 or interval forms. (But these last four types cannot be combined
10537 arbitrarily:@: error forms may not contain modulo forms, for example.)
10538 Finally, all these types of numbers may be combined into vectors,
10539 matrices, or algebraic formulas.
10542 * Integers:: The most basic data type.
10543 * Fractions:: This and above are called @dfn{rationals}.
10544 * Floats:: This and above are called @dfn{reals}.
10545 * Complex Numbers:: This and above are called @dfn{numbers}.
10547 * Vectors and Matrices::
10554 * Incomplete Objects::
10559 @node Integers, Fractions, Data Types, Data Types
10564 The Calculator stores integers to arbitrary precision. Addition,
10565 subtraction, and multiplication of integers always yields an exact
10566 integer result. (If the result of a division or exponentiation of
10567 integers is not an integer, it is expressed in fractional or
10568 floating-point form according to the current Fraction mode.
10569 @xref{Fraction Mode}.)
10571 A decimal integer is represented as an optional sign followed by a
10572 sequence of digits. Grouping (@pxref{Grouping Digits}) can be used to
10573 insert a comma at every third digit for display purposes, but you
10574 must not type commas during the entry of numbers.
10577 A non-decimal integer is represented as an optional sign, a radix
10578 between 2 and 36, a @samp{#} symbol, and one or more digits. For radix 11
10579 and above, the letters A through Z (upper- or lower-case) count as
10580 digits and do not terminate numeric entry mode. @xref{Radix Modes}, for how
10581 to set the default radix for display of integers. Numbers of any radix
10582 may be entered at any time. If you press @kbd{#} at the beginning of a
10583 number, the current display radix is used.
10585 @node Fractions, Floats, Integers, Data Types
10590 A @dfn{fraction} is a ratio of two integers. Fractions are traditionally
10591 written ``2/3'' but Calc uses the notation @samp{2:3}. (The @kbd{/} key
10592 performs RPN division; the following two sequences push the number
10593 @samp{2:3} on the stack: @kbd{2 :@: 3 @key{RET}}, or @kbd{2 @key{RET} 3 /}
10594 assuming Fraction mode has been enabled.)
10595 When the Calculator produces a fractional result it always reduces it to
10596 simplest form, which may in fact be an integer.
10598 Fractions may also be entered in a three-part form, where @samp{2:3:4}
10599 represents two-and-three-quarters. @xref{Fraction Formats}, for fraction
10602 Non-decimal fractions are entered and displayed as
10603 @samp{@var{radix}#@var{num}:@var{denom}} (or in the analogous three-part
10604 form). The numerator and denominator always use the same radix.
10606 @node Floats, Complex Numbers, Fractions, Data Types
10610 @cindex Floating-point numbers
10611 A floating-point number or @dfn{float} is a number stored in scientific
10612 notation. The number of significant digits in the fractional part is
10613 governed by the current floating precision (@pxref{Precision}). The
10614 range of acceptable values is from
10615 @texline @math{10^{-3999999}}
10616 @infoline @expr{10^-3999999}
10618 @texline @math{10^{4000000}}
10619 @infoline @expr{10^4000000}
10620 (exclusive), plus the corresponding negative values and zero.
10622 Calculations that would exceed the allowable range of values (such
10623 as @samp{exp(exp(20))}) are left in symbolic form by Calc. The
10624 messages ``floating-point overflow'' or ``floating-point underflow''
10625 indicate that during the calculation a number would have been produced
10626 that was too large or too close to zero, respectively, to be represented
10627 by Calc. This does not necessarily mean the final result would have
10628 overflowed, just that an overflow occurred while computing the result.
10629 (In fact, it could report an underflow even though the final result
10630 would have overflowed!)
10632 If a rational number and a float are mixed in a calculation, the result
10633 will in general be expressed as a float. Commands that require an integer
10634 value (such as @kbd{k g} [@code{gcd}]) will also accept integer-valued
10635 floats, i.e., floating-point numbers with nothing after the decimal point.
10637 Floats are identified by the presence of a decimal point and/or an
10638 exponent. In general a float consists of an optional sign, digits
10639 including an optional decimal point, and an optional exponent consisting
10640 of an @samp{e}, an optional sign, and up to seven exponent digits.
10641 For example, @samp{23.5e-2} is 23.5 times ten to the minus-second power,
10644 Floating-point numbers are normally displayed in decimal notation with
10645 all significant figures shown. Exceedingly large or small numbers are
10646 displayed in scientific notation. Various other display options are
10647 available. @xref{Float Formats}.
10649 @cindex Accuracy of calculations
10650 Floating-point numbers are stored in decimal, not binary. The result
10651 of each operation is rounded to the nearest value representable in the
10652 number of significant digits specified by the current precision,
10653 rounding away from zero in the case of a tie. Thus (in the default
10654 display mode) what you see is exactly what you get. Some operations such
10655 as square roots and transcendental functions are performed with several
10656 digits of extra precision and then rounded down, in an effort to make the
10657 final result accurate to the full requested precision. However,
10658 accuracy is not rigorously guaranteed. If you suspect the validity of a
10659 result, try doing the same calculation in a higher precision. The
10660 Calculator's arithmetic is not intended to be IEEE-conformant in any
10663 While floats are always @emph{stored} in decimal, they can be entered
10664 and displayed in any radix just like integers and fractions. Since a
10665 float that is entered in a radix other that 10 will be converted to
10666 decimal, the number that Calc stores may not be exactly the number that
10667 was entered, it will be the closest decimal approximation given the
10668 current precison. The notation @samp{@var{radix}#@var{ddd}.@var{ddd}}
10669 is a floating-point number whose digits are in the specified radix.
10670 Note that the @samp{.} is more aptly referred to as a ``radix point''
10671 than as a decimal point in this case. The number @samp{8#123.4567} is
10672 defined as @samp{8#1234567 * 8^-4}. If the radix is 14 or less, you can
10673 use @samp{e} notation to write a non-decimal number in scientific
10674 notation. The exponent is written in decimal, and is considered to be a
10675 power of the radix: @samp{8#1234567e-4}. If the radix is 15 or above,
10676 the letter @samp{e} is a digit, so scientific notation must be written
10677 out, e.g., @samp{16#123.4567*16^2}. The first two exercises of the
10678 Modes Tutorial explore some of the properties of non-decimal floats.
10680 @node Complex Numbers, Infinities, Floats, Data Types
10681 @section Complex Numbers
10684 @cindex Complex numbers
10685 There are two supported formats for complex numbers: rectangular and
10686 polar. The default format is rectangular, displayed in the form
10687 @samp{(@var{real},@var{imag})} where @var{real} is the real part and
10688 @var{imag} is the imaginary part, each of which may be any real number.
10689 Rectangular complex numbers can also be displayed in @samp{@var{a}+@var{b}i}
10690 notation; @pxref{Complex Formats}.
10692 Polar complex numbers are displayed in the form
10693 @texline `@tfn{(}@var{r}@tfn{;}@math{\theta}@tfn{)}'
10694 @infoline `@tfn{(}@var{r}@tfn{;}@var{theta}@tfn{)}'
10695 where @var{r} is the nonnegative magnitude and
10696 @texline @math{\theta}
10697 @infoline @var{theta}
10698 is the argument or phase angle. The range of
10699 @texline @math{\theta}
10700 @infoline @var{theta}
10701 depends on the current angular mode (@pxref{Angular Modes}); it is
10702 generally between @mathit{-180} and @mathit{+180} degrees or the equivalent range
10705 Complex numbers are entered in stages using incomplete objects.
10706 @xref{Incomplete Objects}.
10708 Operations on rectangular complex numbers yield rectangular complex
10709 results, and similarly for polar complex numbers. Where the two types
10710 are mixed, or where new complex numbers arise (as for the square root of
10711 a negative real), the current @dfn{Polar mode} is used to determine the
10712 type. @xref{Polar Mode}.
10714 A complex result in which the imaginary part is zero (or the phase angle
10715 is 0 or 180 degrees or @cpi{} radians) is automatically converted to a real
10718 @node Infinities, Vectors and Matrices, Complex Numbers, Data Types
10719 @section Infinities
10723 @cindex @code{inf} variable
10724 @cindex @code{uinf} variable
10725 @cindex @code{nan} variable
10729 The word @code{inf} represents the mathematical concept of @dfn{infinity}.
10730 Calc actually has three slightly different infinity-like values:
10731 @code{inf}, @code{uinf}, and @code{nan}. These are just regular
10732 variable names (@pxref{Variables}); you should avoid using these
10733 names for your own variables because Calc gives them special
10734 treatment. Infinities, like all variable names, are normally
10735 entered using algebraic entry.
10737 Mathematically speaking, it is not rigorously correct to treat
10738 ``infinity'' as if it were a number, but mathematicians often do
10739 so informally. When they say that @samp{1 / inf = 0}, what they
10740 really mean is that @expr{1 / x}, as @expr{x} becomes larger and
10741 larger, becomes arbitrarily close to zero. So you can imagine
10742 that if @expr{x} got ``all the way to infinity,'' then @expr{1 / x}
10743 would go all the way to zero. Similarly, when they say that
10744 @samp{exp(inf) = inf}, they mean that
10745 @texline @math{e^x}
10746 @infoline @expr{exp(x)}
10747 grows without bound as @expr{x} grows. The symbol @samp{-inf} likewise
10748 stands for an infinitely negative real value; for example, we say that
10749 @samp{exp(-inf) = 0}. You can have an infinity pointing in any
10750 direction on the complex plane: @samp{sqrt(-inf) = i inf}.
10752 The same concept of limits can be used to define @expr{1 / 0}. We
10753 really want the value that @expr{1 / x} approaches as @expr{x}
10754 approaches zero. But if all we have is @expr{1 / 0}, we can't
10755 tell which direction @expr{x} was coming from. If @expr{x} was
10756 positive and decreasing toward zero, then we should say that
10757 @samp{1 / 0 = inf}. But if @expr{x} was negative and increasing
10758 toward zero, the answer is @samp{1 / 0 = -inf}. In fact, @expr{x}
10759 could be an imaginary number, giving the answer @samp{i inf} or
10760 @samp{-i inf}. Calc uses the special symbol @samp{uinf} to mean
10761 @dfn{undirected infinity}, i.e., a value which is infinitely
10762 large but with an unknown sign (or direction on the complex plane).
10764 Calc actually has three modes that say how infinities are handled.
10765 Normally, infinities never arise from calculations that didn't
10766 already have them. Thus, @expr{1 / 0} is treated simply as an
10767 error and left unevaluated. The @kbd{m i} (@code{calc-infinite-mode})
10768 command (@pxref{Infinite Mode}) enables a mode in which
10769 @expr{1 / 0} evaluates to @code{uinf} instead. There is also
10770 an alternative type of infinite mode which says to treat zeros
10771 as if they were positive, so that @samp{1 / 0 = inf}. While this
10772 is less mathematically correct, it may be the answer you want in
10775 Since all infinities are ``as large'' as all others, Calc simplifies,
10776 e.g., @samp{5 inf} to @samp{inf}. Another example is
10777 @samp{5 - inf = -inf}, where the @samp{-inf} is so large that
10778 adding a finite number like five to it does not affect it.
10779 Note that @samp{a - inf} also results in @samp{-inf}; Calc assumes
10780 that variables like @code{a} always stand for finite quantities.
10781 Just to show that infinities really are all the same size,
10782 note that @samp{sqrt(inf) = inf^2 = exp(inf) = inf} in Calc's
10785 It's not so easy to define certain formulas like @samp{0 * inf} and
10786 @samp{inf / inf}. Depending on where these zeros and infinities
10787 came from, the answer could be literally anything. The latter
10788 formula could be the limit of @expr{x / x} (giving a result of one),
10789 or @expr{2 x / x} (giving two), or @expr{x^2 / x} (giving @code{inf}),
10790 or @expr{x / x^2} (giving zero). Calc uses the symbol @code{nan}
10791 to represent such an @dfn{indeterminate} value. (The name ``nan''
10792 comes from analogy with the ``NAN'' concept of IEEE standard
10793 arithmetic; it stands for ``Not A Number.'' This is somewhat of a
10794 misnomer, since @code{nan} @emph{does} stand for some number or
10795 infinity, it's just that @emph{which} number it stands for
10796 cannot be determined.) In Calc's notation, @samp{0 * inf = nan}
10797 and @samp{inf / inf = nan}. A few other common indeterminate
10798 expressions are @samp{inf - inf} and @samp{inf ^ 0}. Also,
10799 @samp{0 / 0 = nan} if you have turned on Infinite mode
10800 (as described above).
10802 Infinities are especially useful as parts of @dfn{intervals}.
10803 @xref{Interval Forms}.
10805 @node Vectors and Matrices, Strings, Infinities, Data Types
10806 @section Vectors and Matrices
10810 @cindex Plain vectors
10812 The @dfn{vector} data type is flexible and general. A vector is simply a
10813 list of zero or more data objects. When these objects are numbers, the
10814 whole is a vector in the mathematical sense. When these objects are
10815 themselves vectors of equal (nonzero) length, the whole is a @dfn{matrix}.
10816 A vector which is not a matrix is referred to here as a @dfn{plain vector}.
10818 A vector is displayed as a list of values separated by commas and enclosed
10819 in square brackets: @samp{[1, 2, 3]}. Thus the following is a 2 row by
10820 3 column matrix: @samp{[[1, 2, 3], [4, 5, 6]]}. Vectors, like complex
10821 numbers, are entered as incomplete objects. @xref{Incomplete Objects}.
10822 During algebraic entry, vectors are entered all at once in the usual
10823 brackets-and-commas form. Matrices may be entered algebraically as nested
10824 vectors, or using the shortcut notation @w{@samp{[1, 2, 3; 4, 5, 6]}},
10825 with rows separated by semicolons. The commas may usually be omitted
10826 when entering vectors: @samp{[1 2 3]}. Curly braces may be used in
10827 place of brackets: @samp{@{1, 2, 3@}}, but the commas are required in
10830 Traditional vector and matrix arithmetic is also supported;
10831 @pxref{Basic Arithmetic} and @pxref{Matrix Functions}.
10832 Many other operations are applied to vectors element-wise. For example,
10833 the complex conjugate of a vector is a vector of the complex conjugates
10840 Algebraic functions for building vectors include @samp{vec(a, b, c)}
10841 to build @samp{[a, b, c]}, @samp{cvec(a, n, m)} to build an
10842 @texline @math{n\times m}
10843 @infoline @var{n}x@var{m}
10844 matrix of @samp{a}s, and @samp{index(n)} to build a vector of integers
10845 from 1 to @samp{n}.
10847 @node Strings, HMS Forms, Vectors and Matrices, Data Types
10853 @cindex Character strings
10854 Character strings are not a special data type in the Calculator.
10855 Rather, a string is represented simply as a vector all of whose
10856 elements are integers in the range 0 to 255 (ASCII codes). You can
10857 enter a string at any time by pressing the @kbd{"} key. Quotation
10858 marks and backslashes are written @samp{\"} and @samp{\\}, respectively,
10859 inside strings. Other notations introduced by backslashes are:
10875 Finally, a backslash followed by three octal digits produces any
10876 character from its ASCII code.
10879 @pindex calc-display-strings
10880 Strings are normally displayed in vector-of-integers form. The
10881 @w{@kbd{d "}} (@code{calc-display-strings}) command toggles a mode in
10882 which any vectors of small integers are displayed as quoted strings
10885 The backslash notations shown above are also used for displaying
10886 strings. Characters 128 and above are not translated by Calc; unless
10887 you have an Emacs modified for 8-bit fonts, these will show up in
10888 backslash-octal-digits notation. For characters below 32, and
10889 for character 127, Calc uses the backslash-letter combination if
10890 there is one, or otherwise uses a @samp{\^} sequence.
10892 The only Calc feature that uses strings is @dfn{compositions};
10893 @pxref{Compositions}. Strings also provide a convenient
10894 way to do conversions between ASCII characters and integers.
10900 There is a @code{string} function which provides a different display
10901 format for strings. Basically, @samp{string(@var{s})}, where @var{s}
10902 is a vector of integers in the proper range, is displayed as the
10903 corresponding string of characters with no surrounding quotation
10904 marks or other modifications. Thus @samp{string("ABC")} (or
10905 @samp{string([65 66 67])}) will look like @samp{ABC} on the stack.
10906 This happens regardless of whether @w{@kbd{d "}} has been used. The
10907 only way to turn it off is to use @kbd{d U} (unformatted language
10908 mode) which will display @samp{string("ABC")} instead.
10910 Control characters are displayed somewhat differently by @code{string}.
10911 Characters below 32, and character 127, are shown using @samp{^} notation
10912 (same as shown above, but without the backslash). The quote and
10913 backslash characters are left alone, as are characters 128 and above.
10919 The @code{bstring} function is just like @code{string} except that
10920 the resulting string is breakable across multiple lines if it doesn't
10921 fit all on one line. Potential break points occur at every space
10922 character in the string.
10924 @node HMS Forms, Date Forms, Strings, Data Types
10928 @cindex Hours-minutes-seconds forms
10929 @cindex Degrees-minutes-seconds forms
10930 @dfn{HMS} stands for Hours-Minutes-Seconds; when used as an angular
10931 argument, the interpretation is Degrees-Minutes-Seconds. All functions
10932 that operate on angles accept HMS forms. These are interpreted as
10933 degrees regardless of the current angular mode. It is also possible to
10934 use HMS as the angular mode so that calculated angles are expressed in
10935 degrees, minutes, and seconds.
10941 @kindex ' (HMS forms)
10945 @kindex " (HMS forms)
10949 @kindex h (HMS forms)
10953 @kindex o (HMS forms)
10957 @kindex m (HMS forms)
10961 @kindex s (HMS forms)
10962 The default format for HMS values is
10963 @samp{@var{hours}@@ @var{mins}' @var{secs}"}. During entry, the letters
10964 @samp{h} (for ``hours'') or
10965 @samp{o} (approximating the ``degrees'' symbol) are accepted as well as
10966 @samp{@@}, @samp{m} is accepted in place of @samp{'}, and @samp{s} is
10967 accepted in place of @samp{"}.
10968 The @var{hours} value is an integer (or integer-valued float).
10969 The @var{mins} value is an integer or integer-valued float between 0 and 59.
10970 The @var{secs} value is a real number between 0 (inclusive) and 60
10971 (exclusive). A positive HMS form is interpreted as @var{hours} +
10972 @var{mins}/60 + @var{secs}/3600. A negative HMS form is interpreted
10973 as @mathit{- @var{hours}} @mathit{-} @var{mins}/60 @mathit{-} @var{secs}/3600.
10974 Display format for HMS forms is quite flexible. @xref{HMS Formats}.
10976 HMS forms can be added and subtracted. When they are added to numbers,
10977 the numbers are interpreted according to the current angular mode. HMS
10978 forms can also be multiplied and divided by real numbers. Dividing
10979 two HMS forms produces a real-valued ratio of the two angles.
10982 @cindex Time of day
10983 Just for kicks, @kbd{M-x calc-time} pushes the current time of day on
10984 the stack as an HMS form.
10986 @node Date Forms, Modulo Forms, HMS Forms, Data Types
10987 @section Date Forms
10991 A @dfn{date form} represents a date and possibly an associated time.
10992 Simple date arithmetic is supported: Adding a number to a date
10993 produces a new date shifted by that many days; adding an HMS form to
10994 a date shifts it by that many hours. Subtracting two date forms
10995 computes the number of days between them (represented as a simple
10996 number). Many other operations, such as multiplying two date forms,
10997 are nonsensical and are not allowed by Calc.
10999 Date forms are entered and displayed enclosed in @samp{< >} brackets.
11000 The default format is, e.g., @samp{<Wed Jan 9, 1991>} for dates,
11001 or @samp{<3:32:20pm Wed Jan 9, 1991>} for dates with times.
11002 Input is flexible; date forms can be entered in any of the usual
11003 notations for dates and times. @xref{Date Formats}.
11005 Date forms are stored internally as numbers, specifically the number
11006 of days since midnight on the morning of January 1 of the year 1 AD.
11007 If the internal number is an integer, the form represents a date only;
11008 if the internal number is a fraction or float, the form represents
11009 a date and time. For example, @samp{<6:00am Wed Jan 9, 1991>}
11010 is represented by the number 726842.25. The standard precision of
11011 12 decimal digits is enough to ensure that a (reasonable) date and
11012 time can be stored without roundoff error.
11014 If the current precision is greater than 12, date forms will keep
11015 additional digits in the seconds position. For example, if the
11016 precision is 15, the seconds will keep three digits after the
11017 decimal point. Decreasing the precision below 12 may cause the
11018 time part of a date form to become inaccurate. This can also happen
11019 if astronomically high years are used, though this will not be an
11020 issue in everyday (or even everymillennium) use. Note that date
11021 forms without times are stored as exact integers, so roundoff is
11022 never an issue for them.
11024 You can use the @kbd{v p} (@code{calc-pack}) and @kbd{v u}
11025 (@code{calc-unpack}) commands to get at the numerical representation
11026 of a date form. @xref{Packing and Unpacking}.
11028 Date forms can go arbitrarily far into the future or past. Negative
11029 year numbers represent years BC. Calc uses a combination of the
11030 Gregorian and Julian calendars, following the history of Great
11031 Britain and the British colonies. This is the same calendar that
11032 is used by the @code{cal} program in most Unix implementations.
11034 @cindex Julian calendar
11035 @cindex Gregorian calendar
11036 Some historical background: The Julian calendar was created by
11037 Julius Caesar in the year 46 BC as an attempt to fix the gradual
11038 drift caused by the lack of leap years in the calendar used
11039 until that time. The Julian calendar introduced an extra day in
11040 all years divisible by four. After some initial confusion, the
11041 calendar was adopted around the year we call 8 AD. Some centuries
11042 later it became apparent that the Julian year of 365.25 days was
11043 itself not quite right. In 1582 Pope Gregory XIII introduced the
11044 Gregorian calendar, which added the new rule that years divisible
11045 by 100, but not by 400, were not to be considered leap years
11046 despite being divisible by four. Many countries delayed adoption
11047 of the Gregorian calendar because of religious differences;
11048 in Britain it was put off until the year 1752, by which time
11049 the Julian calendar had fallen eleven days behind the true
11050 seasons. So the switch to the Gregorian calendar in early
11051 September 1752 introduced a discontinuity: The day after
11052 Sep 2, 1752 is Sep 14, 1752. Calc follows this convention.
11053 To take another example, Russia waited until 1918 before
11054 adopting the new calendar, and thus needed to remove thirteen
11055 days (between Feb 1, 1918 and Feb 14, 1918). This means that
11056 Calc's reckoning will be inconsistent with Russian history between
11057 1752 and 1918, and similarly for various other countries.
11059 Today's timekeepers introduce an occasional ``leap second'' as
11060 well, but Calc does not take these minor effects into account.
11061 (If it did, it would have to report a non-integer number of days
11062 between, say, @samp{<12:00am Mon Jan 1, 1900>} and
11063 @samp{<12:00am Sat Jan 1, 2000>}.)
11065 Calc uses the Julian calendar for all dates before the year 1752,
11066 including dates BC when the Julian calendar technically had not
11067 yet been invented. Thus the claim that day number @mathit{-10000} is
11068 called ``August 16, 28 BC'' should be taken with a grain of salt.
11070 Please note that there is no ``year 0''; the day before
11071 @samp{<Sat Jan 1, +1>} is @samp{<Fri Dec 31, -1>}. These are
11072 days 0 and @mathit{-1} respectively in Calc's internal numbering scheme.
11074 @cindex Julian day counting
11075 Another day counting system in common use is, confusingly, also called
11076 ``Julian.'' The Julian day number is the numbers of days since
11077 12:00 noon (GMT) on Jan 1, 4713 BC, which in Calc's scheme (in GMT)
11078 is @mathit{-1721423.5} (recall that Calc starts at midnight instead
11079 of noon). Thus to convert a Calc date code obtained by unpacking a
11080 date form into a Julian day number, simply add 1721423.5 after
11081 compensating for the time zone difference. The built-in @kbd{t J}
11082 command performs this conversion for you.
11084 The Julian day number is based on the Julian cycle, which was invented
11085 in 1583 by Joseph Justus Scaliger. Scaliger named it the Julian cycle
11086 since it is involves the Julian calendar, but some have suggested that
11087 Scaliger named it in honor of his father, Julius Caesar Scaliger. The
11088 Julian cycle is based it on three other cycles: the indiction cycle,
11089 the Metonic cycle, and the solar cycle. The indiction cycle is a 15
11090 year cycle originally used by the Romans for tax purposes but later
11091 used to date medieval documents. The Metonic cycle is a 19 year
11092 cycle; 19 years is close to being a common multiple of a solar year
11093 and a lunar month, and so every 19 years the phases of the moon will
11094 occur on the same days of the year. The solar cycle is a 28 year
11095 cycle; the Julian calendar repeats itself every 28 years. The
11096 smallest time period which contains multiples of all three cycles is
11097 the least common multiple of 15 years, 19 years and 28 years, which
11098 (since they're pairwise relatively prime) is
11099 @texline @math{15\times 19\times 28 = 7980} years.
11100 @infoline 15*19*28 = 7980 years.
11101 This is the length of a Julian cycle. Working backwards, the previous
11102 year in which all three cycles began was 4713 BC, and so Scalinger
11103 chose that year as the beginning of a Julian cycle. Since at the time
11104 there were no historical records from before 4713 BC, using this year
11105 as a starting point had the advantage of avoiding negative year
11106 numbers. In 1849, the astronomer John Herschel (son of William
11107 Herschel) suggested using the number of days since the beginning of
11108 the Julian cycle as an astronomical dating system; this idea was taken
11109 up by other astronomers. (At the time, noon was the start of the
11110 astronomical day. Herschel originally suggested counting the days
11111 since Jan 1, 4713 BC at noon Alexandria time; this was later amended to
11112 noon GMT.) Julian day numbering is largely used in astronomy.
11114 @cindex Unix time format
11115 The Unix operating system measures time as an integer number of
11116 seconds since midnight, Jan 1, 1970. To convert a Calc date
11117 value into a Unix time stamp, first subtract 719164 (the code
11118 for @samp{<Jan 1, 1970>}), then multiply by 86400 (the number of
11119 seconds in a day) and press @kbd{R} to round to the nearest
11120 integer. If you have a date form, you can simply subtract the
11121 day @samp{<Jan 1, 1970>} instead of unpacking and subtracting
11122 719164. Likewise, divide by 86400 and add @samp{<Jan 1, 1970>}
11123 to convert from Unix time to a Calc date form. (Note that
11124 Unix normally maintains the time in the GMT time zone; you may
11125 need to subtract five hours to get New York time, or eight hours
11126 for California time. The same is usually true of Julian day
11127 counts.) The built-in @kbd{t U} command performs these
11130 @node Modulo Forms, Error Forms, Date Forms, Data Types
11131 @section Modulo Forms
11134 @cindex Modulo forms
11135 A @dfn{modulo form} is a real number which is taken modulo (i.e., within
11136 an integer multiple of) some value @var{M}. Arithmetic modulo @var{M}
11137 often arises in number theory. Modulo forms are written
11138 `@var{a} @tfn{mod} @var{M}',
11139 where @var{a} and @var{M} are real numbers or HMS forms, and
11140 @texline @math{0 \le a < M}.
11141 @infoline @expr{0 <= a < @var{M}}.
11142 In many applications @expr{a} and @expr{M} will be
11143 integers but this is not required.
11148 @kindex M (modulo forms)
11152 @tindex mod (operator)
11153 To create a modulo form during numeric entry, press the shift-@kbd{M}
11154 key to enter the word @samp{mod}. As a special convenience, pressing
11155 shift-@kbd{M} a second time automatically enters the value of @expr{M}
11156 that was most recently used before. During algebraic entry, either
11157 type @samp{mod} by hand or press @kbd{M-m} (that's @kbd{@key{META}-m}).
11158 Once again, pressing this a second time enters the current modulo.
11160 Modulo forms are not to be confused with the modulo operator @samp{%}.
11161 The expression @samp{27 % 10} means to compute 27 modulo 10 to produce
11162 the result 7. Further computations treat this 7 as just a regular integer.
11163 The expression @samp{27 mod 10} produces the result @samp{7 mod 10};
11164 further computations with this value are again reduced modulo 10 so that
11165 the result always lies in the desired range.
11167 When two modulo forms with identical @expr{M}'s are added or multiplied,
11168 the Calculator simply adds or multiplies the values, then reduces modulo
11169 @expr{M}. If one argument is a modulo form and the other a plain number,
11170 the plain number is treated like a compatible modulo form. It is also
11171 possible to raise modulo forms to powers; the result is the value raised
11172 to the power, then reduced modulo @expr{M}. (When all values involved
11173 are integers, this calculation is done much more efficiently than
11174 actually computing the power and then reducing.)
11176 @cindex Modulo division
11177 Two modulo forms `@var{a} @tfn{mod} @var{M}' and `@var{b} @tfn{mod} @var{M}'
11178 can be divided if @expr{a}, @expr{b}, and @expr{M} are all
11179 integers. The result is the modulo form which, when multiplied by
11180 `@var{b} @tfn{mod} @var{M}', produces `@var{a} @tfn{mod} @var{M}'. If
11181 there is no solution to this equation (which can happen only when
11182 @expr{M} is non-prime), or if any of the arguments are non-integers, the
11183 division is left in symbolic form. Other operations, such as square
11184 roots, are not yet supported for modulo forms. (Note that, although
11185 @w{`@tfn{(}@var{a} @tfn{mod} @var{M}@tfn{)^.5}'} will compute a ``modulo square root''
11186 in the sense of reducing
11187 @texline @math{\sqrt a}
11188 @infoline @expr{sqrt(a)}
11189 modulo @expr{M}, this is not a useful definition from the
11190 number-theoretical point of view.)
11192 It is possible to mix HMS forms and modulo forms. For example, an
11193 HMS form modulo 24 could be used to manipulate clock times; an HMS
11194 form modulo 360 would be suitable for angles. Making the modulo @expr{M}
11195 also be an HMS form eliminates troubles that would arise if the angular
11196 mode were inadvertently set to Radians, in which case
11197 @w{@samp{2@@ 0' 0" mod 24}} would be interpreted as two degrees modulo
11200 Modulo forms cannot have variables or formulas for components. If you
11201 enter the formula @samp{(x + 2) mod 5}, Calc propagates the modulus
11202 to each of the coefficients: @samp{(1 mod 5) x + (2 mod 5)}.
11204 You can use @kbd{v p} and @kbd{%} to modify modulo forms.
11205 @xref{Packing and Unpacking}. @xref{Basic Arithmetic}.
11211 The algebraic function @samp{makemod(a, m)} builds the modulo form
11212 @w{@samp{a mod m}}.
11214 @node Error Forms, Interval Forms, Modulo Forms, Data Types
11215 @section Error Forms
11218 @cindex Error forms
11219 @cindex Standard deviations
11220 An @dfn{error form} is a number with an associated standard
11221 deviation, as in @samp{2.3 +/- 0.12}. The notation
11222 @texline `@var{x} @tfn{+/-} @math{\sigma}'
11223 @infoline `@var{x} @tfn{+/-} sigma'
11224 stands for an uncertain value which follows
11225 a normal or Gaussian distribution of mean @expr{x} and standard
11226 deviation or ``error''
11227 @texline @math{\sigma}.
11228 @infoline @expr{sigma}.
11229 Both the mean and the error can be either numbers or
11230 formulas. Generally these are real numbers but the mean may also be
11231 complex. If the error is negative or complex, it is changed to its
11232 absolute value. An error form with zero error is converted to a
11233 regular number by the Calculator.
11235 All arithmetic and transcendental functions accept error forms as input.
11236 Operations on the mean-value part work just like operations on regular
11237 numbers. The error part for any function @expr{f(x)} (such as
11238 @texline @math{\sin x}
11239 @infoline @expr{sin(x)})
11240 is defined by the error of @expr{x} times the derivative of @expr{f}
11241 evaluated at the mean value of @expr{x}. For a two-argument function
11242 @expr{f(x,y)} (such as addition) the error is the square root of the sum
11243 of the squares of the errors due to @expr{x} and @expr{y}.
11246 f(x \hbox{\code{ +/- }} \sigma)
11247 &= f(x) \hbox{\code{ +/- }} \sigma \left| {df(x) \over dx} \right| \cr
11248 f(x \hbox{\code{ +/- }} \sigma_x, y \hbox{\code{ +/- }} \sigma_y)
11249 &= f(x,y) \hbox{\code{ +/- }}
11250 \sqrt{\left(\sigma_x \left| {\partial f(x,y) \over \partial x}
11252 +\left(\sigma_y \left| {\partial f(x,y) \over \partial y}
11253 \right| \right)^2 } \cr
11257 definition assumes the errors in @expr{x} and @expr{y} are uncorrelated.
11258 A side effect of this definition is that @samp{(2 +/- 1) * (2 +/- 1)}
11259 is not the same as @samp{(2 +/- 1)^2}; the former represents the product
11260 of two independent values which happen to have the same probability
11261 distributions, and the latter is the product of one random value with itself.
11262 The former will produce an answer with less error, since on the average
11263 the two independent errors can be expected to cancel out.
11265 Consult a good text on error analysis for a discussion of the proper use
11266 of standard deviations. Actual errors often are neither Gaussian-distributed
11267 nor uncorrelated, and the above formulas are valid only when errors
11268 are small. As an example, the error arising from
11269 @texline `@tfn{sin(}@var{x} @tfn{+/-} @math{\sigma}@tfn{)}'
11270 @infoline `@tfn{sin(}@var{x} @tfn{+/-} @var{sigma}@tfn{)}'
11272 @texline `@math{\sigma} @tfn{abs(cos(}@var{x}@tfn{))}'.
11273 @infoline `@var{sigma} @tfn{abs(cos(}@var{x}@tfn{))}'.
11274 When @expr{x} is close to zero,
11275 @texline @math{\cos x}
11276 @infoline @expr{cos(x)}
11277 is close to one so the error in the sine is close to
11278 @texline @math{\sigma};
11279 @infoline @expr{sigma};
11280 this makes sense, since
11281 @texline @math{\sin x}
11282 @infoline @expr{sin(x)}
11283 is approximately @expr{x} near zero, so a given error in @expr{x} will
11284 produce about the same error in the sine. Likewise, near 90 degrees
11285 @texline @math{\cos x}
11286 @infoline @expr{cos(x)}
11287 is nearly zero and so the computed error is
11288 small: The sine curve is nearly flat in that region, so an error in @expr{x}
11289 has relatively little effect on the value of
11290 @texline @math{\sin x}.
11291 @infoline @expr{sin(x)}.
11292 However, consider @samp{sin(90 +/- 1000)}. The cosine of 90 is zero, so
11293 Calc will report zero error! We get an obviously wrong result because
11294 we have violated the small-error approximation underlying the error
11295 analysis. If the error in @expr{x} had been small, the error in
11296 @texline @math{\sin x}
11297 @infoline @expr{sin(x)}
11298 would indeed have been negligible.
11303 @kindex p (error forms)
11305 To enter an error form during regular numeric entry, use the @kbd{p}
11306 (``plus-or-minus'') key to type the @samp{+/-} symbol. (If you try actually
11307 typing @samp{+/-} the @kbd{+} key will be interpreted as the Calculator's
11308 @kbd{+} command!) Within an algebraic formula, you can press @kbd{M-+} to
11309 type the @samp{+/-} symbol, or type it out by hand.
11311 Error forms and complex numbers can be mixed; the formulas shown above
11312 are used for complex numbers, too; note that if the error part evaluates
11313 to a complex number its absolute value (or the square root of the sum of
11314 the squares of the absolute values of the two error contributions) is
11315 used. Mathematically, this corresponds to a radially symmetric Gaussian
11316 distribution of numbers on the complex plane. However, note that Calc
11317 considers an error form with real components to represent a real number,
11318 not a complex distribution around a real mean.
11320 Error forms may also be composed of HMS forms. For best results, both
11321 the mean and the error should be HMS forms if either one is.
11327 The algebraic function @samp{sdev(a, b)} builds the error form @samp{a +/- b}.
11329 @node Interval Forms, Incomplete Objects, Error Forms, Data Types
11330 @section Interval Forms
11333 @cindex Interval forms
11334 An @dfn{interval} is a subset of consecutive real numbers. For example,
11335 the interval @samp{[2 ..@: 4]} represents all the numbers from 2 to 4,
11336 inclusive. If you multiply it by the interval @samp{[0.5 ..@: 2]} you
11337 obtain @samp{[1 ..@: 8]}. This calculation represents the fact that if
11338 you multiply some number in the range @samp{[2 ..@: 4]} by some other
11339 number in the range @samp{[0.5 ..@: 2]}, your result will lie in the range
11340 from 1 to 8. Interval arithmetic is used to get a worst-case estimate
11341 of the possible range of values a computation will produce, given the
11342 set of possible values of the input.
11345 Calc supports several varieties of intervals, including @dfn{closed}
11346 intervals of the type shown above, @dfn{open} intervals such as
11347 @samp{(2 ..@: 4)}, which represents the range of numbers from 2 to 4
11348 @emph{exclusive}, and @dfn{semi-open} intervals in which one end
11349 uses a round parenthesis and the other a square bracket. In mathematical
11351 @samp{[2 ..@: 4]} means @expr{2 <= x <= 4}, whereas
11352 @samp{[2 ..@: 4)} represents @expr{2 <= x < 4},
11353 @samp{(2 ..@: 4]} represents @expr{2 < x <= 4}, and
11354 @samp{(2 ..@: 4)} represents @expr{2 < x < 4}.
11357 Calc supports several varieties of intervals, including \dfn{closed}
11358 intervals of the type shown above, \dfn{open} intervals such as
11359 \samp{(2 ..\: 4)}, which represents the range of numbers from 2 to 4
11360 \emph{exclusive}, and \dfn{semi-open} intervals in which one end
11361 uses a round parenthesis and the other a square bracket. In mathematical
11364 [2 \hbox{\cite{..}} 4] &\quad\hbox{means}\quad 2 \le x \le 4 \cr
11365 [2 \hbox{\cite{..}} 4) &\quad\hbox{means}\quad 2 \le x < 4 \cr
11366 (2 \hbox{\cite{..}} 4] &\quad\hbox{means}\quad 2 < x \le 4 \cr
11367 (2 \hbox{\cite{..}} 4) &\quad\hbox{means}\quad 2 < x < 4 \cr
11371 The lower and upper limits of an interval must be either real numbers
11372 (or HMS or date forms), or symbolic expressions which are assumed to be
11373 real-valued, or @samp{-inf} and @samp{inf}. In general the lower limit
11374 must be less than the upper limit. A closed interval containing only
11375 one value, @samp{[3 ..@: 3]}, is converted to a plain number (3)
11376 automatically. An interval containing no values at all (such as
11377 @samp{[3 ..@: 2]} or @samp{[2 ..@: 2)}) can be represented but is not
11378 guaranteed to behave well when used in arithmetic. Note that the
11379 interval @samp{[3 .. inf)} represents all real numbers greater than
11380 or equal to 3, and @samp{(-inf .. inf)} represents all real numbers.
11381 In fact, @samp{[-inf .. inf]} represents all real numbers including
11382 the real infinities.
11384 Intervals are entered in the notation shown here, either as algebraic
11385 formulas, or using incomplete forms. (@xref{Incomplete Objects}.)
11386 In algebraic formulas, multiple periods in a row are collected from
11387 left to right, so that @samp{1...1e2} is interpreted as @samp{1.0 ..@: 1e2}
11388 rather than @samp{1 ..@: 0.1e2}. Add spaces or zeros if you want to
11389 get the other interpretation. If you omit the lower or upper limit,
11390 a default of @samp{-inf} or @samp{inf} (respectively) is furnished.
11392 Infinite mode also affects operations on intervals
11393 (@pxref{Infinities}). Calc will always introduce an open infinity,
11394 as in @samp{1 / (0 .. 2] = [0.5 .. inf)}. But closed infinities,
11395 @w{@samp{1 / [0 .. 2] = [0.5 .. inf]}}, arise only in Infinite mode;
11396 otherwise they are left unevaluated. Note that the ``direction'' of
11397 a zero is not an issue in this case since the zero is always assumed
11398 to be continuous with the rest of the interval. For intervals that
11399 contain zero inside them Calc is forced to give the result,
11400 @samp{1 / (-2 .. 2) = [-inf .. inf]}.
11402 While it may seem that intervals and error forms are similar, they are
11403 based on entirely different concepts of inexact quantities. An error
11405 @texline `@var{x} @tfn{+/-} @math{\sigma}'
11406 @infoline `@var{x} @tfn{+/-} @var{sigma}'
11407 means a variable is random, and its value could
11408 be anything but is ``probably'' within one
11409 @texline @math{\sigma}
11410 @infoline @var{sigma}
11411 of the mean value @expr{x}. An interval
11412 `@tfn{[}@var{a} @tfn{..@:} @var{b}@tfn{]}' means a
11413 variable's value is unknown, but guaranteed to lie in the specified
11414 range. Error forms are statistical or ``average case'' approximations;
11415 interval arithmetic tends to produce ``worst case'' bounds on an
11418 Intervals may not contain complex numbers, but they may contain
11419 HMS forms or date forms.
11421 @xref{Set Operations}, for commands that interpret interval forms
11422 as subsets of the set of real numbers.
11428 The algebraic function @samp{intv(n, a, b)} builds an interval form
11429 from @samp{a} to @samp{b}; @samp{n} is an integer code which must
11430 be 0 for @samp{(..)}, 1 for @samp{(..]}, 2 for @samp{[..)}, or
11433 Please note that in fully rigorous interval arithmetic, care would be
11434 taken to make sure that the computation of the lower bound rounds toward
11435 minus infinity, while upper bound computations round toward plus
11436 infinity. Calc's arithmetic always uses a round-to-nearest mode,
11437 which means that roundoff errors could creep into an interval
11438 calculation to produce intervals slightly smaller than they ought to
11439 be. For example, entering @samp{[1..2]} and pressing @kbd{Q 2 ^}
11440 should yield the interval @samp{[1..2]} again, but in fact it yields the
11441 (slightly too small) interval @samp{[1..1.9999999]} due to roundoff
11444 @node Incomplete Objects, Variables, Interval Forms, Data Types
11445 @section Incomplete Objects
11465 @cindex Incomplete vectors
11466 @cindex Incomplete complex numbers
11467 @cindex Incomplete interval forms
11468 When @kbd{(} or @kbd{[} is typed to begin entering a complex number or
11469 vector, respectively, the effect is to push an @dfn{incomplete} complex
11470 number or vector onto the stack. The @kbd{,} key adds the value(s) at
11471 the top of the stack onto the current incomplete object. The @kbd{)}
11472 and @kbd{]} keys ``close'' the incomplete object after adding any values
11473 on the top of the stack in front of the incomplete object.
11475 As a result, the sequence of keystrokes @kbd{[ 2 , 3 @key{RET} 2 * , 9 ]}
11476 pushes the vector @samp{[2, 6, 9]} onto the stack. Likewise, @kbd{( 1 , 2 Q )}
11477 pushes the complex number @samp{(1, 1.414)} (approximately).
11479 If several values lie on the stack in front of the incomplete object,
11480 all are collected and appended to the object. Thus the @kbd{,} key
11481 is redundant: @kbd{[ 2 @key{RET} 3 @key{RET} 2 * 9 ]}. Some people
11482 prefer the equivalent @key{SPC} key to @key{RET}.
11484 As a special case, typing @kbd{,} immediately after @kbd{(}, @kbd{[}, or
11485 @kbd{,} adds a zero or duplicates the preceding value in the list being
11486 formed. Typing @key{DEL} during incomplete entry removes the last item
11490 The @kbd{;} key is used in the same way as @kbd{,} to create polar complex
11491 numbers: @kbd{( 1 ; 2 )}. When entering a vector, @kbd{;} is useful for
11492 creating a matrix. In particular, @kbd{[ [ 1 , 2 ; 3 , 4 ; 5 , 6 ] ]} is
11493 equivalent to @kbd{[ [ 1 , 2 ] , [ 3 , 4 ] , [ 5 , 6 ] ]}.
11497 Incomplete entry is also used to enter intervals. For example,
11498 @kbd{[ 2 ..@: 4 )} enters a semi-open interval. Note that when you type
11499 the first period, it will be interpreted as a decimal point, but when
11500 you type a second period immediately afterward, it is re-interpreted as
11501 part of the interval symbol. Typing @kbd{..} corresponds to executing
11502 the @code{calc-dots} command.
11504 If you find incomplete entry distracting, you may wish to enter vectors
11505 and complex numbers as algebraic formulas by pressing the apostrophe key.
11507 @node Variables, Formulas, Incomplete Objects, Data Types
11511 @cindex Variables, in formulas
11512 A @dfn{variable} is somewhere between a storage register on a conventional
11513 calculator, and a variable in a programming language. (In fact, a Calc
11514 variable is really just an Emacs Lisp variable that contains a Calc number
11515 or formula.) A variable's name is normally composed of letters and digits.
11516 Calc also allows apostrophes and @code{#} signs in variable names.
11517 (The Calc variable @code{foo} corresponds to the Emacs Lisp variable
11518 @code{var-foo}, but unless you access the variable from within Emacs
11519 Lisp, you don't need to worry about it. Variable names in algebraic
11520 formulas implicitly have @samp{var-} prefixed to their names. The
11521 @samp{#} character in variable names used in algebraic formulas
11522 corresponds to a dash @samp{-} in the Lisp variable name. If the name
11523 contains any dashes, the prefix @samp{var-} is @emph{not} automatically
11524 added. Thus the two formulas @samp{foo + 1} and @samp{var#foo + 1} both
11525 refer to the same variable.)
11527 In a command that takes a variable name, you can either type the full
11528 name of a variable, or type a single digit to use one of the special
11529 convenience variables @code{q0} through @code{q9}. For example,
11530 @kbd{3 s s 2} stores the number 3 in variable @code{q2}, and
11531 @w{@kbd{3 s s foo @key{RET}}} stores that number in variable
11534 To push a variable itself (as opposed to the variable's value) on the
11535 stack, enter its name as an algebraic expression using the apostrophe
11539 @pindex calc-evaluate
11540 @cindex Evaluation of variables in a formula
11541 @cindex Variables, evaluation
11542 @cindex Formulas, evaluation
11543 The @kbd{=} (@code{calc-evaluate}) key ``evaluates'' a formula by
11544 replacing all variables in the formula which have been given values by a
11545 @code{calc-store} or @code{calc-let} command by their stored values.
11546 Other variables are left alone. Thus a variable that has not been
11547 stored acts like an abstract variable in algebra; a variable that has
11548 been stored acts more like a register in a traditional calculator.
11549 With a positive numeric prefix argument, @kbd{=} evaluates the top
11550 @var{n} stack entries; with a negative argument, @kbd{=} evaluates
11551 the @var{n}th stack entry.
11553 @cindex @code{e} variable
11554 @cindex @code{pi} variable
11555 @cindex @code{i} variable
11556 @cindex @code{phi} variable
11557 @cindex @code{gamma} variable
11563 A few variables are called @dfn{special constants}. Their names are
11564 @samp{e}, @samp{pi}, @samp{i}, @samp{phi}, and @samp{gamma}.
11565 (@xref{Scientific Functions}.) When they are evaluated with @kbd{=},
11566 their values are calculated if necessary according to the current precision
11567 or complex polar mode. If you wish to use these symbols for other purposes,
11568 simply undefine or redefine them using @code{calc-store}.
11570 The variables @samp{inf}, @samp{uinf}, and @samp{nan} stand for
11571 infinite or indeterminate values. It's best not to use them as
11572 regular variables, since Calc uses special algebraic rules when
11573 it manipulates them. Calc displays a warning message if you store
11574 a value into any of these special variables.
11576 @xref{Store and Recall}, for a discussion of commands dealing with variables.
11578 @node Formulas, , Variables, Data Types
11583 @cindex Expressions
11584 @cindex Operators in formulas
11585 @cindex Precedence of operators
11586 When you press the apostrophe key you may enter any expression or formula
11587 in algebraic form. (Calc uses the terms ``expression'' and ``formula''
11588 interchangeably.) An expression is built up of numbers, variable names,
11589 and function calls, combined with various arithmetic operators.
11591 be used to indicate grouping. Spaces are ignored within formulas, except
11592 that spaces are not permitted within variable names or numbers.
11593 Arithmetic operators, in order from highest to lowest precedence, and
11594 with their equivalent function names, are:
11596 @samp{_} [@code{subscr}] (subscripts);
11598 postfix @samp{%} [@code{percent}] (as in @samp{25% = 0.25});
11600 prefix @samp{!} [@code{lnot}] (logical ``not,'' as in @samp{!x});
11602 @samp{+/-} [@code{sdev}] (the standard deviation symbol) and
11603 @samp{mod} [@code{makemod}] (the symbol for modulo forms);
11605 postfix @samp{!} [@code{fact}] (factorial, as in @samp{n!})
11606 and postfix @samp{!!} [@code{dfact}] (double factorial);
11608 @samp{^} [@code{pow}] (raised-to-the-power-of);
11610 prefix @samp{+} and @samp{-} [@code{neg}] (as in @samp{-x});
11612 @samp{*} [@code{mul}];
11614 @samp{/} [@code{div}], @samp{%} [@code{mod}] (modulo), and
11615 @samp{\} [@code{idiv}] (integer division);
11617 infix @samp{+} [@code{add}] and @samp{-} [@code{sub}] (as in @samp{x-y});
11619 @samp{|} [@code{vconcat}] (vector concatenation);
11621 relations @samp{=} [@code{eq}], @samp{!=} [@code{neq}], @samp{<} [@code{lt}],
11622 @samp{>} [@code{gt}], @samp{<=} [@code{leq}], and @samp{>=} [@code{geq}];
11624 @samp{&&} [@code{land}] (logical ``and'');
11626 @samp{||} [@code{lor}] (logical ``or'');
11628 the C-style ``if'' operator @samp{a?b:c} [@code{if}];
11630 @samp{!!!} [@code{pnot}] (rewrite pattern ``not'');
11632 @samp{&&&} [@code{pand}] (rewrite pattern ``and'');
11634 @samp{|||} [@code{por}] (rewrite pattern ``or'');
11636 @samp{:=} [@code{assign}] (for assignments and rewrite rules);
11638 @samp{::} [@code{condition}] (rewrite pattern condition);
11640 @samp{=>} [@code{evalto}].
11642 Note that, unlike in usual computer notation, multiplication binds more
11643 strongly than division: @samp{a*b/c*d} is equivalent to
11644 @texline @math{a b \over c d}.
11645 @infoline @expr{(a*b)/(c*d)}.
11647 @cindex Multiplication, implicit
11648 @cindex Implicit multiplication
11649 The multiplication sign @samp{*} may be omitted in many cases. In particular,
11650 if the righthand side is a number, variable name, or parenthesized
11651 expression, the @samp{*} may be omitted. Implicit multiplication has the
11652 same precedence as the explicit @samp{*} operator. The one exception to
11653 the rule is that a variable name followed by a parenthesized expression,
11655 is interpreted as a function call, not an implicit @samp{*}. In many
11656 cases you must use a space if you omit the @samp{*}: @samp{2a} is the
11657 same as @samp{2*a}, and @samp{a b} is the same as @samp{a*b}, but @samp{ab}
11658 is a variable called @code{ab}, @emph{not} the product of @samp{a} and
11659 @samp{b}! Also note that @samp{f (x)} is still a function call.
11661 @cindex Implicit comma in vectors
11662 The rules are slightly different for vectors written with square brackets.
11663 In vectors, the space character is interpreted (like the comma) as a
11664 separator of elements of the vector. Thus @w{@samp{[ 2a b+c d ]}} is
11665 equivalent to @samp{[2*a, b+c, d]}, whereas @samp{2a b+c d} is equivalent
11666 to @samp{2*a*b + c*d}.
11667 Note that spaces around the brackets, and around explicit commas, are
11668 ignored. To force spaces to be interpreted as multiplication you can
11669 enclose a formula in parentheses as in @samp{[(a b) 2(c d)]}, which is
11670 interpreted as @samp{[a*b, 2*c*d]}. An implicit comma is also inserted
11671 between @samp{][}, as in the matrix @samp{[[1 2][3 4]]}.
11673 Vectors that contain commas (not embedded within nested parentheses or
11674 brackets) do not treat spaces specially: @samp{[a b, 2 c d]} is a vector
11675 of two elements. Also, if it would be an error to treat spaces as
11676 separators, but not otherwise, then Calc will ignore spaces:
11677 @w{@samp{[a - b]}} is a vector of one element, but @w{@samp{[a -b]}} is
11678 a vector of two elements. Finally, vectors entered with curly braces
11679 instead of square brackets do not give spaces any special treatment.
11680 When Calc displays a vector that does not contain any commas, it will
11681 insert parentheses if necessary to make the meaning clear:
11682 @w{@samp{[(a b)]}}.
11684 The expression @samp{5%-2} is ambiguous; is this five-percent minus two,
11685 or five modulo minus-two? Calc always interprets the leftmost symbol as
11686 an infix operator preferentially (modulo, in this case), so you would
11687 need to write @samp{(5%)-2} to get the former interpretation.
11689 @cindex Function call notation
11690 A function call is, e.g., @samp{sin(1+x)}. (The Calc algebraic function
11691 @code{foo} corresponds to the Emacs Lisp function @code{calcFunc-foo},
11692 but unless you access the function from within Emacs Lisp, you don't
11693 need to worry about it.) Most mathematical Calculator commands like
11694 @code{calc-sin} have function equivalents like @code{sin}.
11695 If no Lisp function is defined for a function called by a formula, the
11696 call is left as it is during algebraic manipulation: @samp{f(x+y)} is
11697 left alone. Beware that many innocent-looking short names like @code{in}
11698 and @code{re} have predefined meanings which could surprise you; however,
11699 single letters or single letters followed by digits are always safe to
11700 use for your own function names. @xref{Function Index}.
11702 In the documentation for particular commands, the notation @kbd{H S}
11703 (@code{calc-sinh}) [@code{sinh}] means that the key sequence @kbd{H S}, the
11704 command @kbd{M-x calc-sinh}, and the algebraic function @code{sinh(x)} all
11705 represent the same operation.
11707 Commands that interpret (``parse'') text as algebraic formulas include
11708 algebraic entry (@kbd{'}), editing commands like @kbd{`} which parse
11709 the contents of the editing buffer when you finish, the @kbd{C-x * g}
11710 and @w{@kbd{C-x * r}} commands, the @kbd{C-y} command, the X window system
11711 ``paste'' mouse operation, and Embedded mode. All of these operations
11712 use the same rules for parsing formulas; in particular, language modes
11713 (@pxref{Language Modes}) affect them all in the same way.
11715 When you read a large amount of text into the Calculator (say a vector
11716 which represents a big set of rewrite rules; @pxref{Rewrite Rules}),
11717 you may wish to include comments in the text. Calc's formula parser
11718 ignores the symbol @samp{%%} and anything following it on a line:
11721 [ a + b, %% the sum of "a" and "b"
11723 %% last line is coming up:
11728 This is parsed exactly the same as @samp{[ a + b, c + d, e + f ]}.
11730 @xref{Syntax Tables}, for a way to create your own operators and other
11731 input notations. @xref{Compositions}, for a way to create new display
11734 @xref{Algebra}, for commands for manipulating formulas symbolically.
11736 @node Stack and Trail, Mode Settings, Data Types, Top
11737 @chapter Stack and Trail Commands
11740 This chapter describes the Calc commands for manipulating objects on the
11741 stack and in the trail buffer. (These commands operate on objects of any
11742 type, such as numbers, vectors, formulas, and incomplete objects.)
11745 * Stack Manipulation::
11746 * Editing Stack Entries::
11751 @node Stack Manipulation, Editing Stack Entries, Stack and Trail, Stack and Trail
11752 @section Stack Manipulation Commands
11758 @cindex Duplicating stack entries
11759 To duplicate the top object on the stack, press @key{RET} or @key{SPC}
11760 (two equivalent keys for the @code{calc-enter} command).
11761 Given a positive numeric prefix argument, these commands duplicate
11762 several elements at the top of the stack.
11763 Given a negative argument,
11764 these commands duplicate the specified element of the stack.
11765 Given an argument of zero, they duplicate the entire stack.
11766 For example, with @samp{10 20 30} on the stack,
11767 @key{RET} creates @samp{10 20 30 30},
11768 @kbd{C-u 2 @key{RET}} creates @samp{10 20 30 20 30},
11769 @kbd{C-u - 2 @key{RET}} creates @samp{10 20 30 20}, and
11770 @kbd{C-u 0 @key{RET}} creates @samp{10 20 30 10 20 30}.
11774 The @key{LFD} (@code{calc-over}) command (on a key marked Line-Feed if you
11775 have it, else on @kbd{C-j}) is like @code{calc-enter}
11776 except that the sign of the numeric prefix argument is interpreted
11777 oppositely. Also, with no prefix argument the default argument is 2.
11778 Thus with @samp{10 20 30} on the stack, @key{LFD} and @kbd{C-u 2 @key{LFD}}
11779 are both equivalent to @kbd{C-u - 2 @key{RET}}, producing
11780 @samp{10 20 30 20}.
11785 @cindex Removing stack entries
11786 @cindex Deleting stack entries
11787 To remove the top element from the stack, press @key{DEL} (@code{calc-pop}).
11788 The @kbd{C-d} key is a synonym for @key{DEL}.
11789 (If the top element is an incomplete object with at least one element, the
11790 last element is removed from it.) Given a positive numeric prefix argument,
11791 several elements are removed. Given a negative argument, the specified
11792 element of the stack is deleted. Given an argument of zero, the entire
11794 For example, with @samp{10 20 30} on the stack,
11795 @key{DEL} leaves @samp{10 20},
11796 @kbd{C-u 2 @key{DEL}} leaves @samp{10},
11797 @kbd{C-u - 2 @key{DEL}} leaves @samp{10 30}, and
11798 @kbd{C-u 0 @key{DEL}} leaves an empty stack.
11800 @kindex M-@key{DEL}
11801 @pindex calc-pop-above
11802 The @kbd{M-@key{DEL}} (@code{calc-pop-above}) command is to @key{DEL} what
11803 @key{LFD} is to @key{RET}: It interprets the sign of the numeric
11804 prefix argument in the opposite way, and the default argument is 2.
11805 Thus @kbd{M-@key{DEL}} by itself removes the second-from-top stack element,
11806 leaving the first, third, fourth, and so on; @kbd{M-3 M-@key{DEL}} deletes
11807 the third stack element.
11810 @pindex calc-roll-down
11811 To exchange the top two elements of the stack, press @key{TAB}
11812 (@code{calc-roll-down}). Given a positive numeric prefix argument, the
11813 specified number of elements at the top of the stack are rotated downward.
11814 Given a negative argument, the entire stack is rotated downward the specified
11815 number of times. Given an argument of zero, the entire stack is reversed
11817 For example, with @samp{10 20 30 40 50} on the stack,
11818 @key{TAB} creates @samp{10 20 30 50 40},
11819 @kbd{C-u 3 @key{TAB}} creates @samp{10 20 50 30 40},
11820 @kbd{C-u - 2 @key{TAB}} creates @samp{40 50 10 20 30}, and
11821 @kbd{C-u 0 @key{TAB}} creates @samp{50 40 30 20 10}.
11823 @kindex M-@key{TAB}
11824 @pindex calc-roll-up
11825 The command @kbd{M-@key{TAB}} (@code{calc-roll-up}) is analogous to @key{TAB}
11826 except that it rotates upward instead of downward. Also, the default
11827 with no prefix argument is to rotate the top 3 elements.
11828 For example, with @samp{10 20 30 40 50} on the stack,
11829 @kbd{M-@key{TAB}} creates @samp{10 20 40 50 30},
11830 @kbd{C-u 4 M-@key{TAB}} creates @samp{10 30 40 50 20},
11831 @kbd{C-u - 2 M-@key{TAB}} creates @samp{30 40 50 10 20}, and
11832 @kbd{C-u 0 M-@key{TAB}} creates @samp{50 40 30 20 10}.
11834 A good way to view the operation of @key{TAB} and @kbd{M-@key{TAB}} is in
11835 terms of moving a particular element to a new position in the stack.
11836 With a positive argument @var{n}, @key{TAB} moves the top stack
11837 element down to level @var{n}, making room for it by pulling all the
11838 intervening stack elements toward the top. @kbd{M-@key{TAB}} moves the
11839 element at level @var{n} up to the top. (Compare with @key{LFD},
11840 which copies instead of moving the element in level @var{n}.)
11842 With a negative argument @mathit{-@var{n}}, @key{TAB} rotates the stack
11843 to move the object in level @var{n} to the deepest place in the
11844 stack, and the object in level @mathit{@var{n}+1} to the top. @kbd{M-@key{TAB}}
11845 rotates the deepest stack element to be in level @mathit{n}, also
11846 putting the top stack element in level @mathit{@var{n}+1}.
11848 @xref{Selecting Subformulas}, for a way to apply these commands to
11849 any portion of a vector or formula on the stack.
11851 @node Editing Stack Entries, Trail Commands, Stack Manipulation, Stack and Trail
11852 @section Editing Stack Entries
11857 @pindex calc-edit-finish
11858 @cindex Editing the stack with Emacs
11859 The @kbd{`} (@code{calc-edit}) command creates a temporary buffer
11860 (@samp{*Calc Edit*}) for editing the top-of-stack value using regular
11861 Emacs commands. Note that @kbd{`} is a backquote, not a quote. With a
11862 numeric prefix argument, it edits the specified number of stack entries
11863 at once. (An argument of zero edits the entire stack; a negative
11864 argument edits one specific stack entry.)
11866 When you are done editing, press @kbd{C-c C-c} to finish and return
11867 to Calc. The @key{RET} and @key{LFD} keys also work to finish most
11868 sorts of editing, though in some cases Calc leaves @key{RET} with its
11869 usual meaning (``insert a newline'') if it's a situation where you
11870 might want to insert new lines into the editing buffer.
11872 When you finish editing, the Calculator parses the lines of text in
11873 the @samp{*Calc Edit*} buffer as numbers or formulas, replaces the
11874 original stack elements in the original buffer with these new values,
11875 then kills the @samp{*Calc Edit*} buffer. The original Calculator buffer
11876 continues to exist during editing, but for best results you should be
11877 careful not to change it until you have finished the edit. You can
11878 also cancel the edit by killing the buffer with @kbd{C-x k}.
11880 The formula is normally reevaluated as it is put onto the stack.
11881 For example, editing @samp{a + 2} to @samp{3 + 2} and pressing
11882 @kbd{C-c C-c} will push 5 on the stack. If you use @key{LFD} to
11883 finish, Calc will put the result on the stack without evaluating it.
11885 If you give a prefix argument to @kbd{C-c C-c},
11886 Calc will not kill the @samp{*Calc Edit*} buffer. You can switch
11887 back to that buffer and continue editing if you wish. However, you
11888 should understand that if you initiated the edit with @kbd{`}, the
11889 @kbd{C-c C-c} operation will be programmed to replace the top of the
11890 stack with the new edited value, and it will do this even if you have
11891 rearranged the stack in the meanwhile. This is not so much of a problem
11892 with other editing commands, though, such as @kbd{s e}
11893 (@code{calc-edit-variable}; @pxref{Operations on Variables}).
11895 If the @code{calc-edit} command involves more than one stack entry,
11896 each line of the @samp{*Calc Edit*} buffer is interpreted as a
11897 separate formula. Otherwise, the entire buffer is interpreted as
11898 one formula, with line breaks ignored. (You can use @kbd{C-o} or
11899 @kbd{C-q C-j} to insert a newline in the buffer without pressing @key{RET}.)
11901 The @kbd{`} key also works during numeric or algebraic entry. The
11902 text entered so far is moved to the @code{*Calc Edit*} buffer for
11903 more extensive editing than is convenient in the minibuffer.
11905 @node Trail Commands, Keep Arguments, Editing Stack Entries, Stack and Trail
11906 @section Trail Commands
11909 @cindex Trail buffer
11910 The commands for manipulating the Calc Trail buffer are two-key sequences
11911 beginning with the @kbd{t} prefix.
11914 @pindex calc-trail-display
11915 The @kbd{t d} (@code{calc-trail-display}) command turns display of the
11916 trail on and off. Normally the trail display is toggled on if it was off,
11917 off if it was on. With a numeric prefix of zero, this command always
11918 turns the trail off; with a prefix of one, it always turns the trail on.
11919 The other trail-manipulation commands described here automatically turn
11920 the trail on. Note that when the trail is off values are still recorded
11921 there; they are simply not displayed. To set Emacs to turn the trail
11922 off by default, type @kbd{t d} and then save the mode settings with
11923 @kbd{m m} (@code{calc-save-modes}).
11926 @pindex calc-trail-in
11928 @pindex calc-trail-out
11929 The @kbd{t i} (@code{calc-trail-in}) and @kbd{t o}
11930 (@code{calc-trail-out}) commands switch the cursor into and out of the
11931 Calc Trail window. In practice they are rarely used, since the commands
11932 shown below are a more convenient way to move around in the
11933 trail, and they work ``by remote control'' when the cursor is still
11934 in the Calculator window.
11936 @cindex Trail pointer
11937 There is a @dfn{trail pointer} which selects some entry of the trail at
11938 any given time. The trail pointer looks like a @samp{>} symbol right
11939 before the selected number. The following commands operate on the
11940 trail pointer in various ways.
11943 @pindex calc-trail-yank
11944 @cindex Retrieving previous results
11945 The @kbd{t y} (@code{calc-trail-yank}) command reads the selected value in
11946 the trail and pushes it onto the Calculator stack. It allows you to
11947 re-use any previously computed value without retyping. With a numeric
11948 prefix argument @var{n}, it yanks the value @var{n} lines above the current
11952 @pindex calc-trail-scroll-left
11954 @pindex calc-trail-scroll-right
11955 The @kbd{t <} (@code{calc-trail-scroll-left}) and @kbd{t >}
11956 (@code{calc-trail-scroll-right}) commands horizontally scroll the trail
11957 window left or right by one half of its width.
11960 @pindex calc-trail-next
11962 @pindex calc-trail-previous
11964 @pindex calc-trail-forward
11966 @pindex calc-trail-backward
11967 The @kbd{t n} (@code{calc-trail-next}) and @kbd{t p}
11968 (@code{calc-trail-previous)} commands move the trail pointer down or up
11969 one line. The @kbd{t f} (@code{calc-trail-forward}) and @kbd{t b}
11970 (@code{calc-trail-backward}) commands move the trail pointer down or up
11971 one screenful at a time. All of these commands accept numeric prefix
11972 arguments to move several lines or screenfuls at a time.
11975 @pindex calc-trail-first
11977 @pindex calc-trail-last
11979 @pindex calc-trail-here
11980 The @kbd{t [} (@code{calc-trail-first}) and @kbd{t ]}
11981 (@code{calc-trail-last}) commands move the trail pointer to the first or
11982 last line of the trail. The @kbd{t h} (@code{calc-trail-here}) command
11983 moves the trail pointer to the cursor position; unlike the other trail
11984 commands, @kbd{t h} works only when Calc Trail is the selected window.
11987 @pindex calc-trail-isearch-forward
11989 @pindex calc-trail-isearch-backward
11991 The @kbd{t s} (@code{calc-trail-isearch-forward}) and @kbd{t r}
11992 (@code{calc-trail-isearch-backward}) commands perform an incremental
11993 search forward or backward through the trail. You can press @key{RET}
11994 to terminate the search; the trail pointer moves to the current line.
11995 If you cancel the search with @kbd{C-g}, the trail pointer stays where
11996 it was when the search began.
11999 The @kbd{t s} (@code{calc-trail-isearch-forward}) and @kbd{t r}
12000 (@code{calc-trail-isearch-backward}) com\-mands perform an incremental
12001 search forward or backward through the trail. You can press @key{RET}
12002 to terminate the search; the trail pointer moves to the current line.
12003 If you cancel the search with @kbd{C-g}, the trail pointer stays where
12004 it was when the search began.
12008 @pindex calc-trail-marker
12009 The @kbd{t m} (@code{calc-trail-marker}) command allows you to enter a
12010 line of text of your own choosing into the trail. The text is inserted
12011 after the line containing the trail pointer; this usually means it is
12012 added to the end of the trail. Trail markers are useful mainly as the
12013 targets for later incremental searches in the trail.
12016 @pindex calc-trail-kill
12017 The @kbd{t k} (@code{calc-trail-kill}) command removes the selected line
12018 from the trail. The line is saved in the Emacs kill ring suitable for
12019 yanking into another buffer, but it is not easy to yank the text back
12020 into the trail buffer. With a numeric prefix argument, this command
12021 kills the @var{n} lines below or above the selected one.
12023 The @kbd{t .} (@code{calc-full-trail-vectors}) command is described
12024 elsewhere; @pxref{Vector and Matrix Formats}.
12026 @node Keep Arguments, , Trail Commands, Stack and Trail
12027 @section Keep Arguments
12031 @pindex calc-keep-args
12032 The @kbd{K} (@code{calc-keep-args}) command acts like a prefix for
12033 the following command. It prevents that command from removing its
12034 arguments from the stack. For example, after @kbd{2 @key{RET} 3 +},
12035 the stack contains the sole number 5, but after @kbd{2 @key{RET} 3 K +},
12036 the stack contains the arguments and the result: @samp{2 3 5}.
12038 With the exception of keyboard macros, this works for all commands that
12039 take arguments off the stack. (To avoid potentially unpleasant behavior,
12040 a @kbd{K} prefix before a keyboard macro will be ignored. A @kbd{K}
12041 prefix called @emph{within} the keyboard macro will still take effect.)
12042 As another example, @kbd{K a s} simplifies a formula, pushing the
12043 simplified version of the formula onto the stack after the original
12044 formula (rather than replacing the original formula). Note that you
12045 could get the same effect by typing @kbd{@key{RET} a s}, copying the
12046 formula and then simplifying the copy. One difference is that for a very
12047 large formula the time taken to format the intermediate copy in
12048 @kbd{@key{RET} a s} could be noticeable; @kbd{K a s} would avoid this
12051 Even stack manipulation commands are affected. @key{TAB} works by
12052 popping two values and pushing them back in the opposite order,
12053 so @kbd{2 @key{RET} 3 K @key{TAB}} produces @samp{2 3 3 2}.
12055 A few Calc commands provide other ways of doing the same thing.
12056 For example, @kbd{' sin($)} replaces the number on the stack with
12057 its sine using algebraic entry; to push the sine and keep the
12058 original argument you could use either @kbd{' sin($1)} or
12059 @kbd{K ' sin($)}. @xref{Algebraic Entry}. Also, the @kbd{s s}
12060 command is effectively the same as @kbd{K s t}. @xref{Storing Variables}.
12062 If you execute a command and then decide you really wanted to keep
12063 the argument, you can press @kbd{M-@key{RET}} (@code{calc-last-args}).
12064 This command pushes the last arguments that were popped by any command
12065 onto the stack. Note that the order of things on the stack will be
12066 different than with @kbd{K}: @kbd{2 @key{RET} 3 + M-@key{RET}} leaves
12067 @samp{5 2 3} on the stack instead of @samp{2 3 5}. @xref{Undo}.
12069 @node Mode Settings, Arithmetic, Stack and Trail, Top
12070 @chapter Mode Settings
12073 This chapter describes commands that set modes in the Calculator.
12074 They do not affect the contents of the stack, although they may change
12075 the @emph{appearance} or @emph{interpretation} of the stack's contents.
12078 * General Mode Commands::
12080 * Inverse and Hyperbolic::
12081 * Calculation Modes::
12082 * Simplification Modes::
12090 @node General Mode Commands, Precision, Mode Settings, Mode Settings
12091 @section General Mode Commands
12095 @pindex calc-save-modes
12096 @cindex Continuous memory
12097 @cindex Saving mode settings
12098 @cindex Permanent mode settings
12099 @cindex Calc init file, mode settings
12100 You can save all of the current mode settings in your Calc init file
12101 (the file given by the variable @code{calc-settings-file}, typically
12102 @file{~/.calc.el}) with the @kbd{m m} (@code{calc-save-modes}) command.
12103 This will cause Emacs to reestablish these modes each time it starts up.
12104 The modes saved in the file include everything controlled by the @kbd{m}
12105 and @kbd{d} prefix keys, the current precision and binary word size,
12106 whether or not the trail is displayed, the current height of the Calc
12107 window, and more. The current interface (used when you type @kbd{C-x * *})
12108 is also saved. If there were already saved mode settings in the
12109 file, they are replaced. Otherwise, the new mode information is
12110 appended to the end of the file.
12113 @pindex calc-mode-record-mode
12114 The @kbd{m R} (@code{calc-mode-record-mode}) command tells Calc to
12115 record all the mode settings (as if by pressing @kbd{m m}) every
12116 time a mode setting changes. If the modes are saved this way, then this
12117 ``automatic mode recording'' mode is also saved.
12118 Type @kbd{m R} again to disable this method of recording the mode
12119 settings. To turn it off permanently, the @kbd{m m} command will also be
12120 necessary. (If Embedded mode is enabled, other options for recording
12121 the modes are available; @pxref{Mode Settings in Embedded Mode}.)
12124 @pindex calc-settings-file-name
12125 The @kbd{m F} (@code{calc-settings-file-name}) command allows you to
12126 choose a different file than the current value of @code{calc-settings-file}
12127 for @kbd{m m}, @kbd{Z P}, and similar commands to save permanent information.
12128 You are prompted for a file name. All Calc modes are then reset to
12129 their default values, then settings from the file you named are loaded
12130 if this file exists, and this file becomes the one that Calc will
12131 use in the future for commands like @kbd{m m}. The default settings
12132 file name is @file{~/.calc.el}. You can see the current file name by
12133 giving a blank response to the @kbd{m F} prompt. See also the
12134 discussion of the @code{calc-settings-file} variable; @pxref{Customizing Calc}.
12136 If the file name you give is your user init file (typically
12137 @file{~/.emacs}), @kbd{m F} will not automatically load the new file. This
12138 is because your user init file may contain other things you don't want
12139 to reread. You can give
12140 a numeric prefix argument of 1 to @kbd{m F} to force it to read the
12141 file no matter what. Conversely, an argument of @mathit{-1} tells
12142 @kbd{m F} @emph{not} to read the new file. An argument of 2 or @mathit{-2}
12143 tells @kbd{m F} not to reset the modes to their defaults beforehand,
12144 which is useful if you intend your new file to have a variant of the
12145 modes present in the file you were using before.
12148 @pindex calc-always-load-extensions
12149 The @kbd{m x} (@code{calc-always-load-extensions}) command enables a mode
12150 in which the first use of Calc loads the entire program, including all
12151 extensions modules. Otherwise, the extensions modules will not be loaded
12152 until the various advanced Calc features are used. Since this mode only
12153 has effect when Calc is first loaded, @kbd{m x} is usually followed by
12154 @kbd{m m} to make the mode-setting permanent. To load all of Calc just
12155 once, rather than always in the future, you can press @kbd{C-x * L}.
12158 @pindex calc-shift-prefix
12159 The @kbd{m S} (@code{calc-shift-prefix}) command enables a mode in which
12160 all of Calc's letter prefix keys may be typed shifted as well as unshifted.
12161 If you are typing, say, @kbd{a S} (@code{calc-solve-for}) quite often
12162 you might find it easier to turn this mode on so that you can type
12163 @kbd{A S} instead. When this mode is enabled, the commands that used to
12164 be on those single shifted letters (e.g., @kbd{A} (@code{calc-abs})) can
12165 now be invoked by pressing the shifted letter twice: @kbd{A A}. Note
12166 that the @kbd{v} prefix key always works both shifted and unshifted, and
12167 the @kbd{z} and @kbd{Z} prefix keys are always distinct. Also, the @kbd{h}
12168 prefix is not affected by this mode. Press @kbd{m S} again to disable
12169 shifted-prefix mode.
12171 @node Precision, Inverse and Hyperbolic, General Mode Commands, Mode Settings
12176 @pindex calc-precision
12177 @cindex Precision of calculations
12178 The @kbd{p} (@code{calc-precision}) command controls the precision to
12179 which floating-point calculations are carried. The precision must be
12180 at least 3 digits and may be arbitrarily high, within the limits of
12181 memory and time. This affects only floats: Integer and rational
12182 calculations are always carried out with as many digits as necessary.
12184 The @kbd{p} key prompts for the current precision. If you wish you
12185 can instead give the precision as a numeric prefix argument.
12187 Many internal calculations are carried to one or two digits higher
12188 precision than normal. Results are rounded down afterward to the
12189 current precision. Unless a special display mode has been selected,
12190 floats are always displayed with their full stored precision, i.e.,
12191 what you see is what you get. Reducing the current precision does not
12192 round values already on the stack, but those values will be rounded
12193 down before being used in any calculation. The @kbd{c 0} through
12194 @kbd{c 9} commands (@pxref{Conversions}) can be used to round an
12195 existing value to a new precision.
12197 @cindex Accuracy of calculations
12198 It is important to distinguish the concepts of @dfn{precision} and
12199 @dfn{accuracy}. In the normal usage of these words, the number
12200 123.4567 has a precision of 7 digits but an accuracy of 4 digits.
12201 The precision is the total number of digits not counting leading
12202 or trailing zeros (regardless of the position of the decimal point).
12203 The accuracy is simply the number of digits after the decimal point
12204 (again not counting trailing zeros). In Calc you control the precision,
12205 not the accuracy of computations. If you were to set the accuracy
12206 instead, then calculations like @samp{exp(100)} would generate many
12207 more digits than you would typically need, while @samp{exp(-100)} would
12208 probably round to zero! In Calc, both these computations give you
12209 exactly 12 (or the requested number of) significant digits.
12211 The only Calc features that deal with accuracy instead of precision
12212 are fixed-point display mode for floats (@kbd{d f}; @pxref{Float Formats}),
12213 and the rounding functions like @code{floor} and @code{round}
12214 (@pxref{Integer Truncation}). Also, @kbd{c 0} through @kbd{c 9}
12215 deal with both precision and accuracy depending on the magnitudes
12216 of the numbers involved.
12218 If you need to work with a particular fixed accuracy (say, dollars and
12219 cents with two digits after the decimal point), one solution is to work
12220 with integers and an ``implied'' decimal point. For example, $8.99
12221 divided by 6 would be entered @kbd{899 @key{RET} 6 /}, yielding 149.833
12222 (actually $1.49833 with our implied decimal point); pressing @kbd{R}
12223 would round this to 150 cents, i.e., $1.50.
12225 @xref{Floats}, for still more on floating-point precision and related
12228 @node Inverse and Hyperbolic, Calculation Modes, Precision, Mode Settings
12229 @section Inverse and Hyperbolic Flags
12233 @pindex calc-inverse
12234 There is no single-key equivalent to the @code{calc-arcsin} function.
12235 Instead, you must first press @kbd{I} (@code{calc-inverse}) to set
12236 the @dfn{Inverse Flag}, then press @kbd{S} (@code{calc-sin}).
12237 The @kbd{I} key actually toggles the Inverse Flag. When this flag
12238 is set, the word @samp{Inv} appears in the mode line.
12241 @pindex calc-hyperbolic
12242 Likewise, the @kbd{H} key (@code{calc-hyperbolic}) sets or clears the
12243 Hyperbolic Flag, which transforms @code{calc-sin} into @code{calc-sinh}.
12244 If both of these flags are set at once, the effect will be
12245 @code{calc-arcsinh}. (The Hyperbolic flag is also used by some
12246 non-trigonometric commands; for example @kbd{H L} computes a base-10,
12247 instead of base-@mathit{e}, logarithm.)
12249 Command names like @code{calc-arcsin} are provided for completeness, and
12250 may be executed with @kbd{x} or @kbd{M-x}. Their effect is simply to
12251 toggle the Inverse and/or Hyperbolic flags and then execute the
12252 corresponding base command (@code{calc-sin} in this case).
12254 The Inverse and Hyperbolic flags apply only to the next Calculator
12255 command, after which they are automatically cleared. (They are also
12256 cleared if the next keystroke is not a Calc command.) Digits you
12257 type after @kbd{I} or @kbd{H} (or @kbd{K}) are treated as prefix
12258 arguments for the next command, not as numeric entries. The same
12259 is true of @kbd{C-u}, but not of the minus sign (@kbd{K -} means to
12260 subtract and keep arguments).
12262 The third Calc prefix flag, @kbd{K} (keep-arguments), is discussed
12263 elsewhere. @xref{Keep Arguments}.
12265 @node Calculation Modes, Simplification Modes, Inverse and Hyperbolic, Mode Settings
12266 @section Calculation Modes
12269 The commands in this section are two-key sequences beginning with
12270 the @kbd{m} prefix. (That's the letter @kbd{m}, not the @key{META} key.)
12271 The @samp{m a} (@code{calc-algebraic-mode}) command is described elsewhere
12272 (@pxref{Algebraic Entry}).
12281 * Automatic Recomputation::
12282 * Working Message::
12285 @node Angular Modes, Polar Mode, Calculation Modes, Calculation Modes
12286 @subsection Angular Modes
12289 @cindex Angular mode
12290 The Calculator supports three notations for angles: radians, degrees,
12291 and degrees-minutes-seconds. When a number is presented to a function
12292 like @code{sin} that requires an angle, the current angular mode is
12293 used to interpret the number as either radians or degrees. If an HMS
12294 form is presented to @code{sin}, it is always interpreted as
12295 degrees-minutes-seconds.
12297 Functions that compute angles produce a number in radians, a number in
12298 degrees, or an HMS form depending on the current angular mode. If the
12299 result is a complex number and the current mode is HMS, the number is
12300 instead expressed in degrees. (Complex-number calculations would
12301 normally be done in Radians mode, though. Complex numbers are converted
12302 to degrees by calculating the complex result in radians and then
12303 multiplying by 180 over @cpi{}.)
12306 @pindex calc-radians-mode
12308 @pindex calc-degrees-mode
12310 @pindex calc-hms-mode
12311 The @kbd{m r} (@code{calc-radians-mode}), @kbd{m d} (@code{calc-degrees-mode}),
12312 and @kbd{m h} (@code{calc-hms-mode}) commands control the angular mode.
12313 The current angular mode is displayed on the Emacs mode line.
12314 The default angular mode is Degrees.
12316 @node Polar Mode, Fraction Mode, Angular Modes, Calculation Modes
12317 @subsection Polar Mode
12321 The Calculator normally ``prefers'' rectangular complex numbers in the
12322 sense that rectangular form is used when the proper form can not be
12323 decided from the input. This might happen by multiplying a rectangular
12324 number by a polar one, by taking the square root of a negative real
12325 number, or by entering @kbd{( 2 @key{SPC} 3 )}.
12328 @pindex calc-polar-mode
12329 The @kbd{m p} (@code{calc-polar-mode}) command toggles complex-number
12330 preference between rectangular and polar forms. In Polar mode, all
12331 of the above example situations would produce polar complex numbers.
12333 @node Fraction Mode, Infinite Mode, Polar Mode, Calculation Modes
12334 @subsection Fraction Mode
12337 @cindex Fraction mode
12338 @cindex Division of integers
12339 Division of two integers normally yields a floating-point number if the
12340 result cannot be expressed as an integer. In some cases you would
12341 rather get an exact fractional answer. One way to accomplish this is
12342 to use the @kbd{:} (@code{calc-fdiv}) [@code{fdiv}] command, which
12343 divides the two integers on the top of the stack to produce a fraction:
12344 @kbd{6 @key{RET} 4 :} produces @expr{3:2} even though
12345 @kbd{6 @key{RET} 4 /} produces @expr{1.5}.
12348 @pindex calc-frac-mode
12349 To set the Calculator to produce fractional results for normal integer
12350 divisions, use the @kbd{m f} (@code{calc-frac-mode}) command.
12351 For example, @expr{8/4} produces @expr{2} in either mode,
12352 but @expr{6/4} produces @expr{3:2} in Fraction mode, @expr{1.5} in
12355 At any time you can use @kbd{c f} (@code{calc-float}) to convert a
12356 fraction to a float, or @kbd{c F} (@code{calc-fraction}) to convert a
12357 float to a fraction. @xref{Conversions}.
12359 @node Infinite Mode, Symbolic Mode, Fraction Mode, Calculation Modes
12360 @subsection Infinite Mode
12363 @cindex Infinite mode
12364 The Calculator normally treats results like @expr{1 / 0} as errors;
12365 formulas like this are left in unsimplified form. But Calc can be
12366 put into a mode where such calculations instead produce ``infinite''
12370 @pindex calc-infinite-mode
12371 The @kbd{m i} (@code{calc-infinite-mode}) command turns this mode
12372 on and off. When the mode is off, infinities do not arise except
12373 in calculations that already had infinities as inputs. (One exception
12374 is that infinite open intervals like @samp{[0 .. inf)} can be
12375 generated; however, intervals closed at infinity (@samp{[0 .. inf]})
12376 will not be generated when Infinite mode is off.)
12378 With Infinite mode turned on, @samp{1 / 0} will generate @code{uinf},
12379 an undirected infinity. @xref{Infinities}, for a discussion of the
12380 difference between @code{inf} and @code{uinf}. Also, @expr{0 / 0}
12381 evaluates to @code{nan}, the ``indeterminate'' symbol. Various other
12382 functions can also return infinities in this mode; for example,
12383 @samp{ln(0) = -inf}, and @samp{gamma(-7) = uinf}. Once again,
12384 note that @samp{exp(inf) = inf} regardless of Infinite mode because
12385 this calculation has infinity as an input.
12387 @cindex Positive Infinite mode
12388 The @kbd{m i} command with a numeric prefix argument of zero,
12389 i.e., @kbd{C-u 0 m i}, turns on a Positive Infinite mode in
12390 which zero is treated as positive instead of being directionless.
12391 Thus, @samp{1 / 0 = inf} and @samp{-1 / 0 = -inf} in this mode.
12392 Note that zero never actually has a sign in Calc; there are no
12393 separate representations for @mathit{+0} and @mathit{-0}. Positive
12394 Infinite mode merely changes the interpretation given to the
12395 single symbol, @samp{0}. One consequence of this is that, while
12396 you might expect @samp{1 / -0 = -inf}, actually @samp{1 / -0}
12397 is equivalent to @samp{1 / 0}, which is equal to positive @code{inf}.
12399 @node Symbolic Mode, Matrix Mode, Infinite Mode, Calculation Modes
12400 @subsection Symbolic Mode
12403 @cindex Symbolic mode
12404 @cindex Inexact results
12405 Calculations are normally performed numerically wherever possible.
12406 For example, the @code{calc-sqrt} command, or @code{sqrt} function in an
12407 algebraic expression, produces a numeric answer if the argument is a
12408 number or a symbolic expression if the argument is an expression:
12409 @kbd{2 Q} pushes 1.4142 but @kbd{@key{'} x+1 @key{RET} Q} pushes @samp{sqrt(x+1)}.
12412 @pindex calc-symbolic-mode
12413 In @dfn{Symbolic mode}, controlled by the @kbd{m s} (@code{calc-symbolic-mode})
12414 command, functions which would produce inexact, irrational results are
12415 left in symbolic form. Thus @kbd{16 Q} pushes 4, but @kbd{2 Q} pushes
12419 @pindex calc-eval-num
12420 The shift-@kbd{N} (@code{calc-eval-num}) command evaluates numerically
12421 the expression at the top of the stack, by temporarily disabling
12422 @code{calc-symbolic-mode} and executing @kbd{=} (@code{calc-evaluate}).
12423 Given a numeric prefix argument, it also
12424 sets the floating-point precision to the specified value for the duration
12427 To evaluate a formula numerically without expanding the variables it
12428 contains, you can use the key sequence @kbd{m s a v m s} (this uses
12429 @code{calc-alg-evaluate}, which resimplifies but doesn't evaluate
12432 @node Matrix Mode, Automatic Recomputation, Symbolic Mode, Calculation Modes
12433 @subsection Matrix and Scalar Modes
12436 @cindex Matrix mode
12437 @cindex Scalar mode
12438 Calc sometimes makes assumptions during algebraic manipulation that
12439 are awkward or incorrect when vectors and matrices are involved.
12440 Calc has two modes, @dfn{Matrix mode} and @dfn{Scalar mode}, which
12441 modify its behavior around vectors in useful ways.
12444 @pindex calc-matrix-mode
12445 Press @kbd{m v} (@code{calc-matrix-mode}) once to enter Matrix mode.
12446 In this mode, all objects are assumed to be matrices unless provably
12447 otherwise. One major effect is that Calc will no longer consider
12448 multiplication to be commutative. (Recall that in matrix arithmetic,
12449 @samp{A*B} is not the same as @samp{B*A}.) This assumption affects
12450 rewrite rules and algebraic simplification. Another effect of this
12451 mode is that calculations that would normally produce constants like
12452 0 and 1 (e.g., @expr{a - a} and @expr{a / a}, respectively) will now
12453 produce function calls that represent ``generic'' zero or identity
12454 matrices: @samp{idn(0)}, @samp{idn(1)}. The @code{idn} function
12455 @samp{idn(@var{a},@var{n})} returns @var{a} times an @var{n}x@var{n}
12456 identity matrix; if @var{n} is omitted, it doesn't know what
12457 dimension to use and so the @code{idn} call remains in symbolic
12458 form. However, if this generic identity matrix is later combined
12459 with a matrix whose size is known, it will be converted into
12460 a true identity matrix of the appropriate size. On the other hand,
12461 if it is combined with a scalar (as in @samp{idn(1) + 2}), Calc
12462 will assume it really was a scalar after all and produce, e.g., 3.
12464 Press @kbd{m v} a second time to get Scalar mode. Here, objects are
12465 assumed @emph{not} to be vectors or matrices unless provably so.
12466 For example, normally adding a variable to a vector, as in
12467 @samp{[x, y, z] + a}, will leave the sum in symbolic form because
12468 as far as Calc knows, @samp{a} could represent either a number or
12469 another 3-vector. In Scalar mode, @samp{a} is assumed to be a
12470 non-vector, and the addition is evaluated to @samp{[x+a, y+a, z+a]}.
12472 Press @kbd{m v} a third time to return to the normal mode of operation.
12474 If you press @kbd{m v} with a numeric prefix argument @var{n}, you
12475 get a special ``dimensioned'' Matrix mode in which matrices of
12476 unknown size are assumed to be @var{n}x@var{n} square matrices.
12477 Then, the function call @samp{idn(1)} will expand into an actual
12478 matrix rather than representing a ``generic'' matrix. Simply typing
12479 @kbd{C-u m v} will get you a square Matrix mode, in which matrices of
12480 unknown size are assumed to be square matrices of unspecified size.
12482 @cindex Declaring scalar variables
12483 Of course these modes are approximations to the true state of
12484 affairs, which is probably that some quantities will be matrices
12485 and others will be scalars. One solution is to ``declare''
12486 certain variables or functions to be scalar-valued.
12487 @xref{Declarations}, to see how to make declarations in Calc.
12489 There is nothing stopping you from declaring a variable to be
12490 scalar and then storing a matrix in it; however, if you do, the
12491 results you get from Calc may not be valid. Suppose you let Calc
12492 get the result @samp{[x+a, y+a, z+a]} shown above, and then stored
12493 @samp{[1, 2, 3]} in @samp{a}. The result would not be the same as
12494 for @samp{[x, y, z] + [1, 2, 3]}, but that's because you have broken
12495 your earlier promise to Calc that @samp{a} would be scalar.
12497 Another way to mix scalars and matrices is to use selections
12498 (@pxref{Selecting Subformulas}). Use Matrix mode when operating on
12499 your formula normally; then, to apply Scalar mode to a certain part
12500 of the formula without affecting the rest just select that part,
12501 change into Scalar mode and press @kbd{=} to resimplify the part
12502 under this mode, then change back to Matrix mode before deselecting.
12504 @node Automatic Recomputation, Working Message, Matrix Mode, Calculation Modes
12505 @subsection Automatic Recomputation
12508 The @dfn{evaluates-to} operator, @samp{=>}, has the special
12509 property that any @samp{=>} formulas on the stack are recomputed
12510 whenever variable values or mode settings that might affect them
12511 are changed. @xref{Evaluates-To Operator}.
12514 @pindex calc-auto-recompute
12515 The @kbd{m C} (@code{calc-auto-recompute}) command turns this
12516 automatic recomputation on and off. If you turn it off, Calc will
12517 not update @samp{=>} operators on the stack (nor those in the
12518 attached Embedded mode buffer, if there is one). They will not
12519 be updated unless you explicitly do so by pressing @kbd{=} or until
12520 you press @kbd{m C} to turn recomputation back on. (While automatic
12521 recomputation is off, you can think of @kbd{m C m C} as a command
12522 to update all @samp{=>} operators while leaving recomputation off.)
12524 To update @samp{=>} operators in an Embedded buffer while
12525 automatic recomputation is off, use @w{@kbd{C-x * u}}.
12526 @xref{Embedded Mode}.
12528 @node Working Message, , Automatic Recomputation, Calculation Modes
12529 @subsection Working Messages
12532 @cindex Performance
12533 @cindex Working messages
12534 Since the Calculator is written entirely in Emacs Lisp, which is not
12535 designed for heavy numerical work, many operations are quite slow.
12536 The Calculator normally displays the message @samp{Working...} in the
12537 echo area during any command that may be slow. In addition, iterative
12538 operations such as square roots and trigonometric functions display the
12539 intermediate result at each step. Both of these types of messages can
12540 be disabled if you find them distracting.
12543 @pindex calc-working
12544 Type @kbd{m w} (@code{calc-working}) with a numeric prefix of 0 to
12545 disable all ``working'' messages. Use a numeric prefix of 1 to enable
12546 only the plain @samp{Working...} message. Use a numeric prefix of 2 to
12547 see intermediate results as well. With no numeric prefix this displays
12550 While it may seem that the ``working'' messages will slow Calc down
12551 considerably, experiments have shown that their impact is actually
12552 quite small. But if your terminal is slow you may find that it helps
12553 to turn the messages off.
12555 @node Simplification Modes, Declarations, Calculation Modes, Mode Settings
12556 @section Simplification Modes
12559 The current @dfn{simplification mode} controls how numbers and formulas
12560 are ``normalized'' when being taken from or pushed onto the stack.
12561 Some normalizations are unavoidable, such as rounding floating-point
12562 results to the current precision, and reducing fractions to simplest
12563 form. Others, such as simplifying a formula like @expr{a+a} (or @expr{2+3}),
12564 are done by default but can be turned off when necessary.
12566 When you press a key like @kbd{+} when @expr{2} and @expr{3} are on the
12567 stack, Calc pops these numbers, normalizes them, creates the formula
12568 @expr{2+3}, normalizes it, and pushes the result. Of course the standard
12569 rules for normalizing @expr{2+3} will produce the result @expr{5}.
12571 Simplification mode commands consist of the lower-case @kbd{m} prefix key
12572 followed by a shifted letter.
12575 @pindex calc-no-simplify-mode
12576 The @kbd{m O} (@code{calc-no-simplify-mode}) command turns off all optional
12577 simplifications. These would leave a formula like @expr{2+3} alone. In
12578 fact, nothing except simple numbers are ever affected by normalization
12582 @pindex calc-num-simplify-mode
12583 The @kbd{m N} (@code{calc-num-simplify-mode}) command turns off simplification
12584 of any formulas except those for which all arguments are constants. For
12585 example, @expr{1+2} is simplified to @expr{3}, and @expr{a+(2-2)} is
12586 simplified to @expr{a+0} but no further, since one argument of the sum
12587 is not a constant. Unfortunately, @expr{(a+2)-2} is @emph{not} simplified
12588 because the top-level @samp{-} operator's arguments are not both
12589 constant numbers (one of them is the formula @expr{a+2}).
12590 A constant is a number or other numeric object (such as a constant
12591 error form or modulo form), or a vector all of whose
12592 elements are constant.
12595 @pindex calc-default-simplify-mode
12596 The @kbd{m D} (@code{calc-default-simplify-mode}) command restores the
12597 default simplifications for all formulas. This includes many easy and
12598 fast algebraic simplifications such as @expr{a+0} to @expr{a}, and
12599 @expr{a + 2 a} to @expr{3 a}, as well as evaluating functions like
12600 @expr{@tfn{deriv}(x^2, x)} to @expr{2 x}.
12603 @pindex calc-bin-simplify-mode
12604 The @kbd{m B} (@code{calc-bin-simplify-mode}) mode applies the default
12605 simplifications to a result and then, if the result is an integer,
12606 uses the @kbd{b c} (@code{calc-clip}) command to clip the integer according
12607 to the current binary word size. @xref{Binary Functions}. Real numbers
12608 are rounded to the nearest integer and then clipped; other kinds of
12609 results (after the default simplifications) are left alone.
12612 @pindex calc-alg-simplify-mode
12613 The @kbd{m A} (@code{calc-alg-simplify-mode}) mode does algebraic
12614 simplification; it applies all the default simplifications, and also
12615 the more powerful (and slower) simplifications made by @kbd{a s}
12616 (@code{calc-simplify}). @xref{Algebraic Simplifications}.
12619 @pindex calc-ext-simplify-mode
12620 The @kbd{m E} (@code{calc-ext-simplify-mode}) mode does ``extended''
12621 algebraic simplification, as by the @kbd{a e} (@code{calc-simplify-extended})
12622 command. @xref{Unsafe Simplifications}.
12625 @pindex calc-units-simplify-mode
12626 The @kbd{m U} (@code{calc-units-simplify-mode}) mode does units
12627 simplification; it applies the command @kbd{u s}
12628 (@code{calc-simplify-units}), which in turn
12629 is a superset of @kbd{a s}. In this mode, variable names which
12630 are identifiable as unit names (like @samp{mm} for ``millimeters'')
12631 are simplified with their unit definitions in mind.
12633 A common technique is to set the simplification mode down to the lowest
12634 amount of simplification you will allow to be applied automatically, then
12635 use manual commands like @kbd{a s} and @kbd{c c} (@code{calc-clean}) to
12636 perform higher types of simplifications on demand. @xref{Algebraic
12637 Definitions}, for another sample use of No-Simplification mode.
12639 @node Declarations, Display Modes, Simplification Modes, Mode Settings
12640 @section Declarations
12643 A @dfn{declaration} is a statement you make that promises you will
12644 use a certain variable or function in a restricted way. This may
12645 give Calc the freedom to do things that it couldn't do if it had to
12646 take the fully general situation into account.
12649 * Declaration Basics::
12650 * Kinds of Declarations::
12651 * Functions for Declarations::
12654 @node Declaration Basics, Kinds of Declarations, Declarations, Declarations
12655 @subsection Declaration Basics
12659 @pindex calc-declare-variable
12660 The @kbd{s d} (@code{calc-declare-variable}) command is the easiest
12661 way to make a declaration for a variable. This command prompts for
12662 the variable name, then prompts for the declaration. The default
12663 at the declaration prompt is the previous declaration, if any.
12664 You can edit this declaration, or press @kbd{C-k} to erase it and
12665 type a new declaration. (Or, erase it and press @key{RET} to clear
12666 the declaration, effectively ``undeclaring'' the variable.)
12668 A declaration is in general a vector of @dfn{type symbols} and
12669 @dfn{range} values. If there is only one type symbol or range value,
12670 you can write it directly rather than enclosing it in a vector.
12671 For example, @kbd{s d foo @key{RET} real @key{RET}} declares @code{foo} to
12672 be a real number, and @kbd{s d bar @key{RET} [int, const, [1..6]] @key{RET}}
12673 declares @code{bar} to be a constant integer between 1 and 6.
12674 (Actually, you can omit the outermost brackets and Calc will
12675 provide them for you: @kbd{s d bar @key{RET} int, const, [1..6] @key{RET}}.)
12677 @cindex @code{Decls} variable
12679 Declarations in Calc are kept in a special variable called @code{Decls}.
12680 This variable encodes the set of all outstanding declarations in
12681 the form of a matrix. Each row has two elements: A variable or
12682 vector of variables declared by that row, and the declaration
12683 specifier as described above. You can use the @kbd{s D} command to
12684 edit this variable if you wish to see all the declarations at once.
12685 @xref{Operations on Variables}, for a description of this command
12686 and the @kbd{s p} command that allows you to save your declarations
12687 permanently if you wish.
12689 Items being declared can also be function calls. The arguments in
12690 the call are ignored; the effect is to say that this function returns
12691 values of the declared type for any valid arguments. The @kbd{s d}
12692 command declares only variables, so if you wish to make a function
12693 declaration you will have to edit the @code{Decls} matrix yourself.
12695 For example, the declaration matrix
12701 [ f(1,2,3), [0 .. inf) ] ]
12706 declares that @code{foo} represents a real number, @code{j}, @code{k}
12707 and @code{n} represent integers, and the function @code{f} always
12708 returns a real number in the interval shown.
12711 If there is a declaration for the variable @code{All}, then that
12712 declaration applies to all variables that are not otherwise declared.
12713 It does not apply to function names. For example, using the row
12714 @samp{[All, real]} says that all your variables are real unless they
12715 are explicitly declared without @code{real} in some other row.
12716 The @kbd{s d} command declares @code{All} if you give a blank
12717 response to the variable-name prompt.
12719 @node Kinds of Declarations, Functions for Declarations, Declaration Basics, Declarations
12720 @subsection Kinds of Declarations
12723 The type-specifier part of a declaration (that is, the second prompt
12724 in the @kbd{s d} command) can be a type symbol, an interval, or a
12725 vector consisting of zero or more type symbols followed by zero or
12726 more intervals or numbers that represent the set of possible values
12731 [ [ a, [1, 2, 3, 4, 5] ]
12733 [ c, [int, 1 .. 5] ] ]
12737 Here @code{a} is declared to contain one of the five integers shown;
12738 @code{b} is any number in the interval from 1 to 5 (any real number
12739 since we haven't specified), and @code{c} is any integer in that
12740 interval. Thus the declarations for @code{a} and @code{c} are
12741 nearly equivalent (see below).
12743 The type-specifier can be the empty vector @samp{[]} to say that
12744 nothing is known about a given variable's value. This is the same
12745 as not declaring the variable at all except that it overrides any
12746 @code{All} declaration which would otherwise apply.
12748 The initial value of @code{Decls} is the empty vector @samp{[]}.
12749 If @code{Decls} has no stored value or if the value stored in it
12750 is not valid, it is ignored and there are no declarations as far
12751 as Calc is concerned. (The @kbd{s d} command will replace such a
12752 malformed value with a fresh empty matrix, @samp{[]}, before recording
12753 the new declaration.) Unrecognized type symbols are ignored.
12755 The following type symbols describe what sorts of numbers will be
12756 stored in a variable:
12762 Numerical integers. (Integers or integer-valued floats.)
12764 Fractions. (Rational numbers which are not integers.)
12766 Rational numbers. (Either integers or fractions.)
12768 Floating-point numbers.
12770 Real numbers. (Integers, fractions, or floats. Actually,
12771 intervals and error forms with real components also count as
12774 Positive real numbers. (Strictly greater than zero.)
12776 Nonnegative real numbers. (Greater than or equal to zero.)
12778 Numbers. (Real or complex.)
12781 Calc uses this information to determine when certain simplifications
12782 of formulas are safe. For example, @samp{(x^y)^z} cannot be
12783 simplified to @samp{x^(y z)} in general; for example,
12784 @samp{((-3)^2)^1:2} is 3, but @samp{(-3)^(2*1:2) = (-3)^1} is @mathit{-3}.
12785 However, this simplification @emph{is} safe if @code{z} is known
12786 to be an integer, or if @code{x} is known to be a nonnegative
12787 real number. If you have given declarations that allow Calc to
12788 deduce either of these facts, Calc will perform this simplification
12791 Calc can apply a certain amount of logic when using declarations.
12792 For example, @samp{(x^y)^(2n+1)} will be simplified if @code{n}
12793 has been declared @code{int}; Calc knows that an integer times an
12794 integer, plus an integer, must always be an integer. (In fact,
12795 Calc would simplify @samp{(-x)^(2n+1)} to @samp{-(x^(2n+1))} since
12796 it is able to determine that @samp{2n+1} must be an odd integer.)
12798 Similarly, @samp{(abs(x)^y)^z} will be simplified to @samp{abs(x)^(y z)}
12799 because Calc knows that the @code{abs} function always returns a
12800 nonnegative real. If you had a @code{myabs} function that also had
12801 this property, you could get Calc to recognize it by adding the row
12802 @samp{[myabs(), nonneg]} to the @code{Decls} matrix.
12804 One instance of this simplification is @samp{sqrt(x^2)} (since the
12805 @code{sqrt} function is effectively a one-half power). Normally
12806 Calc leaves this formula alone. After the command
12807 @kbd{s d x @key{RET} real @key{RET}}, however, it can simplify the formula to
12808 @samp{abs(x)}. And after @kbd{s d x @key{RET} nonneg @key{RET}}, Calc can
12809 simplify this formula all the way to @samp{x}.
12811 If there are any intervals or real numbers in the type specifier,
12812 they comprise the set of possible values that the variable or
12813 function being declared can have. In particular, the type symbol
12814 @code{real} is effectively the same as the range @samp{[-inf .. inf]}
12815 (note that infinity is included in the range of possible values);
12816 @code{pos} is the same as @samp{(0 .. inf]}, and @code{nonneg} is
12817 the same as @samp{[0 .. inf]}. Saying @samp{[real, [-5 .. 5]]} is
12818 redundant because the fact that the variable is real can be
12819 deduced just from the interval, but @samp{[int, [-5 .. 5]]} and
12820 @samp{[rat, [-5 .. 5]]} are useful combinations.
12822 Note that the vector of intervals or numbers is in the same format
12823 used by Calc's set-manipulation commands. @xref{Set Operations}.
12825 The type specifier @samp{[1, 2, 3]} is equivalent to
12826 @samp{[numint, 1, 2, 3]}, @emph{not} to @samp{[int, 1, 2, 3]}.
12827 In other words, the range of possible values means only that
12828 the variable's value must be numerically equal to a number in
12829 that range, but not that it must be equal in type as well.
12830 Calc's set operations act the same way; @samp{in(2, [1., 2., 3.])}
12831 and @samp{in(1.5, [1:2, 3:2, 5:2])} both report ``true.''
12833 If you use a conflicting combination of type specifiers, the
12834 results are unpredictable. An example is @samp{[pos, [0 .. 5]]},
12835 where the interval does not lie in the range described by the
12838 ``Real'' declarations mostly affect simplifications involving powers
12839 like the one described above. Another case where they are used
12840 is in the @kbd{a P} command which returns a list of all roots of a
12841 polynomial; if the variable has been declared real, only the real
12842 roots (if any) will be included in the list.
12844 ``Integer'' declarations are used for simplifications which are valid
12845 only when certain values are integers (such as @samp{(x^y)^z}
12848 Another command that makes use of declarations is @kbd{a s}, when
12849 simplifying equations and inequalities. It will cancel @code{x}
12850 from both sides of @samp{a x = b x} only if it is sure @code{x}
12851 is non-zero, say, because it has a @code{pos} declaration.
12852 To declare specifically that @code{x} is real and non-zero,
12853 use @samp{[[-inf .. 0), (0 .. inf]]}. (There is no way in the
12854 current notation to say that @code{x} is nonzero but not necessarily
12855 real.) The @kbd{a e} command does ``unsafe'' simplifications,
12856 including cancelling @samp{x} from the equation when @samp{x} is
12857 not known to be nonzero.
12859 Another set of type symbols distinguish between scalars and vectors.
12863 The value is not a vector.
12865 The value is a vector.
12867 The value is a matrix (a rectangular vector of vectors).
12869 The value is a square matrix.
12872 These type symbols can be combined with the other type symbols
12873 described above; @samp{[int, matrix]} describes an object which
12874 is a matrix of integers.
12876 Scalar/vector declarations are used to determine whether certain
12877 algebraic operations are safe. For example, @samp{[a, b, c] + x}
12878 is normally not simplified to @samp{[a + x, b + x, c + x]}, but
12879 it will be if @code{x} has been declared @code{scalar}. On the
12880 other hand, multiplication is usually assumed to be commutative,
12881 but the terms in @samp{x y} will never be exchanged if both @code{x}
12882 and @code{y} are known to be vectors or matrices. (Calc currently
12883 never distinguishes between @code{vector} and @code{matrix}
12886 @xref{Matrix Mode}, for a discussion of Matrix mode and
12887 Scalar mode, which are similar to declaring @samp{[All, matrix]}
12888 or @samp{[All, scalar]} but much more convenient.
12890 One more type symbol that is recognized is used with the @kbd{H a d}
12891 command for taking total derivatives of a formula. @xref{Calculus}.
12895 The value is a constant with respect to other variables.
12898 Calc does not check the declarations for a variable when you store
12899 a value in it. However, storing @mathit{-3.5} in a variable that has
12900 been declared @code{pos}, @code{int}, or @code{matrix} may have
12901 unexpected effects; Calc may evaluate @samp{sqrt(x^2)} to @expr{3.5}
12902 if it substitutes the value first, or to @expr{-3.5} if @code{x}
12903 was declared @code{pos} and the formula @samp{sqrt(x^2)} is
12904 simplified to @samp{x} before the value is substituted. Before
12905 using a variable for a new purpose, it is best to use @kbd{s d}
12906 or @kbd{s D} to check to make sure you don't still have an old
12907 declaration for the variable that will conflict with its new meaning.
12909 @node Functions for Declarations, , Kinds of Declarations, Declarations
12910 @subsection Functions for Declarations
12913 Calc has a set of functions for accessing the current declarations
12914 in a convenient manner. These functions return 1 if the argument
12915 can be shown to have the specified property, or 0 if the argument
12916 can be shown @emph{not} to have that property; otherwise they are
12917 left unevaluated. These functions are suitable for use with rewrite
12918 rules (@pxref{Conditional Rewrite Rules}) or programming constructs
12919 (@pxref{Conditionals in Macros}). They can be entered only using
12920 algebraic notation. @xref{Logical Operations}, for functions
12921 that perform other tests not related to declarations.
12923 For example, @samp{dint(17)} returns 1 because 17 is an integer, as
12924 do @samp{dint(n)} and @samp{dint(2 n - 3)} if @code{n} has been declared
12925 @code{int}, but @samp{dint(2.5)} and @samp{dint(n + 0.5)} return 0.
12926 Calc consults knowledge of its own built-in functions as well as your
12927 own declarations: @samp{dint(floor(x))} returns 1.
12941 The @code{dint} function checks if its argument is an integer.
12942 The @code{dnatnum} function checks if its argument is a natural
12943 number, i.e., a nonnegative integer. The @code{dnumint} function
12944 checks if its argument is numerically an integer, i.e., either an
12945 integer or an integer-valued float. Note that these and the other
12946 data type functions also accept vectors or matrices composed of
12947 suitable elements, and that real infinities @samp{inf} and @samp{-inf}
12948 are considered to be integers for the purposes of these functions.
12954 The @code{drat} function checks if its argument is rational, i.e.,
12955 an integer or fraction. Infinities count as rational, but intervals
12956 and error forms do not.
12962 The @code{dreal} function checks if its argument is real. This
12963 includes integers, fractions, floats, real error forms, and intervals.
12969 The @code{dimag} function checks if its argument is imaginary,
12970 i.e., is mathematically equal to a real number times @expr{i}.
12984 The @code{dpos} function checks for positive (but nonzero) reals.
12985 The @code{dneg} function checks for negative reals. The @code{dnonneg}
12986 function checks for nonnegative reals, i.e., reals greater than or
12987 equal to zero. Note that the @kbd{a s} command can simplify an
12988 expression like @expr{x > 0} to 1 or 0 using @code{dpos}, and that
12989 @kbd{a s} is effectively applied to all conditions in rewrite rules,
12990 so the actual functions @code{dpos}, @code{dneg}, and @code{dnonneg}
12991 are rarely necessary.
12997 The @code{dnonzero} function checks that its argument is nonzero.
12998 This includes all nonzero real or complex numbers, all intervals that
12999 do not include zero, all nonzero modulo forms, vectors all of whose
13000 elements are nonzero, and variables or formulas whose values can be
13001 deduced to be nonzero. It does not include error forms, since they
13002 represent values which could be anything including zero. (This is
13003 also the set of objects considered ``true'' in conditional contexts.)
13013 The @code{deven} function returns 1 if its argument is known to be
13014 an even integer (or integer-valued float); it returns 0 if its argument
13015 is known not to be even (because it is known to be odd or a non-integer).
13016 The @kbd{a s} command uses this to simplify a test of the form
13017 @samp{x % 2 = 0}. There is also an analogous @code{dodd} function.
13023 The @code{drange} function returns a set (an interval or a vector
13024 of intervals and/or numbers; @pxref{Set Operations}) that describes
13025 the set of possible values of its argument. If the argument is
13026 a variable or a function with a declaration, the range is copied
13027 from the declaration. Otherwise, the possible signs of the
13028 expression are determined using a method similar to @code{dpos},
13029 etc., and a suitable set like @samp{[0 .. inf]} is returned. If
13030 the expression is not provably real, the @code{drange} function
13031 remains unevaluated.
13037 The @code{dscalar} function returns 1 if its argument is provably
13038 scalar, or 0 if its argument is provably non-scalar. It is left
13039 unevaluated if this cannot be determined. (If Matrix mode or Scalar
13040 mode is in effect, this function returns 1 or 0, respectively,
13041 if it has no other information.) When Calc interprets a condition
13042 (say, in a rewrite rule) it considers an unevaluated formula to be
13043 ``false.'' Thus, @samp{dscalar(a)} is ``true'' only if @code{a} is
13044 provably scalar, and @samp{!dscalar(a)} is ``true'' only if @code{a}
13045 is provably non-scalar; both are ``false'' if there is insufficient
13046 information to tell.
13048 @node Display Modes, Language Modes, Declarations, Mode Settings
13049 @section Display Modes
13052 The commands in this section are two-key sequences beginning with the
13053 @kbd{d} prefix. The @kbd{d l} (@code{calc-line-numbering}) and @kbd{d b}
13054 (@code{calc-line-breaking}) commands are described elsewhere;
13055 @pxref{Stack Basics} and @pxref{Normal Language Modes}, respectively.
13056 Display formats for vectors and matrices are also covered elsewhere;
13057 @pxref{Vector and Matrix Formats}.
13059 One thing all display modes have in common is their treatment of the
13060 @kbd{H} prefix. This prefix causes any mode command that would normally
13061 refresh the stack to leave the stack display alone. The word ``Dirty''
13062 will appear in the mode line when Calc thinks the stack display may not
13063 reflect the latest mode settings.
13065 @kindex d @key{RET}
13066 @pindex calc-refresh-top
13067 The @kbd{d @key{RET}} (@code{calc-refresh-top}) command reformats the
13068 top stack entry according to all the current modes. Positive prefix
13069 arguments reformat the top @var{n} entries; negative prefix arguments
13070 reformat the specified entry, and a prefix of zero is equivalent to
13071 @kbd{d @key{SPC}} (@code{calc-refresh}), which reformats the entire stack.
13072 For example, @kbd{H d s M-2 d @key{RET}} changes to scientific notation
13073 but reformats only the top two stack entries in the new mode.
13075 The @kbd{I} prefix has another effect on the display modes. The mode
13076 is set only temporarily; the top stack entry is reformatted according
13077 to that mode, then the original mode setting is restored. In other
13078 words, @kbd{I d s} is equivalent to @kbd{H d s d @key{RET} H d (@var{old mode})}.
13082 * Grouping Digits::
13084 * Complex Formats::
13085 * Fraction Formats::
13088 * Truncating the Stack::
13093 @node Radix Modes, Grouping Digits, Display Modes, Display Modes
13094 @subsection Radix Modes
13097 @cindex Radix display
13098 @cindex Non-decimal numbers
13099 @cindex Decimal and non-decimal numbers
13100 Calc normally displays numbers in decimal (@dfn{base-10} or @dfn{radix-10})
13101 notation. Calc can actually display in any radix from two (binary) to 36.
13102 When the radix is above 10, the letters @code{A} to @code{Z} are used as
13103 digits. When entering such a number, letter keys are interpreted as
13104 potential digits rather than terminating numeric entry mode.
13110 @cindex Hexadecimal integers
13111 @cindex Octal integers
13112 The key sequences @kbd{d 2}, @kbd{d 8}, @kbd{d 6}, and @kbd{d 0} select
13113 binary, octal, hexadecimal, and decimal as the current display radix,
13114 respectively. Numbers can always be entered in any radix, though the
13115 current radix is used as a default if you press @kbd{#} without any initial
13116 digits. A number entered without a @kbd{#} is @emph{always} interpreted
13121 To set the radix generally, use @kbd{d r} (@code{calc-radix}) and enter
13122 an integer from 2 to 36. You can specify the radix as a numeric prefix
13123 argument; otherwise you will be prompted for it.
13126 @pindex calc-leading-zeros
13127 @cindex Leading zeros
13128 Integers normally are displayed with however many digits are necessary to
13129 represent the integer and no more. The @kbd{d z} (@code{calc-leading-zeros})
13130 command causes integers to be padded out with leading zeros according to the
13131 current binary word size. (@xref{Binary Functions}, for a discussion of
13132 word size.) If the absolute value of the word size is @expr{w}, all integers
13133 are displayed with at least enough digits to represent
13134 @texline @math{2^w-1}
13135 @infoline @expr{(2^w)-1}
13136 in the current radix. (Larger integers will still be displayed in their
13139 @node Grouping Digits, Float Formats, Radix Modes, Display Modes
13140 @subsection Grouping Digits
13144 @pindex calc-group-digits
13145 @cindex Grouping digits
13146 @cindex Digit grouping
13147 Long numbers can be hard to read if they have too many digits. For
13148 example, the factorial of 30 is 33 digits long! Press @kbd{d g}
13149 (@code{calc-group-digits}) to enable @dfn{Grouping} mode, in which digits
13150 are displayed in clumps of 3 or 4 (depending on the current radix)
13151 separated by commas.
13153 The @kbd{d g} command toggles grouping on and off.
13154 With a numeric prefix of 0, this command displays the current state of
13155 the grouping flag; with an argument of minus one it disables grouping;
13156 with a positive argument @expr{N} it enables grouping on every @expr{N}
13157 digits. For floating-point numbers, grouping normally occurs only
13158 before the decimal point. A negative prefix argument @expr{-N} enables
13159 grouping every @expr{N} digits both before and after the decimal point.
13162 @pindex calc-group-char
13163 The @kbd{d ,} (@code{calc-group-char}) command allows you to choose any
13164 character as the grouping separator. The default is the comma character.
13165 If you find it difficult to read vectors of large integers grouped with
13166 commas, you may wish to use spaces or some other character instead.
13167 This command takes the next character you type, whatever it is, and
13168 uses it as the digit separator. As a special case, @kbd{d , \} selects
13169 @samp{\,} (@TeX{}'s thin-space symbol) as the digit separator.
13171 Please note that grouped numbers will not generally be parsed correctly
13172 if re-read in textual form, say by the use of @kbd{C-x * y} and @kbd{C-x * g}.
13173 (@xref{Kill and Yank}, for details on these commands.) One exception is
13174 the @samp{\,} separator, which doesn't interfere with parsing because it
13175 is ignored by @TeX{} language mode.
13177 @node Float Formats, Complex Formats, Grouping Digits, Display Modes
13178 @subsection Float Formats
13181 Floating-point quantities are normally displayed in standard decimal
13182 form, with scientific notation used if the exponent is especially high
13183 or low. All significant digits are normally displayed. The commands
13184 in this section allow you to choose among several alternative display
13185 formats for floats.
13188 @pindex calc-normal-notation
13189 The @kbd{d n} (@code{calc-normal-notation}) command selects the normal
13190 display format. All significant figures in a number are displayed.
13191 With a positive numeric prefix, numbers are rounded if necessary to
13192 that number of significant digits. With a negative numerix prefix,
13193 the specified number of significant digits less than the current
13194 precision is used. (Thus @kbd{C-u -2 d n} displays 10 digits if the
13195 current precision is 12.)
13198 @pindex calc-fix-notation
13199 The @kbd{d f} (@code{calc-fix-notation}) command selects fixed-point
13200 notation. The numeric argument is the number of digits after the
13201 decimal point, zero or more. This format will relax into scientific
13202 notation if a nonzero number would otherwise have been rounded all the
13203 way to zero. Specifying a negative number of digits is the same as
13204 for a positive number, except that small nonzero numbers will be rounded
13205 to zero rather than switching to scientific notation.
13208 @pindex calc-sci-notation
13209 @cindex Scientific notation, display of
13210 The @kbd{d s} (@code{calc-sci-notation}) command selects scientific
13211 notation. A positive argument sets the number of significant figures
13212 displayed, of which one will be before and the rest after the decimal
13213 point. A negative argument works the same as for @kbd{d n} format.
13214 The default is to display all significant digits.
13217 @pindex calc-eng-notation
13218 @cindex Engineering notation, display of
13219 The @kbd{d e} (@code{calc-eng-notation}) command selects engineering
13220 notation. This is similar to scientific notation except that the
13221 exponent is rounded down to a multiple of three, with from one to three
13222 digits before the decimal point. An optional numeric prefix sets the
13223 number of significant digits to display, as for @kbd{d s}.
13225 It is important to distinguish between the current @emph{precision} and
13226 the current @emph{display format}. After the commands @kbd{C-u 10 p}
13227 and @kbd{C-u 6 d n} the Calculator computes all results to ten
13228 significant figures but displays only six. (In fact, intermediate
13229 calculations are often carried to one or two more significant figures,
13230 but values placed on the stack will be rounded down to ten figures.)
13231 Numbers are never actually rounded to the display precision for storage,
13232 except by commands like @kbd{C-k} and @kbd{C-x * y} which operate on the
13233 actual displayed text in the Calculator buffer.
13236 @pindex calc-point-char
13237 The @kbd{d .} (@code{calc-point-char}) command selects the character used
13238 as a decimal point. Normally this is a period; users in some countries
13239 may wish to change this to a comma. Note that this is only a display
13240 style; on entry, periods must always be used to denote floating-point
13241 numbers, and commas to separate elements in a list.
13243 @node Complex Formats, Fraction Formats, Float Formats, Display Modes
13244 @subsection Complex Formats
13248 @pindex calc-complex-notation
13249 There are three supported notations for complex numbers in rectangular
13250 form. The default is as a pair of real numbers enclosed in parentheses
13251 and separated by a comma: @samp{(a,b)}. The @kbd{d c}
13252 (@code{calc-complex-notation}) command selects this style.
13255 @pindex calc-i-notation
13257 @pindex calc-j-notation
13258 The other notations are @kbd{d i} (@code{calc-i-notation}), in which
13259 numbers are displayed in @samp{a+bi} form, and @kbd{d j}
13260 (@code{calc-j-notation}) which displays the form @samp{a+bj} preferred
13261 in some disciplines.
13263 @cindex @code{i} variable
13265 Complex numbers are normally entered in @samp{(a,b)} format.
13266 If you enter @samp{2+3i} as an algebraic formula, it will be stored as
13267 the formula @samp{2 + 3 * i}. However, if you use @kbd{=} to evaluate
13268 this formula and you have not changed the variable @samp{i}, the @samp{i}
13269 will be interpreted as @samp{(0,1)} and the formula will be simplified
13270 to @samp{(2,3)}. Other commands (like @code{calc-sin}) will @emph{not}
13271 interpret the formula @samp{2 + 3 * i} as a complex number.
13272 @xref{Variables}, under ``special constants.''
13274 @node Fraction Formats, HMS Formats, Complex Formats, Display Modes
13275 @subsection Fraction Formats
13279 @pindex calc-over-notation
13280 Display of fractional numbers is controlled by the @kbd{d o}
13281 (@code{calc-over-notation}) command. By default, a number like
13282 eight thirds is displayed in the form @samp{8:3}. The @kbd{d o} command
13283 prompts for a one- or two-character format. If you give one character,
13284 that character is used as the fraction separator. Common separators are
13285 @samp{:} and @samp{/}. (During input of numbers, the @kbd{:} key must be
13286 used regardless of the display format; in particular, the @kbd{/} is used
13287 for RPN-style division, @emph{not} for entering fractions.)
13289 If you give two characters, fractions use ``integer-plus-fractional-part''
13290 notation. For example, the format @samp{+/} would display eight thirds
13291 as @samp{2+2/3}. If two colons are present in a number being entered,
13292 the number is interpreted in this form (so that the entries @kbd{2:2:3}
13293 and @kbd{8:3} are equivalent).
13295 It is also possible to follow the one- or two-character format with
13296 a number. For example: @samp{:10} or @samp{+/3}. In this case,
13297 Calc adjusts all fractions that are displayed to have the specified
13298 denominator, if possible. Otherwise it adjusts the denominator to
13299 be a multiple of the specified value. For example, in @samp{:6} mode
13300 the fraction @expr{1:6} will be unaffected, but @expr{2:3} will be
13301 displayed as @expr{4:6}, @expr{1:2} will be displayed as @expr{3:6},
13302 and @expr{1:8} will be displayed as @expr{3:24}. Integers are also
13303 affected by this mode: 3 is displayed as @expr{18:6}. Note that the
13304 format @samp{:1} writes fractions the same as @samp{:}, but it writes
13305 integers as @expr{n:1}.
13307 The fraction format does not affect the way fractions or integers are
13308 stored, only the way they appear on the screen. The fraction format
13309 never affects floats.
13311 @node HMS Formats, Date Formats, Fraction Formats, Display Modes
13312 @subsection HMS Formats
13316 @pindex calc-hms-notation
13317 The @kbd{d h} (@code{calc-hms-notation}) command controls the display of
13318 HMS (hours-minutes-seconds) forms. It prompts for a string which
13319 consists basically of an ``hours'' marker, optional punctuation, a
13320 ``minutes'' marker, more optional punctuation, and a ``seconds'' marker.
13321 Punctuation is zero or more spaces, commas, or semicolons. The hours
13322 marker is one or more non-punctuation characters. The minutes and
13323 seconds markers must be single non-punctuation characters.
13325 The default HMS format is @samp{@@ ' "}, producing HMS values of the form
13326 @samp{23@@ 30' 15.75"}. The format @samp{deg, ms} would display this same
13327 value as @samp{23deg, 30m15.75s}. During numeric entry, the @kbd{h} or @kbd{o}
13328 keys are recognized as synonyms for @kbd{@@} regardless of display format.
13329 The @kbd{m} and @kbd{s} keys are recognized as synonyms for @kbd{'} and
13330 @kbd{"}, respectively, but only if an @kbd{@@} (or @kbd{h} or @kbd{o}) has
13331 already been typed; otherwise, they have their usual meanings
13332 (@kbd{m-} prefix and @kbd{s-} prefix). Thus, @kbd{5 "}, @kbd{0 @@ 5 "}, and
13333 @kbd{0 h 5 s} are some of the ways to enter the quantity ``five seconds.''
13334 The @kbd{'} key is recognized as ``minutes'' only if @kbd{@@} (or @kbd{h} or
13335 @kbd{o}) has already been pressed; otherwise it means to switch to algebraic
13338 @node Date Formats, Truncating the Stack, HMS Formats, Display Modes
13339 @subsection Date Formats
13343 @pindex calc-date-notation
13344 The @kbd{d d} (@code{calc-date-notation}) command controls the display
13345 of date forms (@pxref{Date Forms}). It prompts for a string which
13346 contains letters that represent the various parts of a date and time.
13347 To show which parts should be omitted when the form represents a pure
13348 date with no time, parts of the string can be enclosed in @samp{< >}
13349 marks. If you don't include @samp{< >} markers in the format, Calc
13350 guesses at which parts, if any, should be omitted when formatting
13353 The default format is: @samp{<H:mm:SSpp >Www Mmm D, YYYY}.
13354 An example string in this format is @samp{3:32pm Wed Jan 9, 1991}.
13355 If you enter a blank format string, this default format is
13358 Calc uses @samp{< >} notation for nameless functions as well as for
13359 dates. @xref{Specifying Operators}. To avoid confusion with nameless
13360 functions, your date formats should avoid using the @samp{#} character.
13363 * Date Formatting Codes::
13364 * Free-Form Dates::
13365 * Standard Date Formats::
13368 @node Date Formatting Codes, Free-Form Dates, Date Formats, Date Formats
13369 @subsubsection Date Formatting Codes
13372 When displaying a date, the current date format is used. All
13373 characters except for letters and @samp{<} and @samp{>} are
13374 copied literally when dates are formatted. The portion between
13375 @samp{< >} markers is omitted for pure dates, or included for
13376 date/time forms. Letters are interpreted according to the table
13379 When dates are read in during algebraic entry, Calc first tries to
13380 match the input string to the current format either with or without
13381 the time part. The punctuation characters (including spaces) must
13382 match exactly; letter fields must correspond to suitable text in
13383 the input. If this doesn't work, Calc checks if the input is a
13384 simple number; if so, the number is interpreted as a number of days
13385 since Jan 1, 1 AD. Otherwise, Calc tries a much more relaxed and
13386 flexible algorithm which is described in the next section.
13388 Weekday names are ignored during reading.
13390 Two-digit year numbers are interpreted as lying in the range
13391 from 1941 to 2039. Years outside that range are always
13392 entered and displayed in full. Year numbers with a leading
13393 @samp{+} sign are always interpreted exactly, allowing the
13394 entry and display of the years 1 through 99 AD.
13396 Here is a complete list of the formatting codes for dates:
13400 Year: ``91'' for 1991, ``7'' for 2007, ``+23'' for 23 AD.
13402 Year: ``91'' for 1991, ``07'' for 2007, ``+23'' for 23 AD.
13404 Year: ``91'' for 1991, `` 7'' for 2007, ``+23'' for 23 AD.
13406 Year: ``1991'' for 1991, ``23'' for 23 AD.
13408 Year: ``1991'' for 1991, ``+23'' for 23 AD.
13410 Year: ``ad'' or blank.
13412 Year: ``AD'' or blank.
13414 Year: ``ad '' or blank. (Note trailing space.)
13416 Year: ``AD '' or blank.
13418 Year: ``a.d.'' or blank.
13420 Year: ``A.D.'' or blank.
13422 Year: ``bc'' or blank.
13424 Year: ``BC'' or blank.
13426 Year: `` bc'' or blank. (Note leading space.)
13428 Year: `` BC'' or blank.
13430 Year: ``b.c.'' or blank.
13432 Year: ``B.C.'' or blank.
13434 Month: ``8'' for August.
13436 Month: ``08'' for August.
13438 Month: `` 8'' for August.
13440 Month: ``AUG'' for August.
13442 Month: ``Aug'' for August.
13444 Month: ``aug'' for August.
13446 Month: ``AUGUST'' for August.
13448 Month: ``August'' for August.
13450 Day: ``7'' for 7th day of month.
13452 Day: ``07'' for 7th day of month.
13454 Day: `` 7'' for 7th day of month.
13456 Weekday: ``0'' for Sunday, ``6'' for Saturday.
13458 Weekday: ``SUN'' for Sunday.
13460 Weekday: ``Sun'' for Sunday.
13462 Weekday: ``sun'' for Sunday.
13464 Weekday: ``SUNDAY'' for Sunday.
13466 Weekday: ``Sunday'' for Sunday.
13468 Day of year: ``34'' for Feb. 3.
13470 Day of year: ``034'' for Feb. 3.
13472 Day of year: `` 34'' for Feb. 3.
13474 Hour: ``5'' for 5 AM; ``17'' for 5 PM.
13476 Hour: ``05'' for 5 AM; ``17'' for 5 PM.
13478 Hour: `` 5'' for 5 AM; ``17'' for 5 PM.
13480 Hour: ``5'' for 5 AM and 5 PM.
13482 Hour: ``05'' for 5 AM and 5 PM.
13484 Hour: `` 5'' for 5 AM and 5 PM.
13486 AM/PM: ``a'' or ``p''.
13488 AM/PM: ``A'' or ``P''.
13490 AM/PM: ``am'' or ``pm''.
13492 AM/PM: ``AM'' or ``PM''.
13494 AM/PM: ``a.m.'' or ``p.m.''.
13496 AM/PM: ``A.M.'' or ``P.M.''.
13498 Minutes: ``7'' for 7.
13500 Minutes: ``07'' for 7.
13502 Minutes: `` 7'' for 7.
13504 Seconds: ``7'' for 7; ``7.23'' for 7.23.
13506 Seconds: ``07'' for 7; ``07.23'' for 7.23.
13508 Seconds: `` 7'' for 7; `` 7.23'' for 7.23.
13510 Optional seconds: ``07'' for 7; blank for 0.
13512 Optional seconds: `` 7'' for 7; blank for 0.
13514 Numeric date/time: ``726842.25'' for 6:00am Wed Jan 9, 1991.
13516 Numeric date: ``726842'' for any time on Wed Jan 9, 1991.
13518 Julian date/time: ``2448265.75'' for 6:00am Wed Jan 9, 1991.
13520 Julian date: ``2448266'' for any time on Wed Jan 9, 1991.
13522 Unix time: ``663400800'' for 6:00am Wed Jan 9, 1991.
13524 Brackets suppression. An ``X'' at the front of the format
13525 causes the surrounding @w{@samp{< >}} delimiters to be omitted
13526 when formatting dates. Note that the brackets are still
13527 required for algebraic entry.
13530 If ``SS'' or ``BS'' (optional seconds) is preceded by a colon, the
13531 colon is also omitted if the seconds part is zero.
13533 If ``bb,'' ``bbb'' or ``bbbb'' or their upper-case equivalents
13534 appear in the format, then negative year numbers are displayed
13535 without a minus sign. Note that ``aa'' and ``bb'' are mutually
13536 exclusive. Some typical usages would be @samp{YYYY AABB};
13537 @samp{AAAYYYYBBB}; @samp{YYYYBBB}.
13539 The formats ``YY,'' ``YYYY,'' ``MM,'' ``DD,'' ``ddd,'' ``hh,'' ``HH,''
13540 ``mm,'' ``ss,'' and ``SS'' actually match any number of digits during
13541 reading unless several of these codes are strung together with no
13542 punctuation in between, in which case the input must have exactly as
13543 many digits as there are letters in the format.
13545 The ``j,'' ``J,'' and ``U'' formats do not make any time zone
13546 adjustment. They effectively use @samp{julian(x,0)} and
13547 @samp{unixtime(x,0)} to make the conversion; @pxref{Date Arithmetic}.
13549 @node Free-Form Dates, Standard Date Formats, Date Formatting Codes, Date Formats
13550 @subsubsection Free-Form Dates
13553 When reading a date form during algebraic entry, Calc falls back
13554 on the algorithm described here if the input does not exactly
13555 match the current date format. This algorithm generally
13556 ``does the right thing'' and you don't have to worry about it,
13557 but it is described here in full detail for the curious.
13559 Calc does not distinguish between upper- and lower-case letters
13560 while interpreting dates.
13562 First, the time portion, if present, is located somewhere in the
13563 text and then removed. The remaining text is then interpreted as
13566 A time is of the form @samp{hh:mm:ss}, possibly with the seconds
13567 part omitted and possibly with an AM/PM indicator added to indicate
13568 12-hour time. If the AM/PM is present, the minutes may also be
13569 omitted. The AM/PM part may be any of the words @samp{am},
13570 @samp{pm}, @samp{noon}, or @samp{midnight}; each of these may be
13571 abbreviated to one letter, and the alternate forms @samp{a.m.},
13572 @samp{p.m.}, and @samp{mid} are also understood. Obviously
13573 @samp{noon} and @samp{midnight} are allowed only on 12:00:00.
13574 The words @samp{noon}, @samp{mid}, and @samp{midnight} are also
13575 recognized with no number attached.
13577 If there is no AM/PM indicator, the time is interpreted in 24-hour
13580 To read the date portion, all words and numbers are isolated
13581 from the string; other characters are ignored. All words must
13582 be either month names or day-of-week names (the latter of which
13583 are ignored). Names can be written in full or as three-letter
13586 Large numbers, or numbers with @samp{+} or @samp{-} signs,
13587 are interpreted as years. If one of the other numbers is
13588 greater than 12, then that must be the day and the remaining
13589 number in the input is therefore the month. Otherwise, Calc
13590 assumes the month, day and year are in the same order that they
13591 appear in the current date format. If the year is omitted, the
13592 current year is taken from the system clock.
13594 If there are too many or too few numbers, or any unrecognizable
13595 words, then the input is rejected.
13597 If there are any large numbers (of five digits or more) other than
13598 the year, they are ignored on the assumption that they are something
13599 like Julian dates that were included along with the traditional
13600 date components when the date was formatted.
13602 One of the words @samp{ad}, @samp{a.d.}, @samp{bc}, or @samp{b.c.}
13603 may optionally be used; the latter two are equivalent to a
13604 minus sign on the year value.
13606 If you always enter a four-digit year, and use a name instead
13607 of a number for the month, there is no danger of ambiguity.
13609 @node Standard Date Formats, , Free-Form Dates, Date Formats
13610 @subsubsection Standard Date Formats
13613 There are actually ten standard date formats, numbered 0 through 9.
13614 Entering a blank line at the @kbd{d d} command's prompt gives
13615 you format number 1, Calc's usual format. You can enter any digit
13616 to select the other formats.
13618 To create your own standard date formats, give a numeric prefix
13619 argument from 0 to 9 to the @w{@kbd{d d}} command. The format you
13620 enter will be recorded as the new standard format of that
13621 number, as well as becoming the new current date format.
13622 You can save your formats permanently with the @w{@kbd{m m}}
13623 command (@pxref{Mode Settings}).
13627 @samp{N} (Numerical format)
13629 @samp{<H:mm:SSpp >Www Mmm D, YYYY} (American format)
13631 @samp{D Mmm YYYY<, h:mm:SS>} (European format)
13633 @samp{Www Mmm BD< hh:mm:ss> YYYY} (Unix written date format)
13635 @samp{M/D/Y< H:mm:SSpp>} (American slashed format)
13637 @samp{D.M.Y< h:mm:SS>} (European dotted format)
13639 @samp{M-D-Y< H:mm:SSpp>} (American dashed format)
13641 @samp{D-M-Y< h:mm:SS>} (European dashed format)
13643 @samp{j<, h:mm:ss>} (Julian day plus time)
13645 @samp{YYddd< hh:mm:ss>} (Year-day format)
13648 @node Truncating the Stack, Justification, Date Formats, Display Modes
13649 @subsection Truncating the Stack
13653 @pindex calc-truncate-stack
13654 @cindex Truncating the stack
13655 @cindex Narrowing the stack
13656 The @kbd{d t} (@code{calc-truncate-stack}) command moves the @samp{.}@:
13657 line that marks the top-of-stack up or down in the Calculator buffer.
13658 The number right above that line is considered to the be at the top of
13659 the stack. Any numbers below that line are ``hidden'' from all stack
13660 operations (although still visible to the user). This is similar to the
13661 Emacs ``narrowing'' feature, except that the values below the @samp{.}
13662 are @emph{visible}, just temporarily frozen. This feature allows you to
13663 keep several independent calculations running at once in different parts
13664 of the stack, or to apply a certain command to an element buried deep in
13667 Pressing @kbd{d t} by itself moves the @samp{.} to the line the cursor
13668 is on. Thus, this line and all those below it become hidden. To un-hide
13669 these lines, move down to the end of the buffer and press @w{@kbd{d t}}.
13670 With a positive numeric prefix argument @expr{n}, @kbd{d t} hides the
13671 bottom @expr{n} values in the buffer. With a negative argument, it hides
13672 all but the top @expr{n} values. With an argument of zero, it hides zero
13673 values, i.e., moves the @samp{.} all the way down to the bottom.
13676 @pindex calc-truncate-up
13678 @pindex calc-truncate-down
13679 The @kbd{d [} (@code{calc-truncate-up}) and @kbd{d ]}
13680 (@code{calc-truncate-down}) commands move the @samp{.} up or down one
13681 line at a time (or several lines with a prefix argument).
13683 @node Justification, Labels, Truncating the Stack, Display Modes
13684 @subsection Justification
13688 @pindex calc-left-justify
13690 @pindex calc-center-justify
13692 @pindex calc-right-justify
13693 Values on the stack are normally left-justified in the window. You can
13694 control this arrangement by typing @kbd{d <} (@code{calc-left-justify}),
13695 @kbd{d >} (@code{calc-right-justify}), or @kbd{d =}
13696 (@code{calc-center-justify}). For example, in Right-Justification mode,
13697 stack entries are displayed flush-right against the right edge of the
13700 If you change the width of the Calculator window you may have to type
13701 @kbd{d @key{SPC}} (@code{calc-refresh}) to re-align right-justified or centered
13704 Right-justification is especially useful together with fixed-point
13705 notation (see @code{d f}; @code{calc-fix-notation}). With these modes
13706 together, the decimal points on numbers will always line up.
13708 With a numeric prefix argument, the justification commands give you
13709 a little extra control over the display. The argument specifies the
13710 horizontal ``origin'' of a display line. It is also possible to
13711 specify a maximum line width using the @kbd{d b} command (@pxref{Normal
13712 Language Modes}). For reference, the precise rules for formatting and
13713 breaking lines are given below. Notice that the interaction between
13714 origin and line width is slightly different in each justification
13717 In Left-Justified mode, the line is indented by a number of spaces
13718 given by the origin (default zero). If the result is longer than the
13719 maximum line width, if given, or too wide to fit in the Calc window
13720 otherwise, then it is broken into lines which will fit; each broken
13721 line is indented to the origin.
13723 In Right-Justified mode, lines are shifted right so that the rightmost
13724 character is just before the origin, or just before the current
13725 window width if no origin was specified. If the line is too long
13726 for this, then it is broken; the current line width is used, if
13727 specified, or else the origin is used as a width if that is
13728 specified, or else the line is broken to fit in the window.
13730 In Centering mode, the origin is the column number of the center of
13731 each stack entry. If a line width is specified, lines will not be
13732 allowed to go past that width; Calc will either indent less or
13733 break the lines if necessary. If no origin is specified, half the
13734 line width or Calc window width is used.
13736 Note that, in each case, if line numbering is enabled the display
13737 is indented an additional four spaces to make room for the line
13738 number. The width of the line number is taken into account when
13739 positioning according to the current Calc window width, but not
13740 when positioning by explicit origins and widths. In the latter
13741 case, the display is formatted as specified, and then uniformly
13742 shifted over four spaces to fit the line numbers.
13744 @node Labels, , Justification, Display Modes
13749 @pindex calc-left-label
13750 The @kbd{d @{} (@code{calc-left-label}) command prompts for a string,
13751 then displays that string to the left of every stack entry. If the
13752 entries are left-justified (@pxref{Justification}), then they will
13753 appear immediately after the label (unless you specified an origin
13754 greater than the length of the label). If the entries are centered
13755 or right-justified, the label appears on the far left and does not
13756 affect the horizontal position of the stack entry.
13758 Give a blank string (with @kbd{d @{ @key{RET}}) to turn the label off.
13761 @pindex calc-right-label
13762 The @kbd{d @}} (@code{calc-right-label}) command similarly adds a
13763 label on the righthand side. It does not affect positioning of
13764 the stack entries unless they are right-justified. Also, if both
13765 a line width and an origin are given in Right-Justified mode, the
13766 stack entry is justified to the origin and the righthand label is
13767 justified to the line width.
13769 One application of labels would be to add equation numbers to
13770 formulas you are manipulating in Calc and then copying into a
13771 document (possibly using Embedded mode). The equations would
13772 typically be centered, and the equation numbers would be on the
13773 left or right as you prefer.
13775 @node Language Modes, Modes Variable, Display Modes, Mode Settings
13776 @section Language Modes
13779 The commands in this section change Calc to use a different notation for
13780 entry and display of formulas, corresponding to the conventions of some
13781 other common language such as Pascal or La@TeX{}. Objects displayed on the
13782 stack or yanked from the Calculator to an editing buffer will be formatted
13783 in the current language; objects entered in algebraic entry or yanked from
13784 another buffer will be interpreted according to the current language.
13786 The current language has no effect on things written to or read from the
13787 trail buffer, nor does it affect numeric entry. Only algebraic entry is
13788 affected. You can make even algebraic entry ignore the current language
13789 and use the standard notation by giving a numeric prefix, e.g., @kbd{C-u '}.
13791 For example, suppose the formula @samp{2*a[1] + atan(a[2])} occurs in a C
13792 program; elsewhere in the program you need the derivatives of this formula
13793 with respect to @samp{a[1]} and @samp{a[2]}. First, type @kbd{d C}
13794 to switch to C notation. Now use @code{C-u C-x * g} to grab the formula
13795 into the Calculator, @kbd{a d a[1] @key{RET}} to differentiate with respect
13796 to the first variable, and @kbd{C-x * y} to yank the formula for the derivative
13797 back into your C program. Press @kbd{U} to undo the differentiation and
13798 repeat with @kbd{a d a[2] @key{RET}} for the other derivative.
13800 Without being switched into C mode first, Calc would have misinterpreted
13801 the brackets in @samp{a[1]} and @samp{a[2]}, would not have known that
13802 @code{atan} was equivalent to Calc's built-in @code{arctan} function,
13803 and would have written the formula back with notations (like implicit
13804 multiplication) which would not have been valid for a C program.
13806 As another example, suppose you are maintaining a C program and a La@TeX{}
13807 document, each of which needs a copy of the same formula. You can grab the
13808 formula from the program in C mode, switch to La@TeX{} mode, and yank the
13809 formula into the document in La@TeX{} math-mode format.
13811 Language modes are selected by typing the letter @kbd{d} followed by a
13812 shifted letter key.
13815 * Normal Language Modes::
13816 * C FORTRAN Pascal::
13817 * TeX and LaTeX Language Modes::
13818 * Eqn Language Mode::
13819 * Yacas Language Mode::
13820 * Maxima Language Mode::
13821 * Giac Language Mode::
13822 * Mathematica Language Mode::
13823 * Maple Language Mode::
13828 @node Normal Language Modes, C FORTRAN Pascal, Language Modes, Language Modes
13829 @subsection Normal Language Modes
13833 @pindex calc-normal-language
13834 The @kbd{d N} (@code{calc-normal-language}) command selects the usual
13835 notation for Calc formulas, as described in the rest of this manual.
13836 Matrices are displayed in a multi-line tabular format, but all other
13837 objects are written in linear form, as they would be typed from the
13841 @pindex calc-flat-language
13842 @cindex Matrix display
13843 The @kbd{d O} (@code{calc-flat-language}) command selects a language
13844 identical with the normal one, except that matrices are written in
13845 one-line form along with everything else. In some applications this
13846 form may be more suitable for yanking data into other buffers.
13849 @pindex calc-line-breaking
13850 @cindex Line breaking
13851 @cindex Breaking up long lines
13852 Even in one-line mode, long formulas or vectors will still be split
13853 across multiple lines if they exceed the width of the Calculator window.
13854 The @kbd{d b} (@code{calc-line-breaking}) command turns this line-breaking
13855 feature on and off. (It works independently of the current language.)
13856 If you give a numeric prefix argument of five or greater to the @kbd{d b}
13857 command, that argument will specify the line width used when breaking
13861 @pindex calc-big-language
13862 The @kbd{d B} (@code{calc-big-language}) command selects a language
13863 which uses textual approximations to various mathematical notations,
13864 such as powers, quotients, and square roots:
13874 in place of @samp{sqrt((a+1)/b + c^2)}.
13876 Subscripts like @samp{a_i} are displayed as actual subscripts in Big
13877 mode. Double subscripts, @samp{a_i_j} (@samp{subscr(subscr(a, i), j)})
13878 are displayed as @samp{a} with subscripts separated by commas:
13879 @samp{i, j}. They must still be entered in the usual underscore
13882 One slight ambiguity of Big notation is that
13891 can represent either the negative rational number @expr{-3:4}, or the
13892 actual expression @samp{-(3/4)}; but the latter formula would normally
13893 never be displayed because it would immediately be evaluated to
13894 @expr{-3:4} or @expr{-0.75}, so this ambiguity is not a problem in
13897 Non-decimal numbers are displayed with subscripts. Thus there is no
13898 way to tell the difference between @samp{16#C2} and @samp{C2_16},
13899 though generally you will know which interpretation is correct.
13900 Logarithms @samp{log(x,b)} and @samp{log10(x)} also use subscripts
13903 In Big mode, stack entries often take up several lines. To aid
13904 readability, stack entries are separated by a blank line in this mode.
13905 You may find it useful to expand the Calc window's height using
13906 @kbd{C-x ^} (@code{enlarge-window}) or to make the Calc window the only
13907 one on the screen with @kbd{C-x 1} (@code{delete-other-windows}).
13909 Long lines are currently not rearranged to fit the window width in
13910 Big mode, so you may need to use the @kbd{<} and @kbd{>} keys
13911 to scroll across a wide formula. For really big formulas, you may
13912 even need to use @kbd{@{} and @kbd{@}} to scroll up and down.
13915 @pindex calc-unformatted-language
13916 The @kbd{d U} (@code{calc-unformatted-language}) command altogether disables
13917 the use of operator notation in formulas. In this mode, the formula
13918 shown above would be displayed:
13921 sqrt(add(div(add(a, 1), b), pow(c, 2)))
13924 These four modes differ only in display format, not in the format
13925 expected for algebraic entry. The standard Calc operators work in
13926 all four modes, and unformatted notation works in any language mode
13927 (except that Mathematica mode expects square brackets instead of
13930 @node C FORTRAN Pascal, TeX and LaTeX Language Modes, Normal Language Modes, Language Modes
13931 @subsection C, FORTRAN, and Pascal Modes
13935 @pindex calc-c-language
13937 The @kbd{d C} (@code{calc-c-language}) command selects the conventions
13938 of the C language for display and entry of formulas. This differs from
13939 the normal language mode in a variety of (mostly minor) ways. In
13940 particular, C language operators and operator precedences are used in
13941 place of Calc's usual ones. For example, @samp{a^b} means @samp{xor(a,b)}
13942 in C mode; a value raised to a power is written as a function call,
13945 In C mode, vectors and matrices use curly braces instead of brackets.
13946 Octal and hexadecimal values are written with leading @samp{0} or @samp{0x}
13947 rather than using the @samp{#} symbol. Array subscripting is
13948 translated into @code{subscr} calls, so that @samp{a[i]} in C
13949 mode is the same as @samp{a_i} in Normal mode. Assignments
13950 turn into the @code{assign} function, which Calc normally displays
13951 using the @samp{:=} symbol.
13953 The variables @code{pi} and @code{e} would be displayed @samp{pi}
13954 and @samp{e} in Normal mode, but in C mode they are displayed as
13955 @samp{M_PI} and @samp{M_E}, corresponding to the names of constants
13956 typically provided in the @file{<math.h>} header. Functions whose
13957 names are different in C are translated automatically for entry and
13958 display purposes. For example, entering @samp{asin(x)} will push the
13959 formula @samp{arcsin(x)} onto the stack; this formula will be displayed
13960 as @samp{asin(x)} as long as C mode is in effect.
13963 @pindex calc-pascal-language
13964 @cindex Pascal language
13965 The @kbd{d P} (@code{calc-pascal-language}) command selects Pascal
13966 conventions. Like C mode, Pascal mode interprets array brackets and uses
13967 a different table of operators. Hexadecimal numbers are entered and
13968 displayed with a preceding dollar sign. (Thus the regular meaning of
13969 @kbd{$2} during algebraic entry does not work in Pascal mode, though
13970 @kbd{$} (and @kbd{$$}, etc.) not followed by digits works the same as
13971 always.) No special provisions are made for other non-decimal numbers,
13972 vectors, and so on, since there is no universally accepted standard way
13973 of handling these in Pascal.
13976 @pindex calc-fortran-language
13977 @cindex FORTRAN language
13978 The @kbd{d F} (@code{calc-fortran-language}) command selects FORTRAN
13979 conventions. Various function names are transformed into FORTRAN
13980 equivalents. Vectors are written as @samp{/1, 2, 3/}, and may be
13981 entered this way or using square brackets. Since FORTRAN uses round
13982 parentheses for both function calls and array subscripts, Calc displays
13983 both in the same way; @samp{a(i)} is interpreted as a function call
13984 upon reading, and subscripts must be entered as @samp{subscr(a, i)}.
13985 If the variable @code{a} has been declared to have type
13986 @code{vector} or @code{matrix}, however, then @samp{a(i)} will be
13987 parsed as a subscript. (@xref{Declarations}.) Usually it doesn't
13988 matter, though; if you enter the subscript expression @samp{a(i)} and
13989 Calc interprets it as a function call, you'll never know the difference
13990 unless you switch to another language mode or replace @code{a} with an
13991 actual vector (or unless @code{a} happens to be the name of a built-in
13994 Underscores are allowed in variable and function names in all of these
13995 language modes. The underscore here is equivalent to the @samp{#} in
13996 Normal mode, or to hyphens in the underlying Emacs Lisp variable names.
13998 FORTRAN and Pascal modes normally do not adjust the case of letters in
13999 formulas. Most built-in Calc names use lower-case letters. If you use a
14000 positive numeric prefix argument with @kbd{d P} or @kbd{d F}, these
14001 modes will use upper-case letters exclusively for display, and will
14002 convert to lower-case on input. With a negative prefix, these modes
14003 convert to lower-case for display and input.
14005 @node TeX and LaTeX Language Modes, Eqn Language Mode, C FORTRAN Pascal, Language Modes
14006 @subsection @TeX{} and La@TeX{} Language Modes
14010 @pindex calc-tex-language
14011 @cindex TeX language
14013 @pindex calc-latex-language
14014 @cindex LaTeX language
14015 The @kbd{d T} (@code{calc-tex-language}) command selects the conventions
14016 of ``math mode'' in Donald Knuth's @TeX{} typesetting language,
14017 and the @kbd{d L} (@code{calc-latex-language}) command selects the
14018 conventions of ``math mode'' in La@TeX{}, a typesetting language that
14019 uses @TeX{} as its formatting engine. Calc's La@TeX{} language mode can
14020 read any formula that the @TeX{} language mode can, although La@TeX{}
14021 mode may display it differently.
14023 Formulas are entered and displayed in the appropriate notation;
14024 @texline @math{\sin(a/b)}
14025 @infoline @expr{sin(a/b)}
14026 will appear as @samp{\sin\left( @{a \over b@} \right)} in @TeX{} mode and
14027 @samp{\sin\left(\frac@{a@}@{b@}\right)} in La@TeX{} mode.
14028 Math formulas are often enclosed by @samp{$ $} signs in @TeX{} and
14029 La@TeX{}; these should be omitted when interfacing with Calc. To Calc,
14030 the @samp{$} sign has the same meaning it always does in algebraic
14031 formulas (a reference to an existing entry on the stack).
14033 Complex numbers are displayed as in @samp{3 + 4i}. Fractions and
14034 quotients are written using @code{\over} in @TeX{} mode (as in
14035 @code{@{a \over b@}}) and @code{\frac} in La@TeX{} mode (as in
14036 @code{\frac@{a@}@{b@}}); binomial coefficients are written with
14037 @code{\choose} in @TeX{} mode (as in @code{@{a \choose b@}}) and
14038 @code{\binom} in La@TeX{} mode (as in @code{\binom@{a@}@{b@}}).
14039 Interval forms are written with @code{\ldots}, and error forms are
14040 written with @code{\pm}. Absolute values are written as in
14041 @samp{|x + 1|}, and the floor and ceiling functions are written with
14042 @code{\lfloor}, @code{\rfloor}, etc. The words @code{\left} and
14043 @code{\right} are ignored when reading formulas in @TeX{} and La@TeX{}
14044 modes. Both @code{inf} and @code{uinf} are written as @code{\infty};
14045 when read, @code{\infty} always translates to @code{inf}.
14047 Function calls are written the usual way, with the function name followed
14048 by the arguments in parentheses. However, functions for which @TeX{}
14049 and La@TeX{} have special names (like @code{\sin}) will use curly braces
14050 instead of parentheses for very simple arguments. During input, curly
14051 braces and parentheses work equally well for grouping, but when the
14052 document is formatted the curly braces will be invisible. Thus the
14054 @texline @math{\sin{2 x}}
14055 @infoline @expr{sin 2x}
14057 @texline @math{\sin(2 + x)}.
14058 @infoline @expr{sin(2 + x)}.
14060 Function and variable names not treated specially by @TeX{} and La@TeX{}
14061 are simply written out as-is, which will cause them to come out in
14062 italic letters in the printed document. If you invoke @kbd{d T} or
14063 @kbd{d L} with a positive numeric prefix argument, names of more than
14064 one character will instead be enclosed in a protective commands that
14065 will prevent them from being typeset in the math italics; they will be
14066 written @samp{\hbox@{@var{name}@}} in @TeX{} mode and
14067 @samp{\text@{@var{name}@}} in La@TeX{} mode. The
14068 @samp{\hbox@{ @}} and @samp{\text@{ @}} notations are ignored during
14069 reading. If you use a negative prefix argument, such function names are
14070 written @samp{\@var{name}}, and function names that begin with @code{\} during
14071 reading have the @code{\} removed. (Note that in this mode, long
14072 variable names are still written with @code{\hbox} or @code{\text}.
14073 However, you can always make an actual variable name like @code{\bar} in
14076 During reading, text of the form @samp{\matrix@{ ...@: @}} is replaced
14077 by @samp{[ ...@: ]}. The same also applies to @code{\pmatrix} and
14078 @code{\bmatrix}. In La@TeX{} mode this also applies to
14079 @samp{\begin@{matrix@} ... \end@{matrix@}},
14080 @samp{\begin@{bmatrix@} ... \end@{bmatrix@}},
14081 @samp{\begin@{pmatrix@} ... \end@{pmatrix@}}, as well as
14082 @samp{\begin@{smallmatrix@} ... \end@{smallmatrix@}}.
14083 The symbol @samp{&} is interpreted as a comma,
14084 and the symbols @samp{\cr} and @samp{\\} are interpreted as semicolons.
14085 During output, matrices are displayed in @samp{\matrix@{ a & b \\ c & d@}}
14086 format in @TeX{} mode and in
14087 @samp{\begin@{pmatrix@} a & b \\ c & d \end@{pmatrix@}} format in
14088 La@TeX{} mode; you may need to edit this afterwards to change to your
14089 preferred matrix form. If you invoke @kbd{d T} or @kbd{d L} with an
14090 argument of 2 or -2, then matrices will be displayed in two-dimensional
14101 This may be convenient for isolated matrices, but could lead to
14102 expressions being displayed like
14105 \begin@{pmatrix@} \times x
14112 While this wouldn't bother Calc, it is incorrect La@TeX{}.
14113 (Similarly for @TeX{}.)
14115 Accents like @code{\tilde} and @code{\bar} translate into function
14116 calls internally (@samp{tilde(x)}, @samp{bar(x)}). The @code{\underline}
14117 sequence is treated as an accent. The @code{\vec} accent corresponds
14118 to the function name @code{Vec}, because @code{vec} is the name of
14119 a built-in Calc function. The following table shows the accents
14120 in Calc, @TeX{}, La@TeX{} and @dfn{eqn} (described in the next section):
14125 @let@calcindexershow=@calcindexernoshow @c Suppress marginal notes
14126 @let@calcindexersh=@calcindexernoshow
14235 acute \acute \acute
14239 breve \breve \breve
14241 check \check \check
14247 dotdot \ddot \ddot dotdot
14250 grave \grave \grave
14255 tilde \tilde \tilde tilde
14257 under \underline \underline under
14262 The @samp{=>} (evaluates-to) operator appears as a @code{\to} symbol:
14263 @samp{@{@var{a} \to @var{b}@}}. @TeX{} defines @code{\to} as an
14264 alias for @code{\rightarrow}. However, if the @samp{=>} is the
14265 top-level expression being formatted, a slightly different notation
14266 is used: @samp{\evalto @var{a} \to @var{b}}. The @code{\evalto}
14267 word is ignored by Calc's input routines, and is undefined in @TeX{}.
14268 You will typically want to include one of the following definitions
14269 at the top of a @TeX{} file that uses @code{\evalto}:
14273 \def\evalto#1\to@{@}
14276 The first definition formats evaluates-to operators in the usual
14277 way. The second causes only the @var{b} part to appear in the
14278 printed document; the @var{a} part and the arrow are hidden.
14279 Another definition you may wish to use is @samp{\let\to=\Rightarrow}
14280 which causes @code{\to} to appear more like Calc's @samp{=>} symbol.
14281 @xref{Evaluates-To Operator}, for a discussion of @code{evalto}.
14283 The complete set of @TeX{} control sequences that are ignored during
14287 \hbox \mbox \text \left \right
14288 \, \> \: \; \! \quad \qquad \hfil \hfill
14289 \displaystyle \textstyle \dsize \tsize
14290 \scriptstyle \scriptscriptstyle \ssize \ssize
14291 \rm \bf \it \sl \roman \bold \italic \slanted
14292 \cal \mit \Cal \Bbb \frak \goth
14296 Note that, because these symbols are ignored, reading a @TeX{} or
14297 La@TeX{} formula into Calc and writing it back out may lose spacing and
14300 Also, the ``discretionary multiplication sign'' @samp{\*} is read
14301 the same as @samp{*}.
14304 The @TeX{} version of this manual includes some printed examples at the
14305 end of this section.
14308 Here are some examples of how various Calc formulas are formatted in @TeX{}:
14313 \sin\left( {a^2 \over b_i} \right)
14317 $$ \sin\left( a^2 \over b_i \right) $$
14323 [(3, 4), 3:4, 3 +/- 4, [3 .. inf)]
14324 [3 + 4i, @{3 \over 4@}, 3 \pm 4, [3 \ldots \infty)]
14329 $$ [3 + 4i, {3 \over 4}, 3 \pm 4, [ 3 \ldots \infty)] $$
14335 [abs(a), abs(a / b), floor(a), ceil(a / b)]
14336 [|a|, \left| a \over b \right|,
14337 \lfloor a \rfloor, \left\lceil a \over b \right\rceil]
14341 $$ [|a|, \left| a \over b \right|,
14342 \lfloor a \rfloor, \left\lceil a \over b \right\rceil] $$
14348 [sin(a), sin(2 a), sin(2 + a), sin(a / b)]
14349 [\sin@{a@}, \sin@{2 a@}, \sin(2 + a),
14350 \sin\left( @{a \over b@} \right)]
14355 $$ [\sin{a}, \sin{2 a}, \sin(2 + a), \sin\left( {a \over b} \right)] $$
14359 First with plain @kbd{d T}, then with @kbd{C-u d T}, then finally with
14360 @kbd{C-u - d T} (using the example definition
14361 @samp{\def\foo#1@{\tilde F(#1)@}}:
14365 [f(a), foo(bar), sin(pi)]
14366 [f(a), foo(bar), \sin{\pi}]
14367 [f(a), \hbox@{foo@}(\hbox@{bar@}), \sin@{\pi@}]
14368 [f(a), \foo@{\hbox@{bar@}@}, \sin@{\pi@}]
14372 $$ [f(a), foo(bar), \sin{\pi}] $$
14373 $$ [f(a), \hbox{foo}(\hbox{bar}), \sin{\pi}] $$
14374 $$ [f(a), \tilde F(\hbox{bar}), \sin{\pi}] $$
14378 First with @samp{\def\evalto@{@}}, then with @samp{\def\evalto#1\to@{@}}:
14383 \evalto 2 + 3 \to 5
14393 First with standard @code{\to}, then with @samp{\let\to\Rightarrow}:
14397 [2 + 3 => 5, a / 2 => (b + c) / 2]
14398 [@{2 + 3 \to 5@}, @{@{a \over 2@} \to @{b + c \over 2@}@}]
14403 $$ [{2 + 3 \to 5}, {{a \over 2} \to {b + c \over 2}}] $$
14404 {\let\to\Rightarrow
14405 $$ [{2 + 3 \to 5}, {{a \over 2} \to {b + c \over 2}}] $$}
14409 Matrices normally, then changing @code{\matrix} to @code{\pmatrix}:
14413 [ [ a / b, 0 ], [ 0, 2^(x + 1) ] ]
14414 \matrix@{ @{a \over b@} & 0 \\ 0 & 2^@{(x + 1)@} @}
14415 \pmatrix@{ @{a \over b@} & 0 \\ 0 & 2^@{(x + 1)@} @}
14420 $$ \matrix{ {a \over b} & 0 \cr 0 & 2^{(x + 1)} } $$
14421 $$ \pmatrix{ {a \over b} & 0 \cr 0 & 2^{(x + 1)} } $$
14426 @node Eqn Language Mode, Yacas Language Mode, TeX and LaTeX Language Modes, Language Modes
14427 @subsection Eqn Language Mode
14431 @pindex calc-eqn-language
14432 @dfn{Eqn} is another popular formatter for math formulas. It is
14433 designed for use with the TROFF text formatter, and comes standard
14434 with many versions of Unix. The @kbd{d E} (@code{calc-eqn-language})
14435 command selects @dfn{eqn} notation.
14437 The @dfn{eqn} language's main idiosyncrasy is that whitespace plays
14438 a significant part in the parsing of the language. For example,
14439 @samp{sqrt x+1 + y} treats @samp{x+1} as the argument of the
14440 @code{sqrt} operator. @dfn{Eqn} also understands more conventional
14441 grouping using curly braces: @samp{sqrt@{x+1@} + y}. Braces are
14442 required only when the argument contains spaces.
14444 In Calc's @dfn{eqn} mode, however, curly braces are required to
14445 delimit arguments of operators like @code{sqrt}. The first of the
14446 above examples would treat only the @samp{x} as the argument of
14447 @code{sqrt}, and in fact @samp{sin x+1} would be interpreted as
14448 @samp{sin * x + 1}, because @code{sin} is not a special operator
14449 in the @dfn{eqn} language. If you always surround the argument
14450 with curly braces, Calc will never misunderstand.
14452 Calc also understands parentheses as grouping characters. Another
14453 peculiarity of @dfn{eqn}'s syntax makes it advisable to separate
14454 words with spaces from any surrounding characters that aren't curly
14455 braces, so Calc writes @samp{sin ( x + y )} in @dfn{eqn} mode.
14456 (The spaces around @code{sin} are important to make @dfn{eqn}
14457 recognize that @code{sin} should be typeset in a roman font, and
14458 the spaces around @code{x} and @code{y} are a good idea just in
14459 case the @dfn{eqn} document has defined special meanings for these
14462 Powers and subscripts are written with the @code{sub} and @code{sup}
14463 operators, respectively. Note that the caret symbol @samp{^} is
14464 treated the same as a space in @dfn{eqn} mode, as is the @samp{~}
14465 symbol (these are used to introduce spaces of various widths into
14466 the typeset output of @dfn{eqn}).
14468 As in La@TeX{} mode, Calc's formatter omits parentheses around the
14469 arguments of functions like @code{ln} and @code{sin} if they are
14470 ``simple-looking''; in this case Calc surrounds the argument with
14471 braces, separated by a @samp{~} from the function name: @samp{sin~@{x@}}.
14473 Font change codes (like @samp{roman @var{x}}) and positioning codes
14474 (like @samp{~} and @samp{down @var{n} @var{x}}) are ignored by the
14475 @dfn{eqn} reader. Also ignored are the words @code{left}, @code{right},
14476 @code{mark}, and @code{lineup}. Quotation marks in @dfn{eqn} mode input
14477 are treated the same as curly braces: @samp{sqrt "1+x"} is equivalent to
14478 @samp{sqrt @{1+x@}}; this is only an approximation to the true meaning
14479 of quotes in @dfn{eqn}, but it is good enough for most uses.
14481 Accent codes (@samp{@var{x} dot}) are handled by treating them as
14482 function calls (@samp{dot(@var{x})}) internally.
14483 @xref{TeX and LaTeX Language Modes}, for a table of these accent
14484 functions. The @code{prime} accent is treated specially if it occurs on
14485 a variable or function name: @samp{f prime prime @w{( x prime )}} is
14486 stored internally as @samp{f'@w{'}(x')}. For example, taking the
14487 derivative of @samp{f(2 x)} with @kbd{a d x} will produce @samp{2 f'(2
14488 x)}, which @dfn{eqn} mode will display as @samp{2 f prime ( 2 x )}.
14490 Assignments are written with the @samp{<-} (left-arrow) symbol,
14491 and @code{evalto} operators are written with @samp{->} or
14492 @samp{evalto ... ->} (@pxref{TeX and LaTeX Language Modes}, for a discussion
14493 of this). The regular Calc symbols @samp{:=} and @samp{=>} are also
14494 recognized for these operators during reading.
14496 Vectors in @dfn{eqn} mode use regular Calc square brackets, but
14497 matrices are formatted as @samp{matrix @{ ccol @{ a above b @} ... @}}.
14498 The words @code{lcol} and @code{rcol} are recognized as synonyms
14499 for @code{ccol} during input, and are generated instead of @code{ccol}
14500 if the matrix justification mode so specifies.
14502 @node Yacas Language Mode, Maxima Language Mode, Eqn Language Mode, Language Modes
14503 @subsection Yacas Language Mode
14507 @pindex calc-yacas-language
14508 @cindex Yacas language
14509 The @kbd{d Y} (@code{calc-yacas-language}) command selects the
14510 conventions of Yacas, a free computer algebra system. While the
14511 operators and functions in Yacas are similar to those of Calc, the names
14512 of built-in functions in Yacas are capitalized. The Calc formula
14513 @samp{sin(2 x)}, for example, is entered and displayed @samp{Sin(2 x)}
14514 in Yacas mode, and `@samp{arcsin(x^2)} is @samp{ArcSin(x^2)} in Yacas
14515 mode. Complex numbers are written are written @samp{3 + 4 I}.
14516 The standard special constants are written @code{Pi}, @code{E},
14517 @code{I}, @code{GoldenRatio} and @code{Gamma}. @code{Infinity}
14518 represents both @code{inf} and @code{uinf}, and @code{Undefined}
14519 represents @code{nan}.
14521 Certain operators on functions, such as @code{D} for differentiation
14522 and @code{Integrate} for integration, take a prefix form in Yacas. For
14523 example, the derivative of @w{@samp{e^x sin(x)}} can be computed with
14524 @w{@samp{D(x) Exp(x)*Sin(x)}}.
14526 Other notable differences between Yacas and standard Calc expressions
14527 are that vectors and matrices use curly braces in Yacas, and subscripts
14528 use square brackets. If, for example, @samp{A} represents the list
14529 @samp{@{a,2,c,4@}}, then @samp{A[3]} would equal @samp{c}.
14532 @node Maxima Language Mode, Giac Language Mode, Yacas Language Mode, Language Modes
14533 @subsection Maxima Language Mode
14537 @pindex calc-maxima-language
14538 @cindex Maxima language
14539 The @kbd{d X} (@code{calc-maxima-language}) command selects the
14540 conventions of Maxima, another free computer algebra system. The
14541 function names in Maxima are similar, but not always identical, to Calc.
14542 For example, instead of @samp{arcsin(x)}, Maxima will use
14543 @samp{asin(x)}. Complex numbers are written @samp{3 + 4 %i}. The
14544 standard special constants are written @code{%pi}, @code{%e},
14545 @code{%i}, @code{%phi} and @code{%gamma}. In Maxima, @code{inf} means
14546 the same as in Calc, but @code{infinity} represents Calc's @code{uinf}.
14548 Underscores as well as percent signs are allowed in function and
14549 variable names in Maxima mode. The underscore again is equivalent to
14550 the @samp{#} in Normal mode, and the percent sign is equivalent to
14553 Maxima uses square brackets for lists and vectors, and matrices are
14554 written as calls to the function @code{matrix}, given the row vectors of
14555 the matrix as arguments. Square brackets are also used as subscripts.
14557 @node Giac Language Mode, Mathematica Language Mode, Maxima Language Mode, Language Modes
14558 @subsection Giac Language Mode
14562 @pindex calc-giac-language
14563 @cindex Giac language
14564 The @kbd{d A} (@code{calc-giac-language}) command selects the
14565 conventions of Giac, another free computer algebra system. The function
14566 names in Giac are similar to Maxima. Complex numbers are written
14567 @samp{3 + 4 i}. The standard special constants in Giac are the same as
14568 in Calc, except that @code{infinity} represents both Calc's @code{inf}
14571 Underscores are allowed in function and variable names in Giac mode.
14572 Brackets are used for subscripts. In Giac, indexing of lists begins at
14573 0, instead of 1 as in Calc. So if @samp{A} represents the list
14574 @samp{[a,2,c,4]}, then @samp{A[2]} would equal @samp{c}. In general,
14575 @samp{A[n]} in Giac mode corresponds to @samp{A_(n+1)} in Normal mode.
14577 The Giac interval notation @samp{2 .. 3} has no surrounding brackets;
14578 Calc reads @samp{2 .. 3} as the closed interval @samp{[2 .. 3]} and
14579 writes any kind of interval as @samp{2 .. 3}. This means you cannot see
14580 the difference between an open and a closed interval while in Giac mode.
14582 @node Mathematica Language Mode, Maple Language Mode, Giac Language Mode, Language Modes
14583 @subsection Mathematica Language Mode
14587 @pindex calc-mathematica-language
14588 @cindex Mathematica language
14589 The @kbd{d M} (@code{calc-mathematica-language}) command selects the
14590 conventions of Mathematica. Notable differences in Mathematica mode
14591 are that the names of built-in functions are capitalized, and function
14592 calls use square brackets instead of parentheses. Thus the Calc
14593 formula @samp{sin(2 x)} is entered and displayed @w{@samp{Sin[2 x]}} in
14596 Vectors and matrices use curly braces in Mathematica. Complex numbers
14597 are written @samp{3 + 4 I}. The standard special constants in Calc are
14598 written @code{Pi}, @code{E}, @code{I}, @code{GoldenRatio}, @code{EulerGamma},
14599 @code{Infinity}, @code{ComplexInfinity}, and @code{Indeterminate} in
14601 Non-decimal numbers are written, e.g., @samp{16^^7fff}. Floating-point
14602 numbers in scientific notation are written @samp{1.23*10.^3}.
14603 Subscripts use double square brackets: @samp{a[[i]]}.
14605 @node Maple Language Mode, Compositions, Mathematica Language Mode, Language Modes
14606 @subsection Maple Language Mode
14610 @pindex calc-maple-language
14611 @cindex Maple language
14612 The @kbd{d W} (@code{calc-maple-language}) command selects the
14613 conventions of Maple.
14615 Maple's language is much like C. Underscores are allowed in symbol
14616 names; square brackets are used for subscripts; explicit @samp{*}s for
14617 multiplications are required. Use either @samp{^} or @samp{**} to
14620 Maple uses square brackets for lists and curly braces for sets. Calc
14621 interprets both notations as vectors, and displays vectors with square
14622 brackets. This means Maple sets will be converted to lists when they
14623 pass through Calc. As a special case, matrices are written as calls
14624 to the function @code{matrix}, given a list of lists as the argument,
14625 and can be read in this form or with all-capitals @code{MATRIX}.
14627 The Maple interval notation @samp{2 .. 3} is like Giac's interval
14628 notation, and is handled the same by Calc.
14630 Maple writes complex numbers as @samp{3 + 4*I}. Its special constants
14631 are @code{Pi}, @code{E}, @code{I}, and @code{infinity} (all three of
14632 @code{inf}, @code{uinf}, and @code{nan} display as @code{infinity}).
14633 Floating-point numbers are written @samp{1.23*10.^3}.
14635 Among things not currently handled by Calc's Maple mode are the
14636 various quote symbols, procedures and functional operators, and
14637 inert (@samp{&}) operators.
14639 @node Compositions, Syntax Tables, Maple Language Mode, Language Modes
14640 @subsection Compositions
14643 @cindex Compositions
14644 There are several @dfn{composition functions} which allow you to get
14645 displays in a variety of formats similar to those in Big language
14646 mode. Most of these functions do not evaluate to anything; they are
14647 placeholders which are left in symbolic form by Calc's evaluator but
14648 are recognized by Calc's display formatting routines.
14650 Two of these, @code{string} and @code{bstring}, are described elsewhere.
14651 @xref{Strings}. For example, @samp{string("ABC")} is displayed as
14652 @samp{ABC}. When viewed on the stack it will be indistinguishable from
14653 the variable @code{ABC}, but internally it will be stored as
14654 @samp{string([65, 66, 67])} and can still be manipulated this way; for
14655 example, the selection and vector commands @kbd{j 1 v v j u} would
14656 select the vector portion of this object and reverse the elements, then
14657 deselect to reveal a string whose characters had been reversed.
14659 The composition functions do the same thing in all language modes
14660 (although their components will of course be formatted in the current
14661 language mode). The one exception is Unformatted mode (@kbd{d U}),
14662 which does not give the composition functions any special treatment.
14663 The functions are discussed here because of their relationship to
14664 the language modes.
14667 * Composition Basics::
14668 * Horizontal Compositions::
14669 * Vertical Compositions::
14670 * Other Compositions::
14671 * Information about Compositions::
14672 * User-Defined Compositions::
14675 @node Composition Basics, Horizontal Compositions, Compositions, Compositions
14676 @subsubsection Composition Basics
14679 Compositions are generally formed by stacking formulas together
14680 horizontally or vertically in various ways. Those formulas are
14681 themselves compositions. @TeX{} users will find this analogous
14682 to @TeX{}'s ``boxes.'' Each multi-line composition has a
14683 @dfn{baseline}; horizontal compositions use the baselines to
14684 decide how formulas should be positioned relative to one another.
14685 For example, in the Big mode formula
14697 the second term of the sum is four lines tall and has line three as
14698 its baseline. Thus when the term is combined with 17, line three
14699 is placed on the same level as the baseline of 17.
14705 Another important composition concept is @dfn{precedence}. This is
14706 an integer that represents the binding strength of various operators.
14707 For example, @samp{*} has higher precedence (195) than @samp{+} (180),
14708 which means that @samp{(a * b) + c} will be formatted without the
14709 parentheses, but @samp{a * (b + c)} will keep the parentheses.
14711 The operator table used by normal and Big language modes has the
14712 following precedences:
14715 _ 1200 @r{(subscripts)}
14716 % 1100 @r{(as in n}%@r{)}
14717 ! 1000 @r{(as in }!@r{n)}
14720 !! 210 @r{(as in n}!!@r{)}
14721 ! 210 @r{(as in n}!@r{)}
14723 - 197 @r{(as in }-@r{n)}
14724 * 195 @r{(or implicit multiplication)}
14726 + - 180 @r{(as in a}+@r{b)}
14728 < = 160 @r{(and other relations)}
14740 The general rule is that if an operator with precedence @expr{n}
14741 occurs as an argument to an operator with precedence @expr{m}, then
14742 the argument is enclosed in parentheses if @expr{n < m}. Top-level
14743 expressions and expressions which are function arguments, vector
14744 components, etc., are formatted with precedence zero (so that they
14745 normally never get additional parentheses).
14747 For binary left-associative operators like @samp{+}, the righthand
14748 argument is actually formatted with one-higher precedence than shown
14749 in the table. This makes sure @samp{(a + b) + c} omits the parentheses,
14750 but the unnatural form @samp{a + (b + c)} keeps its parentheses.
14751 Right-associative operators like @samp{^} format the lefthand argument
14752 with one-higher precedence.
14758 The @code{cprec} function formats an expression with an arbitrary
14759 precedence. For example, @samp{cprec(abc, 185)} will combine into
14760 sums and products as follows: @samp{7 + abc}, @samp{7 (abc)} (because
14761 this @code{cprec} form has higher precedence than addition, but lower
14762 precedence than multiplication).
14768 A final composition issue is @dfn{line breaking}. Calc uses two
14769 different strategies for ``flat'' and ``non-flat'' compositions.
14770 A non-flat composition is anything that appears on multiple lines
14771 (not counting line breaking). Examples would be matrices and Big
14772 mode powers and quotients. Non-flat compositions are displayed
14773 exactly as specified. If they come out wider than the current
14774 window, you must use horizontal scrolling (@kbd{<} and @kbd{>}) to
14777 Flat compositions, on the other hand, will be broken across several
14778 lines if they are too wide to fit the window. Certain points in a
14779 composition are noted internally as @dfn{break points}. Calc's
14780 general strategy is to fill each line as much as possible, then to
14781 move down to the next line starting at the first break point that
14782 didn't fit. However, the line breaker understands the hierarchical
14783 structure of formulas. It will not break an ``inner'' formula if
14784 it can use an earlier break point from an ``outer'' formula instead.
14785 For example, a vector of sums might be formatted as:
14789 [ a + b + c, d + e + f,
14790 g + h + i, j + k + l, m ]
14795 If the @samp{m} can fit, then so, it seems, could the @samp{g}.
14796 But Calc prefers to break at the comma since the comma is part
14797 of a ``more outer'' formula. Calc would break at a plus sign
14798 only if it had to, say, if the very first sum in the vector had
14799 itself been too large to fit.
14801 Of the composition functions described below, only @code{choriz}
14802 generates break points. The @code{bstring} function (@pxref{Strings})
14803 also generates breakable items: A break point is added after every
14804 space (or group of spaces) except for spaces at the very beginning or
14807 Composition functions themselves count as levels in the formula
14808 hierarchy, so a @code{choriz} that is a component of a larger
14809 @code{choriz} will be less likely to be broken. As a special case,
14810 if a @code{bstring} occurs as a component of a @code{choriz} or
14811 @code{choriz}-like object (such as a vector or a list of arguments
14812 in a function call), then the break points in that @code{bstring}
14813 will be on the same level as the break points of the surrounding
14816 @node Horizontal Compositions, Vertical Compositions, Composition Basics, Compositions
14817 @subsubsection Horizontal Compositions
14824 The @code{choriz} function takes a vector of objects and composes
14825 them horizontally. For example, @samp{choriz([17, a b/c, d])} formats
14826 as @w{@samp{17a b / cd}} in Normal language mode, or as
14837 in Big language mode. This is actually one case of the general
14838 function @samp{choriz(@var{vec}, @var{sep}, @var{prec})}, where
14839 either or both of @var{sep} and @var{prec} may be omitted.
14840 @var{Prec} gives the @dfn{precedence} to use when formatting
14841 each of the components of @var{vec}. The default precedence is
14842 the precedence from the surrounding environment.
14844 @var{Sep} is a string (i.e., a vector of character codes as might
14845 be entered with @code{" "} notation) which should separate components
14846 of the composition. Also, if @var{sep} is given, the line breaker
14847 will allow lines to be broken after each occurrence of @var{sep}.
14848 If @var{sep} is omitted, the composition will not be breakable
14849 (unless any of its component compositions are breakable).
14851 For example, @samp{2 choriz([a, b c, d = e], " + ", 180)} is
14852 formatted as @samp{2 a + b c + (d = e)}. To get the @code{choriz}
14853 to have precedence 180 ``outwards'' as well as ``inwards,''
14854 enclose it in a @code{cprec} form: @samp{2 cprec(choriz(...), 180)}
14855 formats as @samp{2 (a + b c + (d = e))}.
14857 The baseline of a horizontal composition is the same as the
14858 baselines of the component compositions, which are all aligned.
14860 @node Vertical Compositions, Other Compositions, Horizontal Compositions, Compositions
14861 @subsubsection Vertical Compositions
14868 The @code{cvert} function makes a vertical composition. Each
14869 component of the vector is centered in a column. The baseline of
14870 the result is by default the top line of the resulting composition.
14871 For example, @samp{f(cvert([a, bb, ccc]), cvert([a^2 + 1, b^2]))}
14872 formats in Big mode as
14887 There are several special composition functions that work only as
14888 components of a vertical composition. The @code{cbase} function
14889 controls the baseline of the vertical composition; the baseline
14890 will be the same as the baseline of whatever component is enclosed
14891 in @code{cbase}. Thus @samp{f(cvert([a, cbase(bb), ccc]),
14892 cvert([a^2 + 1, cbase(b^2)]))} displays as
14912 There are also @code{ctbase} and @code{cbbase} functions which
14913 make the baseline of the vertical composition equal to the top
14914 or bottom line (rather than the baseline) of that component.
14915 Thus @samp{cvert([cbase(a / b)]) + cvert([ctbase(a / b)]) +
14916 cvert([cbbase(a / b)])} gives
14928 There should be only one @code{cbase}, @code{ctbase}, or @code{cbbase}
14929 function in a given vertical composition. These functions can also
14930 be written with no arguments: @samp{ctbase()} is a zero-height object
14931 which means the baseline is the top line of the following item, and
14932 @samp{cbbase()} means the baseline is the bottom line of the preceding
14939 The @code{crule} function builds a ``rule,'' or horizontal line,
14940 across a vertical composition. By itself @samp{crule()} uses @samp{-}
14941 characters to build the rule. You can specify any other character,
14942 e.g., @samp{crule("=")}. The argument must be a character code or
14943 vector of exactly one character code. It is repeated to match the
14944 width of the widest item in the stack. For example, a quotient
14945 with a thick line is @samp{cvert([a + 1, cbase(crule("=")), b^2])}:
14964 Finally, the functions @code{clvert} and @code{crvert} act exactly
14965 like @code{cvert} except that the items are left- or right-justified
14966 in the stack. Thus @samp{clvert([a, bb, ccc]) + crvert([a, bb, ccc])}
14977 Like @code{choriz}, the vertical compositions accept a second argument
14978 which gives the precedence to use when formatting the components.
14979 Vertical compositions do not support separator strings.
14981 @node Other Compositions, Information about Compositions, Vertical Compositions, Compositions
14982 @subsubsection Other Compositions
14989 The @code{csup} function builds a superscripted expression. For
14990 example, @samp{csup(a, b)} looks the same as @samp{a^b} does in Big
14991 language mode. This is essentially a horizontal composition of
14992 @samp{a} and @samp{b}, where @samp{b} is shifted up so that its
14993 bottom line is one above the baseline.
14999 Likewise, the @code{csub} function builds a subscripted expression.
15000 This shifts @samp{b} down so that its top line is one below the
15001 bottom line of @samp{a} (note that this is not quite analogous to
15002 @code{csup}). Other arrangements can be obtained by using
15003 @code{choriz} and @code{cvert} directly.
15009 The @code{cflat} function formats its argument in ``flat'' mode,
15010 as obtained by @samp{d O}, if the current language mode is normal
15011 or Big. It has no effect in other language modes. For example,
15012 @samp{a^(b/c)} is formatted by Big mode like @samp{csup(a, cflat(b/c))}
15013 to improve its readability.
15019 The @code{cspace} function creates horizontal space. For example,
15020 @samp{cspace(4)} is effectively the same as @samp{string(" ")}.
15021 A second string (i.e., vector of characters) argument is repeated
15022 instead of the space character. For example, @samp{cspace(4, "ab")}
15023 looks like @samp{abababab}. If the second argument is not a string,
15024 it is formatted in the normal way and then several copies of that
15025 are composed together: @samp{cspace(4, a^2)} yields
15035 If the number argument is zero, this is a zero-width object.
15041 The @code{cvspace} function creates vertical space, or a vertical
15042 stack of copies of a certain string or formatted object. The
15043 baseline is the center line of the resulting stack. A numerical
15044 argument of zero will produce an object which contributes zero
15045 height if used in a vertical composition.
15055 There are also @code{ctspace} and @code{cbspace} functions which
15056 create vertical space with the baseline the same as the baseline
15057 of the top or bottom copy, respectively, of the second argument.
15058 Thus @samp{cvspace(2, a/b) + ctspace(2, a/b) + cbspace(2, a/b)}
15075 @node Information about Compositions, User-Defined Compositions, Other Compositions, Compositions
15076 @subsubsection Information about Compositions
15079 The functions in this section are actual functions; they compose their
15080 arguments according to the current language and other display modes,
15081 then return a certain measurement of the composition as an integer.
15087 The @code{cwidth} function measures the width, in characters, of a
15088 composition. For example, @samp{cwidth(a + b)} is 5, and
15089 @samp{cwidth(a / b)} is 5 in Normal mode, 1 in Big mode, and 11 in
15090 @TeX{} mode (for @samp{@{a \over b@}}). The argument may involve
15091 the composition functions described in this section.
15097 The @code{cheight} function measures the height of a composition.
15098 This is the total number of lines in the argument's printed form.
15108 The functions @code{cascent} and @code{cdescent} measure the amount
15109 of the height that is above (and including) the baseline, or below
15110 the baseline, respectively. Thus @samp{cascent(@var{x}) + cdescent(@var{x})}
15111 always equals @samp{cheight(@var{x})}. For a one-line formula like
15112 @samp{a + b}, @code{cascent} returns 1 and @code{cdescent} returns 0.
15113 For @samp{a / b} in Big mode, @code{cascent} returns 2 and @code{cdescent}
15114 returns 1. The only formula for which @code{cascent} will return zero
15115 is @samp{cvspace(0)} or equivalents.
15117 @node User-Defined Compositions, , Information about Compositions, Compositions
15118 @subsubsection User-Defined Compositions
15122 @pindex calc-user-define-composition
15123 The @kbd{Z C} (@code{calc-user-define-composition}) command lets you
15124 define the display format for any algebraic function. You provide a
15125 formula containing a certain number of argument variables on the stack.
15126 Any time Calc formats a call to the specified function in the current
15127 language mode and with that number of arguments, Calc effectively
15128 replaces the function call with that formula with the arguments
15131 Calc builds the default argument list by sorting all the variable names
15132 that appear in the formula into alphabetical order. You can edit this
15133 argument list before pressing @key{RET} if you wish. Any variables in
15134 the formula that do not appear in the argument list will be displayed
15135 literally; any arguments that do not appear in the formula will not
15136 affect the display at all.
15138 You can define formats for built-in functions, for functions you have
15139 defined with @kbd{Z F} (@pxref{Algebraic Definitions}), or for functions
15140 which have no definitions but are being used as purely syntactic objects.
15141 You can define different formats for each language mode, and for each
15142 number of arguments, using a succession of @kbd{Z C} commands. When
15143 Calc formats a function call, it first searches for a format defined
15144 for the current language mode (and number of arguments); if there is
15145 none, it uses the format defined for the Normal language mode. If
15146 neither format exists, Calc uses its built-in standard format for that
15147 function (usually just @samp{@var{func}(@var{args})}).
15149 If you execute @kbd{Z C} with the number 0 on the stack instead of a
15150 formula, any defined formats for the function in the current language
15151 mode will be removed. The function will revert to its standard format.
15153 For example, the default format for the binomial coefficient function
15154 @samp{choose(n, m)} in the Big language mode is
15165 You might prefer the notation,
15175 To define this notation, first make sure you are in Big mode,
15176 then put the formula
15179 choriz([cvert([cvspace(1), n]), C, cvert([cvspace(1), m])])
15183 on the stack and type @kbd{Z C}. Answer the first prompt with
15184 @code{choose}. The second prompt will be the default argument list
15185 of @samp{(C m n)}. Edit this list to be @samp{(n m)} and press
15186 @key{RET}. Now, try it out: For example, turn simplification
15187 off with @kbd{m O} and enter @samp{choose(a,b) + choose(7,3)}
15188 as an algebraic entry.
15197 As another example, let's define the usual notation for Stirling
15198 numbers of the first kind, @samp{stir1(n, m)}. This is just like
15199 the regular format for binomial coefficients but with square brackets
15200 instead of parentheses.
15203 choriz([string("["), cvert([n, cbase(cvspace(1)), m]), string("]")])
15206 Now type @kbd{Z C stir1 @key{RET}}, edit the argument list to
15207 @samp{(n m)}, and type @key{RET}.
15209 The formula provided to @kbd{Z C} usually will involve composition
15210 functions, but it doesn't have to. Putting the formula @samp{a + b + c}
15211 onto the stack and typing @kbd{Z C foo @key{RET} @key{RET}} would define
15212 the function @samp{foo(x,y,z)} to display like @samp{x + y + z}.
15213 This ``sum'' will act exactly like a real sum for all formatting
15214 purposes (it will be parenthesized the same, and so on). However
15215 it will be computationally unrelated to a sum. For example, the
15216 formula @samp{2 * foo(1, 2, 3)} will display as @samp{2 (1 + 2 + 3)}.
15217 Operator precedences have caused the ``sum'' to be written in
15218 parentheses, but the arguments have not actually been summed.
15219 (Generally a display format like this would be undesirable, since
15220 it can easily be confused with a real sum.)
15222 The special function @code{eval} can be used inside a @kbd{Z C}
15223 composition formula to cause all or part of the formula to be
15224 evaluated at display time. For example, if the formula is
15225 @samp{a + eval(b + c)}, then @samp{foo(1, 2, 3)} will be displayed
15226 as @samp{1 + 5}. Evaluation will use the default simplifications,
15227 regardless of the current simplification mode. There are also
15228 @code{evalsimp} and @code{evalextsimp} which simplify as if by
15229 @kbd{a s} and @kbd{a e} (respectively). Note that these ``functions''
15230 operate only in the context of composition formulas (and also in
15231 rewrite rules, where they serve a similar purpose; @pxref{Rewrite
15232 Rules}). On the stack, a call to @code{eval} will be left in
15235 It is not a good idea to use @code{eval} except as a last resort.
15236 It can cause the display of formulas to be extremely slow. For
15237 example, while @samp{eval(a + b)} might seem quite fast and simple,
15238 there are several situations where it could be slow. For example,
15239 @samp{a} and/or @samp{b} could be polar complex numbers, in which
15240 case doing the sum requires trigonometry. Or, @samp{a} could be
15241 the factorial @samp{fact(100)} which is unevaluated because you
15242 have typed @kbd{m O}; @code{eval} will evaluate it anyway to
15243 produce a large, unwieldy integer.
15245 You can save your display formats permanently using the @kbd{Z P}
15246 command (@pxref{Creating User Keys}).
15248 @node Syntax Tables, , Compositions, Language Modes
15249 @subsection Syntax Tables
15252 @cindex Syntax tables
15253 @cindex Parsing formulas, customized
15254 Syntax tables do for input what compositions do for output: They
15255 allow you to teach custom notations to Calc's formula parser.
15256 Calc keeps a separate syntax table for each language mode.
15258 (Note that the Calc ``syntax tables'' discussed here are completely
15259 unrelated to the syntax tables described in the Emacs manual.)
15262 @pindex calc-edit-user-syntax
15263 The @kbd{Z S} (@code{calc-edit-user-syntax}) command edits the
15264 syntax table for the current language mode. If you want your
15265 syntax to work in any language, define it in the Normal language
15266 mode. Type @kbd{C-c C-c} to finish editing the syntax table, or
15267 @kbd{C-x k} to cancel the edit. The @kbd{m m} command saves all
15268 the syntax tables along with the other mode settings;
15269 @pxref{General Mode Commands}.
15272 * Syntax Table Basics::
15273 * Precedence in Syntax Tables::
15274 * Advanced Syntax Patterns::
15275 * Conditional Syntax Rules::
15278 @node Syntax Table Basics, Precedence in Syntax Tables, Syntax Tables, Syntax Tables
15279 @subsubsection Syntax Table Basics
15282 @dfn{Parsing} is the process of converting a raw string of characters,
15283 such as you would type in during algebraic entry, into a Calc formula.
15284 Calc's parser works in two stages. First, the input is broken down
15285 into @dfn{tokens}, such as words, numbers, and punctuation symbols
15286 like @samp{+}, @samp{:=}, and @samp{+/-}. Space between tokens is
15287 ignored (except when it serves to separate adjacent words). Next,
15288 the parser matches this string of tokens against various built-in
15289 syntactic patterns, such as ``an expression followed by @samp{+}
15290 followed by another expression'' or ``a name followed by @samp{(},
15291 zero or more expressions separated by commas, and @samp{)}.''
15293 A @dfn{syntax table} is a list of user-defined @dfn{syntax rules},
15294 which allow you to specify new patterns to define your own
15295 favorite input notations. Calc's parser always checks the syntax
15296 table for the current language mode, then the table for the Normal
15297 language mode, before it uses its built-in rules to parse an
15298 algebraic formula you have entered. Each syntax rule should go on
15299 its own line; it consists of a @dfn{pattern}, a @samp{:=} symbol,
15300 and a Calc formula with an optional @dfn{condition}. (Syntax rules
15301 resemble algebraic rewrite rules, but the notation for patterns is
15302 completely different.)
15304 A syntax pattern is a list of tokens, separated by spaces.
15305 Except for a few special symbols, tokens in syntax patterns are
15306 matched literally, from left to right. For example, the rule,
15313 would cause Calc to parse the formula @samp{4+foo()*5} as if it
15314 were @samp{4+(2+3)*5}. Notice that the parentheses were written
15315 as two separate tokens in the rule. As a result, the rule works
15316 for both @samp{foo()} and @w{@samp{foo ( )}}. If we had written
15317 the rule as @samp{foo () := 2+3}, then Calc would treat @samp{()}
15318 as a single, indivisible token, so that @w{@samp{foo( )}} would
15319 not be recognized by the rule. (It would be parsed as a regular
15320 zero-argument function call instead.) In fact, this rule would
15321 also make trouble for the rest of Calc's parser: An unrelated
15322 formula like @samp{bar()} would now be tokenized into @samp{bar ()}
15323 instead of @samp{bar ( )}, so that the standard parser for function
15324 calls would no longer recognize it!
15326 While it is possible to make a token with a mixture of letters
15327 and punctuation symbols, this is not recommended. It is better to
15328 break it into several tokens, as we did with @samp{foo()} above.
15330 The symbol @samp{#} in a syntax pattern matches any Calc expression.
15331 On the righthand side, the things that matched the @samp{#}s can
15332 be referred to as @samp{#1}, @samp{#2}, and so on (where @samp{#1}
15333 matches the leftmost @samp{#} in the pattern). For example, these
15334 rules match a user-defined function, prefix operator, infix operator,
15335 and postfix operator, respectively:
15338 foo ( # ) := myfunc(#1)
15339 foo # := myprefix(#1)
15340 # foo # := myinfix(#1,#2)
15341 # foo := mypostfix(#1)
15344 Thus @samp{foo(3)} will parse as @samp{myfunc(3)}, and @samp{2+3 foo}
15345 will parse as @samp{mypostfix(2+3)}.
15347 It is important to write the first two rules in the order shown,
15348 because Calc tries rules in order from first to last. If the
15349 pattern @samp{foo #} came first, it would match anything that could
15350 match the @samp{foo ( # )} rule, since an expression in parentheses
15351 is itself a valid expression. Thus the @w{@samp{foo ( # )}} rule would
15352 never get to match anything. Likewise, the last two rules must be
15353 written in the order shown or else @samp{3 foo 4} will be parsed as
15354 @samp{mypostfix(3) * 4}. (Of course, the best way to avoid these
15355 ambiguities is not to use the same symbol in more than one way at
15356 the same time! In case you're not convinced, try the following
15357 exercise: How will the above rules parse the input @samp{foo(3,4)},
15358 if at all? Work it out for yourself, then try it in Calc and see.)
15360 Calc is quite flexible about what sorts of patterns are allowed.
15361 The only rule is that every pattern must begin with a literal
15362 token (like @samp{foo} in the first two patterns above), or with
15363 a @samp{#} followed by a literal token (as in the last two
15364 patterns). After that, any mixture is allowed, although putting
15365 two @samp{#}s in a row will not be very useful since two
15366 expressions with nothing between them will be parsed as one
15367 expression that uses implicit multiplication.
15369 As a more practical example, Maple uses the notation
15370 @samp{sum(a(i), i=1..10)} for sums, which Calc's Maple mode doesn't
15371 recognize at present. To handle this syntax, we simply add the
15375 sum ( # , # = # .. # ) := sum(#1,#2,#3,#4)
15379 to the Maple mode syntax table. As another example, C mode can't
15380 read assignment operators like @samp{++} and @samp{*=}. We can
15381 define these operators quite easily:
15384 # *= # := muleq(#1,#2)
15385 # ++ := postinc(#1)
15390 To complete the job, we would use corresponding composition functions
15391 and @kbd{Z C} to cause these functions to display in their respective
15392 Maple and C notations. (Note that the C example ignores issues of
15393 operator precedence, which are discussed in the next section.)
15395 You can enclose any token in quotes to prevent its usual
15396 interpretation in syntax patterns:
15399 # ":=" # := becomes(#1,#2)
15402 Quotes also allow you to include spaces in a token, although once
15403 again it is generally better to use two tokens than one token with
15404 an embedded space. To include an actual quotation mark in a quoted
15405 token, precede it with a backslash. (This also works to include
15406 backslashes in tokens.)
15409 # "bad token" # "/\"\\" # := silly(#1,#2,#3)
15413 This will parse @samp{3 bad token 4 /"\ 5} to @samp{silly(3,4,5)}.
15415 The token @kbd{#} has a predefined meaning in Calc's formula parser;
15416 it is not valid to use @samp{"#"} in a syntax rule. However, longer
15417 tokens that include the @samp{#} character are allowed. Also, while
15418 @samp{"$"} and @samp{"\""} are allowed as tokens, their presence in
15419 the syntax table will prevent those characters from working in their
15420 usual ways (referring to stack entries and quoting strings,
15423 Finally, the notation @samp{%%} anywhere in a syntax table causes
15424 the rest of the line to be ignored as a comment.
15426 @node Precedence in Syntax Tables, Advanced Syntax Patterns, Syntax Table Basics, Syntax Tables
15427 @subsubsection Precedence
15430 Different operators are generally assigned different @dfn{precedences}.
15431 By default, an operator defined by a rule like
15434 # foo # := foo(#1,#2)
15438 will have an extremely low precedence, so that @samp{2*3+4 foo 5 == 6}
15439 will be parsed as @samp{(2*3+4) foo (5 == 6)}. To change the
15440 precedence of an operator, use the notation @samp{#/@var{p}} in
15441 place of @samp{#}, where @var{p} is an integer precedence level.
15442 For example, 185 lies between the precedences for @samp{+} and
15443 @samp{*}, so if we change this rule to
15446 #/185 foo #/186 := foo(#1,#2)
15450 then @samp{2+3 foo 4*5} will be parsed as @samp{2+(3 foo (4*5))}.
15451 Also, because we've given the righthand expression slightly higher
15452 precedence, our new operator will be left-associative:
15453 @samp{1 foo 2 foo 3} will be parsed as @samp{(1 foo 2) foo 3}.
15454 By raising the precedence of the lefthand expression instead, we
15455 can create a right-associative operator.
15457 @xref{Composition Basics}, for a table of precedences of the
15458 standard Calc operators. For the precedences of operators in other
15459 language modes, look in the Calc source file @file{calc-lang.el}.
15461 @node Advanced Syntax Patterns, Conditional Syntax Rules, Precedence in Syntax Tables, Syntax Tables
15462 @subsubsection Advanced Syntax Patterns
15465 To match a function with a variable number of arguments, you could
15469 foo ( # ) := myfunc(#1)
15470 foo ( # , # ) := myfunc(#1,#2)
15471 foo ( # , # , # ) := myfunc(#1,#2,#3)
15475 but this isn't very elegant. To match variable numbers of items,
15476 Calc uses some notations inspired regular expressions and the
15477 ``extended BNF'' style used by some language designers.
15480 foo ( @{ # @}*, ) := apply(myfunc,#1)
15483 The token @samp{@{} introduces a repeated or optional portion.
15484 One of the three tokens @samp{@}*}, @samp{@}+}, or @samp{@}?}
15485 ends the portion. These will match zero or more, one or more,
15486 or zero or one copies of the enclosed pattern, respectively.
15487 In addition, @samp{@}*} and @samp{@}+} can be followed by a
15488 separator token (with no space in between, as shown above).
15489 Thus @samp{@{ # @}*,} matches nothing, or one expression, or
15490 several expressions separated by commas.
15492 A complete @samp{@{ ... @}} item matches as a vector of the
15493 items that matched inside it. For example, the above rule will
15494 match @samp{foo(1,2,3)} to get @samp{apply(myfunc,[1,2,3])}.
15495 The Calc @code{apply} function takes a function name and a vector
15496 of arguments and builds a call to the function with those
15497 arguments, so the net result is the formula @samp{myfunc(1,2,3)}.
15499 If the body of a @samp{@{ ... @}} contains several @samp{#}s
15500 (or nested @samp{@{ ... @}} constructs), then the items will be
15501 strung together into the resulting vector. If the body
15502 does not contain anything but literal tokens, the result will
15503 always be an empty vector.
15506 foo ( @{ # , # @}+, ) := bar(#1)
15507 foo ( @{ @{ # @}*, @}*; ) := matrix(#1)
15511 will parse @samp{foo(1, 2, 3, 4)} as @samp{bar([1, 2, 3, 4])}, and
15512 @samp{foo(1, 2; 3, 4)} as @samp{matrix([[1, 2], [3, 4]])}. Also, after
15513 some thought it's easy to see how this pair of rules will parse
15514 @samp{foo(1, 2, 3)} as @samp{matrix([[1, 2, 3]])}, since the first
15515 rule will only match an even number of arguments. The rule
15518 foo ( # @{ , # , # @}? ) := bar(#1,#2)
15522 will parse @samp{foo(2,3,4)} as @samp{bar(2,[3,4])}, and
15523 @samp{foo(2)} as @samp{bar(2,[])}.
15525 The notation @samp{@{ ... @}?.} (note the trailing period) works
15526 just the same as regular @samp{@{ ... @}?}, except that it does not
15527 count as an argument; the following two rules are equivalent:
15530 foo ( # , @{ also @}? # ) := bar(#1,#3)
15531 foo ( # , @{ also @}?. # ) := bar(#1,#2)
15535 Note that in the first case the optional text counts as @samp{#2},
15536 which will always be an empty vector, but in the second case no
15537 empty vector is produced.
15539 Another variant is @samp{@{ ... @}?$}, which means the body is
15540 optional only at the end of the input formula. All built-in syntax
15541 rules in Calc use this for closing delimiters, so that during
15542 algebraic entry you can type @kbd{[sqrt(2), sqrt(3 @key{RET}}, omitting
15543 the closing parenthesis and bracket. Calc does this automatically
15544 for trailing @samp{)}, @samp{]}, and @samp{>} tokens in syntax
15545 rules, but you can use @samp{@{ ... @}?$} explicitly to get
15546 this effect with any token (such as @samp{"@}"} or @samp{end}).
15547 Like @samp{@{ ... @}?.}, this notation does not count as an
15548 argument. Conversely, you can use quotes, as in @samp{")"}, to
15549 prevent a closing-delimiter token from being automatically treated
15552 Calc's parser does not have full backtracking, which means some
15553 patterns will not work as you might expect:
15556 foo ( @{ # , @}? # , # ) := bar(#1,#2,#3)
15560 Here we are trying to make the first argument optional, so that
15561 @samp{foo(2,3)} parses as @samp{bar([],2,3)}. Unfortunately, Calc
15562 first tries to match @samp{2,} against the optional part of the
15563 pattern, finds a match, and so goes ahead to match the rest of the
15564 pattern. Later on it will fail to match the second comma, but it
15565 doesn't know how to go back and try the other alternative at that
15566 point. One way to get around this would be to use two rules:
15569 foo ( # , # , # ) := bar([#1],#2,#3)
15570 foo ( # , # ) := bar([],#1,#2)
15573 More precisely, when Calc wants to match an optional or repeated
15574 part of a pattern, it scans forward attempting to match that part.
15575 If it reaches the end of the optional part without failing, it
15576 ``finalizes'' its choice and proceeds. If it fails, though, it
15577 backs up and tries the other alternative. Thus Calc has ``partial''
15578 backtracking. A fully backtracking parser would go on to make sure
15579 the rest of the pattern matched before finalizing the choice.
15581 @node Conditional Syntax Rules, , Advanced Syntax Patterns, Syntax Tables
15582 @subsubsection Conditional Syntax Rules
15585 It is possible to attach a @dfn{condition} to a syntax rule. For
15589 foo ( # ) := ifoo(#1) :: integer(#1)
15590 foo ( # ) := gfoo(#1)
15594 will parse @samp{foo(3)} as @samp{ifoo(3)}, but will parse
15595 @samp{foo(3.5)} and @samp{foo(x)} as calls to @code{gfoo}. Any
15596 number of conditions may be attached; all must be true for the
15597 rule to succeed. A condition is ``true'' if it evaluates to a
15598 nonzero number. @xref{Logical Operations}, for a list of Calc
15599 functions like @code{integer} that perform logical tests.
15601 The exact sequence of events is as follows: When Calc tries a
15602 rule, it first matches the pattern as usual. It then substitutes
15603 @samp{#1}, @samp{#2}, etc., in the conditions, if any. Next, the
15604 conditions are simplified and evaluated in order from left to right,
15605 as if by the @w{@kbd{a s}} algebra command (@pxref{Simplifying Formulas}).
15606 Each result is true if it is a nonzero number, or an expression
15607 that can be proven to be nonzero (@pxref{Declarations}). If the
15608 results of all conditions are true, the expression (such as
15609 @samp{ifoo(#1)}) has its @samp{#}s substituted, and that is the
15610 result of the parse. If the result of any condition is false, Calc
15611 goes on to try the next rule in the syntax table.
15613 Syntax rules also support @code{let} conditions, which operate in
15614 exactly the same way as they do in algebraic rewrite rules.
15615 @xref{Other Features of Rewrite Rules}, for details. A @code{let}
15616 condition is always true, but as a side effect it defines a
15617 variable which can be used in later conditions, and also in the
15618 expression after the @samp{:=} sign:
15621 foo ( # ) := hifoo(x) :: let(x := #1 + 0.5) :: dnumint(x)
15625 The @code{dnumint} function tests if a value is numerically an
15626 integer, i.e., either a true integer or an integer-valued float.
15627 This rule will parse @code{foo} with a half-integer argument,
15628 like @samp{foo(3.5)}, to a call like @samp{hifoo(4.)}.
15630 The lefthand side of a syntax rule @code{let} must be a simple
15631 variable, not the arbitrary pattern that is allowed in rewrite
15634 The @code{matches} function is also treated specially in syntax
15635 rule conditions (again, in the same way as in rewrite rules).
15636 @xref{Matching Commands}. If the matching pattern contains
15637 meta-variables, then those meta-variables may be used in later
15638 conditions and in the result expression. The arguments to
15639 @code{matches} are not evaluated in this situation.
15642 sum ( # , # ) := sum(#1,a,b,c) :: matches(#2, a=[b..c])
15646 This is another way to implement the Maple mode @code{sum} notation.
15647 In this approach, we allow @samp{#2} to equal the whole expression
15648 @samp{i=1..10}. Then, we use @code{matches} to break it apart into
15649 its components. If the expression turns out not to match the pattern,
15650 the syntax rule will fail. Note that @kbd{Z S} always uses Calc's
15651 Normal language mode for editing expressions in syntax rules, so we
15652 must use regular Calc notation for the interval @samp{[b..c]} that
15653 will correspond to the Maple mode interval @samp{1..10}.
15655 @node Modes Variable, Calc Mode Line, Language Modes, Mode Settings
15656 @section The @code{Modes} Variable
15660 @pindex calc-get-modes
15661 The @kbd{m g} (@code{calc-get-modes}) command pushes onto the stack
15662 a vector of numbers that describes the various mode settings that
15663 are in effect. With a numeric prefix argument, it pushes only the
15664 @var{n}th mode, i.e., the @var{n}th element of this vector. Keyboard
15665 macros can use the @kbd{m g} command to modify their behavior based
15666 on the current mode settings.
15668 @cindex @code{Modes} variable
15670 The modes vector is also available in the special variable
15671 @code{Modes}. In other words, @kbd{m g} is like @kbd{s r Modes @key{RET}}.
15672 It will not work to store into this variable; in fact, if you do,
15673 @code{Modes} will cease to track the current modes. (The @kbd{m g}
15674 command will continue to work, however.)
15676 In general, each number in this vector is suitable as a numeric
15677 prefix argument to the associated mode-setting command. (Recall
15678 that the @kbd{~} key takes a number from the stack and gives it as
15679 a numeric prefix to the next command.)
15681 The elements of the modes vector are as follows:
15685 Current precision. Default is 12; associated command is @kbd{p}.
15688 Binary word size. Default is 32; associated command is @kbd{b w}.
15691 Stack size (not counting the value about to be pushed by @kbd{m g}).
15692 This is zero if @kbd{m g} is executed with an empty stack.
15695 Number radix. Default is 10; command is @kbd{d r}.
15698 Floating-point format. This is the number of digits, plus the
15699 constant 0 for normal notation, 10000 for scientific notation,
15700 20000 for engineering notation, or 30000 for fixed-point notation.
15701 These codes are acceptable as prefix arguments to the @kbd{d n}
15702 command, but note that this may lose information: For example,
15703 @kbd{d s} and @kbd{C-u 12 d s} have similar (but not quite
15704 identical) effects if the current precision is 12, but they both
15705 produce a code of 10012, which will be treated by @kbd{d n} as
15706 @kbd{C-u 12 d s}. If the precision then changes, the float format
15707 will still be frozen at 12 significant figures.
15710 Angular mode. Default is 1 (degrees). Other values are 2 (radians)
15711 and 3 (HMS). The @kbd{m d} command accepts these prefixes.
15714 Symbolic mode. Value is 0 or 1; default is 0. Command is @kbd{m s}.
15717 Fraction mode. Value is 0 or 1; default is 0. Command is @kbd{m f}.
15720 Polar mode. Value is 0 (rectangular) or 1 (polar); default is 0.
15721 Command is @kbd{m p}.
15724 Matrix/Scalar mode. Default value is @mathit{-1}. Value is 0 for Scalar
15725 mode, @mathit{-2} for Matrix mode, @mathit{-3} for square Matrix mode,
15727 @texline @math{N\times N}
15728 @infoline @var{N}x@var{N}
15729 Matrix mode. Command is @kbd{m v}.
15732 Simplification mode. Default is 1. Value is @mathit{-1} for off (@kbd{m O}),
15733 0 for @kbd{m N}, 2 for @kbd{m B}, 3 for @kbd{m A}, 4 for @kbd{m E},
15734 or 5 for @w{@kbd{m U}}. The @kbd{m D} command accepts these prefixes.
15737 Infinite mode. Default is @mathit{-1} (off). Value is 1 if the mode is on,
15738 or 0 if the mode is on with positive zeros. Command is @kbd{m i}.
15741 For example, the sequence @kbd{M-1 m g @key{RET} 2 + ~ p} increases the
15742 precision by two, leaving a copy of the old precision on the stack.
15743 Later, @kbd{~ p} will restore the original precision using that
15744 stack value. (This sequence might be especially useful inside a
15747 As another example, @kbd{M-3 m g 1 - ~ @key{DEL}} deletes all but the
15748 oldest (bottommost) stack entry.
15750 Yet another example: The HP-48 ``round'' command rounds a number
15751 to the current displayed precision. You could roughly emulate this
15752 in Calc with the sequence @kbd{M-5 m g 10000 % ~ c c}. (This
15753 would not work for fixed-point mode, but it wouldn't be hard to
15754 do a full emulation with the help of the @kbd{Z [} and @kbd{Z ]}
15755 programming commands. @xref{Conditionals in Macros}.)
15757 @node Calc Mode Line, , Modes Variable, Mode Settings
15758 @section The Calc Mode Line
15761 @cindex Mode line indicators
15762 This section is a summary of all symbols that can appear on the
15763 Calc mode line, the highlighted bar that appears under the Calc
15764 stack window (or under an editing window in Embedded mode).
15766 The basic mode line format is:
15769 --%*-Calc: 12 Deg @var{other modes} (Calculator)
15772 The @samp{%*} indicates that the buffer is ``read-only''; it shows that
15773 regular Emacs commands are not allowed to edit the stack buffer
15774 as if it were text.
15776 The word @samp{Calc:} changes to @samp{CalcEmbed:} if Embedded mode
15777 is enabled. The words after this describe the various Calc modes
15778 that are in effect.
15780 The first mode is always the current precision, an integer.
15781 The second mode is always the angular mode, either @code{Deg},
15782 @code{Rad}, or @code{Hms}.
15784 Here is a complete list of the remaining symbols that can appear
15789 Algebraic mode (@kbd{m a}; @pxref{Algebraic Entry}).
15792 Incomplete algebraic mode (@kbd{C-u m a}).
15795 Total algebraic mode (@kbd{m t}).
15798 Symbolic mode (@kbd{m s}; @pxref{Symbolic Mode}).
15801 Matrix mode (@kbd{m v}; @pxref{Matrix Mode}).
15803 @item Matrix@var{n}
15804 Dimensioned Matrix mode (@kbd{C-u @var{n} m v}; @pxref{Matrix Mode}).
15807 Square Matrix mode (@kbd{C-u m v}; @pxref{Matrix Mode}).
15810 Scalar mode (@kbd{m v}; @pxref{Matrix Mode}).
15813 Polar complex mode (@kbd{m p}; @pxref{Polar Mode}).
15816 Fraction mode (@kbd{m f}; @pxref{Fraction Mode}).
15819 Infinite mode (@kbd{m i}; @pxref{Infinite Mode}).
15822 Positive Infinite mode (@kbd{C-u 0 m i}).
15825 Default simplifications off (@kbd{m O}; @pxref{Simplification Modes}).
15828 Default simplifications for numeric arguments only (@kbd{m N}).
15830 @item BinSimp@var{w}
15831 Binary-integer simplification mode; word size @var{w} (@kbd{m B}, @kbd{b w}).
15834 Algebraic simplification mode (@kbd{m A}).
15837 Extended algebraic simplification mode (@kbd{m E}).
15840 Units simplification mode (@kbd{m U}).
15843 Current radix is 2 (@kbd{d 2}; @pxref{Radix Modes}).
15846 Current radix is 8 (@kbd{d 8}).
15849 Current radix is 16 (@kbd{d 6}).
15852 Current radix is @var{n} (@kbd{d r}).
15855 Leading zeros (@kbd{d z}; @pxref{Radix Modes}).
15858 Big language mode (@kbd{d B}; @pxref{Normal Language Modes}).
15861 One-line normal language mode (@kbd{d O}).
15864 Unformatted language mode (@kbd{d U}).
15867 C language mode (@kbd{d C}; @pxref{C FORTRAN Pascal}).
15870 Pascal language mode (@kbd{d P}).
15873 FORTRAN language mode (@kbd{d F}).
15876 @TeX{} language mode (@kbd{d T}; @pxref{TeX and LaTeX Language Modes}).
15879 La@TeX{} language mode (@kbd{d L}; @pxref{TeX and LaTeX Language Modes}).
15882 @dfn{Eqn} language mode (@kbd{d E}; @pxref{Eqn Language Mode}).
15885 Mathematica language mode (@kbd{d M}; @pxref{Mathematica Language Mode}).
15888 Maple language mode (@kbd{d W}; @pxref{Maple Language Mode}).
15891 Normal float mode with @var{n} digits (@kbd{d n}; @pxref{Float Formats}).
15894 Fixed point mode with @var{n} digits after the point (@kbd{d f}).
15897 Scientific notation mode (@kbd{d s}).
15900 Scientific notation with @var{n} digits (@kbd{d s}).
15903 Engineering notation mode (@kbd{d e}).
15906 Engineering notation with @var{n} digits (@kbd{d e}).
15909 Left-justified display indented by @var{n} (@kbd{d <}; @pxref{Justification}).
15912 Right-justified display (@kbd{d >}).
15915 Right-justified display with width @var{n} (@kbd{d >}).
15918 Centered display (@kbd{d =}).
15920 @item Center@var{n}
15921 Centered display with center column @var{n} (@kbd{d =}).
15924 Line breaking with width @var{n} (@kbd{d b}; @pxref{Normal Language Modes}).
15927 No line breaking (@kbd{d b}).
15930 Selections show deep structure (@kbd{j b}; @pxref{Making Selections}).
15933 Record modes in @file{~/.calc.el} (@kbd{m R}; @pxref{General Mode Commands}).
15936 Record modes in Embedded buffer (@kbd{m R}).
15939 Record modes as editing-only in Embedded buffer (@kbd{m R}).
15942 Record modes as permanent-only in Embedded buffer (@kbd{m R}).
15945 Record modes as global in Embedded buffer (@kbd{m R}).
15948 Automatic recomputation turned off (@kbd{m C}; @pxref{Automatic
15952 GNUPLOT process is alive in background (@pxref{Graphics}).
15955 Top-of-stack has a selection (Embedded only; @pxref{Making Selections}).
15958 The stack display may not be up-to-date (@pxref{Display Modes}).
15961 ``Inverse'' prefix was pressed (@kbd{I}; @pxref{Inverse and Hyperbolic}).
15964 ``Hyperbolic'' prefix was pressed (@kbd{H}).
15967 ``Keep-arguments'' prefix was pressed (@kbd{K}).
15970 Stack is truncated (@kbd{d t}; @pxref{Truncating the Stack}).
15973 In addition, the symbols @code{Active} and @code{~Active} can appear
15974 as minor modes on an Embedded buffer's mode line. @xref{Embedded Mode}.
15976 @node Arithmetic, Scientific Functions, Mode Settings, Top
15977 @chapter Arithmetic Functions
15980 This chapter describes the Calc commands for doing simple calculations
15981 on numbers, such as addition, absolute value, and square roots. These
15982 commands work by removing the top one or two values from the stack,
15983 performing the desired operation, and pushing the result back onto the
15984 stack. If the operation cannot be performed, the result pushed is a
15985 formula instead of a number, such as @samp{2/0} (because division by zero
15986 is invalid) or @samp{sqrt(x)} (because the argument @samp{x} is a formula).
15988 Most of the commands described here can be invoked by a single keystroke.
15989 Some of the more obscure ones are two-letter sequences beginning with
15990 the @kbd{f} (``functions'') prefix key.
15992 @xref{Prefix Arguments}, for a discussion of the effect of numeric
15993 prefix arguments on commands in this chapter which do not otherwise
15994 interpret a prefix argument.
15997 * Basic Arithmetic::
15998 * Integer Truncation::
15999 * Complex Number Functions::
16001 * Date Arithmetic::
16002 * Financial Functions::
16003 * Binary Functions::
16006 @node Basic Arithmetic, Integer Truncation, Arithmetic, Arithmetic
16007 @section Basic Arithmetic
16016 The @kbd{+} (@code{calc-plus}) command adds two numbers. The numbers may
16017 be any of the standard Calc data types. The resulting sum is pushed back
16020 If both arguments of @kbd{+} are vectors or matrices (of matching dimensions),
16021 the result is a vector or matrix sum. If one argument is a vector and the
16022 other a scalar (i.e., a non-vector), the scalar is added to each of the
16023 elements of the vector to form a new vector. If the scalar is not a
16024 number, the operation is left in symbolic form: Suppose you added @samp{x}
16025 to the vector @samp{[1,2]}. You may want the result @samp{[1+x,2+x]}, or
16026 you may plan to substitute a 2-vector for @samp{x} in the future. Since
16027 the Calculator can't tell which interpretation you want, it makes the
16028 safest assumption. @xref{Reducing and Mapping}, for a way to add @samp{x}
16029 to every element of a vector.
16031 If either argument of @kbd{+} is a complex number, the result will in general
16032 be complex. If one argument is in rectangular form and the other polar,
16033 the current Polar mode determines the form of the result. If Symbolic
16034 mode is enabled, the sum may be left as a formula if the necessary
16035 conversions for polar addition are non-trivial.
16037 If both arguments of @kbd{+} are HMS forms, the forms are added according to
16038 the usual conventions of hours-minutes-seconds notation. If one argument
16039 is an HMS form and the other is a number, that number is converted from
16040 degrees or radians (depending on the current Angular mode) to HMS format
16041 and then the two HMS forms are added.
16043 If one argument of @kbd{+} is a date form, the other can be either a
16044 real number, which advances the date by a certain number of days, or
16045 an HMS form, which advances the date by a certain amount of time.
16046 Subtracting two date forms yields the number of days between them.
16047 Adding two date forms is meaningless, but Calc interprets it as the
16048 subtraction of one date form and the negative of the other. (The
16049 negative of a date form can be understood by remembering that dates
16050 are stored as the number of days before or after Jan 1, 1 AD.)
16052 If both arguments of @kbd{+} are error forms, the result is an error form
16053 with an appropriately computed standard deviation. If one argument is an
16054 error form and the other is a number, the number is taken to have zero error.
16055 Error forms may have symbolic formulas as their mean and/or error parts;
16056 adding these will produce a symbolic error form result. However, adding an
16057 error form to a plain symbolic formula (as in @samp{(a +/- b) + c}) will not
16058 work, for the same reasons just mentioned for vectors. Instead you must
16059 write @samp{(a +/- b) + (c +/- 0)}.
16061 If both arguments of @kbd{+} are modulo forms with equal values of @expr{M},
16062 or if one argument is a modulo form and the other a plain number, the
16063 result is a modulo form which represents the sum, modulo @expr{M}, of
16066 If both arguments of @kbd{+} are intervals, the result is an interval
16067 which describes all possible sums of the possible input values. If
16068 one argument is a plain number, it is treated as the interval
16069 @w{@samp{[x ..@: x]}}.
16071 If one argument of @kbd{+} is an infinity and the other is not, the
16072 result is that same infinity. If both arguments are infinite and in
16073 the same direction, the result is the same infinity, but if they are
16074 infinite in different directions the result is @code{nan}.
16082 The @kbd{-} (@code{calc-minus}) command subtracts two values. The top
16083 number on the stack is subtracted from the one behind it, so that the
16084 computation @kbd{5 @key{RET} 2 -} produces 3, not @mathit{-3}. All options
16085 available for @kbd{+} are available for @kbd{-} as well.
16093 The @kbd{*} (@code{calc-times}) command multiplies two numbers. If one
16094 argument is a vector and the other a scalar, the scalar is multiplied by
16095 the elements of the vector to produce a new vector. If both arguments
16096 are vectors, the interpretation depends on the dimensions of the
16097 vectors: If both arguments are matrices, a matrix multiplication is
16098 done. If one argument is a matrix and the other a plain vector, the
16099 vector is interpreted as a row vector or column vector, whichever is
16100 dimensionally correct. If both arguments are plain vectors, the result
16101 is a single scalar number which is the dot product of the two vectors.
16103 If one argument of @kbd{*} is an HMS form and the other a number, the
16104 HMS form is multiplied by that amount. It is an error to multiply two
16105 HMS forms together, or to attempt any multiplication involving date
16106 forms. Error forms, modulo forms, and intervals can be multiplied;
16107 see the comments for addition of those forms. When two error forms
16108 or intervals are multiplied they are considered to be statistically
16109 independent; thus, @samp{[-2 ..@: 3] * [-2 ..@: 3]} is @samp{[-6 ..@: 9]},
16110 whereas @w{@samp{[-2 ..@: 3] ^ 2}} is @samp{[0 ..@: 9]}.
16113 @pindex calc-divide
16118 The @kbd{/} (@code{calc-divide}) command divides two numbers.
16120 When combining multiplication and division in an algebraic formula, it
16121 is good style to use parentheses to distinguish between possible
16122 interpretations; the expression @samp{a/b*c} should be written
16123 @samp{(a/b)*c} or @samp{a/(b*c)}, as appropriate. Without the
16124 parentheses, Calc will interpret @samp{a/b*c} as @samp{a/(b*c)}, since
16125 in algebraic entry Calc gives division a lower precedence than
16126 multiplication. (This is not standard across all computer languages, and
16127 Calc may change the precedence depending on the language mode being used.
16128 @xref{Language Modes}.) This default ordering can be changed by setting
16129 the customizable variable @code{calc-multiplication-has-precedence} to
16130 @code{nil} (@pxref{Customizing Calc}); this will give multiplication and
16131 division equal precedences. Note that Calc's default choice of
16132 precedence allows @samp{a b / c d} to be used as a shortcut for
16141 When dividing a scalar @expr{B} by a square matrix @expr{A}, the
16142 computation performed is @expr{B} times the inverse of @expr{A}. This
16143 also occurs if @expr{B} is itself a vector or matrix, in which case the
16144 effect is to solve the set of linear equations represented by @expr{B}.
16145 If @expr{B} is a matrix with the same number of rows as @expr{A}, or a
16146 plain vector (which is interpreted here as a column vector), then the
16147 equation @expr{A X = B} is solved for the vector or matrix @expr{X}.
16148 Otherwise, if @expr{B} is a non-square matrix with the same number of
16149 @emph{columns} as @expr{A}, the equation @expr{X A = B} is solved. If
16150 you wish a vector @expr{B} to be interpreted as a row vector to be
16151 solved as @expr{X A = B}, make it into a one-row matrix with @kbd{C-u 1
16152 v p} first. To force a left-handed solution with a square matrix
16153 @expr{B}, transpose @expr{A} and @expr{B} before dividing, then
16154 transpose the result.
16156 HMS forms can be divided by real numbers or by other HMS forms. Error
16157 forms can be divided in any combination of ways. Modulo forms where both
16158 values and the modulo are integers can be divided to get an integer modulo
16159 form result. Intervals can be divided; dividing by an interval that
16160 encompasses zero or has zero as a limit will result in an infinite
16169 The @kbd{^} (@code{calc-power}) command raises a number to a power. If
16170 the power is an integer, an exact result is computed using repeated
16171 multiplications. For non-integer powers, Calc uses Newton's method or
16172 logarithms and exponentials. Square matrices can be raised to integer
16173 powers. If either argument is an error (or interval or modulo) form,
16174 the result is also an error (or interval or modulo) form.
16178 If you press the @kbd{I} (inverse) key first, the @kbd{I ^} command
16179 computes an Nth root: @kbd{125 @key{RET} 3 I ^} computes the number 5.
16180 (This is entirely equivalent to @kbd{125 @key{RET} 1:3 ^}.)
16189 The @kbd{\} (@code{calc-idiv}) command divides two numbers on the stack
16190 to produce an integer result. It is equivalent to dividing with
16191 @key{/}, then rounding down with @kbd{F} (@code{calc-floor}), only a bit
16192 more convenient and efficient. Also, since it is an all-integer
16193 operation when the arguments are integers, it avoids problems that
16194 @kbd{/ F} would have with floating-point roundoff.
16202 The @kbd{%} (@code{calc-mod}) command performs a ``modulo'' (or ``remainder'')
16203 operation. Mathematically, @samp{a%b = a - (a\b)*b}, and is defined
16204 for all real numbers @expr{a} and @expr{b} (except @expr{b=0}). For
16205 positive @expr{b}, the result will always be between 0 (inclusive) and
16206 @expr{b} (exclusive). Modulo does not work for HMS forms and error forms.
16207 If @expr{a} is a modulo form, its modulo is changed to @expr{b}, which
16208 must be positive real number.
16213 The @kbd{:} (@code{calc-fdiv}) [@code{fdiv}] command
16214 divides the two integers on the top of the stack to produce a fractional
16215 result. This is a convenient shorthand for enabling Fraction mode (with
16216 @kbd{m f}) temporarily and using @samp{/}. Note that during numeric entry
16217 the @kbd{:} key is interpreted as a fraction separator, so to divide 8 by 6
16218 you would have to type @kbd{8 @key{RET} 6 @key{RET} :}. (Of course, in
16219 this case, it would be much easier simply to enter the fraction directly
16220 as @kbd{8:6 @key{RET}}!)
16223 @pindex calc-change-sign
16224 The @kbd{n} (@code{calc-change-sign}) command negates the number on the top
16225 of the stack. It works on numbers, vectors and matrices, HMS forms, date
16226 forms, error forms, intervals, and modulo forms.
16231 The @kbd{A} (@code{calc-abs}) [@code{abs}] command computes the absolute
16232 value of a number. The result of @code{abs} is always a nonnegative
16233 real number: With a complex argument, it computes the complex magnitude.
16234 With a vector or matrix argument, it computes the Frobenius norm, i.e.,
16235 the square root of the sum of the squares of the absolute values of the
16236 elements. The absolute value of an error form is defined by replacing
16237 the mean part with its absolute value and leaving the error part the same.
16238 The absolute value of a modulo form is undefined. The absolute value of
16239 an interval is defined in the obvious way.
16242 @pindex calc-abssqr
16244 The @kbd{f A} (@code{calc-abssqr}) [@code{abssqr}] command computes the
16245 absolute value squared of a number, vector or matrix, or error form.
16250 The @kbd{f s} (@code{calc-sign}) [@code{sign}] command returns 1 if its
16251 argument is positive, @mathit{-1} if its argument is negative, or 0 if its
16252 argument is zero. In algebraic form, you can also write @samp{sign(a,x)}
16253 which evaluates to @samp{x * sign(a)}, i.e., either @samp{x}, @samp{-x}, or
16254 zero depending on the sign of @samp{a}.
16260 The @kbd{&} (@code{calc-inv}) [@code{inv}] command computes the
16261 reciprocal of a number, i.e., @expr{1 / x}. Operating on a square
16262 matrix, it computes the inverse of that matrix.
16267 The @kbd{Q} (@code{calc-sqrt}) [@code{sqrt}] command computes the square
16268 root of a number. For a negative real argument, the result will be a
16269 complex number whose form is determined by the current Polar mode.
16274 The @kbd{f h} (@code{calc-hypot}) [@code{hypot}] command computes the square
16275 root of the sum of the squares of two numbers. That is, @samp{hypot(a,b)}
16276 is the length of the hypotenuse of a right triangle with sides @expr{a}
16277 and @expr{b}. If the arguments are complex numbers, their squared
16278 magnitudes are used.
16283 The @kbd{f Q} (@code{calc-isqrt}) [@code{isqrt}] command computes the
16284 integer square root of an integer. This is the true square root of the
16285 number, rounded down to an integer. For example, @samp{isqrt(10)}
16286 produces 3. Note that, like @kbd{\} [@code{idiv}], this uses exact
16287 integer arithmetic throughout to avoid roundoff problems. If the input
16288 is a floating-point number or other non-integer value, this is exactly
16289 the same as @samp{floor(sqrt(x))}.
16297 The @kbd{f n} (@code{calc-min}) [@code{min}] and @kbd{f x} (@code{calc-max})
16298 [@code{max}] commands take the minimum or maximum of two real numbers,
16299 respectively. These commands also work on HMS forms, date forms,
16300 intervals, and infinities. (In algebraic expressions, these functions
16301 take any number of arguments and return the maximum or minimum among
16302 all the arguments.)
16306 @pindex calc-mant-part
16308 @pindex calc-xpon-part
16310 The @kbd{f M} (@code{calc-mant-part}) [@code{mant}] function extracts
16311 the ``mantissa'' part @expr{m} of its floating-point argument; @kbd{f X}
16312 (@code{calc-xpon-part}) [@code{xpon}] extracts the ``exponent'' part
16313 @expr{e}. The original number is equal to
16314 @texline @math{m \times 10^e},
16315 @infoline @expr{m * 10^e},
16316 where @expr{m} is in the interval @samp{[1.0 ..@: 10.0)} except that
16317 @expr{m=e=0} if the original number is zero. For integers
16318 and fractions, @code{mant} returns the number unchanged and @code{xpon}
16319 returns zero. The @kbd{v u} (@code{calc-unpack}) command can also be
16320 used to ``unpack'' a floating-point number; this produces an integer
16321 mantissa and exponent, with the constraint that the mantissa is not
16322 a multiple of ten (again except for the @expr{m=e=0} case).
16325 @pindex calc-scale-float
16327 The @kbd{f S} (@code{calc-scale-float}) [@code{scf}] function scales a number
16328 by a given power of ten. Thus, @samp{scf(mant(x), xpon(x)) = x} for any
16329 real @samp{x}. The second argument must be an integer, but the first
16330 may actually be any numeric value. For example, @samp{scf(5,-2) = 0.05}
16331 or @samp{1:20} depending on the current Fraction mode.
16335 @pindex calc-decrement
16336 @pindex calc-increment
16339 The @kbd{f [} (@code{calc-decrement}) [@code{decr}] and @kbd{f ]}
16340 (@code{calc-increment}) [@code{incr}] functions decrease or increase
16341 a number by one unit. For integers, the effect is obvious. For
16342 floating-point numbers, the change is by one unit in the last place.
16343 For example, incrementing @samp{12.3456} when the current precision
16344 is 6 digits yields @samp{12.3457}. If the current precision had been
16345 8 digits, the result would have been @samp{12.345601}. Incrementing
16346 @samp{0.0} produces
16347 @texline @math{10^{-p}},
16348 @infoline @expr{10^-p},
16349 where @expr{p} is the current
16350 precision. These operations are defined only on integers and floats.
16351 With numeric prefix arguments, they change the number by @expr{n} units.
16353 Note that incrementing followed by decrementing, or vice-versa, will
16354 almost but not quite always cancel out. Suppose the precision is
16355 6 digits and the number @samp{9.99999} is on the stack. Incrementing
16356 will produce @samp{10.0000}; decrementing will produce @samp{9.9999}.
16357 One digit has been dropped. This is an unavoidable consequence of the
16358 way floating-point numbers work.
16360 Incrementing a date/time form adjusts it by a certain number of seconds.
16361 Incrementing a pure date form adjusts it by a certain number of days.
16363 @node Integer Truncation, Complex Number Functions, Basic Arithmetic, Arithmetic
16364 @section Integer Truncation
16367 There are four commands for truncating a real number to an integer,
16368 differing mainly in their treatment of negative numbers. All of these
16369 commands have the property that if the argument is an integer, the result
16370 is the same integer. An integer-valued floating-point argument is converted
16373 If you press @kbd{H} (@code{calc-hyperbolic}) first, the result will be
16374 expressed as an integer-valued floating-point number.
16376 @cindex Integer part of a number
16385 The @kbd{F} (@code{calc-floor}) [@code{floor} or @code{ffloor}] command
16386 truncates a real number to the next lower integer, i.e., toward minus
16387 infinity. Thus @kbd{3.6 F} produces 3, but @kbd{_3.6 F} produces
16391 @pindex calc-ceiling
16398 The @kbd{I F} (@code{calc-ceiling}) [@code{ceil} or @code{fceil}]
16399 command truncates toward positive infinity. Thus @kbd{3.6 I F} produces
16400 4, and @kbd{_3.6 I F} produces @mathit{-3}.
16410 The @kbd{R} (@code{calc-round}) [@code{round} or @code{fround}] command
16411 rounds to the nearest integer. When the fractional part is .5 exactly,
16412 this command rounds away from zero. (All other rounding in the
16413 Calculator uses this convention as well.) Thus @kbd{3.5 R} produces 4
16414 but @kbd{3.4 R} produces 3; @kbd{_3.5 R} produces @mathit{-4}.
16424 The @kbd{I R} (@code{calc-trunc}) [@code{trunc} or @code{ftrunc}]
16425 command truncates toward zero. In other words, it ``chops off''
16426 everything after the decimal point. Thus @kbd{3.6 I R} produces 3 and
16427 @kbd{_3.6 I R} produces @mathit{-3}.
16429 These functions may not be applied meaningfully to error forms, but they
16430 do work for intervals. As a convenience, applying @code{floor} to a
16431 modulo form floors the value part of the form. Applied to a vector,
16432 these functions operate on all elements of the vector one by one.
16433 Applied to a date form, they operate on the internal numerical
16434 representation of dates, converting a date/time form into a pure date.
16452 There are two more rounding functions which can only be entered in
16453 algebraic notation. The @code{roundu} function is like @code{round}
16454 except that it rounds up, toward plus infinity, when the fractional
16455 part is .5. This distinction matters only for negative arguments.
16456 Also, @code{rounde} rounds to an even number in the case of a tie,
16457 rounding up or down as necessary. For example, @samp{rounde(3.5)} and
16458 @samp{rounde(4.5)} both return 4, but @samp{rounde(5.5)} returns 6.
16459 The advantage of round-to-even is that the net error due to rounding
16460 after a long calculation tends to cancel out to zero. An important
16461 subtle point here is that the number being fed to @code{rounde} will
16462 already have been rounded to the current precision before @code{rounde}
16463 begins. For example, @samp{rounde(2.500001)} with a current precision
16464 of 6 will incorrectly, or at least surprisingly, yield 2 because the
16465 argument will first have been rounded down to @expr{2.5} (which
16466 @code{rounde} sees as an exact tie between 2 and 3).
16468 Each of these functions, when written in algebraic formulas, allows
16469 a second argument which specifies the number of digits after the
16470 decimal point to keep. For example, @samp{round(123.4567, 2)} will
16471 produce the answer 123.46, and @samp{round(123.4567, -1)} will
16472 produce 120 (i.e., the cutoff is one digit to the @emph{left} of
16473 the decimal point). A second argument of zero is equivalent to
16474 no second argument at all.
16476 @cindex Fractional part of a number
16477 To compute the fractional part of a number (i.e., the amount which, when
16478 added to `@tfn{floor(}@var{n}@tfn{)}', will produce @var{n}) just take @var{n}
16479 modulo 1 using the @code{%} command.
16481 Note also the @kbd{\} (integer quotient), @kbd{f I} (integer logarithm),
16482 and @kbd{f Q} (integer square root) commands, which are analogous to
16483 @kbd{/}, @kbd{B}, and @kbd{Q}, respectively, except that they take integer
16484 arguments and return the result rounded down to an integer.
16486 @node Complex Number Functions, Conversions, Integer Truncation, Arithmetic
16487 @section Complex Number Functions
16493 The @kbd{J} (@code{calc-conj}) [@code{conj}] command computes the
16494 complex conjugate of a number. For complex number @expr{a+bi}, the
16495 complex conjugate is @expr{a-bi}. If the argument is a real number,
16496 this command leaves it the same. If the argument is a vector or matrix,
16497 this command replaces each element by its complex conjugate.
16500 @pindex calc-argument
16502 The @kbd{G} (@code{calc-argument}) [@code{arg}] command computes the
16503 ``argument'' or polar angle of a complex number. For a number in polar
16504 notation, this is simply the second component of the pair
16505 @texline `@tfn{(}@var{r}@tfn{;}@math{\theta}@tfn{)}'.
16506 @infoline `@tfn{(}@var{r}@tfn{;}@var{theta}@tfn{)}'.
16507 The result is expressed according to the current angular mode and will
16508 be in the range @mathit{-180} degrees (exclusive) to @mathit{+180} degrees
16509 (inclusive), or the equivalent range in radians.
16511 @pindex calc-imaginary
16512 The @code{calc-imaginary} command multiplies the number on the
16513 top of the stack by the imaginary number @expr{i = (0,1)}. This
16514 command is not normally bound to a key in Calc, but it is available
16515 on the @key{IMAG} button in Keypad mode.
16520 The @kbd{f r} (@code{calc-re}) [@code{re}] command replaces a complex number
16521 by its real part. This command has no effect on real numbers. (As an
16522 added convenience, @code{re} applied to a modulo form extracts
16528 The @kbd{f i} (@code{calc-im}) [@code{im}] command replaces a complex number
16529 by its imaginary part; real numbers are converted to zero. With a vector
16530 or matrix argument, these functions operate element-wise.
16535 @kindex v p (complex)
16537 The @kbd{v p} (@code{calc-pack}) command can pack the top two numbers on
16538 the stack into a composite object such as a complex number. With
16539 a prefix argument of @mathit{-1}, it produces a rectangular complex number;
16540 with an argument of @mathit{-2}, it produces a polar complex number.
16541 (Also, @pxref{Building Vectors}.)
16546 @kindex v u (complex)
16547 @pindex calc-unpack
16548 The @kbd{v u} (@code{calc-unpack}) command takes the complex number
16549 (or other composite object) on the top of the stack and unpacks it
16550 into its separate components.
16552 @node Conversions, Date Arithmetic, Complex Number Functions, Arithmetic
16553 @section Conversions
16556 The commands described in this section convert numbers from one form
16557 to another; they are two-key sequences beginning with the letter @kbd{c}.
16562 The @kbd{c f} (@code{calc-float}) [@code{pfloat}] command converts the
16563 number on the top of the stack to floating-point form. For example,
16564 @expr{23} is converted to @expr{23.0}, @expr{3:2} is converted to
16565 @expr{1.5}, and @expr{2.3} is left the same. If the value is a composite
16566 object such as a complex number or vector, each of the components is
16567 converted to floating-point. If the value is a formula, all numbers
16568 in the formula are converted to floating-point. Note that depending
16569 on the current floating-point precision, conversion to floating-point
16570 format may lose information.
16572 As a special exception, integers which appear as powers or subscripts
16573 are not floated by @kbd{c f}. If you really want to float a power,
16574 you can use a @kbd{j s} command to select the power followed by @kbd{c f}.
16575 Because @kbd{c f} cannot examine the formula outside of the selection,
16576 it does not notice that the thing being floated is a power.
16577 @xref{Selecting Subformulas}.
16579 The normal @kbd{c f} command is ``pervasive'' in the sense that it
16580 applies to all numbers throughout the formula. The @code{pfloat}
16581 algebraic function never stays around in a formula; @samp{pfloat(a + 1)}
16582 changes to @samp{a + 1.0} as soon as it is evaluated.
16586 With the Hyperbolic flag, @kbd{H c f} [@code{float}] operates
16587 only on the number or vector of numbers at the top level of its
16588 argument. Thus, @samp{float(1)} is 1.0, but @samp{float(a + 1)}
16589 is left unevaluated because its argument is not a number.
16591 You should use @kbd{H c f} if you wish to guarantee that the final
16592 value, once all the variables have been assigned, is a float; you
16593 would use @kbd{c f} if you wish to do the conversion on the numbers
16594 that appear right now.
16597 @pindex calc-fraction
16599 The @kbd{c F} (@code{calc-fraction}) [@code{pfrac}] command converts a
16600 floating-point number into a fractional approximation. By default, it
16601 produces a fraction whose decimal representation is the same as the
16602 input number, to within the current precision. You can also give a
16603 numeric prefix argument to specify a tolerance, either directly, or,
16604 if the prefix argument is zero, by using the number on top of the stack
16605 as the tolerance. If the tolerance is a positive integer, the fraction
16606 is correct to within that many significant figures. If the tolerance is
16607 a non-positive integer, it specifies how many digits fewer than the current
16608 precision to use. If the tolerance is a floating-point number, the
16609 fraction is correct to within that absolute amount.
16613 The @code{pfrac} function is pervasive, like @code{pfloat}.
16614 There is also a non-pervasive version, @kbd{H c F} [@code{frac}],
16615 which is analogous to @kbd{H c f} discussed above.
16618 @pindex calc-to-degrees
16620 The @kbd{c d} (@code{calc-to-degrees}) [@code{deg}] command converts a
16621 number into degrees form. The value on the top of the stack may be an
16622 HMS form (interpreted as degrees-minutes-seconds), or a real number which
16623 will be interpreted in radians regardless of the current angular mode.
16626 @pindex calc-to-radians
16628 The @kbd{c r} (@code{calc-to-radians}) [@code{rad}] command converts an
16629 HMS form or angle in degrees into an angle in radians.
16632 @pindex calc-to-hms
16634 The @kbd{c h} (@code{calc-to-hms}) [@code{hms}] command converts a real
16635 number, interpreted according to the current angular mode, to an HMS
16636 form describing the same angle. In algebraic notation, the @code{hms}
16637 function also accepts three arguments: @samp{hms(@var{h}, @var{m}, @var{s})}.
16638 (The three-argument version is independent of the current angular mode.)
16640 @pindex calc-from-hms
16641 The @code{calc-from-hms} command converts the HMS form on the top of the
16642 stack into a real number according to the current angular mode.
16649 The @kbd{c p} (@code{calc-polar}) command converts the complex number on
16650 the top of the stack from polar to rectangular form, or from rectangular
16651 to polar form, whichever is appropriate. Real numbers are left the same.
16652 This command is equivalent to the @code{rect} or @code{polar}
16653 functions in algebraic formulas, depending on the direction of
16654 conversion. (It uses @code{polar}, except that if the argument is
16655 already a polar complex number, it uses @code{rect} instead. The
16656 @kbd{I c p} command always uses @code{rect}.)
16661 The @kbd{c c} (@code{calc-clean}) [@code{pclean}] command ``cleans'' the
16662 number on the top of the stack. Floating point numbers are re-rounded
16663 according to the current precision. Polar numbers whose angular
16664 components have strayed from the @mathit{-180} to @mathit{+180} degree range
16665 are normalized. (Note that results will be undesirable if the current
16666 angular mode is different from the one under which the number was
16667 produced!) Integers and fractions are generally unaffected by this
16668 operation. Vectors and formulas are cleaned by cleaning each component
16669 number (i.e., pervasively).
16671 If the simplification mode is set below the default level, it is raised
16672 to the default level for the purposes of this command. Thus, @kbd{c c}
16673 applies the default simplifications even if their automatic application
16674 is disabled. @xref{Simplification Modes}.
16676 @cindex Roundoff errors, correcting
16677 A numeric prefix argument to @kbd{c c} sets the floating-point precision
16678 to that value for the duration of the command. A positive prefix (of at
16679 least 3) sets the precision to the specified value; a negative or zero
16680 prefix decreases the precision by the specified amount.
16683 @pindex calc-clean-num
16684 The keystroke sequences @kbd{c 0} through @kbd{c 9} are equivalent
16685 to @kbd{c c} with the corresponding negative prefix argument. If roundoff
16686 errors have changed 2.0 into 1.999999, typing @kbd{c 1} to clip off one
16687 decimal place often conveniently does the trick.
16689 The @kbd{c c} command with a numeric prefix argument, and the @kbd{c 0}
16690 through @kbd{c 9} commands, also ``clip'' very small floating-point
16691 numbers to zero. If the exponent is less than or equal to the negative
16692 of the specified precision, the number is changed to 0.0. For example,
16693 if the current precision is 12, then @kbd{c 2} changes the vector
16694 @samp{[1e-8, 1e-9, 1e-10, 1e-11]} to @samp{[1e-8, 1e-9, 0, 0]}.
16695 Numbers this small generally arise from roundoff noise.
16697 If the numbers you are using really are legitimately this small,
16698 you should avoid using the @kbd{c 0} through @kbd{c 9} commands.
16699 (The plain @kbd{c c} command rounds to the current precision but
16700 does not clip small numbers.)
16702 One more property of @kbd{c 0} through @kbd{c 9}, and of @kbd{c c} with
16703 a prefix argument, is that integer-valued floats are converted to
16704 plain integers, so that @kbd{c 1} on @samp{[1., 1.5, 2., 2.5, 3.]}
16705 produces @samp{[1, 1.5, 2, 2.5, 3]}. This is not done for huge
16706 numbers (@samp{1e100} is technically an integer-valued float, but
16707 you wouldn't want it automatically converted to a 100-digit integer).
16712 With the Hyperbolic flag, @kbd{H c c} and @kbd{H c 0} through @kbd{H c 9}
16713 operate non-pervasively [@code{clean}].
16715 @node Date Arithmetic, Financial Functions, Conversions, Arithmetic
16716 @section Date Arithmetic
16719 @cindex Date arithmetic, additional functions
16720 The commands described in this section perform various conversions
16721 and calculations involving date forms (@pxref{Date Forms}). They
16722 use the @kbd{t} (for time/date) prefix key followed by shifted
16725 The simplest date arithmetic is done using the regular @kbd{+} and @kbd{-}
16726 commands. In particular, adding a number to a date form advances the
16727 date form by a certain number of days; adding an HMS form to a date
16728 form advances the date by a certain amount of time; and subtracting two
16729 date forms produces a difference measured in days. The commands
16730 described here provide additional, more specialized operations on dates.
16732 Many of these commands accept a numeric prefix argument; if you give
16733 plain @kbd{C-u} as the prefix, these commands will instead take the
16734 additional argument from the top of the stack.
16737 * Date Conversions::
16743 @node Date Conversions, Date Functions, Date Arithmetic, Date Arithmetic
16744 @subsection Date Conversions
16750 The @kbd{t D} (@code{calc-date}) [@code{date}] command converts a
16751 date form into a number, measured in days since Jan 1, 1 AD. The
16752 result will be an integer if @var{date} is a pure date form, or a
16753 fraction or float if @var{date} is a date/time form. Or, if its
16754 argument is a number, it converts this number into a date form.
16756 With a numeric prefix argument, @kbd{t D} takes that many objects
16757 (up to six) from the top of the stack and interprets them in one
16758 of the following ways:
16760 The @samp{date(@var{year}, @var{month}, @var{day})} function
16761 builds a pure date form out of the specified year, month, and
16762 day, which must all be integers. @var{Year} is a year number,
16763 such as 1991 (@emph{not} the same as 91!). @var{Month} must be
16764 an integer in the range 1 to 12; @var{day} must be in the range
16765 1 to 31. If the specified month has fewer than 31 days and
16766 @var{day} is too large, the equivalent day in the following
16767 month will be used.
16769 The @samp{date(@var{month}, @var{day})} function builds a
16770 pure date form using the current year, as determined by the
16773 The @samp{date(@var{year}, @var{month}, @var{day}, @var{hms})}
16774 function builds a date/time form using an @var{hms} form.
16776 The @samp{date(@var{year}, @var{month}, @var{day}, @var{hour},
16777 @var{minute}, @var{second})} function builds a date/time form.
16778 @var{hour} should be an integer in the range 0 to 23;
16779 @var{minute} should be an integer in the range 0 to 59;
16780 @var{second} should be any real number in the range @samp{[0 .. 60)}.
16781 The last two arguments default to zero if omitted.
16784 @pindex calc-julian
16786 @cindex Julian day counts, conversions
16787 The @kbd{t J} (@code{calc-julian}) [@code{julian}] command converts
16788 a date form into a Julian day count, which is the number of days
16789 since noon (GMT) on Jan 1, 4713 BC. A pure date is converted to an
16790 integer Julian count representing noon of that day. A date/time form
16791 is converted to an exact floating-point Julian count, adjusted to
16792 interpret the date form in the current time zone but the Julian
16793 day count in Greenwich Mean Time. A numeric prefix argument allows
16794 you to specify the time zone; @pxref{Time Zones}. Use a prefix of
16795 zero to suppress the time zone adjustment. Note that pure date forms
16796 are never time-zone adjusted.
16798 This command can also do the opposite conversion, from a Julian day
16799 count (either an integer day, or a floating-point day and time in
16800 the GMT zone), into a pure date form or a date/time form in the
16801 current or specified time zone.
16804 @pindex calc-unix-time
16806 @cindex Unix time format, conversions
16807 The @kbd{t U} (@code{calc-unix-time}) [@code{unixtime}] command
16808 converts a date form into a Unix time value, which is the number of
16809 seconds since midnight on Jan 1, 1970, or vice-versa. The numeric result
16810 will be an integer if the current precision is 12 or less; for higher
16811 precisions, the result may be a float with (@var{precision}@minus{}12)
16812 digits after the decimal. Just as for @kbd{t J}, the numeric time
16813 is interpreted in the GMT time zone and the date form is interpreted
16814 in the current or specified zone. Some systems use Unix-like
16815 numbering but with the local time zone; give a prefix of zero to
16816 suppress the adjustment if so.
16819 @pindex calc-convert-time-zones
16821 @cindex Time Zones, converting between
16822 The @kbd{t C} (@code{calc-convert-time-zones}) [@code{tzconv}]
16823 command converts a date form from one time zone to another. You
16824 are prompted for each time zone name in turn; you can answer with
16825 any suitable Calc time zone expression (@pxref{Time Zones}).
16826 If you answer either prompt with a blank line, the local time
16827 zone is used for that prompt. You can also answer the first
16828 prompt with @kbd{$} to take the two time zone names from the
16829 stack (and the date to be converted from the third stack level).
16831 @node Date Functions, Business Days, Date Conversions, Date Arithmetic
16832 @subsection Date Functions
16838 The @kbd{t N} (@code{calc-now}) [@code{now}] command pushes the
16839 current date and time on the stack as a date form. The time is
16840 reported in terms of the specified time zone; with no numeric prefix
16841 argument, @kbd{t N} reports for the current time zone.
16844 @pindex calc-date-part
16845 The @kbd{t P} (@code{calc-date-part}) command extracts one part
16846 of a date form. The prefix argument specifies the part; with no
16847 argument, this command prompts for a part code from 1 to 9.
16848 The various part codes are described in the following paragraphs.
16851 The @kbd{M-1 t P} [@code{year}] function extracts the year number
16852 from a date form as an integer, e.g., 1991. This and the
16853 following functions will also accept a real number for an
16854 argument, which is interpreted as a standard Calc day number.
16855 Note that this function will never return zero, since the year
16856 1 BC immediately precedes the year 1 AD.
16859 The @kbd{M-2 t P} [@code{month}] function extracts the month number
16860 from a date form as an integer in the range 1 to 12.
16863 The @kbd{M-3 t P} [@code{day}] function extracts the day number
16864 from a date form as an integer in the range 1 to 31.
16867 The @kbd{M-4 t P} [@code{hour}] function extracts the hour from
16868 a date form as an integer in the range 0 (midnight) to 23. Note
16869 that 24-hour time is always used. This returns zero for a pure
16870 date form. This function (and the following two) also accept
16871 HMS forms as input.
16874 The @kbd{M-5 t P} [@code{minute}] function extracts the minute
16875 from a date form as an integer in the range 0 to 59.
16878 The @kbd{M-6 t P} [@code{second}] function extracts the second
16879 from a date form. If the current precision is 12 or less,
16880 the result is an integer in the range 0 to 59. For higher
16881 precisions, the result may instead be a floating-point number.
16884 The @kbd{M-7 t P} [@code{weekday}] function extracts the weekday
16885 number from a date form as an integer in the range 0 (Sunday)
16889 The @kbd{M-8 t P} [@code{yearday}] function extracts the day-of-year
16890 number from a date form as an integer in the range 1 (January 1)
16891 to 366 (December 31 of a leap year).
16894 The @kbd{M-9 t P} [@code{time}] function extracts the time portion
16895 of a date form as an HMS form. This returns @samp{0@@ 0' 0"}
16896 for a pure date form.
16899 @pindex calc-new-month
16901 The @kbd{t M} (@code{calc-new-month}) [@code{newmonth}] command
16902 computes a new date form that represents the first day of the month
16903 specified by the input date. The result is always a pure date
16904 form; only the year and month numbers of the input are retained.
16905 With a numeric prefix argument @var{n} in the range from 1 to 31,
16906 @kbd{t M} computes the @var{n}th day of the month. (If @var{n}
16907 is greater than the actual number of days in the month, or if
16908 @var{n} is zero, the last day of the month is used.)
16911 @pindex calc-new-year
16913 The @kbd{t Y} (@code{calc-new-year}) [@code{newyear}] command
16914 computes a new pure date form that represents the first day of
16915 the year specified by the input. The month, day, and time
16916 of the input date form are lost. With a numeric prefix argument
16917 @var{n} in the range from 1 to 366, @kbd{t Y} computes the
16918 @var{n}th day of the year (366 is treated as 365 in non-leap
16919 years). A prefix argument of 0 computes the last day of the
16920 year (December 31). A negative prefix argument from @mathit{-1} to
16921 @mathit{-12} computes the first day of the @var{n}th month of the year.
16924 @pindex calc-new-week
16926 The @kbd{t W} (@code{calc-new-week}) [@code{newweek}] command
16927 computes a new pure date form that represents the Sunday on or before
16928 the input date. With a numeric prefix argument, it can be made to
16929 use any day of the week as the starting day; the argument must be in
16930 the range from 0 (Sunday) to 6 (Saturday). This function always
16931 subtracts between 0 and 6 days from the input date.
16933 Here's an example use of @code{newweek}: Find the date of the next
16934 Wednesday after a given date. Using @kbd{M-3 t W} or @samp{newweek(d, 3)}
16935 will give you the @emph{preceding} Wednesday, so @samp{newweek(d+7, 3)}
16936 will give you the following Wednesday. A further look at the definition
16937 of @code{newweek} shows that if the input date is itself a Wednesday,
16938 this formula will return the Wednesday one week in the future. An
16939 exercise for the reader is to modify this formula to yield the same day
16940 if the input is already a Wednesday. Another interesting exercise is
16941 to preserve the time-of-day portion of the input (@code{newweek} resets
16942 the time to midnight; hint:@: how can @code{newweek} be defined in terms
16943 of the @code{weekday} function?).
16949 The @samp{pwday(@var{date})} function (not on any key) computes the
16950 day-of-month number of the Sunday on or before @var{date}. With
16951 two arguments, @samp{pwday(@var{date}, @var{day})} computes the day
16952 number of the Sunday on or before day number @var{day} of the month
16953 specified by @var{date}. The @var{day} must be in the range from
16954 7 to 31; if the day number is greater than the actual number of days
16955 in the month, the true number of days is used instead. Thus
16956 @samp{pwday(@var{date}, 7)} finds the first Sunday of the month, and
16957 @samp{pwday(@var{date}, 31)} finds the last Sunday of the month.
16958 With a third @var{weekday} argument, @code{pwday} can be made to look
16959 for any day of the week instead of Sunday.
16962 @pindex calc-inc-month
16964 The @kbd{t I} (@code{calc-inc-month}) [@code{incmonth}] command
16965 increases a date form by one month, or by an arbitrary number of
16966 months specified by a numeric prefix argument. The time portion,
16967 if any, of the date form stays the same. The day also stays the
16968 same, except that if the new month has fewer days the day
16969 number may be reduced to lie in the valid range. For example,
16970 @samp{incmonth(<Jan 31, 1991>)} produces @samp{<Feb 28, 1991>}.
16971 Because of this, @kbd{t I t I} and @kbd{M-2 t I} do not always give
16972 the same results (@samp{<Mar 28, 1991>} versus @samp{<Mar 31, 1991>}
16979 The @samp{incyear(@var{date}, @var{step})} function increases
16980 a date form by the specified number of years, which may be
16981 any positive or negative integer. Note that @samp{incyear(d, n)}
16982 is equivalent to @w{@samp{incmonth(d, 12*n)}}, but these do not have
16983 simple equivalents in terms of day arithmetic because
16984 months and years have varying lengths. If the @var{step}
16985 argument is omitted, 1 year is assumed. There is no keyboard
16986 command for this function; use @kbd{C-u 12 t I} instead.
16988 There is no @code{newday} function at all because @kbd{F} [@code{floor}]
16989 serves this purpose. Similarly, instead of @code{incday} and
16990 @code{incweek} simply use @expr{d + n} or @expr{d + 7 n}.
16992 @xref{Basic Arithmetic}, for the @kbd{f ]} [@code{incr}] command
16993 which can adjust a date/time form by a certain number of seconds.
16995 @node Business Days, Time Zones, Date Functions, Date Arithmetic
16996 @subsection Business Days
16999 Often time is measured in ``business days'' or ``working days,''
17000 where weekends and holidays are skipped. Calc's normal date
17001 arithmetic functions use calendar days, so that subtracting two
17002 consecutive Mondays will yield a difference of 7 days. By contrast,
17003 subtracting two consecutive Mondays would yield 5 business days
17004 (assuming two-day weekends and the absence of holidays).
17010 @pindex calc-business-days-plus
17011 @pindex calc-business-days-minus
17012 The @kbd{t +} (@code{calc-business-days-plus}) [@code{badd}]
17013 and @kbd{t -} (@code{calc-business-days-minus}) [@code{bsub}]
17014 commands perform arithmetic using business days. For @kbd{t +},
17015 one argument must be a date form and the other must be a real
17016 number (positive or negative). If the number is not an integer,
17017 then a certain amount of time is added as well as a number of
17018 days; for example, adding 0.5 business days to a time in Friday
17019 evening will produce a time in Monday morning. It is also
17020 possible to add an HMS form; adding @samp{12@@ 0' 0"} also adds
17021 half a business day. For @kbd{t -}, the arguments are either a
17022 date form and a number or HMS form, or two date forms, in which
17023 case the result is the number of business days between the two
17026 @cindex @code{Holidays} variable
17028 By default, Calc considers any day that is not a Saturday or
17029 Sunday to be a business day. You can define any number of
17030 additional holidays by editing the variable @code{Holidays}.
17031 (There is an @w{@kbd{s H}} convenience command for editing this
17032 variable.) Initially, @code{Holidays} contains the vector
17033 @samp{[sat, sun]}. Entries in the @code{Holidays} vector may
17034 be any of the following kinds of objects:
17038 Date forms (pure dates, not date/time forms). These specify
17039 particular days which are to be treated as holidays.
17042 Intervals of date forms. These specify a range of days, all of
17043 which are holidays (e.g., Christmas week). @xref{Interval Forms}.
17046 Nested vectors of date forms. Each date form in the vector is
17047 considered to be a holiday.
17050 Any Calc formula which evaluates to one of the above three things.
17051 If the formula involves the variable @expr{y}, it stands for a
17052 yearly repeating holiday; @expr{y} will take on various year
17053 numbers like 1992. For example, @samp{date(y, 12, 25)} specifies
17054 Christmas day, and @samp{newweek(date(y, 11, 7), 4) + 21} specifies
17055 Thanksgiving (which is held on the fourth Thursday of November).
17056 If the formula involves the variable @expr{m}, that variable
17057 takes on month numbers from 1 to 12: @samp{date(y, m, 15)} is
17058 a holiday that takes place on the 15th of every month.
17061 A weekday name, such as @code{sat} or @code{sun}. This is really
17062 a variable whose name is a three-letter, lower-case day name.
17065 An interval of year numbers (integers). This specifies the span of
17066 years over which this holiday list is to be considered valid. Any
17067 business-day arithmetic that goes outside this range will result
17068 in an error message. Use this if you are including an explicit
17069 list of holidays, rather than a formula to generate them, and you
17070 want to make sure you don't accidentally go beyond the last point
17071 where the holidays you entered are complete. If there is no
17072 limiting interval in the @code{Holidays} vector, the default
17073 @samp{[1 .. 2737]} is used. (This is the absolute range of years
17074 for which Calc's business-day algorithms will operate.)
17077 An interval of HMS forms. This specifies the span of hours that
17078 are to be considered one business day. For example, if this
17079 range is @samp{[9@@ 0' 0" .. 17@@ 0' 0"]} (i.e., 9am to 5pm), then
17080 the business day is only eight hours long, so that @kbd{1.5 t +}
17081 on @samp{<4:00pm Fri Dec 13, 1991>} will add one business day and
17082 four business hours to produce @samp{<12:00pm Tue Dec 17, 1991>}.
17083 Likewise, @kbd{t -} will now express differences in time as
17084 fractions of an eight-hour day. Times before 9am will be treated
17085 as 9am by business date arithmetic, and times at or after 5pm will
17086 be treated as 4:59:59pm. If there is no HMS interval in @code{Holidays},
17087 the full 24-hour day @samp{[0@ 0' 0" .. 24@ 0' 0"]} is assumed.
17088 (Regardless of the type of bounds you specify, the interval is
17089 treated as inclusive on the low end and exclusive on the high end,
17090 so that the work day goes from 9am up to, but not including, 5pm.)
17093 If the @code{Holidays} vector is empty, then @kbd{t +} and
17094 @kbd{t -} will act just like @kbd{+} and @kbd{-} because there will
17095 then be no difference between business days and calendar days.
17097 Calc expands the intervals and formulas you give into a complete
17098 list of holidays for internal use. This is done mainly to make
17099 sure it can detect multiple holidays. (For example,
17100 @samp{<Jan 1, 1989>} is both New Year's Day and a Sunday, but
17101 Calc's algorithms take care to count it only once when figuring
17102 the number of holidays between two dates.)
17104 Since the complete list of holidays for all the years from 1 to
17105 2737 would be huge, Calc actually computes only the part of the
17106 list between the smallest and largest years that have been involved
17107 in business-day calculations so far. Normally, you won't have to
17108 worry about this. Keep in mind, however, that if you do one
17109 calculation for 1992, and another for 1792, even if both involve
17110 only a small range of years, Calc will still work out all the
17111 holidays that fall in that 200-year span.
17113 If you add a (positive) number of days to a date form that falls on a
17114 weekend or holiday, the date form is treated as if it were the most
17115 recent business day. (Thus adding one business day to a Friday,
17116 Saturday, or Sunday will all yield the following Monday.) If you
17117 subtract a number of days from a weekend or holiday, the date is
17118 effectively on the following business day. (So subtracting one business
17119 day from Saturday, Sunday, or Monday yields the preceding Friday.) The
17120 difference between two dates one or both of which fall on holidays
17121 equals the number of actual business days between them. These
17122 conventions are consistent in the sense that, if you add @var{n}
17123 business days to any date, the difference between the result and the
17124 original date will come out to @var{n} business days. (It can't be
17125 completely consistent though; a subtraction followed by an addition
17126 might come out a bit differently, since @kbd{t +} is incapable of
17127 producing a date that falls on a weekend or holiday.)
17133 There is a @code{holiday} function, not on any keys, that takes
17134 any date form and returns 1 if that date falls on a weekend or
17135 holiday, as defined in @code{Holidays}, or 0 if the date is a
17138 @node Time Zones, , Business Days, Date Arithmetic
17139 @subsection Time Zones
17143 @cindex Daylight saving time
17144 Time zones and daylight saving time are a complicated business.
17145 The conversions to and from Julian and Unix-style dates automatically
17146 compute the correct time zone and daylight saving adjustment to use,
17147 provided they can figure out this information. This section describes
17148 Calc's time zone adjustment algorithm in detail, in case you want to
17149 do conversions in different time zones or in case Calc's algorithms
17150 can't determine the right correction to use.
17152 Adjustments for time zones and daylight saving time are done by
17153 @kbd{t U}, @kbd{t J}, @kbd{t N}, and @kbd{t C}, but not by any other
17154 commands. In particular, @samp{<may 1 1991> - <apr 1 1991>} evaluates
17155 to exactly 30 days even though there is a daylight-saving
17156 transition in between. This is also true for Julian pure dates:
17157 @samp{julian(<may 1 1991>) - julian(<apr 1 1991>)}. But Julian
17158 and Unix date/times will adjust for daylight saving time: using Calc's
17159 default daylight saving time rule (see the explanation below),
17160 @samp{julian(<12am may 1 1991>) - julian(<12am apr 1 1991>)}
17161 evaluates to @samp{29.95833} (that's 29 days and 23 hours)
17162 because one hour was lost when daylight saving commenced on
17165 In brief, the idiom @samp{julian(@var{date1}) - julian(@var{date2})}
17166 computes the actual number of 24-hour periods between two dates, whereas
17167 @samp{@var{date1} - @var{date2}} computes the number of calendar
17168 days between two dates without taking daylight saving into account.
17170 @pindex calc-time-zone
17175 The @code{calc-time-zone} [@code{tzone}] command converts the time
17176 zone specified by its numeric prefix argument into a number of
17177 seconds difference from Greenwich mean time (GMT). If the argument
17178 is a number, the result is simply that value multiplied by 3600.
17179 Typical arguments for North America are 5 (Eastern) or 8 (Pacific). If
17180 Daylight Saving time is in effect, one hour should be subtracted from
17181 the normal difference.
17183 If you give a prefix of plain @kbd{C-u}, @code{calc-time-zone} (like other
17184 date arithmetic commands that include a time zone argument) takes the
17185 zone argument from the top of the stack. (In the case of @kbd{t J}
17186 and @kbd{t U}, the normal argument is then taken from the second-to-top
17187 stack position.) This allows you to give a non-integer time zone
17188 adjustment. The time-zone argument can also be an HMS form, or
17189 it can be a variable which is a time zone name in upper- or lower-case.
17190 For example @samp{tzone(PST) = tzone(8)} and @samp{tzone(pdt) = tzone(7)}
17191 (for Pacific standard and daylight saving times, respectively).
17193 North American and European time zone names are defined as follows;
17194 note that for each time zone there is one name for standard time,
17195 another for daylight saving time, and a third for ``generalized'' time
17196 in which the daylight saving adjustment is computed from context.
17200 YST PST MST CST EST AST NST GMT WET MET MEZ
17201 9 8 7 6 5 4 3.5 0 -1 -2 -2
17203 YDT PDT MDT CDT EDT ADT NDT BST WETDST METDST MESZ
17204 8 7 6 5 4 3 2.5 -1 -2 -3 -3
17206 YGT PGT MGT CGT EGT AGT NGT BGT WEGT MEGT MEGZ
17207 9/8 8/7 7/6 6/5 5/4 4/3 3.5/2.5 0/-1 -1/-2 -2/-3 -2/-3
17211 @vindex math-tzone-names
17212 To define time zone names that do not appear in the above table,
17213 you must modify the Lisp variable @code{math-tzone-names}. This
17214 is a list of lists describing the different time zone names; its
17215 structure is best explained by an example. The three entries for
17216 Pacific Time look like this:
17220 ( ( "PST" 8 0 ) ; Name as an upper-case string, then standard
17221 ( "PDT" 8 -1 ) ; adjustment, then daylight saving adjustment.
17222 ( "PGT" 8 "PST" "PDT" ) ) ; Generalized time zone.
17226 @cindex @code{TimeZone} variable
17228 With no arguments, @code{calc-time-zone} or @samp{tzone()} will by
17229 default get the time zone and daylight saving information from the
17230 calendar (@pxref{Daylight Saving,Calendar/Diary,The Calendar and the Diary,
17231 emacs,The GNU Emacs Manual}). To use a different time zone, or if the
17232 calendar does not give the desired result, you can set the Calc variable
17233 @code{TimeZone} (which is by default @code{nil}) to an appropriate
17234 time zone name. (The easiest way to do this is to edit the
17235 @code{TimeZone} variable using Calc's @kbd{s T} command, then use the
17236 @kbd{s p} (@code{calc-permanent-variable}) command to save the value of
17237 @code{TimeZone} permanently.)
17238 If the time zone given by @code{TimeZone} is a generalized time zone,
17239 e.g., @code{EGT}, Calc examines the date being converted to tell whether
17240 to use standard or daylight saving time. But if the current time zone
17241 is explicit, e.g., @code{EST} or @code{EDT}, then that adjustment is
17242 used exactly and Calc's daylight saving algorithm is not consulted.
17243 The special time zone name @code{local}
17244 is equivalent to no argument; i.e., it uses the information obtained
17247 The @kbd{t J} and @code{t U} commands with no numeric prefix
17248 arguments do the same thing as @samp{tzone()}; namely, use the
17249 information from the calendar if @code{TimeZone} is @code{nil},
17250 otherwise use the time zone given by @code{TimeZone}.
17252 @vindex math-daylight-savings-hook
17253 @findex math-std-daylight-savings
17254 When Calc computes the daylight saving information itself (i.e., when
17255 the @code{TimeZone} variable is set), it will by default consider
17256 daylight saving time to begin at 2 a.m.@: on the second Sunday of March
17257 (for years from 2007 on) or on the last Sunday in April (for years
17258 before 2007), and to end at 2 a.m.@: on the first Sunday of
17259 November. (for years from 2007 on) or the last Sunday in October (for
17260 years before 2007). These are the rules that have been in effect in
17261 much of North America since 1966 and take into account the rule change
17262 that began in 2007. If you are in a country that uses different rules
17263 for computing daylight saving time, you have two choices: Write your own
17264 daylight saving hook, or control time zones explicitly by setting the
17265 @code{TimeZone} variable and/or always giving a time-zone argument for
17266 the conversion functions.
17268 The Lisp variable @code{math-daylight-savings-hook} holds the
17269 name of a function that is used to compute the daylight saving
17270 adjustment for a given date. The default is
17271 @code{math-std-daylight-savings}, which computes an adjustment
17272 (either 0 or @mathit{-1}) using the North American rules given above.
17274 The daylight saving hook function is called with four arguments:
17275 The date, as a floating-point number in standard Calc format;
17276 a six-element list of the date decomposed into year, month, day,
17277 hour, minute, and second, respectively; a string which contains
17278 the generalized time zone name in upper-case, e.g., @code{"WEGT"};
17279 and a special adjustment to be applied to the hour value when
17280 converting into a generalized time zone (see below).
17282 @findex math-prev-weekday-in-month
17283 The Lisp function @code{math-prev-weekday-in-month} is useful for
17284 daylight saving computations. This is an internal version of
17285 the user-level @code{pwday} function described in the previous
17286 section. It takes four arguments: The floating-point date value,
17287 the corresponding six-element date list, the day-of-month number,
17288 and the weekday number (0-6).
17290 The default daylight saving hook ignores the time zone name, but a
17291 more sophisticated hook could use different algorithms for different
17292 time zones. It would also be possible to use different algorithms
17293 depending on the year number, but the default hook always uses the
17294 algorithm for 1987 and later. Here is a listing of the default
17295 daylight saving hook:
17298 (defun math-std-daylight-savings (date dt zone bump)
17299 (cond ((< (nth 1 dt) 4) 0)
17301 (let ((sunday (math-prev-weekday-in-month date dt 7 0)))
17302 (cond ((< (nth 2 dt) sunday) 0)
17303 ((= (nth 2 dt) sunday)
17304 (if (>= (nth 3 dt) (+ 3 bump)) -1 0))
17306 ((< (nth 1 dt) 10) -1)
17308 (let ((sunday (math-prev-weekday-in-month date dt 31 0)))
17309 (cond ((< (nth 2 dt) sunday) -1)
17310 ((= (nth 2 dt) sunday)
17311 (if (>= (nth 3 dt) (+ 2 bump)) 0 -1))
17318 The @code{bump} parameter is equal to zero when Calc is converting
17319 from a date form in a generalized time zone into a GMT date value.
17320 It is @mathit{-1} when Calc is converting in the other direction. The
17321 adjustments shown above ensure that the conversion behaves correctly
17322 and reasonably around the 2 a.m.@: transition in each direction.
17324 There is a ``missing'' hour between 2 a.m.@: and 3 a.m.@: at the
17325 beginning of daylight saving time; converting a date/time form that
17326 falls in this hour results in a time value for the following hour,
17327 from 3 a.m.@: to 4 a.m. At the end of daylight saving time, the
17328 hour from 1 a.m.@: to 2 a.m.@: repeats itself; converting a date/time
17329 form that falls in this hour results in a time value for the first
17330 manifestation of that time (@emph{not} the one that occurs one hour
17333 If @code{math-daylight-savings-hook} is @code{nil}, then the
17334 daylight saving adjustment is always taken to be zero.
17336 In algebraic formulas, @samp{tzone(@var{zone}, @var{date})}
17337 computes the time zone adjustment for a given zone name at a
17338 given date. The @var{date} is ignored unless @var{zone} is a
17339 generalized time zone. If @var{date} is a date form, the
17340 daylight saving computation is applied to it as it appears.
17341 If @var{date} is a numeric date value, it is adjusted for the
17342 daylight-saving version of @var{zone} before being given to
17343 the daylight saving hook. This odd-sounding rule ensures
17344 that the daylight-saving computation is always done in
17345 local time, not in the GMT time that a numeric @var{date}
17346 is typically represented in.
17352 The @samp{dsadj(@var{date}, @var{zone})} function computes the
17353 daylight saving adjustment that is appropriate for @var{date} in
17354 time zone @var{zone}. If @var{zone} is explicitly in or not in
17355 daylight saving time (e.g., @code{PDT} or @code{PST}) the
17356 @var{date} is ignored. If @var{zone} is a generalized time zone,
17357 the algorithms described above are used. If @var{zone} is omitted,
17358 the computation is done for the current time zone.
17360 @node Financial Functions, Binary Functions, Date Arithmetic, Arithmetic
17361 @section Financial Functions
17364 Calc's financial or business functions use the @kbd{b} prefix
17365 key followed by a shifted letter. (The @kbd{b} prefix followed by
17366 a lower-case letter is used for operations on binary numbers.)
17368 Note that the rate and the number of intervals given to these
17369 functions must be on the same time scale, e.g., both months or
17370 both years. Mixing an annual interest rate with a time expressed
17371 in months will give you very wrong answers!
17373 It is wise to compute these functions to a higher precision than
17374 you really need, just to make sure your answer is correct to the
17375 last penny; also, you may wish to check the definitions at the end
17376 of this section to make sure the functions have the meaning you expect.
17382 * Related Financial Functions::
17383 * Depreciation Functions::
17384 * Definitions of Financial Functions::
17387 @node Percentages, Future Value, Financial Functions, Financial Functions
17388 @subsection Percentages
17391 @pindex calc-percent
17394 The @kbd{M-%} (@code{calc-percent}) command takes a percentage value,
17395 say 5.4, and converts it to an equivalent actual number. For example,
17396 @kbd{5.4 M-%} enters 0.054 on the stack. (That's the @key{META} or
17397 @key{ESC} key combined with @kbd{%}.)
17399 Actually, @kbd{M-%} creates a formula of the form @samp{5.4%}.
17400 You can enter @samp{5.4%} yourself during algebraic entry. The
17401 @samp{%} operator simply means, ``the preceding value divided by
17402 100.'' The @samp{%} operator has very high precedence, so that
17403 @samp{1+8%} is interpreted as @samp{1+(8%)}, not as @samp{(1+8)%}.
17404 (The @samp{%} operator is just a postfix notation for the
17405 @code{percent} function, just like @samp{20!} is the notation for
17406 @samp{fact(20)}, or twenty-factorial.)
17408 The formula @samp{5.4%} would normally evaluate immediately to
17409 0.054, but the @kbd{M-%} command suppresses evaluation as it puts
17410 the formula onto the stack. However, the next Calc command that
17411 uses the formula @samp{5.4%} will evaluate it as its first step.
17412 The net effect is that you get to look at @samp{5.4%} on the stack,
17413 but Calc commands see it as @samp{0.054}, which is what they expect.
17415 In particular, @samp{5.4%} and @samp{0.054} are suitable values
17416 for the @var{rate} arguments of the various financial functions,
17417 but the number @samp{5.4} is probably @emph{not} suitable---it
17418 represents a rate of 540 percent!
17420 The key sequence @kbd{M-% *} effectively means ``percent-of.''
17421 For example, @kbd{68 @key{RET} 25 M-% *} computes 17, which is 25% of
17422 68 (and also 68% of 25, which comes out to the same thing).
17425 @pindex calc-convert-percent
17426 The @kbd{c %} (@code{calc-convert-percent}) command converts the
17427 value on the top of the stack from numeric to percentage form.
17428 For example, if 0.08 is on the stack, @kbd{c %} converts it to
17429 @samp{8%}. The quantity is the same, it's just represented
17430 differently. (Contrast this with @kbd{M-%}, which would convert
17431 this number to @samp{0.08%}.) The @kbd{=} key is a convenient way
17432 to convert a formula like @samp{8%} back to numeric form, 0.08.
17434 To compute what percentage one quantity is of another quantity,
17435 use @kbd{/ c %}. For example, @w{@kbd{17 @key{RET} 68 / c %}} displays
17439 @pindex calc-percent-change
17441 The @kbd{b %} (@code{calc-percent-change}) [@code{relch}] command
17442 calculates the percentage change from one number to another.
17443 For example, @kbd{40 @key{RET} 50 b %} produces the answer @samp{25%},
17444 since 50 is 25% larger than 40. A negative result represents a
17445 decrease: @kbd{50 @key{RET} 40 b %} produces @samp{-20%}, since 40 is
17446 20% smaller than 50. (The answers are different in magnitude
17447 because, in the first case, we're increasing by 25% of 40, but
17448 in the second case, we're decreasing by 20% of 50.) The effect
17449 of @kbd{40 @key{RET} 50 b %} is to compute @expr{(50-40)/40}, converting
17450 the answer to percentage form as if by @kbd{c %}.
17452 @node Future Value, Present Value, Percentages, Financial Functions
17453 @subsection Future Value
17457 @pindex calc-fin-fv
17459 The @kbd{b F} (@code{calc-fin-fv}) [@code{fv}] command computes
17460 the future value of an investment. It takes three arguments
17461 from the stack: @samp{fv(@var{rate}, @var{n}, @var{payment})}.
17462 If you give payments of @var{payment} every year for @var{n}
17463 years, and the money you have paid earns interest at @var{rate} per
17464 year, then this function tells you what your investment would be
17465 worth at the end of the period. (The actual interval doesn't
17466 have to be years, as long as @var{n} and @var{rate} are expressed
17467 in terms of the same intervals.) This function assumes payments
17468 occur at the @emph{end} of each interval.
17472 The @kbd{I b F} [@code{fvb}] command does the same computation,
17473 but assuming your payments are at the beginning of each interval.
17474 Suppose you plan to deposit $1000 per year in a savings account
17475 earning 5.4% interest, starting right now. How much will be
17476 in the account after five years? @code{fvb(5.4%, 5, 1000) = 5870.73}.
17477 Thus you will have earned $870 worth of interest over the years.
17478 Using the stack, this calculation would have been
17479 @kbd{5.4 M-% 5 @key{RET} 1000 I b F}. Note that the rate is expressed
17480 as a number between 0 and 1, @emph{not} as a percentage.
17484 The @kbd{H b F} [@code{fvl}] command computes the future value
17485 of an initial lump sum investment. Suppose you could deposit
17486 those five thousand dollars in the bank right now; how much would
17487 they be worth in five years? @code{fvl(5.4%, 5, 5000) = 6503.89}.
17489 The algebraic functions @code{fv} and @code{fvb} accept an optional
17490 fourth argument, which is used as an initial lump sum in the sense
17491 of @code{fvl}. In other words, @code{fv(@var{rate}, @var{n},
17492 @var{payment}, @var{initial}) = fv(@var{rate}, @var{n}, @var{payment})
17493 + fvl(@var{rate}, @var{n}, @var{initial})}.
17495 To illustrate the relationships between these functions, we could
17496 do the @code{fvb} calculation ``by hand'' using @code{fvl}. The
17497 final balance will be the sum of the contributions of our five
17498 deposits at various times. The first deposit earns interest for
17499 five years: @code{fvl(5.4%, 5, 1000) = 1300.78}. The second
17500 deposit only earns interest for four years: @code{fvl(5.4%, 4, 1000) =
17501 1234.13}. And so on down to the last deposit, which earns one
17502 year's interest: @code{fvl(5.4%, 1, 1000) = 1054.00}. The sum of
17503 these five values is, sure enough, $5870.73, just as was computed
17504 by @code{fvb} directly.
17506 What does @code{fv(5.4%, 5, 1000) = 5569.96} mean? The payments
17507 are now at the ends of the periods. The end of one year is the same
17508 as the beginning of the next, so what this really means is that we've
17509 lost the payment at year zero (which contributed $1300.78), but we're
17510 now counting the payment at year five (which, since it didn't have
17511 a chance to earn interest, counts as $1000). Indeed, @expr{5569.96 =
17512 5870.73 - 1300.78 + 1000} (give or take a bit of roundoff error).
17514 @node Present Value, Related Financial Functions, Future Value, Financial Functions
17515 @subsection Present Value
17519 @pindex calc-fin-pv
17521 The @kbd{b P} (@code{calc-fin-pv}) [@code{pv}] command computes
17522 the present value of an investment. Like @code{fv}, it takes
17523 three arguments: @code{pv(@var{rate}, @var{n}, @var{payment})}.
17524 It computes the present value of a series of regular payments.
17525 Suppose you have the chance to make an investment that will
17526 pay $2000 per year over the next four years; as you receive
17527 these payments you can put them in the bank at 9% interest.
17528 You want to know whether it is better to make the investment, or
17529 to keep the money in the bank where it earns 9% interest right
17530 from the start. The calculation @code{pv(9%, 4, 2000)} gives the
17531 result 6479.44. If your initial investment must be less than this,
17532 say, $6000, then the investment is worthwhile. But if you had to
17533 put up $7000, then it would be better just to leave it in the bank.
17535 Here is the interpretation of the result of @code{pv}: You are
17536 trying to compare the return from the investment you are
17537 considering, which is @code{fv(9%, 4, 2000) = 9146.26}, with
17538 the return from leaving the money in the bank, which is
17539 @code{fvl(9%, 4, @var{x})} where @var{x} is the amount of money
17540 you would have to put up in advance. The @code{pv} function
17541 finds the break-even point, @expr{x = 6479.44}, at which
17542 @code{fvl(9%, 4, 6479.44)} is also equal to 9146.26. This is
17543 the largest amount you should be willing to invest.
17547 The @kbd{I b P} [@code{pvb}] command solves the same problem,
17548 but with payments occurring at the beginning of each interval.
17549 It has the same relationship to @code{fvb} as @code{pv} has
17550 to @code{fv}. For example @code{pvb(9%, 4, 2000) = 7062.59},
17551 a larger number than @code{pv} produced because we get to start
17552 earning interest on the return from our investment sooner.
17556 The @kbd{H b P} [@code{pvl}] command computes the present value of
17557 an investment that will pay off in one lump sum at the end of the
17558 period. For example, if we get our $8000 all at the end of the
17559 four years, @code{pvl(9%, 4, 8000) = 5667.40}. This is much
17560 less than @code{pv} reported, because we don't earn any interest
17561 on the return from this investment. Note that @code{pvl} and
17562 @code{fvl} are simple inverses: @code{fvl(9%, 4, 5667.40) = 8000}.
17564 You can give an optional fourth lump-sum argument to @code{pv}
17565 and @code{pvb}; this is handled in exactly the same way as the
17566 fourth argument for @code{fv} and @code{fvb}.
17569 @pindex calc-fin-npv
17571 The @kbd{b N} (@code{calc-fin-npv}) [@code{npv}] command computes
17572 the net present value of a series of irregular investments.
17573 The first argument is the interest rate. The second argument is
17574 a vector which represents the expected return from the investment
17575 at the end of each interval. For example, if the rate represents
17576 a yearly interest rate, then the vector elements are the return
17577 from the first year, second year, and so on.
17579 Thus, @code{npv(9%, [2000,2000,2000,2000]) = pv(9%, 4, 2000) = 6479.44}.
17580 Obviously this function is more interesting when the payments are
17583 The @code{npv} function can actually have two or more arguments.
17584 Multiple arguments are interpreted in the same way as for the
17585 vector statistical functions like @code{vsum}.
17586 @xref{Single-Variable Statistics}. Basically, if there are several
17587 payment arguments, each either a vector or a plain number, all these
17588 values are collected left-to-right into the complete list of payments.
17589 A numeric prefix argument on the @kbd{b N} command says how many
17590 payment values or vectors to take from the stack.
17594 The @kbd{I b N} [@code{npvb}] command computes the net present
17595 value where payments occur at the beginning of each interval
17596 rather than at the end.
17598 @node Related Financial Functions, Depreciation Functions, Present Value, Financial Functions
17599 @subsection Related Financial Functions
17602 The functions in this section are basically inverses of the
17603 present value functions with respect to the various arguments.
17606 @pindex calc-fin-pmt
17608 The @kbd{b M} (@code{calc-fin-pmt}) [@code{pmt}] command computes
17609 the amount of periodic payment necessary to amortize a loan.
17610 Thus @code{pmt(@var{rate}, @var{n}, @var{amount})} equals the
17611 value of @var{payment} such that @code{pv(@var{rate}, @var{n},
17612 @var{payment}) = @var{amount}}.
17616 The @kbd{I b M} [@code{pmtb}] command does the same computation
17617 but using @code{pvb} instead of @code{pv}. Like @code{pv} and
17618 @code{pvb}, these functions can also take a fourth argument which
17619 represents an initial lump-sum investment.
17622 The @kbd{H b M} key just invokes the @code{fvl} function, which is
17623 the inverse of @code{pvl}. There is no explicit @code{pmtl} function.
17626 @pindex calc-fin-nper
17628 The @kbd{b #} (@code{calc-fin-nper}) [@code{nper}] command computes
17629 the number of regular payments necessary to amortize a loan.
17630 Thus @code{nper(@var{rate}, @var{payment}, @var{amount})} equals
17631 the value of @var{n} such that @code{pv(@var{rate}, @var{n},
17632 @var{payment}) = @var{amount}}. If @var{payment} is too small
17633 ever to amortize a loan for @var{amount} at interest rate @var{rate},
17634 the @code{nper} function is left in symbolic form.
17638 The @kbd{I b #} [@code{nperb}] command does the same computation
17639 but using @code{pvb} instead of @code{pv}. You can give a fourth
17640 lump-sum argument to these functions, but the computation will be
17641 rather slow in the four-argument case.
17645 The @kbd{H b #} [@code{nperl}] command does the same computation
17646 using @code{pvl}. By exchanging @var{payment} and @var{amount} you
17647 can also get the solution for @code{fvl}. For example,
17648 @code{nperl(8%, 2000, 1000) = 9.006}, so if you place $1000 in a
17649 bank account earning 8%, it will take nine years to grow to $2000.
17652 @pindex calc-fin-rate
17654 The @kbd{b T} (@code{calc-fin-rate}) [@code{rate}] command computes
17655 the rate of return on an investment. This is also an inverse of @code{pv}:
17656 @code{rate(@var{n}, @var{payment}, @var{amount})} computes the value of
17657 @var{rate} such that @code{pv(@var{rate}, @var{n}, @var{payment}) =
17658 @var{amount}}. The result is expressed as a formula like @samp{6.3%}.
17664 The @kbd{I b T} [@code{rateb}] and @kbd{H b T} [@code{ratel}]
17665 commands solve the analogous equations with @code{pvb} or @code{pvl}
17666 in place of @code{pv}. Also, @code{rate} and @code{rateb} can
17667 accept an optional fourth argument just like @code{pv} and @code{pvb}.
17668 To redo the above example from a different perspective,
17669 @code{ratel(9, 2000, 1000) = 8.00597%}, which says you will need an
17670 interest rate of 8% in order to double your account in nine years.
17673 @pindex calc-fin-irr
17675 The @kbd{b I} (@code{calc-fin-irr}) [@code{irr}] command is the
17676 analogous function to @code{rate} but for net present value.
17677 Its argument is a vector of payments. Thus @code{irr(@var{payments})}
17678 computes the @var{rate} such that @code{npv(@var{rate}, @var{payments}) = 0};
17679 this rate is known as the @dfn{internal rate of return}.
17683 The @kbd{I b I} [@code{irrb}] command computes the internal rate of
17684 return assuming payments occur at the beginning of each period.
17686 @node Depreciation Functions, Definitions of Financial Functions, Related Financial Functions, Financial Functions
17687 @subsection Depreciation Functions
17690 The functions in this section calculate @dfn{depreciation}, which is
17691 the amount of value that a possession loses over time. These functions
17692 are characterized by three parameters: @var{cost}, the original cost
17693 of the asset; @var{salvage}, the value the asset will have at the end
17694 of its expected ``useful life''; and @var{life}, the number of years
17695 (or other periods) of the expected useful life.
17697 There are several methods for calculating depreciation that differ in
17698 the way they spread the depreciation over the lifetime of the asset.
17701 @pindex calc-fin-sln
17703 The @kbd{b S} (@code{calc-fin-sln}) [@code{sln}] command computes the
17704 ``straight-line'' depreciation. In this method, the asset depreciates
17705 by the same amount every year (or period). For example,
17706 @samp{sln(12000, 2000, 5)} returns 2000. The asset costs $12000
17707 initially and will be worth $2000 after five years; it loses $2000
17711 @pindex calc-fin-syd
17713 The @kbd{b Y} (@code{calc-fin-syd}) [@code{syd}] command computes the
17714 accelerated ``sum-of-years'-digits'' depreciation. Here the depreciation
17715 is higher during the early years of the asset's life. Since the
17716 depreciation is different each year, @kbd{b Y} takes a fourth @var{period}
17717 parameter which specifies which year is requested, from 1 to @var{life}.
17718 If @var{period} is outside this range, the @code{syd} function will
17722 @pindex calc-fin-ddb
17724 The @kbd{b D} (@code{calc-fin-ddb}) [@code{ddb}] command computes an
17725 accelerated depreciation using the double-declining balance method.
17726 It also takes a fourth @var{period} parameter.
17728 For symmetry, the @code{sln} function will accept a @var{period}
17729 parameter as well, although it will ignore its value except that the
17730 return value will as usual be zero if @var{period} is out of range.
17732 For example, pushing the vector @expr{[1,2,3,4,5]} (perhaps with @kbd{v x 5})
17733 and then mapping @kbd{V M ' [sln(12000,2000,5,$), syd(12000,2000,5,$),
17734 ddb(12000,2000,5,$)] @key{RET}} produces a matrix that allows us to compare
17735 the three depreciation methods:
17739 [ [ 2000, 3333, 4800 ]
17740 [ 2000, 2667, 2880 ]
17741 [ 2000, 2000, 1728 ]
17742 [ 2000, 1333, 592 ]
17748 (Values have been rounded to nearest integers in this figure.)
17749 We see that @code{sln} depreciates by the same amount each year,
17750 @kbd{syd} depreciates more at the beginning and less at the end,
17751 and @kbd{ddb} weights the depreciation even more toward the beginning.
17753 Summing columns with @kbd{V R : +} yields @expr{[10000, 10000, 10000]};
17754 the total depreciation in any method is (by definition) the
17755 difference between the cost and the salvage value.
17757 @node Definitions of Financial Functions, , Depreciation Functions, Financial Functions
17758 @subsection Definitions
17761 For your reference, here are the actual formulas used to compute
17762 Calc's financial functions.
17764 Calc will not evaluate a financial function unless the @var{rate} or
17765 @var{n} argument is known. However, @var{payment} or @var{amount} can
17766 be a variable. Calc expands these functions according to the
17767 formulas below for symbolic arguments only when you use the @kbd{a "}
17768 (@code{calc-expand-formula}) command, or when taking derivatives or
17769 integrals or solving equations involving the functions.
17772 These formulas are shown using the conventions of Big display
17773 mode (@kbd{d B}); for example, the formula for @code{fv} written
17774 linearly is @samp{pmt * ((1 + rate)^n) - 1) / rate}.
17779 fv(rate, n, pmt) = pmt * ---------------
17783 ((1 + rate) - 1) (1 + rate)
17784 fvb(rate, n, pmt) = pmt * ----------------------------
17788 fvl(rate, n, pmt) = pmt * (1 + rate)
17792 pv(rate, n, pmt) = pmt * ----------------
17796 (1 - (1 + rate) ) (1 + rate)
17797 pvb(rate, n, pmt) = pmt * -----------------------------
17801 pvl(rate, n, pmt) = pmt * (1 + rate)
17804 npv(rate, [a, b, c]) = a*(1 + rate) + b*(1 + rate) + c*(1 + rate)
17807 npvb(rate, [a, b, c]) = a + b*(1 + rate) + c*(1 + rate)
17810 (amt - x * (1 + rate) ) * rate
17811 pmt(rate, n, amt, x) = -------------------------------
17816 (amt - x * (1 + rate) ) * rate
17817 pmtb(rate, n, amt, x) = -------------------------------
17819 (1 - (1 + rate) ) (1 + rate)
17822 nper(rate, pmt, amt) = - log(1 - ------------, 1 + rate)
17826 nperb(rate, pmt, amt) = - log(1 - ---------------, 1 + rate)
17830 nperl(rate, pmt, amt) = - log(---, 1 + rate)
17835 ratel(n, pmt, amt) = ------ - 1
17840 sln(cost, salv, life) = -----------
17843 (cost - salv) * (life - per + 1)
17844 syd(cost, salv, life, per) = --------------------------------
17845 life * (life + 1) / 2
17848 ddb(cost, salv, life, per) = --------, book = cost - depreciation so far
17854 $$ \code{fv}(r, n, p) = p { (1 + r)^n - 1 \over r } $$
17855 $$ \code{fvb}(r, n, p) = p { ((1 + r)^n - 1) (1 + r) \over r } $$
17856 $$ \code{fvl}(r, n, p) = p (1 + r)^n $$
17857 $$ \code{pv}(r, n, p) = p { 1 - (1 + r)^{-n} \over r } $$
17858 $$ \code{pvb}(r, n, p) = p { (1 - (1 + r)^{-n}) (1 + r) \over r } $$
17859 $$ \code{pvl}(r, n, p) = p (1 + r)^{-n} $$
17860 $$ \code{npv}(r, [a,b,c]) = a (1 + r)^{-1} + b (1 + r)^{-2} + c (1 + r)^{-3} $$
17861 $$ \code{npvb}(r, [a,b,c]) = a + b (1 + r)^{-1} + c (1 + r)^{-2} $$
17862 $$ \code{pmt}(r, n, a, x) = { (a - x (1 + r)^{-n}) r \over 1 - (1 + r)^{-n} }$$
17863 $$ \code{pmtb}(r, n, a, x) = { (a - x (1 + r)^{-n}) r \over
17864 (1 - (1 + r)^{-n}) (1 + r) } $$
17865 $$ \code{nper}(r, p, a) = -\code{log}(1 - { a r \over p }, 1 + r) $$
17866 $$ \code{nperb}(r, p, a) = -\code{log}(1 - { a r \over p (1 + r) }, 1 + r) $$
17867 $$ \code{nperl}(r, p, a) = -\code{log}({a \over p}, 1 + r) $$
17868 $$ \code{ratel}(n, p, a) = { p^{1/n} \over a^{1/n} } - 1 $$
17869 $$ \code{sln}(c, s, l) = { c - s \over l } $$
17870 $$ \code{syd}(c, s, l, p) = { (c - s) (l - p + 1) \over l (l+1) / 2 } $$
17871 $$ \code{ddb}(c, s, l, p) = { 2 (c - \hbox{depreciation so far}) \over l } $$
17875 In @code{pmt} and @code{pmtb}, @expr{x=0} if omitted.
17877 These functions accept any numeric objects, including error forms,
17878 intervals, and even (though not very usefully) complex numbers. The
17879 above formulas specify exactly the behavior of these functions with
17880 all sorts of inputs.
17882 Note that if the first argument to the @code{log} in @code{nper} is
17883 negative, @code{nper} leaves itself in symbolic form rather than
17884 returning a (financially meaningless) complex number.
17886 @samp{rate(num, pmt, amt)} solves the equation
17887 @samp{pv(rate, num, pmt) = amt} for @samp{rate} using @kbd{H a R}
17888 (@code{calc-find-root}), with the interval @samp{[.01% .. 100%]}
17889 for an initial guess. The @code{rateb} function is the same except
17890 that it uses @code{pvb}. Note that @code{ratel} can be solved
17891 directly; its formula is shown in the above list.
17893 Similarly, @samp{irr(pmts)} solves the equation @samp{npv(rate, pmts) = 0}
17896 If you give a fourth argument to @code{nper} or @code{nperb}, Calc
17897 will also use @kbd{H a R} to solve the equation using an initial
17898 guess interval of @samp{[0 .. 100]}.
17900 A fourth argument to @code{fv} simply sums the two components
17901 calculated from the above formulas for @code{fv} and @code{fvl}.
17902 The same is true of @code{fvb}, @code{pv}, and @code{pvb}.
17904 The @kbd{ddb} function is computed iteratively; the ``book'' value
17905 starts out equal to @var{cost}, and decreases according to the above
17906 formula for the specified number of periods. If the book value
17907 would decrease below @var{salvage}, it only decreases to @var{salvage}
17908 and the depreciation is zero for all subsequent periods. The @code{ddb}
17909 function returns the amount the book value decreased in the specified
17912 @node Binary Functions, , Financial Functions, Arithmetic
17913 @section Binary Number Functions
17916 The commands in this chapter all use two-letter sequences beginning with
17917 the @kbd{b} prefix.
17919 @cindex Binary numbers
17920 The ``binary'' operations actually work regardless of the currently
17921 displayed radix, although their results make the most sense in a radix
17922 like 2, 8, or 16 (as obtained by the @kbd{d 2}, @kbd{d 8}, or @w{@kbd{d 6}}
17923 commands, respectively). You may also wish to enable display of leading
17924 zeros with @kbd{d z}. @xref{Radix Modes}.
17926 @cindex Word size for binary operations
17927 The Calculator maintains a current @dfn{word size} @expr{w}, an
17928 arbitrary positive or negative integer. For a positive word size, all
17929 of the binary operations described here operate modulo @expr{2^w}. In
17930 particular, negative arguments are converted to positive integers modulo
17931 @expr{2^w} by all binary functions.
17933 If the word size is negative, binary operations produce 2's complement
17935 @texline @math{-2^{-w-1}}
17936 @infoline @expr{-(2^(-w-1))}
17938 @texline @math{2^{-w-1}-1}
17939 @infoline @expr{2^(-w-1)-1}
17940 inclusive. Either mode accepts inputs in any range; the sign of
17941 @expr{w} affects only the results produced.
17946 The @kbd{b c} (@code{calc-clip})
17947 [@code{clip}] command can be used to clip a number by reducing it modulo
17948 @expr{2^w}. The commands described in this chapter automatically clip
17949 their results to the current word size. Note that other operations like
17950 addition do not use the current word size, since integer addition
17951 generally is not ``binary.'' (However, @pxref{Simplification Modes},
17952 @code{calc-bin-simplify-mode}.) For example, with a word size of 8
17953 bits @kbd{b c} converts a number to the range 0 to 255; with a word
17954 size of @mathit{-8} @kbd{b c} converts to the range @mathit{-128} to 127.
17957 @pindex calc-word-size
17958 The default word size is 32 bits. All operations except the shifts and
17959 rotates allow you to specify a different word size for that one
17960 operation by giving a numeric prefix argument: @kbd{C-u 8 b c} clips the
17961 top of stack to the range 0 to 255 regardless of the current word size.
17962 To set the word size permanently, use @kbd{b w} (@code{calc-word-size}).
17963 This command displays a prompt with the current word size; press @key{RET}
17964 immediately to keep this word size, or type a new word size at the prompt.
17966 When the binary operations are written in symbolic form, they take an
17967 optional second (or third) word-size parameter. When a formula like
17968 @samp{and(a,b)} is finally evaluated, the word size current at that time
17969 will be used, but when @samp{and(a,b,-8)} is evaluated, a word size of
17970 @mathit{-8} will always be used. A symbolic binary function will be left
17971 in symbolic form unless the all of its argument(s) are integers or
17972 integer-valued floats.
17974 If either or both arguments are modulo forms for which @expr{M} is a
17975 power of two, that power of two is taken as the word size unless a
17976 numeric prefix argument overrides it. The current word size is never
17977 consulted when modulo-power-of-two forms are involved.
17982 The @kbd{b a} (@code{calc-and}) [@code{and}] command computes the bitwise
17983 AND of the two numbers on the top of the stack. In other words, for each
17984 of the @expr{w} binary digits of the two numbers (pairwise), the corresponding
17985 bit of the result is 1 if and only if both input bits are 1:
17986 @samp{and(2#1100, 2#1010) = 2#1000}.
17991 The @kbd{b o} (@code{calc-or}) [@code{or}] command computes the bitwise
17992 inclusive OR of two numbers. A bit is 1 if either of the input bits, or
17993 both, are 1: @samp{or(2#1100, 2#1010) = 2#1110}.
17998 The @kbd{b x} (@code{calc-xor}) [@code{xor}] command computes the bitwise
17999 exclusive OR of two numbers. A bit is 1 if exactly one of the input bits
18000 is 1: @samp{xor(2#1100, 2#1010) = 2#0110}.
18005 The @kbd{b d} (@code{calc-diff}) [@code{diff}] command computes the bitwise
18006 difference of two numbers; this is defined by @samp{diff(a,b) = and(a,not(b))},
18007 so that @samp{diff(2#1100, 2#1010) = 2#0100}.
18012 The @kbd{b n} (@code{calc-not}) [@code{not}] command computes the bitwise
18013 NOT of a number. A bit is 1 if the input bit is 0 and vice-versa.
18016 @pindex calc-lshift-binary
18018 The @kbd{b l} (@code{calc-lshift-binary}) [@code{lsh}] command shifts a
18019 number left by one bit, or by the number of bits specified in the numeric
18020 prefix argument. A negative prefix argument performs a logical right shift,
18021 in which zeros are shifted in on the left. In symbolic form, @samp{lsh(a)}
18022 is short for @samp{lsh(a,1)}, which in turn is short for @samp{lsh(a,n,w)}.
18023 Bits shifted ``off the end,'' according to the current word size, are lost.
18039 The @kbd{H b l} command also does a left shift, but it takes two arguments
18040 from the stack (the value to shift, and, at top-of-stack, the number of
18041 bits to shift). This version interprets the prefix argument just like
18042 the regular binary operations, i.e., as a word size. The Hyperbolic flag
18043 has a similar effect on the rest of the binary shift and rotate commands.
18046 @pindex calc-rshift-binary
18048 The @kbd{b r} (@code{calc-rshift-binary}) [@code{rsh}] command shifts a
18049 number right by one bit, or by the number of bits specified in the numeric
18050 prefix argument: @samp{rsh(a,n) = lsh(a,-n)}.
18053 @pindex calc-lshift-arith
18055 The @kbd{b L} (@code{calc-lshift-arith}) [@code{ash}] command shifts a
18056 number left. It is analogous to @code{lsh}, except that if the shift
18057 is rightward (the prefix argument is negative), an arithmetic shift
18058 is performed as described below.
18061 @pindex calc-rshift-arith
18063 The @kbd{b R} (@code{calc-rshift-arith}) [@code{rash}] command performs
18064 an ``arithmetic'' shift to the right, in which the leftmost bit (according
18065 to the current word size) is duplicated rather than shifting in zeros.
18066 This corresponds to dividing by a power of two where the input is interpreted
18067 as a signed, twos-complement number. (The distinction between the @samp{rsh}
18068 and @samp{rash} operations is totally independent from whether the word
18069 size is positive or negative.) With a negative prefix argument, this
18070 performs a standard left shift.
18073 @pindex calc-rotate-binary
18075 The @kbd{b t} (@code{calc-rotate-binary}) [@code{rot}] command rotates a
18076 number one bit to the left. The leftmost bit (according to the current
18077 word size) is dropped off the left and shifted in on the right. With a
18078 numeric prefix argument, the number is rotated that many bits to the left
18081 @xref{Set Operations}, for the @kbd{b p} and @kbd{b u} commands that
18082 pack and unpack binary integers into sets. (For example, @kbd{b u}
18083 unpacks the number @samp{2#11001} to the set of bit-numbers
18084 @samp{[0, 3, 4]}.) Type @kbd{b u V #} to count the number of ``1''
18085 bits in a binary integer.
18087 Another interesting use of the set representation of binary integers
18088 is to reverse the bits in, say, a 32-bit integer. Type @kbd{b u} to
18089 unpack; type @kbd{31 @key{TAB} -} to replace each bit-number in the set
18090 with 31 minus that bit-number; type @kbd{b p} to pack the set back
18091 into a binary integer.
18093 @node Scientific Functions, Matrix Functions, Arithmetic, Top
18094 @chapter Scientific Functions
18097 The functions described here perform trigonometric and other transcendental
18098 calculations. They generally produce floating-point answers correct to the
18099 full current precision. The @kbd{H} (Hyperbolic) and @kbd{I} (Inverse)
18100 flag keys must be used to get some of these functions from the keyboard.
18104 @cindex @code{pi} variable
18107 @cindex @code{e} variable
18110 @cindex @code{gamma} variable
18112 @cindex Gamma constant, Euler's
18113 @cindex Euler's gamma constant
18115 @cindex @code{phi} variable
18116 @cindex Phi, golden ratio
18117 @cindex Golden ratio
18118 One miscellaneous command is shift-@kbd{P} (@code{calc-pi}), which pushes
18119 the value of @cpi{} (at the current precision) onto the stack. With the
18120 Hyperbolic flag, it pushes the value @expr{e}, the base of natural logarithms.
18121 With the Inverse flag, it pushes Euler's constant
18122 @texline @math{\gamma}
18123 @infoline @expr{gamma}
18124 (about 0.5772). With both Inverse and Hyperbolic, it
18125 pushes the ``golden ratio''
18126 @texline @math{\phi}
18127 @infoline @expr{phi}
18128 (about 1.618). (At present, Euler's constant is not available
18129 to unlimited precision; Calc knows only the first 100 digits.)
18130 In Symbolic mode, these commands push the
18131 actual variables @samp{pi}, @samp{e}, @samp{gamma}, and @samp{phi},
18132 respectively, instead of their values; @pxref{Symbolic Mode}.
18142 The @kbd{Q} (@code{calc-sqrt}) [@code{sqrt}] function is described elsewhere;
18143 @pxref{Basic Arithmetic}. With the Inverse flag [@code{sqr}], this command
18144 computes the square of the argument.
18146 @xref{Prefix Arguments}, for a discussion of the effect of numeric
18147 prefix arguments on commands in this chapter which do not otherwise
18148 interpret a prefix argument.
18151 * Logarithmic Functions::
18152 * Trigonometric and Hyperbolic Functions::
18153 * Advanced Math Functions::
18156 * Combinatorial Functions::
18157 * Probability Distribution Functions::
18160 @node Logarithmic Functions, Trigonometric and Hyperbolic Functions, Scientific Functions, Scientific Functions
18161 @section Logarithmic Functions
18171 The shift-@kbd{L} (@code{calc-ln}) [@code{ln}] command computes the natural
18172 logarithm of the real or complex number on the top of the stack. With
18173 the Inverse flag it computes the exponential function instead, although
18174 this is redundant with the @kbd{E} command.
18183 The shift-@kbd{E} (@code{calc-exp}) [@code{exp}] command computes the
18184 exponential, i.e., @expr{e} raised to the power of the number on the stack.
18185 The meanings of the Inverse and Hyperbolic flags follow from those for
18186 the @code{calc-ln} command.
18201 The @kbd{H L} (@code{calc-log10}) [@code{log10}] command computes the common
18202 (base-10) logarithm of a number. (With the Inverse flag [@code{exp10}],
18203 it raises ten to a given power.) Note that the common logarithm of a
18204 complex number is computed by taking the natural logarithm and dividing
18206 @texline @math{\ln10}.
18207 @infoline @expr{ln(10)}.
18214 The @kbd{B} (@code{calc-log}) [@code{log}] command computes a logarithm
18215 to any base. For example, @kbd{1024 @key{RET} 2 B} produces 10, since
18216 @texline @math{2^{10} = 1024}.
18217 @infoline @expr{2^10 = 1024}.
18218 In certain cases like @samp{log(3,9)}, the result
18219 will be either @expr{1:2} or @expr{0.5} depending on the current Fraction
18220 mode setting. With the Inverse flag [@code{alog}], this command is
18221 similar to @kbd{^} except that the order of the arguments is reversed.
18226 The @kbd{f I} (@code{calc-ilog}) [@code{ilog}] command computes the
18227 integer logarithm of a number to any base. The number and the base must
18228 themselves be positive integers. This is the true logarithm, rounded
18229 down to an integer. Thus @kbd{ilog(x,10)} is 3 for all @expr{x} in the
18230 range from 1000 to 9999. If both arguments are positive integers, exact
18231 integer arithmetic is used; otherwise, this is equivalent to
18232 @samp{floor(log(x,b))}.
18237 The @kbd{f E} (@code{calc-expm1}) [@code{expm1}] command computes
18238 @texline @math{e^x - 1},
18239 @infoline @expr{exp(x)-1},
18240 but using an algorithm that produces a more accurate
18241 answer when the result is close to zero, i.e., when
18242 @texline @math{e^x}
18243 @infoline @expr{exp(x)}
18249 The @kbd{f L} (@code{calc-lnp1}) [@code{lnp1}] command computes
18250 @texline @math{\ln(x+1)},
18251 @infoline @expr{ln(x+1)},
18252 producing a more accurate answer when @expr{x} is close to zero.
18254 @node Trigonometric and Hyperbolic Functions, Advanced Math Functions, Logarithmic Functions, Scientific Functions
18255 @section Trigonometric/Hyperbolic Functions
18261 The shift-@kbd{S} (@code{calc-sin}) [@code{sin}] command computes the sine
18262 of an angle or complex number. If the input is an HMS form, it is interpreted
18263 as degrees-minutes-seconds; otherwise, the input is interpreted according
18264 to the current angular mode. It is best to use Radians mode when operating
18265 on complex numbers.
18267 Calc's ``units'' mechanism includes angular units like @code{deg},
18268 @code{rad}, and @code{grad}. While @samp{sin(45 deg)} is not evaluated
18269 all the time, the @kbd{u s} (@code{calc-simplify-units}) command will
18270 simplify @samp{sin(45 deg)} by taking the sine of 45 degrees, regardless
18271 of the current angular mode. @xref{Basic Operations on Units}.
18273 Also, the symbolic variable @code{pi} is not ordinarily recognized in
18274 arguments to trigonometric functions, as in @samp{sin(3 pi / 4)}, but
18275 the @kbd{a s} (@code{calc-simplify}) command recognizes many such
18276 formulas when the current angular mode is Radians @emph{and} Symbolic
18277 mode is enabled; this example would be replaced by @samp{sqrt(2) / 2}.
18278 @xref{Symbolic Mode}. Beware, this simplification occurs even if you
18279 have stored a different value in the variable @samp{pi}; this is one
18280 reason why changing built-in variables is a bad idea. Arguments of
18281 the form @expr{x} plus a multiple of @cpiover{2} are also simplified.
18282 Calc includes similar formulas for @code{cos} and @code{tan}.
18284 The @kbd{a s} command knows all angles which are integer multiples of
18285 @cpiover{12}, @cpiover{10}, or @cpiover{8} radians. In Degrees mode,
18286 analogous simplifications occur for integer multiples of 15 or 18
18287 degrees, and for arguments plus multiples of 90 degrees.
18290 @pindex calc-arcsin
18292 With the Inverse flag, @code{calc-sin} computes an arcsine. This is also
18293 available as the @code{calc-arcsin} command or @code{arcsin} algebraic
18294 function. The returned argument is converted to degrees, radians, or HMS
18295 notation depending on the current angular mode.
18301 @pindex calc-arcsinh
18303 With the Hyperbolic flag, @code{calc-sin} computes the hyperbolic
18304 sine, also available as @code{calc-sinh} [@code{sinh}]. With the
18305 Hyperbolic and Inverse flags, it computes the hyperbolic arcsine
18306 (@code{calc-arcsinh}) [@code{arcsinh}].
18315 @pindex calc-arccos
18333 @pindex calc-arccosh
18351 @pindex calc-arctan
18369 @pindex calc-arctanh
18374 The shift-@kbd{C} (@code{calc-cos}) [@code{cos}] command computes the cosine
18375 of an angle or complex number, and shift-@kbd{T} (@code{calc-tan}) [@code{tan}]
18376 computes the tangent, along with all the various inverse and hyperbolic
18377 variants of these functions.
18380 @pindex calc-arctan2
18382 The @kbd{f T} (@code{calc-arctan2}) [@code{arctan2}] command takes two
18383 numbers from the stack and computes the arc tangent of their ratio. The
18384 result is in the full range from @mathit{-180} (exclusive) to @mathit{+180}
18385 (inclusive) degrees, or the analogous range in radians. A similar
18386 result would be obtained with @kbd{/} followed by @kbd{I T}, but the
18387 value would only be in the range from @mathit{-90} to @mathit{+90} degrees
18388 since the division loses information about the signs of the two
18389 components, and an error might result from an explicit division by zero
18390 which @code{arctan2} would avoid. By (arbitrary) definition,
18391 @samp{arctan2(0,0)=0}.
18393 @pindex calc-sincos
18405 The @code{calc-sincos} [@code{sincos}] command computes the sine and
18406 cosine of a number, returning them as a vector of the form
18407 @samp{[@var{cos}, @var{sin}]}.
18408 With the Inverse flag [@code{arcsincos}], this command takes a two-element
18409 vector as an argument and computes @code{arctan2} of the elements.
18410 (This command does not accept the Hyperbolic flag.)
18424 The remaining trigonometric functions, @code{calc-sec} [@code{sec}],
18425 @code{calc-csc} [@code{csc}] and @code{calc-cot} [@code{cot}], are also
18426 available. With the Hyperbolic flag, these compute their hyperbolic
18427 counterparts, which are also available separately as @code{calc-sech}
18428 [@code{sech}], @code{calc-csch} [@code{csch}] and @code{calc-coth}
18429 [@code{coth}]. (These commands do not accept the Inverse flag.)
18431 @node Advanced Math Functions, Branch Cuts, Trigonometric and Hyperbolic Functions, Scientific Functions
18432 @section Advanced Mathematical Functions
18435 Calc can compute a variety of less common functions that arise in
18436 various branches of mathematics. All of the functions described in
18437 this section allow arbitrary complex arguments and, except as noted,
18438 will work to arbitrarily large precisions. They can not at present
18439 handle error forms or intervals as arguments.
18441 NOTE: These functions are still experimental. In particular, their
18442 accuracy is not guaranteed in all domains. It is advisable to set the
18443 current precision comfortably higher than you actually need when
18444 using these functions. Also, these functions may be impractically
18445 slow for some values of the arguments.
18450 The @kbd{f g} (@code{calc-gamma}) [@code{gamma}] command computes the Euler
18451 gamma function. For positive integer arguments, this is related to the
18452 factorial function: @samp{gamma(n+1) = fact(n)}. For general complex
18453 arguments the gamma function can be defined by the following definite
18455 @texline @math{\Gamma(a) = \int_0^\infty t^{a-1} e^t dt}.
18456 @infoline @expr{gamma(a) = integ(t^(a-1) exp(t), t, 0, inf)}.
18457 (The actual implementation uses far more efficient computational methods.)
18473 @pindex calc-inc-gamma
18486 The @kbd{f G} (@code{calc-inc-gamma}) [@code{gammaP}] command computes
18487 the incomplete gamma function, denoted @samp{P(a,x)}. This is defined by
18489 @texline @math{P(a,x) = \left( \int_0^x t^{a-1} e^t dt \right) / \Gamma(a)}.
18490 @infoline @expr{gammaP(a,x) = integ(t^(a-1) exp(t), t, 0, x) / gamma(a)}.
18491 This implies that @samp{gammaP(a,inf) = 1} for any @expr{a} (see the
18492 definition of the normal gamma function).
18494 Several other varieties of incomplete gamma function are defined.
18495 The complement of @expr{P(a,x)}, called @expr{Q(a,x) = 1-P(a,x)} by
18496 some authors, is computed by the @kbd{I f G} [@code{gammaQ}] command.
18497 You can think of this as taking the other half of the integral, from
18498 @expr{x} to infinity.
18501 The functions corresponding to the integrals that define @expr{P(a,x)}
18502 and @expr{Q(a,x)} but without the normalizing @expr{1/gamma(a)}
18503 factor are called @expr{g(a,x)} and @expr{G(a,x)}, respectively
18504 (where @expr{g} and @expr{G} represent the lower- and upper-case Greek
18505 letter gamma). You can obtain these using the @kbd{H f G} [@code{gammag}]
18506 and @kbd{H I f G} [@code{gammaG}] commands.
18510 The functions corresponding to the integrals that define $P(a,x)$
18511 and $Q(a,x)$ but without the normalizing $1/\Gamma(a)$
18512 factor are called $\gamma(a,x)$ and $\Gamma(a,x)$, respectively.
18513 You can obtain these using the \kbd{H f G} [\code{gammag}] and
18514 \kbd{I H f G} [\code{gammaG}] commands.
18520 The @kbd{f b} (@code{calc-beta}) [@code{beta}] command computes the
18521 Euler beta function, which is defined in terms of the gamma function as
18522 @texline @math{B(a,b) = \Gamma(a) \Gamma(b) / \Gamma(a+b)},
18523 @infoline @expr{beta(a,b) = gamma(a) gamma(b) / gamma(a+b)},
18525 @texline @math{B(a,b) = \int_0^1 t^{a-1} (1-t)^{b-1} dt}.
18526 @infoline @expr{beta(a,b) = integ(t^(a-1) (1-t)^(b-1), t, 0, 1)}.
18530 @pindex calc-inc-beta
18533 The @kbd{f B} (@code{calc-inc-beta}) [@code{betaI}] command computes
18534 the incomplete beta function @expr{I(x,a,b)}. It is defined by
18535 @texline @math{I(x,a,b) = \left( \int_0^x t^{a-1} (1-t)^{b-1} dt \right) / B(a,b)}.
18536 @infoline @expr{betaI(x,a,b) = integ(t^(a-1) (1-t)^(b-1), t, 0, x) / beta(a,b)}.
18537 Once again, the @kbd{H} (hyperbolic) prefix gives the corresponding
18538 un-normalized version [@code{betaB}].
18545 The @kbd{f e} (@code{calc-erf}) [@code{erf}] command computes the
18547 @texline @math{\hbox{erf}(x) = {2 \over \sqrt{\pi}} \int_0^x e^{-t^2} dt}.
18548 @infoline @expr{erf(x) = 2 integ(exp(-(t^2)), t, 0, x) / sqrt(pi)}.
18549 The complementary error function @kbd{I f e} (@code{calc-erfc}) [@code{erfc}]
18550 is the corresponding integral from @samp{x} to infinity; the sum
18551 @texline @math{\hbox{erf}(x) + \hbox{erfc}(x) = 1}.
18552 @infoline @expr{erf(x) + erfc(x) = 1}.
18556 @pindex calc-bessel-J
18557 @pindex calc-bessel-Y
18560 The @kbd{f j} (@code{calc-bessel-J}) [@code{besJ}] and @kbd{f y}
18561 (@code{calc-bessel-Y}) [@code{besY}] commands compute the Bessel
18562 functions of the first and second kinds, respectively.
18563 In @samp{besJ(n,x)} and @samp{besY(n,x)} the ``order'' parameter
18564 @expr{n} is often an integer, but is not required to be one.
18565 Calc's implementation of the Bessel functions currently limits the
18566 precision to 8 digits, and may not be exact even to that precision.
18569 @node Branch Cuts, Random Numbers, Advanced Math Functions, Scientific Functions
18570 @section Branch Cuts and Principal Values
18573 @cindex Branch cuts
18574 @cindex Principal values
18575 All of the logarithmic, trigonometric, and other scientific functions are
18576 defined for complex numbers as well as for reals.
18577 This section describes the values
18578 returned in cases where the general result is a family of possible values.
18579 Calc follows section 12.5.3 of Steele's @dfn{Common Lisp, the Language},
18580 second edition, in these matters. This section will describe each
18581 function briefly; for a more detailed discussion (including some nifty
18582 diagrams), consult Steele's book.
18584 Note that the branch cuts for @code{arctan} and @code{arctanh} were
18585 changed between the first and second editions of Steele. Recent
18586 versions of Calc follow the second edition.
18588 The new branch cuts exactly match those of the HP-28/48 calculators.
18589 They also match those of Mathematica 1.2, except that Mathematica's
18590 @code{arctan} cut is always in the right half of the complex plane,
18591 and its @code{arctanh} cut is always in the top half of the plane.
18592 Calc's cuts are continuous with quadrants I and III for @code{arctan},
18593 or II and IV for @code{arctanh}.
18595 Note: The current implementations of these functions with complex arguments
18596 are designed with proper behavior around the branch cuts in mind, @emph{not}
18597 efficiency or accuracy. You may need to increase the floating precision
18598 and wait a while to get suitable answers from them.
18600 For @samp{sqrt(a+bi)}: When @expr{a<0} and @expr{b} is small but positive
18601 or zero, the result is close to the @expr{+i} axis. For @expr{b} small and
18602 negative, the result is close to the @expr{-i} axis. The result always lies
18603 in the right half of the complex plane.
18605 For @samp{ln(a+bi)}: The real part is defined as @samp{ln(abs(a+bi))}.
18606 The imaginary part is defined as @samp{arg(a+bi) = arctan2(b,a)}.
18607 Thus the branch cuts for @code{sqrt} and @code{ln} both lie on the
18608 negative real axis.
18610 The following table describes these branch cuts in another way.
18611 If the real and imaginary parts of @expr{z} are as shown, then
18612 the real and imaginary parts of @expr{f(z)} will be as shown.
18613 Here @code{eps} stands for a small positive value; each
18614 occurrence of @code{eps} may stand for a different small value.
18618 ----------------------------------------
18621 -, +eps +eps, + +eps, +
18622 -, -eps +eps, - +eps, -
18625 For @samp{z1^z2}: This is defined by @samp{exp(ln(z1)*z2)}.
18626 One interesting consequence of this is that @samp{(-8)^1:3} does
18627 not evaluate to @mathit{-2} as you might expect, but to the complex
18628 number @expr{(1., 1.732)}. Both of these are valid cube roots
18629 of @mathit{-8} (as is @expr{(1., -1.732)}); Calc chooses a perhaps
18630 less-obvious root for the sake of mathematical consistency.
18632 For @samp{arcsin(z)}: This is defined by @samp{-i*ln(i*z + sqrt(1-z^2))}.
18633 The branch cuts are on the real axis, less than @mathit{-1} and greater than 1.
18635 For @samp{arccos(z)}: This is defined by @samp{-i*ln(z + i*sqrt(1-z^2))},
18636 or equivalently by @samp{pi/2 - arcsin(z)}. The branch cuts are on
18637 the real axis, less than @mathit{-1} and greater than 1.
18639 For @samp{arctan(z)}: This is defined by
18640 @samp{(ln(1+i*z) - ln(1-i*z)) / (2*i)}. The branch cuts are on the
18641 imaginary axis, below @expr{-i} and above @expr{i}.
18643 For @samp{arcsinh(z)}: This is defined by @samp{ln(z + sqrt(1+z^2))}.
18644 The branch cuts are on the imaginary axis, below @expr{-i} and
18647 For @samp{arccosh(z)}: This is defined by
18648 @samp{ln(z + (z+1)*sqrt((z-1)/(z+1)))}. The branch cut is on the
18649 real axis less than 1.
18651 For @samp{arctanh(z)}: This is defined by @samp{(ln(1+z) - ln(1-z)) / 2}.
18652 The branch cuts are on the real axis, less than @mathit{-1} and greater than 1.
18654 The following tables for @code{arcsin}, @code{arccos}, and
18655 @code{arctan} assume the current angular mode is Radians. The
18656 hyperbolic functions operate independently of the angular mode.
18659 z arcsin(z) arccos(z)
18660 -------------------------------------------------------
18661 (-1..1), 0 (-pi/2..pi/2), 0 (0..pi), 0
18662 (-1..1), +eps (-pi/2..pi/2), +eps (0..pi), -eps
18663 (-1..1), -eps (-pi/2..pi/2), -eps (0..pi), +eps
18664 <-1, 0 -pi/2, + pi, -
18665 <-1, +eps -pi/2 + eps, + pi - eps, -
18666 <-1, -eps -pi/2 + eps, - pi - eps, +
18668 >1, +eps pi/2 - eps, + +eps, -
18669 >1, -eps pi/2 - eps, - +eps, +
18673 z arccosh(z) arctanh(z)
18674 -----------------------------------------------------
18675 (-1..1), 0 0, (0..pi) any, 0
18676 (-1..1), +eps +eps, (0..pi) any, +eps
18677 (-1..1), -eps +eps, (-pi..0) any, -eps
18678 <-1, 0 +, pi -, pi/2
18679 <-1, +eps +, pi - eps -, pi/2 - eps
18680 <-1, -eps +, -pi + eps -, -pi/2 + eps
18681 >1, 0 +, 0 +, -pi/2
18682 >1, +eps +, +eps +, pi/2 - eps
18683 >1, -eps +, -eps +, -pi/2 + eps
18687 z arcsinh(z) arctan(z)
18688 -----------------------------------------------------
18689 0, (-1..1) 0, (-pi/2..pi/2) 0, any
18690 0, <-1 -, -pi/2 -pi/2, -
18691 +eps, <-1 +, -pi/2 + eps pi/2 - eps, -
18692 -eps, <-1 -, -pi/2 + eps -pi/2 + eps, -
18693 0, >1 +, pi/2 pi/2, +
18694 +eps, >1 +, pi/2 - eps pi/2 - eps, +
18695 -eps, >1 -, pi/2 - eps -pi/2 + eps, +
18698 Finally, the following identities help to illustrate the relationship
18699 between the complex trigonometric and hyperbolic functions. They
18700 are valid everywhere, including on the branch cuts.
18703 sin(i*z) = i*sinh(z) arcsin(i*z) = i*arcsinh(z)
18704 cos(i*z) = cosh(z) arcsinh(i*z) = i*arcsin(z)
18705 tan(i*z) = i*tanh(z) arctan(i*z) = i*arctanh(z)
18706 sinh(i*z) = i*sin(z) cosh(i*z) = cos(z)
18709 The ``advanced math'' functions (gamma, Bessel, etc.@:) are also defined
18710 for general complex arguments, but their branch cuts and principal values
18711 are not rigorously specified at present.
18713 @node Random Numbers, Combinatorial Functions, Branch Cuts, Scientific Functions
18714 @section Random Numbers
18718 @pindex calc-random
18720 The @kbd{k r} (@code{calc-random}) [@code{random}] command produces
18721 random numbers of various sorts.
18723 Given a positive numeric prefix argument @expr{M}, it produces a random
18724 integer @expr{N} in the range
18725 @texline @math{0 \le N < M}.
18726 @infoline @expr{0 <= N < M}.
18727 Each possible value @expr{N} appears with equal probability.
18729 With no numeric prefix argument, the @kbd{k r} command takes its argument
18730 from the stack instead. Once again, if this is a positive integer @expr{M}
18731 the result is a random integer less than @expr{M}. However, note that
18732 while numeric prefix arguments are limited to six digits or so, an @expr{M}
18733 taken from the stack can be arbitrarily large. If @expr{M} is negative,
18734 the result is a random integer in the range
18735 @texline @math{M < N \le 0}.
18736 @infoline @expr{M < N <= 0}.
18738 If the value on the stack is a floating-point number @expr{M}, the result
18739 is a random floating-point number @expr{N} in the range
18740 @texline @math{0 \le N < M}
18741 @infoline @expr{0 <= N < M}
18743 @texline @math{M < N \le 0},
18744 @infoline @expr{M < N <= 0},
18745 according to the sign of @expr{M}.
18747 If @expr{M} is zero, the result is a Gaussian-distributed random real
18748 number; the distribution has a mean of zero and a standard deviation
18749 of one. The algorithm used generates random numbers in pairs; thus,
18750 every other call to this function will be especially fast.
18752 If @expr{M} is an error form
18753 @texline @math{m} @code{+/-} @math{\sigma}
18754 @infoline @samp{m +/- s}
18756 @texline @math{\sigma}
18758 are both real numbers, the result uses a Gaussian distribution with mean
18759 @var{m} and standard deviation
18760 @texline @math{\sigma}.
18763 If @expr{M} is an interval form, the lower and upper bounds specify the
18764 acceptable limits of the random numbers. If both bounds are integers,
18765 the result is a random integer in the specified range. If either bound
18766 is floating-point, the result is a random real number in the specified
18767 range. If the interval is open at either end, the result will be sure
18768 not to equal that end value. (This makes a big difference for integer
18769 intervals, but for floating-point intervals it's relatively minor:
18770 with a precision of 6, @samp{random([1.0..2.0))} will return any of one
18771 million numbers from 1.00000 to 1.99999; @samp{random([1.0..2.0])} may
18772 additionally return 2.00000, but the probability of this happening is
18775 If @expr{M} is a vector, the result is one element taken at random from
18776 the vector. All elements of the vector are given equal probabilities.
18779 The sequence of numbers produced by @kbd{k r} is completely random by
18780 default, i.e., the sequence is seeded each time you start Calc using
18781 the current time and other information. You can get a reproducible
18782 sequence by storing a particular ``seed value'' in the Calc variable
18783 @code{RandSeed}. Any integer will do for a seed; integers of from 1
18784 to 12 digits are good. If you later store a different integer into
18785 @code{RandSeed}, Calc will switch to a different pseudo-random
18786 sequence. If you ``unstore'' @code{RandSeed}, Calc will re-seed itself
18787 from the current time. If you store the same integer that you used
18788 before back into @code{RandSeed}, you will get the exact same sequence
18789 of random numbers as before.
18791 @pindex calc-rrandom
18792 The @code{calc-rrandom} command (not on any key) produces a random real
18793 number between zero and one. It is equivalent to @samp{random(1.0)}.
18796 @pindex calc-random-again
18797 The @kbd{k a} (@code{calc-random-again}) command produces another random
18798 number, re-using the most recent value of @expr{M}. With a numeric
18799 prefix argument @var{n}, it produces @var{n} more random numbers using
18800 that value of @expr{M}.
18803 @pindex calc-shuffle
18805 The @kbd{k h} (@code{calc-shuffle}) command produces a vector of several
18806 random values with no duplicates. The value on the top of the stack
18807 specifies the set from which the random values are drawn, and may be any
18808 of the @expr{M} formats described above. The numeric prefix argument
18809 gives the length of the desired list. (If you do not provide a numeric
18810 prefix argument, the length of the list is taken from the top of the
18811 stack, and @expr{M} from second-to-top.)
18813 If @expr{M} is a floating-point number, zero, or an error form (so
18814 that the random values are being drawn from the set of real numbers)
18815 there is little practical difference between using @kbd{k h} and using
18816 @kbd{k r} several times. But if the set of possible values consists
18817 of just a few integers, or the elements of a vector, then there is
18818 a very real chance that multiple @kbd{k r}'s will produce the same
18819 number more than once. The @kbd{k h} command produces a vector whose
18820 elements are always distinct. (Actually, there is a slight exception:
18821 If @expr{M} is a vector, no given vector element will be drawn more
18822 than once, but if several elements of @expr{M} are equal, they may
18823 each make it into the result vector.)
18825 One use of @kbd{k h} is to rearrange a list at random. This happens
18826 if the prefix argument is equal to the number of values in the list:
18827 @kbd{[1, 1.5, 2, 2.5, 3] 5 k h} might produce the permuted list
18828 @samp{[2.5, 1, 1.5, 3, 2]}. As a convenient feature, if the argument
18829 @var{n} is negative it is replaced by the size of the set represented
18830 by @expr{M}. Naturally, this is allowed only when @expr{M} specifies
18831 a small discrete set of possibilities.
18833 To do the equivalent of @kbd{k h} but with duplications allowed,
18834 given @expr{M} on the stack and with @var{n} just entered as a numeric
18835 prefix, use @kbd{v b} to build a vector of copies of @expr{M}, then use
18836 @kbd{V M k r} to ``map'' the normal @kbd{k r} function over the
18837 elements of this vector. @xref{Matrix Functions}.
18840 * Random Number Generator:: (Complete description of Calc's algorithm)
18843 @node Random Number Generator, , Random Numbers, Random Numbers
18844 @subsection Random Number Generator
18846 Calc's random number generator uses several methods to ensure that
18847 the numbers it produces are highly random. Knuth's @emph{Art of
18848 Computer Programming}, Volume II, contains a thorough description
18849 of the theory of random number generators and their measurement and
18852 If @code{RandSeed} has no stored value, Calc calls Emacs' built-in
18853 @code{random} function to get a stream of random numbers, which it
18854 then treats in various ways to avoid problems inherent in the simple
18855 random number generators that many systems use to implement @code{random}.
18857 When Calc's random number generator is first invoked, it ``seeds''
18858 the low-level random sequence using the time of day, so that the
18859 random number sequence will be different every time you use Calc.
18861 Since Emacs Lisp doesn't specify the range of values that will be
18862 returned by its @code{random} function, Calc exercises the function
18863 several times to estimate the range. When Calc subsequently uses
18864 the @code{random} function, it takes only 10 bits of the result
18865 near the most-significant end. (It avoids at least the bottom
18866 four bits, preferably more, and also tries to avoid the top two
18867 bits.) This strategy works well with the linear congruential
18868 generators that are typically used to implement @code{random}.
18870 If @code{RandSeed} contains an integer, Calc uses this integer to
18871 seed an ``additive congruential'' method (Knuth's algorithm 3.2.2A,
18873 @texline @math{X_{n-55} - X_{n-24}}.
18874 @infoline @expr{X_n-55 - X_n-24}).
18875 This method expands the seed
18876 value into a large table which is maintained internally; the variable
18877 @code{RandSeed} is changed from, e.g., 42 to the vector @expr{[42]}
18878 to indicate that the seed has been absorbed into this table. When
18879 @code{RandSeed} contains a vector, @kbd{k r} and related commands
18880 continue to use the same internal table as last time. There is no
18881 way to extract the complete state of the random number generator
18882 so that you can restart it from any point; you can only restart it
18883 from the same initial seed value. A simple way to restart from the
18884 same seed is to type @kbd{s r RandSeed} to get the seed vector,
18885 @kbd{v u} to unpack it back into a number, then @kbd{s t RandSeed}
18886 to reseed the generator with that number.
18888 Calc uses a ``shuffling'' method as described in algorithm 3.2.2B
18889 of Knuth. It fills a table with 13 random 10-bit numbers. Then,
18890 to generate a new random number, it uses the previous number to
18891 index into the table, picks the value it finds there as the new
18892 random number, then replaces that table entry with a new value
18893 obtained from a call to the base random number generator (either
18894 the additive congruential generator or the @code{random} function
18895 supplied by the system). If there are any flaws in the base
18896 generator, shuffling will tend to even them out. But if the system
18897 provides an excellent @code{random} function, shuffling will not
18898 damage its randomness.
18900 To create a random integer of a certain number of digits, Calc
18901 builds the integer three decimal digits at a time. For each group
18902 of three digits, Calc calls its 10-bit shuffling random number generator
18903 (which returns a value from 0 to 1023); if the random value is 1000
18904 or more, Calc throws it out and tries again until it gets a suitable
18907 To create a random floating-point number with precision @var{p}, Calc
18908 simply creates a random @var{p}-digit integer and multiplies by
18909 @texline @math{10^{-p}}.
18910 @infoline @expr{10^-p}.
18911 The resulting random numbers should be very clean, but note
18912 that relatively small numbers will have few significant random digits.
18913 In other words, with a precision of 12, you will occasionally get
18914 numbers on the order of
18915 @texline @math{10^{-9}}
18916 @infoline @expr{10^-9}
18918 @texline @math{10^{-10}},
18919 @infoline @expr{10^-10},
18920 but those numbers will only have two or three random digits since they
18921 correspond to small integers times
18922 @texline @math{10^{-12}}.
18923 @infoline @expr{10^-12}.
18925 To create a random integer in the interval @samp{[0 .. @var{m})}, Calc
18926 counts the digits in @var{m}, creates a random integer with three
18927 additional digits, then reduces modulo @var{m}. Unless @var{m} is a
18928 power of ten the resulting values will be very slightly biased toward
18929 the lower numbers, but this bias will be less than 0.1%. (For example,
18930 if @var{m} is 42, Calc will reduce a random integer less than 100000
18931 modulo 42 to get a result less than 42. It is easy to show that the
18932 numbers 40 and 41 will be only 2380/2381 as likely to result from this
18933 modulo operation as numbers 39 and below.) If @var{m} is a power of
18934 ten, however, the numbers should be completely unbiased.
18936 The Gaussian random numbers generated by @samp{random(0.0)} use the
18937 ``polar'' method described in Knuth section 3.4.1C. This method
18938 generates a pair of Gaussian random numbers at a time, so only every
18939 other call to @samp{random(0.0)} will require significant calculations.
18941 @node Combinatorial Functions, Probability Distribution Functions, Random Numbers, Scientific Functions
18942 @section Combinatorial Functions
18945 Commands relating to combinatorics and number theory begin with the
18946 @kbd{k} key prefix.
18951 The @kbd{k g} (@code{calc-gcd}) [@code{gcd}] command computes the
18952 Greatest Common Divisor of two integers. It also accepts fractions;
18953 the GCD of two fractions is defined by taking the GCD of the
18954 numerators, and the LCM of the denominators. This definition is
18955 consistent with the idea that @samp{a / gcd(a,x)} should yield an
18956 integer for any @samp{a} and @samp{x}. For other types of arguments,
18957 the operation is left in symbolic form.
18962 The @kbd{k l} (@code{calc-lcm}) [@code{lcm}] command computes the
18963 Least Common Multiple of two integers or fractions. The product of
18964 the LCM and GCD of two numbers is equal to the product of the
18968 @pindex calc-extended-gcd
18970 The @kbd{k E} (@code{calc-extended-gcd}) [@code{egcd}] command computes
18971 the GCD of two integers @expr{x} and @expr{y} and returns a vector
18972 @expr{[g, a, b]} where
18973 @texline @math{g = \gcd(x,y) = a x + b y}.
18974 @infoline @expr{g = gcd(x,y) = a x + b y}.
18977 @pindex calc-factorial
18983 The @kbd{!} (@code{calc-factorial}) [@code{fact}] command computes the
18984 factorial of the number at the top of the stack. If the number is an
18985 integer, the result is an exact integer. If the number is an
18986 integer-valued float, the result is a floating-point approximation. If
18987 the number is a non-integral real number, the generalized factorial is used,
18988 as defined by the Euler Gamma function. Please note that computation of
18989 large factorials can be slow; using floating-point format will help
18990 since fewer digits must be maintained. The same is true of many of
18991 the commands in this section.
18994 @pindex calc-double-factorial
19000 The @kbd{k d} (@code{calc-double-factorial}) [@code{dfact}] command
19001 computes the ``double factorial'' of an integer. For an even integer,
19002 this is the product of even integers from 2 to @expr{N}. For an odd
19003 integer, this is the product of odd integers from 3 to @expr{N}. If
19004 the argument is an integer-valued float, the result is a floating-point
19005 approximation. This function is undefined for negative even integers.
19006 The notation @expr{N!!} is also recognized for double factorials.
19009 @pindex calc-choose
19011 The @kbd{k c} (@code{calc-choose}) [@code{choose}] command computes the
19012 binomial coefficient @expr{N}-choose-@expr{M}, where @expr{M} is the number
19013 on the top of the stack and @expr{N} is second-to-top. If both arguments
19014 are integers, the result is an exact integer. Otherwise, the result is a
19015 floating-point approximation. The binomial coefficient is defined for all
19017 @texline @math{N! \over M! (N-M)!\,}.
19018 @infoline @expr{N! / M! (N-M)!}.
19024 The @kbd{H k c} (@code{calc-perm}) [@code{perm}] command computes the
19025 number-of-permutations function @expr{N! / (N-M)!}.
19028 The \kbd{H k c} (\code{calc-perm}) [\code{perm}] command computes the
19029 number-of-perm\-utations function $N! \over (N-M)!\,$.
19034 @pindex calc-bernoulli-number
19036 The @kbd{k b} (@code{calc-bernoulli-number}) [@code{bern}] command
19037 computes a given Bernoulli number. The value at the top of the stack
19038 is a nonnegative integer @expr{n} that specifies which Bernoulli number
19039 is desired. The @kbd{H k b} command computes a Bernoulli polynomial,
19040 taking @expr{n} from the second-to-top position and @expr{x} from the
19041 top of the stack. If @expr{x} is a variable or formula the result is
19042 a polynomial in @expr{x}; if @expr{x} is a number the result is a number.
19046 @pindex calc-euler-number
19048 The @kbd{k e} (@code{calc-euler-number}) [@code{euler}] command similarly
19049 computes an Euler number, and @w{@kbd{H k e}} computes an Euler polynomial.
19050 Bernoulli and Euler numbers occur in the Taylor expansions of several
19055 @pindex calc-stirling-number
19058 The @kbd{k s} (@code{calc-stirling-number}) [@code{stir1}] command
19059 computes a Stirling number of the first
19060 @texline kind@tie{}@math{n \brack m},
19062 given two integers @expr{n} and @expr{m} on the stack. The @kbd{H k s}
19063 [@code{stir2}] command computes a Stirling number of the second
19064 @texline kind@tie{}@math{n \brace m}.
19066 These are the number of @expr{m}-cycle permutations of @expr{n} objects,
19067 and the number of ways to partition @expr{n} objects into @expr{m}
19068 non-empty sets, respectively.
19071 @pindex calc-prime-test
19073 The @kbd{k p} (@code{calc-prime-test}) command checks if the integer on
19074 the top of the stack is prime. For integers less than eight million, the
19075 answer is always exact and reasonably fast. For larger integers, a
19076 probabilistic method is used (see Knuth vol. II, section 4.5.4, algorithm P).
19077 The number is first checked against small prime factors (up to 13). Then,
19078 any number of iterations of the algorithm are performed. Each step either
19079 discovers that the number is non-prime, or substantially increases the
19080 certainty that the number is prime. After a few steps, the chance that
19081 a number was mistakenly described as prime will be less than one percent.
19082 (Indeed, this is a worst-case estimate of the probability; in practice
19083 even a single iteration is quite reliable.) After the @kbd{k p} command,
19084 the number will be reported as definitely prime or non-prime if possible,
19085 or otherwise ``probably'' prime with a certain probability of error.
19091 The normal @kbd{k p} command performs one iteration of the primality
19092 test. Pressing @kbd{k p} repeatedly for the same integer will perform
19093 additional iterations. Also, @kbd{k p} with a numeric prefix performs
19094 the specified number of iterations. There is also an algebraic function
19095 @samp{prime(n)} or @samp{prime(n,iters)} which returns 1 if @expr{n}
19096 is (probably) prime and 0 if not.
19099 @pindex calc-prime-factors
19101 The @kbd{k f} (@code{calc-prime-factors}) [@code{prfac}] command
19102 attempts to decompose an integer into its prime factors. For numbers up
19103 to 25 million, the answer is exact although it may take some time. The
19104 result is a vector of the prime factors in increasing order. For larger
19105 inputs, prime factors above 5000 may not be found, in which case the
19106 last number in the vector will be an unfactored integer greater than 25
19107 million (with a warning message). For negative integers, the first
19108 element of the list will be @mathit{-1}. For inputs @mathit{-1}, @mathit{0}, and
19109 @mathit{1}, the result is a list of the same number.
19112 @pindex calc-next-prime
19114 @mindex nextpr@idots
19117 The @kbd{k n} (@code{calc-next-prime}) [@code{nextprime}] command finds
19118 the next prime above a given number. Essentially, it searches by calling
19119 @code{calc-prime-test} on successive integers until it finds one that
19120 passes the test. This is quite fast for integers less than eight million,
19121 but once the probabilistic test comes into play the search may be rather
19122 slow. Ordinarily this command stops for any prime that passes one iteration
19123 of the primality test. With a numeric prefix argument, a number must pass
19124 the specified number of iterations before the search stops. (This only
19125 matters when searching above eight million.) You can always use additional
19126 @kbd{k p} commands to increase your certainty that the number is indeed
19130 @pindex calc-prev-prime
19132 @mindex prevpr@idots
19135 The @kbd{I k n} (@code{calc-prev-prime}) [@code{prevprime}] command
19136 analogously finds the next prime less than a given number.
19139 @pindex calc-totient
19141 The @kbd{k t} (@code{calc-totient}) [@code{totient}] command computes the
19143 @texline function@tie{}@math{\phi(n)},
19144 @infoline function,
19145 the number of integers less than @expr{n} which
19146 are relatively prime to @expr{n}.
19149 @pindex calc-moebius
19151 The @kbd{k m} (@code{calc-moebius}) [@code{moebius}] command computes the
19152 @texline M@"obius @math{\mu}
19153 @infoline Moebius ``mu''
19154 function. If the input number is a product of @expr{k}
19155 distinct factors, this is @expr{(-1)^k}. If the input number has any
19156 duplicate factors (i.e., can be divided by the same prime more than once),
19157 the result is zero.
19159 @node Probability Distribution Functions, , Combinatorial Functions, Scientific Functions
19160 @section Probability Distribution Functions
19163 The functions in this section compute various probability distributions.
19164 For continuous distributions, this is the integral of the probability
19165 density function from @expr{x} to infinity. (These are the ``upper
19166 tail'' distribution functions; there are also corresponding ``lower
19167 tail'' functions which integrate from minus infinity to @expr{x}.)
19168 For discrete distributions, the upper tail function gives the sum
19169 from @expr{x} to infinity; the lower tail function gives the sum
19170 from minus infinity up to, but not including,@w{ }@expr{x}.
19172 To integrate from @expr{x} to @expr{y}, just use the distribution
19173 function twice and subtract. For example, the probability that a
19174 Gaussian random variable with mean 2 and standard deviation 1 will
19175 lie in the range from 2.5 to 2.8 is @samp{utpn(2.5,2,1) - utpn(2.8,2,1)}
19176 (``the probability that it is greater than 2.5, but not greater than 2.8''),
19177 or equivalently @samp{ltpn(2.8,2,1) - ltpn(2.5,2,1)}.
19184 The @kbd{k B} (@code{calc-utpb}) [@code{utpb}] function uses the
19185 binomial distribution. Push the parameters @var{n}, @var{p}, and
19186 then @var{x} onto the stack; the result (@samp{utpb(x,n,p)}) is the
19187 probability that an event will occur @var{x} or more times out
19188 of @var{n} trials, if its probability of occurring in any given
19189 trial is @var{p}. The @kbd{I k B} [@code{ltpb}] function is
19190 the probability that the event will occur fewer than @var{x} times.
19192 The other probability distribution functions similarly take the
19193 form @kbd{k @var{X}} (@code{calc-utp@var{x}}) [@code{utp@var{x}}]
19194 and @kbd{I k @var{X}} [@code{ltp@var{x}}], for various letters
19195 @var{x}. The arguments to the algebraic functions are the value of
19196 the random variable first, then whatever other parameters define the
19197 distribution. Note these are among the few Calc functions where the
19198 order of the arguments in algebraic form differs from the order of
19199 arguments as found on the stack. (The random variable comes last on
19200 the stack, so that you can type, e.g., @kbd{2 @key{RET} 1 @key{RET} 2.5
19201 k N M-@key{RET} @key{DEL} 2.8 k N -}, using @kbd{M-@key{RET} @key{DEL}} to
19202 recover the original arguments but substitute a new value for @expr{x}.)
19215 The @samp{utpc(x,v)} function uses the chi-square distribution with
19216 @texline @math{\nu}
19218 degrees of freedom. It is the probability that a model is
19219 correct if its chi-square statistic is @expr{x}.
19232 The @samp{utpf(F,v1,v2)} function uses the F distribution, used in
19233 various statistical tests. The parameters
19234 @texline @math{\nu_1}
19235 @infoline @expr{v1}
19237 @texline @math{\nu_2}
19238 @infoline @expr{v2}
19239 are the degrees of freedom in the numerator and denominator,
19240 respectively, used in computing the statistic @expr{F}.
19253 The @samp{utpn(x,m,s)} function uses a normal (Gaussian) distribution
19254 with mean @expr{m} and standard deviation
19255 @texline @math{\sigma}.
19256 @infoline @expr{s}.
19257 It is the probability that such a normal-distributed random variable
19258 would exceed @expr{x}.
19271 The @samp{utpp(n,x)} function uses a Poisson distribution with
19272 mean @expr{x}. It is the probability that @expr{n} or more such
19273 Poisson random events will occur.
19286 The @samp{utpt(t,v)} function uses the Student's ``t'' distribution
19288 @texline @math{\nu}
19290 degrees of freedom. It is the probability that a
19291 t-distributed random variable will be greater than @expr{t}.
19292 (Note: This computes the distribution function
19293 @texline @math{A(t|\nu)}
19294 @infoline @expr{A(t|v)}
19296 @texline @math{A(0|\nu) = 1}
19297 @infoline @expr{A(0|v) = 1}
19299 @texline @math{A(\infty|\nu) \to 0}.
19300 @infoline @expr{A(inf|v) -> 0}.
19301 The @code{UTPT} operation on the HP-48 uses a different definition which
19302 returns half of Calc's value: @samp{UTPT(t,v) = .5*utpt(t,v)}.)
19304 While Calc does not provide inverses of the probability distribution
19305 functions, the @kbd{a R} command can be used to solve for the inverse.
19306 Since the distribution functions are monotonic, @kbd{a R} is guaranteed
19307 to be able to find a solution given any initial guess.
19308 @xref{Numerical Solutions}.
19310 @node Matrix Functions, Algebra, Scientific Functions, Top
19311 @chapter Vector/Matrix Functions
19314 Many of the commands described here begin with the @kbd{v} prefix.
19315 (For convenience, the shift-@kbd{V} prefix is equivalent to @kbd{v}.)
19316 The commands usually apply to both plain vectors and matrices; some
19317 apply only to matrices or only to square matrices. If the argument
19318 has the wrong dimensions the operation is left in symbolic form.
19320 Vectors are entered and displayed using @samp{[a,b,c]} notation.
19321 Matrices are vectors of which all elements are vectors of equal length.
19322 (Though none of the standard Calc commands use this concept, a
19323 three-dimensional matrix or rank-3 tensor could be defined as a
19324 vector of matrices, and so on.)
19327 * Packing and Unpacking::
19328 * Building Vectors::
19329 * Extracting Elements::
19330 * Manipulating Vectors::
19331 * Vector and Matrix Arithmetic::
19333 * Statistical Operations::
19334 * Reducing and Mapping::
19335 * Vector and Matrix Formats::
19338 @node Packing and Unpacking, Building Vectors, Matrix Functions, Matrix Functions
19339 @section Packing and Unpacking
19342 Calc's ``pack'' and ``unpack'' commands collect stack entries to build
19343 composite objects such as vectors and complex numbers. They are
19344 described in this chapter because they are most often used to build
19349 The @kbd{v p} (@code{calc-pack}) [@code{pack}] command collects several
19350 elements from the stack into a matrix, complex number, HMS form, error
19351 form, etc. It uses a numeric prefix argument to specify the kind of
19352 object to be built; this argument is referred to as the ``packing mode.''
19353 If the packing mode is a nonnegative integer, a vector of that
19354 length is created. For example, @kbd{C-u 5 v p} will pop the top
19355 five stack elements and push back a single vector of those five
19356 elements. (@kbd{C-u 0 v p} simply creates an empty vector.)
19358 The same effect can be had by pressing @kbd{[} to push an incomplete
19359 vector on the stack, using @key{TAB} (@code{calc-roll-down}) to sneak
19360 the incomplete object up past a certain number of elements, and
19361 then pressing @kbd{]} to complete the vector.
19363 Negative packing modes create other kinds of composite objects:
19367 Two values are collected to build a complex number. For example,
19368 @kbd{5 @key{RET} 7 C-u -1 v p} creates the complex number
19369 @expr{(5, 7)}. The result is always a rectangular complex
19370 number. The two input values must both be real numbers,
19371 i.e., integers, fractions, or floats. If they are not, Calc
19372 will instead build a formula like @samp{a + (0, 1) b}. (The
19373 other packing modes also create a symbolic answer if the
19374 components are not suitable.)
19377 Two values are collected to build a polar complex number.
19378 The first is the magnitude; the second is the phase expressed
19379 in either degrees or radians according to the current angular
19383 Three values are collected into an HMS form. The first
19384 two values (hours and minutes) must be integers or
19385 integer-valued floats. The third value may be any real
19389 Two values are collected into an error form. The inputs
19390 may be real numbers or formulas.
19393 Two values are collected into a modulo form. The inputs
19394 must be real numbers.
19397 Two values are collected into the interval @samp{[a .. b]}.
19398 The inputs may be real numbers, HMS or date forms, or formulas.
19401 Two values are collected into the interval @samp{[a .. b)}.
19404 Two values are collected into the interval @samp{(a .. b]}.
19407 Two values are collected into the interval @samp{(a .. b)}.
19410 Two integer values are collected into a fraction.
19413 Two values are collected into a floating-point number.
19414 The first is the mantissa; the second, which must be an
19415 integer, is the exponent. The result is the mantissa
19416 times ten to the power of the exponent.
19419 This is treated the same as @mathit{-11} by the @kbd{v p} command.
19420 When unpacking, @mathit{-12} specifies that a floating-point mantissa
19424 A real number is converted into a date form.
19427 Three numbers (year, month, day) are packed into a pure date form.
19430 Six numbers are packed into a date/time form.
19433 With any of the two-input negative packing modes, either or both
19434 of the inputs may be vectors. If both are vectors of the same
19435 length, the result is another vector made by packing corresponding
19436 elements of the input vectors. If one input is a vector and the
19437 other is a plain number, the number is packed along with each vector
19438 element to produce a new vector. For example, @kbd{C-u -4 v p}
19439 could be used to convert a vector of numbers and a vector of errors
19440 into a single vector of error forms; @kbd{C-u -5 v p} could convert
19441 a vector of numbers and a single number @var{M} into a vector of
19442 numbers modulo @var{M}.
19444 If you don't give a prefix argument to @kbd{v p}, it takes
19445 the packing mode from the top of the stack. The elements to
19446 be packed then begin at stack level 2. Thus
19447 @kbd{1 @key{RET} 2 @key{RET} 4 n v p} is another way to
19448 enter the error form @samp{1 +/- 2}.
19450 If the packing mode taken from the stack is a vector, the result is a
19451 matrix with the dimensions specified by the elements of the vector,
19452 which must each be integers. For example, if the packing mode is
19453 @samp{[2, 3]}, then six numbers will be taken from the stack and
19454 returned in the form @samp{[@w{[a, b, c]}, [d, e, f]]}.
19456 If any elements of the vector are negative, other kinds of
19457 packing are done at that level as described above. For
19458 example, @samp{[2, 3, -4]} takes 12 objects and creates a
19459 @texline @math{2\times3}
19461 matrix of error forms: @samp{[[a +/- b, c +/- d ... ]]}.
19462 Also, @samp{[-4, -10]} will convert four integers into an
19463 error form consisting of two fractions: @samp{a:b +/- c:d}.
19469 There is an equivalent algebraic function,
19470 @samp{pack(@var{mode}, @var{items})} where @var{mode} is a
19471 packing mode (an integer or a vector of integers) and @var{items}
19472 is a vector of objects to be packed (re-packed, really) according
19473 to that mode. For example, @samp{pack([3, -4], [a,b,c,d,e,f])}
19474 yields @samp{[a +/- b, @w{c +/- d}, e +/- f]}. The function is
19475 left in symbolic form if the packing mode is invalid, or if the
19476 number of data items does not match the number of items required
19480 @pindex calc-unpack
19481 The @kbd{v u} (@code{calc-unpack}) command takes the vector, complex
19482 number, HMS form, or other composite object on the top of the stack and
19483 ``unpacks'' it, pushing each of its elements onto the stack as separate
19484 objects. Thus, it is the ``inverse'' of @kbd{v p}. If the value
19485 at the top of the stack is a formula, @kbd{v u} unpacks it by pushing
19486 each of the arguments of the top-level operator onto the stack.
19488 You can optionally give a numeric prefix argument to @kbd{v u}
19489 to specify an explicit (un)packing mode. If the packing mode is
19490 negative and the input is actually a vector or matrix, the result
19491 will be two or more similar vectors or matrices of the elements.
19492 For example, given the vector @samp{[@w{a +/- b}, c^2, d +/- 7]},
19493 the result of @kbd{C-u -4 v u} will be the two vectors
19494 @samp{[a, c^2, d]} and @w{@samp{[b, 0, 7]}}.
19496 Note that the prefix argument can have an effect even when the input is
19497 not a vector. For example, if the input is the number @mathit{-5}, then
19498 @kbd{c-u -1 v u} yields @mathit{-5} and 0 (the components of @mathit{-5}
19499 when viewed as a rectangular complex number); @kbd{C-u -2 v u} yields 5
19500 and 180 (assuming Degrees mode); and @kbd{C-u -10 v u} yields @mathit{-5}
19501 and 1 (the numerator and denominator of @mathit{-5}, viewed as a rational
19502 number). Plain @kbd{v u} with this input would complain that the input
19503 is not a composite object.
19505 Unpacking mode @mathit{-11} converts a float into an integer mantissa and
19506 an integer exponent, where the mantissa is not divisible by 10
19507 (except that 0.0 is represented by a mantissa and exponent of 0).
19508 Unpacking mode @mathit{-12} converts a float into a floating-point mantissa
19509 and integer exponent, where the mantissa (for non-zero numbers)
19510 is guaranteed to lie in the range [1 .. 10). In both cases,
19511 the mantissa is shifted left or right (and the exponent adjusted
19512 to compensate) in order to satisfy these constraints.
19514 Positive unpacking modes are treated differently than for @kbd{v p}.
19515 A mode of 1 is much like plain @kbd{v u} with no prefix argument,
19516 except that in addition to the components of the input object,
19517 a suitable packing mode to re-pack the object is also pushed.
19518 Thus, @kbd{C-u 1 v u} followed by @kbd{v p} will re-build the
19521 A mode of 2 unpacks two levels of the object; the resulting
19522 re-packing mode will be a vector of length 2. This might be used
19523 to unpack a matrix, say, or a vector of error forms. Higher
19524 unpacking modes unpack the input even more deeply.
19530 There are two algebraic functions analogous to @kbd{v u}.
19531 The @samp{unpack(@var{mode}, @var{item})} function unpacks the
19532 @var{item} using the given @var{mode}, returning the result as
19533 a vector of components. Here the @var{mode} must be an
19534 integer, not a vector. For example, @samp{unpack(-4, a +/- b)}
19535 returns @samp{[a, b]}, as does @samp{unpack(1, a +/- b)}.
19541 The @code{unpackt} function is like @code{unpack} but instead
19542 of returning a simple vector of items, it returns a vector of
19543 two things: The mode, and the vector of items. For example,
19544 @samp{unpackt(1, 2:3 +/- 1:4)} returns @samp{[-4, [2:3, 1:4]]},
19545 and @samp{unpackt(2, 2:3 +/- 1:4)} returns @samp{[[-4, -10], [2, 3, 1, 4]]}.
19546 The identity for re-building the original object is
19547 @samp{apply(pack, unpackt(@var{n}, @var{x})) = @var{x}}. (The
19548 @code{apply} function builds a function call given the function
19549 name and a vector of arguments.)
19551 @cindex Numerator of a fraction, extracting
19552 Subscript notation is a useful way to extract a particular part
19553 of an object. For example, to get the numerator of a rational
19554 number, you can use @samp{unpack(-10, @var{x})_1}.
19556 @node Building Vectors, Extracting Elements, Packing and Unpacking, Matrix Functions
19557 @section Building Vectors
19560 Vectors and matrices can be added,
19561 subtracted, multiplied, and divided; @pxref{Basic Arithmetic}.
19564 @pindex calc-concat
19569 The @kbd{|} (@code{calc-concat}) [@code{vconcat}] command ``concatenates'' two vectors
19570 into one. For example, after @kbd{@w{[ 1 , 2 ]} [ 3 , 4 ] |}, the stack
19571 will contain the single vector @samp{[1, 2, 3, 4]}. If the arguments
19572 are matrices, the rows of the first matrix are concatenated with the
19573 rows of the second. (In other words, two matrices are just two vectors
19574 of row-vectors as far as @kbd{|} is concerned.)
19576 If either argument to @kbd{|} is a scalar (a non-vector), it is treated
19577 like a one-element vector for purposes of concatenation: @kbd{1 [ 2 , 3 ] |}
19578 produces the vector @samp{[1, 2, 3]}. Likewise, if one argument is a
19579 matrix and the other is a plain vector, the vector is treated as a
19584 The @kbd{H |} (@code{calc-append}) [@code{append}] command concatenates
19585 two vectors without any special cases. Both inputs must be vectors.
19586 Whether or not they are matrices is not taken into account. If either
19587 argument is a scalar, the @code{append} function is left in symbolic form.
19588 See also @code{cons} and @code{rcons} below.
19592 The @kbd{I |} and @kbd{H I |} commands are similar, but they use their
19593 two stack arguments in the opposite order. Thus @kbd{I |} is equivalent
19594 to @kbd{@key{TAB} |}, but possibly more convenient and also a bit faster.
19599 The @kbd{v d} (@code{calc-diag}) [@code{diag}] function builds a diagonal
19600 square matrix. The optional numeric prefix gives the number of rows
19601 and columns in the matrix. If the value at the top of the stack is a
19602 vector, the elements of the vector are used as the diagonal elements; the
19603 prefix, if specified, must match the size of the vector. If the value on
19604 the stack is a scalar, it is used for each element on the diagonal, and
19605 the prefix argument is required.
19607 To build a constant square matrix, e.g., a
19608 @texline @math{3\times3}
19610 matrix filled with ones, use @kbd{0 M-3 v d 1 +}, i.e., build a zero
19611 matrix first and then add a constant value to that matrix. (Another
19612 alternative would be to use @kbd{v b} and @kbd{v a}; see below.)
19617 The @kbd{v i} (@code{calc-ident}) [@code{idn}] function builds an identity
19618 matrix of the specified size. It is a convenient form of @kbd{v d}
19619 where the diagonal element is always one. If no prefix argument is given,
19620 this command prompts for one.
19622 In algebraic notation, @samp{idn(a,n)} acts much like @samp{diag(a,n)},
19623 except that @expr{a} is required to be a scalar (non-vector) quantity.
19624 If @expr{n} is omitted, @samp{idn(a)} represents @expr{a} times an
19625 identity matrix of unknown size. Calc can operate algebraically on
19626 such generic identity matrices, and if one is combined with a matrix
19627 whose size is known, it is converted automatically to an identity
19628 matrix of a suitable matching size. The @kbd{v i} command with an
19629 argument of zero creates a generic identity matrix, @samp{idn(1)}.
19630 Note that in dimensioned Matrix mode (@pxref{Matrix Mode}), generic
19631 identity matrices are immediately expanded to the current default
19637 The @kbd{v x} (@code{calc-index}) [@code{index}] function builds a vector
19638 of consecutive integers from 1 to @var{n}, where @var{n} is the numeric
19639 prefix argument. If you do not provide a prefix argument, you will be
19640 prompted to enter a suitable number. If @var{n} is negative, the result
19641 is a vector of negative integers from @var{n} to @mathit{-1}.
19643 With a prefix argument of just @kbd{C-u}, the @kbd{v x} command takes
19644 three values from the stack: @var{n}, @var{start}, and @var{incr} (with
19645 @var{incr} at top-of-stack). Counting starts at @var{start} and increases
19646 by @var{incr} for successive vector elements. If @var{start} or @var{n}
19647 is in floating-point format, the resulting vector elements will also be
19648 floats. Note that @var{start} and @var{incr} may in fact be any kind
19649 of numbers or formulas.
19651 When @var{start} and @var{incr} are specified, a negative @var{n} has a
19652 different interpretation: It causes a geometric instead of arithmetic
19653 sequence to be generated. For example, @samp{index(-3, a, b)} produces
19654 @samp{[a, a b, a b^2]}. If you omit @var{incr} in the algebraic form,
19655 @samp{index(@var{n}, @var{start})}, the default value for @var{incr}
19656 is one for positive @var{n} or two for negative @var{n}.
19659 @pindex calc-build-vector
19661 The @kbd{v b} (@code{calc-build-vector}) [@code{cvec}] function builds a
19662 vector of @var{n} copies of the value on the top of the stack, where @var{n}
19663 is the numeric prefix argument. In algebraic formulas, @samp{cvec(x,n,m)}
19664 can also be used to build an @var{n}-by-@var{m} matrix of copies of @var{x}.
19665 (Interactively, just use @kbd{v b} twice: once to build a row, then again
19666 to build a matrix of copies of that row.)
19674 The @kbd{v h} (@code{calc-head}) [@code{head}] function returns the first
19675 element of a vector. The @kbd{I v h} (@code{calc-tail}) [@code{tail}]
19676 function returns the vector with its first element removed. In both
19677 cases, the argument must be a non-empty vector.
19682 The @kbd{v k} (@code{calc-cons}) [@code{cons}] function takes a value @var{h}
19683 and a vector @var{t} from the stack, and produces the vector whose head is
19684 @var{h} and whose tail is @var{t}. This is similar to @kbd{|}, except
19685 if @var{h} is itself a vector, @kbd{|} will concatenate the two vectors
19686 whereas @code{cons} will insert @var{h} at the front of the vector @var{t}.
19706 Each of these three functions also accepts the Hyperbolic flag [@code{rhead},
19707 @code{rtail}, @code{rcons}] in which case @var{t} instead represents
19708 the @emph{last} single element of the vector, with @var{h}
19709 representing the remainder of the vector. Thus the vector
19710 @samp{[a, b, c, d] = cons(a, [b, c, d]) = rcons([a, b, c], d)}.
19711 Also, @samp{head([a, b, c, d]) = a}, @samp{tail([a, b, c, d]) = [b, c, d]},
19712 @samp{rhead([a, b, c, d]) = [a, b, c]}, and @samp{rtail([a, b, c, d]) = d}.
19714 @node Extracting Elements, Manipulating Vectors, Building Vectors, Matrix Functions
19715 @section Extracting Vector Elements
19721 The @kbd{v r} (@code{calc-mrow}) [@code{mrow}] command extracts one row of
19722 the matrix on the top of the stack, or one element of the plain vector on
19723 the top of the stack. The row or element is specified by the numeric
19724 prefix argument; the default is to prompt for the row or element number.
19725 The matrix or vector is replaced by the specified row or element in the
19726 form of a vector or scalar, respectively.
19728 @cindex Permutations, applying
19729 With a prefix argument of @kbd{C-u} only, @kbd{v r} takes the index of
19730 the element or row from the top of the stack, and the vector or matrix
19731 from the second-to-top position. If the index is itself a vector of
19732 integers, the result is a vector of the corresponding elements of the
19733 input vector, or a matrix of the corresponding rows of the input matrix.
19734 This command can be used to obtain any permutation of a vector.
19736 With @kbd{C-u}, if the index is an interval form with integer components,
19737 it is interpreted as a range of indices and the corresponding subvector or
19738 submatrix is returned.
19740 @cindex Subscript notation
19742 @pindex calc-subscript
19745 Subscript notation in algebraic formulas (@samp{a_b}) stands for the
19746 Calc function @code{subscr}, which is synonymous with @code{mrow}.
19747 Thus, @samp{[x, y, z]_k} produces @expr{x}, @expr{y}, or @expr{z} if
19748 @expr{k} is one, two, or three, respectively. A double subscript
19749 (@samp{M_i_j}, equivalent to @samp{subscr(subscr(M, i), j)}) will
19750 access the element at row @expr{i}, column @expr{j} of a matrix.
19751 The @kbd{a _} (@code{calc-subscript}) command creates a subscript
19752 formula @samp{a_b} out of two stack entries. (It is on the @kbd{a}
19753 ``algebra'' prefix because subscripted variables are often used
19754 purely as an algebraic notation.)
19757 Given a negative prefix argument, @kbd{v r} instead deletes one row or
19758 element from the matrix or vector on the top of the stack. Thus
19759 @kbd{C-u 2 v r} replaces a matrix with its second row, but @kbd{C-u -2 v r}
19760 replaces the matrix with the same matrix with its second row removed.
19761 In algebraic form this function is called @code{mrrow}.
19764 Given a prefix argument of zero, @kbd{v r} extracts the diagonal elements
19765 of a square matrix in the form of a vector. In algebraic form this
19766 function is called @code{getdiag}.
19772 The @kbd{v c} (@code{calc-mcol}) [@code{mcol} or @code{mrcol}] command is
19773 the analogous operation on columns of a matrix. Given a plain vector
19774 it extracts (or removes) one element, just like @kbd{v r}. If the
19775 index in @kbd{C-u v c} is an interval or vector and the argument is a
19776 matrix, the result is a submatrix with only the specified columns
19777 retained (and possibly permuted in the case of a vector index).
19779 To extract a matrix element at a given row and column, use @kbd{v r} to
19780 extract the row as a vector, then @kbd{v c} to extract the column element
19781 from that vector. In algebraic formulas, it is often more convenient to
19782 use subscript notation: @samp{m_i_j} gives row @expr{i}, column @expr{j}
19783 of matrix @expr{m}.
19786 @pindex calc-subvector
19788 The @kbd{v s} (@code{calc-subvector}) [@code{subvec}] command extracts
19789 a subvector of a vector. The arguments are the vector, the starting
19790 index, and the ending index, with the ending index in the top-of-stack
19791 position. The starting index indicates the first element of the vector
19792 to take. The ending index indicates the first element @emph{past} the
19793 range to be taken. Thus, @samp{subvec([a, b, c, d, e], 2, 4)} produces
19794 the subvector @samp{[b, c]}. You could get the same result using
19795 @samp{mrow([a, b, c, d, e], @w{[2 .. 4)})}.
19797 If either the start or the end index is zero or negative, it is
19798 interpreted as relative to the end of the vector. Thus
19799 @samp{subvec([a, b, c, d, e], 2, -2)} also produces @samp{[b, c]}. In
19800 the algebraic form, the end index can be omitted in which case it
19801 is taken as zero, i.e., elements from the starting element to the
19802 end of the vector are used. The infinity symbol, @code{inf}, also
19803 has this effect when used as the ending index.
19807 With the Inverse flag, @kbd{I v s} [@code{rsubvec}] removes a subvector
19808 from a vector. The arguments are interpreted the same as for the
19809 normal @kbd{v s} command. Thus, @samp{rsubvec([a, b, c, d, e], 2, 4)}
19810 produces @samp{[a, d, e]}. It is always true that @code{subvec} and
19811 @code{rsubvec} return complementary parts of the input vector.
19813 @xref{Selecting Subformulas}, for an alternative way to operate on
19814 vectors one element at a time.
19816 @node Manipulating Vectors, Vector and Matrix Arithmetic, Extracting Elements, Matrix Functions
19817 @section Manipulating Vectors
19821 @pindex calc-vlength
19823 The @kbd{v l} (@code{calc-vlength}) [@code{vlen}] command computes the
19824 length of a vector. The length of a non-vector is considered to be zero.
19825 Note that matrices are just vectors of vectors for the purposes of this
19830 With the Hyperbolic flag, @kbd{H v l} [@code{mdims}] computes a vector
19831 of the dimensions of a vector, matrix, or higher-order object. For
19832 example, @samp{mdims([[a,b,c],[d,e,f]])} returns @samp{[2, 3]} since
19834 @texline @math{2\times3}
19839 @pindex calc-vector-find
19841 The @kbd{v f} (@code{calc-vector-find}) [@code{find}] command searches
19842 along a vector for the first element equal to a given target. The target
19843 is on the top of the stack; the vector is in the second-to-top position.
19844 If a match is found, the result is the index of the matching element.
19845 Otherwise, the result is zero. The numeric prefix argument, if given,
19846 allows you to select any starting index for the search.
19849 @pindex calc-arrange-vector
19851 @cindex Arranging a matrix
19852 @cindex Reshaping a matrix
19853 @cindex Flattening a matrix
19854 The @kbd{v a} (@code{calc-arrange-vector}) [@code{arrange}] command
19855 rearranges a vector to have a certain number of columns and rows. The
19856 numeric prefix argument specifies the number of columns; if you do not
19857 provide an argument, you will be prompted for the number of columns.
19858 The vector or matrix on the top of the stack is @dfn{flattened} into a
19859 plain vector. If the number of columns is nonzero, this vector is
19860 then formed into a matrix by taking successive groups of @var{n} elements.
19861 If the number of columns does not evenly divide the number of elements
19862 in the vector, the last row will be short and the result will not be
19863 suitable for use as a matrix. For example, with the matrix
19864 @samp{[[1, 2], @w{[3, 4]}]} on the stack, @kbd{v a 4} produces
19865 @samp{[[1, 2, 3, 4]]} (a
19866 @texline @math{1\times4}
19868 matrix), @kbd{v a 1} produces @samp{[[1], [2], [3], [4]]} (a
19869 @texline @math{4\times1}
19871 matrix), @kbd{v a 2} produces @samp{[[1, 2], [3, 4]]} (the original
19872 @texline @math{2\times2}
19874 matrix), @w{@kbd{v a 3}} produces @samp{[[1, 2, 3], [4]]} (not a
19875 matrix), and @kbd{v a 0} produces the flattened list
19876 @samp{[1, 2, @w{3, 4}]}.
19878 @cindex Sorting data
19884 The @kbd{V S} (@code{calc-sort}) [@code{sort}] command sorts the elements of
19885 a vector into increasing order. Real numbers, real infinities, and
19886 constant interval forms come first in this ordering; next come other
19887 kinds of numbers, then variables (in alphabetical order), then finally
19888 come formulas and other kinds of objects; these are sorted according
19889 to a kind of lexicographic ordering with the useful property that
19890 one vector is less or greater than another if the first corresponding
19891 unequal elements are less or greater, respectively. Since quoted strings
19892 are stored by Calc internally as vectors of ASCII character codes
19893 (@pxref{Strings}), this means vectors of strings are also sorted into
19894 alphabetical order by this command.
19896 The @kbd{I V S} [@code{rsort}] command sorts a vector into decreasing order.
19898 @cindex Permutation, inverse of
19899 @cindex Inverse of permutation
19900 @cindex Index tables
19901 @cindex Rank tables
19907 The @kbd{V G} (@code{calc-grade}) [@code{grade}, @code{rgrade}] command
19908 produces an index table or permutation vector which, if applied to the
19909 input vector (as the index of @kbd{C-u v r}, say), would sort the vector.
19910 A permutation vector is just a vector of integers from 1 to @var{n}, where
19911 each integer occurs exactly once. One application of this is to sort a
19912 matrix of data rows using one column as the sort key; extract that column,
19913 grade it with @kbd{V G}, then use the result to reorder the original matrix
19914 with @kbd{C-u v r}. Another interesting property of the @code{V G} command
19915 is that, if the input is itself a permutation vector, the result will
19916 be the inverse of the permutation. The inverse of an index table is
19917 a rank table, whose @var{k}th element says where the @var{k}th original
19918 vector element will rest when the vector is sorted. To get a rank
19919 table, just use @kbd{V G V G}.
19921 With the Inverse flag, @kbd{I V G} produces an index table that would
19922 sort the input into decreasing order. Note that @kbd{V S} and @kbd{V G}
19923 use a ``stable'' sorting algorithm, i.e., any two elements which are equal
19924 will not be moved out of their original order. Generally there is no way
19925 to tell with @kbd{V S}, since two elements which are equal look the same,
19926 but with @kbd{V G} this can be an important issue. In the matrix-of-rows
19927 example, suppose you have names and telephone numbers as two columns and
19928 you wish to sort by phone number primarily, and by name when the numbers
19929 are equal. You can sort the data matrix by names first, and then again
19930 by phone numbers. Because the sort is stable, any two rows with equal
19931 phone numbers will remain sorted by name even after the second sort.
19935 @pindex calc-histogram
19937 @mindex histo@idots
19940 The @kbd{V H} (@code{calc-histogram}) [@code{histogram}] command builds a
19941 histogram of a vector of numbers. Vector elements are assumed to be
19942 integers or real numbers in the range [0..@var{n}) for some ``number of
19943 bins'' @var{n}, which is the numeric prefix argument given to the
19944 command. The result is a vector of @var{n} counts of how many times
19945 each value appeared in the original vector. Non-integers in the input
19946 are rounded down to integers. Any vector elements outside the specified
19947 range are ignored. (You can tell if elements have been ignored by noting
19948 that the counts in the result vector don't add up to the length of the
19952 With the Hyperbolic flag, @kbd{H V H} pulls two vectors from the stack.
19953 The second-to-top vector is the list of numbers as before. The top
19954 vector is an equal-sized list of ``weights'' to attach to the elements
19955 of the data vector. For example, if the first data element is 4.2 and
19956 the first weight is 10, then 10 will be added to bin 4 of the result
19957 vector. Without the hyperbolic flag, every element has a weight of one.
19960 @pindex calc-transpose
19962 The @kbd{v t} (@code{calc-transpose}) [@code{trn}] command computes
19963 the transpose of the matrix at the top of the stack. If the argument
19964 is a plain vector, it is treated as a row vector and transposed into
19965 a one-column matrix.
19968 @pindex calc-reverse-vector
19970 The @kbd{v v} (@code{calc-reverse-vector}) [@code{rev}] command reverses
19971 a vector end-for-end. Given a matrix, it reverses the order of the rows.
19972 (To reverse the columns instead, just use @kbd{v t v v v t}. The same
19973 principle can be used to apply other vector commands to the columns of
19977 @pindex calc-mask-vector
19979 The @kbd{v m} (@code{calc-mask-vector}) [@code{vmask}] command uses
19980 one vector as a mask to extract elements of another vector. The mask
19981 is in the second-to-top position; the target vector is on the top of
19982 the stack. These vectors must have the same length. The result is
19983 the same as the target vector, but with all elements which correspond
19984 to zeros in the mask vector deleted. Thus, for example,
19985 @samp{vmask([1, 0, 1, 0, 1], [a, b, c, d, e])} produces @samp{[a, c, e]}.
19986 @xref{Logical Operations}.
19989 @pindex calc-expand-vector
19991 The @kbd{v e} (@code{calc-expand-vector}) [@code{vexp}] command
19992 expands a vector according to another mask vector. The result is a
19993 vector the same length as the mask, but with nonzero elements replaced
19994 by successive elements from the target vector. The length of the target
19995 vector is normally the number of nonzero elements in the mask. If the
19996 target vector is longer, its last few elements are lost. If the target
19997 vector is shorter, the last few nonzero mask elements are left
19998 unreplaced in the result. Thus @samp{vexp([2, 0, 3, 0, 7], [a, b])}
19999 produces @samp{[a, 0, b, 0, 7]}.
20002 With the Hyperbolic flag, @kbd{H v e} takes a filler value from the
20003 top of the stack; the mask and target vectors come from the third and
20004 second elements of the stack. This filler is used where the mask is
20005 zero: @samp{vexp([2, 0, 3, 0, 7], [a, b], z)} produces
20006 @samp{[a, z, c, z, 7]}. If the filler value is itself a vector,
20007 then successive values are taken from it, so that the effect is to
20008 interleave two vectors according to the mask:
20009 @samp{vexp([2, 0, 3, 7, 0, 0], [a, b], [x, y])} produces
20010 @samp{[a, x, b, 7, y, 0]}.
20012 Another variation on the masking idea is to combine @samp{[a, b, c, d, e]}
20013 with the mask @samp{[1, 0, 1, 0, 1]} to produce @samp{[a, 0, c, 0, e]}.
20014 You can accomplish this with @kbd{V M a &}, mapping the logical ``and''
20015 operation across the two vectors. @xref{Logical Operations}. Note that
20016 the @code{? :} operation also discussed there allows other types of
20017 masking using vectors.
20019 @node Vector and Matrix Arithmetic, Set Operations, Manipulating Vectors, Matrix Functions
20020 @section Vector and Matrix Arithmetic
20023 Basic arithmetic operations like addition and multiplication are defined
20024 for vectors and matrices as well as for numbers. Division of matrices, in
20025 the sense of multiplying by the inverse, is supported. (Division by a
20026 matrix actually uses LU-decomposition for greater accuracy and speed.)
20027 @xref{Basic Arithmetic}.
20029 The following functions are applied element-wise if their arguments are
20030 vectors or matrices: @code{change-sign}, @code{conj}, @code{arg},
20031 @code{re}, @code{im}, @code{polar}, @code{rect}, @code{clean},
20032 @code{float}, @code{frac}. @xref{Function Index}.
20035 @pindex calc-conj-transpose
20037 The @kbd{V J} (@code{calc-conj-transpose}) [@code{ctrn}] command computes
20038 the conjugate transpose of its argument, i.e., @samp{conj(trn(x))}.
20043 @kindex A (vectors)
20044 @pindex calc-abs (vectors)
20048 @tindex abs (vectors)
20049 The @kbd{A} (@code{calc-abs}) [@code{abs}] command computes the
20050 Frobenius norm of a vector or matrix argument. This is the square
20051 root of the sum of the squares of the absolute values of the
20052 elements of the vector or matrix. If the vector is interpreted as
20053 a point in two- or three-dimensional space, this is the distance
20054 from that point to the origin.
20059 The @kbd{v n} (@code{calc-rnorm}) [@code{rnorm}] command computes the
20060 infinity-norm of a vector, or the row norm of a matrix. For a plain
20061 vector, this is the maximum of the absolute values of the elements. For
20062 a matrix, this is the maximum of the row-absolute-value-sums, i.e., of
20063 the sums of the absolute values of the elements along the various rows.
20068 The @kbd{V N} (@code{calc-cnorm}) [@code{cnorm}] command computes
20069 the one-norm of a vector, or column norm of a matrix. For a plain
20070 vector, this is the sum of the absolute values of the elements.
20071 For a matrix, this is the maximum of the column-absolute-value-sums.
20072 General @expr{k}-norms for @expr{k} other than one or infinity are
20073 not provided. However, the 2-norm (or Frobenius norm) is provided for
20074 vectors by the @kbd{A} (@code{calc-abs}) command.
20079 The @kbd{V C} (@code{calc-cross}) [@code{cross}] command computes the
20080 right-handed cross product of two vectors, each of which must have
20081 exactly three elements.
20086 @kindex & (matrices)
20087 @pindex calc-inv (matrices)
20091 @tindex inv (matrices)
20092 The @kbd{&} (@code{calc-inv}) [@code{inv}] command computes the
20093 inverse of a square matrix. If the matrix is singular, the inverse
20094 operation is left in symbolic form. Matrix inverses are recorded so
20095 that once an inverse (or determinant) of a particular matrix has been
20096 computed, the inverse and determinant of the matrix can be recomputed
20097 quickly in the future.
20099 If the argument to @kbd{&} is a plain number @expr{x}, this
20100 command simply computes @expr{1/x}. This is okay, because the
20101 @samp{/} operator also does a matrix inversion when dividing one
20107 The @kbd{V D} (@code{calc-mdet}) [@code{det}] command computes the
20108 determinant of a square matrix.
20113 The @kbd{V L} (@code{calc-mlud}) [@code{lud}] command computes the
20114 LU decomposition of a matrix. The result is a list of three matrices
20115 which, when multiplied together left-to-right, form the original matrix.
20116 The first is a permutation matrix that arises from pivoting in the
20117 algorithm, the second is lower-triangular with ones on the diagonal,
20118 and the third is upper-triangular.
20121 @pindex calc-mtrace
20123 The @kbd{V T} (@code{calc-mtrace}) [@code{tr}] command computes the
20124 trace of a square matrix. This is defined as the sum of the diagonal
20125 elements of the matrix.
20130 The @kbd{V K} (@code{calc-kron}) [@code{kron}] command computes
20131 the Kronecker product of two matrices.
20133 @node Set Operations, Statistical Operations, Vector and Matrix Arithmetic, Matrix Functions
20134 @section Set Operations using Vectors
20137 @cindex Sets, as vectors
20138 Calc includes several commands which interpret vectors as @dfn{sets} of
20139 objects. A set is a collection of objects; any given object can appear
20140 only once in the set. Calc stores sets as vectors of objects in
20141 sorted order. Objects in a Calc set can be any of the usual things,
20142 such as numbers, variables, or formulas. Two set elements are considered
20143 equal if they are identical, except that numerically equal numbers like
20144 the integer 4 and the float 4.0 are considered equal even though they
20145 are not ``identical.'' Variables are treated like plain symbols without
20146 attached values by the set operations; subtracting the set @samp{[b]}
20147 from @samp{[a, b]} always yields the set @samp{[a]} even though if
20148 the variables @samp{a} and @samp{b} both equaled 17, you might
20149 expect the answer @samp{[]}.
20151 If a set contains interval forms, then it is assumed to be a set of
20152 real numbers. In this case, all set operations require the elements
20153 of the set to be only things that are allowed in intervals: Real
20154 numbers, plus and minus infinity, HMS forms, and date forms. If
20155 there are variables or other non-real objects present in a real set,
20156 all set operations on it will be left in unevaluated form.
20158 If the input to a set operation is a plain number or interval form
20159 @var{a}, it is treated like the one-element vector @samp{[@var{a}]}.
20160 The result is always a vector, except that if the set consists of a
20161 single interval, the interval itself is returned instead.
20163 @xref{Logical Operations}, for the @code{in} function which tests if
20164 a certain value is a member of a given set. To test if the set @expr{A}
20165 is a subset of the set @expr{B}, use @samp{vdiff(A, B) = []}.
20168 @pindex calc-remove-duplicates
20170 The @kbd{V +} (@code{calc-remove-duplicates}) [@code{rdup}] command
20171 converts an arbitrary vector into set notation. It works by sorting
20172 the vector as if by @kbd{V S}, then removing duplicates. (For example,
20173 @kbd{[a, 5, 4, a, 4.0]} is sorted to @samp{[4, 4.0, 5, a, a]} and then
20174 reduced to @samp{[4, 5, a]}). Overlapping intervals are merged as
20175 necessary. You rarely need to use @kbd{V +} explicitly, since all the
20176 other set-based commands apply @kbd{V +} to their inputs before using
20180 @pindex calc-set-union
20182 The @kbd{V V} (@code{calc-set-union}) [@code{vunion}] command computes
20183 the union of two sets. An object is in the union of two sets if and
20184 only if it is in either (or both) of the input sets. (You could
20185 accomplish the same thing by concatenating the sets with @kbd{|},
20186 then using @kbd{V +}.)
20189 @pindex calc-set-intersect
20191 The @kbd{V ^} (@code{calc-set-intersect}) [@code{vint}] command computes
20192 the intersection of two sets. An object is in the intersection if
20193 and only if it is in both of the input sets. Thus if the input
20194 sets are disjoint, i.e., if they share no common elements, the result
20195 will be the empty vector @samp{[]}. Note that the characters @kbd{V}
20196 and @kbd{^} were chosen to be close to the conventional mathematical
20198 @texline union@tie{}(@math{A \cup B})
20201 @texline intersection@tie{}(@math{A \cap B}).
20202 @infoline intersection.
20205 @pindex calc-set-difference
20207 The @kbd{V -} (@code{calc-set-difference}) [@code{vdiff}] command computes
20208 the difference between two sets. An object is in the difference
20209 @expr{A - B} if and only if it is in @expr{A} but not in @expr{B}.
20210 Thus subtracting @samp{[y,z]} from a set will remove the elements
20211 @samp{y} and @samp{z} if they are present. You can also think of this
20212 as a general @dfn{set complement} operator; if @expr{A} is the set of
20213 all possible values, then @expr{A - B} is the ``complement'' of @expr{B}.
20214 Obviously this is only practical if the set of all possible values in
20215 your problem is small enough to list in a Calc vector (or simple
20216 enough to express in a few intervals).
20219 @pindex calc-set-xor
20221 The @kbd{V X} (@code{calc-set-xor}) [@code{vxor}] command computes
20222 the ``exclusive-or,'' or ``symmetric difference'' of two sets.
20223 An object is in the symmetric difference of two sets if and only
20224 if it is in one, but @emph{not} both, of the sets. Objects that
20225 occur in both sets ``cancel out.''
20228 @pindex calc-set-complement
20230 The @kbd{V ~} (@code{calc-set-complement}) [@code{vcompl}] command
20231 computes the complement of a set with respect to the real numbers.
20232 Thus @samp{vcompl(x)} is equivalent to @samp{vdiff([-inf .. inf], x)}.
20233 For example, @samp{vcompl([2, (3 .. 4]])} evaluates to
20234 @samp{[[-inf .. 2), (2 .. 3], (4 .. inf]]}.
20237 @pindex calc-set-floor
20239 The @kbd{V F} (@code{calc-set-floor}) [@code{vfloor}] command
20240 reinterprets a set as a set of integers. Any non-integer values,
20241 and intervals that do not enclose any integers, are removed. Open
20242 intervals are converted to equivalent closed intervals. Successive
20243 integers are converted into intervals of integers. For example, the
20244 complement of the set @samp{[2, 6, 7, 8]} is messy, but if you wanted
20245 the complement with respect to the set of integers you could type
20246 @kbd{V ~ V F} to get @samp{[[-inf .. 1], [3 .. 5], [9 .. inf]]}.
20249 @pindex calc-set-enumerate
20251 The @kbd{V E} (@code{calc-set-enumerate}) [@code{venum}] command
20252 converts a set of integers into an explicit vector. Intervals in
20253 the set are expanded out to lists of all integers encompassed by
20254 the intervals. This only works for finite sets (i.e., sets which
20255 do not involve @samp{-inf} or @samp{inf}).
20258 @pindex calc-set-span
20260 The @kbd{V :} (@code{calc-set-span}) [@code{vspan}] command converts any
20261 set of reals into an interval form that encompasses all its elements.
20262 The lower limit will be the smallest element in the set; the upper
20263 limit will be the largest element. For an empty set, @samp{vspan([])}
20264 returns the empty interval @w{@samp{[0 .. 0)}}.
20267 @pindex calc-set-cardinality
20269 The @kbd{V #} (@code{calc-set-cardinality}) [@code{vcard}] command counts
20270 the number of integers in a set. The result is the length of the vector
20271 that would be produced by @kbd{V E}, although the computation is much
20272 more efficient than actually producing that vector.
20274 @cindex Sets, as binary numbers
20275 Another representation for sets that may be more appropriate in some
20276 cases is binary numbers. If you are dealing with sets of integers
20277 in the range 0 to 49, you can use a 50-bit binary number where a
20278 particular bit is 1 if the corresponding element is in the set.
20279 @xref{Binary Functions}, for a list of commands that operate on
20280 binary numbers. Note that many of the above set operations have
20281 direct equivalents in binary arithmetic: @kbd{b o} (@code{calc-or}),
20282 @kbd{b a} (@code{calc-and}), @kbd{b d} (@code{calc-diff}),
20283 @kbd{b x} (@code{calc-xor}), and @kbd{b n} (@code{calc-not}),
20284 respectively. You can use whatever representation for sets is most
20289 @pindex calc-pack-bits
20290 @pindex calc-unpack-bits
20293 The @kbd{b u} (@code{calc-unpack-bits}) [@code{vunpack}] command
20294 converts an integer that represents a set in binary into a set
20295 in vector/interval notation. For example, @samp{vunpack(67)}
20296 returns @samp{[[0 .. 1], 6]}. If the input is negative, the set
20297 it represents is semi-infinite: @samp{vunpack(-4) = [2 .. inf)}.
20298 Use @kbd{V E} afterwards to expand intervals to individual
20299 values if you wish. Note that this command uses the @kbd{b}
20300 (binary) prefix key.
20302 The @kbd{b p} (@code{calc-pack-bits}) [@code{vpack}] command
20303 converts the other way, from a vector or interval representing
20304 a set of nonnegative integers into a binary integer describing
20305 the same set. The set may include positive infinity, but must
20306 not include any negative numbers. The input is interpreted as a
20307 set of integers in the sense of @kbd{V F} (@code{vfloor}). Beware
20308 that a simple input like @samp{[100]} can result in a huge integer
20310 @texline (@math{2^{100}}, a 31-digit integer, in this case).
20311 @infoline (@expr{2^100}, a 31-digit integer, in this case).
20313 @node Statistical Operations, Reducing and Mapping, Set Operations, Matrix Functions
20314 @section Statistical Operations on Vectors
20317 @cindex Statistical functions
20318 The commands in this section take vectors as arguments and compute
20319 various statistical measures on the data stored in the vectors. The
20320 references used in the definitions of these functions are Bevington's
20321 @emph{Data Reduction and Error Analysis for the Physical Sciences},
20322 and @emph{Numerical Recipes} by Press, Flannery, Teukolsky and
20325 The statistical commands use the @kbd{u} prefix key followed by
20326 a shifted letter or other character.
20328 @xref{Manipulating Vectors}, for a description of @kbd{V H}
20329 (@code{calc-histogram}).
20331 @xref{Curve Fitting}, for the @kbd{a F} command for doing
20332 least-squares fits to statistical data.
20334 @xref{Probability Distribution Functions}, for several common
20335 probability distribution functions.
20338 * Single-Variable Statistics::
20339 * Paired-Sample Statistics::
20342 @node Single-Variable Statistics, Paired-Sample Statistics, Statistical Operations, Statistical Operations
20343 @subsection Single-Variable Statistics
20346 These functions do various statistical computations on single
20347 vectors. Given a numeric prefix argument, they actually pop
20348 @var{n} objects from the stack and combine them into a data
20349 vector. Each object may be either a number or a vector; if a
20350 vector, any sub-vectors inside it are ``flattened'' as if by
20351 @kbd{v a 0}; @pxref{Manipulating Vectors}. By default one object
20352 is popped, which (in order to be useful) is usually a vector.
20354 If an argument is a variable name, and the value stored in that
20355 variable is a vector, then the stored vector is used. This method
20356 has the advantage that if your data vector is large, you can avoid
20357 the slow process of manipulating it directly on the stack.
20359 These functions are left in symbolic form if any of their arguments
20360 are not numbers or vectors, e.g., if an argument is a formula, or
20361 a non-vector variable. However, formulas embedded within vector
20362 arguments are accepted; the result is a symbolic representation
20363 of the computation, based on the assumption that the formula does
20364 not itself represent a vector. All varieties of numbers such as
20365 error forms and interval forms are acceptable.
20367 Some of the functions in this section also accept a single error form
20368 or interval as an argument. They then describe a property of the
20369 normal or uniform (respectively) statistical distribution described
20370 by the argument. The arguments are interpreted in the same way as
20371 the @var{M} argument of the random number function @kbd{k r}. In
20372 particular, an interval with integer limits is considered an integer
20373 distribution, so that @samp{[2 .. 6)} is the same as @samp{[2 .. 5]}.
20374 An interval with at least one floating-point limit is a continuous
20375 distribution: @samp{[2.0 .. 6.0)} is @emph{not} the same as
20376 @samp{[2.0 .. 5.0]}!
20379 @pindex calc-vector-count
20381 The @kbd{u #} (@code{calc-vector-count}) [@code{vcount}] command
20382 computes the number of data values represented by the inputs.
20383 For example, @samp{vcount(1, [2, 3], [[4, 5], [], x, y])} returns 7.
20384 If the argument is a single vector with no sub-vectors, this
20385 simply computes the length of the vector.
20389 @pindex calc-vector-sum
20390 @pindex calc-vector-prod
20393 @cindex Summations (statistical)
20394 The @kbd{u +} (@code{calc-vector-sum}) [@code{vsum}] command
20395 computes the sum of the data values. The @kbd{u *}
20396 (@code{calc-vector-prod}) [@code{vprod}] command computes the
20397 product of the data values. If the input is a single flat vector,
20398 these are the same as @kbd{V R +} and @kbd{V R *}
20399 (@pxref{Reducing and Mapping}).
20403 @pindex calc-vector-max
20404 @pindex calc-vector-min
20407 The @kbd{u X} (@code{calc-vector-max}) [@code{vmax}] command
20408 computes the maximum of the data values, and the @kbd{u N}
20409 (@code{calc-vector-min}) [@code{vmin}] command computes the minimum.
20410 If the argument is an interval, this finds the minimum or maximum
20411 value in the interval. (Note that @samp{vmax([2..6)) = 5} as
20412 described above.) If the argument is an error form, this returns
20413 plus or minus infinity.
20416 @pindex calc-vector-mean
20418 @cindex Mean of data values
20419 The @kbd{u M} (@code{calc-vector-mean}) [@code{vmean}] command
20420 computes the average (arithmetic mean) of the data values.
20421 If the inputs are error forms
20422 @texline @math{x \pm \sigma},
20423 @infoline @samp{x +/- s},
20424 this is the weighted mean of the @expr{x} values with weights
20425 @texline @math{1 /\sigma^2}.
20426 @infoline @expr{1 / s^2}.
20429 $$ \mu = { \displaystyle \sum { x_i \over \sigma_i^2 } \over
20430 \displaystyle \sum { 1 \over \sigma_i^2 } } $$
20432 If the inputs are not error forms, this is simply the sum of the
20433 values divided by the count of the values.
20435 Note that a plain number can be considered an error form with
20437 @texline @math{\sigma = 0}.
20438 @infoline @expr{s = 0}.
20439 If the input to @kbd{u M} is a mixture of
20440 plain numbers and error forms, the result is the mean of the
20441 plain numbers, ignoring all values with non-zero errors. (By the
20442 above definitions it's clear that a plain number effectively
20443 has an infinite weight, next to which an error form with a finite
20444 weight is completely negligible.)
20446 This function also works for distributions (error forms or
20447 intervals). The mean of an error form `@var{a} @tfn{+/-} @var{b}' is simply
20448 @expr{a}. The mean of an interval is the mean of the minimum
20449 and maximum values of the interval.
20452 @pindex calc-vector-mean-error
20454 The @kbd{I u M} (@code{calc-vector-mean-error}) [@code{vmeane}]
20455 command computes the mean of the data points expressed as an
20456 error form. This includes the estimated error associated with
20457 the mean. If the inputs are error forms, the error is the square
20458 root of the reciprocal of the sum of the reciprocals of the squares
20459 of the input errors. (I.e., the variance is the reciprocal of the
20460 sum of the reciprocals of the variances.)
20463 $$ \sigma_\mu^2 = {1 \over \displaystyle \sum {1 \over \sigma_i^2}} $$
20465 If the inputs are plain
20466 numbers, the error is equal to the standard deviation of the values
20467 divided by the square root of the number of values. (This works
20468 out to be equivalent to calculating the standard deviation and
20469 then assuming each value's error is equal to this standard
20473 $$ \sigma_\mu^2 = {\sigma^2 \over N} $$
20477 @pindex calc-vector-median
20479 @cindex Median of data values
20480 The @kbd{H u M} (@code{calc-vector-median}) [@code{vmedian}]
20481 command computes the median of the data values. The values are
20482 first sorted into numerical order; the median is the middle
20483 value after sorting. (If the number of data values is even,
20484 the median is taken to be the average of the two middle values.)
20485 The median function is different from the other functions in
20486 this section in that the arguments must all be real numbers;
20487 variables are not accepted even when nested inside vectors.
20488 (Otherwise it is not possible to sort the data values.) If
20489 any of the input values are error forms, their error parts are
20492 The median function also accepts distributions. For both normal
20493 (error form) and uniform (interval) distributions, the median is
20494 the same as the mean.
20497 @pindex calc-vector-harmonic-mean
20499 @cindex Harmonic mean
20500 The @kbd{H I u M} (@code{calc-vector-harmonic-mean}) [@code{vhmean}]
20501 command computes the harmonic mean of the data values. This is
20502 defined as the reciprocal of the arithmetic mean of the reciprocals
20506 $$ { N \over \displaystyle \sum {1 \over x_i} } $$
20510 @pindex calc-vector-geometric-mean
20512 @cindex Geometric mean
20513 The @kbd{u G} (@code{calc-vector-geometric-mean}) [@code{vgmean}]
20514 command computes the geometric mean of the data values. This
20515 is the @var{n}th root of the product of the values. This is also
20516 equal to the @code{exp} of the arithmetic mean of the logarithms
20517 of the data values.
20520 $$ \exp \left ( \sum { \ln x_i } \right ) =
20521 \left ( \prod { x_i } \right)^{1 / N} $$
20526 The @kbd{H u G} [@code{agmean}] command computes the ``arithmetic-geometric
20527 mean'' of two numbers taken from the stack. This is computed by
20528 replacing the two numbers with their arithmetic mean and geometric
20529 mean, then repeating until the two values converge.
20532 $$ a_{i+1} = { a_i + b_i \over 2 } , \qquad b_{i+1} = \sqrt{a_i b_i} $$
20535 @cindex Root-mean-square
20536 Another commonly used mean, the RMS (root-mean-square), can be computed
20537 for a vector of numbers simply by using the @kbd{A} command.
20540 @pindex calc-vector-sdev
20542 @cindex Standard deviation
20543 @cindex Sample statistics
20544 The @kbd{u S} (@code{calc-vector-sdev}) [@code{vsdev}] command
20545 computes the standard
20546 @texline deviation@tie{}@math{\sigma}
20547 @infoline deviation
20548 of the data values. If the values are error forms, the errors are used
20549 as weights just as for @kbd{u M}. This is the @emph{sample} standard
20550 deviation, whose value is the square root of the sum of the squares of
20551 the differences between the values and the mean of the @expr{N} values,
20552 divided by @expr{N-1}.
20555 $$ \sigma^2 = {1 \over N - 1} \sum (x_i - \mu)^2 $$
20558 This function also applies to distributions. The standard deviation
20559 of a single error form is simply the error part. The standard deviation
20560 of a continuous interval happens to equal the difference between the
20562 @texline @math{\sqrt{12}}.
20563 @infoline @expr{sqrt(12)}.
20564 The standard deviation of an integer interval is the same as the
20565 standard deviation of a vector of those integers.
20568 @pindex calc-vector-pop-sdev
20570 @cindex Population statistics
20571 The @kbd{I u S} (@code{calc-vector-pop-sdev}) [@code{vpsdev}]
20572 command computes the @emph{population} standard deviation.
20573 It is defined by the same formula as above but dividing
20574 by @expr{N} instead of by @expr{N-1}. The population standard
20575 deviation is used when the input represents the entire set of
20576 data values in the distribution; the sample standard deviation
20577 is used when the input represents a sample of the set of all
20578 data values, so that the mean computed from the input is itself
20579 only an estimate of the true mean.
20582 $$ \sigma^2 = {1 \over N} \sum (x_i - \mu)^2 $$
20585 For error forms and continuous intervals, @code{vpsdev} works
20586 exactly like @code{vsdev}. For integer intervals, it computes the
20587 population standard deviation of the equivalent vector of integers.
20591 @pindex calc-vector-variance
20592 @pindex calc-vector-pop-variance
20595 @cindex Variance of data values
20596 The @kbd{H u S} (@code{calc-vector-variance}) [@code{vvar}] and
20597 @kbd{H I u S} (@code{calc-vector-pop-variance}) [@code{vpvar}]
20598 commands compute the variance of the data values. The variance
20600 @texline square@tie{}@math{\sigma^2}
20602 of the standard deviation, i.e., the sum of the
20603 squares of the deviations of the data values from the mean.
20604 (This definition also applies when the argument is a distribution.)
20610 The @code{vflat} algebraic function returns a vector of its
20611 arguments, interpreted in the same way as the other functions
20612 in this section. For example, @samp{vflat(1, [2, [3, 4]], 5)}
20613 returns @samp{[1, 2, 3, 4, 5]}.
20615 @node Paired-Sample Statistics, , Single-Variable Statistics, Statistical Operations
20616 @subsection Paired-Sample Statistics
20619 The functions in this section take two arguments, which must be
20620 vectors of equal size. The vectors are each flattened in the same
20621 way as by the single-variable statistical functions. Given a numeric
20622 prefix argument of 1, these functions instead take one object from
20623 the stack, which must be an
20624 @texline @math{N\times2}
20626 matrix of data values. Once again, variable names can be used in place
20627 of actual vectors and matrices.
20630 @pindex calc-vector-covariance
20633 The @kbd{u C} (@code{calc-vector-covariance}) [@code{vcov}] command
20634 computes the sample covariance of two vectors. The covariance
20635 of vectors @var{x} and @var{y} is the sum of the products of the
20636 differences between the elements of @var{x} and the mean of @var{x}
20637 times the differences between the corresponding elements of @var{y}
20638 and the mean of @var{y}, all divided by @expr{N-1}. Note that
20639 the variance of a vector is just the covariance of the vector
20640 with itself. Once again, if the inputs are error forms the
20641 errors are used as weight factors. If both @var{x} and @var{y}
20642 are composed of error forms, the error for a given data point
20643 is taken as the square root of the sum of the squares of the two
20647 $$ \sigma_{x\!y}^2 = {1 \over N-1} \sum (x_i - \mu_x) (y_i - \mu_y) $$
20648 $$ \sigma_{x\!y}^2 =
20649 {\displaystyle {1 \over N-1}
20650 \sum {(x_i - \mu_x) (y_i - \mu_y) \over \sigma_i^2}
20651 \over \displaystyle {1 \over N} \sum {1 \over \sigma_i^2}}
20656 @pindex calc-vector-pop-covariance
20658 The @kbd{I u C} (@code{calc-vector-pop-covariance}) [@code{vpcov}]
20659 command computes the population covariance, which is the same as the
20660 sample covariance computed by @kbd{u C} except dividing by @expr{N}
20661 instead of @expr{N-1}.
20664 @pindex calc-vector-correlation
20666 @cindex Correlation coefficient
20667 @cindex Linear correlation
20668 The @kbd{H u C} (@code{calc-vector-correlation}) [@code{vcorr}]
20669 command computes the linear correlation coefficient of two vectors.
20670 This is defined by the covariance of the vectors divided by the
20671 product of their standard deviations. (There is no difference
20672 between sample or population statistics here.)
20675 $$ r_{x\!y} = { \sigma_{x\!y}^2 \over \sigma_x^2 \sigma_y^2 } $$
20678 @node Reducing and Mapping, Vector and Matrix Formats, Statistical Operations, Matrix Functions
20679 @section Reducing and Mapping Vectors
20682 The commands in this section allow for more general operations on the
20683 elements of vectors.
20688 The simplest of these operations is @kbd{V A} (@code{calc-apply})
20689 [@code{apply}], which applies a given operator to the elements of a vector.
20690 For example, applying the hypothetical function @code{f} to the vector
20691 @w{@samp{[1, 2, 3]}} would produce the function call @samp{f(1, 2, 3)}.
20692 Applying the @code{+} function to the vector @samp{[a, b]} gives
20693 @samp{a + b}. Applying @code{+} to the vector @samp{[a, b, c]} is an
20694 error, since the @code{+} function expects exactly two arguments.
20696 While @kbd{V A} is useful in some cases, you will usually find that either
20697 @kbd{V R} or @kbd{V M}, described below, is closer to what you want.
20700 * Specifying Operators::
20703 * Nesting and Fixed Points::
20704 * Generalized Products::
20707 @node Specifying Operators, Mapping, Reducing and Mapping, Reducing and Mapping
20708 @subsection Specifying Operators
20711 Commands in this section (like @kbd{V A}) prompt you to press the key
20712 corresponding to the desired operator. Press @kbd{?} for a partial
20713 list of the available operators. Generally, an operator is any key or
20714 sequence of keys that would normally take one or more arguments from
20715 the stack and replace them with a result. For example, @kbd{V A H C}
20716 uses the hyperbolic cosine operator, @code{cosh}. (Since @code{cosh}
20717 expects one argument, @kbd{V A H C} requires a vector with a single
20718 element as its argument.)
20720 You can press @kbd{x} at the operator prompt to select any algebraic
20721 function by name to use as the operator. This includes functions you
20722 have defined yourself using the @kbd{Z F} command. (@xref{Algebraic
20723 Definitions}.) If you give a name for which no function has been
20724 defined, the result is left in symbolic form, as in @samp{f(1, 2, 3)}.
20725 Calc will prompt for the number of arguments the function takes if it
20726 can't figure it out on its own (say, because you named a function that
20727 is currently undefined). It is also possible to type a digit key before
20728 the function name to specify the number of arguments, e.g.,
20729 @kbd{V M 3 x f @key{RET}} calls @code{f} with three arguments even if it
20730 looks like it ought to have only two. This technique may be necessary
20731 if the function allows a variable number of arguments. For example,
20732 the @kbd{v e} [@code{vexp}] function accepts two or three arguments;
20733 if you want to map with the three-argument version, you will have to
20734 type @kbd{V M 3 v e}.
20736 It is also possible to apply any formula to a vector by treating that
20737 formula as a function. When prompted for the operator to use, press
20738 @kbd{'} (the apostrophe) and type your formula as an algebraic entry.
20739 You will then be prompted for the argument list, which defaults to a
20740 list of all variables that appear in the formula, sorted into alphabetic
20741 order. For example, suppose you enter the formula @w{@samp{x + 2y^x}}.
20742 The default argument list would be @samp{(x y)}, which means that if
20743 this function is applied to the arguments @samp{[3, 10]} the result will
20744 be @samp{3 + 2*10^3}. (If you plan to use a certain formula in this
20745 way often, you might consider defining it as a function with @kbd{Z F}.)
20747 Another way to specify the arguments to the formula you enter is with
20748 @kbd{$}, @kbd{$$}, and so on. For example, @kbd{V A ' $$ + 2$^$$}
20749 has the same effect as the previous example. The argument list is
20750 automatically taken to be @samp{($$ $)}. (The order of the arguments
20751 may seem backwards, but it is analogous to the way normal algebraic
20752 entry interacts with the stack.)
20754 If you press @kbd{$} at the operator prompt, the effect is similar to
20755 the apostrophe except that the relevant formula is taken from top-of-stack
20756 instead. The actual vector arguments of the @kbd{V A $} or related command
20757 then start at the second-to-top stack position. You will still be
20758 prompted for an argument list.
20760 @cindex Nameless functions
20761 @cindex Generic functions
20762 A function can be written without a name using the notation @samp{<#1 - #2>},
20763 which means ``a function of two arguments that computes the first
20764 argument minus the second argument.'' The symbols @samp{#1} and @samp{#2}
20765 are placeholders for the arguments. You can use any names for these
20766 placeholders if you wish, by including an argument list followed by a
20767 colon: @samp{<x, y : x - y>}. When you type @kbd{V A ' $$ + 2$^$$ @key{RET}},
20768 Calc builds the nameless function @samp{<#1 + 2 #2^#1>} as the function
20769 to map across the vectors. When you type @kbd{V A ' x + 2y^x @key{RET} @key{RET}},
20770 Calc builds the nameless function @w{@samp{<x, y : x + 2 y^x>}}. In both
20771 cases, Calc also writes the nameless function to the Trail so that you
20772 can get it back later if you wish.
20774 If there is only one argument, you can write @samp{#} in place of @samp{#1}.
20775 (Note that @samp{< >} notation is also used for date forms. Calc tells
20776 that @samp{<@var{stuff}>} is a nameless function by the presence of
20777 @samp{#} signs inside @var{stuff}, or by the fact that @var{stuff}
20778 begins with a list of variables followed by a colon.)
20780 You can type a nameless function directly to @kbd{V A '}, or put one on
20781 the stack and use it with @w{@kbd{V A $}}. Calc will not prompt for an
20782 argument list in this case, since the nameless function specifies the
20783 argument list as well as the function itself. In @kbd{V A '}, you can
20784 omit the @samp{< >} marks if you use @samp{#} notation for the arguments,
20785 so that @kbd{V A ' #1+#2 @key{RET}} is the same as @kbd{V A ' <#1+#2> @key{RET}},
20786 which in turn is the same as @kbd{V A ' $$+$ @key{RET}}.
20788 @cindex Lambda expressions
20793 The internal format for @samp{<x, y : x + y>} is @samp{lambda(x, y, x + y)}.
20794 (The word @code{lambda} derives from Lisp notation and the theory of
20795 functions.) The internal format for @samp{<#1 + #2>} is @samp{lambda(ArgA,
20796 ArgB, ArgA + ArgB)}. Note that there is no actual Calc function called
20797 @code{lambda}; the whole point is that the @code{lambda} expression is
20798 used in its symbolic form, not evaluated for an answer until it is applied
20799 to specific arguments by a command like @kbd{V A} or @kbd{V M}.
20801 (Actually, @code{lambda} does have one special property: Its arguments
20802 are never evaluated; for example, putting @samp{<(2/3) #>} on the stack
20803 will not simplify the @samp{2/3} until the nameless function is actually
20832 As usual, commands like @kbd{V A} have algebraic function name equivalents.
20833 For example, @kbd{V A k g} with an argument of @samp{v} is equivalent to
20834 @samp{apply(gcd, v)}. The first argument specifies the operator name,
20835 and is either a variable whose name is the same as the function name,
20836 or a nameless function like @samp{<#^3+1>}. Operators that are normally
20837 written as algebraic symbols have the names @code{add}, @code{sub},
20838 @code{mul}, @code{div}, @code{pow}, @code{neg}, @code{mod}, and
20845 The @code{call} function builds a function call out of several arguments:
20846 @samp{call(gcd, x, y)} is the same as @samp{apply(gcd, [x, y])}, which
20847 in turn is the same as @samp{gcd(x, y)}. The first argument of @code{call},
20848 like the other functions described here, may be either a variable naming a
20849 function, or a nameless function (@samp{call(<#1+2#2>, x, y)} is the same
20852 (Experts will notice that it's not quite proper to use a variable to name
20853 a function, since the name @code{gcd} corresponds to the Lisp variable
20854 @code{var-gcd} but to the Lisp function @code{calcFunc-gcd}. Calc
20855 automatically makes this translation, so you don't have to worry
20858 @node Mapping, Reducing, Specifying Operators, Reducing and Mapping
20859 @subsection Mapping
20865 The @kbd{V M} (@code{calc-map}) [@code{map}] command applies a given
20866 operator elementwise to one or more vectors. For example, mapping
20867 @code{A} [@code{abs}] produces a vector of the absolute values of the
20868 elements in the input vector. Mapping @code{+} pops two vectors from
20869 the stack, which must be of equal length, and produces a vector of the
20870 pairwise sums of the elements. If either argument is a non-vector, it
20871 is duplicated for each element of the other vector. For example,
20872 @kbd{[1,2,3] 2 V M ^} squares the elements of the specified vector.
20873 With the 2 listed first, it would have computed a vector of powers of
20874 two. Mapping a user-defined function pops as many arguments from the
20875 stack as the function requires. If you give an undefined name, you will
20876 be prompted for the number of arguments to use.
20878 If any argument to @kbd{V M} is a matrix, the operator is normally mapped
20879 across all elements of the matrix. For example, given the matrix
20880 @expr{[[1, -2, 3], [-4, 5, -6]]}, @kbd{V M A} takes six absolute values to
20882 @texline @math{3\times2}
20884 matrix, @expr{[[1, 2, 3], [4, 5, 6]]}.
20887 The command @kbd{V M _} [@code{mapr}] (i.e., type an underscore at the
20888 operator prompt) maps by rows instead. For example, @kbd{V M _ A} views
20889 the above matrix as a vector of two 3-element row vectors. It produces
20890 a new vector which contains the absolute values of those row vectors,
20891 namely @expr{[3.74, 8.77]}. (Recall, the absolute value of a vector is
20892 defined as the square root of the sum of the squares of the elements.)
20893 Some operators accept vectors and return new vectors; for example,
20894 @kbd{v v} reverses a vector, so @kbd{V M _ v v} would reverse each row
20895 of the matrix to get a new matrix, @expr{[[3, -2, 1], [-6, 5, -4]]}.
20897 Sometimes a vector of vectors (representing, say, strings, sets, or lists)
20898 happens to look like a matrix. If so, remember to use @kbd{V M _} if you
20899 want to map a function across the whole strings or sets rather than across
20900 their individual elements.
20903 The command @kbd{V M :} [@code{mapc}] maps by columns. Basically, it
20904 transposes the input matrix, maps by rows, and then, if the result is a
20905 matrix, transposes again. For example, @kbd{V M : A} takes the absolute
20906 values of the three columns of the matrix, treating each as a 2-vector,
20907 and @kbd{V M : v v} reverses the columns to get the matrix
20908 @expr{[[-4, 5, -6], [1, -2, 3]]}.
20910 (The symbols @kbd{_} and @kbd{:} were chosen because they had row-like
20911 and column-like appearances, and were not already taken by useful
20912 operators. Also, they appear shifted on most keyboards so they are easy
20913 to type after @kbd{V M}.)
20915 The @kbd{_} and @kbd{:} modifiers have no effect on arguments that are
20916 not matrices (so if none of the arguments are matrices, they have no
20917 effect at all). If some of the arguments are matrices and others are
20918 plain numbers, the plain numbers are held constant for all rows of the
20919 matrix (so that @kbd{2 V M _ ^} squares every row of a matrix; squaring
20920 a vector takes a dot product of the vector with itself).
20922 If some of the arguments are vectors with the same lengths as the
20923 rows (for @kbd{V M _}) or columns (for @kbd{V M :}) of the matrix
20924 arguments, those vectors are also held constant for every row or
20927 Sometimes it is useful to specify another mapping command as the operator
20928 to use with @kbd{V M}. For example, @kbd{V M _ V A +} applies @kbd{V A +}
20929 to each row of the input matrix, which in turn adds the two values on that
20930 row. If you give another vector-operator command as the operator for
20931 @kbd{V M}, it automatically uses map-by-rows mode if you don't specify
20932 otherwise; thus @kbd{V M V A +} is equivalent to @kbd{V M _ V A +}. (If
20933 you really want to map-by-elements another mapping command, you can use
20934 a triple-nested mapping command: @kbd{V M V M V A +} means to map
20935 @kbd{V M V A +} over the rows of the matrix; in turn, @kbd{V A +} is
20936 mapped over the elements of each row.)
20940 Previous versions of Calc had ``map across'' and ``map down'' modes
20941 that are now considered obsolete; the old ``map across'' is now simply
20942 @kbd{V M V A}, and ``map down'' is now @kbd{V M : V A}. The algebraic
20943 functions @code{mapa} and @code{mapd} are still supported, though.
20944 Note also that, while the old mapping modes were persistent (once you
20945 set the mode, it would apply to later mapping commands until you reset
20946 it), the new @kbd{:} and @kbd{_} modifiers apply only to the current
20947 mapping command. The default @kbd{V M} always means map-by-elements.
20949 @xref{Algebraic Manipulation}, for the @kbd{a M} command, which is like
20950 @kbd{V M} but for equations and inequalities instead of vectors.
20951 @xref{Storing Variables}, for the @kbd{s m} command which modifies a
20952 variable's stored value using a @kbd{V M}-like operator.
20954 @node Reducing, Nesting and Fixed Points, Mapping, Reducing and Mapping
20955 @subsection Reducing
20959 @pindex calc-reduce
20961 The @kbd{V R} (@code{calc-reduce}) [@code{reduce}] command applies a given
20962 binary operator across all the elements of a vector. A binary operator is
20963 a function such as @code{+} or @code{max} which takes two arguments. For
20964 example, reducing @code{+} over a vector computes the sum of the elements
20965 of the vector. Reducing @code{-} computes the first element minus each of
20966 the remaining elements. Reducing @code{max} computes the maximum element
20967 and so on. In general, reducing @code{f} over the vector @samp{[a, b, c, d]}
20968 produces @samp{f(f(f(a, b), c), d)}.
20972 The @kbd{I V R} [@code{rreduce}] command is similar to @kbd{V R} except
20973 that works from right to left through the vector. For example, plain
20974 @kbd{V R -} on the vector @samp{[a, b, c, d]} produces @samp{a - b - c - d}
20975 but @kbd{I V R -} on the same vector produces @samp{a - (b - (c - d))},
20976 or @samp{a - b + c - d}. This ``alternating sum'' occurs frequently
20977 in power series expansions.
20981 The @kbd{V U} (@code{calc-accumulate}) [@code{accum}] command does an
20982 accumulation operation. Here Calc does the corresponding reduction
20983 operation, but instead of producing only the final result, it produces
20984 a vector of all the intermediate results. Accumulating @code{+} over
20985 the vector @samp{[a, b, c, d]} produces the vector
20986 @samp{[a, a + b, a + b + c, a + b + c + d]}.
20990 The @kbd{I V U} [@code{raccum}] command does a right-to-left accumulation.
20991 For example, @kbd{I V U -} on the vector @samp{[a, b, c, d]} produces the
20992 vector @samp{[a - b + c - d, b - c + d, c - d, d]}.
20998 As for @kbd{V M}, @kbd{V R} normally reduces a matrix elementwise. For
20999 example, given the matrix @expr{[[a, b, c], [d, e, f]]}, @kbd{V R +} will
21000 compute @expr{a + b + c + d + e + f}. You can type @kbd{V R _} or
21001 @kbd{V R :} to modify this behavior. The @kbd{V R _} [@code{reducea}]
21002 command reduces ``across'' the matrix; it reduces each row of the matrix
21003 as a vector, then collects the results. Thus @kbd{V R _ +} of this
21004 matrix would produce @expr{[a + b + c, d + e + f]}. Similarly, @kbd{V R :}
21005 [@code{reduced}] reduces down; @kbd{V R : +} would produce @expr{[a + d,
21010 There is a third ``by rows'' mode for reduction that is occasionally
21011 useful; @kbd{V R =} [@code{reducer}] simply reduces the operator over
21012 the rows of the matrix themselves. Thus @kbd{V R = +} on the above
21013 matrix would get the same result as @kbd{V R : +}, since adding two
21014 row vectors is equivalent to adding their elements. But @kbd{V R = *}
21015 would multiply the two rows (to get a single number, their dot product),
21016 while @kbd{V R : *} would produce a vector of the products of the columns.
21018 These three matrix reduction modes work with @kbd{V R} and @kbd{I V R},
21019 but they are not currently supported with @kbd{V U} or @kbd{I V U}.
21023 The obsolete reduce-by-columns function, @code{reducec}, is still
21024 supported but there is no way to get it through the @kbd{V R} command.
21026 The commands @kbd{C-x * :} and @kbd{C-x * _} are equivalent to typing
21027 @kbd{C-x * r} to grab a rectangle of data into Calc, and then typing
21028 @kbd{V R : +} or @kbd{V R _ +}, respectively, to sum the columns or
21029 rows of the matrix. @xref{Grabbing From Buffers}.
21031 @node Nesting and Fixed Points, Generalized Products, Reducing, Reducing and Mapping
21032 @subsection Nesting and Fixed Points
21037 The @kbd{H V R} [@code{nest}] command applies a function to a given
21038 argument repeatedly. It takes two values, @samp{a} and @samp{n}, from
21039 the stack, where @samp{n} must be an integer. It then applies the
21040 function nested @samp{n} times; if the function is @samp{f} and @samp{n}
21041 is 3, the result is @samp{f(f(f(a)))}. The number @samp{n} may be
21042 negative if Calc knows an inverse for the function @samp{f}; for
21043 example, @samp{nest(sin, a, -2)} returns @samp{arcsin(arcsin(a))}.
21047 The @kbd{H V U} [@code{anest}] command is an accumulating version of
21048 @code{nest}: It returns a vector of @samp{n+1} values, e.g.,
21049 @samp{[a, f(a), f(f(a)), f(f(f(a)))]}. If @samp{n} is negative and
21050 @samp{F} is the inverse of @samp{f}, then the result is of the
21051 form @samp{[a, F(a), F(F(a)), F(F(F(a)))]}.
21055 @cindex Fixed points
21056 The @kbd{H I V R} [@code{fixp}] command is like @kbd{H V R}, except
21057 that it takes only an @samp{a} value from the stack; the function is
21058 applied until it reaches a ``fixed point,'' i.e., until the result
21063 The @kbd{H I V U} [@code{afixp}] command is an accumulating @code{fixp}.
21064 The first element of the return vector will be the initial value @samp{a};
21065 the last element will be the final result that would have been returned
21068 For example, 0.739085 is a fixed point of the cosine function (in radians):
21069 @samp{cos(0.739085) = 0.739085}. You can find this value by putting, say,
21070 1.0 on the stack and typing @kbd{H I V U C}. (We use the accumulating
21071 version so we can see the intermediate results: @samp{[1, 0.540302, 0.857553,
21072 0.65329, ...]}. With a precision of six, this command will take 36 steps
21073 to converge to 0.739085.)
21075 Newton's method for finding roots is a classic example of iteration
21076 to a fixed point. To find the square root of five starting with an
21077 initial guess, Newton's method would look for a fixed point of the
21078 function @samp{(x + 5/x) / 2}. Putting a guess of 1 on the stack
21079 and typing @kbd{H I V R ' ($ + 5/$)/2 @key{RET}} quickly yields the result
21080 2.23607. This is equivalent to using the @kbd{a R} (@code{calc-find-root})
21081 command to find a root of the equation @samp{x^2 = 5}.
21083 These examples used numbers for @samp{a} values. Calc keeps applying
21084 the function until two successive results are equal to within the
21085 current precision. For complex numbers, both the real parts and the
21086 imaginary parts must be equal to within the current precision. If
21087 @samp{a} is a formula (say, a variable name), then the function is
21088 applied until two successive results are exactly the same formula.
21089 It is up to you to ensure that the function will eventually converge;
21090 if it doesn't, you may have to press @kbd{C-g} to stop the Calculator.
21092 The algebraic @code{fixp} function takes two optional arguments, @samp{n}
21093 and @samp{tol}. The first is the maximum number of steps to be allowed,
21094 and must be either an integer or the symbol @samp{inf} (infinity, the
21095 default). The second is a convergence tolerance. If a tolerance is
21096 specified, all results during the calculation must be numbers, not
21097 formulas, and the iteration stops when the magnitude of the difference
21098 between two successive results is less than or equal to the tolerance.
21099 (This implies that a tolerance of zero iterates until the results are
21102 Putting it all together, @samp{fixp(<(# + A/#)/2>, B, 20, 1e-10)}
21103 computes the square root of @samp{A} given the initial guess @samp{B},
21104 stopping when the result is correct within the specified tolerance, or
21105 when 20 steps have been taken, whichever is sooner.
21107 @node Generalized Products, , Nesting and Fixed Points, Reducing and Mapping
21108 @subsection Generalized Products
21111 @pindex calc-outer-product
21113 The @kbd{V O} (@code{calc-outer-product}) [@code{outer}] command applies
21114 a given binary operator to all possible pairs of elements from two
21115 vectors, to produce a matrix. For example, @kbd{V O *} with @samp{[a, b]}
21116 and @samp{[x, y, z]} on the stack produces a multiplication table:
21117 @samp{[[a x, a y, a z], [b x, b y, b z]]}. Element @var{r},@var{c} of
21118 the result matrix is obtained by applying the operator to element @var{r}
21119 of the lefthand vector and element @var{c} of the righthand vector.
21122 @pindex calc-inner-product
21124 The @kbd{V I} (@code{calc-inner-product}) [@code{inner}] command computes
21125 the generalized inner product of two vectors or matrices, given a
21126 ``multiplicative'' operator and an ``additive'' operator. These can each
21127 actually be any binary operators; if they are @samp{*} and @samp{+},
21128 respectively, the result is a standard matrix multiplication. Element
21129 @var{r},@var{c} of the result matrix is obtained by mapping the
21130 multiplicative operator across row @var{r} of the lefthand matrix and
21131 column @var{c} of the righthand matrix, and then reducing with the additive
21132 operator. Just as for the standard @kbd{*} command, this can also do a
21133 vector-matrix or matrix-vector inner product, or a vector-vector
21134 generalized dot product.
21136 Since @kbd{V I} requires two operators, it prompts twice. In each case,
21137 you can use any of the usual methods for entering the operator. If you
21138 use @kbd{$} twice to take both operator formulas from the stack, the
21139 first (multiplicative) operator is taken from the top of the stack
21140 and the second (additive) operator is taken from second-to-top.
21142 @node Vector and Matrix Formats, , Reducing and Mapping, Matrix Functions
21143 @section Vector and Matrix Display Formats
21146 Commands for controlling vector and matrix display use the @kbd{v} prefix
21147 instead of the usual @kbd{d} prefix. But they are display modes; in
21148 particular, they are influenced by the @kbd{I} and @kbd{H} prefix keys
21149 in the same way (@pxref{Display Modes}). Matrix display is also
21150 influenced by the @kbd{d O} (@code{calc-flat-language}) mode;
21151 @pxref{Normal Language Modes}.
21154 @pindex calc-matrix-left-justify
21156 @pindex calc-matrix-center-justify
21158 @pindex calc-matrix-right-justify
21159 The commands @kbd{v <} (@code{calc-matrix-left-justify}), @kbd{v >}
21160 (@code{calc-matrix-right-justify}), and @w{@kbd{v =}}
21161 (@code{calc-matrix-center-justify}) control whether matrix elements
21162 are justified to the left, right, or center of their columns.
21165 @pindex calc-vector-brackets
21167 @pindex calc-vector-braces
21169 @pindex calc-vector-parens
21170 The @kbd{v [} (@code{calc-vector-brackets}) command turns the square
21171 brackets that surround vectors and matrices displayed in the stack on
21172 and off. The @kbd{v @{} (@code{calc-vector-braces}) and @kbd{v (}
21173 (@code{calc-vector-parens}) commands use curly braces or parentheses,
21174 respectively, instead of square brackets. For example, @kbd{v @{} might
21175 be used in preparation for yanking a matrix into a buffer running
21176 Mathematica. (In fact, the Mathematica language mode uses this mode;
21177 @pxref{Mathematica Language Mode}.) Note that, regardless of the
21178 display mode, either brackets or braces may be used to enter vectors,
21179 and parentheses may never be used for this purpose.
21182 @pindex calc-matrix-brackets
21183 The @kbd{v ]} (@code{calc-matrix-brackets}) command controls the
21184 ``big'' style display of matrices. It prompts for a string of code
21185 letters; currently implemented letters are @code{R}, which enables
21186 brackets on each row of the matrix; @code{O}, which enables outer
21187 brackets in opposite corners of the matrix; and @code{C}, which
21188 enables commas or semicolons at the ends of all rows but the last.
21189 The default format is @samp{RO}. (Before Calc 2.00, the format
21190 was fixed at @samp{ROC}.) Here are some example matrices:
21194 [ [ 123, 0, 0 ] [ [ 123, 0, 0 ],
21195 [ 0, 123, 0 ] [ 0, 123, 0 ],
21196 [ 0, 0, 123 ] ] [ 0, 0, 123 ] ]
21205 [ 123, 0, 0 [ 123, 0, 0 ;
21206 0, 123, 0 0, 123, 0 ;
21207 0, 0, 123 ] 0, 0, 123 ]
21216 [ 123, 0, 0 ] 123, 0, 0
21217 [ 0, 123, 0 ] 0, 123, 0
21218 [ 0, 0, 123 ] 0, 0, 123
21225 Note that of the formats shown here, @samp{RO}, @samp{ROC}, and
21226 @samp{OC} are all recognized as matrices during reading, while
21227 the others are useful for display only.
21230 @pindex calc-vector-commas
21231 The @kbd{v ,} (@code{calc-vector-commas}) command turns commas on and
21232 off in vector and matrix display.
21234 In vectors of length one, and in all vectors when commas have been
21235 turned off, Calc adds extra parentheses around formulas that might
21236 otherwise be ambiguous. For example, @samp{[a b]} could be a vector
21237 of the one formula @samp{a b}, or it could be a vector of two
21238 variables with commas turned off. Calc will display the former
21239 case as @samp{[(a b)]}. You can disable these extra parentheses
21240 (to make the output less cluttered at the expense of allowing some
21241 ambiguity) by adding the letter @code{P} to the control string you
21242 give to @kbd{v ]} (as described above).
21245 @pindex calc-full-vectors
21246 The @kbd{v .} (@code{calc-full-vectors}) command turns abbreviated
21247 display of long vectors on and off. In this mode, vectors of six
21248 or more elements, or matrices of six or more rows or columns, will
21249 be displayed in an abbreviated form that displays only the first
21250 three elements and the last element: @samp{[a, b, c, ..., z]}.
21251 When very large vectors are involved this will substantially
21252 improve Calc's display speed.
21255 @pindex calc-full-trail-vectors
21256 The @kbd{t .} (@code{calc-full-trail-vectors}) command controls a
21257 similar mode for recording vectors in the Trail. If you turn on
21258 this mode, vectors of six or more elements and matrices of six or
21259 more rows or columns will be abbreviated when they are put in the
21260 Trail. The @kbd{t y} (@code{calc-trail-yank}) command will be
21261 unable to recover those vectors. If you are working with very
21262 large vectors, this mode will improve the speed of all operations
21263 that involve the trail.
21266 @pindex calc-break-vectors
21267 The @kbd{v /} (@code{calc-break-vectors}) command turns multi-line
21268 vector display on and off. Normally, matrices are displayed with one
21269 row per line but all other types of vectors are displayed in a single
21270 line. This mode causes all vectors, whether matrices or not, to be
21271 displayed with a single element per line. Sub-vectors within the
21272 vectors will still use the normal linear form.
21274 @node Algebra, Units, Matrix Functions, Top
21278 This section covers the Calc features that help you work with
21279 algebraic formulas. First, the general sub-formula selection
21280 mechanism is described; this works in conjunction with any Calc
21281 commands. Then, commands for specific algebraic operations are
21282 described. Finally, the flexible @dfn{rewrite rule} mechanism
21285 The algebraic commands use the @kbd{a} key prefix; selection
21286 commands use the @kbd{j} (for ``just a letter that wasn't used
21287 for anything else'') prefix.
21289 @xref{Editing Stack Entries}, to see how to manipulate formulas
21290 using regular Emacs editing commands.
21292 When doing algebraic work, you may find several of the Calculator's
21293 modes to be helpful, including Algebraic Simplification mode (@kbd{m A})
21294 or No-Simplification mode (@kbd{m O}),
21295 Algebraic entry mode (@kbd{m a}), Fraction mode (@kbd{m f}), and
21296 Symbolic mode (@kbd{m s}). @xref{Mode Settings}, for discussions
21297 of these modes. You may also wish to select Big display mode (@kbd{d B}).
21298 @xref{Normal Language Modes}.
21301 * Selecting Subformulas::
21302 * Algebraic Manipulation::
21303 * Simplifying Formulas::
21306 * Solving Equations::
21307 * Numerical Solutions::
21310 * Logical Operations::
21314 @node Selecting Subformulas, Algebraic Manipulation, Algebra, Algebra
21315 @section Selecting Sub-Formulas
21319 @cindex Sub-formulas
21320 @cindex Parts of formulas
21321 When working with an algebraic formula it is often necessary to
21322 manipulate a portion of the formula rather than the formula as a
21323 whole. Calc allows you to ``select'' a portion of any formula on
21324 the stack. Commands which would normally operate on that stack
21325 entry will now operate only on the sub-formula, leaving the
21326 surrounding part of the stack entry alone.
21328 One common non-algebraic use for selection involves vectors. To work
21329 on one element of a vector in-place, simply select that element as a
21330 ``sub-formula'' of the vector.
21333 * Making Selections::
21334 * Changing Selections::
21335 * Displaying Selections::
21336 * Operating on Selections::
21337 * Rearranging with Selections::
21340 @node Making Selections, Changing Selections, Selecting Subformulas, Selecting Subformulas
21341 @subsection Making Selections
21345 @pindex calc-select-here
21346 To select a sub-formula, move the Emacs cursor to any character in that
21347 sub-formula, and press @w{@kbd{j s}} (@code{calc-select-here}). Calc will
21348 highlight the smallest portion of the formula that contains that
21349 character. By default the sub-formula is highlighted by blanking out
21350 all of the rest of the formula with dots. Selection works in any
21351 display mode but is perhaps easiest in Big mode (@kbd{d B}).
21352 Suppose you enter the following formula:
21364 (by typing @kbd{' ((a+b)^3 + sqrt(c)) / (2x+1)}). If you move the
21365 cursor to the letter @samp{b} and press @w{@kbd{j s}}, the display changes
21378 Every character not part of the sub-formula @samp{b} has been changed
21379 to a dot. The @samp{*} next to the line number is to remind you that
21380 the formula has a portion of it selected. (In this case, it's very
21381 obvious, but it might not always be. If Embedded mode is enabled,
21382 the word @samp{Sel} also appears in the mode line because the stack
21383 may not be visible. @pxref{Embedded Mode}.)
21385 If you had instead placed the cursor on the parenthesis immediately to
21386 the right of the @samp{b}, the selection would have been:
21398 The portion selected is always large enough to be considered a complete
21399 formula all by itself, so selecting the parenthesis selects the whole
21400 formula that it encloses. Putting the cursor on the @samp{+} sign
21401 would have had the same effect.
21403 (Strictly speaking, the Emacs cursor is really the manifestation of
21404 the Emacs ``point,'' which is a position @emph{between} two characters
21405 in the buffer. So purists would say that Calc selects the smallest
21406 sub-formula which contains the character to the right of ``point.'')
21408 If you supply a numeric prefix argument @var{n}, the selection is
21409 expanded to the @var{n}th enclosing sub-formula. Thus, positioning
21410 the cursor on the @samp{b} and typing @kbd{C-u 1 j s} will select
21411 @samp{a + b}; typing @kbd{C-u 2 j s} will select @samp{(a + b)^3},
21414 If the cursor is not on any part of the formula, or if you give a
21415 numeric prefix that is too large, the entire formula is selected.
21417 If the cursor is on the @samp{.} line that marks the top of the stack
21418 (i.e., its normal ``rest position''), this command selects the entire
21419 formula at stack level 1. Most selection commands similarly operate
21420 on the formula at the top of the stack if you haven't positioned the
21421 cursor on any stack entry.
21424 @pindex calc-select-additional
21425 The @kbd{j a} (@code{calc-select-additional}) command enlarges the
21426 current selection to encompass the cursor. To select the smallest
21427 sub-formula defined by two different points, move to the first and
21428 press @kbd{j s}, then move to the other and press @kbd{j a}. This
21429 is roughly analogous to using @kbd{C-@@} (@code{set-mark-command}) to
21430 select the two ends of a region of text during normal Emacs editing.
21433 @pindex calc-select-once
21434 The @kbd{j o} (@code{calc-select-once}) command selects a formula in
21435 exactly the same way as @kbd{j s}, except that the selection will
21436 last only as long as the next command that uses it. For example,
21437 @kbd{j o 1 +} is a handy way to add one to the sub-formula indicated
21440 (A somewhat more precise definition: The @kbd{j o} command sets a flag
21441 such that the next command involving selected stack entries will clear
21442 the selections on those stack entries afterwards. All other selection
21443 commands except @kbd{j a} and @kbd{j O} clear this flag.)
21447 @pindex calc-select-here-maybe
21448 @pindex calc-select-once-maybe
21449 The @kbd{j S} (@code{calc-select-here-maybe}) and @kbd{j O}
21450 (@code{calc-select-once-maybe}) commands are equivalent to @kbd{j s}
21451 and @kbd{j o}, respectively, except that if the formula already
21452 has a selection they have no effect. This is analogous to the
21453 behavior of some commands such as @kbd{j r} (@code{calc-rewrite-selection};
21454 @pxref{Selections with Rewrite Rules}) and is mainly intended to be
21455 used in keyboard macros that implement your own selection-oriented
21458 Selection of sub-formulas normally treats associative terms like
21459 @samp{a + b - c + d} and @samp{x * y * z} as single levels of the formula.
21460 If you place the cursor anywhere inside @samp{a + b - c + d} except
21461 on one of the variable names and use @kbd{j s}, you will select the
21462 entire four-term sum.
21465 @pindex calc-break-selections
21466 The @kbd{j b} (@code{calc-break-selections}) command controls a mode
21467 in which the ``deep structure'' of these associative formulas shows
21468 through. Calc actually stores the above formulas as
21469 @samp{((a + b) - c) + d} and @samp{x * (y * z)}. (Note that for certain
21470 obscure reasons, by default Calc treats multiplication as
21471 right-associative.) Once you have enabled @kbd{j b} mode, selecting
21472 with the cursor on the @samp{-} sign would only select the @samp{a + b -
21473 c} portion, which makes sense when the deep structure of the sum is
21474 considered. There is no way to select the @samp{b - c + d} portion;
21475 although this might initially look like just as legitimate a sub-formula
21476 as @samp{a + b - c}, the deep structure shows that it isn't. The @kbd{d
21477 U} command can be used to view the deep structure of any formula
21478 (@pxref{Normal Language Modes}).
21480 When @kbd{j b} mode has not been enabled, the deep structure is
21481 generally hidden by the selection commands---what you see is what
21485 @pindex calc-unselect
21486 The @kbd{j u} (@code{calc-unselect}) command unselects the formula
21487 that the cursor is on. If there was no selection in the formula,
21488 this command has no effect. With a numeric prefix argument, it
21489 unselects the @var{n}th stack element rather than using the cursor
21493 @pindex calc-clear-selections
21494 The @kbd{j c} (@code{calc-clear-selections}) command unselects all
21497 @node Changing Selections, Displaying Selections, Making Selections, Selecting Subformulas
21498 @subsection Changing Selections
21502 @pindex calc-select-more
21503 Once you have selected a sub-formula, you can expand it using the
21504 @w{@kbd{j m}} (@code{calc-select-more}) command. If @samp{a + b} is
21505 selected, pressing @w{@kbd{j m}} repeatedly works as follows:
21510 (a + b) . . . (a + b) + V c (a + b) + V c
21511 1* ............... 1* ............... 1* ---------------
21512 . . . . . . . . 2 x + 1
21517 In the last example, the entire formula is selected. This is roughly
21518 the same as having no selection at all, but because there are subtle
21519 differences the @samp{*} character is still there on the line number.
21521 With a numeric prefix argument @var{n}, @kbd{j m} expands @var{n}
21522 times (or until the entire formula is selected). Note that @kbd{j s}
21523 with argument @var{n} is equivalent to plain @kbd{j s} followed by
21524 @kbd{j m} with argument @var{n}. If @w{@kbd{j m}} is used when there
21525 is no current selection, it is equivalent to @w{@kbd{j s}}.
21527 Even though @kbd{j m} does not explicitly use the location of the
21528 cursor within the formula, it nevertheless uses the cursor to determine
21529 which stack element to operate on. As usual, @kbd{j m} when the cursor
21530 is not on any stack element operates on the top stack element.
21533 @pindex calc-select-less
21534 The @kbd{j l} (@code{calc-select-less}) command reduces the current
21535 selection around the cursor position. That is, it selects the
21536 immediate sub-formula of the current selection which contains the
21537 cursor, the opposite of @kbd{j m}. If the cursor is not inside the
21538 current selection, the command de-selects the formula.
21541 @pindex calc-select-part
21542 The @kbd{j 1} through @kbd{j 9} (@code{calc-select-part}) commands
21543 select the @var{n}th sub-formula of the current selection. They are
21544 like @kbd{j l} (@code{calc-select-less}) except they use counting
21545 rather than the cursor position to decide which sub-formula to select.
21546 For example, if the current selection is @kbd{a + b + c} or
21547 @kbd{f(a, b, c)} or @kbd{[a, b, c]}, then @kbd{j 1} selects @samp{a},
21548 @kbd{j 2} selects @samp{b}, and @kbd{j 3} selects @samp{c}; in each of
21549 these cases, @kbd{j 4} through @kbd{j 9} would be errors.
21551 If there is no current selection, @kbd{j 1} through @kbd{j 9} select
21552 the @var{n}th top-level sub-formula. (In other words, they act as if
21553 the entire stack entry were selected first.) To select the @var{n}th
21554 sub-formula where @var{n} is greater than nine, you must instead invoke
21555 @w{@kbd{j 1}} with @var{n} as a numeric prefix argument.
21559 @pindex calc-select-next
21560 @pindex calc-select-previous
21561 The @kbd{j n} (@code{calc-select-next}) and @kbd{j p}
21562 (@code{calc-select-previous}) commands change the current selection
21563 to the next or previous sub-formula at the same level. For example,
21564 if @samp{b} is selected in @w{@samp{2 + a*b*c + x}}, then @kbd{j n}
21565 selects @samp{c}. Further @kbd{j n} commands would be in error because,
21566 even though there is something to the right of @samp{c} (namely, @samp{x}),
21567 it is not at the same level; in this case, it is not a term of the
21568 same product as @samp{b} and @samp{c}. However, @kbd{j m} (to select
21569 the whole product @samp{a*b*c} as a term of the sum) followed by
21570 @w{@kbd{j n}} would successfully select the @samp{x}.
21572 Similarly, @kbd{j p} moves the selection from the @samp{b} in this
21573 sample formula to the @samp{a}. Both commands accept numeric prefix
21574 arguments to move several steps at a time.
21576 It is interesting to compare Calc's selection commands with the
21577 Emacs Info system's commands for navigating through hierarchically
21578 organized documentation. Calc's @kbd{j n} command is completely
21579 analogous to Info's @kbd{n} command. Likewise, @kbd{j p} maps to
21580 @kbd{p}, @kbd{j 2} maps to @kbd{2}, and Info's @kbd{u} is like @kbd{j m}.
21581 (Note that @kbd{j u} stands for @code{calc-unselect}, not ``up''.)
21582 The Info @kbd{m} command is somewhat similar to Calc's @kbd{j s} and
21583 @kbd{j l}; in each case, you can jump directly to a sub-component
21584 of the hierarchy simply by pointing to it with the cursor.
21586 @node Displaying Selections, Operating on Selections, Changing Selections, Selecting Subformulas
21587 @subsection Displaying Selections
21591 @pindex calc-show-selections
21592 The @kbd{j d} (@code{calc-show-selections}) command controls how
21593 selected sub-formulas are displayed. One of the alternatives is
21594 illustrated in the above examples; if we press @kbd{j d} we switch
21595 to the other style in which the selected portion itself is obscured
21601 (a + b) . . . ## # ## + V c
21602 1* ............... 1* ---------------
21607 @node Operating on Selections, Rearranging with Selections, Displaying Selections, Selecting Subformulas
21608 @subsection Operating on Selections
21611 Once a selection is made, all Calc commands that manipulate items
21612 on the stack will operate on the selected portions of the items
21613 instead. (Note that several stack elements may have selections
21614 at once, though there can be only one selection at a time in any
21615 given stack element.)
21618 @pindex calc-enable-selections
21619 The @kbd{j e} (@code{calc-enable-selections}) command disables the
21620 effect that selections have on Calc commands. The current selections
21621 still exist, but Calc commands operate on whole stack elements anyway.
21622 This mode can be identified by the fact that the @samp{*} markers on
21623 the line numbers are gone, even though selections are visible. To
21624 reactivate the selections, press @kbd{j e} again.
21626 To extract a sub-formula as a new formula, simply select the
21627 sub-formula and press @key{RET}. This normally duplicates the top
21628 stack element; here it duplicates only the selected portion of that
21631 To replace a sub-formula with something different, you can enter the
21632 new value onto the stack and press @key{TAB}. This normally exchanges
21633 the top two stack elements; here it swaps the value you entered into
21634 the selected portion of the formula, returning the old selected
21635 portion to the top of the stack.
21640 (a + b) . . . 17 x y . . . 17 x y + V c
21641 2* ............... 2* ............. 2: -------------
21642 . . . . . . . . 2 x + 1
21645 1: 17 x y 1: (a + b) 1: (a + b)
21649 In this example we select a sub-formula of our original example,
21650 enter a new formula, @key{TAB} it into place, then deselect to see
21651 the complete, edited formula.
21653 If you want to swap whole formulas around even though they contain
21654 selections, just use @kbd{j e} before and after.
21657 @pindex calc-enter-selection
21658 The @kbd{j '} (@code{calc-enter-selection}) command is another way
21659 to replace a selected sub-formula. This command does an algebraic
21660 entry just like the regular @kbd{'} key. When you press @key{RET},
21661 the formula you type replaces the original selection. You can use
21662 the @samp{$} symbol in the formula to refer to the original
21663 selection. If there is no selection in the formula under the cursor,
21664 the cursor is used to make a temporary selection for the purposes of
21665 the command. Thus, to change a term of a formula, all you have to
21666 do is move the Emacs cursor to that term and press @kbd{j '}.
21669 @pindex calc-edit-selection
21670 The @kbd{j `} (@code{calc-edit-selection}) command is a similar
21671 analogue of the @kbd{`} (@code{calc-edit}) command. It edits the
21672 selected sub-formula in a separate buffer. If there is no
21673 selection, it edits the sub-formula indicated by the cursor.
21675 To delete a sub-formula, press @key{DEL}. This generally replaces
21676 the sub-formula with the constant zero, but in a few suitable contexts
21677 it uses the constant one instead. The @key{DEL} key automatically
21678 deselects and re-simplifies the entire formula afterwards. Thus:
21683 17 x y + # # 17 x y 17 # y 17 y
21684 1* ------------- 1: ------- 1* ------- 1: -------
21685 2 x + 1 2 x + 1 2 x + 1 2 x + 1
21689 In this example, we first delete the @samp{sqrt(c)} term; Calc
21690 accomplishes this by replacing @samp{sqrt(c)} with zero and
21691 resimplifying. We then delete the @kbd{x} in the numerator;
21692 since this is part of a product, Calc replaces it with @samp{1}
21695 If you select an element of a vector and press @key{DEL}, that
21696 element is deleted from the vector. If you delete one side of
21697 an equation or inequality, only the opposite side remains.
21699 @kindex j @key{DEL}
21700 @pindex calc-del-selection
21701 The @kbd{j @key{DEL}} (@code{calc-del-selection}) command is like
21702 @key{DEL} but with the auto-selecting behavior of @kbd{j '} and
21703 @kbd{j `}. It deletes the selected portion of the formula
21704 indicated by the cursor, or, in the absence of a selection, it
21705 deletes the sub-formula indicated by the cursor position.
21707 @kindex j @key{RET}
21708 @pindex calc-grab-selection
21709 (There is also an auto-selecting @kbd{j @key{RET}} (@code{calc-copy-selection})
21712 Normal arithmetic operations also apply to sub-formulas. Here we
21713 select the denominator, press @kbd{5 -} to subtract five from the
21714 denominator, press @kbd{n} to negate the denominator, then
21715 press @kbd{Q} to take the square root.
21719 .. . .. . .. . .. .
21720 1* ....... 1* ....... 1* ....... 1* ..........
21721 2 x + 1 2 x - 4 4 - 2 x _________
21726 Certain types of operations on selections are not allowed. For
21727 example, for an arithmetic function like @kbd{-} no more than one of
21728 the arguments may be a selected sub-formula. (As the above example
21729 shows, the result of the subtraction is spliced back into the argument
21730 which had the selection; if there were more than one selection involved,
21731 this would not be well-defined.) If you try to subtract two selections,
21732 the command will abort with an error message.
21734 Operations on sub-formulas sometimes leave the formula as a whole
21735 in an ``un-natural'' state. Consider negating the @samp{2 x} term
21736 of our sample formula by selecting it and pressing @kbd{n}
21737 (@code{calc-change-sign}).
21742 1* .......... 1* ...........
21743 ......... ..........
21744 . . . 2 x . . . -2 x
21748 Unselecting the sub-formula reveals that the minus sign, which would
21749 normally have cancelled out with the subtraction automatically, has
21750 not been able to do so because the subtraction was not part of the
21751 selected portion. Pressing @kbd{=} (@code{calc-evaluate}) or doing
21752 any other mathematical operation on the whole formula will cause it
21758 1: ----------- 1: ----------
21759 __________ _________
21760 V 4 - -2 x V 4 + 2 x
21764 @node Rearranging with Selections, , Operating on Selections, Selecting Subformulas
21765 @subsection Rearranging Formulas using Selections
21769 @pindex calc-commute-right
21770 The @kbd{j R} (@code{calc-commute-right}) command moves the selected
21771 sub-formula to the right in its surrounding formula. Generally the
21772 selection is one term of a sum or product; the sum or product is
21773 rearranged according to the commutative laws of algebra.
21775 As with @kbd{j '} and @kbd{j @key{DEL}}, the term under the cursor is used
21776 if there is no selection in the current formula. All commands described
21777 in this section share this property. In this example, we place the
21778 cursor on the @samp{a} and type @kbd{j R}, then repeat.
21781 1: a + b - c 1: b + a - c 1: b - c + a
21785 Note that in the final step above, the @samp{a} is switched with
21786 the @samp{c} but the signs are adjusted accordingly. When moving
21787 terms of sums and products, @kbd{j R} will never change the
21788 mathematical meaning of the formula.
21790 The selected term may also be an element of a vector or an argument
21791 of a function. The term is exchanged with the one to its right.
21792 In this case, the ``meaning'' of the vector or function may of
21793 course be drastically changed.
21796 1: [a, b, c] 1: [b, a, c] 1: [b, c, a]
21798 1: f(a, b, c) 1: f(b, a, c) 1: f(b, c, a)
21802 @pindex calc-commute-left
21803 The @kbd{j L} (@code{calc-commute-left}) command is like @kbd{j R}
21804 except that it swaps the selected term with the one to its left.
21806 With numeric prefix arguments, these commands move the selected
21807 term several steps at a time. It is an error to try to move a
21808 term left or right past the end of its enclosing formula.
21809 With numeric prefix arguments of zero, these commands move the
21810 selected term as far as possible in the given direction.
21813 @pindex calc-sel-distribute
21814 The @kbd{j D} (@code{calc-sel-distribute}) command mixes the selected
21815 sum or product into the surrounding formula using the distributive
21816 law. For example, in @samp{a * (b - c)} with the @samp{b - c}
21817 selected, the result is @samp{a b - a c}. This also distributes
21818 products or quotients into surrounding powers, and can also do
21819 transformations like @samp{exp(a + b)} to @samp{exp(a) exp(b)},
21820 where @samp{a + b} is the selected term, and @samp{ln(a ^ b)}
21821 to @samp{ln(a) b}, where @samp{a ^ b} is the selected term.
21823 For multiple-term sums or products, @kbd{j D} takes off one term
21824 at a time: @samp{a * (b + c - d)} goes to @samp{a * (c - d) + a b}
21825 with the @samp{c - d} selected so that you can type @kbd{j D}
21826 repeatedly to expand completely. The @kbd{j D} command allows a
21827 numeric prefix argument which specifies the maximum number of
21828 times to expand at once; the default is one time only.
21830 @vindex DistribRules
21831 The @kbd{j D} command is implemented using rewrite rules.
21832 @xref{Selections with Rewrite Rules}. The rules are stored in
21833 the Calc variable @code{DistribRules}. A convenient way to view
21834 these rules is to use @kbd{s e} (@code{calc-edit-variable}) which
21835 displays and edits the stored value of a variable. Press @kbd{C-c C-c}
21836 to return from editing mode; be careful not to make any actual changes
21837 or else you will affect the behavior of future @kbd{j D} commands!
21839 To extend @kbd{j D} to handle new cases, just edit @code{DistribRules}
21840 as described above. You can then use the @kbd{s p} command to save
21841 this variable's value permanently for future Calc sessions.
21842 @xref{Operations on Variables}.
21845 @pindex calc-sel-merge
21847 The @kbd{j M} (@code{calc-sel-merge}) command is the complement
21848 of @kbd{j D}; given @samp{a b - a c} with either @samp{a b} or
21849 @samp{a c} selected, the result is @samp{a * (b - c)}. Once
21850 again, @kbd{j M} can also merge calls to functions like @code{exp}
21851 and @code{ln}; examine the variable @code{MergeRules} to see all
21852 the relevant rules.
21855 @pindex calc-sel-commute
21856 @vindex CommuteRules
21857 The @kbd{j C} (@code{calc-sel-commute}) command swaps the arguments
21858 of the selected sum, product, or equation. It always behaves as
21859 if @kbd{j b} mode were in effect, i.e., the sum @samp{a + b + c} is
21860 treated as the nested sums @samp{(a + b) + c} by this command.
21861 If you put the cursor on the first @samp{+}, the result is
21862 @samp{(b + a) + c}; if you put the cursor on the second @samp{+}, the
21863 result is @samp{c + (a + b)} (which the default simplifications
21864 will rearrange to @samp{(c + a) + b}). The relevant rules are stored
21865 in the variable @code{CommuteRules}.
21867 You may need to turn default simplifications off (with the @kbd{m O}
21868 command) in order to get the full benefit of @kbd{j C}. For example,
21869 commuting @samp{a - b} produces @samp{-b + a}, but the default
21870 simplifications will ``simplify'' this right back to @samp{a - b} if
21871 you don't turn them off. The same is true of some of the other
21872 manipulations described in this section.
21875 @pindex calc-sel-negate
21876 @vindex NegateRules
21877 The @kbd{j N} (@code{calc-sel-negate}) command replaces the selected
21878 term with the negative of that term, then adjusts the surrounding
21879 formula in order to preserve the meaning. For example, given
21880 @samp{exp(a - b)} where @samp{a - b} is selected, the result is
21881 @samp{1 / exp(b - a)}. By contrast, selecting a term and using the
21882 regular @kbd{n} (@code{calc-change-sign}) command negates the
21883 term without adjusting the surroundings, thus changing the meaning
21884 of the formula as a whole. The rules variable is @code{NegateRules}.
21887 @pindex calc-sel-invert
21888 @vindex InvertRules
21889 The @kbd{j &} (@code{calc-sel-invert}) command is similar to @kbd{j N}
21890 except it takes the reciprocal of the selected term. For example,
21891 given @samp{a - ln(b)} with @samp{b} selected, the result is
21892 @samp{a + ln(1/b)}. The rules variable is @code{InvertRules}.
21895 @pindex calc-sel-jump-equals
21897 The @kbd{j E} (@code{calc-sel-jump-equals}) command moves the
21898 selected term from one side of an equation to the other. Given
21899 @samp{a + b = c + d} with @samp{c} selected, the result is
21900 @samp{a + b - c = d}. This command also works if the selected
21901 term is part of a @samp{*}, @samp{/}, or @samp{^} formula. The
21902 relevant rules variable is @code{JumpRules}.
21906 @pindex calc-sel-isolate
21907 The @kbd{j I} (@code{calc-sel-isolate}) command isolates the
21908 selected term on its side of an equation. It uses the @kbd{a S}
21909 (@code{calc-solve-for}) command to solve the equation, and the
21910 Hyperbolic flag affects it in the same way. @xref{Solving Equations}.
21911 When it applies, @kbd{j I} is often easier to use than @kbd{j E}.
21912 It understands more rules of algebra, and works for inequalities
21913 as well as equations.
21917 @pindex calc-sel-mult-both-sides
21918 @pindex calc-sel-div-both-sides
21919 The @kbd{j *} (@code{calc-sel-mult-both-sides}) command prompts for a
21920 formula using algebraic entry, then multiplies both sides of the
21921 selected quotient or equation by that formula. It simplifies each
21922 side with @kbd{a s} (@code{calc-simplify}) before re-forming the
21923 quotient or equation. You can suppress this simplification by
21924 providing any numeric prefix argument. There is also a @kbd{j /}
21925 (@code{calc-sel-div-both-sides}) which is similar to @kbd{j *} but
21926 dividing instead of multiplying by the factor you enter.
21928 As a special feature, if the numerator of the quotient is 1, then
21929 the denominator is expanded at the top level using the distributive
21930 law (i.e., using the @kbd{C-u -1 a x} command). Suppose the
21931 formula on the stack is @samp{1 / (sqrt(a) + 1)}, and you wish
21932 to eliminate the square root in the denominator by multiplying both
21933 sides by @samp{sqrt(a) - 1}. Calc's default simplifications would
21934 change the result @samp{(sqrt(a) - 1) / (sqrt(a) - 1) (sqrt(a) + 1)}
21935 right back to the original form by cancellation; Calc expands the
21936 denominator to @samp{sqrt(a) (sqrt(a) - 1) + sqrt(a) - 1} to prevent
21937 this. (You would now want to use an @kbd{a x} command to expand
21938 the rest of the way, whereupon the denominator would cancel out to
21939 the desired form, @samp{a - 1}.) When the numerator is not 1, this
21940 initial expansion is not necessary because Calc's default
21941 simplifications will not notice the potential cancellation.
21943 If the selection is an inequality, @kbd{j *} and @kbd{j /} will
21944 accept any factor, but will warn unless they can prove the factor
21945 is either positive or negative. (In the latter case the direction
21946 of the inequality will be switched appropriately.) @xref{Declarations},
21947 for ways to inform Calc that a given variable is positive or
21948 negative. If Calc can't tell for sure what the sign of the factor
21949 will be, it will assume it is positive and display a warning
21952 For selections that are not quotients, equations, or inequalities,
21953 these commands pull out a multiplicative factor: They divide (or
21954 multiply) by the entered formula, simplify, then multiply (or divide)
21955 back by the formula.
21959 @pindex calc-sel-add-both-sides
21960 @pindex calc-sel-sub-both-sides
21961 The @kbd{j +} (@code{calc-sel-add-both-sides}) and @kbd{j -}
21962 (@code{calc-sel-sub-both-sides}) commands analogously add to or
21963 subtract from both sides of an equation or inequality. For other
21964 types of selections, they extract an additive factor. A numeric
21965 prefix argument suppresses simplification of the intermediate
21969 @pindex calc-sel-unpack
21970 The @kbd{j U} (@code{calc-sel-unpack}) command replaces the
21971 selected function call with its argument. For example, given
21972 @samp{a + sin(x^2)} with @samp{sin(x^2)} selected, the result
21973 is @samp{a + x^2}. (The @samp{x^2} will remain selected; if you
21974 wanted to change the @code{sin} to @code{cos}, just press @kbd{C}
21975 now to take the cosine of the selected part.)
21978 @pindex calc-sel-evaluate
21979 The @kbd{j v} (@code{calc-sel-evaluate}) command performs the
21980 normal default simplifications on the selected sub-formula.
21981 These are the simplifications that are normally done automatically
21982 on all results, but which may have been partially inhibited by
21983 previous selection-related operations, or turned off altogether
21984 by the @kbd{m O} command. This command is just an auto-selecting
21985 version of the @w{@kbd{a v}} command (@pxref{Algebraic Manipulation}).
21987 With a numeric prefix argument of 2, @kbd{C-u 2 j v} applies
21988 the @kbd{a s} (@code{calc-simplify}) command to the selected
21989 sub-formula. With a prefix argument of 3 or more, e.g., @kbd{C-u j v}
21990 applies the @kbd{a e} (@code{calc-simplify-extended}) command.
21991 @xref{Simplifying Formulas}. With a negative prefix argument
21992 it simplifies at the top level only, just as with @kbd{a v}.
21993 Here the ``top'' level refers to the top level of the selected
21997 @pindex calc-sel-expand-formula
21998 The @kbd{j "} (@code{calc-sel-expand-formula}) command is to @kbd{a "}
21999 (@pxref{Algebraic Manipulation}) what @kbd{j v} is to @kbd{a v}.
22001 You can use the @kbd{j r} (@code{calc-rewrite-selection}) command
22002 to define other algebraic operations on sub-formulas. @xref{Rewrite Rules}.
22004 @node Algebraic Manipulation, Simplifying Formulas, Selecting Subformulas, Algebra
22005 @section Algebraic Manipulation
22008 The commands in this section perform general-purpose algebraic
22009 manipulations. They work on the whole formula at the top of the
22010 stack (unless, of course, you have made a selection in that
22013 Many algebra commands prompt for a variable name or formula. If you
22014 answer the prompt with a blank line, the variable or formula is taken
22015 from top-of-stack, and the normal argument for the command is taken
22016 from the second-to-top stack level.
22019 @pindex calc-alg-evaluate
22020 The @kbd{a v} (@code{calc-alg-evaluate}) command performs the normal
22021 default simplifications on a formula; for example, @samp{a - -b} is
22022 changed to @samp{a + b}. These simplifications are normally done
22023 automatically on all Calc results, so this command is useful only if
22024 you have turned default simplifications off with an @kbd{m O}
22025 command. @xref{Simplification Modes}.
22027 It is often more convenient to type @kbd{=}, which is like @kbd{a v}
22028 but which also substitutes stored values for variables in the formula.
22029 Use @kbd{a v} if you want the variables to ignore their stored values.
22031 If you give a numeric prefix argument of 2 to @kbd{a v}, it simplifies
22032 as if in Algebraic Simplification mode. This is equivalent to typing
22033 @kbd{a s}; @pxref{Simplifying Formulas}. If you give a numeric prefix
22034 of 3 or more, it uses Extended Simplification mode (@kbd{a e}).
22036 If you give a negative prefix argument @mathit{-1}, @mathit{-2}, or @mathit{-3},
22037 it simplifies in the corresponding mode but only works on the top-level
22038 function call of the formula. For example, @samp{(2 + 3) * (2 + 3)} will
22039 simplify to @samp{(2 + 3)^2}, without simplifying the sub-formulas
22040 @samp{2 + 3}. As another example, typing @kbd{V R +} to sum the vector
22041 @samp{[1, 2, 3, 4]} produces the formula @samp{reduce(add, [1, 2, 3, 4])}
22042 in No-Simplify mode. Using @kbd{a v} will evaluate this all the way to
22043 10; using @kbd{C-u - a v} will evaluate it only to @samp{1 + 2 + 3 + 4}.
22044 (@xref{Reducing and Mapping}.)
22048 The @kbd{=} command corresponds to the @code{evalv} function, and
22049 the related @kbd{N} command, which is like @kbd{=} but temporarily
22050 disables Symbolic mode (@kbd{m s}) during the evaluation, corresponds
22051 to the @code{evalvn} function. (These commands interpret their prefix
22052 arguments differently than @kbd{a v}; @kbd{=} treats the prefix as
22053 the number of stack elements to evaluate at once, and @kbd{N} treats
22054 it as a temporary different working precision.)
22056 The @code{evalvn} function can take an alternate working precision
22057 as an optional second argument. This argument can be either an
22058 integer, to set the precision absolutely, or a vector containing
22059 a single integer, to adjust the precision relative to the current
22060 precision. Note that @code{evalvn} with a larger than current
22061 precision will do the calculation at this higher precision, but the
22062 result will as usual be rounded back down to the current precision
22063 afterward. For example, @samp{evalvn(pi - 3.1415)} at a precision
22064 of 12 will return @samp{9.265359e-5}; @samp{evalvn(pi - 3.1415, 30)}
22065 will return @samp{9.26535897932e-5} (computing a 25-digit result which
22066 is then rounded down to 12); and @samp{evalvn(pi - 3.1415, [-2])}
22067 will return @samp{9.2654e-5}.
22070 @pindex calc-expand-formula
22071 The @kbd{a "} (@code{calc-expand-formula}) command expands functions
22072 into their defining formulas wherever possible. For example,
22073 @samp{deg(x^2)} is changed to @samp{180 x^2 / pi}. Most functions,
22074 like @code{sin} and @code{gcd}, are not defined by simple formulas
22075 and so are unaffected by this command. One important class of
22076 functions which @emph{can} be expanded is the user-defined functions
22077 created by the @kbd{Z F} command. @xref{Algebraic Definitions}.
22078 Other functions which @kbd{a "} can expand include the probability
22079 distribution functions, most of the financial functions, and the
22080 hyperbolic and inverse hyperbolic functions. A numeric prefix argument
22081 affects @kbd{a "} in the same way as it does @kbd{a v}: A positive
22082 argument expands all functions in the formula and then simplifies in
22083 various ways; a negative argument expands and simplifies only the
22084 top-level function call.
22087 @pindex calc-map-equation
22089 The @kbd{a M} (@code{calc-map-equation}) [@code{mapeq}] command applies
22090 a given function or operator to one or more equations. It is analogous
22091 to @kbd{V M}, which operates on vectors instead of equations.
22092 @pxref{Reducing and Mapping}. For example, @kbd{a M S} changes
22093 @samp{x = y+1} to @samp{sin(x) = sin(y+1)}, and @kbd{a M +} with
22094 @samp{x = y+1} and @expr{6} on the stack produces @samp{x+6 = y+7}.
22095 With two equations on the stack, @kbd{a M +} would add the lefthand
22096 sides together and the righthand sides together to get the two
22097 respective sides of a new equation.
22099 Mapping also works on inequalities. Mapping two similar inequalities
22100 produces another inequality of the same type. Mapping an inequality
22101 with an equation produces an inequality of the same type. Mapping a
22102 @samp{<=} with a @samp{<} or @samp{!=} (not-equal) produces a @samp{<}.
22103 If inequalities with opposite direction (e.g., @samp{<} and @samp{>})
22104 are mapped, the direction of the second inequality is reversed to
22105 match the first: Using @kbd{a M +} on @samp{a < b} and @samp{a > 2}
22106 reverses the latter to get @samp{2 < a}, which then allows the
22107 combination @samp{a + 2 < b + a}, which the @kbd{a s} command can
22108 then simplify to get @samp{2 < b}.
22110 Using @kbd{a M *}, @kbd{a M /}, @kbd{a M n}, or @kbd{a M &} to negate
22111 or invert an inequality will reverse the direction of the inequality.
22112 Other adjustments to inequalities are @emph{not} done automatically;
22113 @kbd{a M S} will change @w{@samp{x < y}} to @samp{sin(x) < sin(y)} even
22114 though this is not true for all values of the variables.
22118 With the Hyperbolic flag, @kbd{H a M} [@code{mapeqp}] does a plain
22119 mapping operation without reversing the direction of any inequalities.
22120 Thus, @kbd{H a M &} would change @kbd{x > 2} to @kbd{1/x > 0.5}.
22121 (This change is mathematically incorrect, but perhaps you were
22122 fixing an inequality which was already incorrect.)
22126 With the Inverse flag, @kbd{I a M} [@code{mapeqr}] always reverses
22127 the direction of the inequality. You might use @kbd{I a M C} to
22128 change @samp{x < y} to @samp{cos(x) > cos(y)} if you know you are
22129 working with small positive angles.
22132 @pindex calc-substitute
22134 The @kbd{a b} (@code{calc-substitute}) [@code{subst}] command substitutes
22136 of some variable or sub-expression of an expression with a new
22137 sub-expression. For example, substituting @samp{sin(x)} with @samp{cos(y)}
22138 in @samp{2 sin(x)^2 + x sin(x) + sin(2 x)} produces
22139 @samp{2 cos(y)^2 + x cos(y) + @w{sin(2 x)}}.
22140 Note that this is a purely structural substitution; the lone @samp{x} and
22141 the @samp{sin(2 x)} stayed the same because they did not look like
22142 @samp{sin(x)}. @xref{Rewrite Rules}, for a more general method for
22143 doing substitutions.
22145 The @kbd{a b} command normally prompts for two formulas, the old
22146 one and the new one. If you enter a blank line for the first
22147 prompt, all three arguments are taken from the stack (new, then old,
22148 then target expression). If you type an old formula but then enter a
22149 blank line for the new one, the new formula is taken from top-of-stack
22150 and the target from second-to-top. If you answer both prompts, the
22151 target is taken from top-of-stack as usual.
22153 Note that @kbd{a b} has no understanding of commutativity or
22154 associativity. The pattern @samp{x+y} will not match the formula
22155 @samp{y+x}. Also, @samp{y+z} will not match inside the formula @samp{x+y+z}
22156 because the @samp{+} operator is left-associative, so the ``deep
22157 structure'' of that formula is @samp{(x+y) + z}. Use @kbd{d U}
22158 (@code{calc-unformatted-language}) mode to see the true structure of
22159 a formula. The rewrite rule mechanism, discussed later, does not have
22162 As an algebraic function, @code{subst} takes three arguments:
22163 Target expression, old, new. Note that @code{subst} is always
22164 evaluated immediately, even if its arguments are variables, so if
22165 you wish to put a call to @code{subst} onto the stack you must
22166 turn the default simplifications off first (with @kbd{m O}).
22168 @node Simplifying Formulas, Polynomials, Algebraic Manipulation, Algebra
22169 @section Simplifying Formulas
22173 @pindex calc-simplify
22175 The @kbd{a s} (@code{calc-simplify}) [@code{simplify}] command applies
22176 various algebraic rules to simplify a formula. This includes rules which
22177 are not part of the default simplifications because they may be too slow
22178 to apply all the time, or may not be desirable all of the time. For
22179 example, non-adjacent terms of sums are combined, as in @samp{a + b + 2 a}
22180 to @samp{b + 3 a}, and some formulas like @samp{sin(arcsin(x))} are
22181 simplified to @samp{x}.
22183 The sections below describe all the various kinds of algebraic
22184 simplifications Calc provides in full detail. None of Calc's
22185 simplification commands are designed to pull rabbits out of hats;
22186 they simply apply certain specific rules to put formulas into
22187 less redundant or more pleasing forms. Serious algebra in Calc
22188 must be done manually, usually with a combination of selections
22189 and rewrite rules. @xref{Rearranging with Selections}.
22190 @xref{Rewrite Rules}.
22192 @xref{Simplification Modes}, for commands to control what level of
22193 simplification occurs automatically. Normally only the ``default
22194 simplifications'' occur.
22197 * Default Simplifications::
22198 * Algebraic Simplifications::
22199 * Unsafe Simplifications::
22200 * Simplification of Units::
22203 @node Default Simplifications, Algebraic Simplifications, Simplifying Formulas, Simplifying Formulas
22204 @subsection Default Simplifications
22207 @cindex Default simplifications
22208 This section describes the ``default simplifications,'' those which are
22209 normally applied to all results. For example, if you enter the variable
22210 @expr{x} on the stack twice and push @kbd{+}, Calc's default
22211 simplifications automatically change @expr{x + x} to @expr{2 x}.
22213 The @kbd{m O} command turns off the default simplifications, so that
22214 @expr{x + x} will remain in this form unless you give an explicit
22215 ``simplify'' command like @kbd{=} or @kbd{a v}. @xref{Algebraic
22216 Manipulation}. The @kbd{m D} command turns the default simplifications
22219 The most basic default simplification is the evaluation of functions.
22220 For example, @expr{2 + 3} is evaluated to @expr{5}, and @expr{@tfn{sqrt}(9)}
22221 is evaluated to @expr{3}. Evaluation does not occur if the arguments
22222 to a function are somehow of the wrong type @expr{@tfn{tan}([2,3,4])}),
22223 range (@expr{@tfn{tan}(90)}), or number (@expr{@tfn{tan}(3,5)}),
22224 or if the function name is not recognized (@expr{@tfn{f}(5)}), or if
22225 Symbolic mode (@pxref{Symbolic Mode}) prevents evaluation
22226 (@expr{@tfn{sqrt}(2)}).
22228 Calc simplifies (evaluates) the arguments to a function before it
22229 simplifies the function itself. Thus @expr{@tfn{sqrt}(5+4)} is
22230 simplified to @expr{@tfn{sqrt}(9)} before the @code{sqrt} function
22231 itself is applied. There are very few exceptions to this rule:
22232 @code{quote}, @code{lambda}, and @code{condition} (the @code{::}
22233 operator) do not evaluate their arguments, @code{if} (the @code{? :}
22234 operator) does not evaluate all of its arguments, and @code{evalto}
22235 does not evaluate its lefthand argument.
22237 Most commands apply the default simplifications to all arguments they
22238 take from the stack, perform a particular operation, then simplify
22239 the result before pushing it back on the stack. In the common special
22240 case of regular arithmetic commands like @kbd{+} and @kbd{Q} [@code{sqrt}],
22241 the arguments are simply popped from the stack and collected into a
22242 suitable function call, which is then simplified (the arguments being
22243 simplified first as part of the process, as described above).
22245 The default simplifications are too numerous to describe completely
22246 here, but this section will describe the ones that apply to the
22247 major arithmetic operators. This list will be rather technical in
22248 nature, and will probably be interesting to you only if you are
22249 a serious user of Calc's algebra facilities.
22255 As well as the simplifications described here, if you have stored
22256 any rewrite rules in the variable @code{EvalRules} then these rules
22257 will also be applied before any built-in default simplifications.
22258 @xref{Automatic Rewrites}, for details.
22264 And now, on with the default simplifications:
22266 Arithmetic operators like @kbd{+} and @kbd{*} always take two
22267 arguments in Calc's internal form. Sums and products of three or
22268 more terms are arranged by the associative law of algebra into
22269 a left-associative form for sums, @expr{((a + b) + c) + d}, and
22270 (by default) a right-associative form for products,
22271 @expr{a * (b * (c * d))}. Formulas like @expr{(a + b) + (c + d)} are
22272 rearranged to left-associative form, though this rarely matters since
22273 Calc's algebra commands are designed to hide the inner structure of sums
22274 and products as much as possible. Sums and products in their proper
22275 associative form will be written without parentheses in the examples
22278 Sums and products are @emph{not} rearranged according to the
22279 commutative law (@expr{a + b} to @expr{b + a}) except in a few
22280 special cases described below. Some algebra programs always
22281 rearrange terms into a canonical order, which enables them to
22282 see that @expr{a b + b a} can be simplified to @expr{2 a b}.
22283 Calc assumes you have put the terms into the order you want
22284 and generally leaves that order alone, with the consequence
22285 that formulas like the above will only be simplified if you
22286 explicitly give the @kbd{a s} command. @xref{Algebraic
22289 Differences @expr{a - b} are treated like sums @expr{a + (-b)}
22290 for purposes of simplification; one of the default simplifications
22291 is to rewrite @expr{a + (-b)} or @expr{(-b) + a}, where @expr{-b}
22292 represents a ``negative-looking'' term, into @expr{a - b} form.
22293 ``Negative-looking'' means negative numbers, negated formulas like
22294 @expr{-x}, and products or quotients in which either term is
22297 Other simplifications involving negation are @expr{-(-x)} to @expr{x};
22298 @expr{-(a b)} or @expr{-(a/b)} where either @expr{a} or @expr{b} is
22299 negative-looking, simplified by negating that term, or else where
22300 @expr{a} or @expr{b} is any number, by negating that number;
22301 @expr{-(a + b)} to @expr{-a - b}, and @expr{-(b - a)} to @expr{a - b}.
22302 (This, and rewriting @expr{(-b) + a} to @expr{a - b}, are the only
22303 cases where the order of terms in a sum is changed by the default
22306 The distributive law is used to simplify sums in some cases:
22307 @expr{a x + b x} to @expr{(a + b) x}, where @expr{a} represents
22308 a number or an implicit 1 or @mathit{-1} (as in @expr{x} or @expr{-x})
22309 and similarly for @expr{b}. Use the @kbd{a c}, @w{@kbd{a f}}, or
22310 @kbd{j M} commands to merge sums with non-numeric coefficients
22311 using the distributive law.
22313 The distributive law is only used for sums of two terms, or
22314 for adjacent terms in a larger sum. Thus @expr{a + b + b + c}
22315 is simplified to @expr{a + 2 b + c}, but @expr{a + b + c + b}
22316 is not simplified. The reason is that comparing all terms of a
22317 sum with one another would require time proportional to the
22318 square of the number of terms; Calc relegates potentially slow
22319 operations like this to commands that have to be invoked
22320 explicitly, like @kbd{a s}.
22322 Finally, @expr{a + 0} and @expr{0 + a} are simplified to @expr{a}.
22323 A consequence of the above rules is that @expr{0 - a} is simplified
22330 The products @expr{1 a} and @expr{a 1} are simplified to @expr{a};
22331 @expr{(-1) a} and @expr{a (-1)} are simplified to @expr{-a};
22332 @expr{0 a} and @expr{a 0} are simplified to @expr{0}, except that
22333 in Matrix mode where @expr{a} is not provably scalar the result
22334 is the generic zero matrix @samp{idn(0)}, and that if @expr{a} is
22335 infinite the result is @samp{nan}.
22337 Also, @expr{(-a) b} and @expr{a (-b)} are simplified to @expr{-(a b)},
22338 where this occurs for negated formulas but not for regular negative
22341 Products are commuted only to move numbers to the front:
22342 @expr{a b 2} is commuted to @expr{2 a b}.
22344 The product @expr{a (b + c)} is distributed over the sum only if
22345 @expr{a} and at least one of @expr{b} and @expr{c} are numbers:
22346 @expr{2 (x + 3)} goes to @expr{2 x + 6}. The formula
22347 @expr{(-a) (b - c)}, where @expr{-a} is a negative number, is
22348 rewritten to @expr{a (c - b)}.
22350 The distributive law of products and powers is used for adjacent
22351 terms of the product: @expr{x^a x^b} goes to
22352 @texline @math{x^{a+b}}
22353 @infoline @expr{x^(a+b)}
22354 where @expr{a} is a number, or an implicit 1 (as in @expr{x}),
22355 or the implicit one-half of @expr{@tfn{sqrt}(x)}, and similarly for
22356 @expr{b}. The result is written using @samp{sqrt} or @samp{1/sqrt}
22357 if the sum of the powers is @expr{1/2} or @expr{-1/2}, respectively.
22358 If the sum of the powers is zero, the product is simplified to
22359 @expr{1} or to @samp{idn(1)} if Matrix mode is enabled.
22361 The product of a negative power times anything but another negative
22362 power is changed to use division:
22363 @texline @math{x^{-2} y}
22364 @infoline @expr{x^(-2) y}
22365 goes to @expr{y / x^2} unless Matrix mode is
22366 in effect and neither @expr{x} nor @expr{y} are scalar (in which
22367 case it is considered unsafe to rearrange the order of the terms).
22369 Finally, @expr{a (b/c)} is rewritten to @expr{(a b)/c}, and also
22370 @expr{(a/b) c} is changed to @expr{(a c)/b} unless in Matrix mode.
22376 Simplifications for quotients are analogous to those for products.
22377 The quotient @expr{0 / x} is simplified to @expr{0}, with the same
22378 exceptions that were noted for @expr{0 x}. Likewise, @expr{x / 1}
22379 and @expr{x / (-1)} are simplified to @expr{x} and @expr{-x},
22382 The quotient @expr{x / 0} is left unsimplified or changed to an
22383 infinite quantity, as directed by the current infinite mode.
22384 @xref{Infinite Mode}.
22387 @texline @math{a / b^{-c}}
22388 @infoline @expr{a / b^(-c)}
22389 is changed to @expr{a b^c}, where @expr{-c} is any negative-looking
22390 power. Also, @expr{1 / b^c} is changed to
22391 @texline @math{b^{-c}}
22392 @infoline @expr{b^(-c)}
22393 for any power @expr{c}.
22395 Also, @expr{(-a) / b} and @expr{a / (-b)} go to @expr{-(a/b)};
22396 @expr{(a/b) / c} goes to @expr{a / (b c)}; and @expr{a / (b/c)}
22397 goes to @expr{(a c) / b} unless Matrix mode prevents this
22398 rearrangement. Similarly, @expr{a / (b:c)} is simplified to
22399 @expr{(c:b) a} for any fraction @expr{b:c}.
22401 The distributive law is applied to @expr{(a + b) / c} only if
22402 @expr{c} and at least one of @expr{a} and @expr{b} are numbers.
22403 Quotients of powers and square roots are distributed just as
22404 described for multiplication.
22406 Quotients of products cancel only in the leading terms of the
22407 numerator and denominator. In other words, @expr{a x b / a y b}
22408 is cancelled to @expr{x b / y b} but not to @expr{x / y}. Once
22409 again this is because full cancellation can be slow; use @kbd{a s}
22410 to cancel all terms of the quotient.
22412 Quotients of negative-looking values are simplified according
22413 to @expr{(-a) / (-b)} to @expr{a / b}, @expr{(-a) / (b - c)}
22414 to @expr{a / (c - b)}, and @expr{(a - b) / (-c)} to @expr{(b - a) / c}.
22420 The formula @expr{x^0} is simplified to @expr{1}, or to @samp{idn(1)}
22421 in Matrix mode. The formula @expr{0^x} is simplified to @expr{0}
22422 unless @expr{x} is a negative number, complex number or zero.
22423 If @expr{x} is negative, complex or @expr{0.0}, @expr{0^x} is an
22424 infinity or an unsimplified formula according to the current infinite
22425 mode. The expression @expr{0^0} is simplified to @expr{1}.
22427 Powers of products or quotients @expr{(a b)^c}, @expr{(a/b)^c}
22428 are distributed to @expr{a^c b^c}, @expr{a^c / b^c} only if @expr{c}
22429 is an integer, or if either @expr{a} or @expr{b} are nonnegative
22430 real numbers. Powers of powers @expr{(a^b)^c} are simplified to
22431 @texline @math{a^{b c}}
22432 @infoline @expr{a^(b c)}
22433 only when @expr{c} is an integer and @expr{b c} also
22434 evaluates to an integer. Without these restrictions these simplifications
22435 would not be safe because of problems with principal values.
22437 @texline @math{((-3)^{1/2})^2}
22438 @infoline @expr{((-3)^1:2)^2}
22439 is safe to simplify, but
22440 @texline @math{((-3)^2)^{1/2}}
22441 @infoline @expr{((-3)^2)^1:2}
22442 is not.) @xref{Declarations}, for ways to inform Calc that your
22443 variables satisfy these requirements.
22445 As a special case of this rule, @expr{@tfn{sqrt}(x)^n} is simplified to
22446 @texline @math{x^{n/2}}
22447 @infoline @expr{x^(n/2)}
22448 only for even integers @expr{n}.
22450 If @expr{a} is known to be real, @expr{b} is an even integer, and
22451 @expr{c} is a half- or quarter-integer, then @expr{(a^b)^c} is
22452 simplified to @expr{@tfn{abs}(a^(b c))}.
22454 Also, @expr{(-a)^b} is simplified to @expr{a^b} if @expr{b} is an
22455 even integer, or to @expr{-(a^b)} if @expr{b} is an odd integer,
22456 for any negative-looking expression @expr{-a}.
22458 Square roots @expr{@tfn{sqrt}(x)} generally act like one-half powers
22459 @texline @math{x^{1:2}}
22460 @infoline @expr{x^1:2}
22461 for the purposes of the above-listed simplifications.
22464 @texline @math{1 / x^{1:2}}
22465 @infoline @expr{1 / x^1:2}
22467 @texline @math{x^{-1:2}},
22468 @infoline @expr{x^(-1:2)},
22469 but @expr{1 / @tfn{sqrt}(x)} is left alone.
22475 Generic identity matrices (@pxref{Matrix Mode}) are simplified by the
22476 following rules: @expr{@tfn{idn}(a) + b} to @expr{a + b} if @expr{b}
22477 is provably scalar, or expanded out if @expr{b} is a matrix;
22478 @expr{@tfn{idn}(a) + @tfn{idn}(b)} to @expr{@tfn{idn}(a + b)};
22479 @expr{-@tfn{idn}(a)} to @expr{@tfn{idn}(-a)}; @expr{a @tfn{idn}(b)} to
22480 @expr{@tfn{idn}(a b)} if @expr{a} is provably scalar, or to @expr{a b}
22481 if @expr{a} is provably non-scalar; @expr{@tfn{idn}(a) @tfn{idn}(b)} to
22482 @expr{@tfn{idn}(a b)}; analogous simplifications for quotients involving
22483 @code{idn}; and @expr{@tfn{idn}(a)^n} to @expr{@tfn{idn}(a^n)} where
22484 @expr{n} is an integer.
22490 The @code{floor} function and other integer truncation functions
22491 vanish if the argument is provably integer-valued, so that
22492 @expr{@tfn{floor}(@tfn{round}(x))} simplifies to @expr{@tfn{round}(x)}.
22493 Also, combinations of @code{float}, @code{floor} and its friends,
22494 and @code{ffloor} and its friends, are simplified in appropriate
22495 ways. @xref{Integer Truncation}.
22497 The expression @expr{@tfn{abs}(-x)} changes to @expr{@tfn{abs}(x)}.
22498 The expression @expr{@tfn{abs}(@tfn{abs}(x))} changes to
22499 @expr{@tfn{abs}(x)}; in fact, @expr{@tfn{abs}(x)} changes to @expr{x} or
22500 @expr{-x} if @expr{x} is provably nonnegative or nonpositive
22501 (@pxref{Declarations}).
22503 While most functions do not recognize the variable @code{i} as an
22504 imaginary number, the @code{arg} function does handle the two cases
22505 @expr{@tfn{arg}(@tfn{i})} and @expr{@tfn{arg}(-@tfn{i})} just for convenience.
22507 The expression @expr{@tfn{conj}(@tfn{conj}(x))} simplifies to @expr{x}.
22508 Various other expressions involving @code{conj}, @code{re}, and
22509 @code{im} are simplified, especially if some of the arguments are
22510 provably real or involve the constant @code{i}. For example,
22511 @expr{@tfn{conj}(a + b i)} is changed to
22512 @expr{@tfn{conj}(a) - @tfn{conj}(b) i}, or to @expr{a - b i} if @expr{a}
22513 and @expr{b} are known to be real.
22515 Functions like @code{sin} and @code{arctan} generally don't have
22516 any default simplifications beyond simply evaluating the functions
22517 for suitable numeric arguments and infinity. The @kbd{a s} command
22518 described in the next section does provide some simplifications for
22519 these functions, though.
22521 One important simplification that does occur is that
22522 @expr{@tfn{ln}(@tfn{e})} is simplified to 1, and @expr{@tfn{ln}(@tfn{e}^x)} is
22523 simplified to @expr{x} for any @expr{x}. This occurs even if you have
22524 stored a different value in the Calc variable @samp{e}; but this would
22525 be a bad idea in any case if you were also using natural logarithms!
22527 Among the logical functions, @tfn{!(@var{a} <= @var{b})} changes to
22528 @tfn{@var{a} > @var{b}} and so on. Equations and inequalities where both sides
22529 are either negative-looking or zero are simplified by negating both sides
22530 and reversing the inequality. While it might seem reasonable to simplify
22531 @expr{!!x} to @expr{x}, this would not be valid in general because
22532 @expr{!!2} is 1, not 2.
22534 Most other Calc functions have few if any default simplifications
22535 defined, aside of course from evaluation when the arguments are
22538 @node Algebraic Simplifications, Unsafe Simplifications, Default Simplifications, Simplifying Formulas
22539 @subsection Algebraic Simplifications
22542 @cindex Algebraic simplifications
22543 The @kbd{a s} command makes simplifications that may be too slow to
22544 do all the time, or that may not be desirable all of the time.
22545 If you find these simplifications are worthwhile, you can type
22546 @kbd{m A} to have Calc apply them automatically.
22548 This section describes all simplifications that are performed by
22549 the @kbd{a s} command. Note that these occur in addition to the
22550 default simplifications; even if the default simplifications have
22551 been turned off by an @kbd{m O} command, @kbd{a s} will turn them
22552 back on temporarily while it simplifies the formula.
22554 There is a variable, @code{AlgSimpRules}, in which you can put rewrites
22555 to be applied by @kbd{a s}. Its use is analogous to @code{EvalRules},
22556 but without the special restrictions. Basically, the simplifier does
22557 @samp{@w{a r} AlgSimpRules} with an infinite repeat count on the whole
22558 expression being simplified, then it traverses the expression applying
22559 the built-in rules described below. If the result is different from
22560 the original expression, the process repeats with the default
22561 simplifications (including @code{EvalRules}), then @code{AlgSimpRules},
22562 then the built-in simplifications, and so on.
22568 Sums are simplified in two ways. Constant terms are commuted to the
22569 end of the sum, so that @expr{a + 2 + b} changes to @expr{a + b + 2}.
22570 The only exception is that a constant will not be commuted away
22571 from the first position of a difference, i.e., @expr{2 - x} is not
22572 commuted to @expr{-x + 2}.
22574 Also, terms of sums are combined by the distributive law, as in
22575 @expr{x + y + 2 x} to @expr{y + 3 x}. This always occurs for
22576 adjacent terms, but @kbd{a s} compares all pairs of terms including
22583 Products are sorted into a canonical order using the commutative
22584 law. For example, @expr{b c a} is commuted to @expr{a b c}.
22585 This allows easier comparison of products; for example, the default
22586 simplifications will not change @expr{x y + y x} to @expr{2 x y},
22587 but @kbd{a s} will; it first rewrites the sum to @expr{x y + x y},
22588 and then the default simplifications are able to recognize a sum
22589 of identical terms.
22591 The canonical ordering used to sort terms of products has the
22592 property that real-valued numbers, interval forms and infinities
22593 come first, and are sorted into increasing order. The @kbd{V S}
22594 command uses the same ordering when sorting a vector.
22596 Sorting of terms of products is inhibited when Matrix mode is
22597 turned on; in this case, Calc will never exchange the order of
22598 two terms unless it knows at least one of the terms is a scalar.
22600 Products of powers are distributed by comparing all pairs of
22601 terms, using the same method that the default simplifications
22602 use for adjacent terms of products.
22604 Even though sums are not sorted, the commutative law is still
22605 taken into account when terms of a product are being compared.
22606 Thus @expr{(x + y) (y + x)} will be simplified to @expr{(x + y)^2}.
22607 A subtle point is that @expr{(x - y) (y - x)} will @emph{not}
22608 be simplified to @expr{-(x - y)^2}; Calc does not notice that
22609 one term can be written as a constant times the other, even if
22610 that constant is @mathit{-1}.
22612 A fraction times any expression, @expr{(a:b) x}, is changed to
22613 a quotient involving integers: @expr{a x / b}. This is not
22614 done for floating-point numbers like @expr{0.5}, however. This
22615 is one reason why you may find it convenient to turn Fraction mode
22616 on while doing algebra; @pxref{Fraction Mode}.
22622 Quotients are simplified by comparing all terms in the numerator
22623 with all terms in the denominator for possible cancellation using
22624 the distributive law. For example, @expr{a x^2 b / c x^3 d} will
22625 cancel @expr{x^2} from the top and bottom to get @expr{a b / c x d}.
22626 (The terms in the denominator will then be rearranged to @expr{c d x}
22627 as described above.) If there is any common integer or fractional
22628 factor in the numerator and denominator, it is cancelled out;
22629 for example, @expr{(4 x + 6) / 8 x} simplifies to @expr{(2 x + 3) / 4 x}.
22631 Non-constant common factors are not found even by @kbd{a s}. To
22632 cancel the factor @expr{a} in @expr{(a x + a) / a^2} you could first
22633 use @kbd{j M} on the product @expr{a x} to Merge the numerator to
22634 @expr{a (1+x)}, which can then be simplified successfully.
22640 Integer powers of the variable @code{i} are simplified according
22641 to the identity @expr{i^2 = -1}. If you store a new value other
22642 than the complex number @expr{(0,1)} in @code{i}, this simplification
22643 will no longer occur. This is done by @kbd{a s} instead of by default
22644 in case someone (unwisely) uses the name @code{i} for a variable
22645 unrelated to complex numbers; it would be unfortunate if Calc
22646 quietly and automatically changed this formula for reasons the
22647 user might not have been thinking of.
22649 Square roots of integer or rational arguments are simplified in
22650 several ways. (Note that these will be left unevaluated only in
22651 Symbolic mode.) First, square integer or rational factors are
22652 pulled out so that @expr{@tfn{sqrt}(8)} is rewritten as
22653 @texline @math{2\,@tfn{sqrt}(2)}.
22654 @infoline @expr{2 sqrt(2)}.
22655 Conceptually speaking this implies factoring the argument into primes
22656 and moving pairs of primes out of the square root, but for reasons of
22657 efficiency Calc only looks for primes up to 29.
22659 Square roots in the denominator of a quotient are moved to the
22660 numerator: @expr{1 / @tfn{sqrt}(3)} changes to @expr{@tfn{sqrt}(3) / 3}.
22661 The same effect occurs for the square root of a fraction:
22662 @expr{@tfn{sqrt}(2:3)} changes to @expr{@tfn{sqrt}(6) / 3}.
22668 The @code{%} (modulo) operator is simplified in several ways
22669 when the modulus @expr{M} is a positive real number. First, if
22670 the argument is of the form @expr{x + n} for some real number
22671 @expr{n}, then @expr{n} is itself reduced modulo @expr{M}. For
22672 example, @samp{(x - 23) % 10} is simplified to @samp{(x + 7) % 10}.
22674 If the argument is multiplied by a constant, and this constant
22675 has a common integer divisor with the modulus, then this factor is
22676 cancelled out. For example, @samp{12 x % 15} is changed to
22677 @samp{3 (4 x % 5)} by factoring out 3. Also, @samp{(12 x + 1) % 15}
22678 is changed to @samp{3 ((4 x + 1:3) % 5)}. While these forms may
22679 not seem ``simpler,'' they allow Calc to discover useful information
22680 about modulo forms in the presence of declarations.
22682 If the modulus is 1, then Calc can use @code{int} declarations to
22683 evaluate the expression. For example, the idiom @samp{x % 2} is
22684 often used to check whether a number is odd or even. As described
22685 above, @w{@samp{2 n % 2}} and @samp{(2 n + 1) % 2} are simplified to
22686 @samp{2 (n % 1)} and @samp{2 ((n + 1:2) % 1)}, respectively; Calc
22687 can simplify these to 0 and 1 (respectively) if @code{n} has been
22688 declared to be an integer.
22694 Trigonometric functions are simplified in several ways. Whenever a
22695 products of two trigonometric functions can be replaced by a single
22696 function, the replacement is made; for example,
22697 @expr{@tfn{tan}(x) @tfn{cos}(x)} is simplified to @expr{@tfn{sin}(x)}.
22698 Reciprocals of trigonometric functions are replaced by their reciprocal
22699 function; for example, @expr{1/@tfn{sec}(x)} is simplified to
22700 @expr{@tfn{cos}(x)}. The corresponding simplifications for the
22701 hyperbolic functions are also handled.
22703 Trigonometric functions of their inverse functions are
22704 simplified. The expression @expr{@tfn{sin}(@tfn{arcsin}(x))} is
22705 simplified to @expr{x}, and similarly for @code{cos} and @code{tan}.
22706 Trigonometric functions of inverses of different trigonometric
22707 functions can also be simplified, as in @expr{@tfn{sin}(@tfn{arccos}(x))}
22708 to @expr{@tfn{sqrt}(1 - x^2)}.
22710 If the argument to @code{sin} is negative-looking, it is simplified to
22711 @expr{-@tfn{sin}(x)}, and similarly for @code{cos} and @code{tan}.
22712 Finally, certain special values of the argument are recognized;
22713 @pxref{Trigonometric and Hyperbolic Functions}.
22715 Hyperbolic functions of their inverses and of negative-looking
22716 arguments are also handled, as are exponentials of inverse
22717 hyperbolic functions.
22719 No simplifications for inverse trigonometric and hyperbolic
22720 functions are known, except for negative arguments of @code{arcsin},
22721 @code{arctan}, @code{arcsinh}, and @code{arctanh}. Note that
22722 @expr{@tfn{arcsin}(@tfn{sin}(x))} can @emph{not} safely change to
22723 @expr{x}, since this only correct within an integer multiple of
22724 @texline @math{2 \pi}
22725 @infoline @expr{2 pi}
22726 radians or 360 degrees. However, @expr{@tfn{arcsinh}(@tfn{sinh}(x))} is
22727 simplified to @expr{x} if @expr{x} is known to be real.
22729 Several simplifications that apply to logarithms and exponentials
22730 are that @expr{@tfn{exp}(@tfn{ln}(x))},
22731 @texline @tfn{e}@math{^{\ln(x)}},
22732 @infoline @expr{e^@tfn{ln}(x)},
22734 @texline @math{10^{{\rm log10}(x)}}
22735 @infoline @expr{10^@tfn{log10}(x)}
22736 all reduce to @expr{x}. Also, @expr{@tfn{ln}(@tfn{exp}(x))}, etc., can
22737 reduce to @expr{x} if @expr{x} is provably real. The form
22738 @expr{@tfn{exp}(x)^y} is simplified to @expr{@tfn{exp}(x y)}. If @expr{x}
22739 is a suitable multiple of
22740 @texline @math{\pi i}
22741 @infoline @expr{pi i}
22742 (as described above for the trigonometric functions), then
22743 @expr{@tfn{exp}(x)} or @expr{e^x} will be expanded. Finally,
22744 @expr{@tfn{ln}(x)} is simplified to a form involving @code{pi} and
22745 @code{i} where @expr{x} is provably negative, positive imaginary, or
22746 negative imaginary.
22748 The error functions @code{erf} and @code{erfc} are simplified when
22749 their arguments are negative-looking or are calls to the @code{conj}
22756 Equations and inequalities are simplified by cancelling factors
22757 of products, quotients, or sums on both sides. Inequalities
22758 change sign if a negative multiplicative factor is cancelled.
22759 Non-constant multiplicative factors as in @expr{a b = a c} are
22760 cancelled from equations only if they are provably nonzero (generally
22761 because they were declared so; @pxref{Declarations}). Factors
22762 are cancelled from inequalities only if they are nonzero and their
22765 Simplification also replaces an equation or inequality with
22766 1 or 0 (``true'' or ``false'') if it can through the use of
22767 declarations. If @expr{x} is declared to be an integer greater
22768 than 5, then @expr{x < 3}, @expr{x = 3}, and @expr{x = 7.5} are
22769 all simplified to 0, but @expr{x > 3} is simplified to 1.
22770 By a similar analysis, @expr{abs(x) >= 0} is simplified to 1,
22771 as is @expr{x^2 >= 0} if @expr{x} is known to be real.
22773 @node Unsafe Simplifications, Simplification of Units, Algebraic Simplifications, Simplifying Formulas
22774 @subsection ``Unsafe'' Simplifications
22777 @cindex Unsafe simplifications
22778 @cindex Extended simplification
22780 @pindex calc-simplify-extended
22782 @mindex esimpl@idots
22785 The @kbd{a e} (@code{calc-simplify-extended}) [@code{esimplify}] command
22787 except that it applies some additional simplifications which are not
22788 ``safe'' in all cases. Use this only if you know the values in your
22789 formula lie in the restricted ranges for which these simplifications
22790 are valid. The symbolic integrator uses @kbd{a e};
22791 one effect of this is that the integrator's results must be used with
22792 caution. Where an integral table will often attach conditions like
22793 ``for positive @expr{a} only,'' Calc (like most other symbolic
22794 integration programs) will simply produce an unqualified result.
22796 Because @kbd{a e}'s simplifications are unsafe, it is sometimes better
22797 to type @kbd{C-u -3 a v}, which does extended simplification only
22798 on the top level of the formula without affecting the sub-formulas.
22799 In fact, @kbd{C-u -3 j v} allows you to target extended simplification
22800 to any specific part of a formula.
22802 The variable @code{ExtSimpRules} contains rewrites to be applied by
22803 the @kbd{a e} command. These are applied in addition to
22804 @code{EvalRules} and @code{AlgSimpRules}. (The @kbd{a r AlgSimpRules}
22805 step described above is simply followed by an @kbd{a r ExtSimpRules} step.)
22807 Following is a complete list of ``unsafe'' simplifications performed
22814 Inverse trigonometric or hyperbolic functions, called with their
22815 corresponding non-inverse functions as arguments, are simplified
22816 by @kbd{a e}. For example, @expr{@tfn{arcsin}(@tfn{sin}(x))} changes
22817 to @expr{x}. Also, @expr{@tfn{arcsin}(@tfn{cos}(x))} and
22818 @expr{@tfn{arccos}(@tfn{sin}(x))} both change to @expr{@tfn{pi}/2 - x}.
22819 These simplifications are unsafe because they are valid only for
22820 values of @expr{x} in a certain range; outside that range, values
22821 are folded down to the 360-degree range that the inverse trigonometric
22822 functions always produce.
22824 Powers of powers @expr{(x^a)^b} are simplified to
22825 @texline @math{x^{a b}}
22826 @infoline @expr{x^(a b)}
22827 for all @expr{a} and @expr{b}. These results will be valid only
22828 in a restricted range of @expr{x}; for example, in
22829 @texline @math{(x^2)^{1:2}}
22830 @infoline @expr{(x^2)^1:2}
22831 the powers cancel to get @expr{x}, which is valid for positive values
22832 of @expr{x} but not for negative or complex values.
22834 Similarly, @expr{@tfn{sqrt}(x^a)} and @expr{@tfn{sqrt}(x)^a} are both
22835 simplified (possibly unsafely) to
22836 @texline @math{x^{a/2}}.
22837 @infoline @expr{x^(a/2)}.
22839 Forms like @expr{@tfn{sqrt}(1 - sin(x)^2)} are simplified to, e.g.,
22840 @expr{@tfn{cos}(x)}. Calc has identities of this sort for @code{sin},
22841 @code{cos}, @code{tan}, @code{sinh}, and @code{cosh}.
22843 Arguments of square roots are partially factored to look for
22844 squared terms that can be extracted. For example,
22845 @expr{@tfn{sqrt}(a^2 b^3 + a^3 b^2)} simplifies to
22846 @expr{a b @tfn{sqrt}(a+b)}.
22848 The simplifications of @expr{@tfn{ln}(@tfn{exp}(x))},
22849 @expr{@tfn{ln}(@tfn{e}^x)}, and @expr{@tfn{log10}(10^x)} to @expr{x} are also
22850 unsafe because of problems with principal values (although these
22851 simplifications are safe if @expr{x} is known to be real).
22853 Common factors are cancelled from products on both sides of an
22854 equation, even if those factors may be zero: @expr{a x / b x}
22855 to @expr{a / b}. Such factors are never cancelled from
22856 inequalities: Even @kbd{a e} is not bold enough to reduce
22857 @expr{a x < b x} to @expr{a < b} (or @expr{a > b}, depending
22858 on whether you believe @expr{x} is positive or negative).
22859 The @kbd{a M /} command can be used to divide a factor out of
22860 both sides of an inequality.
22862 @node Simplification of Units, , Unsafe Simplifications, Simplifying Formulas
22863 @subsection Simplification of Units
22866 The simplifications described in this section are applied by the
22867 @kbd{u s} (@code{calc-simplify-units}) command. These are in addition
22868 to the regular @kbd{a s} (but not @kbd{a e}) simplifications described
22869 earlier. @xref{Basic Operations on Units}.
22871 The variable @code{UnitSimpRules} contains rewrites to be applied by
22872 the @kbd{u s} command. These are applied in addition to @code{EvalRules}
22873 and @code{AlgSimpRules}.
22875 Scalar mode is automatically put into effect when simplifying units.
22876 @xref{Matrix Mode}.
22878 Sums @expr{a + b} involving units are simplified by extracting the
22879 units of @expr{a} as if by the @kbd{u x} command (call the result
22880 @expr{u_a}), then simplifying the expression @expr{b / u_a}
22881 using @kbd{u b} and @kbd{u s}. If the result has units then the sum
22882 is inconsistent and is left alone. Otherwise, it is rewritten
22883 in terms of the units @expr{u_a}.
22885 If units auto-ranging mode is enabled, products or quotients in
22886 which the first argument is a number which is out of range for the
22887 leading unit are modified accordingly.
22889 When cancelling and combining units in products and quotients,
22890 Calc accounts for unit names that differ only in the prefix letter.
22891 For example, @samp{2 km m} is simplified to @samp{2000 m^2}.
22892 However, compatible but different units like @code{ft} and @code{in}
22893 are not combined in this way.
22895 Quotients @expr{a / b} are simplified in three additional ways. First,
22896 if @expr{b} is a number or a product beginning with a number, Calc
22897 computes the reciprocal of this number and moves it to the numerator.
22899 Second, for each pair of unit names from the numerator and denominator
22900 of a quotient, if the units are compatible (e.g., they are both
22901 units of area) then they are replaced by the ratio between those
22902 units. For example, in @samp{3 s in N / kg cm} the units
22903 @samp{in / cm} will be replaced by @expr{2.54}.
22905 Third, if the units in the quotient exactly cancel out, so that
22906 a @kbd{u b} command on the quotient would produce a dimensionless
22907 number for an answer, then the quotient simplifies to that number.
22909 For powers and square roots, the ``unsafe'' simplifications
22910 @expr{(a b)^c} to @expr{a^c b^c}, @expr{(a/b)^c} to @expr{a^c / b^c},
22911 and @expr{(a^b)^c} to
22912 @texline @math{a^{b c}}
22913 @infoline @expr{a^(b c)}
22914 are done if the powers are real numbers. (These are safe in the context
22915 of units because all numbers involved can reasonably be assumed to be
22918 Also, if a unit name is raised to a fractional power, and the
22919 base units in that unit name all occur to powers which are a
22920 multiple of the denominator of the power, then the unit name
22921 is expanded out into its base units, which can then be simplified
22922 according to the previous paragraph. For example, @samp{acre^1.5}
22923 is simplified by noting that @expr{1.5 = 3:2}, that @samp{acre}
22924 is defined in terms of @samp{m^2}, and that the 2 in the power of
22925 @code{m} is a multiple of 2 in @expr{3:2}. Thus, @code{acre^1.5} is
22926 replaced by approximately
22927 @texline @math{(4046 m^2)^{1.5}}
22928 @infoline @expr{(4046 m^2)^1.5},
22929 which is then changed to
22930 @texline @math{4046^{1.5} \, (m^2)^{1.5}},
22931 @infoline @expr{4046^1.5 (m^2)^1.5},
22932 then to @expr{257440 m^3}.
22934 The functions @code{float}, @code{frac}, @code{clean}, @code{abs},
22935 as well as @code{floor} and the other integer truncation functions,
22936 applied to unit names or products or quotients involving units, are
22937 simplified. For example, @samp{round(1.6 in)} is changed to
22938 @samp{round(1.6) round(in)}; the lefthand term evaluates to 2,
22939 and the righthand term simplifies to @code{in}.
22941 The functions @code{sin}, @code{cos}, and @code{tan} with arguments
22942 that have angular units like @code{rad} or @code{arcmin} are
22943 simplified by converting to base units (radians), then evaluating
22944 with the angular mode temporarily set to radians.
22946 @node Polynomials, Calculus, Simplifying Formulas, Algebra
22947 @section Polynomials
22949 A @dfn{polynomial} is a sum of terms which are coefficients times
22950 various powers of a ``base'' variable. For example, @expr{2 x^2 + 3 x - 4}
22951 is a polynomial in @expr{x}. Some formulas can be considered
22952 polynomials in several different variables: @expr{1 + 2 x + 3 y + 4 x y^2}
22953 is a polynomial in both @expr{x} and @expr{y}. Polynomial coefficients
22954 are often numbers, but they may in general be any formulas not
22955 involving the base variable.
22958 @pindex calc-factor
22960 The @kbd{a f} (@code{calc-factor}) [@code{factor}] command factors a
22961 polynomial into a product of terms. For example, the polynomial
22962 @expr{x^3 + 2 x^2 + x} is factored into @samp{x*(x+1)^2}. As another
22963 example, @expr{a c + b d + b c + a d} is factored into the product
22964 @expr{(a + b) (c + d)}.
22966 Calc currently has three algorithms for factoring. Formulas which are
22967 linear in several variables, such as the second example above, are
22968 merged according to the distributive law. Formulas which are
22969 polynomials in a single variable, with constant integer or fractional
22970 coefficients, are factored into irreducible linear and/or quadratic
22971 terms. The first example above factors into three linear terms
22972 (@expr{x}, @expr{x+1}, and @expr{x+1} again). Finally, formulas
22973 which do not fit the above criteria are handled by the algebraic
22976 Calc's polynomial factorization algorithm works by using the general
22977 root-finding command (@w{@kbd{a P}}) to solve for the roots of the
22978 polynomial. It then looks for roots which are rational numbers
22979 or complex-conjugate pairs, and converts these into linear and
22980 quadratic terms, respectively. Because it uses floating-point
22981 arithmetic, it may be unable to find terms that involve large
22982 integers (whose number of digits approaches the current precision).
22983 Also, irreducible factors of degree higher than quadratic are not
22984 found, and polynomials in more than one variable are not treated.
22985 (A more robust factorization algorithm may be included in a future
22988 @vindex FactorRules
23000 The rewrite-based factorization method uses rules stored in the variable
23001 @code{FactorRules}. @xref{Rewrite Rules}, for a discussion of the
23002 operation of rewrite rules. The default @code{FactorRules} are able
23003 to factor quadratic forms symbolically into two linear terms,
23004 @expr{(a x + b) (c x + d)}. You can edit these rules to include other
23005 cases if you wish. To use the rules, Calc builds the formula
23006 @samp{thecoefs(x, [a, b, c, ...])} where @code{x} is the polynomial
23007 base variable and @code{a}, @code{b}, etc., are polynomial coefficients
23008 (which may be numbers or formulas). The constant term is written first,
23009 i.e., in the @code{a} position. When the rules complete, they should have
23010 changed the formula into the form @samp{thefactors(x, [f1, f2, f3, ...])}
23011 where each @code{fi} should be a factored term, e.g., @samp{x - ai}.
23012 Calc then multiplies these terms together to get the complete
23013 factored form of the polynomial. If the rules do not change the
23014 @code{thecoefs} call to a @code{thefactors} call, @kbd{a f} leaves the
23015 polynomial alone on the assumption that it is unfactorable. (Note that
23016 the function names @code{thecoefs} and @code{thefactors} are used only
23017 as placeholders; there are no actual Calc functions by those names.)
23021 The @kbd{H a f} [@code{factors}] command also factors a polynomial,
23022 but it returns a list of factors instead of an expression which is the
23023 product of the factors. Each factor is represented by a sub-vector
23024 of the factor, and the power with which it appears. For example,
23025 @expr{x^5 + x^4 - 33 x^3 + 63 x^2} factors to @expr{(x + 7) x^2 (x - 3)^2}
23026 in @kbd{a f}, or to @expr{[ [x, 2], [x+7, 1], [x-3, 2] ]} in @kbd{H a f}.
23027 If there is an overall numeric factor, it always comes first in the list.
23028 The functions @code{factor} and @code{factors} allow a second argument
23029 when written in algebraic form; @samp{factor(x,v)} factors @expr{x} with
23030 respect to the specific variable @expr{v}. The default is to factor with
23031 respect to all the variables that appear in @expr{x}.
23034 @pindex calc-collect
23036 The @kbd{a c} (@code{calc-collect}) [@code{collect}] command rearranges a
23038 polynomial in a given variable, ordered in decreasing powers of that
23039 variable. For example, given @expr{1 + 2 x + 3 y + 4 x y^2} on
23040 the stack, @kbd{a c x} would produce @expr{(2 + 4 y^2) x + (1 + 3 y)},
23041 and @kbd{a c y} would produce @expr{(4 x) y^2 + 3 y + (1 + 2 x)}.
23042 The polynomial will be expanded out using the distributive law as
23043 necessary: Collecting @expr{x} in @expr{(x - 1)^3} produces
23044 @expr{x^3 - 3 x^2 + 3 x - 1}. Terms not involving @expr{x} will
23047 The ``variable'' you specify at the prompt can actually be any
23048 expression: @kbd{a c ln(x+1)} will collect together all terms multiplied
23049 by @samp{ln(x+1)} or integer powers thereof. If @samp{x} also appears
23050 in the formula in a context other than @samp{ln(x+1)}, @kbd{a c} will
23051 treat those occurrences as unrelated to @samp{ln(x+1)}, i.e., as constants.
23054 @pindex calc-expand
23056 The @kbd{a x} (@code{calc-expand}) [@code{expand}] command expands an
23057 expression by applying the distributive law everywhere. It applies to
23058 products, quotients, and powers involving sums. By default, it fully
23059 distributes all parts of the expression. With a numeric prefix argument,
23060 the distributive law is applied only the specified number of times, then
23061 the partially expanded expression is left on the stack.
23063 The @kbd{a x} and @kbd{j D} commands are somewhat redundant. Use
23064 @kbd{a x} if you want to expand all products of sums in your formula.
23065 Use @kbd{j D} if you want to expand a particular specified term of
23066 the formula. There is an exactly analogous correspondence between
23067 @kbd{a f} and @kbd{j M}. (The @kbd{j D} and @kbd{j M} commands
23068 also know many other kinds of expansions, such as
23069 @samp{exp(a + b) = exp(a) exp(b)}, which @kbd{a x} and @kbd{a f}
23072 Calc's automatic simplifications will sometimes reverse a partial
23073 expansion. For example, the first step in expanding @expr{(x+1)^3} is
23074 to write @expr{(x+1) (x+1)^2}. If @kbd{a x} stops there and tries
23075 to put this formula onto the stack, though, Calc will automatically
23076 simplify it back to @expr{(x+1)^3} form. The solution is to turn
23077 simplification off first (@pxref{Simplification Modes}), or to run
23078 @kbd{a x} without a numeric prefix argument so that it expands all
23079 the way in one step.
23084 The @kbd{a a} (@code{calc-apart}) [@code{apart}] command expands a
23085 rational function by partial fractions. A rational function is the
23086 quotient of two polynomials; @code{apart} pulls this apart into a
23087 sum of rational functions with simple denominators. In algebraic
23088 notation, the @code{apart} function allows a second argument that
23089 specifies which variable to use as the ``base''; by default, Calc
23090 chooses the base variable automatically.
23093 @pindex calc-normalize-rat
23095 The @kbd{a n} (@code{calc-normalize-rat}) [@code{nrat}] command
23096 attempts to arrange a formula into a quotient of two polynomials.
23097 For example, given @expr{1 + (a + b/c) / d}, the result would be
23098 @expr{(b + a c + c d) / c d}. The quotient is reduced, so that
23099 @kbd{a n} will simplify @expr{(x^2 + 2x + 1) / (x^2 - 1)} by dividing
23100 out the common factor @expr{x + 1}, yielding @expr{(x + 1) / (x - 1)}.
23103 @pindex calc-poly-div
23105 The @kbd{a \} (@code{calc-poly-div}) [@code{pdiv}] command divides
23106 two polynomials @expr{u} and @expr{v}, yielding a new polynomial
23107 @expr{q}. If several variables occur in the inputs, the inputs are
23108 considered multivariate polynomials. (Calc divides by the variable
23109 with the largest power in @expr{u} first, or, in the case of equal
23110 powers, chooses the variables in alphabetical order.) For example,
23111 dividing @expr{x^2 + 3 x + 2} by @expr{x + 2} yields @expr{x + 1}.
23112 The remainder from the division, if any, is reported at the bottom
23113 of the screen and is also placed in the Trail along with the quotient.
23115 Using @code{pdiv} in algebraic notation, you can specify the particular
23116 variable to be used as the base: @code{pdiv(@var{a},@var{b},@var{x})}.
23117 If @code{pdiv} is given only two arguments (as is always the case with
23118 the @kbd{a \} command), then it does a multivariate division as outlined
23122 @pindex calc-poly-rem
23124 The @kbd{a %} (@code{calc-poly-rem}) [@code{prem}] command divides
23125 two polynomials and keeps the remainder @expr{r}. The quotient
23126 @expr{q} is discarded. For any formulas @expr{a} and @expr{b}, the
23127 results of @kbd{a \} and @kbd{a %} satisfy @expr{a = q b + r}.
23128 (This is analogous to plain @kbd{\} and @kbd{%}, which compute the
23129 integer quotient and remainder from dividing two numbers.)
23133 @pindex calc-poly-div-rem
23136 The @kbd{a /} (@code{calc-poly-div-rem}) [@code{pdivrem}] command
23137 divides two polynomials and reports both the quotient and the
23138 remainder as a vector @expr{[q, r]}. The @kbd{H a /} [@code{pdivide}]
23139 command divides two polynomials and constructs the formula
23140 @expr{q + r/b} on the stack. (Naturally if the remainder is zero,
23141 this will immediately simplify to @expr{q}.)
23144 @pindex calc-poly-gcd
23146 The @kbd{a g} (@code{calc-poly-gcd}) [@code{pgcd}] command computes
23147 the greatest common divisor of two polynomials. (The GCD actually
23148 is unique only to within a constant multiplier; Calc attempts to
23149 choose a GCD which will be unsurprising.) For example, the @kbd{a n}
23150 command uses @kbd{a g} to take the GCD of the numerator and denominator
23151 of a quotient, then divides each by the result using @kbd{a \}. (The
23152 definition of GCD ensures that this division can take place without
23153 leaving a remainder.)
23155 While the polynomials used in operations like @kbd{a /} and @kbd{a g}
23156 often have integer coefficients, this is not required. Calc can also
23157 deal with polynomials over the rationals or floating-point reals.
23158 Polynomials with modulo-form coefficients are also useful in many
23159 applications; if you enter @samp{(x^2 + 3 x - 1) mod 5}, Calc
23160 automatically transforms this into a polynomial over the field of
23161 integers mod 5: @samp{(1 mod 5) x^2 + (3 mod 5) x + (4 mod 5)}.
23163 Congratulations and thanks go to Ove Ewerlid
23164 (@code{ewerlid@@mizar.DoCS.UU.SE}), who contributed many of the
23165 polynomial routines used in the above commands.
23167 @xref{Decomposing Polynomials}, for several useful functions for
23168 extracting the individual coefficients of a polynomial.
23170 @node Calculus, Solving Equations, Polynomials, Algebra
23174 The following calculus commands do not automatically simplify their
23175 inputs or outputs using @code{calc-simplify}. You may find it helps
23176 to do this by hand by typing @kbd{a s} or @kbd{a e}. It may also help
23177 to use @kbd{a x} and/or @kbd{a c} to arrange a result in the most
23181 * Differentiation::
23183 * Customizing the Integrator::
23184 * Numerical Integration::
23188 @node Differentiation, Integration, Calculus, Calculus
23189 @subsection Differentiation
23194 @pindex calc-derivative
23197 The @kbd{a d} (@code{calc-derivative}) [@code{deriv}] command computes
23198 the derivative of the expression on the top of the stack with respect to
23199 some variable, which it will prompt you to enter. Normally, variables
23200 in the formula other than the specified differentiation variable are
23201 considered constant, i.e., @samp{deriv(y,x)} is reduced to zero. With
23202 the Hyperbolic flag, the @code{tderiv} (total derivative) operation is used
23203 instead, in which derivatives of variables are not reduced to zero
23204 unless those variables are known to be ``constant,'' i.e., independent
23205 of any other variables. (The built-in special variables like @code{pi}
23206 are considered constant, as are variables that have been declared
23207 @code{const}; @pxref{Declarations}.)
23209 With a numeric prefix argument @var{n}, this command computes the
23210 @var{n}th derivative.
23212 When working with trigonometric functions, it is best to switch to
23213 Radians mode first (with @w{@kbd{m r}}). The derivative of @samp{sin(x)}
23214 in degrees is @samp{(pi/180) cos(x)}, probably not the expected
23217 If you use the @code{deriv} function directly in an algebraic formula,
23218 you can write @samp{deriv(f,x,x0)} which represents the derivative
23219 of @expr{f} with respect to @expr{x}, evaluated at the point
23220 @texline @math{x=x_0}.
23221 @infoline @expr{x=x0}.
23223 If the formula being differentiated contains functions which Calc does
23224 not know, the derivatives of those functions are produced by adding
23225 primes (apostrophe characters). For example, @samp{deriv(f(2x), x)}
23226 produces @samp{2 f'(2 x)}, where the function @code{f'} represents the
23227 derivative of @code{f}.
23229 For functions you have defined with the @kbd{Z F} command, Calc expands
23230 the functions according to their defining formulas unless you have
23231 also defined @code{f'} suitably. For example, suppose we define
23232 @samp{sinc(x) = sin(x)/x} using @kbd{Z F}. If we then differentiate
23233 the formula @samp{sinc(2 x)}, the formula will be expanded to
23234 @samp{sin(2 x) / (2 x)} and differentiated. However, if we also
23235 define @samp{sinc'(x) = dsinc(x)}, say, then Calc will write the
23236 result as @samp{2 dsinc(2 x)}. @xref{Algebraic Definitions}.
23238 For multi-argument functions @samp{f(x,y,z)}, the derivative with respect
23239 to the first argument is written @samp{f'(x,y,z)}; derivatives with
23240 respect to the other arguments are @samp{f'2(x,y,z)} and @samp{f'3(x,y,z)}.
23241 Various higher-order derivatives can be formed in the obvious way, e.g.,
23242 @samp{f'@var{}'(x)} (the second derivative of @code{f}) or
23243 @samp{f'@var{}'2'3(x,y,z)} (@code{f} differentiated with respect to each
23246 @node Integration, Customizing the Integrator, Differentiation, Calculus
23247 @subsection Integration
23251 @pindex calc-integral
23253 The @kbd{a i} (@code{calc-integral}) [@code{integ}] command computes the
23254 indefinite integral of the expression on the top of the stack with
23255 respect to a prompted-for variable. The integrator is not guaranteed to
23256 work for all integrable functions, but it is able to integrate several
23257 large classes of formulas. In particular, any polynomial or rational
23258 function (a polynomial divided by a polynomial) is acceptable.
23259 (Rational functions don't have to be in explicit quotient form, however;
23260 @texline @math{x/(1+x^{-2})}
23261 @infoline @expr{x/(1+x^-2)}
23262 is not strictly a quotient of polynomials, but it is equivalent to
23263 @expr{x^3/(x^2+1)}, which is.) Also, square roots of terms involving
23264 @expr{x} and @expr{x^2} may appear in rational functions being
23265 integrated. Finally, rational functions involving trigonometric or
23266 hyperbolic functions can be integrated.
23268 With an argument (@kbd{C-u a i}), this command will compute the definite
23269 integral of the expression on top of the stack. In this case, the
23270 command will again prompt for an integration variable, then prompt for a
23271 lower limit and an upper limit.
23274 If you use the @code{integ} function directly in an algebraic formula,
23275 you can also write @samp{integ(f,x,v)} which expresses the resulting
23276 indefinite integral in terms of variable @code{v} instead of @code{x}.
23277 With four arguments, @samp{integ(f(x),x,a,b)} represents a definite
23278 integral from @code{a} to @code{b}.
23281 If you use the @code{integ} function directly in an algebraic formula,
23282 you can also write @samp{integ(f,x,v)} which expresses the resulting
23283 indefinite integral in terms of variable @code{v} instead of @code{x}.
23284 With four arguments, @samp{integ(f(x),x,a,b)} represents a definite
23285 integral $\int_a^b f(x) \, dx$.
23288 Please note that the current implementation of Calc's integrator sometimes
23289 produces results that are significantly more complex than they need to
23290 be. For example, the integral Calc finds for
23291 @texline @math{1/(x+\sqrt{x^2+1})}
23292 @infoline @expr{1/(x+sqrt(x^2+1))}
23293 is several times more complicated than the answer Mathematica
23294 returns for the same input, although the two forms are numerically
23295 equivalent. Also, any indefinite integral should be considered to have
23296 an arbitrary constant of integration added to it, although Calc does not
23297 write an explicit constant of integration in its result. For example,
23298 Calc's solution for
23299 @texline @math{1/(1+\tan x)}
23300 @infoline @expr{1/(1+tan(x))}
23301 differs from the solution given in the @emph{CRC Math Tables} by a
23303 @texline @math{\pi i / 2}
23304 @infoline @expr{pi i / 2},
23305 due to a different choice of constant of integration.
23307 The Calculator remembers all the integrals it has done. If conditions
23308 change in a way that would invalidate the old integrals, say, a switch
23309 from Degrees to Radians mode, then they will be thrown out. If you
23310 suspect this is not happening when it should, use the
23311 @code{calc-flush-caches} command; @pxref{Caches}.
23314 Calc normally will pursue integration by substitution or integration by
23315 parts up to 3 nested times before abandoning an approach as fruitless.
23316 If the integrator is taking too long, you can lower this limit by storing
23317 a number (like 2) in the variable @code{IntegLimit}. (The @kbd{s I}
23318 command is a convenient way to edit @code{IntegLimit}.) If this variable
23319 has no stored value or does not contain a nonnegative integer, a limit
23320 of 3 is used. The lower this limit is, the greater the chance that Calc
23321 will be unable to integrate a function it could otherwise handle. Raising
23322 this limit allows the Calculator to solve more integrals, though the time
23323 it takes may grow exponentially. You can monitor the integrator's actions
23324 by creating an Emacs buffer called @code{*Trace*}. If such a buffer
23325 exists, the @kbd{a i} command will write a log of its actions there.
23327 If you want to manipulate integrals in a purely symbolic way, you can
23328 set the integration nesting limit to 0 to prevent all but fast
23329 table-lookup solutions of integrals. You might then wish to define
23330 rewrite rules for integration by parts, various kinds of substitutions,
23331 and so on. @xref{Rewrite Rules}.
23333 @node Customizing the Integrator, Numerical Integration, Integration, Calculus
23334 @subsection Customizing the Integrator
23338 Calc has two built-in rewrite rules called @code{IntegRules} and
23339 @code{IntegAfterRules} which you can edit to define new integration
23340 methods. @xref{Rewrite Rules}. At each step of the integration process,
23341 Calc wraps the current integrand in a call to the fictitious function
23342 @samp{integtry(@var{expr},@var{var})}, where @var{expr} is the
23343 integrand and @var{var} is the integration variable. If your rules
23344 rewrite this to be a plain formula (not a call to @code{integtry}), then
23345 Calc will use this formula as the integral of @var{expr}. For example,
23346 the rule @samp{integtry(mysin(x),x) := -mycos(x)} would define a rule to
23347 integrate a function @code{mysin} that acts like the sine function.
23348 Then, putting @samp{4 mysin(2y+1)} on the stack and typing @kbd{a i y}
23349 will produce the integral @samp{-2 mycos(2y+1)}. Note that Calc has
23350 automatically made various transformations on the integral to allow it
23351 to use your rule; integral tables generally give rules for
23352 @samp{mysin(a x + b)}, but you don't need to use this much generality
23353 in your @code{IntegRules}.
23355 @cindex Exponential integral Ei(x)
23360 As a more serious example, the expression @samp{exp(x)/x} cannot be
23361 integrated in terms of the standard functions, so the ``exponential
23362 integral'' function
23363 @texline @math{{\rm Ei}(x)}
23364 @infoline @expr{Ei(x)}
23365 was invented to describe it.
23366 We can get Calc to do this integral in terms of a made-up @code{Ei}
23367 function by adding the rule @samp{[integtry(exp(x)/x, x) := Ei(x)]}
23368 to @code{IntegRules}. Now entering @samp{exp(2x)/x} on the stack
23369 and typing @kbd{a i x} yields @samp{Ei(2 x)}. This new rule will
23370 work with Calc's various built-in integration methods (such as
23371 integration by substitution) to solve a variety of other problems
23372 involving @code{Ei}: For example, now Calc will also be able to
23373 integrate @samp{exp(exp(x))} and @samp{ln(ln(x))} (to get @samp{Ei(exp(x))}
23374 and @samp{x ln(ln(x)) - Ei(ln(x))}, respectively).
23376 Your rule may do further integration by calling @code{integ}. For
23377 example, @samp{integtry(twice(u),x) := twice(integ(u))} allows Calc
23378 to integrate @samp{twice(sin(x))} to get @samp{twice(-cos(x))}.
23379 Note that @code{integ} was called with only one argument. This notation
23380 is allowed only within @code{IntegRules}; it means ``integrate this
23381 with respect to the same integration variable.'' If Calc is unable
23382 to integrate @code{u}, the integration that invoked @code{IntegRules}
23383 also fails. Thus integrating @samp{twice(f(x))} fails, returning the
23384 unevaluated integral @samp{integ(twice(f(x)), x)}. It is still valid
23385 to call @code{integ} with two or more arguments, however; in this case,
23386 if @code{u} is not integrable, @code{twice} itself will still be
23387 integrated: If the above rule is changed to @samp{... := twice(integ(u,x))},
23388 then integrating @samp{twice(f(x))} will yield @samp{twice(integ(f(x),x))}.
23390 If a rule instead produces the formula @samp{integsubst(@var{sexpr},
23391 @var{svar})}, either replacing the top-level @code{integtry} call or
23392 nested anywhere inside the expression, then Calc will apply the
23393 substitution @samp{@var{u} = @var{sexpr}(@var{svar})} to try to
23394 integrate the original @var{expr}. For example, the rule
23395 @samp{sqrt(a) := integsubst(sqrt(x),x)} says that if Calc ever finds
23396 a square root in the integrand, it should attempt the substitution
23397 @samp{u = sqrt(x)}. (This particular rule is unnecessary because
23398 Calc always tries ``obvious'' substitutions where @var{sexpr} actually
23399 appears in the integrand.) The variable @var{svar} may be the same
23400 as the @var{var} that appeared in the call to @code{integtry}, but
23403 When integrating according to an @code{integsubst}, Calc uses the
23404 equation solver to find the inverse of @var{sexpr} (if the integrand
23405 refers to @var{var} anywhere except in subexpressions that exactly
23406 match @var{sexpr}). It uses the differentiator to find the derivative
23407 of @var{sexpr} and/or its inverse (it has two methods that use one
23408 derivative or the other). You can also specify these items by adding
23409 extra arguments to the @code{integsubst} your rules construct; the
23410 general form is @samp{integsubst(@var{sexpr}, @var{svar}, @var{sinv},
23411 @var{sprime})}, where @var{sinv} is the inverse of @var{sexpr} (still
23412 written as a function of @var{svar}), and @var{sprime} is the
23413 derivative of @var{sexpr} with respect to @var{svar}. If you don't
23414 specify these things, and Calc is not able to work them out on its
23415 own with the information it knows, then your substitution rule will
23416 work only in very specific, simple cases.
23418 Calc applies @code{IntegRules} as if by @kbd{C-u 1 a r IntegRules};
23419 in other words, Calc stops rewriting as soon as any rule in your rule
23420 set succeeds. (If it weren't for this, the @samp{integsubst(sqrt(x),x)}
23421 example above would keep on adding layers of @code{integsubst} calls
23424 @vindex IntegSimpRules
23425 Another set of rules, stored in @code{IntegSimpRules}, are applied
23426 every time the integrator uses @kbd{a s} to simplify an intermediate
23427 result. For example, putting the rule @samp{twice(x) := 2 x} into
23428 @code{IntegSimpRules} would tell Calc to convert the @code{twice}
23429 function into a form it knows whenever integration is attempted.
23431 One more way to influence the integrator is to define a function with
23432 the @kbd{Z F} command (@pxref{Algebraic Definitions}). Calc's
23433 integrator automatically expands such functions according to their
23434 defining formulas, even if you originally asked for the function to
23435 be left unevaluated for symbolic arguments. (Certain other Calc
23436 systems, such as the differentiator and the equation solver, also
23439 @vindex IntegAfterRules
23440 Sometimes Calc is able to find a solution to your integral, but it
23441 expresses the result in a way that is unnecessarily complicated. If
23442 this happens, you can either use @code{integsubst} as described
23443 above to try to hint at a more direct path to the desired result, or
23444 you can use @code{IntegAfterRules}. This is an extra rule set that
23445 runs after the main integrator returns its result; basically, Calc does
23446 an @kbd{a r IntegAfterRules} on the result before showing it to you.
23447 (It also does an @kbd{a s}, without @code{IntegSimpRules}, after that
23448 to further simplify the result.) For example, Calc's integrator
23449 sometimes produces expressions of the form @samp{ln(1+x) - ln(1-x)};
23450 the default @code{IntegAfterRules} rewrite this into the more readable
23451 form @samp{2 arctanh(x)}. Note that, unlike @code{IntegRules},
23452 @code{IntegSimpRules} and @code{IntegAfterRules} are applied any number
23453 of times until no further changes are possible. Rewriting by
23454 @code{IntegAfterRules} occurs only after the main integrator has
23455 finished, not at every step as for @code{IntegRules} and
23456 @code{IntegSimpRules}.
23458 @node Numerical Integration, Taylor Series, Customizing the Integrator, Calculus
23459 @subsection Numerical Integration
23463 @pindex calc-num-integral
23465 If you want a purely numerical answer to an integration problem, you can
23466 use the @kbd{a I} (@code{calc-num-integral}) [@code{ninteg}] command. This
23467 command prompts for an integration variable, a lower limit, and an
23468 upper limit. Except for the integration variable, all other variables
23469 that appear in the integrand formula must have stored values. (A stored
23470 value, if any, for the integration variable itself is ignored.)
23472 Numerical integration works by evaluating your formula at many points in
23473 the specified interval. Calc uses an ``open Romberg'' method; this means
23474 that it does not evaluate the formula actually at the endpoints (so that
23475 it is safe to integrate @samp{sin(x)/x} from zero, for example). Also,
23476 the Romberg method works especially well when the function being
23477 integrated is fairly smooth. If the function is not smooth, Calc will
23478 have to evaluate it at quite a few points before it can accurately
23479 determine the value of the integral.
23481 Integration is much faster when the current precision is small. It is
23482 best to set the precision to the smallest acceptable number of digits
23483 before you use @kbd{a I}. If Calc appears to be taking too long, press
23484 @kbd{C-g} to halt it and try a lower precision. If Calc still appears
23485 to need hundreds of evaluations, check to make sure your function is
23486 well-behaved in the specified interval.
23488 It is possible for the lower integration limit to be @samp{-inf} (minus
23489 infinity). Likewise, the upper limit may be plus infinity. Calc
23490 internally transforms the integral into an equivalent one with finite
23491 limits. However, integration to or across singularities is not supported:
23492 The integral of @samp{1/sqrt(x)} from 0 to 1 exists (it can be found
23493 by Calc's symbolic integrator, for example), but @kbd{a I} will fail
23494 because the integrand goes to infinity at one of the endpoints.
23496 @node Taylor Series, , Numerical Integration, Calculus
23497 @subsection Taylor Series
23501 @pindex calc-taylor
23503 The @kbd{a t} (@code{calc-taylor}) [@code{taylor}] command computes a
23504 power series expansion or Taylor series of a function. You specify the
23505 variable and the desired number of terms. You may give an expression of
23506 the form @samp{@var{var} = @var{a}} or @samp{@var{var} - @var{a}} instead
23507 of just a variable to produce a Taylor expansion about the point @var{a}.
23508 You may specify the number of terms with a numeric prefix argument;
23509 otherwise the command will prompt you for the number of terms. Note that
23510 many series expansions have coefficients of zero for some terms, so you
23511 may appear to get fewer terms than you asked for.
23513 If the @kbd{a i} command is unable to find a symbolic integral for a
23514 function, you can get an approximation by integrating the function's
23517 @node Solving Equations, Numerical Solutions, Calculus, Algebra
23518 @section Solving Equations
23522 @pindex calc-solve-for
23524 @cindex Equations, solving
23525 @cindex Solving equations
23526 The @kbd{a S} (@code{calc-solve-for}) [@code{solve}] command rearranges
23527 an equation to solve for a specific variable. An equation is an
23528 expression of the form @expr{L = R}. For example, the command @kbd{a S x}
23529 will rearrange @expr{y = 3x + 6} to the form, @expr{x = y/3 - 2}. If the
23530 input is not an equation, it is treated like an equation of the
23533 This command also works for inequalities, as in @expr{y < 3x + 6}.
23534 Some inequalities cannot be solved where the analogous equation could
23535 be; for example, solving
23536 @texline @math{a < b \, c}
23537 @infoline @expr{a < b c}
23538 for @expr{b} is impossible
23539 without knowing the sign of @expr{c}. In this case, @kbd{a S} will
23541 @texline @math{b \mathbin{\hbox{\code{!=}}} a/c}
23542 @infoline @expr{b != a/c}
23543 (using the not-equal-to operator) to signify that the direction of the
23544 inequality is now unknown. The inequality
23545 @texline @math{a \le b \, c}
23546 @infoline @expr{a <= b c}
23547 is not even partially solved. @xref{Declarations}, for a way to tell
23548 Calc that the signs of the variables in a formula are in fact known.
23550 Two useful commands for working with the result of @kbd{a S} are
23551 @kbd{a .} (@pxref{Logical Operations}), which converts @expr{x = y/3 - 2}
23552 to @expr{y/3 - 2}, and @kbd{s l} (@pxref{Let Command}) which evaluates
23553 another formula with @expr{x} set equal to @expr{y/3 - 2}.
23556 * Multiple Solutions::
23557 * Solving Systems of Equations::
23558 * Decomposing Polynomials::
23561 @node Multiple Solutions, Solving Systems of Equations, Solving Equations, Solving Equations
23562 @subsection Multiple Solutions
23567 Some equations have more than one solution. The Hyperbolic flag
23568 (@code{H a S}) [@code{fsolve}] tells the solver to report the fully
23569 general family of solutions. It will invent variables @code{n1},
23570 @code{n2}, @dots{}, which represent independent arbitrary integers, and
23571 @code{s1}, @code{s2}, @dots{}, which represent independent arbitrary
23572 signs (either @mathit{+1} or @mathit{-1}). If you don't use the Hyperbolic
23573 flag, Calc will use zero in place of all arbitrary integers, and plus
23574 one in place of all arbitrary signs. Note that variables like @code{n1}
23575 and @code{s1} are not given any special interpretation in Calc except by
23576 the equation solver itself. As usual, you can use the @w{@kbd{s l}}
23577 (@code{calc-let}) command to obtain solutions for various actual values
23578 of these variables.
23580 For example, @kbd{' x^2 = y @key{RET} H a S x @key{RET}} solves to
23581 get @samp{x = s1 sqrt(y)}, indicating that the two solutions to the
23582 equation are @samp{sqrt(y)} and @samp{-sqrt(y)}. Another way to
23583 think about it is that the square-root operation is really a
23584 two-valued function; since every Calc function must return a
23585 single result, @code{sqrt} chooses to return the positive result.
23586 Then @kbd{H a S} doctors this result using @code{s1} to indicate
23587 the full set of possible values of the mathematical square-root.
23589 There is a similar phenomenon going the other direction: Suppose
23590 we solve @samp{sqrt(y) = x} for @code{y}. Calc squares both sides
23591 to get @samp{y = x^2}. This is correct, except that it introduces
23592 some dubious solutions. Consider solving @samp{sqrt(y) = -3}:
23593 Calc will report @expr{y = 9} as a valid solution, which is true
23594 in the mathematical sense of square-root, but false (there is no
23595 solution) for the actual Calc positive-valued @code{sqrt}. This
23596 happens for both @kbd{a S} and @kbd{H a S}.
23598 @cindex @code{GenCount} variable
23608 If you store a positive integer in the Calc variable @code{GenCount},
23609 then Calc will generate formulas of the form @samp{as(@var{n})} for
23610 arbitrary signs, and @samp{an(@var{n})} for arbitrary integers,
23611 where @var{n} represents successive values taken by incrementing
23612 @code{GenCount} by one. While the normal arbitrary sign and
23613 integer symbols start over at @code{s1} and @code{n1} with each
23614 new Calc command, the @code{GenCount} approach will give each
23615 arbitrary value a name that is unique throughout the entire Calc
23616 session. Also, the arbitrary values are function calls instead
23617 of variables, which is advantageous in some cases. For example,
23618 you can make a rewrite rule that recognizes all arbitrary signs
23619 using a pattern like @samp{as(n)}. The @kbd{s l} command only works
23620 on variables, but you can use the @kbd{a b} (@code{calc-substitute})
23621 command to substitute actual values for function calls like @samp{as(3)}.
23623 The @kbd{s G} (@code{calc-edit-GenCount}) command is a convenient
23624 way to create or edit this variable. Press @kbd{C-c C-c} to finish.
23626 If you have not stored a value in @code{GenCount}, or if the value
23627 in that variable is not a positive integer, the regular
23628 @code{s1}/@code{n1} notation is used.
23634 With the Inverse flag, @kbd{I a S} [@code{finv}] treats the expression
23635 on top of the stack as a function of the specified variable and solves
23636 to find the inverse function, written in terms of the same variable.
23637 For example, @kbd{I a S x} inverts @expr{2x + 6} to @expr{x/2 - 3}.
23638 You can use both Inverse and Hyperbolic [@code{ffinv}] to obtain a
23639 fully general inverse, as described above.
23642 @pindex calc-poly-roots
23644 Some equations, specifically polynomials, have a known, finite number
23645 of solutions. The @kbd{a P} (@code{calc-poly-roots}) [@code{roots}]
23646 command uses @kbd{H a S} to solve an equation in general form, then, for
23647 all arbitrary-sign variables like @code{s1}, and all arbitrary-integer
23648 variables like @code{n1} for which @code{n1} only usefully varies over
23649 a finite range, it expands these variables out to all their possible
23650 values. The results are collected into a vector, which is returned.
23651 For example, @samp{roots(x^4 = 1, x)} returns the four solutions
23652 @samp{[1, -1, (0, 1), (0, -1)]}. Generally an @var{n}th degree
23653 polynomial will always have @var{n} roots on the complex plane.
23654 (If you have given a @code{real} declaration for the solution
23655 variable, then only the real-valued solutions, if any, will be
23656 reported; @pxref{Declarations}.)
23658 Note that because @kbd{a P} uses @kbd{H a S}, it is able to deliver
23659 symbolic solutions if the polynomial has symbolic coefficients. Also
23660 note that Calc's solver is not able to get exact symbolic solutions
23661 to all polynomials. Polynomials containing powers up to @expr{x^4}
23662 can always be solved exactly; polynomials of higher degree sometimes
23663 can be: @expr{x^6 + x^3 + 1} is converted to @expr{(x^3)^2 + (x^3) + 1},
23664 which can be solved for @expr{x^3} using the quadratic equation, and then
23665 for @expr{x} by taking cube roots. But in many cases, like
23666 @expr{x^6 + x + 1}, Calc does not know how to rewrite the polynomial
23667 into a form it can solve. The @kbd{a P} command can still deliver a
23668 list of numerical roots, however, provided that Symbolic mode (@kbd{m s})
23669 is not turned on. (If you work with Symbolic mode on, recall that the
23670 @kbd{N} (@code{calc-eval-num}) key is a handy way to reevaluate the
23671 formula on the stack with Symbolic mode temporarily off.) Naturally,
23672 @kbd{a P} can only provide numerical roots if the polynomial coefficients
23673 are all numbers (real or complex).
23675 @node Solving Systems of Equations, Decomposing Polynomials, Multiple Solutions, Solving Equations
23676 @subsection Solving Systems of Equations
23679 @cindex Systems of equations, symbolic
23680 You can also use the commands described above to solve systems of
23681 simultaneous equations. Just create a vector of equations, then
23682 specify a vector of variables for which to solve. (You can omit
23683 the surrounding brackets when entering the vector of variables
23686 For example, putting @samp{[x + y = a, x - y = b]} on the stack
23687 and typing @kbd{a S x,y @key{RET}} produces the vector of solutions
23688 @samp{[x = a - (a-b)/2, y = (a-b)/2]}. The result vector will
23689 have the same length as the variables vector, and the variables
23690 will be listed in the same order there. Note that the solutions
23691 are not always simplified as far as possible; the solution for
23692 @expr{x} here could be improved by an application of the @kbd{a n}
23695 Calc's algorithm works by trying to eliminate one variable at a
23696 time by solving one of the equations for that variable and then
23697 substituting into the other equations. Calc will try all the
23698 possibilities, but you can speed things up by noting that Calc
23699 first tries to eliminate the first variable with the first
23700 equation, then the second variable with the second equation,
23701 and so on. It also helps to put the simpler (e.g., more linear)
23702 equations toward the front of the list. Calc's algorithm will
23703 solve any system of linear equations, and also many kinds of
23710 Normally there will be as many variables as equations. If you
23711 give fewer variables than equations (an ``over-determined'' system
23712 of equations), Calc will find a partial solution. For example,
23713 typing @kbd{a S y @key{RET}} with the above system of equations
23714 would produce @samp{[y = a - x]}. There are now several ways to
23715 express this solution in terms of the original variables; Calc uses
23716 the first one that it finds. You can control the choice by adding
23717 variable specifiers of the form @samp{elim(@var{v})} to the
23718 variables list. This says that @var{v} should be eliminated from
23719 the equations; the variable will not appear at all in the solution.
23720 For example, typing @kbd{a S y,elim(x)} would yield
23721 @samp{[y = a - (b+a)/2]}.
23723 If the variables list contains only @code{elim} specifiers,
23724 Calc simply eliminates those variables from the equations
23725 and then returns the resulting set of equations. For example,
23726 @kbd{a S elim(x)} produces @samp{[a - 2 y = b]}. Every variable
23727 eliminated will reduce the number of equations in the system
23730 Again, @kbd{a S} gives you one solution to the system of
23731 equations. If there are several solutions, you can use @kbd{H a S}
23732 to get a general family of solutions, or, if there is a finite
23733 number of solutions, you can use @kbd{a P} to get a list. (In
23734 the latter case, the result will take the form of a matrix where
23735 the rows are different solutions and the columns correspond to the
23736 variables you requested.)
23738 Another way to deal with certain kinds of overdetermined systems of
23739 equations is the @kbd{a F} command, which does least-squares fitting
23740 to satisfy the equations. @xref{Curve Fitting}.
23742 @node Decomposing Polynomials, , Solving Systems of Equations, Solving Equations
23743 @subsection Decomposing Polynomials
23750 The @code{poly} function takes a polynomial and a variable as
23751 arguments, and returns a vector of polynomial coefficients (constant
23752 coefficient first). For example, @samp{poly(x^3 + 2 x, x)} returns
23753 @expr{[0, 2, 0, 1]}. If the input is not a polynomial in @expr{x},
23754 the call to @code{poly} is left in symbolic form. If the input does
23755 not involve the variable @expr{x}, the input is returned in a list
23756 of length one, representing a polynomial with only a constant
23757 coefficient. The call @samp{poly(x, x)} returns the vector @expr{[0, 1]}.
23758 The last element of the returned vector is guaranteed to be nonzero;
23759 note that @samp{poly(0, x)} returns the empty vector @expr{[]}.
23760 Note also that @expr{x} may actually be any formula; for example,
23761 @samp{poly(sin(x)^2 - sin(x) + 3, sin(x))} returns @expr{[3, -1, 1]}.
23763 @cindex Coefficients of polynomial
23764 @cindex Degree of polynomial
23765 To get the @expr{x^k} coefficient of polynomial @expr{p}, use
23766 @samp{poly(p, x)_(k+1)}. To get the degree of polynomial @expr{p},
23767 use @samp{vlen(poly(p, x)) - 1}. For example, @samp{poly((x+1)^4, x)}
23768 returns @samp{[1, 4, 6, 4, 1]}, so @samp{poly((x+1)^4, x)_(2+1)}
23769 gives the @expr{x^2} coefficient of this polynomial, 6.
23775 One important feature of the solver is its ability to recognize
23776 formulas which are ``essentially'' polynomials. This ability is
23777 made available to the user through the @code{gpoly} function, which
23778 is used just like @code{poly}: @samp{gpoly(@var{expr}, @var{var})}.
23779 If @var{expr} is a polynomial in some term which includes @var{var}, then
23780 this function will return a vector @samp{[@var{x}, @var{c}, @var{a}]}
23781 where @var{x} is the term that depends on @var{var}, @var{c} is a
23782 vector of polynomial coefficients (like the one returned by @code{poly}),
23783 and @var{a} is a multiplier which is usually 1. Basically,
23784 @samp{@var{expr} = @var{a}*(@var{c}_1 + @var{c}_2 @var{x} +
23785 @var{c}_3 @var{x}^2 + ...)}. The last element of @var{c} is
23786 guaranteed to be non-zero, and @var{c} will not equal @samp{[1]}
23787 (i.e., the trivial decomposition @var{expr} = @var{x} is not
23788 considered a polynomial). One side effect is that @samp{gpoly(x, x)}
23789 and @samp{gpoly(6, x)}, both of which might be expected to recognize
23790 their arguments as polynomials, will not because the decomposition
23791 is considered trivial.
23793 For example, @samp{gpoly((x-2)^2, x)} returns @samp{[x, [4, -4, 1], 1]},
23794 since the expanded form of this polynomial is @expr{4 - 4 x + x^2}.
23796 The term @var{x} may itself be a polynomial in @var{var}. This is
23797 done to reduce the size of the @var{c} vector. For example,
23798 @samp{gpoly(x^4 + x^2 - 1, x)} returns @samp{[x^2, [-1, 1, 1], 1]},
23799 since a quadratic polynomial in @expr{x^2} is easier to solve than
23800 a quartic polynomial in @expr{x}.
23802 A few more examples of the kinds of polynomials @code{gpoly} can
23806 sin(x) - 1 [sin(x), [-1, 1], 1]
23807 x + 1/x - 1 [x, [1, -1, 1], 1/x]
23808 x + 1/x [x^2, [1, 1], 1/x]
23809 x^3 + 2 x [x^2, [2, 1], x]
23810 x + x^2:3 + sqrt(x) [x^1:6, [1, 1, 0, 1], x^1:2]
23811 x^(2a) + 2 x^a + 5 [x^a, [5, 2, 1], 1]
23812 (exp(-x) + exp(x)) / 2 [e^(2 x), [0.5, 0.5], e^-x]
23815 The @code{poly} and @code{gpoly} functions accept a third integer argument
23816 which specifies the largest degree of polynomial that is acceptable.
23817 If this is @expr{n}, then only @var{c} vectors of length @expr{n+1}
23818 or less will be returned. Otherwise, the @code{poly} or @code{gpoly}
23819 call will remain in symbolic form. For example, the equation solver
23820 can handle quartics and smaller polynomials, so it calls
23821 @samp{gpoly(@var{expr}, @var{var}, 4)} to discover whether @var{expr}
23822 can be treated by its linear, quadratic, cubic, or quartic formulas.
23828 The @code{pdeg} function computes the degree of a polynomial;
23829 @samp{pdeg(p,x)} is the highest power of @code{x} that appears in
23830 @code{p}. This is the same as @samp{vlen(poly(p,x))-1}, but is
23831 much more efficient. If @code{p} is constant with respect to @code{x},
23832 then @samp{pdeg(p,x) = 0}. If @code{p} is not a polynomial in @code{x}
23833 (e.g., @samp{pdeg(2 cos(x), x)}, the function remains unevaluated.
23834 It is possible to omit the second argument @code{x}, in which case
23835 @samp{pdeg(p)} returns the highest total degree of any term of the
23836 polynomial, counting all variables that appear in @code{p}. Note
23837 that @code{pdeg(c) = pdeg(c,x) = 0} for any nonzero constant @code{c};
23838 the degree of the constant zero is considered to be @code{-inf}
23845 The @code{plead} function finds the leading term of a polynomial.
23846 Thus @samp{plead(p,x)} is equivalent to @samp{poly(p,x)_vlen(poly(p,x))},
23847 though again more efficient. In particular, @samp{plead((2x+1)^10, x)}
23848 returns 1024 without expanding out the list of coefficients. The
23849 value of @code{plead(p,x)} will be zero only if @expr{p = 0}.
23855 The @code{pcont} function finds the @dfn{content} of a polynomial. This
23856 is the greatest common divisor of all the coefficients of the polynomial.
23857 With two arguments, @code{pcont(p,x)} effectively uses @samp{poly(p,x)}
23858 to get a list of coefficients, then uses @code{pgcd} (the polynomial
23859 GCD function) to combine these into an answer. For example,
23860 @samp{pcont(4 x y^2 + 6 x^2 y, x)} is @samp{2 y}. The content is
23861 basically the ``biggest'' polynomial that can be divided into @code{p}
23862 exactly. The sign of the content is the same as the sign of the leading
23865 With only one argument, @samp{pcont(p)} computes the numerical
23866 content of the polynomial, i.e., the @code{gcd} of the numerical
23867 coefficients of all the terms in the formula. Note that @code{gcd}
23868 is defined on rational numbers as well as integers; it computes
23869 the @code{gcd} of the numerators and the @code{lcm} of the
23870 denominators. Thus @samp{pcont(4:3 x y^2 + 6 x^2 y)} returns 2:3.
23871 Dividing the polynomial by this number will clear all the
23872 denominators, as well as dividing by any common content in the
23873 numerators. The numerical content of a polynomial is negative only
23874 if all the coefficients in the polynomial are negative.
23880 The @code{pprim} function finds the @dfn{primitive part} of a
23881 polynomial, which is simply the polynomial divided (using @code{pdiv}
23882 if necessary) by its content. If the input polynomial has rational
23883 coefficients, the result will have integer coefficients in simplest
23886 @node Numerical Solutions, Curve Fitting, Solving Equations, Algebra
23887 @section Numerical Solutions
23890 Not all equations can be solved symbolically. The commands in this
23891 section use numerical algorithms that can find a solution to a specific
23892 instance of an equation to any desired accuracy. Note that the
23893 numerical commands are slower than their algebraic cousins; it is a
23894 good idea to try @kbd{a S} before resorting to these commands.
23896 (@xref{Curve Fitting}, for some other, more specialized, operations
23897 on numerical data.)
23902 * Numerical Systems of Equations::
23905 @node Root Finding, Minimization, Numerical Solutions, Numerical Solutions
23906 @subsection Root Finding
23910 @pindex calc-find-root
23912 @cindex Newton's method
23913 @cindex Roots of equations
23914 @cindex Numerical root-finding
23915 The @kbd{a R} (@code{calc-find-root}) [@code{root}] command finds a
23916 numerical solution (or @dfn{root}) of an equation. (This command treats
23917 inequalities the same as equations. If the input is any other kind
23918 of formula, it is interpreted as an equation of the form @expr{X = 0}.)
23920 The @kbd{a R} command requires an initial guess on the top of the
23921 stack, and a formula in the second-to-top position. It prompts for a
23922 solution variable, which must appear in the formula. All other variables
23923 that appear in the formula must have assigned values, i.e., when
23924 a value is assigned to the solution variable and the formula is
23925 evaluated with @kbd{=}, it should evaluate to a number. Any assigned
23926 value for the solution variable itself is ignored and unaffected by
23929 When the command completes, the initial guess is replaced on the stack
23930 by a vector of two numbers: The value of the solution variable that
23931 solves the equation, and the difference between the lefthand and
23932 righthand sides of the equation at that value. Ordinarily, the second
23933 number will be zero or very nearly zero. (Note that Calc uses a
23934 slightly higher precision while finding the root, and thus the second
23935 number may be slightly different from the value you would compute from
23936 the equation yourself.)
23938 The @kbd{v h} (@code{calc-head}) command is a handy way to extract
23939 the first element of the result vector, discarding the error term.
23941 The initial guess can be a real number, in which case Calc searches
23942 for a real solution near that number, or a complex number, in which
23943 case Calc searches the whole complex plane near that number for a
23944 solution, or it can be an interval form which restricts the search
23945 to real numbers inside that interval.
23947 Calc tries to use @kbd{a d} to take the derivative of the equation.
23948 If this succeeds, it uses Newton's method. If the equation is not
23949 differentiable Calc uses a bisection method. (If Newton's method
23950 appears to be going astray, Calc switches over to bisection if it
23951 can, or otherwise gives up. In this case it may help to try again
23952 with a slightly different initial guess.) If the initial guess is a
23953 complex number, the function must be differentiable.
23955 If the formula (or the difference between the sides of an equation)
23956 is negative at one end of the interval you specify and positive at
23957 the other end, the root finder is guaranteed to find a root.
23958 Otherwise, Calc subdivides the interval into small parts looking for
23959 positive and negative values to bracket the root. When your guess is
23960 an interval, Calc will not look outside that interval for a root.
23964 The @kbd{H a R} [@code{wroot}] command is similar to @kbd{a R}, except
23965 that if the initial guess is an interval for which the function has
23966 the same sign at both ends, then rather than subdividing the interval
23967 Calc attempts to widen it to enclose a root. Use this mode if
23968 you are not sure if the function has a root in your interval.
23970 If the function is not differentiable, and you give a simple number
23971 instead of an interval as your initial guess, Calc uses this widening
23972 process even if you did not type the Hyperbolic flag. (If the function
23973 @emph{is} differentiable, Calc uses Newton's method which does not
23974 require a bounding interval in order to work.)
23976 If Calc leaves the @code{root} or @code{wroot} function in symbolic
23977 form on the stack, it will normally display an explanation for why
23978 no root was found. If you miss this explanation, press @kbd{w}
23979 (@code{calc-why}) to get it back.
23981 @node Minimization, Numerical Systems of Equations, Root Finding, Numerical Solutions
23982 @subsection Minimization
23989 @pindex calc-find-minimum
23990 @pindex calc-find-maximum
23993 @cindex Minimization, numerical
23994 The @kbd{a N} (@code{calc-find-minimum}) [@code{minimize}] command
23995 finds a minimum value for a formula. It is very similar in operation
23996 to @kbd{a R} (@code{calc-find-root}): You give the formula and an initial
23997 guess on the stack, and are prompted for the name of a variable. The guess
23998 may be either a number near the desired minimum, or an interval enclosing
23999 the desired minimum. The function returns a vector containing the
24000 value of the variable which minimizes the formula's value, along
24001 with the minimum value itself.
24003 Note that this command looks for a @emph{local} minimum. Many functions
24004 have more than one minimum; some, like
24005 @texline @math{x \sin x},
24006 @infoline @expr{x sin(x)},
24007 have infinitely many. In fact, there is no easy way to define the
24008 ``global'' minimum of
24009 @texline @math{x \sin x}
24010 @infoline @expr{x sin(x)}
24011 but Calc can still locate any particular local minimum
24012 for you. Calc basically goes downhill from the initial guess until it
24013 finds a point at which the function's value is greater both to the left
24014 and to the right. Calc does not use derivatives when minimizing a function.
24016 If your initial guess is an interval and it looks like the minimum
24017 occurs at one or the other endpoint of the interval, Calc will return
24018 that endpoint only if that endpoint is closed; thus, minimizing @expr{17 x}
24019 over @expr{[2..3]} will return @expr{[2, 38]}, but minimizing over
24020 @expr{(2..3]} would report no minimum found. In general, you should
24021 use closed intervals to find literally the minimum value in that
24022 range of @expr{x}, or open intervals to find the local minimum, if
24023 any, that happens to lie in that range.
24025 Most functions are smooth and flat near their minimum values. Because
24026 of this flatness, if the current precision is, say, 12 digits, the
24027 variable can only be determined meaningfully to about six digits. Thus
24028 you should set the precision to twice as many digits as you need in your
24039 The @kbd{H a N} [@code{wminimize}] command, analogously to @kbd{H a R},
24040 expands the guess interval to enclose a minimum rather than requiring
24041 that the minimum lie inside the interval you supply.
24043 The @kbd{a X} (@code{calc-find-maximum}) [@code{maximize}] and
24044 @kbd{H a X} [@code{wmaximize}] commands effectively minimize the
24045 negative of the formula you supply.
24047 The formula must evaluate to a real number at all points inside the
24048 interval (or near the initial guess if the guess is a number). If
24049 the initial guess is a complex number the variable will be minimized
24050 over the complex numbers; if it is real or an interval it will
24051 be minimized over the reals.
24053 @node Numerical Systems of Equations, , Minimization, Numerical Solutions
24054 @subsection Systems of Equations
24057 @cindex Systems of equations, numerical
24058 The @kbd{a R} command can also solve systems of equations. In this
24059 case, the equation should instead be a vector of equations, the
24060 guess should instead be a vector of numbers (intervals are not
24061 supported), and the variable should be a vector of variables. You
24062 can omit the brackets while entering the list of variables. Each
24063 equation must be differentiable by each variable for this mode to
24064 work. The result will be a vector of two vectors: The variable
24065 values that solved the system of equations, and the differences
24066 between the sides of the equations with those variable values.
24067 There must be the same number of equations as variables. Since
24068 only plain numbers are allowed as guesses, the Hyperbolic flag has
24069 no effect when solving a system of equations.
24071 It is also possible to minimize over many variables with @kbd{a N}
24072 (or maximize with @kbd{a X}). Once again the variable name should
24073 be replaced by a vector of variables, and the initial guess should
24074 be an equal-sized vector of initial guesses. But, unlike the case of
24075 multidimensional @kbd{a R}, the formula being minimized should
24076 still be a single formula, @emph{not} a vector. Beware that
24077 multidimensional minimization is currently @emph{very} slow.
24079 @node Curve Fitting, Summations, Numerical Solutions, Algebra
24080 @section Curve Fitting
24083 The @kbd{a F} command fits a set of data to a @dfn{model formula},
24084 such as @expr{y = m x + b} where @expr{m} and @expr{b} are parameters
24085 to be determined. For a typical set of measured data there will be
24086 no single @expr{m} and @expr{b} that exactly fit the data; in this
24087 case, Calc chooses values of the parameters that provide the closest
24088 possible fit. The model formula can be entered in various ways after
24089 the key sequence @kbd{a F} is pressed.
24091 If the letter @kbd{P} is pressed after @kbd{a F} but before the model
24092 description is entered, the data as well as the model formula will be
24093 plotted after the formula is determined. This will be indicated by a
24094 ``P'' in the minibuffer after the help message.
24098 * Polynomial and Multilinear Fits::
24099 * Error Estimates for Fits::
24100 * Standard Nonlinear Models::
24101 * Curve Fitting Details::
24105 @node Linear Fits, Polynomial and Multilinear Fits, Curve Fitting, Curve Fitting
24106 @subsection Linear Fits
24110 @pindex calc-curve-fit
24112 @cindex Linear regression
24113 @cindex Least-squares fits
24114 The @kbd{a F} (@code{calc-curve-fit}) [@code{fit}] command attempts
24115 to fit a set of data (@expr{x} and @expr{y} vectors of numbers) to a
24116 straight line, polynomial, or other function of @expr{x}. For the
24117 moment we will consider only the case of fitting to a line, and we
24118 will ignore the issue of whether or not the model was in fact a good
24121 In a standard linear least-squares fit, we have a set of @expr{(x,y)}
24122 data points that we wish to fit to the model @expr{y = m x + b}
24123 by adjusting the parameters @expr{m} and @expr{b} to make the @expr{y}
24124 values calculated from the formula be as close as possible to the actual
24125 @expr{y} values in the data set. (In a polynomial fit, the model is
24126 instead, say, @expr{y = a x^3 + b x^2 + c x + d}. In a multilinear fit,
24127 we have data points of the form @expr{(x_1,x_2,x_3,y)} and our model is
24128 @expr{y = a x_1 + b x_2 + c x_3 + d}. These will be discussed later.)
24130 In the model formula, variables like @expr{x} and @expr{x_2} are called
24131 the @dfn{independent variables}, and @expr{y} is the @dfn{dependent
24132 variable}. Variables like @expr{m}, @expr{a}, and @expr{b} are called
24133 the @dfn{parameters} of the model.
24135 The @kbd{a F} command takes the data set to be fitted from the stack.
24136 By default, it expects the data in the form of a matrix. For example,
24137 for a linear or polynomial fit, this would be a
24138 @texline @math{2\times N}
24140 matrix where the first row is a list of @expr{x} values and the second
24141 row has the corresponding @expr{y} values. For the multilinear fit
24142 shown above, the matrix would have four rows (@expr{x_1}, @expr{x_2},
24143 @expr{x_3}, and @expr{y}, respectively).
24145 If you happen to have an
24146 @texline @math{N\times2}
24148 matrix instead of a
24149 @texline @math{2\times N}
24151 matrix, just press @kbd{v t} first to transpose the matrix.
24153 After you type @kbd{a F}, Calc prompts you to select a model. For a
24154 linear fit, press the digit @kbd{1}.
24156 Calc then prompts for you to name the variables. By default it chooses
24157 high letters like @expr{x} and @expr{y} for independent variables and
24158 low letters like @expr{a} and @expr{b} for parameters. (The dependent
24159 variable doesn't need a name.) The two kinds of variables are separated
24160 by a semicolon. Since you generally care more about the names of the
24161 independent variables than of the parameters, Calc also allows you to
24162 name only those and let the parameters use default names.
24164 For example, suppose the data matrix
24169 [ [ 1, 2, 3, 4, 5 ]
24170 [ 5, 7, 9, 11, 13 ] ]
24178 $$ \pmatrix{ 1 & 2 & 3 & 4 & 5 \cr
24179 5 & 7 & 9 & 11 & 13 }
24185 is on the stack and we wish to do a simple linear fit. Type
24186 @kbd{a F}, then @kbd{1} for the model, then @key{RET} to use
24187 the default names. The result will be the formula @expr{3. + 2. x}
24188 on the stack. Calc has created the model expression @kbd{a + b x},
24189 then found the optimal values of @expr{a} and @expr{b} to fit the
24190 data. (In this case, it was able to find an exact fit.) Calc then
24191 substituted those values for @expr{a} and @expr{b} in the model
24194 The @kbd{a F} command puts two entries in the trail. One is, as
24195 always, a copy of the result that went to the stack; the other is
24196 a vector of the actual parameter values, written as equations:
24197 @expr{[a = 3, b = 2]}, in case you'd rather read them in a list
24198 than pick them out of the formula. (You can type @kbd{t y}
24199 to move this vector to the stack; see @ref{Trail Commands}.
24201 Specifying a different independent variable name will affect the
24202 resulting formula: @kbd{a F 1 k @key{RET}} produces @kbd{3 + 2 k}.
24203 Changing the parameter names (say, @kbd{a F 1 k;b,m @key{RET}}) will affect
24204 the equations that go into the trail.
24210 To see what happens when the fit is not exact, we could change
24211 the number 13 in the data matrix to 14 and try the fit again.
24218 Evaluating this formula, say with @kbd{v x 5 @key{RET} @key{TAB} V M $ @key{RET}}, shows
24219 a reasonably close match to the y-values in the data.
24222 [4.8, 7., 9.2, 11.4, 13.6]
24225 Since there is no line which passes through all the @var{n} data points,
24226 Calc has chosen a line that best approximates the data points using
24227 the method of least squares. The idea is to define the @dfn{chi-square}
24232 chi^2 = sum((y_i - (a + b x_i))^2, i, 1, N)
24238 $$ \chi^2 = \sum_{i=1}^N (y_i - (a + b x_i))^2 $$
24243 which is clearly zero if @expr{a + b x} exactly fits all data points,
24244 and increases as various @expr{a + b x_i} values fail to match the
24245 corresponding @expr{y_i} values. There are several reasons why the
24246 summand is squared, one of them being to ensure that
24247 @texline @math{\chi^2 \ge 0}.
24248 @infoline @expr{chi^2 >= 0}.
24249 Least-squares fitting simply chooses the values of @expr{a} and @expr{b}
24250 for which the error
24251 @texline @math{\chi^2}
24252 @infoline @expr{chi^2}
24253 is as small as possible.
24255 Other kinds of models do the same thing but with a different model
24256 formula in place of @expr{a + b x_i}.
24262 A numeric prefix argument causes the @kbd{a F} command to take the
24263 data in some other form than one big matrix. A positive argument @var{n}
24264 will take @var{N} items from the stack, corresponding to the @var{n} rows
24265 of a data matrix. In the linear case, @var{n} must be 2 since there
24266 is always one independent variable and one dependent variable.
24268 A prefix of zero or plain @kbd{C-u} is a compromise; Calc takes two
24269 items from the stack, an @var{n}-row matrix of @expr{x} values, and a
24270 vector of @expr{y} values. If there is only one independent variable,
24271 the @expr{x} values can be either a one-row matrix or a plain vector,
24272 in which case the @kbd{C-u} prefix is the same as a @w{@kbd{C-u 2}} prefix.
24274 @node Polynomial and Multilinear Fits, Error Estimates for Fits, Linear Fits, Curve Fitting
24275 @subsection Polynomial and Multilinear Fits
24278 To fit the data to higher-order polynomials, just type one of the
24279 digits @kbd{2} through @kbd{9} when prompted for a model. For example,
24280 we could fit the original data matrix from the previous section
24281 (with 13, not 14) to a parabola instead of a line by typing
24282 @kbd{a F 2 @key{RET}}.
24285 2.00000000001 x - 1.5e-12 x^2 + 2.99999999999
24288 Note that since the constant and linear terms are enough to fit the
24289 data exactly, it's no surprise that Calc chose a tiny contribution
24290 for @expr{x^2}. (The fact that it's not exactly zero is due only
24291 to roundoff error. Since our data are exact integers, we could get
24292 an exact answer by typing @kbd{m f} first to get Fraction mode.
24293 Then the @expr{x^2} term would vanish altogether. Usually, though,
24294 the data being fitted will be approximate floats so Fraction mode
24297 Doing the @kbd{a F 2} fit on the data set with 14 instead of 13
24298 gives a much larger @expr{x^2} contribution, as Calc bends the
24299 line slightly to improve the fit.
24302 0.142857142855 x^2 + 1.34285714287 x + 3.59999999998
24305 An important result from the theory of polynomial fitting is that it
24306 is always possible to fit @var{n} data points exactly using a polynomial
24307 of degree @mathit{@var{n}-1}, sometimes called an @dfn{interpolating polynomial}.
24308 Using the modified (14) data matrix, a model number of 4 gives
24309 a polynomial that exactly matches all five data points:
24312 0.04167 x^4 - 0.4167 x^3 + 1.458 x^2 - 0.08333 x + 4.
24315 The actual coefficients we get with a precision of 12, like
24316 @expr{0.0416666663588}, clearly suffer from loss of precision.
24317 It is a good idea to increase the working precision to several
24318 digits beyond what you need when you do a fitting operation.
24319 Or, if your data are exact, use Fraction mode to get exact
24322 You can type @kbd{i} instead of a digit at the model prompt to fit
24323 the data exactly to a polynomial. This just counts the number of
24324 columns of the data matrix to choose the degree of the polynomial
24327 Fitting data ``exactly'' to high-degree polynomials is not always
24328 a good idea, though. High-degree polynomials have a tendency to
24329 wiggle uncontrollably in between the fitting data points. Also,
24330 if the exact-fit polynomial is going to be used to interpolate or
24331 extrapolate the data, it is numerically better to use the @kbd{a p}
24332 command described below. @xref{Interpolation}.
24338 Another generalization of the linear model is to assume the
24339 @expr{y} values are a sum of linear contributions from several
24340 @expr{x} values. This is a @dfn{multilinear} fit, and it is also
24341 selected by the @kbd{1} digit key. (Calc decides whether the fit
24342 is linear or multilinear by counting the rows in the data matrix.)
24344 Given the data matrix,
24348 [ [ 1, 2, 3, 4, 5 ]
24350 [ 14.5, 15, 18.5, 22.5, 24 ] ]
24355 the command @kbd{a F 1 @key{RET}} will call the first row @expr{x} and the
24356 second row @expr{y}, and will fit the values in the third row to the
24357 model @expr{a + b x + c y}.
24363 Calc can do multilinear fits with any number of independent variables
24364 (i.e., with any number of data rows).
24370 Yet another variation is @dfn{homogeneous} linear models, in which
24371 the constant term is known to be zero. In the linear case, this
24372 means the model formula is simply @expr{a x}; in the multilinear
24373 case, the model might be @expr{a x + b y + c z}; and in the polynomial
24374 case, the model could be @expr{a x + b x^2 + c x^3}. You can get
24375 a homogeneous linear or multilinear model by pressing the letter
24376 @kbd{h} followed by a regular model key, like @kbd{1} or @kbd{2}.
24377 This will be indicated by an ``h'' in the minibuffer after the help
24380 It is certainly possible to have other constrained linear models,
24381 like @expr{2.3 + a x} or @expr{a - 4 x}. While there is no single
24382 key to select models like these, a later section shows how to enter
24383 any desired model by hand. In the first case, for example, you
24384 would enter @kbd{a F ' 2.3 + a x}.
24386 Another class of models that will work but must be entered by hand
24387 are multinomial fits, e.g., @expr{a + b x + c y + d x^2 + e y^2 + f x y}.
24389 @node Error Estimates for Fits, Standard Nonlinear Models, Polynomial and Multilinear Fits, Curve Fitting
24390 @subsection Error Estimates for Fits
24395 With the Hyperbolic flag, @kbd{H a F} [@code{efit}] performs the same
24396 fitting operation as @kbd{a F}, but reports the coefficients as error
24397 forms instead of plain numbers. Fitting our two data matrices (first
24398 with 13, then with 14) to a line with @kbd{H a F} gives the results,
24402 2.6 +/- 0.382970843103 + 2.2 +/- 0.115470053838 x
24405 In the first case the estimated errors are zero because the linear
24406 fit is perfect. In the second case, the errors are nonzero but
24407 moderately small, because the data are still very close to linear.
24409 It is also possible for the @emph{input} to a fitting operation to
24410 contain error forms. The data values must either all include errors
24411 or all be plain numbers. Error forms can go anywhere but generally
24412 go on the numbers in the last row of the data matrix. If the last
24413 row contains error forms
24414 @texline `@var{y_i}@w{ @tfn{+/-} }@math{\sigma_i}',
24415 @infoline `@var{y_i}@w{ @tfn{+/-} }@var{sigma_i}',
24417 @texline @math{\chi^2}
24418 @infoline @expr{chi^2}
24423 chi^2 = sum(((y_i - (a + b x_i)) / sigma_i)^2, i, 1, N)
24429 $$ \chi^2 = \sum_{i=1}^N \left(y_i - (a + b x_i) \over \sigma_i\right)^2 $$
24434 so that data points with larger error estimates contribute less to
24435 the fitting operation.
24437 If there are error forms on other rows of the data matrix, all the
24438 errors for a given data point are combined; the square root of the
24439 sum of the squares of the errors forms the
24440 @texline @math{\sigma_i}
24441 @infoline @expr{sigma_i}
24442 used for the data point.
24444 Both @kbd{a F} and @kbd{H a F} can accept error forms in the input
24445 matrix, although if you are concerned about error analysis you will
24446 probably use @kbd{H a F} so that the output also contains error
24449 If the input contains error forms but all the
24450 @texline @math{\sigma_i}
24451 @infoline @expr{sigma_i}
24452 values are the same, it is easy to see that the resulting fitted model
24453 will be the same as if the input did not have error forms at all
24454 @texline (@math{\chi^2}
24455 @infoline (@expr{chi^2}
24456 is simply scaled uniformly by
24457 @texline @math{1 / \sigma^2},
24458 @infoline @expr{1 / sigma^2},
24459 which doesn't affect where it has a minimum). But there @emph{will} be
24460 a difference in the estimated errors of the coefficients reported by
24463 Consult any text on statistical modeling of data for a discussion
24464 of where these error estimates come from and how they should be
24473 With the Inverse flag, @kbd{I a F} [@code{xfit}] produces even more
24474 information. The result is a vector of six items:
24478 The model formula with error forms for its coefficients or
24479 parameters. This is the result that @kbd{H a F} would have
24483 A vector of ``raw'' parameter values for the model. These are the
24484 polynomial coefficients or other parameters as plain numbers, in the
24485 same order as the parameters appeared in the final prompt of the
24486 @kbd{I a F} command. For polynomials of degree @expr{d}, this vector
24487 will have length @expr{M = d+1} with the constant term first.
24490 The covariance matrix @expr{C} computed from the fit. This is
24491 an @var{m}x@var{m} symmetric matrix; the diagonal elements
24492 @texline @math{C_{jj}}
24493 @infoline @expr{C_j_j}
24495 @texline @math{\sigma_j^2}
24496 @infoline @expr{sigma_j^2}
24497 of the parameters. The other elements are covariances
24498 @texline @math{\sigma_{ij}^2}
24499 @infoline @expr{sigma_i_j^2}
24500 that describe the correlation between pairs of parameters. (A related
24501 set of numbers, the @dfn{linear correlation coefficients}
24502 @texline @math{r_{ij}},
24503 @infoline @expr{r_i_j},
24505 @texline @math{\sigma_{ij}^2 / \sigma_i \, \sigma_j}.)
24506 @infoline @expr{sigma_i_j^2 / sigma_i sigma_j}.)
24509 A vector of @expr{M} ``parameter filter'' functions whose
24510 meanings are described below. If no filters are necessary this
24511 will instead be an empty vector; this is always the case for the
24512 polynomial and multilinear fits described so far.
24516 @texline @math{\chi^2}
24517 @infoline @expr{chi^2}
24518 for the fit, calculated by the formulas shown above. This gives a
24519 measure of the quality of the fit; statisticians consider
24520 @texline @math{\chi^2 \approx N - M}
24521 @infoline @expr{chi^2 = N - M}
24522 to indicate a moderately good fit (where again @expr{N} is the number of
24523 data points and @expr{M} is the number of parameters).
24526 A measure of goodness of fit expressed as a probability @expr{Q}.
24527 This is computed from the @code{utpc} probability distribution
24529 @texline @math{\chi^2}
24530 @infoline @expr{chi^2}
24531 with @expr{N - M} degrees of freedom. A
24532 value of 0.5 implies a good fit; some texts recommend that often
24533 @expr{Q = 0.1} or even 0.001 can signify an acceptable fit. In
24535 @texline @math{\chi^2}
24536 @infoline @expr{chi^2}
24537 statistics assume the errors in your inputs
24538 follow a normal (Gaussian) distribution; if they don't, you may
24539 have to accept smaller values of @expr{Q}.
24541 The @expr{Q} value is computed only if the input included error
24542 estimates. Otherwise, Calc will report the symbol @code{nan}
24543 for @expr{Q}. The reason is that in this case the
24544 @texline @math{\chi^2}
24545 @infoline @expr{chi^2}
24546 value has effectively been used to estimate the original errors
24547 in the input, and thus there is no redundant information left
24548 over to use for a confidence test.
24551 @node Standard Nonlinear Models, Curve Fitting Details, Error Estimates for Fits, Curve Fitting
24552 @subsection Standard Nonlinear Models
24555 The @kbd{a F} command also accepts other kinds of models besides
24556 lines and polynomials. Some common models have quick single-key
24557 abbreviations; others must be entered by hand as algebraic formulas.
24559 Here is a complete list of the standard models recognized by @kbd{a F}:
24563 Linear or multilinear. @mathit{a + b x + c y + d z}.
24565 Polynomials. @mathit{a + b x + c x^2 + d x^3}.
24567 Exponential. @mathit{a} @tfn{exp}@mathit{(b x)} @tfn{exp}@mathit{(c y)}.
24569 Base-10 exponential. @mathit{a} @tfn{10^}@mathit{(b x)} @tfn{10^}@mathit{(c y)}.
24571 Exponential (alternate notation). @tfn{exp}@mathit{(a + b x + c y)}.
24573 Base-10 exponential (alternate). @tfn{10^}@mathit{(a + b x + c y)}.
24575 Logarithmic. @mathit{a + b} @tfn{ln}@mathit{(x) + c} @tfn{ln}@mathit{(y)}.
24577 Base-10 logarithmic. @mathit{a + b} @tfn{log10}@mathit{(x) + c} @tfn{log10}@mathit{(y)}.
24579 General exponential. @mathit{a b^x c^y}.
24581 Power law. @mathit{a x^b y^c}.
24583 Quadratic. @mathit{a + b (x-c)^2 + d (x-e)^2}.
24586 @texline @math{{a \over b \sqrt{2 \pi}} \exp\left( -{1 \over 2} \left( x - c \over b \right)^2 \right)}.
24587 @infoline @mathit{(a / b sqrt(2 pi)) exp(-0.5*((x-c)/b)^2)}.
24589 Logistic @emph{s} curve.
24590 @texline @math{a/(1+e^{b(x-c)})}.
24591 @infoline @mathit{a/(1 + exp(b (x - c)))}.
24593 Logistic bell curve.
24594 @texline @math{ae^{b(x-c)}/(1+e^{b(x-c)})^2}.
24595 @infoline @mathit{a exp(b (x - c))/(1 + exp(b (x - c)))^2}.
24597 Hubbert linearization.
24598 @texline @math{{y \over x} = a(1-x/b)}.
24599 @infoline @mathit{(y/x) = a (1 - x/b)}.
24602 All of these models are used in the usual way; just press the appropriate
24603 letter at the model prompt, and choose variable names if you wish. The
24604 result will be a formula as shown in the above table, with the best-fit
24605 values of the parameters substituted. (You may find it easier to read
24606 the parameter values from the vector that is placed in the trail.)
24608 All models except Gaussian, logistics, Hubbert and polynomials can
24609 generalize as shown to any number of independent variables. Also, all
24610 the built-in models except for the logistic and Hubbert curves have an
24611 additive or multiplicative parameter shown as @expr{a} in the above table
24612 which can be replaced by zero or one, as appropriate, by typing @kbd{h}
24613 before the model key.
24615 Note that many of these models are essentially equivalent, but express
24616 the parameters slightly differently. For example, @expr{a b^x} and
24617 the other two exponential models are all algebraic rearrangements of
24618 each other. Also, the ``quadratic'' model is just a degree-2 polynomial
24619 with the parameters expressed differently. Use whichever form best
24620 matches the problem.
24622 The HP-28/48 calculators support four different models for curve
24623 fitting, called @code{LIN}, @code{LOG}, @code{EXP}, and @code{PWR}.
24624 These correspond to Calc models @samp{a + b x}, @samp{a + b ln(x)},
24625 @samp{a exp(b x)}, and @samp{a x^b}, respectively. In each case,
24626 @expr{a} is what the HP-48 identifies as the ``intercept,'' and
24627 @expr{b} is what it calls the ``slope.''
24633 If the model you want doesn't appear on this list, press @kbd{'}
24634 (the apostrophe key) at the model prompt to enter any algebraic
24635 formula, such as @kbd{m x - b}, as the model. (Not all models
24636 will work, though---see the next section for details.)
24638 The model can also be an equation like @expr{y = m x + b}.
24639 In this case, Calc thinks of all the rows of the data matrix on
24640 equal terms; this model effectively has two parameters
24641 (@expr{m} and @expr{b}) and two independent variables (@expr{x}
24642 and @expr{y}), with no ``dependent'' variables. Model equations
24643 do not need to take this @expr{y =} form. For example, the
24644 implicit line equation @expr{a x + b y = 1} works fine as a
24647 When you enter a model, Calc makes an alphabetical list of all
24648 the variables that appear in the model. These are used for the
24649 default parameters, independent variables, and dependent variable
24650 (in that order). If you enter a plain formula (not an equation),
24651 Calc assumes the dependent variable does not appear in the formula
24652 and thus does not need a name.
24654 For example, if the model formula has the variables @expr{a,mu,sigma,t,x},
24655 and the data matrix has three rows (meaning two independent variables),
24656 Calc will use @expr{a,mu,sigma} as the default parameters, and the
24657 data rows will be named @expr{t} and @expr{x}, respectively. If you
24658 enter an equation instead of a plain formula, Calc will use @expr{a,mu}
24659 as the parameters, and @expr{sigma,t,x} as the three independent
24662 You can, of course, override these choices by entering something
24663 different at the prompt. If you leave some variables out of the list,
24664 those variables must have stored values and those stored values will
24665 be used as constants in the model. (Stored values for the parameters
24666 and independent variables are ignored by the @kbd{a F} command.)
24667 If you list only independent variables, all the remaining variables
24668 in the model formula will become parameters.
24670 If there are @kbd{$} signs in the model you type, they will stand
24671 for parameters and all other variables (in alphabetical order)
24672 will be independent. Use @kbd{$} for one parameter, @kbd{$$} for
24673 another, and so on. Thus @kbd{$ x + $$} is another way to describe
24676 If you type a @kbd{$} instead of @kbd{'} at the model prompt itself,
24677 Calc will take the model formula from the stack. (The data must then
24678 appear at the second stack level.) The same conventions are used to
24679 choose which variables in the formula are independent by default and
24680 which are parameters.
24682 Models taken from the stack can also be expressed as vectors of
24683 two or three elements, @expr{[@var{model}, @var{vars}]} or
24684 @expr{[@var{model}, @var{vars}, @var{params}]}. Each of @var{vars}
24685 and @var{params} may be either a variable or a vector of variables.
24686 (If @var{params} is omitted, all variables in @var{model} except
24687 those listed as @var{vars} are parameters.)
24689 When you enter a model manually with @kbd{'}, Calc puts a 3-vector
24690 describing the model in the trail so you can get it back if you wish.
24698 Finally, you can store a model in one of the Calc variables
24699 @code{Model1} or @code{Model2}, then use this model by typing
24700 @kbd{a F u} or @kbd{a F U} (respectively). The value stored in
24701 the variable can be any of the formats that @kbd{a F $} would
24702 accept for a model on the stack.
24708 Calc uses the principal values of inverse functions like @code{ln}
24709 and @code{arcsin} when doing fits. For example, when you enter
24710 the model @samp{y = sin(a t + b)} Calc actually uses the easier
24711 form @samp{arcsin(y) = a t + b}. The @code{arcsin} function always
24712 returns results in the range from @mathit{-90} to 90 degrees (or the
24713 equivalent range in radians). Suppose you had data that you
24714 believed to represent roughly three oscillations of a sine wave,
24715 so that the argument of the sine might go from zero to
24716 @texline @math{3\times360}
24717 @infoline @mathit{3*360}
24719 The above model would appear to be a good way to determine the
24720 true frequency and phase of the sine wave, but in practice it
24721 would fail utterly. The righthand side of the actual model
24722 @samp{arcsin(y) = a t + b} will grow smoothly with @expr{t}, but
24723 the lefthand side will bounce back and forth between @mathit{-90} and 90.
24724 No values of @expr{a} and @expr{b} can make the two sides match,
24725 even approximately.
24727 There is no good solution to this problem at present. You could
24728 restrict your data to small enough ranges so that the above problem
24729 doesn't occur (i.e., not straddling any peaks in the sine wave).
24730 Or, in this case, you could use a totally different method such as
24731 Fourier analysis, which is beyond the scope of the @kbd{a F} command.
24732 (Unfortunately, Calc does not currently have any facilities for
24733 taking Fourier and related transforms.)
24735 @node Curve Fitting Details, Interpolation, Standard Nonlinear Models, Curve Fitting
24736 @subsection Curve Fitting Details
24739 Calc's internal least-squares fitter can only handle multilinear
24740 models. More precisely, it can handle any model of the form
24741 @expr{a f(x,y,z) + b g(x,y,z) + c h(x,y,z)}, where @expr{a,b,c}
24742 are the parameters and @expr{x,y,z} are the independent variables
24743 (of course there can be any number of each, not just three).
24745 In a simple multilinear or polynomial fit, it is easy to see how
24746 to convert the model into this form. For example, if the model
24747 is @expr{a + b x + c x^2}, then @expr{f(x) = 1}, @expr{g(x) = x},
24748 and @expr{h(x) = x^2} are suitable functions.
24750 For most other models, Calc uses a variety of algebraic manipulations
24751 to try to put the problem into the form
24754 Y(x,y,z) = A(a,b,c) F(x,y,z) + B(a,b,c) G(x,y,z) + C(a,b,c) H(x,y,z)
24758 where @expr{Y,A,B,C,F,G,H} are arbitrary functions. It computes
24759 @expr{Y}, @expr{F}, @expr{G}, and @expr{H} for all the data points,
24760 does a standard linear fit to find the values of @expr{A}, @expr{B},
24761 and @expr{C}, then uses the equation solver to solve for @expr{a,b,c}
24762 in terms of @expr{A,B,C}.
24764 A remarkable number of models can be cast into this general form.
24765 We'll look at two examples here to see how it works. The power-law
24766 model @expr{y = a x^b} with two independent variables and two parameters
24767 can be rewritten as follows:
24772 y = exp(ln(a) + b ln(x))
24773 ln(y) = ln(a) + b ln(x)
24777 which matches the desired form with
24778 @texline @math{Y = \ln(y)},
24779 @infoline @expr{Y = ln(y)},
24780 @texline @math{A = \ln(a)},
24781 @infoline @expr{A = ln(a)},
24782 @expr{F = 1}, @expr{B = b}, and
24783 @texline @math{G = \ln(x)}.
24784 @infoline @expr{G = ln(x)}.
24785 Calc thus computes the logarithms of your @expr{y} and @expr{x} values,
24786 does a linear fit for @expr{A} and @expr{B}, then solves to get
24787 @texline @math{a = \exp(A)}
24788 @infoline @expr{a = exp(A)}
24791 Another interesting example is the ``quadratic'' model, which can
24792 be handled by expanding according to the distributive law.
24795 y = a + b*(x - c)^2
24796 y = a + b c^2 - 2 b c x + b x^2
24800 which matches with @expr{Y = y}, @expr{A = a + b c^2}, @expr{F = 1},
24801 @expr{B = -2 b c}, @expr{G = x} (the @mathit{-2} factor could just as easily
24802 have been put into @expr{G} instead of @expr{B}), @expr{C = b}, and
24805 The Gaussian model looks quite complicated, but a closer examination
24806 shows that it's actually similar to the quadratic model but with an
24807 exponential that can be brought to the top and moved into @expr{Y}.
24809 The logistic models cannot be put into general linear form. For these
24810 models, and the Hubbert linearization, Calc computes a rough
24811 approximation for the parameters, then uses the Levenberg-Marquardt
24812 iterative method to refine the approximations.
24814 Another model that cannot be put into general linear
24815 form is a Gaussian with a constant background added on, i.e.,
24816 @expr{d} + the regular Gaussian formula. If you have a model like
24817 this, your best bet is to replace enough of your parameters with
24818 constants to make the model linearizable, then adjust the constants
24819 manually by doing a series of fits. You can compare the fits by
24820 graphing them, by examining the goodness-of-fit measures returned by
24821 @kbd{I a F}, or by some other method suitable to your application.
24822 Note that some models can be linearized in several ways. The
24823 Gaussian-plus-@var{d} model can be linearized by setting @expr{d}
24824 (the background) to a constant, or by setting @expr{b} (the standard
24825 deviation) and @expr{c} (the mean) to constants.
24827 To fit a model with constants substituted for some parameters, just
24828 store suitable values in those parameter variables, then omit them
24829 from the list of parameters when you answer the variables prompt.
24835 A last desperate step would be to use the general-purpose
24836 @code{minimize} function rather than @code{fit}. After all, both
24837 functions solve the problem of minimizing an expression (the
24838 @texline @math{\chi^2}
24839 @infoline @expr{chi^2}
24840 sum) by adjusting certain parameters in the expression. The @kbd{a F}
24841 command is able to use a vastly more efficient algorithm due to its
24842 special knowledge about linear chi-square sums, but the @kbd{a N}
24843 command can do the same thing by brute force.
24845 A compromise would be to pick out a few parameters without which the
24846 fit is linearizable, and use @code{minimize} on a call to @code{fit}
24847 which efficiently takes care of the rest of the parameters. The thing
24848 to be minimized would be the value of
24849 @texline @math{\chi^2}
24850 @infoline @expr{chi^2}
24851 returned as the fifth result of the @code{xfit} function:
24854 minimize(xfit(gaus(a,b,c,d,x), x, [a,b,c], data)_5, d, guess)
24858 where @code{gaus} represents the Gaussian model with background,
24859 @code{data} represents the data matrix, and @code{guess} represents
24860 the initial guess for @expr{d} that @code{minimize} requires.
24861 This operation will only be, shall we say, extraordinarily slow
24862 rather than astronomically slow (as would be the case if @code{minimize}
24863 were used by itself to solve the problem).
24869 The @kbd{I a F} [@code{xfit}] command is somewhat trickier when
24870 nonlinear models are used. The second item in the result is the
24871 vector of ``raw'' parameters @expr{A}, @expr{B}, @expr{C}. The
24872 covariance matrix is written in terms of those raw parameters.
24873 The fifth item is a vector of @dfn{filter} expressions. This
24874 is the empty vector @samp{[]} if the raw parameters were the same
24875 as the requested parameters, i.e., if @expr{A = a}, @expr{B = b},
24876 and so on (which is always true if the model is already linear
24877 in the parameters as written, e.g., for polynomial fits). If the
24878 parameters had to be rearranged, the fifth item is instead a vector
24879 of one formula per parameter in the original model. The raw
24880 parameters are expressed in these ``filter'' formulas as
24881 @samp{fitdummy(1)} for @expr{A}, @samp{fitdummy(2)} for @expr{B},
24884 When Calc needs to modify the model to return the result, it replaces
24885 @samp{fitdummy(1)} in all the filters with the first item in the raw
24886 parameters list, and so on for the other raw parameters, then
24887 evaluates the resulting filter formulas to get the actual parameter
24888 values to be substituted into the original model. In the case of
24889 @kbd{H a F} and @kbd{I a F} where the parameters must be error forms,
24890 Calc uses the square roots of the diagonal entries of the covariance
24891 matrix as error values for the raw parameters, then lets Calc's
24892 standard error-form arithmetic take it from there.
24894 If you use @kbd{I a F} with a nonlinear model, be sure to remember
24895 that the covariance matrix is in terms of the raw parameters,
24896 @emph{not} the actual requested parameters. It's up to you to
24897 figure out how to interpret the covariances in the presence of
24898 nontrivial filter functions.
24900 Things are also complicated when the input contains error forms.
24901 Suppose there are three independent and dependent variables, @expr{x},
24902 @expr{y}, and @expr{z}, one or more of which are error forms in the
24903 data. Calc combines all the error values by taking the square root
24904 of the sum of the squares of the errors. It then changes @expr{x}
24905 and @expr{y} to be plain numbers, and makes @expr{z} into an error
24906 form with this combined error. The @expr{Y(x,y,z)} part of the
24907 linearized model is evaluated, and the result should be an error
24908 form. The error part of that result is used for
24909 @texline @math{\sigma_i}
24910 @infoline @expr{sigma_i}
24911 for the data point. If for some reason @expr{Y(x,y,z)} does not return
24912 an error form, the combined error from @expr{z} is used directly for
24913 @texline @math{\sigma_i}.
24914 @infoline @expr{sigma_i}.
24915 Finally, @expr{z} is also stripped of its error
24916 for use in computing @expr{F(x,y,z)}, @expr{G(x,y,z)} and so on;
24917 the righthand side of the linearized model is computed in regular
24918 arithmetic with no error forms.
24920 (While these rules may seem complicated, they are designed to do
24921 the most reasonable thing in the typical case that @expr{Y(x,y,z)}
24922 depends only on the dependent variable @expr{z}, and in fact is
24923 often simply equal to @expr{z}. For common cases like polynomials
24924 and multilinear models, the combined error is simply used as the
24925 @texline @math{\sigma}
24926 @infoline @expr{sigma}
24927 for the data point with no further ado.)
24934 It may be the case that the model you wish to use is linearizable,
24935 but Calc's built-in rules are unable to figure it out. Calc uses
24936 its algebraic rewrite mechanism to linearize a model. The rewrite
24937 rules are kept in the variable @code{FitRules}. You can edit this
24938 variable using the @kbd{s e FitRules} command; in fact, there is
24939 a special @kbd{s F} command just for editing @code{FitRules}.
24940 @xref{Operations on Variables}.
24942 @xref{Rewrite Rules}, for a discussion of rewrite rules.
24976 Calc uses @code{FitRules} as follows. First, it converts the model
24977 to an equation if necessary and encloses the model equation in a
24978 call to the function @code{fitmodel} (which is not actually a defined
24979 function in Calc; it is only used as a placeholder by the rewrite rules).
24980 Parameter variables are renamed to function calls @samp{fitparam(1)},
24981 @samp{fitparam(2)}, and so on, and independent variables are renamed
24982 to @samp{fitvar(1)}, @samp{fitvar(2)}, etc. The dependent variable
24983 is the highest-numbered @code{fitvar}. For example, the power law
24984 model @expr{a x^b} is converted to @expr{y = a x^b}, then to
24988 fitmodel(fitvar(2) = fitparam(1) fitvar(1)^fitparam(2))
24992 Calc then applies the rewrites as if by @samp{C-u 0 a r FitRules}.
24993 (The zero prefix means that rewriting should continue until no further
24994 changes are possible.)
24996 When rewriting is complete, the @code{fitmodel} call should have
24997 been replaced by a @code{fitsystem} call that looks like this:
25000 fitsystem(@var{Y}, @var{FGH}, @var{abc})
25004 where @var{Y} is a formula that describes the function @expr{Y(x,y,z)},
25005 @var{FGH} is the vector of formulas @expr{[F(x,y,z), G(x,y,z), H(x,y,z)]},
25006 and @var{abc} is the vector of parameter filters which refer to the
25007 raw parameters as @samp{fitdummy(1)} for @expr{A}, @samp{fitdummy(2)}
25008 for @expr{B}, etc. While the number of raw parameters (the length of
25009 the @var{FGH} vector) is usually the same as the number of original
25010 parameters (the length of the @var{abc} vector), this is not required.
25012 The power law model eventually boils down to
25016 fitsystem(ln(fitvar(2)),
25017 [1, ln(fitvar(1))],
25018 [exp(fitdummy(1)), fitdummy(2)])
25022 The actual implementation of @code{FitRules} is complicated; it
25023 proceeds in four phases. First, common rearrangements are done
25024 to try to bring linear terms together and to isolate functions like
25025 @code{exp} and @code{ln} either all the way ``out'' (so that they
25026 can be put into @var{Y}) or all the way ``in'' (so that they can
25027 be put into @var{abc} or @var{FGH}). In particular, all
25028 non-constant powers are converted to logs-and-exponentials form,
25029 and the distributive law is used to expand products of sums.
25030 Quotients are rewritten to use the @samp{fitinv} function, where
25031 @samp{fitinv(x)} represents @expr{1/x} while the @code{FitRules}
25032 are operating. (The use of @code{fitinv} makes recognition of
25033 linear-looking forms easier.) If you modify @code{FitRules}, you
25034 will probably only need to modify the rules for this phase.
25036 Phase two, whose rules can actually also apply during phases one
25037 and three, first rewrites @code{fitmodel} to a two-argument
25038 form @samp{fitmodel(@var{Y}, @var{model})}, where @var{Y} is
25039 initially zero and @var{model} has been changed from @expr{a=b}
25040 to @expr{a-b} form. It then tries to peel off invertible functions
25041 from the outside of @var{model} and put them into @var{Y} instead,
25042 calling the equation solver to invert the functions. Finally, when
25043 this is no longer possible, the @code{fitmodel} is changed to a
25044 four-argument @code{fitsystem}, where the fourth argument is
25045 @var{model} and the @var{FGH} and @var{abc} vectors are initially
25046 empty. (The last vector is really @var{ABC}, corresponding to
25047 raw parameters, for now.)
25049 Phase three converts a sum of items in the @var{model} to a sum
25050 of @samp{fitpart(@var{a}, @var{b}, @var{c})} terms which represent
25051 terms @samp{@var{a}*@var{b}*@var{c}} of the sum, where @var{a}
25052 is all factors that do not involve any variables, @var{b} is all
25053 factors that involve only parameters, and @var{c} is the factors
25054 that involve only independent variables. (If this decomposition
25055 is not possible, the rule set will not complete and Calc will
25056 complain that the model is too complex.) Then @code{fitpart}s
25057 with equal @var{b} or @var{c} components are merged back together
25058 using the distributive law in order to minimize the number of
25059 raw parameters needed.
25061 Phase four moves the @code{fitpart} terms into the @var{FGH} and
25062 @var{ABC} vectors. Also, some of the algebraic expansions that
25063 were done in phase 1 are undone now to make the formulas more
25064 computationally efficient. Finally, it calls the solver one more
25065 time to convert the @var{ABC} vector to an @var{abc} vector, and
25066 removes the fourth @var{model} argument (which by now will be zero)
25067 to obtain the three-argument @code{fitsystem} that the linear
25068 least-squares solver wants to see.
25074 @mindex hasfit@idots
25076 @tindex hasfitparams
25084 Two functions which are useful in connection with @code{FitRules}
25085 are @samp{hasfitparams(x)} and @samp{hasfitvars(x)}, which check
25086 whether @expr{x} refers to any parameters or independent variables,
25087 respectively. Specifically, these functions return ``true'' if the
25088 argument contains any @code{fitparam} (or @code{fitvar}) function
25089 calls, and ``false'' otherwise. (Recall that ``true'' means a
25090 nonzero number, and ``false'' means zero. The actual nonzero number
25091 returned is the largest @var{n} from all the @samp{fitparam(@var{n})}s
25092 or @samp{fitvar(@var{n})}s, respectively, that appear in the formula.)
25098 The @code{fit} function in algebraic notation normally takes four
25099 arguments, @samp{fit(@var{model}, @var{vars}, @var{params}, @var{data})},
25100 where @var{model} is the model formula as it would be typed after
25101 @kbd{a F '}, @var{vars} is the independent variable or a vector of
25102 independent variables, @var{params} likewise gives the parameter(s),
25103 and @var{data} is the data matrix. Note that the length of @var{vars}
25104 must be equal to the number of rows in @var{data} if @var{model} is
25105 an equation, or one less than the number of rows if @var{model} is
25106 a plain formula. (Actually, a name for the dependent variable is
25107 allowed but will be ignored in the plain-formula case.)
25109 If @var{params} is omitted, the parameters are all variables in
25110 @var{model} except those that appear in @var{vars}. If @var{vars}
25111 is also omitted, Calc sorts all the variables that appear in
25112 @var{model} alphabetically and uses the higher ones for @var{vars}
25113 and the lower ones for @var{params}.
25115 Alternatively, @samp{fit(@var{modelvec}, @var{data})} is allowed
25116 where @var{modelvec} is a 2- or 3-vector describing the model
25117 and variables, as discussed previously.
25119 If Calc is unable to do the fit, the @code{fit} function is left
25120 in symbolic form, ordinarily with an explanatory message. The
25121 message will be ``Model expression is too complex'' if the
25122 linearizer was unable to put the model into the required form.
25124 The @code{efit} (corresponding to @kbd{H a F}) and @code{xfit}
25125 (for @kbd{I a F}) functions are completely analogous.
25127 @node Interpolation, , Curve Fitting Details, Curve Fitting
25128 @subsection Polynomial Interpolation
25131 @pindex calc-poly-interp
25133 The @kbd{a p} (@code{calc-poly-interp}) [@code{polint}] command does
25134 a polynomial interpolation at a particular @expr{x} value. It takes
25135 two arguments from the stack: A data matrix of the sort used by
25136 @kbd{a F}, and a single number which represents the desired @expr{x}
25137 value. Calc effectively does an exact polynomial fit as if by @kbd{a F i},
25138 then substitutes the @expr{x} value into the result in order to get an
25139 approximate @expr{y} value based on the fit. (Calc does not actually
25140 use @kbd{a F i}, however; it uses a direct method which is both more
25141 efficient and more numerically stable.)
25143 The result of @kbd{a p} is actually a vector of two values: The @expr{y}
25144 value approximation, and an error measure @expr{dy} that reflects Calc's
25145 estimation of the probable error of the approximation at that value of
25146 @expr{x}. If the input @expr{x} is equal to any of the @expr{x} values
25147 in the data matrix, the output @expr{y} will be the corresponding @expr{y}
25148 value from the matrix, and the output @expr{dy} will be exactly zero.
25150 A prefix argument of 2 causes @kbd{a p} to take separate x- and
25151 y-vectors from the stack instead of one data matrix.
25153 If @expr{x} is a vector of numbers, @kbd{a p} will return a matrix of
25154 interpolated results for each of those @expr{x} values. (The matrix will
25155 have two columns, the @expr{y} values and the @expr{dy} values.)
25156 If @expr{x} is a formula instead of a number, the @code{polint} function
25157 remains in symbolic form; use the @kbd{a "} command to expand it out to
25158 a formula that describes the fit in symbolic terms.
25160 In all cases, the @kbd{a p} command leaves the data vectors or matrix
25161 on the stack. Only the @expr{x} value is replaced by the result.
25165 The @kbd{H a p} [@code{ratint}] command does a rational function
25166 interpolation. It is used exactly like @kbd{a p}, except that it
25167 uses as its model the quotient of two polynomials. If there are
25168 @expr{N} data points, the numerator and denominator polynomials will
25169 each have degree @expr{N/2} (if @expr{N} is odd, the denominator will
25170 have degree one higher than the numerator).
25172 Rational approximations have the advantage that they can accurately
25173 describe functions that have poles (points at which the function's value
25174 goes to infinity, so that the denominator polynomial of the approximation
25175 goes to zero). If @expr{x} corresponds to a pole of the fitted rational
25176 function, then the result will be a division by zero. If Infinite mode
25177 is enabled, the result will be @samp{[uinf, uinf]}.
25179 There is no way to get the actual coefficients of the rational function
25180 used by @kbd{H a p}. (The algorithm never generates these coefficients
25181 explicitly, and quotients of polynomials are beyond @w{@kbd{a F}}'s
25182 capabilities to fit.)
25184 @node Summations, Logical Operations, Curve Fitting, Algebra
25185 @section Summations
25188 @cindex Summation of a series
25190 @pindex calc-summation
25192 The @kbd{a +} (@code{calc-summation}) [@code{sum}] command computes
25193 the sum of a formula over a certain range of index values. The formula
25194 is taken from the top of the stack; the command prompts for the
25195 name of the summation index variable, the lower limit of the
25196 sum (any formula), and the upper limit of the sum. If you
25197 enter a blank line at any of these prompts, that prompt and
25198 any later ones are answered by reading additional elements from
25199 the stack. Thus, @kbd{' k^2 @key{RET} ' k @key{RET} 1 @key{RET} 5 @key{RET} a + @key{RET}}
25200 produces the result 55.
25203 $$ \sum_{k=1}^5 k^2 = 55 $$
25206 The choice of index variable is arbitrary, but it's best not to
25207 use a variable with a stored value. In particular, while
25208 @code{i} is often a favorite index variable, it should be avoided
25209 in Calc because @code{i} has the imaginary constant @expr{(0, 1)}
25210 as a value. If you pressed @kbd{=} on a sum over @code{i}, it would
25211 be changed to a nonsensical sum over the ``variable'' @expr{(0, 1)}!
25212 If you really want to use @code{i} as an index variable, use
25213 @w{@kbd{s u i @key{RET}}} first to ``unstore'' this variable.
25214 (@xref{Storing Variables}.)
25216 A numeric prefix argument steps the index by that amount rather
25217 than by one. Thus @kbd{' a_k @key{RET} C-u -2 a + k @key{RET} 10 @key{RET} 0 @key{RET}}
25218 yields @samp{a_10 + a_8 + a_6 + a_4 + a_2 + a_0}. A prefix
25219 argument of plain @kbd{C-u} causes @kbd{a +} to prompt for the
25220 step value, in which case you can enter any formula or enter
25221 a blank line to take the step value from the stack. With the
25222 @kbd{C-u} prefix, @kbd{a +} can take up to five arguments from
25223 the stack: The formula, the variable, the lower limit, the
25224 upper limit, and (at the top of the stack), the step value.
25226 Calc knows how to do certain sums in closed form. For example,
25227 @samp{sum(6 k^2, k, 1, n) = @w{2 n^3} + 3 n^2 + n}. In particular,
25228 this is possible if the formula being summed is polynomial or
25229 exponential in the index variable. Sums of logarithms are
25230 transformed into logarithms of products. Sums of trigonometric
25231 and hyperbolic functions are transformed to sums of exponentials
25232 and then done in closed form. Also, of course, sums in which the
25233 lower and upper limits are both numbers can always be evaluated
25234 just by grinding them out, although Calc will use closed forms
25235 whenever it can for the sake of efficiency.
25237 The notation for sums in algebraic formulas is
25238 @samp{sum(@var{expr}, @var{var}, @var{low}, @var{high}, @var{step})}.
25239 If @var{step} is omitted, it defaults to one. If @var{high} is
25240 omitted, @var{low} is actually the upper limit and the lower limit
25241 is one. If @var{low} is also omitted, the limits are @samp{-inf}
25242 and @samp{inf}, respectively.
25244 Infinite sums can sometimes be evaluated: @samp{sum(.5^k, k, 1, inf)}
25245 returns @expr{1}. This is done by evaluating the sum in closed
25246 form (to @samp{1. - 0.5^n} in this case), then evaluating this
25247 formula with @code{n} set to @code{inf}. Calc's usual rules
25248 for ``infinite'' arithmetic can find the answer from there. If
25249 infinite arithmetic yields a @samp{nan}, or if the sum cannot be
25250 solved in closed form, Calc leaves the @code{sum} function in
25251 symbolic form. @xref{Infinities}.
25253 As a special feature, if the limits are infinite (or omitted, as
25254 described above) but the formula includes vectors subscripted by
25255 expressions that involve the iteration variable, Calc narrows
25256 the limits to include only the range of integers which result in
25257 valid subscripts for the vector. For example, the sum
25258 @samp{sum(k [a,b,c,d,e,f,g]_(2k),k)} evaluates to @samp{b + 2 d + 3 f}.
25260 The limits of a sum do not need to be integers. For example,
25261 @samp{sum(a_k, k, 0, 2 n, n)} produces @samp{a_0 + a_n + a_(2 n)}.
25262 Calc computes the number of iterations using the formula
25263 @samp{1 + (@var{high} - @var{low}) / @var{step}}, which must,
25264 after simplification as if by @kbd{a s}, evaluate to an integer.
25266 If the number of iterations according to the above formula does
25267 not come out to an integer, the sum is invalid and will be left
25268 in symbolic form. However, closed forms are still supplied, and
25269 you are on your honor not to misuse the resulting formulas by
25270 substituting mismatched bounds into them. For example,
25271 @samp{sum(k, k, 1, 10, 2)} is invalid, but Calc will go ahead and
25272 evaluate the closed form solution for the limits 1 and 10 to get
25273 the rather dubious answer, 29.25.
25275 If the lower limit is greater than the upper limit (assuming a
25276 positive step size), the result is generally zero. However,
25277 Calc only guarantees a zero result when the upper limit is
25278 exactly one step less than the lower limit, i.e., if the number
25279 of iterations is @mathit{-1}. Thus @samp{sum(f(k), k, n, n-1)} is zero
25280 but the sum from @samp{n} to @samp{n-2} may report a nonzero value
25281 if Calc used a closed form solution.
25283 Calc's logical predicates like @expr{a < b} return 1 for ``true''
25284 and 0 for ``false.'' @xref{Logical Operations}. This can be
25285 used to advantage for building conditional sums. For example,
25286 @samp{sum(prime(k)*k^2, k, 1, 20)} is the sum of the squares of all
25287 prime numbers from 1 to 20; the @code{prime} predicate returns 1 if
25288 its argument is prime and 0 otherwise. You can read this expression
25289 as ``the sum of @expr{k^2}, where @expr{k} is prime.'' Indeed,
25290 @samp{sum(prime(k)*k^2, k)} would represent the sum of @emph{all} primes
25291 squared, since the limits default to plus and minus infinity, but
25292 there are no such sums that Calc's built-in rules can do in
25295 As another example, @samp{sum((k != k_0) * f(k), k, 1, n)} is the
25296 sum of @expr{f(k)} for all @expr{k} from 1 to @expr{n}, excluding
25297 one value @expr{k_0}. Slightly more tricky is the summand
25298 @samp{(k != k_0) / (k - k_0)}, which is an attempt to describe
25299 the sum of all @expr{1/(k-k_0)} except at @expr{k = k_0}, where
25300 this would be a division by zero. But at @expr{k = k_0}, this
25301 formula works out to the indeterminate form @expr{0 / 0}, which
25302 Calc will not assume is zero. Better would be to use
25303 @samp{(k != k_0) ? 1/(k-k_0) : 0}; the @samp{? :} operator does
25304 an ``if-then-else'' test: This expression says, ``if
25305 @texline @math{k \ne k_0},
25306 @infoline @expr{k != k_0},
25307 then @expr{1/(k-k_0)}, else zero.'' Now the formula @expr{1/(k-k_0)}
25308 will not even be evaluated by Calc when @expr{k = k_0}.
25310 @cindex Alternating sums
25312 @pindex calc-alt-summation
25314 The @kbd{a -} (@code{calc-alt-summation}) [@code{asum}] command
25315 computes an alternating sum. Successive terms of the sequence
25316 are given alternating signs, with the first term (corresponding
25317 to the lower index value) being positive. Alternating sums
25318 are converted to normal sums with an extra term of the form
25319 @samp{(-1)^(k-@var{low})}. This formula is adjusted appropriately
25320 if the step value is other than one. For example, the Taylor
25321 series for the sine function is @samp{asum(x^k / k!, k, 1, inf, 2)}.
25322 (Calc cannot evaluate this infinite series, but it can approximate
25323 it if you replace @code{inf} with any particular odd number.)
25324 Calc converts this series to a regular sum with a step of one,
25325 namely @samp{sum((-1)^k x^(2k+1) / (2k+1)!, k, 0, inf)}.
25327 @cindex Product of a sequence
25329 @pindex calc-product
25331 The @kbd{a *} (@code{calc-product}) [@code{prod}] command is
25332 the analogous way to take a product of many terms. Calc also knows
25333 some closed forms for products, such as @samp{prod(k, k, 1, n) = n!}.
25334 Conditional products can be written @samp{prod(k^prime(k), k, 1, n)}
25335 or @samp{prod(prime(k) ? k : 1, k, 1, n)}.
25338 @pindex calc-tabulate
25340 The @kbd{a T} (@code{calc-tabulate}) [@code{table}] command
25341 evaluates a formula at a series of iterated index values, just
25342 like @code{sum} and @code{prod}, but its result is simply a
25343 vector of the results. For example, @samp{table(a_i, i, 1, 7, 2)}
25344 produces @samp{[a_1, a_3, a_5, a_7]}.
25346 @node Logical Operations, Rewrite Rules, Summations, Algebra
25347 @section Logical Operations
25350 The following commands and algebraic functions return true/false values,
25351 where 1 represents ``true'' and 0 represents ``false.'' In cases where
25352 a truth value is required (such as for the condition part of a rewrite
25353 rule, or as the condition for a @w{@kbd{Z [ Z ]}} control structure), any
25354 nonzero value is accepted to mean ``true.'' (Specifically, anything
25355 for which @code{dnonzero} returns 1 is ``true,'' and anything for
25356 which @code{dnonzero} returns 0 or cannot decide is assumed ``false.''
25357 Note that this means that @w{@kbd{Z [ Z ]}} will execute the ``then''
25358 portion if its condition is provably true, but it will execute the
25359 ``else'' portion for any condition like @expr{a = b} that is not
25360 provably true, even if it might be true. Algebraic functions that
25361 have conditions as arguments, like @code{? :} and @code{&&}, remain
25362 unevaluated if the condition is neither provably true nor provably
25363 false. @xref{Declarations}.)
25366 @pindex calc-equal-to
25370 The @kbd{a =} (@code{calc-equal-to}) command, or @samp{eq(a,b)} function
25371 (which can also be written @samp{a = b} or @samp{a == b} in an algebraic
25372 formula) is true if @expr{a} and @expr{b} are equal, either because they
25373 are identical expressions, or because they are numbers which are
25374 numerically equal. (Thus the integer 1 is considered equal to the float
25375 1.0.) If the equality of @expr{a} and @expr{b} cannot be determined,
25376 the comparison is left in symbolic form. Note that as a command, this
25377 operation pops two values from the stack and pushes back either a 1 or
25378 a 0, or a formula @samp{a = b} if the values' equality cannot be determined.
25380 Many Calc commands use @samp{=} formulas to represent @dfn{equations}.
25381 For example, the @kbd{a S} (@code{calc-solve-for}) command rearranges
25382 an equation to solve for a given variable. The @kbd{a M}
25383 (@code{calc-map-equation}) command can be used to apply any
25384 function to both sides of an equation; for example, @kbd{2 a M *}
25385 multiplies both sides of the equation by two. Note that just
25386 @kbd{2 *} would not do the same thing; it would produce the formula
25387 @samp{2 (a = b)} which represents 2 if the equality is true or
25390 The @code{eq} function with more than two arguments (e.g., @kbd{C-u 3 a =}
25391 or @samp{a = b = c}) tests if all of its arguments are equal. In
25392 algebraic notation, the @samp{=} operator is unusual in that it is
25393 neither left- nor right-associative: @samp{a = b = c} is not the
25394 same as @samp{(a = b) = c} or @samp{a = (b = c)} (which each compare
25395 one variable with the 1 or 0 that results from comparing two other
25399 @pindex calc-not-equal-to
25402 The @kbd{a #} (@code{calc-not-equal-to}) command, or @samp{neq(a,b)} or
25403 @samp{a != b} function, is true if @expr{a} and @expr{b} are not equal.
25404 This also works with more than two arguments; @samp{a != b != c != d}
25405 tests that all four of @expr{a}, @expr{b}, @expr{c}, and @expr{d} are
25422 @pindex calc-less-than
25423 @pindex calc-greater-than
25424 @pindex calc-less-equal
25425 @pindex calc-greater-equal
25454 The @kbd{a <} (@code{calc-less-than}) [@samp{lt(a,b)} or @samp{a < b}]
25455 operation is true if @expr{a} is less than @expr{b}. Similar functions
25456 are @kbd{a >} (@code{calc-greater-than}) [@samp{gt(a,b)} or @samp{a > b}],
25457 @kbd{a [} (@code{calc-less-equal}) [@samp{leq(a,b)} or @samp{a <= b}], and
25458 @kbd{a ]} (@code{calc-greater-equal}) [@samp{geq(a,b)} or @samp{a >= b}].
25460 While the inequality functions like @code{lt} do not accept more
25461 than two arguments, the syntax @w{@samp{a <= b < c}} is translated to an
25462 equivalent expression involving intervals: @samp{b in [a .. c)}.
25463 (See the description of @code{in} below.) All four combinations
25464 of @samp{<} and @samp{<=} are allowed, or any of the four combinations
25465 of @samp{>} and @samp{>=}. Four-argument constructions like
25466 @samp{a < b < c < d}, and mixtures like @w{@samp{a < b = c}} that
25467 involve both equalities and inequalities, are not allowed.
25470 @pindex calc-remove-equal
25472 The @kbd{a .} (@code{calc-remove-equal}) [@code{rmeq}] command extracts
25473 the righthand side of the equation or inequality on the top of the
25474 stack. It also works elementwise on vectors. For example, if
25475 @samp{[x = 2.34, y = z / 2]} is on the stack, then @kbd{a .} produces
25476 @samp{[2.34, z / 2]}. As a special case, if the righthand side is a
25477 variable and the lefthand side is a number (as in @samp{2.34 = x}), then
25478 Calc keeps the lefthand side instead. Finally, this command works with
25479 assignments @samp{x := 2.34} as well as equations, always taking the
25480 righthand side, and for @samp{=>} (evaluates-to) operators, always
25481 taking the lefthand side.
25484 @pindex calc-logical-and
25487 The @kbd{a &} (@code{calc-logical-and}) [@samp{land(a,b)} or @samp{a && b}]
25488 function is true if both of its arguments are true, i.e., are
25489 non-zero numbers. In this case, the result will be either @expr{a} or
25490 @expr{b}, chosen arbitrarily. If either argument is zero, the result is
25491 zero. Otherwise, the formula is left in symbolic form.
25494 @pindex calc-logical-or
25497 The @kbd{a |} (@code{calc-logical-or}) [@samp{lor(a,b)} or @samp{a || b}]
25498 function is true if either or both of its arguments are true (nonzero).
25499 The result is whichever argument was nonzero, choosing arbitrarily if both
25500 are nonzero. If both @expr{a} and @expr{b} are zero, the result is
25504 @pindex calc-logical-not
25507 The @kbd{a !} (@code{calc-logical-not}) [@samp{lnot(a)} or @samp{!@: a}]
25508 function is true if @expr{a} is false (zero), or false if @expr{a} is
25509 true (nonzero). It is left in symbolic form if @expr{a} is not a
25513 @pindex calc-logical-if
25523 @cindex Arguments, not evaluated
25524 The @kbd{a :} (@code{calc-logical-if}) [@samp{if(a,b,c)} or @samp{a ? b :@: c}]
25525 function is equal to either @expr{b} or @expr{c} if @expr{a} is a nonzero
25526 number or zero, respectively. If @expr{a} is not a number, the test is
25527 left in symbolic form and neither @expr{b} nor @expr{c} is evaluated in
25528 any way. In algebraic formulas, this is one of the few Calc functions
25529 whose arguments are not automatically evaluated when the function itself
25530 is evaluated. The others are @code{lambda}, @code{quote}, and
25533 One minor surprise to watch out for is that the formula @samp{a?3:4}
25534 will not work because the @samp{3:4} is parsed as a fraction instead of
25535 as three separate symbols. Type something like @samp{a ? 3 : 4} or
25536 @samp{a?(3):4} instead.
25538 As a special case, if @expr{a} evaluates to a vector, then both @expr{b}
25539 and @expr{c} are evaluated; the result is a vector of the same length
25540 as @expr{a} whose elements are chosen from corresponding elements of
25541 @expr{b} and @expr{c} according to whether each element of @expr{a}
25542 is zero or nonzero. Each of @expr{b} and @expr{c} must be either a
25543 vector of the same length as @expr{a}, or a non-vector which is matched
25544 with all elements of @expr{a}.
25547 @pindex calc-in-set
25549 The @kbd{a @{} (@code{calc-in-set}) [@samp{in(a,b)}] function is true if
25550 the number @expr{a} is in the set of numbers represented by @expr{b}.
25551 If @expr{b} is an interval form, @expr{a} must be one of the values
25552 encompassed by the interval. If @expr{b} is a vector, @expr{a} must be
25553 equal to one of the elements of the vector. (If any vector elements are
25554 intervals, @expr{a} must be in any of the intervals.) If @expr{b} is a
25555 plain number, @expr{a} must be numerically equal to @expr{b}.
25556 @xref{Set Operations}, for a group of commands that manipulate sets
25563 The @samp{typeof(a)} function produces an integer or variable which
25564 characterizes @expr{a}. If @expr{a} is a number, vector, or variable,
25565 the result will be one of the following numbers:
25570 3 Floating-point number
25572 5 Rectangular complex number
25573 6 Polar complex number
25579 12 Infinity (inf, uinf, or nan)
25581 101 Vector (but not a matrix)
25585 Otherwise, @expr{a} is a formula, and the result is a variable which
25586 represents the name of the top-level function call.
25600 The @samp{integer(a)} function returns true if @expr{a} is an integer.
25601 The @samp{real(a)} function
25602 is true if @expr{a} is a real number, either integer, fraction, or
25603 float. The @samp{constant(a)} function returns true if @expr{a} is
25604 any of the objects for which @code{typeof} would produce an integer
25605 code result except for variables, and provided that the components of
25606 an object like a vector or error form are themselves constant.
25607 Note that infinities do not satisfy any of these tests, nor do
25608 special constants like @code{pi} and @code{e}.
25610 @xref{Declarations}, for a set of similar functions that recognize
25611 formulas as well as actual numbers. For example, @samp{dint(floor(x))}
25612 is true because @samp{floor(x)} is provably integer-valued, but
25613 @samp{integer(floor(x))} does not because @samp{floor(x)} is not
25614 literally an integer constant.
25620 The @samp{refers(a,b)} function is true if the variable (or sub-expression)
25621 @expr{b} appears in @expr{a}, or false otherwise. Unlike the other
25622 tests described here, this function returns a definite ``no'' answer
25623 even if its arguments are still in symbolic form. The only case where
25624 @code{refers} will be left unevaluated is if @expr{a} is a plain
25625 variable (different from @expr{b}).
25631 The @samp{negative(a)} function returns true if @expr{a} ``looks'' negative,
25632 because it is a negative number, because it is of the form @expr{-x},
25633 or because it is a product or quotient with a term that looks negative.
25634 This is most useful in rewrite rules. Beware that @samp{negative(a)}
25635 evaluates to 1 or 0 for @emph{any} argument @expr{a}, so it can only
25636 be stored in a formula if the default simplifications are turned off
25637 first with @kbd{m O} (or if it appears in an unevaluated context such
25638 as a rewrite rule condition).
25644 The @samp{variable(a)} function is true if @expr{a} is a variable,
25645 or false if not. If @expr{a} is a function call, this test is left
25646 in symbolic form. Built-in variables like @code{pi} and @code{inf}
25647 are considered variables like any others by this test.
25653 The @samp{nonvar(a)} function is true if @expr{a} is a non-variable.
25654 If its argument is a variable it is left unsimplified; it never
25655 actually returns zero. However, since Calc's condition-testing
25656 commands consider ``false'' anything not provably true, this is
25675 @cindex Linearity testing
25676 The functions @code{lin}, @code{linnt}, @code{islin}, and @code{islinnt}
25677 check if an expression is ``linear,'' i.e., can be written in the form
25678 @expr{a + b x} for some constants @expr{a} and @expr{b}, and some
25679 variable or subformula @expr{x}. The function @samp{islin(f,x)} checks
25680 if formula @expr{f} is linear in @expr{x}, returning 1 if so. For
25681 example, @samp{islin(x,x)}, @samp{islin(-x,x)}, @samp{islin(3,x)}, and
25682 @samp{islin(x y / 3 - 2, x)} all return 1. The @samp{lin(f,x)} function
25683 is similar, except that instead of returning 1 it returns the vector
25684 @expr{[a, b, x]}. For the above examples, this vector would be
25685 @expr{[0, 1, x]}, @expr{[0, -1, x]}, @expr{[3, 0, x]}, and
25686 @expr{[-2, y/3, x]}, respectively. Both @code{lin} and @code{islin}
25687 generally remain unevaluated for expressions which are not linear,
25688 e.g., @samp{lin(2 x^2, x)} and @samp{lin(sin(x), x)}. The second
25689 argument can also be a formula; @samp{islin(2 + 3 sin(x), sin(x))}
25692 The @code{linnt} and @code{islinnt} functions perform a similar check,
25693 but require a ``non-trivial'' linear form, which means that the
25694 @expr{b} coefficient must be non-zero. For example, @samp{lin(2,x)}
25695 returns @expr{[2, 0, x]} and @samp{lin(y,x)} returns @expr{[y, 0, x]},
25696 but @samp{linnt(2,x)} and @samp{linnt(y,x)} are left unevaluated
25697 (in other words, these formulas are considered to be only ``trivially''
25698 linear in @expr{x}).
25700 All four linearity-testing functions allow you to omit the second
25701 argument, in which case the input may be linear in any non-constant
25702 formula. Here, the @expr{a=0}, @expr{b=1} case is also considered
25703 trivial, and only constant values for @expr{a} and @expr{b} are
25704 recognized. Thus, @samp{lin(2 x y)} returns @expr{[0, 2, x y]},
25705 @samp{lin(2 - x y)} returns @expr{[2, -1, x y]}, and @samp{lin(x y)}
25706 returns @expr{[0, 1, x y]}. The @code{linnt} function would allow the
25707 first two cases but not the third. Also, neither @code{lin} nor
25708 @code{linnt} accept plain constants as linear in the one-argument
25709 case: @samp{islin(2,x)} is true, but @samp{islin(2)} is false.
25715 The @samp{istrue(a)} function returns 1 if @expr{a} is a nonzero
25716 number or provably nonzero formula, or 0 if @expr{a} is anything else.
25717 Calls to @code{istrue} can only be manipulated if @kbd{m O} mode is
25718 used to make sure they are not evaluated prematurely. (Note that
25719 declarations are used when deciding whether a formula is true;
25720 @code{istrue} returns 1 when @code{dnonzero} would return 1, and
25721 it returns 0 when @code{dnonzero} would return 0 or leave itself
25724 @node Rewrite Rules, , Logical Operations, Algebra
25725 @section Rewrite Rules
25728 @cindex Rewrite rules
25729 @cindex Transformations
25730 @cindex Pattern matching
25732 @pindex calc-rewrite
25734 The @kbd{a r} (@code{calc-rewrite}) [@code{rewrite}] command makes
25735 substitutions in a formula according to a specified pattern or patterns
25736 known as @dfn{rewrite rules}. Whereas @kbd{a b} (@code{calc-substitute})
25737 matches literally, so that substituting @samp{sin(x)} with @samp{cos(x)}
25738 matches only the @code{sin} function applied to the variable @code{x},
25739 rewrite rules match general kinds of formulas; rewriting using the rule
25740 @samp{sin(x) := cos(x)} matches @code{sin} of any argument and replaces
25741 it with @code{cos} of that same argument. The only significance of the
25742 name @code{x} is that the same name is used on both sides of the rule.
25744 Rewrite rules rearrange formulas already in Calc's memory.
25745 @xref{Syntax Tables}, to read about @dfn{syntax rules}, which are
25746 similar to algebraic rewrite rules but operate when new algebraic
25747 entries are being parsed, converting strings of characters into
25751 * Entering Rewrite Rules::
25752 * Basic Rewrite Rules::
25753 * Conditional Rewrite Rules::
25754 * Algebraic Properties of Rewrite Rules::
25755 * Other Features of Rewrite Rules::
25756 * Composing Patterns in Rewrite Rules::
25757 * Nested Formulas with Rewrite Rules::
25758 * Multi-Phase Rewrite Rules::
25759 * Selections with Rewrite Rules::
25760 * Matching Commands::
25761 * Automatic Rewrites::
25762 * Debugging Rewrites::
25763 * Examples of Rewrite Rules::
25766 @node Entering Rewrite Rules, Basic Rewrite Rules, Rewrite Rules, Rewrite Rules
25767 @subsection Entering Rewrite Rules
25770 Rewrite rules normally use the ``assignment'' operator
25771 @samp{@var{old} := @var{new}}.
25772 This operator is equivalent to the function call @samp{assign(old, new)}.
25773 The @code{assign} function is undefined by itself in Calc, so an
25774 assignment formula such as a rewrite rule will be left alone by ordinary
25775 Calc commands. But certain commands, like the rewrite system, interpret
25776 assignments in special ways.
25778 For example, the rule @samp{sin(x)^2 := 1-cos(x)^2} says to replace
25779 every occurrence of the sine of something, squared, with one minus the
25780 square of the cosine of that same thing. All by itself as a formula
25781 on the stack it does nothing, but when given to the @kbd{a r} command
25782 it turns that command into a sine-squared-to-cosine-squared converter.
25784 To specify a set of rules to be applied all at once, make a vector of
25787 When @kbd{a r} prompts you to enter the rewrite rules, you can answer
25792 With a rule: @kbd{f(x) := g(x) @key{RET}}.
25794 With a vector of rules: @kbd{[f1(x) := g1(x), f2(x) := g2(x)] @key{RET}}.
25795 (You can omit the enclosing square brackets if you wish.)
25797 With the name of a variable that contains the rule or rules vector:
25798 @kbd{myrules @key{RET}}.
25800 With any formula except a rule, a vector, or a variable name; this
25801 will be interpreted as the @var{old} half of a rewrite rule,
25802 and you will be prompted a second time for the @var{new} half:
25803 @kbd{f(x) @key{RET} g(x) @key{RET}}.
25805 With a blank line, in which case the rule, rules vector, or variable
25806 will be taken from the top of the stack (and the formula to be
25807 rewritten will come from the second-to-top position).
25810 If you enter the rules directly (as opposed to using rules stored
25811 in a variable), those rules will be put into the Trail so that you
25812 can retrieve them later. @xref{Trail Commands}.
25814 It is most convenient to store rules you use often in a variable and
25815 invoke them by giving the variable name. The @kbd{s e}
25816 (@code{calc-edit-variable}) command is an easy way to create or edit a
25817 rule set stored in a variable. You may also wish to use @kbd{s p}
25818 (@code{calc-permanent-variable}) to save your rules permanently;
25819 @pxref{Operations on Variables}.
25821 Rewrite rules are compiled into a special internal form for faster
25822 matching. If you enter a rule set directly it must be recompiled
25823 every time. If you store the rules in a variable and refer to them
25824 through that variable, they will be compiled once and saved away
25825 along with the variable for later reference. This is another good
25826 reason to store your rules in a variable.
25828 Calc also accepts an obsolete notation for rules, as vectors
25829 @samp{[@var{old}, @var{new}]}. But because it is easily confused with a
25830 vector of two rules, the use of this notation is no longer recommended.
25832 @node Basic Rewrite Rules, Conditional Rewrite Rules, Entering Rewrite Rules, Rewrite Rules
25833 @subsection Basic Rewrite Rules
25836 To match a particular formula @expr{x} with a particular rewrite rule
25837 @samp{@var{old} := @var{new}}, Calc compares the structure of @expr{x} with
25838 the structure of @var{old}. Variables that appear in @var{old} are
25839 treated as @dfn{meta-variables}; the corresponding positions in @expr{x}
25840 may contain any sub-formulas. For example, the pattern @samp{f(x,y)}
25841 would match the expression @samp{f(12, a+1)} with the meta-variable
25842 @samp{x} corresponding to 12 and with @samp{y} corresponding to
25843 @samp{a+1}. However, this pattern would not match @samp{f(12)} or
25844 @samp{g(12, a+1)}, since there is no assignment of the meta-variables
25845 that will make the pattern match these expressions. Notice that if
25846 the pattern is a single meta-variable, it will match any expression.
25848 If a given meta-variable appears more than once in @var{old}, the
25849 corresponding sub-formulas of @expr{x} must be identical. Thus
25850 the pattern @samp{f(x,x)} would match @samp{f(12, 12)} and
25851 @samp{f(a+1, a+1)} but not @samp{f(12, a+1)} or @samp{f(a+b, b+a)}.
25852 (@xref{Conditional Rewrite Rules}, for a way to match the latter.)
25854 Things other than variables must match exactly between the pattern
25855 and the target formula. To match a particular variable exactly, use
25856 the pseudo-function @samp{quote(v)} in the pattern. For example, the
25857 pattern @samp{x+quote(y)} matches @samp{x+y}, @samp{2+y}, or
25860 The special variable names @samp{e}, @samp{pi}, @samp{i}, @samp{phi},
25861 @samp{gamma}, @samp{inf}, @samp{uinf}, and @samp{nan} always match
25862 literally. Thus the pattern @samp{sin(d + e + f)} acts exactly like
25863 @samp{sin(d + quote(e) + f)}.
25865 If the @var{old} pattern is found to match a given formula, that
25866 formula is replaced by @var{new}, where any occurrences in @var{new}
25867 of meta-variables from the pattern are replaced with the sub-formulas
25868 that they matched. Thus, applying the rule @samp{f(x,y) := g(y+x,x)}
25869 to @samp{f(12, a+1)} would produce @samp{g(a+13, 12)}.
25871 The normal @kbd{a r} command applies rewrite rules over and over
25872 throughout the target formula until no further changes are possible
25873 (up to a limit of 100 times). Use @kbd{C-u 1 a r} to make only one
25876 @node Conditional Rewrite Rules, Algebraic Properties of Rewrite Rules, Basic Rewrite Rules, Rewrite Rules
25877 @subsection Conditional Rewrite Rules
25880 A rewrite rule can also be @dfn{conditional}, written in the form
25881 @samp{@var{old} := @var{new} :: @var{cond}}. (There is also the obsolete
25882 form @samp{[@var{old}, @var{new}, @var{cond}]}.) If a @var{cond} part
25884 rule, this is an additional condition that must be satisfied before
25885 the rule is accepted. Once @var{old} has been successfully matched
25886 to the target expression, @var{cond} is evaluated (with all the
25887 meta-variables substituted for the values they matched) and simplified
25888 with @kbd{a s} (@code{calc-simplify}). If the result is a nonzero
25889 number or any other object known to be nonzero (@pxref{Declarations}),
25890 the rule is accepted. If the result is zero or if it is a symbolic
25891 formula that is not known to be nonzero, the rule is rejected.
25892 @xref{Logical Operations}, for a number of functions that return
25893 1 or 0 according to the results of various tests.
25895 For example, the formula @samp{n > 0} simplifies to 1 or 0 if @expr{n}
25896 is replaced by a positive or nonpositive number, respectively (or if
25897 @expr{n} has been declared to be positive or nonpositive). Thus,
25898 the rule @samp{f(x,y) := g(y+x,x) :: x+y > 0} would apply to
25899 @samp{f(0, 4)} but not to @samp{f(-3, 2)} or @samp{f(12, a+1)}
25900 (assuming no outstanding declarations for @expr{a}). In the case of
25901 @samp{f(-3, 2)}, the condition can be shown not to be satisfied; in
25902 the case of @samp{f(12, a+1)}, the condition merely cannot be shown
25903 to be satisfied, but that is enough to reject the rule.
25905 While Calc will use declarations to reason about variables in the
25906 formula being rewritten, declarations do not apply to meta-variables.
25907 For example, the rule @samp{f(a) := g(a+1)} will match for any values
25908 of @samp{a}, such as complex numbers, vectors, or formulas, even if
25909 @samp{a} has been declared to be real or scalar. If you want the
25910 meta-variable @samp{a} to match only literal real numbers, use
25911 @samp{f(a) := g(a+1) :: real(a)}. If you want @samp{a} to match only
25912 reals and formulas which are provably real, use @samp{dreal(a)} as
25915 The @samp{::} operator is a shorthand for the @code{condition}
25916 function; @samp{@var{old} := @var{new} :: @var{cond}} is equivalent to
25917 the formula @samp{condition(assign(@var{old}, @var{new}), @var{cond})}.
25919 If you have several conditions, you can use @samp{... :: c1 :: c2 :: c3}
25920 or @samp{... :: c1 && c2 && c3}. The two are entirely equivalent.
25922 It is also possible to embed conditions inside the pattern:
25923 @samp{f(x :: x>0, y) := g(y+x, x)}. This is purely a notational
25924 convenience, though; where a condition appears in a rule has no
25925 effect on when it is tested. The rewrite-rule compiler automatically
25926 decides when it is best to test each condition while a rule is being
25929 Certain conditions are handled as special cases by the rewrite rule
25930 system and are tested very efficiently: Where @expr{x} is any
25931 meta-variable, these conditions are @samp{integer(x)}, @samp{real(x)},
25932 @samp{constant(x)}, @samp{negative(x)}, @samp{x >= y} where @expr{y}
25933 is either a constant or another meta-variable and @samp{>=} may be
25934 replaced by any of the six relational operators, and @samp{x % a = b}
25935 where @expr{a} and @expr{b} are constants. Other conditions, like
25936 @samp{x >= y+1} or @samp{dreal(x)}, will be less efficient to check
25937 since Calc must bring the whole evaluator and simplifier into play.
25939 An interesting property of @samp{::} is that neither of its arguments
25940 will be touched by Calc's default simplifications. This is important
25941 because conditions often are expressions that cannot safely be
25942 evaluated early. For example, the @code{typeof} function never
25943 remains in symbolic form; entering @samp{typeof(a)} will put the
25944 number 100 (the type code for variables like @samp{a}) on the stack.
25945 But putting the condition @samp{... :: typeof(a) = 6} on the stack
25946 is safe since @samp{::} prevents the @code{typeof} from being
25947 evaluated until the condition is actually used by the rewrite system.
25949 Since @samp{::} protects its lefthand side, too, you can use a dummy
25950 condition to protect a rule that must itself not evaluate early.
25951 For example, it's not safe to put @samp{a(f,x) := apply(f, [x])} on
25952 the stack because it will immediately evaluate to @samp{a(f,x) := f(x)},
25953 where the meta-variable-ness of @code{f} on the righthand side has been
25954 lost. But @samp{a(f,x) := apply(f, [x]) :: 1} is safe, and of course
25955 the condition @samp{1} is always true (nonzero) so it has no effect on
25956 the functioning of the rule. (The rewrite compiler will ensure that
25957 it doesn't even impact the speed of matching the rule.)
25959 @node Algebraic Properties of Rewrite Rules, Other Features of Rewrite Rules, Conditional Rewrite Rules, Rewrite Rules
25960 @subsection Algebraic Properties of Rewrite Rules
25963 The rewrite mechanism understands the algebraic properties of functions
25964 like @samp{+} and @samp{*}. In particular, pattern matching takes
25965 the associativity and commutativity of the following functions into
25969 + - * = != && || and or xor vint vunion vxor gcd lcm max min beta
25972 For example, the rewrite rule:
25975 a x + b x := (a + b) x
25979 will match formulas of the form,
25982 a x + b x, x a + x b, a x + x b, x a + b x
25985 Rewrites also understand the relationship between the @samp{+} and @samp{-}
25986 operators. The above rewrite rule will also match the formulas,
25989 a x - b x, x a - x b, a x - x b, x a - b x
25993 by matching @samp{b} in the pattern to @samp{-b} from the formula.
25995 Applied to a sum of many terms like @samp{r + a x + s + b x + t}, this
25996 pattern will check all pairs of terms for possible matches. The rewrite
25997 will take whichever suitable pair it discovers first.
25999 In general, a pattern using an associative operator like @samp{a + b}
26000 will try @var{2 n} different ways to match a sum of @var{n} terms
26001 like @samp{x + y + z - w}. First, @samp{a} is matched against each
26002 of @samp{x}, @samp{y}, @samp{z}, and @samp{-w} in turn, with @samp{b}
26003 being matched to the remainders @samp{y + z - w}, @samp{x + z - w}, etc.
26004 If none of these succeed, then @samp{b} is matched against each of the
26005 four terms with @samp{a} matching the remainder. Half-and-half matches,
26006 like @samp{(x + y) + (z - w)}, are not tried.
26008 Note that @samp{*} is not commutative when applied to matrices, but
26009 rewrite rules pretend that it is. If you type @kbd{m v} to enable
26010 Matrix mode (@pxref{Matrix Mode}), rewrite rules will match @samp{*}
26011 literally, ignoring its usual commutativity property. (In the
26012 current implementation, the associativity also vanishes---it is as
26013 if the pattern had been enclosed in a @code{plain} marker; see below.)
26014 If you are applying rewrites to formulas with matrices, it's best to
26015 enable Matrix mode first to prevent algebraically incorrect rewrites
26018 The pattern @samp{-x} will actually match any expression. For example,
26026 will rewrite @samp{f(a)} to @samp{-f(-a)}. To avoid this, either use
26027 a @code{plain} marker as described below, or add a @samp{negative(x)}
26028 condition. The @code{negative} function is true if its argument
26029 ``looks'' negative, for example, because it is a negative number or
26030 because it is a formula like @samp{-x}. The new rule using this
26034 f(x) := -f(-x) :: negative(x) @r{or, equivalently,}
26035 f(-x) := -f(x) :: negative(-x)
26038 In the same way, the pattern @samp{x - y} will match the sum @samp{a + b}
26039 by matching @samp{y} to @samp{-b}.
26041 The pattern @samp{a b} will also match the formula @samp{x/y} if
26042 @samp{y} is a number. Thus the rule @samp{a x + @w{b x} := (a+b) x}
26043 will also convert @samp{a x + x / 2} to @samp{(a + 0.5) x} (or
26044 @samp{(a + 1:2) x}, depending on the current fraction mode).
26046 Calc will @emph{not} take other liberties with @samp{*}, @samp{/}, and
26047 @samp{^}. For example, the pattern @samp{f(a b)} will not match
26048 @samp{f(x^2)}, and @samp{f(a + b)} will not match @samp{f(2 x)}, even
26049 though conceivably these patterns could match with @samp{a = b = x}.
26050 Nor will @samp{f(a b)} match @samp{f(x / y)} if @samp{y} is not a
26051 constant, even though it could be considered to match with @samp{a = x}
26052 and @samp{b = 1/y}. The reasons are partly for efficiency, and partly
26053 because while few mathematical operations are substantively different
26054 for addition and subtraction, often it is preferable to treat the cases
26055 of multiplication, division, and integer powers separately.
26057 Even more subtle is the rule set
26060 [ f(a) + f(b) := f(a + b), -f(a) := f(-a) ]
26064 attempting to match @samp{f(x) - f(y)}. You might think that Calc
26065 will view this subtraction as @samp{f(x) + (-f(y))} and then apply
26066 the above two rules in turn, but actually this will not work because
26067 Calc only does this when considering rules for @samp{+} (like the
26068 first rule in this set). So it will see first that @samp{f(x) + (-f(y))}
26069 does not match @samp{f(a) + f(b)} for any assignments of the
26070 meta-variables, and then it will see that @samp{f(x) - f(y)} does
26071 not match @samp{-f(a)} for any assignment of @samp{a}. Because Calc
26072 tries only one rule at a time, it will not be able to rewrite
26073 @samp{f(x) - f(y)} with this rule set. An explicit @samp{f(a) - f(b)}
26074 rule will have to be added.
26076 Another thing patterns will @emph{not} do is break up complex numbers.
26077 The pattern @samp{myconj(a + @w{b i)} := a - b i} will work for formulas
26078 involving the special constant @samp{i} (such as @samp{3 - 4 i}), but
26079 it will not match actual complex numbers like @samp{(3, -4)}. A version
26080 of the above rule for complex numbers would be
26083 myconj(a) := re(a) - im(a) (0,1) :: im(a) != 0
26087 (Because the @code{re} and @code{im} functions understand the properties
26088 of the special constant @samp{i}, this rule will also work for
26089 @samp{3 - 4 i}. In fact, this particular rule would probably be better
26090 without the @samp{im(a) != 0} condition, since if @samp{im(a) = 0} the
26091 righthand side of the rule will still give the correct answer for the
26092 conjugate of a real number.)
26094 It is also possible to specify optional arguments in patterns. The rule
26097 opt(a) x + opt(b) (x^opt(c) + opt(d)) := f(a, b, c, d)
26101 will match the formula
26108 in a fairly straightforward manner, but it will also match reduced
26112 x + x^2, 2(x + 1) - x, x + x
26116 producing, respectively,
26119 f(1, 1, 2, 0), f(-1, 2, 1, 1), f(1, 1, 1, 0)
26122 (The latter two formulas can be entered only if default simplifications
26123 have been turned off with @kbd{m O}.)
26125 The default value for a term of a sum is zero. The default value
26126 for a part of a product, for a power, or for the denominator of a
26127 quotient, is one. Also, @samp{-x} matches the pattern @samp{opt(a) b}
26128 with @samp{a = -1}.
26130 In particular, the distributive-law rule can be refined to
26133 opt(a) x + opt(b) x := (a + b) x
26137 so that it will convert, e.g., @samp{a x - x}, to @samp{(a - 1) x}.
26139 The pattern @samp{opt(a) + opt(b) x} matches almost any formulas which
26140 are linear in @samp{x}. You can also use the @code{lin} and @code{islin}
26141 functions with rewrite conditions to test for this; @pxref{Logical
26142 Operations}. These functions are not as convenient to use in rewrite
26143 rules, but they recognize more kinds of formulas as linear:
26144 @samp{x/z} is considered linear with @expr{b = 1/z} by @code{lin},
26145 but it will not match the above pattern because that pattern calls
26146 for a multiplication, not a division.
26148 As another example, the obvious rule to replace @samp{sin(x)^2 + cos(x)^2}
26152 sin(x)^2 + cos(x)^2 := 1
26156 misses many cases because the sine and cosine may both be multiplied by
26157 an equal factor. Here's a more successful rule:
26160 opt(a) sin(x)^2 + opt(a) cos(x)^2 := a
26163 Note that this rule will @emph{not} match @samp{sin(x)^2 + 6 cos(x)^2}
26164 because one @expr{a} would have ``matched'' 1 while the other matched 6.
26166 Calc automatically converts a rule like
26176 f(temp, x) := g(x) :: temp = x-1
26180 (where @code{temp} stands for a new, invented meta-variable that
26181 doesn't actually have a name). This modified rule will successfully
26182 match @samp{f(6, 7)}, binding @samp{temp} and @samp{x} to 6 and 7,
26183 respectively, then verifying that they differ by one even though
26184 @samp{6} does not superficially look like @samp{x-1}.
26186 However, Calc does not solve equations to interpret a rule. The
26190 f(x-1, x+1) := g(x)
26194 will not work. That is, it will match @samp{f(a - 1 + b, a + 1 + b)}
26195 but not @samp{f(6, 8)}. Calc always interprets at least one occurrence
26196 of a variable by literal matching. If the variable appears ``isolated''
26197 then Calc is smart enough to use it for literal matching. But in this
26198 last example, Calc is forced to rewrite the rule to @samp{f(x-1, temp)
26199 := g(x) :: temp = x+1} where the @samp{x-1} term must correspond to an
26200 actual ``something-minus-one'' in the target formula.
26202 A successful way to write this would be @samp{f(x, x+2) := g(x+1)}.
26203 You could make this resemble the original form more closely by using
26204 @code{let} notation, which is described in the next section:
26207 f(xm1, x+1) := g(x) :: let(x := xm1+1)
26210 Calc does this rewriting or ``conditionalizing'' for any sub-pattern
26211 which involves only the functions in the following list, operating
26212 only on constants and meta-variables which have already been matched
26213 elsewhere in the pattern. When matching a function call, Calc is
26214 careful to match arguments which are plain variables before arguments
26215 which are calls to any of the functions below, so that a pattern like
26216 @samp{f(x-1, x)} can be conditionalized even though the isolated
26217 @samp{x} comes after the @samp{x-1}.
26220 + - * / \ % ^ abs sign round rounde roundu trunc floor ceil
26221 max min re im conj arg
26224 You can suppress all of the special treatments described in this
26225 section by surrounding a function call with a @code{plain} marker.
26226 This marker causes the function call which is its argument to be
26227 matched literally, without regard to commutativity, associativity,
26228 negation, or conditionalization. When you use @code{plain}, the
26229 ``deep structure'' of the formula being matched can show through.
26233 plain(a - a b) := f(a, b)
26237 will match only literal subtractions. However, the @code{plain}
26238 marker does not affect its arguments' arguments. In this case,
26239 commutativity and associativity is still considered while matching
26240 the @w{@samp{a b}} sub-pattern, so the whole pattern will match
26241 @samp{x - y x} as well as @samp{x - x y}. We could go still
26245 plain(a - plain(a b)) := f(a, b)
26249 which would do a completely strict match for the pattern.
26251 By contrast, the @code{quote} marker means that not only the
26252 function name but also the arguments must be literally the same.
26253 The above pattern will match @samp{x - x y} but
26256 quote(a - a b) := f(a, b)
26260 will match only the single formula @samp{a - a b}. Also,
26263 quote(a - quote(a b)) := f(a, b)
26267 will match only @samp{a - quote(a b)}---probably not the desired
26270 A certain amount of algebra is also done when substituting the
26271 meta-variables on the righthand side of a rule. For example,
26275 a + f(b) := f(a + b)
26279 matching @samp{f(x) - y} would produce @samp{f((-y) + x)} if
26280 taken literally, but the rewrite mechanism will simplify the
26281 righthand side to @samp{f(x - y)} automatically. (Of course,
26282 the default simplifications would do this anyway, so this
26283 special simplification is only noticeable if you have turned the
26284 default simplifications off.) This rewriting is done only when
26285 a meta-variable expands to a ``negative-looking'' expression.
26286 If this simplification is not desirable, you can use a @code{plain}
26287 marker on the righthand side:
26290 a + f(b) := f(plain(a + b))
26294 In this example, we are still allowing the pattern-matcher to
26295 use all the algebra it can muster, but the righthand side will
26296 always simplify to a literal addition like @samp{f((-y) + x)}.
26298 @node Other Features of Rewrite Rules, Composing Patterns in Rewrite Rules, Algebraic Properties of Rewrite Rules, Rewrite Rules
26299 @subsection Other Features of Rewrite Rules
26302 Certain ``function names'' serve as markers in rewrite rules.
26303 Here is a complete list of these markers. First are listed the
26304 markers that work inside a pattern; then come the markers that
26305 work in the righthand side of a rule.
26311 One kind of marker, @samp{import(x)}, takes the place of a whole
26312 rule. Here @expr{x} is the name of a variable containing another
26313 rule set; those rules are ``spliced into'' the rule set that
26314 imports them. For example, if @samp{[f(a+b) := f(a) + f(b),
26315 f(a b) := a f(b) :: real(a)]} is stored in variable @samp{linearF},
26316 then the rule set @samp{[f(0) := 0, import(linearF)]} will apply
26317 all three rules. It is possible to modify the imported rules
26318 slightly: @samp{import(x, v1, x1, v2, x2, @dots{})} imports
26319 the rule set @expr{x} with all occurrences of
26320 @texline @math{v_1},
26321 @infoline @expr{v1},
26322 as either a variable name or a function name, replaced with
26323 @texline @math{x_1}
26324 @infoline @expr{x1}
26326 @texline @math{v_1}
26327 @infoline @expr{v1}
26328 is used as a function name, then
26329 @texline @math{x_1}
26330 @infoline @expr{x1}
26331 must be either a function name itself or a @w{@samp{< >}} nameless
26332 function; @pxref{Specifying Operators}.) For example, @samp{[g(0) := 0,
26333 import(linearF, f, g)]} applies the linearity rules to the function
26334 @samp{g} instead of @samp{f}. Imports can be nested, but the
26335 import-with-renaming feature may fail to rename sub-imports properly.
26337 The special functions allowed in patterns are:
26345 This pattern matches exactly @expr{x}; variable names in @expr{x} are
26346 not interpreted as meta-variables. The only flexibility is that
26347 numbers are compared for numeric equality, so that the pattern
26348 @samp{f(quote(12))} will match both @samp{f(12)} and @samp{f(12.0)}.
26349 (Numbers are always treated this way by the rewrite mechanism:
26350 The rule @samp{f(x,x) := g(x)} will match @samp{f(12, 12.0)}.
26351 The rewrite may produce either @samp{g(12)} or @samp{g(12.0)}
26352 as a result in this case.)
26359 Here @expr{x} must be a function call @samp{f(x1,x2,@dots{})}. This
26360 pattern matches a call to function @expr{f} with the specified
26361 argument patterns. No special knowledge of the properties of the
26362 function @expr{f} is used in this case; @samp{+} is not commutative or
26363 associative. Unlike @code{quote}, the arguments @samp{x1,x2,@dots{}}
26364 are treated as patterns. If you wish them to be treated ``plainly''
26365 as well, you must enclose them with more @code{plain} markers:
26366 @samp{plain(plain(@w{-a}) + plain(b c))}.
26373 Here @expr{x} must be a variable name. This must appear as an
26374 argument to a function or an element of a vector; it specifies that
26375 the argument or element is optional.
26376 As an argument to @samp{+}, @samp{-}, @samp{*}, @samp{&&}, or @samp{||},
26377 or as the second argument to @samp{/} or @samp{^}, the value @var{def}
26378 may be omitted. The pattern @samp{x + opt(y)} matches a sum by
26379 binding one summand to @expr{x} and the other to @expr{y}, and it
26380 matches anything else by binding the whole expression to @expr{x} and
26381 zero to @expr{y}. The other operators above work similarly.
26383 For general miscellaneous functions, the default value @code{def}
26384 must be specified. Optional arguments are dropped starting with
26385 the rightmost one during matching. For example, the pattern
26386 @samp{f(opt(a,0), b, opt(c,b))} will match @samp{f(b)}, @samp{f(a,b)},
26387 or @samp{f(a,b,c)}. Default values of zero and @expr{b} are
26388 supplied in this example for the omitted arguments. Note that
26389 the literal variable @expr{b} will be the default in the latter
26390 case, @emph{not} the value that matched the meta-variable @expr{b}.
26391 In other words, the default @var{def} is effectively quoted.
26393 @item condition(x,c)
26399 This matches the pattern @expr{x}, with the attached condition
26400 @expr{c}. It is the same as @samp{x :: c}.
26408 This matches anything that matches both pattern @expr{x} and
26409 pattern @expr{y}. It is the same as @samp{x &&& y}.
26410 @pxref{Composing Patterns in Rewrite Rules}.
26418 This matches anything that matches either pattern @expr{x} or
26419 pattern @expr{y}. It is the same as @w{@samp{x ||| y}}.
26427 This matches anything that does not match pattern @expr{x}.
26428 It is the same as @samp{!!! x}.
26434 @tindex cons (rewrites)
26435 This matches any vector of one or more elements. The first
26436 element is matched to @expr{h}; a vector of the remaining
26437 elements is matched to @expr{t}. Note that vectors of fixed
26438 length can also be matched as actual vectors: The rule
26439 @samp{cons(a,cons(b,[])) := cons(a+b,[])} is equivalent
26440 to the rule @samp{[a,b] := [a+b]}.
26446 @tindex rcons (rewrites)
26447 This is like @code{cons}, except that the @emph{last} element
26448 is matched to @expr{h}, with the remaining elements matched
26451 @item apply(f,args)
26455 @tindex apply (rewrites)
26456 This matches any function call. The name of the function, in
26457 the form of a variable, is matched to @expr{f}. The arguments
26458 of the function, as a vector of zero or more objects, are
26459 matched to @samp{args}. Constants, variables, and vectors
26460 do @emph{not} match an @code{apply} pattern. For example,
26461 @samp{apply(f,x)} matches any function call, @samp{apply(quote(f),x)}
26462 matches any call to the function @samp{f}, @samp{apply(f,[a,b])}
26463 matches any function call with exactly two arguments, and
26464 @samp{apply(quote(f), cons(a,cons(b,x)))} matches any call
26465 to the function @samp{f} with two or more arguments. Another
26466 way to implement the latter, if the rest of the rule does not
26467 need to refer to the first two arguments of @samp{f} by name,
26468 would be @samp{apply(quote(f), x :: vlen(x) >= 2)}.
26469 Here's a more interesting sample use of @code{apply}:
26472 apply(f,[x+n]) := n + apply(f,[x])
26473 :: in(f, [floor,ceil,round,trunc]) :: integer(n)
26476 Note, however, that this will be slower to match than a rule
26477 set with four separate rules. The reason is that Calc sorts
26478 the rules of a rule set according to top-level function name;
26479 if the top-level function is @code{apply}, Calc must try the
26480 rule for every single formula and sub-formula. If the top-level
26481 function in the pattern is, say, @code{floor}, then Calc invokes
26482 the rule only for sub-formulas which are calls to @code{floor}.
26484 Formulas normally written with operators like @code{+} are still
26485 considered function calls: @code{apply(f,x)} matches @samp{a+b}
26486 with @samp{f = add}, @samp{x = [a,b]}.
26488 You must use @code{apply} for meta-variables with function names
26489 on both sides of a rewrite rule: @samp{apply(f, [x]) := f(x+1)}
26490 is @emph{not} correct, because it rewrites @samp{spam(6)} into
26491 @samp{f(7)}. The righthand side should be @samp{apply(f, [x+1])}.
26492 Also note that you will have to use No-Simplify mode (@kbd{m O})
26493 when entering this rule so that the @code{apply} isn't
26494 evaluated immediately to get the new rule @samp{f(x) := f(x+1)}.
26495 Or, use @kbd{s e} to enter the rule without going through the stack,
26496 or enter the rule as @samp{apply(f, [x]) := apply(f, [x+1]) @w{:: 1}}.
26497 @xref{Conditional Rewrite Rules}.
26504 This is used for applying rules to formulas with selections;
26505 @pxref{Selections with Rewrite Rules}.
26508 Special functions for the righthand sides of rules are:
26512 The notation @samp{quote(x)} is changed to @samp{x} when the
26513 righthand side is used. As far as the rewrite rule is concerned,
26514 @code{quote} is invisible. However, @code{quote} has the special
26515 property in Calc that its argument is not evaluated. Thus,
26516 while it will not work to put the rule @samp{t(a) := typeof(a)}
26517 on the stack because @samp{typeof(a)} is evaluated immediately
26518 to produce @samp{t(a) := 100}, you can use @code{quote} to
26519 protect the righthand side: @samp{t(a) := quote(typeof(a))}.
26520 (@xref{Conditional Rewrite Rules}, for another trick for
26521 protecting rules from evaluation.)
26524 Special properties of and simplifications for the function call
26525 @expr{x} are not used. One interesting case where @code{plain}
26526 is useful is the rule, @samp{q(x) := quote(x)}, trying to expand a
26527 shorthand notation for the @code{quote} function. This rule will
26528 not work as shown; instead of replacing @samp{q(foo)} with
26529 @samp{quote(foo)}, it will replace it with @samp{foo}! The correct
26530 rule would be @samp{q(x) := plain(quote(x))}.
26533 Where @expr{t} is a vector, this is converted into an expanded
26534 vector during rewrite processing. Note that @code{cons} is a regular
26535 Calc function which normally does this anyway; the only way @code{cons}
26536 is treated specially by rewrites is that @code{cons} on the righthand
26537 side of a rule will be evaluated even if default simplifications
26538 have been turned off.
26541 Analogous to @code{cons} except putting @expr{h} at the @emph{end} of
26542 the vector @expr{t}.
26544 @item apply(f,args)
26545 Where @expr{f} is a variable and @var{args} is a vector, this
26546 is converted to a function call. Once again, note that @code{apply}
26547 is also a regular Calc function.
26554 The formula @expr{x} is handled in the usual way, then the
26555 default simplifications are applied to it even if they have
26556 been turned off normally. This allows you to treat any function
26557 similarly to the way @code{cons} and @code{apply} are always
26558 treated. However, there is a slight difference: @samp{cons(2+3, [])}
26559 with default simplifications off will be converted to @samp{[2+3]},
26560 whereas @samp{eval(cons(2+3, []))} will be converted to @samp{[5]}.
26567 The formula @expr{x} has meta-variables substituted in the usual
26568 way, then algebraically simplified as if by the @kbd{a s} command.
26570 @item evalextsimp(x)
26574 @tindex evalextsimp
26575 The formula @expr{x} has meta-variables substituted in the normal
26576 way, then ``extendedly'' simplified as if by the @kbd{a e} command.
26579 @xref{Selections with Rewrite Rules}.
26582 There are also some special functions you can use in conditions.
26590 The expression @expr{x} is evaluated with meta-variables substituted.
26591 The @kbd{a s} command's simplifications are @emph{not} applied by
26592 default, but @expr{x} can include calls to @code{evalsimp} or
26593 @code{evalextsimp} as described above to invoke higher levels
26594 of simplification. The
26595 result of @expr{x} is then bound to the meta-variable @expr{v}. As
26596 usual, if this meta-variable has already been matched to something
26597 else the two values must be equal; if the meta-variable is new then
26598 it is bound to the result of the expression. This variable can then
26599 appear in later conditions, and on the righthand side of the rule.
26600 In fact, @expr{v} may be any pattern in which case the result of
26601 evaluating @expr{x} is matched to that pattern, binding any
26602 meta-variables that appear in that pattern. Note that @code{let}
26603 can only appear by itself as a condition, or as one term of an
26604 @samp{&&} which is a whole condition: It cannot be inside
26605 an @samp{||} term or otherwise buried.
26607 The alternate, equivalent form @samp{let(v, x)} is also recognized.
26608 Note that the use of @samp{:=} by @code{let}, while still being
26609 assignment-like in character, is unrelated to the use of @samp{:=}
26610 in the main part of a rewrite rule.
26612 As an example, @samp{f(a) := g(ia) :: let(ia := 1/a) :: constant(ia)}
26613 replaces @samp{f(a)} with @samp{g} of the inverse of @samp{a}, if
26614 that inverse exists and is constant. For example, if @samp{a} is a
26615 singular matrix the operation @samp{1/a} is left unsimplified and
26616 @samp{constant(ia)} fails, but if @samp{a} is an invertible matrix
26617 then the rule succeeds. Without @code{let} there would be no way
26618 to express this rule that didn't have to invert the matrix twice.
26619 Note that, because the meta-variable @samp{ia} is otherwise unbound
26620 in this rule, the @code{let} condition itself always ``succeeds''
26621 because no matter what @samp{1/a} evaluates to, it can successfully
26622 be bound to @code{ia}.
26624 Here's another example, for integrating cosines of linear
26625 terms: @samp{myint(cos(y),x) := sin(y)/b :: let([a,b,x] := lin(y,x))}.
26626 The @code{lin} function returns a 3-vector if its argument is linear,
26627 or leaves itself unevaluated if not. But an unevaluated @code{lin}
26628 call will not match the 3-vector on the lefthand side of the @code{let},
26629 so this @code{let} both verifies that @code{y} is linear, and binds
26630 the coefficients @code{a} and @code{b} for use elsewhere in the rule.
26631 (It would have been possible to use @samp{sin(a x + b)/b} for the
26632 righthand side instead, but using @samp{sin(y)/b} avoids gratuitous
26633 rearrangement of the argument of the sine.)
26639 Similarly, here is a rule that implements an inverse-@code{erf}
26640 function. It uses @code{root} to search for a solution. If
26641 @code{root} succeeds, it will return a vector of two numbers
26642 where the first number is the desired solution. If no solution
26643 is found, @code{root} remains in symbolic form. So we use
26644 @code{let} to check that the result was indeed a vector.
26647 ierf(x) := y :: let([y,z] := root(erf(a) = x, a, .5))
26651 The meta-variable @var{v}, which must already have been matched
26652 to something elsewhere in the rule, is compared against pattern
26653 @var{p}. Since @code{matches} is a standard Calc function, it
26654 can appear anywhere in a condition. But if it appears alone or
26655 as a term of a top-level @samp{&&}, then you get the special
26656 extra feature that meta-variables which are bound to things
26657 inside @var{p} can be used elsewhere in the surrounding rewrite
26660 The only real difference between @samp{let(p := v)} and
26661 @samp{matches(v, p)} is that the former evaluates @samp{v} using
26662 the default simplifications, while the latter does not.
26666 This is actually a variable, not a function. If @code{remember}
26667 appears as a condition in a rule, then when that rule succeeds
26668 the original expression and rewritten expression are added to the
26669 front of the rule set that contained the rule. If the rule set
26670 was not stored in a variable, @code{remember} is ignored. The
26671 lefthand side is enclosed in @code{quote} in the added rule if it
26672 contains any variables.
26674 For example, the rule @samp{f(n) := n f(n-1) :: remember} applied
26675 to @samp{f(7)} will add the rule @samp{f(7) := 7 f(6)} to the front
26676 of the rule set. The rule set @code{EvalRules} works slightly
26677 differently: There, the evaluation of @samp{f(6)} will complete before
26678 the result is added to the rule set, in this case as @samp{f(7) := 5040}.
26679 Thus @code{remember} is most useful inside @code{EvalRules}.
26681 It is up to you to ensure that the optimization performed by
26682 @code{remember} is safe. For example, the rule @samp{foo(n) := n
26683 :: evalv(eatfoo) > 0 :: remember} is a bad idea (@code{evalv} is
26684 the function equivalent of the @kbd{=} command); if the variable
26685 @code{eatfoo} ever contains 1, rules like @samp{foo(7) := 7} will
26686 be added to the rule set and will continue to operate even if
26687 @code{eatfoo} is later changed to 0.
26694 Remember the match as described above, but only if condition @expr{c}
26695 is true. For example, @samp{remember(n % 4 = 0)} in the above factorial
26696 rule remembers only every fourth result. Note that @samp{remember(1)}
26697 is equivalent to @samp{remember}, and @samp{remember(0)} has no effect.
26700 @node Composing Patterns in Rewrite Rules, Nested Formulas with Rewrite Rules, Other Features of Rewrite Rules, Rewrite Rules
26701 @subsection Composing Patterns in Rewrite Rules
26704 There are three operators, @samp{&&&}, @samp{|||}, and @samp{!!!},
26705 that combine rewrite patterns to make larger patterns. The
26706 combinations are ``and,'' ``or,'' and ``not,'' respectively, and
26707 these operators are the pattern equivalents of @samp{&&}, @samp{||}
26708 and @samp{!} (which operate on zero-or-nonzero logical values).
26710 Note that @samp{&&&}, @samp{|||}, and @samp{!!!} are left in symbolic
26711 form by all regular Calc features; they have special meaning only in
26712 the context of rewrite rule patterns.
26714 The pattern @samp{@var{p1} &&& @var{p2}} matches anything that
26715 matches both @var{p1} and @var{p2}. One especially useful case is
26716 when one of @var{p1} or @var{p2} is a meta-variable. For example,
26717 here is a rule that operates on error forms:
26720 f(x &&& a +/- b, x) := g(x)
26723 This does the same thing, but is arguably simpler than, the rule
26726 f(a +/- b, a +/- b) := g(a +/- b)
26733 Here's another interesting example:
26736 ends(cons(a, x) &&& rcons(y, b)) := [a, b]
26740 which effectively clips out the middle of a vector leaving just
26741 the first and last elements. This rule will change a one-element
26742 vector @samp{[a]} to @samp{[a, a]}. The similar rule
26745 ends(cons(a, rcons(y, b))) := [a, b]
26749 would do the same thing except that it would fail to match a
26750 one-element vector.
26756 The pattern @samp{@var{p1} ||| @var{p2}} matches anything that
26757 matches either @var{p1} or @var{p2}. Calc first tries matching
26758 against @var{p1}; if that fails, it goes on to try @var{p2}.
26764 A simple example of @samp{|||} is
26767 curve(inf ||| -inf) := 0
26771 which converts both @samp{curve(inf)} and @samp{curve(-inf)} to zero.
26773 Here is a larger example:
26776 log(a, b) ||| (ln(a) :: let(b := e)) := mylog(a, b)
26779 This matches both generalized and natural logarithms in a single rule.
26780 Note that the @samp{::} term must be enclosed in parentheses because
26781 that operator has lower precedence than @samp{|||} or @samp{:=}.
26783 (In practice this rule would probably include a third alternative,
26784 omitted here for brevity, to take care of @code{log10}.)
26786 While Calc generally treats interior conditions exactly the same as
26787 conditions on the outside of a rule, it does guarantee that if all the
26788 variables in the condition are special names like @code{e}, or already
26789 bound in the pattern to which the condition is attached (say, if
26790 @samp{a} had appeared in this condition), then Calc will process this
26791 condition right after matching the pattern to the left of the @samp{::}.
26792 Thus, we know that @samp{b} will be bound to @samp{e} only if the
26793 @code{ln} branch of the @samp{|||} was taken.
26795 Note that this rule was careful to bind the same set of meta-variables
26796 on both sides of the @samp{|||}. Calc does not check this, but if
26797 you bind a certain meta-variable only in one branch and then use that
26798 meta-variable elsewhere in the rule, results are unpredictable:
26801 f(a,b) ||| g(b) := h(a,b)
26804 Here if the pattern matches @samp{g(17)}, Calc makes no promises about
26805 the value that will be substituted for @samp{a} on the righthand side.
26811 The pattern @samp{!!! @var{pat}} matches anything that does not
26812 match @var{pat}. Any meta-variables that are bound while matching
26813 @var{pat} remain unbound outside of @var{pat}.
26818 f(x &&& !!! a +/- b, !!![]) := g(x)
26822 converts @code{f} whose first argument is anything @emph{except} an
26823 error form, and whose second argument is not the empty vector, into
26824 a similar call to @code{g} (but without the second argument).
26826 If we know that the second argument will be a vector (empty or not),
26827 then an equivalent rule would be:
26830 f(x, y) := g(x) :: typeof(x) != 7 :: vlen(y) > 0
26834 where of course 7 is the @code{typeof} code for error forms.
26835 Another final condition, that works for any kind of @samp{y},
26836 would be @samp{!istrue(y == [])}. (The @code{istrue} function
26837 returns an explicit 0 if its argument was left in symbolic form;
26838 plain @samp{!(y == [])} or @samp{y != []} would not work to replace
26839 @samp{!!![]} since these would be left unsimplified, and thus cause
26840 the rule to fail, if @samp{y} was something like a variable name.)
26842 It is possible for a @samp{!!!} to refer to meta-variables bound
26843 elsewhere in the pattern. For example,
26850 matches any call to @code{f} with different arguments, changing
26851 this to @code{g} with only the first argument.
26853 If a function call is to be matched and one of the argument patterns
26854 contains a @samp{!!!} somewhere inside it, that argument will be
26862 will be careful to bind @samp{a} to the second argument of @code{f}
26863 before testing the first argument. If Calc had tried to match the
26864 first argument of @code{f} first, the results would have been
26865 disastrous: since @code{a} was unbound so far, the pattern @samp{a}
26866 would have matched anything at all, and the pattern @samp{!!!a}
26867 therefore would @emph{not} have matched anything at all!
26869 @node Nested Formulas with Rewrite Rules, Multi-Phase Rewrite Rules, Composing Patterns in Rewrite Rules, Rewrite Rules
26870 @subsection Nested Formulas with Rewrite Rules
26873 When @kbd{a r} (@code{calc-rewrite}) is used, it takes an expression from
26874 the top of the stack and attempts to match any of the specified rules
26875 to any part of the expression, starting with the whole expression
26876 and then, if that fails, trying deeper and deeper sub-expressions.
26877 For each part of the expression, the rules are tried in the order
26878 they appear in the rules vector. The first rule to match the first
26879 sub-expression wins; it replaces the matched sub-expression according
26880 to the @var{new} part of the rule.
26882 Often, the rule set will match and change the formula several times.
26883 The top-level formula is first matched and substituted repeatedly until
26884 it no longer matches the pattern; then, sub-formulas are tried, and
26885 so on. Once every part of the formula has gotten its chance, the
26886 rewrite mechanism starts over again with the top-level formula
26887 (in case a substitution of one of its arguments has caused it again
26888 to match). This continues until no further matches can be made
26889 anywhere in the formula.
26891 It is possible for a rule set to get into an infinite loop. The
26892 most obvious case, replacing a formula with itself, is not a problem
26893 because a rule is not considered to ``succeed'' unless the righthand
26894 side actually comes out to something different than the original
26895 formula or sub-formula that was matched. But if you accidentally
26896 had both @samp{ln(a b) := ln(a) + ln(b)} and the reverse
26897 @samp{ln(a) + ln(b) := ln(a b)} in your rule set, Calc would
26898 run forever switching a formula back and forth between the two
26901 To avoid disaster, Calc normally stops after 100 changes have been
26902 made to the formula. This will be enough for most multiple rewrites,
26903 but it will keep an endless loop of rewrites from locking up the
26904 computer forever. (On most systems, you can also type @kbd{C-g} to
26905 halt any Emacs command prematurely.)
26907 To change this limit, give a positive numeric prefix argument.
26908 In particular, @kbd{M-1 a r} applies only one rewrite at a time,
26909 useful when you are first testing your rule (or just if repeated
26910 rewriting is not what is called for by your application).
26919 You can also put a ``function call'' @samp{iterations(@var{n})}
26920 in place of a rule anywhere in your rules vector (but usually at
26921 the top). Then, @var{n} will be used instead of 100 as the default
26922 number of iterations for this rule set. You can use
26923 @samp{iterations(inf)} if you want no iteration limit by default.
26924 A prefix argument will override the @code{iterations} limit in the
26932 More precisely, the limit controls the number of ``iterations,''
26933 where each iteration is a successful matching of a rule pattern whose
26934 righthand side, after substituting meta-variables and applying the
26935 default simplifications, is different from the original sub-formula
26938 A prefix argument of zero sets the limit to infinity. Use with caution!
26940 Given a negative numeric prefix argument, @kbd{a r} will match and
26941 substitute the top-level expression up to that many times, but
26942 will not attempt to match the rules to any sub-expressions.
26944 In a formula, @code{rewrite(@var{expr}, @var{rules}, @var{n})}
26945 does a rewriting operation. Here @var{expr} is the expression
26946 being rewritten, @var{rules} is the rule, vector of rules, or
26947 variable containing the rules, and @var{n} is the optional
26948 iteration limit, which may be a positive integer, a negative
26949 integer, or @samp{inf} or @samp{-inf}. If @var{n} is omitted
26950 the @code{iterations} value from the rule set is used; if both
26951 are omitted, 100 is used.
26953 @node Multi-Phase Rewrite Rules, Selections with Rewrite Rules, Nested Formulas with Rewrite Rules, Rewrite Rules
26954 @subsection Multi-Phase Rewrite Rules
26957 It is possible to separate a rewrite rule set into several @dfn{phases}.
26958 During each phase, certain rules will be enabled while certain others
26959 will be disabled. A @dfn{phase schedule} controls the order in which
26960 phases occur during the rewriting process.
26967 If a call to the marker function @code{phase} appears in the rules
26968 vector in place of a rule, all rules following that point will be
26969 members of the phase(s) identified in the arguments to @code{phase}.
26970 Phases are given integer numbers. The markers @samp{phase()} and
26971 @samp{phase(all)} both mean the following rules belong to all phases;
26972 this is the default at the start of the rule set.
26974 If you do not explicitly schedule the phases, Calc sorts all phase
26975 numbers that appear in the rule set and executes the phases in
26976 ascending order. For example, the rule set
26993 has three phases, 1 through 3. Phase 1 consists of the @code{f0},
26994 @code{f1}, and @code{f4} rules (in that order). Phase 2 consists of
26995 @code{f0}, @code{f2}, and @code{f4}. Phase 3 consists of @code{f0}
26998 When Calc rewrites a formula using this rule set, it first rewrites
26999 the formula using only the phase 1 rules until no further changes are
27000 possible. Then it switches to the phase 2 rule set and continues
27001 until no further changes occur, then finally rewrites with phase 3.
27002 When no more phase 3 rules apply, rewriting finishes. (This is
27003 assuming @kbd{a r} with a large enough prefix argument to allow the
27004 rewriting to run to completion; the sequence just described stops
27005 early if the number of iterations specified in the prefix argument,
27006 100 by default, is reached.)
27008 During each phase, Calc descends through the nested levels of the
27009 formula as described previously. (@xref{Nested Formulas with Rewrite
27010 Rules}.) Rewriting starts at the top of the formula, then works its
27011 way down to the parts, then goes back to the top and works down again.
27012 The phase 2 rules do not begin until no phase 1 rules apply anywhere
27019 A @code{schedule} marker appearing in the rule set (anywhere, but
27020 conventionally at the top) changes the default schedule of phases.
27021 In the simplest case, @code{schedule} has a sequence of phase numbers
27022 for arguments; each phase number is invoked in turn until the
27023 arguments to @code{schedule} are exhausted. Thus adding
27024 @samp{schedule(3,2,1)} at the top of the above rule set would
27025 reverse the order of the phases; @samp{schedule(1,2,3)} would have
27026 no effect since this is the default schedule; and @samp{schedule(1,2,1,3)}
27027 would give phase 1 a second chance after phase 2 has completed, before
27028 moving on to phase 3.
27030 Any argument to @code{schedule} can instead be a vector of phase
27031 numbers (or even of sub-vectors). Then the sub-sequence of phases
27032 described by the vector are tried repeatedly until no change occurs
27033 in any phase in the sequence. For example, @samp{schedule([1, 2], 3)}
27034 tries phase 1, then phase 2, then, if either phase made any changes
27035 to the formula, repeats these two phases until they can make no
27036 further progress. Finally, it goes on to phase 3 for finishing
27039 Also, items in @code{schedule} can be variable names as well as
27040 numbers. A variable name is interpreted as the name of a function
27041 to call on the whole formula. For example, @samp{schedule(1, simplify)}
27042 says to apply the phase-1 rules (presumably, all of them), then to
27043 call @code{simplify} which is the function name equivalent of @kbd{a s}.
27044 Likewise, @samp{schedule([1, simplify])} says to alternate between
27045 phase 1 and @kbd{a s} until no further changes occur.
27047 Phases can be used purely to improve efficiency; if it is known that
27048 a certain group of rules will apply only at the beginning of rewriting,
27049 and a certain other group will apply only at the end, then rewriting
27050 will be faster if these groups are identified as separate phases.
27051 Once the phase 1 rules are done, Calc can put them aside and no longer
27052 spend any time on them while it works on phase 2.
27054 There are also some problems that can only be solved with several
27055 rewrite phases. For a real-world example of a multi-phase rule set,
27056 examine the set @code{FitRules}, which is used by the curve-fitting
27057 command to convert a model expression to linear form.
27058 @xref{Curve Fitting Details}. This set is divided into four phases.
27059 The first phase rewrites certain kinds of expressions to be more
27060 easily linearizable, but less computationally efficient. After the
27061 linear components have been picked out, the final phase includes the
27062 opposite rewrites to put each component back into an efficient form.
27063 If both sets of rules were included in one big phase, Calc could get
27064 into an infinite loop going back and forth between the two forms.
27066 Elsewhere in @code{FitRules}, the components are first isolated,
27067 then recombined where possible to reduce the complexity of the linear
27068 fit, then finally packaged one component at a time into vectors.
27069 If the packaging rules were allowed to begin before the recombining
27070 rules were finished, some components might be put away into vectors
27071 before they had a chance to recombine. By putting these rules in
27072 two separate phases, this problem is neatly avoided.
27074 @node Selections with Rewrite Rules, Matching Commands, Multi-Phase Rewrite Rules, Rewrite Rules
27075 @subsection Selections with Rewrite Rules
27078 If a sub-formula of the current formula is selected (as by @kbd{j s};
27079 @pxref{Selecting Subformulas}), the @kbd{a r} (@code{calc-rewrite})
27080 command applies only to that sub-formula. Together with a negative
27081 prefix argument, you can use this fact to apply a rewrite to one
27082 specific part of a formula without affecting any other parts.
27085 @pindex calc-rewrite-selection
27086 The @kbd{j r} (@code{calc-rewrite-selection}) command allows more
27087 sophisticated operations on selections. This command prompts for
27088 the rules in the same way as @kbd{a r}, but it then applies those
27089 rules to the whole formula in question even though a sub-formula
27090 of it has been selected. However, the selected sub-formula will
27091 first have been surrounded by a @samp{select( )} function call.
27092 (Calc's evaluator does not understand the function name @code{select};
27093 this is only a tag used by the @kbd{j r} command.)
27095 For example, suppose the formula on the stack is @samp{2 (a + b)^2}
27096 and the sub-formula @samp{a + b} is selected. This formula will
27097 be rewritten to @samp{2 select(a + b)^2} and then the rewrite
27098 rules will be applied in the usual way. The rewrite rules can
27099 include references to @code{select} to tell where in the pattern
27100 the selected sub-formula should appear.
27102 If there is still exactly one @samp{select( )} function call in
27103 the formula after rewriting is done, it indicates which part of
27104 the formula should be selected afterwards. Otherwise, the
27105 formula will be unselected.
27107 You can make @kbd{j r} act much like @kbd{a r} by enclosing both parts
27108 of the rewrite rule with @samp{select()}. However, @kbd{j r}
27109 allows you to use the current selection in more flexible ways.
27110 Suppose you wished to make a rule which removed the exponent from
27111 the selected term; the rule @samp{select(a)^x := select(a)} would
27112 work. In the above example, it would rewrite @samp{2 select(a + b)^2}
27113 to @samp{2 select(a + b)}. This would then be returned to the
27114 stack as @samp{2 (a + b)} with the @samp{a + b} selected.
27116 The @kbd{j r} command uses one iteration by default, unlike
27117 @kbd{a r} which defaults to 100 iterations. A numeric prefix
27118 argument affects @kbd{j r} in the same way as @kbd{a r}.
27119 @xref{Nested Formulas with Rewrite Rules}.
27121 As with other selection commands, @kbd{j r} operates on the stack
27122 entry that contains the cursor. (If the cursor is on the top-of-stack
27123 @samp{.} marker, it works as if the cursor were on the formula
27126 If you don't specify a set of rules, the rules are taken from the
27127 top of the stack, just as with @kbd{a r}. In this case, the
27128 cursor must indicate stack entry 2 or above as the formula to be
27129 rewritten (otherwise the same formula would be used as both the
27130 target and the rewrite rules).
27132 If the indicated formula has no selection, the cursor position within
27133 the formula temporarily selects a sub-formula for the purposes of this
27134 command. If the cursor is not on any sub-formula (e.g., it is in
27135 the line-number area to the left of the formula), the @samp{select( )}
27136 markers are ignored by the rewrite mechanism and the rules are allowed
27137 to apply anywhere in the formula.
27139 As a special feature, the normal @kbd{a r} command also ignores
27140 @samp{select( )} calls in rewrite rules. For example, if you used the
27141 above rule @samp{select(a)^x := select(a)} with @kbd{a r}, it would apply
27142 the rule as if it were @samp{a^x := a}. Thus, you can write general
27143 purpose rules with @samp{select( )} hints inside them so that they
27144 will ``do the right thing'' in both @kbd{a r} and @kbd{j r},
27145 both with and without selections.
27147 @node Matching Commands, Automatic Rewrites, Selections with Rewrite Rules, Rewrite Rules
27148 @subsection Matching Commands
27154 The @kbd{a m} (@code{calc-match}) [@code{match}] function takes a
27155 vector of formulas and a rewrite-rule-style pattern, and produces
27156 a vector of all formulas which match the pattern. The command
27157 prompts you to enter the pattern; as for @kbd{a r}, you can enter
27158 a single pattern (i.e., a formula with meta-variables), or a
27159 vector of patterns, or a variable which contains patterns, or
27160 you can give a blank response in which case the patterns are taken
27161 from the top of the stack. The pattern set will be compiled once
27162 and saved if it is stored in a variable. If there are several
27163 patterns in the set, vector elements are kept if they match any
27166 For example, @samp{match(a+b, [x, x+y, x-y, 7, x+y+z])}
27167 will return @samp{[x+y, x-y, x+y+z]}.
27169 The @code{import} mechanism is not available for pattern sets.
27171 The @kbd{a m} command can also be used to extract all vector elements
27172 which satisfy any condition: The pattern @samp{x :: x>0} will select
27173 all the positive vector elements.
27177 With the Inverse flag [@code{matchnot}], this command extracts all
27178 vector elements which do @emph{not} match the given pattern.
27184 There is also a function @samp{matches(@var{x}, @var{p})} which
27185 evaluates to 1 if expression @var{x} matches pattern @var{p}, or
27186 to 0 otherwise. This is sometimes useful for including into the
27187 conditional clauses of other rewrite rules.
27193 The function @code{vmatches} is just like @code{matches}, except
27194 that if the match succeeds it returns a vector of assignments to
27195 the meta-variables instead of the number 1. For example,
27196 @samp{vmatches(f(1,2), f(a,b))} returns @samp{[a := 1, b := 2]}.
27197 If the match fails, the function returns the number 0.
27199 @node Automatic Rewrites, Debugging Rewrites, Matching Commands, Rewrite Rules
27200 @subsection Automatic Rewrites
27203 @cindex @code{EvalRules} variable
27205 It is possible to get Calc to apply a set of rewrite rules on all
27206 results, effectively adding to the built-in set of default
27207 simplifications. To do this, simply store your rule set in the
27208 variable @code{EvalRules}. There is a convenient @kbd{s E} command
27209 for editing @code{EvalRules}; @pxref{Operations on Variables}.
27211 For example, suppose you want @samp{sin(a + b)} to be expanded out
27212 to @samp{sin(b) cos(a) + cos(b) sin(a)} wherever it appears, and
27213 similarly for @samp{cos(a + b)}. The corresponding rewrite rule
27218 [ sin(a + b) := cos(a) sin(b) + sin(a) cos(b),
27219 cos(a + b) := cos(a) cos(b) - sin(a) sin(b) ]
27223 To apply these manually, you could put them in a variable called
27224 @code{trigexp} and then use @kbd{a r trigexp} every time you wanted
27225 to expand trig functions. But if instead you store them in the
27226 variable @code{EvalRules}, they will automatically be applied to all
27227 sines and cosines of sums. Then, with @samp{2 x} and @samp{45} on
27228 the stack, typing @kbd{+ S} will (assuming Degrees mode) result in
27229 @samp{0.7071 sin(2 x) + 0.7071 cos(2 x)} automatically.
27231 As each level of a formula is evaluated, the rules from
27232 @code{EvalRules} are applied before the default simplifications.
27233 Rewriting continues until no further @code{EvalRules} apply.
27234 Note that this is different from the usual order of application of
27235 rewrite rules: @code{EvalRules} works from the bottom up, simplifying
27236 the arguments to a function before the function itself, while @kbd{a r}
27237 applies rules from the top down.
27239 Because the @code{EvalRules} are tried first, you can use them to
27240 override the normal behavior of any built-in Calc function.
27242 It is important not to write a rule that will get into an infinite
27243 loop. For example, the rule set @samp{[f(0) := 1, f(n) := n f(n-1)]}
27244 appears to be a good definition of a factorial function, but it is
27245 unsafe. Imagine what happens if @samp{f(2.5)} is simplified. Calc
27246 will continue to subtract 1 from this argument forever without reaching
27247 zero. A safer second rule would be @samp{f(n) := n f(n-1) :: n>0}.
27248 Another dangerous rule is @samp{g(x, y) := g(y, x)}. Rewriting
27249 @samp{g(2, 4)}, this would bounce back and forth between that and
27250 @samp{g(4, 2)} forever. If an infinite loop in @code{EvalRules}
27251 occurs, Emacs will eventually stop with a ``Computation got stuck
27252 or ran too long'' message.
27254 Another subtle difference between @code{EvalRules} and regular rewrites
27255 concerns rules that rewrite a formula into an identical formula. For
27256 example, @samp{f(n) := f(floor(n))} ``fails to match'' when @expr{n} is
27257 already an integer. But in @code{EvalRules} this case is detected only
27258 if the righthand side literally becomes the original formula before any
27259 further simplification. This means that @samp{f(n) := f(floor(n))} will
27260 get into an infinite loop if it occurs in @code{EvalRules}. Calc will
27261 replace @samp{f(6)} with @samp{f(floor(6))}, which is different from
27262 @samp{f(6)}, so it will consider the rule to have matched and will
27263 continue simplifying that formula; first the argument is simplified
27264 to get @samp{f(6)}, then the rule matches again to get @samp{f(floor(6))}
27265 again, ad infinitum. A much safer rule would check its argument first,
27266 say, with @samp{f(n) := f(floor(n)) :: !dint(n)}.
27268 (What really happens is that the rewrite mechanism substitutes the
27269 meta-variables in the righthand side of a rule, compares to see if the
27270 result is the same as the original formula and fails if so, then uses
27271 the default simplifications to simplify the result and compares again
27272 (and again fails if the formula has simplified back to its original
27273 form). The only special wrinkle for the @code{EvalRules} is that the
27274 same rules will come back into play when the default simplifications
27275 are used. What Calc wants to do is build @samp{f(floor(6))}, see that
27276 this is different from the original formula, simplify to @samp{f(6)},
27277 see that this is the same as the original formula, and thus halt the
27278 rewriting. But while simplifying, @samp{f(6)} will again trigger
27279 the same @code{EvalRules} rule and Calc will get into a loop inside
27280 the rewrite mechanism itself.)
27282 The @code{phase}, @code{schedule}, and @code{iterations} markers do
27283 not work in @code{EvalRules}. If the rule set is divided into phases,
27284 only the phase 1 rules are applied, and the schedule is ignored.
27285 The rules are always repeated as many times as possible.
27287 The @code{EvalRules} are applied to all function calls in a formula,
27288 but not to numbers (and other number-like objects like error forms),
27289 nor to vectors or individual variable names. (Though they will apply
27290 to @emph{components} of vectors and error forms when appropriate.) You
27291 might try to make a variable @code{phihat} which automatically expands
27292 to its definition without the need to press @kbd{=} by writing the
27293 rule @samp{quote(phihat) := (1-sqrt(5))/2}, but unfortunately this rule
27294 will not work as part of @code{EvalRules}.
27296 Finally, another limitation is that Calc sometimes calls its built-in
27297 functions directly rather than going through the default simplifications.
27298 When it does this, @code{EvalRules} will not be able to override those
27299 functions. For example, when you take the absolute value of the complex
27300 number @expr{(2, 3)}, Calc computes @samp{sqrt(2*2 + 3*3)} by calling
27301 the multiplication, addition, and square root functions directly rather
27302 than applying the default simplifications to this formula. So an
27303 @code{EvalRules} rule that (perversely) rewrites @samp{sqrt(13) := 6}
27304 would not apply. (However, if you put Calc into Symbolic mode so that
27305 @samp{sqrt(13)} will be left in symbolic form by the built-in square
27306 root function, your rule will be able to apply. But if the complex
27307 number were @expr{(3,4)}, so that @samp{sqrt(25)} must be calculated,
27308 then Symbolic mode will not help because @samp{sqrt(25)} can be
27309 evaluated exactly to 5.)
27311 One subtle restriction that normally only manifests itself with
27312 @code{EvalRules} is that while a given rewrite rule is in the process
27313 of being checked, that same rule cannot be recursively applied. Calc
27314 effectively removes the rule from its rule set while checking the rule,
27315 then puts it back once the match succeeds or fails. (The technical
27316 reason for this is that compiled pattern programs are not reentrant.)
27317 For example, consider the rule @samp{foo(x) := x :: foo(x/2) > 0}
27318 attempting to match @samp{foo(8)}. This rule will be inactive while
27319 the condition @samp{foo(4) > 0} is checked, even though it might be
27320 an integral part of evaluating that condition. Note that this is not
27321 a problem for the more usual recursive type of rule, such as
27322 @samp{foo(x) := foo(x/2)}, because there the rule has succeeded and
27323 been reactivated by the time the righthand side is evaluated.
27325 If @code{EvalRules} has no stored value (its default state), or if
27326 anything but a vector is stored in it, then it is ignored.
27328 Even though Calc's rewrite mechanism is designed to compare rewrite
27329 rules to formulas as quickly as possible, storing rules in
27330 @code{EvalRules} may make Calc run substantially slower. This is
27331 particularly true of rules where the top-level call is a commonly used
27332 function, or is not fixed. The rule @samp{f(n) := n f(n-1) :: n>0} will
27333 only activate the rewrite mechanism for calls to the function @code{f},
27334 but @samp{lg(n) + lg(m) := lg(n m)} will check every @samp{+} operator.
27337 apply(f, [a*b]) := apply(f, [a]) + apply(f, [b]) :: in(f, [ln, log10])
27341 may seem more ``efficient'' than two separate rules for @code{ln} and
27342 @code{log10}, but actually it is vastly less efficient because rules
27343 with @code{apply} as the top-level pattern must be tested against
27344 @emph{every} function call that is simplified.
27346 @cindex @code{AlgSimpRules} variable
27347 @vindex AlgSimpRules
27348 Suppose you want @samp{sin(a + b)} to be expanded out not all the time,
27349 but only when @kbd{a s} is used to simplify the formula. The variable
27350 @code{AlgSimpRules} holds rules for this purpose. The @kbd{a s} command
27351 will apply @code{EvalRules} and @code{AlgSimpRules} to the formula, as
27352 well as all of its built-in simplifications.
27354 Most of the special limitations for @code{EvalRules} don't apply to
27355 @code{AlgSimpRules}. Calc simply does an @kbd{a r AlgSimpRules}
27356 command with an infinite repeat count as the first step of @kbd{a s}.
27357 It then applies its own built-in simplifications throughout the
27358 formula, and then repeats these two steps (along with applying the
27359 default simplifications) until no further changes are possible.
27361 @cindex @code{ExtSimpRules} variable
27362 @cindex @code{UnitSimpRules} variable
27363 @vindex ExtSimpRules
27364 @vindex UnitSimpRules
27365 There are also @code{ExtSimpRules} and @code{UnitSimpRules} variables
27366 that are used by @kbd{a e} and @kbd{u s}, respectively; these commands
27367 also apply @code{EvalRules} and @code{AlgSimpRules}. The variable
27368 @code{IntegSimpRules} contains simplification rules that are used
27369 only during integration by @kbd{a i}.
27371 @node Debugging Rewrites, Examples of Rewrite Rules, Automatic Rewrites, Rewrite Rules
27372 @subsection Debugging Rewrites
27375 If a buffer named @samp{*Trace*} exists, the rewrite mechanism will
27376 record some useful information there as it operates. The original
27377 formula is written there, as is the result of each successful rewrite,
27378 and the final result of the rewriting. All phase changes are also
27381 Calc always appends to @samp{*Trace*}. You must empty this buffer
27382 yourself periodically if it is in danger of growing unwieldy.
27384 Note that the rewriting mechanism is substantially slower when the
27385 @samp{*Trace*} buffer exists, even if the buffer is not visible on
27386 the screen. Once you are done, you will probably want to kill this
27387 buffer (with @kbd{C-x k *Trace* @key{RET}}). If you leave it in
27388 existence and forget about it, all your future rewrite commands will
27389 be needlessly slow.
27391 @node Examples of Rewrite Rules, , Debugging Rewrites, Rewrite Rules
27392 @subsection Examples of Rewrite Rules
27395 Returning to the example of substituting the pattern
27396 @samp{sin(x)^2 + cos(x)^2} with 1, we saw that the rule
27397 @samp{opt(a) sin(x)^2 + opt(a) cos(x)^2 := a} does a good job of
27398 finding suitable cases. Another solution would be to use the rule
27399 @samp{cos(x)^2 := 1 - sin(x)^2}, followed by algebraic simplification
27400 if necessary. This rule will be the most effective way to do the job,
27401 but at the expense of making some changes that you might not desire.
27403 Another algebraic rewrite rule is @samp{exp(x+y) := exp(x) exp(y)}.
27404 To make this work with the @w{@kbd{j r}} command so that it can be
27405 easily targeted to a particular exponential in a large formula,
27406 you might wish to write the rule as @samp{select(exp(x+y)) :=
27407 select(exp(x) exp(y))}. The @samp{select} markers will be
27408 ignored by the regular @kbd{a r} command
27409 (@pxref{Selections with Rewrite Rules}).
27411 A surprisingly useful rewrite rule is @samp{a/(b-c) := a*(b+c)/(b^2-c^2)}.
27412 This will simplify the formula whenever @expr{b} and/or @expr{c} can
27413 be made simpler by squaring. For example, applying this rule to
27414 @samp{2 / (sqrt(2) + 3)} yields @samp{6:7 - 2:7 sqrt(2)} (assuming
27415 Symbolic mode has been enabled to keep the square root from being
27416 evaluated to a floating-point approximation). This rule is also
27417 useful when working with symbolic complex numbers, e.g.,
27418 @samp{(a + b i) / (c + d i)}.
27420 As another example, we could define our own ``triangular numbers'' function
27421 with the rules @samp{[tri(0) := 0, tri(n) := n + tri(n-1) :: n>0]}. Enter
27422 this vector and store it in a variable: @kbd{@w{s t} trirules}. Now, given
27423 a suitable formula like @samp{tri(5)} on the stack, type @samp{a r trirules}
27424 to apply these rules repeatedly. After six applications, @kbd{a r} will
27425 stop with 15 on the stack. Once these rules are debugged, it would probably
27426 be most useful to add them to @code{EvalRules} so that Calc will evaluate
27427 the new @code{tri} function automatically. We could then use @kbd{Z K} on
27428 the keyboard macro @kbd{' tri($) @key{RET}} to make a command that applies
27429 @code{tri} to the value on the top of the stack. @xref{Programming}.
27431 @cindex Quaternions
27432 The following rule set, contributed by
27433 @texline Fran\c cois
27435 Pinard, implements @dfn{quaternions}, a generalization of the concept of
27436 complex numbers. Quaternions have four components, and are here
27437 represented by function calls @samp{quat(@var{w}, [@var{x}, @var{y},
27438 @var{z}])} with ``real part'' @var{w} and the three ``imaginary'' parts
27439 collected into a vector. Various arithmetical operations on quaternions
27440 are supported. To use these rules, either add them to @code{EvalRules},
27441 or create a command based on @kbd{a r} for simplifying quaternion
27442 formulas. A convenient way to enter quaternions would be a command
27443 defined by a keyboard macro containing: @kbd{' quat($$$$, [$$$, $$, $])
27447 [ quat(w, x, y, z) := quat(w, [x, y, z]),
27448 quat(w, [0, 0, 0]) := w,
27449 abs(quat(w, v)) := hypot(w, v),
27450 -quat(w, v) := quat(-w, -v),
27451 r + quat(w, v) := quat(r + w, v) :: real(r),
27452 r - quat(w, v) := quat(r - w, -v) :: real(r),
27453 quat(w1, v1) + quat(w2, v2) := quat(w1 + w2, v1 + v2),
27454 r * quat(w, v) := quat(r * w, r * v) :: real(r),
27455 plain(quat(w1, v1) * quat(w2, v2))
27456 := quat(w1 * w2 - v1 * v2, w1 * v2 + w2 * v1 + cross(v1, v2)),
27457 quat(w1, v1) / r := quat(w1 / r, v1 / r) :: real(r),
27458 z / quat(w, v) := z * quatinv(quat(w, v)),
27459 quatinv(quat(w, v)) := quat(w, -v) / (w^2 + v^2),
27460 quatsqr(quat(w, v)) := quat(w^2 - v^2, 2 * w * v),
27461 quat(w, v)^k := quatsqr(quat(w, v)^(k / 2))
27462 :: integer(k) :: k > 0 :: k % 2 = 0,
27463 quat(w, v)^k := quatsqr(quat(w, v)^((k - 1) / 2)) * quat(w, v)
27464 :: integer(k) :: k > 2,
27465 quat(w, v)^-k := quatinv(quat(w, v)^k) :: integer(k) :: k > 0 ]
27468 Quaternions, like matrices, have non-commutative multiplication.
27469 In other words, @expr{q1 * q2 = q2 * q1} is not necessarily true if
27470 @expr{q1} and @expr{q2} are @code{quat} forms. The @samp{quat*quat}
27471 rule above uses @code{plain} to prevent Calc from rearranging the
27472 product. It may also be wise to add the line @samp{[quat(), matrix]}
27473 to the @code{Decls} matrix, to ensure that Calc's other algebraic
27474 operations will not rearrange a quaternion product. @xref{Declarations}.
27476 These rules also accept a four-argument @code{quat} form, converting
27477 it to the preferred form in the first rule. If you would rather see
27478 results in the four-argument form, just append the two items
27479 @samp{phase(2), quat(w, [x, y, z]) := quat(w, x, y, z)} to the end
27480 of the rule set. (But remember that multi-phase rule sets don't work
27481 in @code{EvalRules}.)
27483 @node Units, Store and Recall, Algebra, Top
27484 @chapter Operating on Units
27487 One special interpretation of algebraic formulas is as numbers with units.
27488 For example, the formula @samp{5 m / s^2} can be read ``five meters
27489 per second squared.'' The commands in this chapter help you
27490 manipulate units expressions in this form. Units-related commands
27491 begin with the @kbd{u} prefix key.
27494 * Basic Operations on Units::
27495 * The Units Table::
27496 * Predefined Units::
27497 * User-Defined Units::
27500 @node Basic Operations on Units, The Units Table, Units, Units
27501 @section Basic Operations on Units
27504 A @dfn{units expression} is a formula which is basically a number
27505 multiplied and/or divided by one or more @dfn{unit names}, which may
27506 optionally be raised to integer powers. Actually, the value part need not
27507 be a number; any product or quotient involving unit names is a units
27508 expression. Many of the units commands will also accept any formula,
27509 where the command applies to all units expressions which appear in the
27512 A unit name is a variable whose name appears in the @dfn{unit table},
27513 or a variable whose name is a prefix character like @samp{k} (for ``kilo'')
27514 or @samp{u} (for ``micro'') followed by a name in the unit table.
27515 A substantial table of built-in units is provided with Calc;
27516 @pxref{Predefined Units}. You can also define your own unit names;
27517 @pxref{User-Defined Units}.
27519 Note that if the value part of a units expression is exactly @samp{1},
27520 it will be removed by the Calculator's automatic algebra routines: The
27521 formula @samp{1 mm} is ``simplified'' to @samp{mm}. This is only a
27522 display anomaly, however; @samp{mm} will work just fine as a
27523 representation of one millimeter.
27525 You may find that Algebraic mode (@pxref{Algebraic Entry}) makes working
27526 with units expressions easier. Otherwise, you will have to remember
27527 to hit the apostrophe key every time you wish to enter units.
27530 @pindex calc-simplify-units
27532 @mindex usimpl@idots
27535 The @kbd{u s} (@code{calc-simplify-units}) [@code{usimplify}] command
27537 expression. It uses @kbd{a s} (@code{calc-simplify}) to simplify the
27538 expression first as a regular algebraic formula; it then looks for
27539 features that can be further simplified by converting one object's units
27540 to be compatible with another's. For example, @samp{5 m + 23 mm} will
27541 simplify to @samp{5.023 m}. When different but compatible units are
27542 added, the righthand term's units are converted to match those of the
27543 lefthand term. @xref{Simplification Modes}, for a way to have this done
27544 automatically at all times.
27546 Units simplification also handles quotients of two units with the same
27547 dimensionality, as in @w{@samp{2 in s/L cm}} to @samp{5.08 s/L}; fractional
27548 powers of unit expressions, as in @samp{sqrt(9 mm^2)} to @samp{3 mm} and
27549 @samp{sqrt(9 acre)} to a quantity in meters; and @code{floor},
27550 @code{ceil}, @code{round}, @code{rounde}, @code{roundu}, @code{trunc},
27551 @code{float}, @code{frac}, @code{abs}, and @code{clean}
27552 applied to units expressions, in which case
27553 the operation in question is applied only to the numeric part of the
27554 expression. Finally, trigonometric functions of quantities with units
27555 of angle are evaluated, regardless of the current angular mode.
27558 @pindex calc-convert-units
27559 The @kbd{u c} (@code{calc-convert-units}) command converts a units
27560 expression to new, compatible units. For example, given the units
27561 expression @samp{55 mph}, typing @kbd{u c m/s @key{RET}} produces
27562 @samp{24.5872 m/s}. If you have previously converted a units expression
27563 with the same type of units (in this case, distance over time), you will
27564 be offered the previous choice of new units as a default. Continuing
27565 the above example, entering the units expression @samp{100 km/hr} and
27566 typing @kbd{u c @key{RET}} (without specifying new units) produces
27567 @samp{27.7777777778 m/s}.
27569 While many of Calc's conversion factors are exact, some are necessarily
27570 approximate. If Calc is in fraction mode (@pxref{Fraction Mode}), then
27571 unit conversions will try to give exact, rational conversions, but it
27572 isn't always possible. Given @samp{55 mph} in fraction mode, typing
27573 @kbd{u c m/s @key{RET}} produces @samp{15367:625 m/s}, for example,
27574 while typing @kbd{u c au/yr @key{RET}} produces
27575 @samp{5.18665819999e-3 au/yr}.
27577 If the units you request are inconsistent with the original units, the
27578 number will be converted into your units times whatever ``remainder''
27579 units are left over. For example, converting @samp{55 mph} into acres
27580 produces @samp{6.08e-3 acre / m s}. (Recall that multiplication binds
27581 more strongly than division in Calc formulas, so the units here are
27582 acres per meter-second.) Remainder units are expressed in terms of
27583 ``fundamental'' units like @samp{m} and @samp{s}, regardless of the
27586 One special exception is that if you specify a single unit name, and
27587 a compatible unit appears somewhere in the units expression, then
27588 that compatible unit will be converted to the new unit and the
27589 remaining units in the expression will be left alone. For example,
27590 given the input @samp{980 cm/s^2}, the command @kbd{u c ms} will
27591 change the @samp{s} to @samp{ms} to get @samp{9.8e-4 cm/ms^2}.
27592 The ``remainder unit'' @samp{cm} is left alone rather than being
27593 changed to the base unit @samp{m}.
27595 You can use explicit unit conversion instead of the @kbd{u s} command
27596 to gain more control over the units of the result of an expression.
27597 For example, given @samp{5 m + 23 mm}, you can type @kbd{u c m} or
27598 @kbd{u c mm} to express the result in either meters or millimeters.
27599 (For that matter, you could type @kbd{u c fath} to express the result
27600 in fathoms, if you preferred!)
27602 In place of a specific set of units, you can also enter one of the
27603 units system names @code{si}, @code{mks} (equivalent), or @code{cgs}.
27604 For example, @kbd{u c si @key{RET}} converts the expression into
27605 International System of Units (SI) base units. Also, @kbd{u c base}
27606 converts to Calc's base units, which are the same as @code{si} units
27607 except that @code{base} uses @samp{g} as the fundamental unit of mass
27608 whereas @code{si} uses @samp{kg}.
27610 @cindex Composite units
27611 The @kbd{u c} command also accepts @dfn{composite units}, which
27612 are expressed as the sum of several compatible unit names. For
27613 example, converting @samp{30.5 in} to units @samp{mi+ft+in} (miles,
27614 feet, and inches) produces @samp{2 ft + 6.5 in}. Calc first
27615 sorts the unit names into order of decreasing relative size.
27616 It then accounts for as much of the input quantity as it can
27617 using an integer number times the largest unit, then moves on
27618 to the next smaller unit, and so on. Only the smallest unit
27619 may have a non-integer amount attached in the result. A few
27620 standard unit names exist for common combinations, such as
27621 @code{mfi} for @samp{mi+ft+in}, and @code{tpo} for @samp{ton+lb+oz}.
27622 Composite units are expanded as if by @kbd{a x}, so that
27623 @samp{(ft+in)/hr} is first converted to @samp{ft/hr+in/hr}.
27625 If the value on the stack does not contain any units, @kbd{u c} will
27626 prompt first for the old units which this value should be considered
27627 to have, then for the new units. Assuming the old and new units you
27628 give are consistent with each other, the result also will not contain
27629 any units. For example, @kbd{@w{u c} cm @key{RET} in @key{RET}}
27630 converts the number 2 on the stack to 5.08.
27633 @pindex calc-base-units
27634 The @kbd{u b} (@code{calc-base-units}) command is shorthand for
27635 @kbd{u c base}; it converts the units expression on the top of the
27636 stack into @code{base} units. If @kbd{u s} does not simplify a
27637 units expression as far as you would like, try @kbd{u b}.
27639 The @kbd{u c} and @kbd{u b} commands treat temperature units (like
27640 @samp{degC} and @samp{K}) as relative temperatures. For example,
27641 @kbd{u c} converts @samp{10 degC} to @samp{18 degF}: A change of 10
27642 degrees Celsius corresponds to a change of 18 degrees Fahrenheit.
27645 @pindex calc-convert-temperature
27646 @cindex Temperature conversion
27647 The @kbd{u t} (@code{calc-convert-temperature}) command converts
27648 absolute temperatures. The value on the stack must be a simple units
27649 expression with units of temperature only. This command would convert
27650 @samp{10 degC} to @samp{50 degF}, the equivalent temperature on the
27654 @pindex calc-remove-units
27656 @pindex calc-extract-units
27657 The @kbd{u r} (@code{calc-remove-units}) command removes units from the
27658 formula at the top of the stack. The @kbd{u x}
27659 (@code{calc-extract-units}) command extracts only the units portion of a
27660 formula. These commands essentially replace every term of the formula
27661 that does or doesn't (respectively) look like a unit name by the
27662 constant 1, then resimplify the formula.
27665 @pindex calc-autorange-units
27666 The @kbd{u a} (@code{calc-autorange-units}) command turns on and off a
27667 mode in which unit prefixes like @code{k} (``kilo'') are automatically
27668 applied to keep the numeric part of a units expression in a reasonable
27669 range. This mode affects @kbd{u s} and all units conversion commands
27670 except @kbd{u b}. For example, with autoranging on, @samp{12345 Hz}
27671 will be simplified to @samp{12.345 kHz}. Autoranging is useful for
27672 some kinds of units (like @code{Hz} and @code{m}), but is probably
27673 undesirable for non-metric units like @code{ft} and @code{tbsp}.
27674 (Composite units are more appropriate for those; see above.)
27676 Autoranging always applies the prefix to the leftmost unit name.
27677 Calc chooses the largest prefix that causes the number to be greater
27678 than or equal to 1.0. Thus an increasing sequence of adjusted times
27679 would be @samp{1 ms, 10 ms, 100 ms, 1 s, 10 s, 100 s, 1 ks}.
27680 Generally the rule of thumb is that the number will be adjusted
27681 to be in the interval @samp{[1 .. 1000)}, although there are several
27682 exceptions to this rule. First, if the unit has a power then this
27683 is not possible; @samp{0.1 s^2} simplifies to @samp{100000 ms^2}.
27684 Second, the ``centi-'' prefix is allowed to form @code{cm} (centimeters),
27685 but will not apply to other units. The ``deci-,'' ``deka-,'' and
27686 ``hecto-'' prefixes are never used. Thus the allowable interval is
27687 @samp{[1 .. 10)} for millimeters and @samp{[1 .. 100)} for centimeters.
27688 Finally, a prefix will not be added to a unit if the resulting name
27689 is also the actual name of another unit; @samp{1e-15 t} would normally
27690 be considered a ``femto-ton,'' but it is written as @samp{1000 at}
27691 (1000 atto-tons) instead because @code{ft} would be confused with feet.
27693 @node The Units Table, Predefined Units, Basic Operations on Units, Units
27694 @section The Units Table
27698 @pindex calc-enter-units-table
27699 The @kbd{u v} (@code{calc-enter-units-table}) command displays the units table
27700 in another buffer called @code{*Units Table*}. Each entry in this table
27701 gives the unit name as it would appear in an expression, the definition
27702 of the unit in terms of simpler units, and a full name or description of
27703 the unit. Fundamental units are defined as themselves; these are the
27704 units produced by the @kbd{u b} command. The fundamental units are
27705 meters, seconds, grams, kelvins, amperes, candelas, moles, radians,
27708 The Units Table buffer also displays the Unit Prefix Table. Note that
27709 two prefixes, ``kilo'' and ``hecto,'' accept either upper- or lower-case
27710 prefix letters. @samp{Meg} is also accepted as a synonym for the @samp{M}
27711 prefix. Whenever a unit name can be interpreted as either a built-in name
27712 or a prefix followed by another built-in name, the former interpretation
27713 wins. For example, @samp{2 pt} means two pints, not two pico-tons.
27715 The Units Table buffer, once created, is not rebuilt unless you define
27716 new units. To force the buffer to be rebuilt, give any numeric prefix
27717 argument to @kbd{u v}.
27720 @pindex calc-view-units-table
27721 The @kbd{u V} (@code{calc-view-units-table}) command is like @kbd{u v} except
27722 that the cursor is not moved into the Units Table buffer. You can
27723 type @kbd{u V} again to remove the Units Table from the display. To
27724 return from the Units Table buffer after a @kbd{u v}, type @kbd{C-x * c}
27725 again or use the regular Emacs @w{@kbd{C-x o}} (@code{other-window})
27726 command. You can also kill the buffer with @kbd{C-x k} if you wish;
27727 the actual units table is safely stored inside the Calculator.
27730 @pindex calc-get-unit-definition
27731 The @kbd{u g} (@code{calc-get-unit-definition}) command retrieves a unit's
27732 defining expression and pushes it onto the Calculator stack. For example,
27733 @kbd{u g in} will produce the expression @samp{2.54 cm}. This is the
27734 same definition for the unit that would appear in the Units Table buffer.
27735 Note that this command works only for actual unit names; @kbd{u g km}
27736 will report that no such unit exists, for example, because @code{km} is
27737 really the unit @code{m} with a @code{k} (``kilo'') prefix. To see a
27738 definition of a unit in terms of base units, it is easier to push the
27739 unit name on the stack and then reduce it to base units with @kbd{u b}.
27742 @pindex calc-explain-units
27743 The @kbd{u e} (@code{calc-explain-units}) command displays an English
27744 description of the units of the expression on the stack. For example,
27745 for the expression @samp{62 km^2 g / s^2 mol K}, the description is
27746 ``Square-Kilometer Gram per (Second-squared Mole Degree-Kelvin).'' This
27747 command uses the English descriptions that appear in the righthand
27748 column of the Units Table.
27750 @node Predefined Units, User-Defined Units, The Units Table, Units
27751 @section Predefined Units
27754 The definitions of many units have changed over the years. For example,
27755 the meter was originally defined in 1791 as one ten-millionth of the
27756 distance from the equator to the north pole. In order to be more
27757 precise, the definition was adjusted several times, and now a meter is
27758 defined as the distance that light will travel in a vacuum in
27759 1/299792458 of a second; consequently, the speed of light in a
27760 vacuum is exactly 299792458 m/s. Many other units have been
27761 redefined in terms of fundamental physical processes; a second, for
27762 example, is currently defined as 9192631770 periods of a certain
27763 radiation related to the cesium-133 atom. The only SI unit that is not
27764 based on a fundamental physical process (although there are efforts to
27765 change this) is the kilogram, which was originally defined as the mass
27766 of one liter of water, but is now defined as the mass of the
27767 International Prototype Kilogram (IPK), a cylinder of platinum-iridium
27768 kept at the Bureau International des Poids et Mesures in S@`evres,
27769 France. (There are several copies of the IPK throughout the world.)
27770 The British imperial units, once defined in terms of physical objects,
27771 were redefined in 1963 in terms of SI units. The US customary units,
27772 which were the same as British units until the British imperial system
27773 was created in 1824, were also defined in terms of the SI units in 1893.
27774 Because of these redefinitions, conversions between metric, British
27775 Imperial, and US customary units can often be done precisely.
27777 Since the exact definitions of many kinds of units have evolved over the
27778 years, and since certain countries sometimes have local differences in
27779 their definitions, it is a good idea to examine Calc's definition of a
27780 unit before depending on its exact value. For example, there are three
27781 different units for gallons, corresponding to the US (@code{gal}),
27782 Canadian (@code{galC}), and British (@code{galUK}) definitions. Also,
27783 note that @code{oz} is a standard ounce of mass, @code{ozt} is a Troy
27784 ounce, and @code{ozfl} is a fluid ounce.
27786 The temperature units corresponding to degrees Kelvin and Centigrade
27787 (Celsius) are the same in this table, since most units commands treat
27788 temperatures as being relative. The @code{calc-convert-temperature}
27789 command has special rules for handling the different absolute magnitudes
27790 of the various temperature scales.
27792 The unit of volume ``liters'' can be referred to by either the lower-case
27793 @code{l} or the upper-case @code{L}.
27795 The unit @code{A} stands for Amperes; the name @code{Ang} is used
27803 The unit @code{pt} stands for pints; the name @code{point} stands for
27804 a typographical point, defined by @samp{72 point = 1 in}. This is
27805 slightly different than the point defined by the American Typefounder's
27806 Association in 1886, but the point used by Calc has become standard
27807 largely due to its use by the PostScript page description language.
27808 There is also @code{texpt}, which stands for a printer's point as
27809 defined by the @TeX{} typesetting system: @samp{72.27 texpt = 1 in}.
27810 Other units used by @TeX{} are available; they are @code{texpc} (a pica),
27811 @code{texbp} (a ``big point'', equal to a standard point which is larger
27812 than the point used by @TeX{}), @code{texdd} (a Didot point),
27813 @code{texcc} (a Cicero) and @code{texsp} (a scaled @TeX{} point,
27814 all dimensions representable in @TeX{} are multiples of this value).
27816 The unit @code{e} stands for the elementary (electron) unit of charge;
27817 because algebra command could mistake this for the special constant
27818 @expr{e}, Calc provides the alternate unit name @code{ech} which is
27819 preferable to @code{e}.
27821 The name @code{g} stands for one gram of mass; there is also @code{gf},
27822 one gram of force. (Likewise for @kbd{lb}, pounds, and @kbd{lbf}.)
27823 Meanwhile, one ``@expr{g}'' of acceleration is denoted @code{ga}.
27825 The unit @code{ton} is a U.S. ton of @samp{2000 lb}, and @code{t} is
27826 a metric ton of @samp{1000 kg}.
27828 The names @code{s} (or @code{sec}) and @code{min} refer to units of
27829 time; @code{arcsec} and @code{arcmin} are units of angle.
27831 Some ``units'' are really physical constants; for example, @code{c}
27832 represents the speed of light, and @code{h} represents Planck's
27833 constant. You can use these just like other units: converting
27834 @samp{.5 c} to @samp{m/s} expresses one-half the speed of light in
27835 meters per second. You can also use this merely as a handy reference;
27836 the @kbd{u g} command gets the definition of one of these constants
27837 in its normal terms, and @kbd{u b} expresses the definition in base
27840 Two units, @code{pi} and @code{alpha} (the fine structure constant,
27841 approximately @mathit{1/137}) are dimensionless. The units simplification
27842 commands simply treat these names as equivalent to their corresponding
27843 values. However you can, for example, use @kbd{u c} to convert a pure
27844 number into multiples of the fine structure constant, or @kbd{u b} to
27845 convert this back into a pure number. (When @kbd{u c} prompts for the
27846 ``old units,'' just enter a blank line to signify that the value
27847 really is unitless.)
27849 @c Describe angular units, luminosity vs. steradians problem.
27851 @node User-Defined Units, , Predefined Units, Units
27852 @section User-Defined Units
27855 Calc provides ways to get quick access to your selected ``favorite''
27856 units, as well as ways to define your own new units.
27859 @pindex calc-quick-units
27861 @cindex @code{Units} variable
27862 @cindex Quick units
27863 To select your favorite units, store a vector of unit names or
27864 expressions in the Calc variable @code{Units}. The @kbd{u 1}
27865 through @kbd{u 9} commands (@code{calc-quick-units}) provide access
27866 to these units. If the value on the top of the stack is a plain
27867 number (with no units attached), then @kbd{u 1} gives it the
27868 specified units. (Basically, it multiplies the number by the
27869 first item in the @code{Units} vector.) If the number on the
27870 stack @emph{does} have units, then @kbd{u 1} converts that number
27871 to the new units. For example, suppose the vector @samp{[in, ft]}
27872 is stored in @code{Units}. Then @kbd{30 u 1} will create the
27873 expression @samp{30 in}, and @kbd{u 2} will convert that expression
27876 The @kbd{u 0} command accesses the tenth element of @code{Units}.
27877 Only ten quick units may be defined at a time. If the @code{Units}
27878 variable has no stored value (the default), or if its value is not
27879 a vector, then the quick-units commands will not function. The
27880 @kbd{s U} command is a convenient way to edit the @code{Units}
27881 variable; @pxref{Operations on Variables}.
27884 @pindex calc-define-unit
27885 @cindex User-defined units
27886 The @kbd{u d} (@code{calc-define-unit}) command records the units
27887 expression on the top of the stack as the definition for a new,
27888 user-defined unit. For example, putting @samp{16.5 ft} on the stack and
27889 typing @kbd{u d rod} defines the new unit @samp{rod} to be equivalent to
27890 16.5 feet. The unit conversion and simplification commands will now
27891 treat @code{rod} just like any other unit of length. You will also be
27892 prompted for an optional English description of the unit, which will
27893 appear in the Units Table. If you wish the definition of this unit to
27894 be displayed in a special way in the Units Table buffer (such as with an
27895 asterisk to indicate an approximate value), then you can call this
27896 command with an argument, @kbd{C-u u d}; you will then also be prompted
27897 for a string that will be used to display the definition.
27900 @pindex calc-undefine-unit
27901 The @kbd{u u} (@code{calc-undefine-unit}) command removes a user-defined
27902 unit. It is not possible to remove one of the predefined units,
27905 If you define a unit with an existing unit name, your new definition
27906 will replace the original definition of that unit. If the unit was a
27907 predefined unit, the old definition will not be replaced, only
27908 ``shadowed.'' The built-in definition will reappear if you later use
27909 @kbd{u u} to remove the shadowing definition.
27911 To create a new fundamental unit, use either 1 or the unit name itself
27912 as the defining expression. Otherwise the expression can involve any
27913 other units that you like (except for composite units like @samp{mfi}).
27914 You can create a new composite unit with a sum of other units as the
27915 defining expression. The next unit operation like @kbd{u c} or @kbd{u v}
27916 will rebuild the internal unit table incorporating your modifications.
27917 Note that erroneous definitions (such as two units defined in terms of
27918 each other) will not be detected until the unit table is next rebuilt;
27919 @kbd{u v} is a convenient way to force this to happen.
27921 Temperature units are treated specially inside the Calculator; it is not
27922 possible to create user-defined temperature units.
27925 @pindex calc-permanent-units
27926 @cindex Calc init file, user-defined units
27927 The @kbd{u p} (@code{calc-permanent-units}) command stores the user-defined
27928 units in your Calc init file (the file given by the variable
27929 @code{calc-settings-file}, typically @file{~/.calc.el}), so that the
27930 units will still be available in subsequent Emacs sessions. If there
27931 was already a set of user-defined units in your Calc init file, it
27932 is replaced by the new set. (@xref{General Mode Commands}, for a way to
27933 tell Calc to use a different file for the Calc init file.)
27935 @node Store and Recall, Graphics, Units, Top
27936 @chapter Storing and Recalling
27939 Calculator variables are really just Lisp variables that contain numbers
27940 or formulas in a form that Calc can understand. The commands in this
27941 section allow you to manipulate variables conveniently. Commands related
27942 to variables use the @kbd{s} prefix key.
27945 * Storing Variables::
27946 * Recalling Variables::
27947 * Operations on Variables::
27949 * Evaluates-To Operator::
27952 @node Storing Variables, Recalling Variables, Store and Recall, Store and Recall
27953 @section Storing Variables
27958 @cindex Storing variables
27959 @cindex Quick variables
27962 The @kbd{s s} (@code{calc-store}) command stores the value at the top of
27963 the stack into a specified variable. It prompts you to enter the
27964 name of the variable. If you press a single digit, the value is stored
27965 immediately in one of the ``quick'' variables @code{q0} through
27966 @code{q9}. Or you can enter any variable name.
27969 @pindex calc-store-into
27970 The @kbd{s s} command leaves the stored value on the stack. There is
27971 also an @kbd{s t} (@code{calc-store-into}) command, which removes a
27972 value from the stack and stores it in a variable.
27974 If the top of stack value is an equation @samp{a = 7} or assignment
27975 @samp{a := 7} with a variable on the lefthand side, then Calc will
27976 assign that variable with that value by default, i.e., if you type
27977 @kbd{s s @key{RET}} or @kbd{s t @key{RET}}. In this example, the
27978 value 7 would be stored in the variable @samp{a}. (If you do type
27979 a variable name at the prompt, the top-of-stack value is stored in
27980 its entirety, even if it is an equation: @samp{s s b @key{RET}}
27981 with @samp{a := 7} on the stack stores @samp{a := 7} in @code{b}.)
27983 In fact, the top of stack value can be a vector of equations or
27984 assignments with different variables on their lefthand sides; the
27985 default will be to store all the variables with their corresponding
27986 righthand sides simultaneously.
27988 It is also possible to type an equation or assignment directly at
27989 the prompt for the @kbd{s s} or @kbd{s t} command: @kbd{s s foo = 7}.
27990 In this case the expression to the right of the @kbd{=} or @kbd{:=}
27991 symbol is evaluated as if by the @kbd{=} command, and that value is
27992 stored in the variable. No value is taken from the stack; @kbd{s s}
27993 and @kbd{s t} are equivalent when used in this way.
27997 The prefix keys @kbd{s} and @kbd{t} may be followed immediately by a
27998 digit; @kbd{s 9} is equivalent to @kbd{s s 9}, and @kbd{t 9} is
27999 equivalent to @kbd{s t 9}. (The @kbd{t} prefix is otherwise used
28000 for trail and time/date commands.)
28036 @pindex calc-store-plus
28037 @pindex calc-store-minus
28038 @pindex calc-store-times
28039 @pindex calc-store-div
28040 @pindex calc-store-power
28041 @pindex calc-store-concat
28042 @pindex calc-store-neg
28043 @pindex calc-store-inv
28044 @pindex calc-store-decr
28045 @pindex calc-store-incr
28046 There are also several ``arithmetic store'' commands. For example,
28047 @kbd{s +} removes a value from the stack and adds it to the specified
28048 variable. The other arithmetic stores are @kbd{s -}, @kbd{s *}, @kbd{s /},
28049 @kbd{s ^}, and @w{@kbd{s |}} (vector concatenation), plus @kbd{s n} and
28050 @kbd{s &} which negate or invert the value in a variable, and @w{@kbd{s [}}
28051 and @kbd{s ]} which decrease or increase a variable by one.
28053 All the arithmetic stores accept the Inverse prefix to reverse the
28054 order of the operands. If @expr{v} represents the contents of the
28055 variable, and @expr{a} is the value drawn from the stack, then regular
28056 @w{@kbd{s -}} assigns
28057 @texline @math{v \coloneq v - a},
28058 @infoline @expr{v := v - a},
28059 but @kbd{I s -} assigns
28060 @texline @math{v \coloneq a - v}.
28061 @infoline @expr{v := a - v}.
28062 While @kbd{I s *} might seem pointless, it is
28063 useful if matrix multiplication is involved. Actually, all the
28064 arithmetic stores use formulas designed to behave usefully both
28065 forwards and backwards:
28069 s + v := v + a v := a + v
28070 s - v := v - a v := a - v
28071 s * v := v * a v := a * v
28072 s / v := v / a v := a / v
28073 s ^ v := v ^ a v := a ^ v
28074 s | v := v | a v := a | v
28075 s n v := v / (-1) v := (-1) / v
28076 s & v := v ^ (-1) v := (-1) ^ v
28077 s [ v := v - 1 v := 1 - v
28078 s ] v := v - (-1) v := (-1) - v
28082 In the last four cases, a numeric prefix argument will be used in
28083 place of the number one. (For example, @kbd{M-2 s ]} increases
28084 a variable by 2, and @kbd{M-2 I s ]} replaces a variable by
28085 minus-two minus the variable.
28087 The first six arithmetic stores can also be typed @kbd{s t +}, @kbd{s t -},
28088 etc. The commands @kbd{s s +}, @kbd{s s -}, and so on are analogous
28089 arithmetic stores that don't remove the value @expr{a} from the stack.
28091 All arithmetic stores report the new value of the variable in the
28092 Trail for your information. They signal an error if the variable
28093 previously had no stored value. If default simplifications have been
28094 turned off, the arithmetic stores temporarily turn them on for numeric
28095 arguments only (i.e., they temporarily do an @kbd{m N} command).
28096 @xref{Simplification Modes}. Large vectors put in the trail by
28097 these commands always use abbreviated (@kbd{t .}) mode.
28100 @pindex calc-store-map
28101 The @kbd{s m} command is a general way to adjust a variable's value
28102 using any Calc function. It is a ``mapping'' command analogous to
28103 @kbd{V M}, @kbd{V R}, etc. @xref{Reducing and Mapping}, to see
28104 how to specify a function for a mapping command. Basically,
28105 all you do is type the Calc command key that would invoke that
28106 function normally. For example, @kbd{s m n} applies the @kbd{n}
28107 key to negate the contents of the variable, so @kbd{s m n} is
28108 equivalent to @kbd{s n}. Also, @kbd{s m Q} takes the square root
28109 of the value stored in a variable, @kbd{s m v v} uses @kbd{v v} to
28110 reverse the vector stored in the variable, and @kbd{s m H I S}
28111 takes the hyperbolic arcsine of the variable contents.
28113 If the mapping function takes two or more arguments, the additional
28114 arguments are taken from the stack; the old value of the variable
28115 is provided as the first argument. Thus @kbd{s m -} with @expr{a}
28116 on the stack computes @expr{v - a}, just like @kbd{s -}. With the
28117 Inverse prefix, the variable's original value becomes the @emph{last}
28118 argument instead of the first. Thus @kbd{I s m -} is also
28119 equivalent to @kbd{I s -}.
28122 @pindex calc-store-exchange
28123 The @kbd{s x} (@code{calc-store-exchange}) command exchanges the value
28124 of a variable with the value on the top of the stack. Naturally, the
28125 variable must already have a stored value for this to work.
28127 You can type an equation or assignment at the @kbd{s x} prompt. The
28128 command @kbd{s x a=6} takes no values from the stack; instead, it
28129 pushes the old value of @samp{a} on the stack and stores @samp{a = 6}.
28132 @pindex calc-unstore
28133 @cindex Void variables
28134 @cindex Un-storing variables
28135 Until you store something in them, most variables are ``void,'' that is,
28136 they contain no value at all. If they appear in an algebraic formula
28137 they will be left alone even if you press @kbd{=} (@code{calc-evaluate}).
28138 The @kbd{s u} (@code{calc-unstore}) command returns a variable to the
28142 @pindex calc-copy-variable
28143 The @kbd{s c} (@code{calc-copy-variable}) command copies the stored
28144 value of one variable to another. One way it differs from a simple
28145 @kbd{s r} followed by an @kbd{s t} (aside from saving keystrokes) is
28146 that the value never goes on the stack and thus is never rounded,
28147 evaluated, or simplified in any way; it is not even rounded down to the
28150 The only variables with predefined values are the ``special constants''
28151 @code{pi}, @code{e}, @code{i}, @code{phi}, and @code{gamma}. You are free
28152 to unstore these variables or to store new values into them if you like,
28153 although some of the algebraic-manipulation functions may assume these
28154 variables represent their standard values. Calc displays a warning if
28155 you change the value of one of these variables, or of one of the other
28156 special variables @code{inf}, @code{uinf}, and @code{nan} (which are
28159 Note that @code{pi} doesn't actually have 3.14159265359 stored in it,
28160 but rather a special magic value that evaluates to @cpi{} at the current
28161 precision. Likewise @code{e}, @code{i}, and @code{phi} evaluate
28162 according to the current precision or polar mode. If you recall a value
28163 from @code{pi} and store it back, this magic property will be lost. The
28164 magic property is preserved, however, when a variable is copied with
28168 @pindex calc-copy-special-constant
28169 If one of the ``special constants'' is redefined (or undefined) so that
28170 it no longer has its magic property, the property can be restored with
28171 @kbd{s k} (@code{calc-copy-special-constant}). This command will prompt
28172 for a special constant and a variable to store it in, and so a special
28173 constant can be stored in any variable. Here, the special constant that
28174 you enter doesn't depend on the value of the corresponding variable;
28175 @code{pi} will represent 3.14159@dots{} regardless of what is currently
28176 stored in the Calc variable @code{pi}. If one of the other special
28177 variables, @code{inf}, @code{uinf} or @code{nan}, is given a value, its
28178 original behavior can be restored by voiding it with @kbd{s u}.
28180 @node Recalling Variables, Operations on Variables, Storing Variables, Store and Recall
28181 @section Recalling Variables
28185 @pindex calc-recall
28186 @cindex Recalling variables
28187 The most straightforward way to extract the stored value from a variable
28188 is to use the @kbd{s r} (@code{calc-recall}) command. This command prompts
28189 for a variable name (similarly to @code{calc-store}), looks up the value
28190 of the specified variable, and pushes that value onto the stack. It is
28191 an error to try to recall a void variable.
28193 It is also possible to recall the value from a variable by evaluating a
28194 formula containing that variable. For example, @kbd{' a @key{RET} =} is
28195 the same as @kbd{s r a @key{RET}} except that if the variable is void, the
28196 former will simply leave the formula @samp{a} on the stack whereas the
28197 latter will produce an error message.
28200 The @kbd{r} prefix may be followed by a digit, so that @kbd{r 9} is
28201 equivalent to @kbd{s r 9}.
28203 @node Operations on Variables, Let Command, Recalling Variables, Store and Recall
28204 @section Other Operations on Variables
28208 @pindex calc-edit-variable
28209 The @kbd{s e} (@code{calc-edit-variable}) command edits the stored
28210 value of a variable without ever putting that value on the stack
28211 or simplifying or evaluating the value. It prompts for the name of
28212 the variable to edit. If the variable has no stored value, the
28213 editing buffer will start out empty. If the editing buffer is
28214 empty when you press @kbd{C-c C-c} to finish, the variable will
28215 be made void. @xref{Editing Stack Entries}, for a general
28216 description of editing.
28218 The @kbd{s e} command is especially useful for creating and editing
28219 rewrite rules which are stored in variables. Sometimes these rules
28220 contain formulas which must not be evaluated until the rules are
28221 actually used. (For example, they may refer to @samp{deriv(x,y)},
28222 where @code{x} will someday become some expression involving @code{y};
28223 if you let Calc evaluate the rule while you are defining it, Calc will
28224 replace @samp{deriv(x,y)} with 0 because the formula @code{x} does
28225 not itself refer to @code{y}.) By contrast, recalling the variable,
28226 editing with @kbd{`}, and storing will evaluate the variable's value
28227 as a side effect of putting the value on the stack.
28275 @pindex calc-store-AlgSimpRules
28276 @pindex calc-store-Decls
28277 @pindex calc-store-EvalRules
28278 @pindex calc-store-FitRules
28279 @pindex calc-store-GenCount
28280 @pindex calc-store-Holidays
28281 @pindex calc-store-IntegLimit
28282 @pindex calc-store-LineStyles
28283 @pindex calc-store-PointStyles
28284 @pindex calc-store-PlotRejects
28285 @pindex calc-store-TimeZone
28286 @pindex calc-store-Units
28287 @pindex calc-store-ExtSimpRules
28288 There are several special-purpose variable-editing commands that
28289 use the @kbd{s} prefix followed by a shifted letter:
28293 Edit @code{AlgSimpRules}. @xref{Algebraic Simplifications}.
28295 Edit @code{Decls}. @xref{Declarations}.
28297 Edit @code{EvalRules}. @xref{Default Simplifications}.
28299 Edit @code{FitRules}. @xref{Curve Fitting}.
28301 Edit @code{GenCount}. @xref{Solving Equations}.
28303 Edit @code{Holidays}. @xref{Business Days}.
28305 Edit @code{IntegLimit}. @xref{Calculus}.
28307 Edit @code{LineStyles}. @xref{Graphics}.
28309 Edit @code{PointStyles}. @xref{Graphics}.
28311 Edit @code{PlotRejects}. @xref{Graphics}.
28313 Edit @code{TimeZone}. @xref{Time Zones}.
28315 Edit @code{Units}. @xref{User-Defined Units}.
28317 Edit @code{ExtSimpRules}. @xref{Unsafe Simplifications}.
28320 These commands are just versions of @kbd{s e} that use fixed variable
28321 names rather than prompting for the variable name.
28324 @pindex calc-permanent-variable
28325 @cindex Storing variables
28326 @cindex Permanent variables
28327 @cindex Calc init file, variables
28328 The @kbd{s p} (@code{calc-permanent-variable}) command saves a
28329 variable's value permanently in your Calc init file (the file given by
28330 the variable @code{calc-settings-file}, typically @file{~/.calc.el}), so
28331 that its value will still be available in future Emacs sessions. You
28332 can re-execute @w{@kbd{s p}} later on to update the saved value, but the
28333 only way to remove a saved variable is to edit your calc init file
28334 by hand. (@xref{General Mode Commands}, for a way to tell Calc to
28335 use a different file for the Calc init file.)
28337 If you do not specify the name of a variable to save (i.e.,
28338 @kbd{s p @key{RET}}), all Calc variables with defined values
28339 are saved except for the special constants @code{pi}, @code{e},
28340 @code{i}, @code{phi}, and @code{gamma}; the variables @code{TimeZone}
28341 and @code{PlotRejects};
28342 @code{FitRules}, @code{DistribRules}, and other built-in rewrite
28343 rules; and @code{PlotData@var{n}} variables generated
28344 by the graphics commands. (You can still save these variables by
28345 explicitly naming them in an @kbd{s p} command.)
28348 @pindex calc-insert-variables
28349 The @kbd{s i} (@code{calc-insert-variables}) command writes
28350 the values of all Calc variables into a specified buffer.
28351 The variables are written with the prefix @code{var-} in the form of
28352 Lisp @code{setq} commands
28353 which store the values in string form. You can place these commands
28354 in your Calc init file (or @file{.emacs}) if you wish, though in this case it
28355 would be easier to use @kbd{s p @key{RET}}. (Note that @kbd{s i}
28356 omits the same set of variables as @w{@kbd{s p @key{RET}}}; the difference
28357 is that @kbd{s i} will store the variables in any buffer, and it also
28358 stores in a more human-readable format.)
28360 @node Let Command, Evaluates-To Operator, Operations on Variables, Store and Recall
28361 @section The Let Command
28366 @cindex Variables, temporary assignment
28367 @cindex Temporary assignment to variables
28368 If you have an expression like @samp{a+b^2} on the stack and you wish to
28369 compute its value where @expr{b=3}, you can simply store 3 in @expr{b} and
28370 then press @kbd{=} to reevaluate the formula. This has the side-effect
28371 of leaving the stored value of 3 in @expr{b} for future operations.
28373 The @kbd{s l} (@code{calc-let}) command evaluates a formula under a
28374 @emph{temporary} assignment of a variable. It stores the value on the
28375 top of the stack into the specified variable, then evaluates the
28376 second-to-top stack entry, then restores the original value (or lack of one)
28377 in the variable. Thus after @kbd{'@w{ }a+b^2 @key{RET} 3 s l b @key{RET}},
28378 the stack will contain the formula @samp{a + 9}. The subsequent command
28379 @kbd{@w{5 s l a} @key{RET}} will replace this formula with the number 14.
28380 The variables @samp{a} and @samp{b} are not permanently affected in any way
28383 The value on the top of the stack may be an equation or assignment, or
28384 a vector of equations or assignments, in which case the default will be
28385 analogous to the case of @kbd{s t @key{RET}}. @xref{Storing Variables}.
28387 Also, you can answer the variable-name prompt with an equation or
28388 assignment: @kbd{s l b=3 @key{RET}} is the same as storing 3 on the stack
28389 and typing @kbd{s l b @key{RET}}.
28391 The @kbd{a b} (@code{calc-substitute}) command is another way to substitute
28392 a variable with a value in a formula. It does an actual substitution
28393 rather than temporarily assigning the variable and evaluating. For
28394 example, letting @expr{n=2} in @samp{f(n pi)} with @kbd{a b} will
28395 produce @samp{f(2 pi)}, whereas @kbd{s l} would give @samp{f(6.28)}
28396 since the evaluation step will also evaluate @code{pi}.
28398 @node Evaluates-To Operator, , Let Command, Store and Recall
28399 @section The Evaluates-To Operator
28404 @cindex Evaluates-to operator
28405 @cindex @samp{=>} operator
28406 The special algebraic symbol @samp{=>} is known as the @dfn{evaluates-to
28407 operator}. (It will show up as an @code{evalto} function call in
28408 other language modes like Pascal and La@TeX{}.) This is a binary
28409 operator, that is, it has a lefthand and a righthand argument,
28410 although it can be entered with the righthand argument omitted.
28412 A formula like @samp{@var{a} => @var{b}} is evaluated by Calc as
28413 follows: First, @var{a} is not simplified or modified in any
28414 way. The previous value of argument @var{b} is thrown away; the
28415 formula @var{a} is then copied and evaluated as if by the @kbd{=}
28416 command according to all current modes and stored variable values,
28417 and the result is installed as the new value of @var{b}.
28419 For example, suppose you enter the algebraic formula @samp{2 + 3 => 17}.
28420 The number 17 is ignored, and the lefthand argument is left in its
28421 unevaluated form; the result is the formula @samp{2 + 3 => 5}.
28424 @pindex calc-evalto
28425 You can enter an @samp{=>} formula either directly using algebraic
28426 entry (in which case the righthand side may be omitted since it is
28427 going to be replaced right away anyhow), or by using the @kbd{s =}
28428 (@code{calc-evalto}) command, which takes @var{a} from the stack
28429 and replaces it with @samp{@var{a} => @var{b}}.
28431 Calc keeps track of all @samp{=>} operators on the stack, and
28432 recomputes them whenever anything changes that might affect their
28433 values, i.e., a mode setting or variable value. This occurs only
28434 if the @samp{=>} operator is at the top level of the formula, or
28435 if it is part of a top-level vector. In other words, pushing
28436 @samp{2 + (a => 17)} will change the 17 to the actual value of
28437 @samp{a} when you enter the formula, but the result will not be
28438 dynamically updated when @samp{a} is changed later because the
28439 @samp{=>} operator is buried inside a sum. However, a vector
28440 of @samp{=>} operators will be recomputed, since it is convenient
28441 to push a vector like @samp{[a =>, b =>, c =>]} on the stack to
28442 make a concise display of all the variables in your problem.
28443 (Another way to do this would be to use @samp{[a, b, c] =>},
28444 which provides a slightly different format of display. You
28445 can use whichever you find easiest to read.)
28448 @pindex calc-auto-recompute
28449 The @kbd{m C} (@code{calc-auto-recompute}) command allows you to
28450 turn this automatic recomputation on or off. If you turn
28451 recomputation off, you must explicitly recompute an @samp{=>}
28452 operator on the stack in one of the usual ways, such as by
28453 pressing @kbd{=}. Turning recomputation off temporarily can save
28454 a lot of time if you will be changing several modes or variables
28455 before you look at the @samp{=>} entries again.
28457 Most commands are not especially useful with @samp{=>} operators
28458 as arguments. For example, given @samp{x + 2 => 17}, it won't
28459 work to type @kbd{1 +} to get @samp{x + 3 => 18}. If you want
28460 to operate on the lefthand side of the @samp{=>} operator on
28461 the top of the stack, type @kbd{j 1} (that's the digit ``one'')
28462 to select the lefthand side, execute your commands, then type
28463 @kbd{j u} to unselect.
28465 All current modes apply when an @samp{=>} operator is computed,
28466 including the current simplification mode. Recall that the
28467 formula @samp{x + y + x} is not handled by Calc's default
28468 simplifications, but the @kbd{a s} command will reduce it to
28469 the simpler form @samp{y + 2 x}. You can also type @kbd{m A}
28470 to enable an Algebraic Simplification mode in which the
28471 equivalent of @kbd{a s} is used on all of Calc's results.
28472 If you enter @samp{x + y + x =>} normally, the result will
28473 be @samp{x + y + x => x + y + x}. If you change to
28474 Algebraic Simplification mode, the result will be
28475 @samp{x + y + x => y + 2 x}. However, just pressing @kbd{a s}
28476 once will have no effect on @samp{x + y + x => x + y + x},
28477 because the righthand side depends only on the lefthand side
28478 and the current mode settings, and the lefthand side is not
28479 affected by commands like @kbd{a s}.
28481 The ``let'' command (@kbd{s l}) has an interesting interaction
28482 with the @samp{=>} operator. The @kbd{s l} command evaluates the
28483 second-to-top stack entry with the top stack entry supplying
28484 a temporary value for a given variable. As you might expect,
28485 if that stack entry is an @samp{=>} operator its righthand
28486 side will temporarily show this value for the variable. In
28487 fact, all @samp{=>}s on the stack will be updated if they refer
28488 to that variable. But this change is temporary in the sense
28489 that the next command that causes Calc to look at those stack
28490 entries will make them revert to the old variable value.
28494 2: a => a 2: a => 17 2: a => a
28495 1: a + 1 => a + 1 1: a + 1 => 18 1: a + 1 => a + 1
28498 17 s l a @key{RET} p 8 @key{RET}
28502 Here the @kbd{p 8} command changes the current precision,
28503 thus causing the @samp{=>} forms to be recomputed after the
28504 influence of the ``let'' is gone. The @kbd{d @key{SPC}} command
28505 (@code{calc-refresh}) is a handy way to force the @samp{=>}
28506 operators on the stack to be recomputed without any other
28510 @pindex calc-assign
28513 Embedded mode also uses @samp{=>} operators. In Embedded mode,
28514 the lefthand side of an @samp{=>} operator can refer to variables
28515 assigned elsewhere in the file by @samp{:=} operators. The
28516 assignment operator @samp{a := 17} does not actually do anything
28517 by itself. But Embedded mode recognizes it and marks it as a sort
28518 of file-local definition of the variable. You can enter @samp{:=}
28519 operators in Algebraic mode, or by using the @kbd{s :}
28520 (@code{calc-assign}) [@code{assign}] command which takes a variable
28521 and value from the stack and replaces them with an assignment.
28523 @xref{TeX and LaTeX Language Modes}, for the way @samp{=>} appears in
28524 @TeX{} language output. The @dfn{eqn} mode gives similar
28525 treatment to @samp{=>}.
28527 @node Graphics, Kill and Yank, Store and Recall, Top
28531 The commands for graphing data begin with the @kbd{g} prefix key. Calc
28532 uses GNUPLOT 2.0 or later to do graphics. These commands will only work
28533 if GNUPLOT is available on your system. (While GNUPLOT sounds like
28534 a relative of GNU Emacs, it is actually completely unrelated.
28535 However, it is free software. It can be obtained from
28536 @samp{http://www.gnuplot.info}.)
28538 @vindex calc-gnuplot-name
28539 If you have GNUPLOT installed on your system but Calc is unable to
28540 find it, you may need to set the @code{calc-gnuplot-name} variable
28541 in your Calc init file or @file{.emacs}. You may also need to set some Lisp
28542 variables to show Calc how to run GNUPLOT on your system; these
28543 are described under @kbd{g D} and @kbd{g O} below. If you are
28544 using the X window system, Calc will configure GNUPLOT for you
28545 automatically. If you have GNUPLOT 3.0 or later and you are not using X,
28546 Calc will configure GNUPLOT to display graphs using simple character
28547 graphics that will work on any terminal.
28551 * Three Dimensional Graphics::
28552 * Managing Curves::
28553 * Graphics Options::
28557 @node Basic Graphics, Three Dimensional Graphics, Graphics, Graphics
28558 @section Basic Graphics
28562 @pindex calc-graph-fast
28563 The easiest graphics command is @kbd{g f} (@code{calc-graph-fast}).
28564 This command takes two vectors of equal length from the stack.
28565 The vector at the top of the stack represents the ``y'' values of
28566 the various data points. The vector in the second-to-top position
28567 represents the corresponding ``x'' values. This command runs
28568 GNUPLOT (if it has not already been started by previous graphing
28569 commands) and displays the set of data points. The points will
28570 be connected by lines, and there will also be some kind of symbol
28571 to indicate the points themselves.
28573 The ``x'' entry may instead be an interval form, in which case suitable
28574 ``x'' values are interpolated between the minimum and maximum values of
28575 the interval (whether the interval is open or closed is ignored).
28577 The ``x'' entry may also be a number, in which case Calc uses the
28578 sequence of ``x'' values @expr{x}, @expr{x+1}, @expr{x+2}, etc.
28579 (Generally the number 0 or 1 would be used for @expr{x} in this case.)
28581 The ``y'' entry may be any formula instead of a vector. Calc effectively
28582 uses @kbd{N} (@code{calc-eval-num}) to evaluate variables in the formula;
28583 the result of this must be a formula in a single (unassigned) variable.
28584 The formula is plotted with this variable taking on the various ``x''
28585 values. Graphs of formulas by default use lines without symbols at the
28586 computed data points. Note that if neither ``x'' nor ``y'' is a vector,
28587 Calc guesses at a reasonable number of data points to use. See the
28588 @kbd{g N} command below. (The ``x'' values must be either a vector
28589 or an interval if ``y'' is a formula.)
28595 If ``y'' is (or evaluates to) a formula of the form
28596 @samp{xy(@var{x}, @var{y})} then the result is a
28597 parametric plot. The two arguments of the fictitious @code{xy} function
28598 are used as the ``x'' and ``y'' coordinates of the curve, respectively.
28599 In this case the ``x'' vector or interval you specified is not directly
28600 visible in the graph. For example, if ``x'' is the interval @samp{[0..360]}
28601 and ``y'' is the formula @samp{xy(sin(t), cos(t))}, the resulting graph
28604 Also, ``x'' and ``y'' may each be variable names, in which case Calc
28605 looks for suitable vectors, intervals, or formulas stored in those
28608 The ``x'' and ``y'' values for the data points (as pulled from the vectors,
28609 calculated from the formulas, or interpolated from the intervals) should
28610 be real numbers (integers, fractions, or floats). One exception to this
28611 is that the ``y'' entry can consist of a vector of numbers combined with
28612 error forms, in which case the points will be plotted with the
28613 appropriate error bars. Other than this, if either the ``x''
28614 value or the ``y'' value of a given data point is not a real number, that
28615 data point will be omitted from the graph. The points on either side
28616 of the invalid point will @emph{not} be connected by a line.
28618 See the documentation for @kbd{g a} below for a description of the way
28619 numeric prefix arguments affect @kbd{g f}.
28621 @cindex @code{PlotRejects} variable
28622 @vindex PlotRejects
28623 If you store an empty vector in the variable @code{PlotRejects}
28624 (i.e., @kbd{[ ] s t PlotRejects}), Calc will append information to
28625 this vector for every data point which was rejected because its
28626 ``x'' or ``y'' values were not real numbers. The result will be
28627 a matrix where each row holds the curve number, data point number,
28628 ``x'' value, and ``y'' value for a rejected data point.
28629 @xref{Evaluates-To Operator}, for a handy way to keep tabs on the
28630 current value of @code{PlotRejects}. @xref{Operations on Variables},
28631 for the @kbd{s R} command which is another easy way to examine
28632 @code{PlotRejects}.
28635 @pindex calc-graph-clear
28636 To clear the graphics display, type @kbd{g c} (@code{calc-graph-clear}).
28637 If the GNUPLOT output device is an X window, the window will go away.
28638 Effects on other kinds of output devices will vary. You don't need
28639 to use @kbd{g c} if you don't want to---if you give another @kbd{g f}
28640 or @kbd{g p} command later on, it will reuse the existing graphics
28641 window if there is one.
28643 @node Three Dimensional Graphics, Managing Curves, Basic Graphics, Graphics
28644 @section Three-Dimensional Graphics
28647 @pindex calc-graph-fast-3d
28648 The @kbd{g F} (@code{calc-graph-fast-3d}) command makes a three-dimensional
28649 graph. It works only if you have GNUPLOT 3.0 or later; with GNUPLOT 2.0,
28650 you will see a GNUPLOT error message if you try this command.
28652 The @kbd{g F} command takes three values from the stack, called ``x'',
28653 ``y'', and ``z'', respectively. As was the case for 2D graphs, there
28654 are several options for these values.
28656 In the first case, ``x'' and ``y'' are each vectors (not necessarily of
28657 the same length); either or both may instead be interval forms. The
28658 ``z'' value must be a matrix with the same number of rows as elements
28659 in ``x'', and the same number of columns as elements in ``y''. The
28660 result is a surface plot where
28661 @texline @math{z_{ij}}
28662 @infoline @expr{z_ij}
28663 is the height of the point
28664 at coordinate @expr{(x_i, y_j)} on the surface. The 3D graph will
28665 be displayed from a certain default viewpoint; you can change this
28666 viewpoint by adding a @samp{set view} to the @samp{*Gnuplot Commands*}
28667 buffer as described later. See the GNUPLOT documentation for a
28668 description of the @samp{set view} command.
28670 Each point in the matrix will be displayed as a dot in the graph,
28671 and these points will be connected by a grid of lines (@dfn{isolines}).
28673 In the second case, ``x'', ``y'', and ``z'' are all vectors of equal
28674 length. The resulting graph displays a 3D line instead of a surface,
28675 where the coordinates of points along the line are successive triplets
28676 of values from the input vectors.
28678 In the third case, ``x'' and ``y'' are vectors or interval forms, and
28679 ``z'' is any formula involving two variables (not counting variables
28680 with assigned values). These variables are sorted into alphabetical
28681 order; the first takes on values from ``x'' and the second takes on
28682 values from ``y'' to form a matrix of results that are graphed as a
28689 If the ``z'' formula evaluates to a call to the fictitious function
28690 @samp{xyz(@var{x}, @var{y}, @var{z})}, then the result is a
28691 ``parametric surface.'' In this case, the axes of the graph are
28692 taken from the @var{x} and @var{y} values in these calls, and the
28693 ``x'' and ``y'' values from the input vectors or intervals are used only
28694 to specify the range of inputs to the formula. For example, plotting
28695 @samp{[0..360], [0..180], xyz(sin(x)*sin(y), cos(x)*sin(y), cos(y))}
28696 will draw a sphere. (Since the default resolution for 3D plots is
28697 5 steps in each of ``x'' and ``y'', this will draw a very crude
28698 sphere. You could use the @kbd{g N} command, described below, to
28699 increase this resolution, or specify the ``x'' and ``y'' values as
28700 vectors with more than 5 elements.
28702 It is also possible to have a function in a regular @kbd{g f} plot
28703 evaluate to an @code{xyz} call. Since @kbd{g f} plots a line, not
28704 a surface, the result will be a 3D parametric line. For example,
28705 @samp{[[0..720], xyz(sin(x), cos(x), x)]} will plot two turns of a
28706 helix (a three-dimensional spiral).
28708 As for @kbd{g f}, each of ``x'', ``y'', and ``z'' may instead be
28709 variables containing the relevant data.
28711 @node Managing Curves, Graphics Options, Three Dimensional Graphics, Graphics
28712 @section Managing Curves
28715 The @kbd{g f} command is really shorthand for the following commands:
28716 @kbd{C-u g d g a g p}. Likewise, @w{@kbd{g F}} is shorthand for
28717 @kbd{C-u g d g A g p}. You can gain more control over your graph
28718 by using these commands directly.
28721 @pindex calc-graph-add
28722 The @kbd{g a} (@code{calc-graph-add}) command adds the ``curve''
28723 represented by the two values on the top of the stack to the current
28724 graph. You can have any number of curves in the same graph. When
28725 you give the @kbd{g p} command, all the curves will be drawn superimposed
28728 The @kbd{g a} command (and many others that affect the current graph)
28729 will cause a special buffer, @samp{*Gnuplot Commands*}, to be displayed
28730 in another window. This buffer is a template of the commands that will
28731 be sent to GNUPLOT when it is time to draw the graph. The first
28732 @kbd{g a} command adds a @code{plot} command to this buffer. Succeeding
28733 @kbd{g a} commands add extra curves onto that @code{plot} command.
28734 Other graph-related commands put other GNUPLOT commands into this
28735 buffer. In normal usage you never need to work with this buffer
28736 directly, but you can if you wish. The only constraint is that there
28737 must be only one @code{plot} command, and it must be the last command
28738 in the buffer. If you want to save and later restore a complete graph
28739 configuration, you can use regular Emacs commands to save and restore
28740 the contents of the @samp{*Gnuplot Commands*} buffer.
28744 If the values on the stack are not variable names, @kbd{g a} will invent
28745 variable names for them (of the form @samp{PlotData@var{n}}) and store
28746 the values in those variables. The ``x'' and ``y'' variables are what
28747 go into the @code{plot} command in the template. If you add a curve
28748 that uses a certain variable and then later change that variable, you
28749 can replot the graph without having to delete and re-add the curve.
28750 That's because the variable name, not the vector, interval or formula
28751 itself, is what was added by @kbd{g a}.
28753 A numeric prefix argument on @kbd{g a} or @kbd{g f} changes the way
28754 stack entries are interpreted as curves. With a positive prefix
28755 argument @expr{n}, the top @expr{n} stack entries are ``y'' values
28756 for @expr{n} different curves which share a common ``x'' value in
28757 the @expr{n+1}st stack entry. (Thus @kbd{g a} with no prefix
28758 argument is equivalent to @kbd{C-u 1 g a}.)
28760 A prefix of zero or plain @kbd{C-u} means to take two stack entries,
28761 ``x'' and ``y'' as usual, but to interpret ``y'' as a vector of
28762 ``y'' values for several curves that share a common ``x''.
28764 A negative prefix argument tells Calc to read @expr{n} vectors from
28765 the stack; each vector @expr{[x, y]} describes an independent curve.
28766 This is the only form of @kbd{g a} that creates several curves at once
28767 that don't have common ``x'' values. (Of course, the range of ``x''
28768 values covered by all the curves ought to be roughly the same if
28769 they are to look nice on the same graph.)
28771 For example, to plot
28772 @texline @math{\sin n x}
28773 @infoline @expr{sin(n x)}
28774 for integers @expr{n}
28775 from 1 to 5, you could use @kbd{v x} to create a vector of integers
28776 (@expr{n}), then @kbd{V M '} or @kbd{V M $} to map @samp{sin(n x)}
28777 across this vector. The resulting vector of formulas is suitable
28778 for use as the ``y'' argument to a @kbd{C-u g a} or @kbd{C-u g f}
28782 @pindex calc-graph-add-3d
28783 The @kbd{g A} (@code{calc-graph-add-3d}) command adds a 3D curve
28784 to the graph. It is not valid to intermix 2D and 3D curves in a
28785 single graph. This command takes three arguments, ``x'', ``y'',
28786 and ``z'', from the stack. With a positive prefix @expr{n}, it
28787 takes @expr{n+2} arguments (common ``x'' and ``y'', plus @expr{n}
28788 separate ``z''s). With a zero prefix, it takes three stack entries
28789 but the ``z'' entry is a vector of curve values. With a negative
28790 prefix @expr{-n}, it takes @expr{n} vectors of the form @expr{[x, y, z]}.
28791 The @kbd{g A} command works by adding a @code{splot} (surface-plot)
28792 command to the @samp{*Gnuplot Commands*} buffer.
28794 (Although @kbd{g a} adds a 2D @code{plot} command to the
28795 @samp{*Gnuplot Commands*} buffer, Calc changes this to @code{splot}
28796 before sending it to GNUPLOT if it notices that the data points are
28797 evaluating to @code{xyz} calls. It will not work to mix 2D and 3D
28798 @kbd{g a} curves in a single graph, although Calc does not currently
28802 @pindex calc-graph-delete
28803 The @kbd{g d} (@code{calc-graph-delete}) command deletes the most
28804 recently added curve from the graph. It has no effect if there are
28805 no curves in the graph. With a numeric prefix argument of any kind,
28806 it deletes all of the curves from the graph.
28809 @pindex calc-graph-hide
28810 The @kbd{g H} (@code{calc-graph-hide}) command ``hides'' or ``unhides''
28811 the most recently added curve. A hidden curve will not appear in
28812 the actual plot, but information about it such as its name and line and
28813 point styles will be retained.
28816 @pindex calc-graph-juggle
28817 The @kbd{g j} (@code{calc-graph-juggle}) command moves the curve
28818 at the end of the list (the ``most recently added curve'') to the
28819 front of the list. The next-most-recent curve is thus exposed for
28820 @w{@kbd{g d}} or similar commands to use. With @kbd{g j} you can work
28821 with any curve in the graph even though curve-related commands only
28822 affect the last curve in the list.
28825 @pindex calc-graph-plot
28826 The @kbd{g p} (@code{calc-graph-plot}) command uses GNUPLOT to draw
28827 the graph described in the @samp{*Gnuplot Commands*} buffer. Any
28828 GNUPLOT parameters which are not defined by commands in this buffer
28829 are reset to their default values. The variables named in the @code{plot}
28830 command are written to a temporary data file and the variable names
28831 are then replaced by the file name in the template. The resulting
28832 plotting commands are fed to the GNUPLOT program. See the documentation
28833 for the GNUPLOT program for more specific information. All temporary
28834 files are removed when Emacs or GNUPLOT exits.
28836 If you give a formula for ``y'', Calc will remember all the values that
28837 it calculates for the formula so that later plots can reuse these values.
28838 Calc throws out these saved values when you change any circumstances
28839 that may affect the data, such as switching from Degrees to Radians
28840 mode, or changing the value of a parameter in the formula. You can
28841 force Calc to recompute the data from scratch by giving a negative
28842 numeric prefix argument to @kbd{g p}.
28844 Calc uses a fairly rough step size when graphing formulas over intervals.
28845 This is to ensure quick response. You can ``refine'' a plot by giving
28846 a positive numeric prefix argument to @kbd{g p}. Calc goes through
28847 the data points it has computed and saved from previous plots of the
28848 function, and computes and inserts a new data point midway between
28849 each of the existing points. You can refine a plot any number of times,
28850 but beware that the amount of calculation involved doubles each time.
28852 Calc does not remember computed values for 3D graphs. This means the
28853 numerix prefix argument, if any, to @kbd{g p} is effectively ignored if
28854 the current graph is three-dimensional.
28857 @pindex calc-graph-print
28858 The @kbd{g P} (@code{calc-graph-print}) command is like @kbd{g p},
28859 except that it sends the output to a printer instead of to the
28860 screen. More precisely, @kbd{g p} looks for @samp{set terminal}
28861 or @samp{set output} commands in the @samp{*Gnuplot Commands*} buffer;
28862 lacking these it uses the default settings. However, @kbd{g P}
28863 ignores @samp{set terminal} and @samp{set output} commands and
28864 uses a different set of default values. All of these values are
28865 controlled by the @kbd{g D} and @kbd{g O} commands discussed below.
28866 Provided everything is set up properly, @kbd{g p} will plot to
28867 the screen unless you have specified otherwise and @kbd{g P} will
28868 always plot to the printer.
28870 @node Graphics Options, Devices, Managing Curves, Graphics
28871 @section Graphics Options
28875 @pindex calc-graph-grid
28876 The @kbd{g g} (@code{calc-graph-grid}) command turns the ``grid''
28877 on and off. It is off by default; tick marks appear only at the
28878 edges of the graph. With the grid turned on, dotted lines appear
28879 across the graph at each tick mark. Note that this command only
28880 changes the setting in @samp{*Gnuplot Commands*}; to see the effects
28881 of the change you must give another @kbd{g p} command.
28884 @pindex calc-graph-border
28885 The @kbd{g b} (@code{calc-graph-border}) command turns the border
28886 (the box that surrounds the graph) on and off. It is on by default.
28887 This command will only work with GNUPLOT 3.0 and later versions.
28890 @pindex calc-graph-key
28891 The @kbd{g k} (@code{calc-graph-key}) command turns the ``key''
28892 on and off. The key is a chart in the corner of the graph that
28893 shows the correspondence between curves and line styles. It is
28894 off by default, and is only really useful if you have several
28895 curves on the same graph.
28898 @pindex calc-graph-num-points
28899 The @kbd{g N} (@code{calc-graph-num-points}) command allows you
28900 to select the number of data points in the graph. This only affects
28901 curves where neither ``x'' nor ``y'' is specified as a vector.
28902 Enter a blank line to revert to the default value (initially 15).
28903 With no prefix argument, this command affects only the current graph.
28904 With a positive prefix argument this command changes or, if you enter
28905 a blank line, displays the default number of points used for all
28906 graphs created by @kbd{g a} that don't specify the resolution explicitly.
28907 With a negative prefix argument, this command changes or displays
28908 the default value (initially 5) used for 3D graphs created by @kbd{g A}.
28909 Note that a 3D setting of 5 means that a total of @expr{5^2 = 25} points
28910 will be computed for the surface.
28912 Data values in the graph of a function are normally computed to a
28913 precision of five digits, regardless of the current precision at the
28914 time. This is usually more than adequate, but there are cases where
28915 it will not be. For example, plotting @expr{1 + x} with @expr{x} in the
28916 interval @samp{[0 ..@: 1e-6]} will round all the data points down
28917 to 1.0! Putting the command @samp{set precision @var{n}} in the
28918 @samp{*Gnuplot Commands*} buffer will cause the data to be computed
28919 at precision @var{n} instead of 5. Since this is such a rare case,
28920 there is no keystroke-based command to set the precision.
28923 @pindex calc-graph-header
28924 The @kbd{g h} (@code{calc-graph-header}) command sets the title
28925 for the graph. This will show up centered above the graph.
28926 The default title is blank (no title).
28929 @pindex calc-graph-name
28930 The @kbd{g n} (@code{calc-graph-name}) command sets the title of an
28931 individual curve. Like the other curve-manipulating commands, it
28932 affects the most recently added curve, i.e., the last curve on the
28933 list in the @samp{*Gnuplot Commands*} buffer. To set the title of
28934 the other curves you must first juggle them to the end of the list
28935 with @kbd{g j}, or edit the @samp{*Gnuplot Commands*} buffer by hand.
28936 Curve titles appear in the key; if the key is turned off they are
28941 @pindex calc-graph-title-x
28942 @pindex calc-graph-title-y
28943 The @kbd{g t} (@code{calc-graph-title-x}) and @kbd{g T}
28944 (@code{calc-graph-title-y}) commands set the titles on the ``x''
28945 and ``y'' axes, respectively. These titles appear next to the
28946 tick marks on the left and bottom edges of the graph, respectively.
28947 Calc does not have commands to control the tick marks themselves,
28948 but you can edit them into the @samp{*Gnuplot Commands*} buffer if
28949 you wish. See the GNUPLOT documentation for details.
28953 @pindex calc-graph-range-x
28954 @pindex calc-graph-range-y
28955 The @kbd{g r} (@code{calc-graph-range-x}) and @kbd{g R}
28956 (@code{calc-graph-range-y}) commands set the range of values on the
28957 ``x'' and ``y'' axes, respectively. You are prompted to enter a
28958 suitable range. This should be either a pair of numbers of the
28959 form, @samp{@var{min}:@var{max}}, or a blank line to revert to the
28960 default behavior of setting the range based on the range of values
28961 in the data, or @samp{$} to take the range from the top of the stack.
28962 Ranges on the stack can be represented as either interval forms or
28963 vectors: @samp{[@var{min} ..@: @var{max}]} or @samp{[@var{min}, @var{max}]}.
28967 @pindex calc-graph-log-x
28968 @pindex calc-graph-log-y
28969 The @kbd{g l} (@code{calc-graph-log-x}) and @kbd{g L} (@code{calc-graph-log-y})
28970 commands allow you to set either or both of the axes of the graph to
28971 be logarithmic instead of linear.
28976 @pindex calc-graph-log-z
28977 @pindex calc-graph-range-z
28978 @pindex calc-graph-title-z
28979 For 3D plots, @kbd{g C-t}, @kbd{g C-r}, and @kbd{g C-l} (those are
28980 letters with the Control key held down) are the corresponding commands
28981 for the ``z'' axis.
28985 @pindex calc-graph-zero-x
28986 @pindex calc-graph-zero-y
28987 The @kbd{g z} (@code{calc-graph-zero-x}) and @kbd{g Z}
28988 (@code{calc-graph-zero-y}) commands control whether a dotted line is
28989 drawn to indicate the ``x'' and/or ``y'' zero axes. (These are the same
28990 dotted lines that would be drawn there anyway if you used @kbd{g g} to
28991 turn the ``grid'' feature on.) Zero-axis lines are on by default, and
28992 may be turned off only in GNUPLOT 3.0 and later versions. They are
28993 not available for 3D plots.
28996 @pindex calc-graph-line-style
28997 The @kbd{g s} (@code{calc-graph-line-style}) command turns the connecting
28998 lines on or off for the most recently added curve, and optionally selects
28999 the style of lines to be used for that curve. Plain @kbd{g s} simply
29000 toggles the lines on and off. With a numeric prefix argument, @kbd{g s}
29001 turns lines on and sets a particular line style. Line style numbers
29002 start at one and their meanings vary depending on the output device.
29003 GNUPLOT guarantees that there will be at least six different line styles
29004 available for any device.
29007 @pindex calc-graph-point-style
29008 The @kbd{g S} (@code{calc-graph-point-style}) command similarly turns
29009 the symbols at the data points on or off, or sets the point style.
29010 If you turn both lines and points off, the data points will show as
29011 tiny dots. If the ``y'' values being plotted contain error forms and
29012 the connecting lines are turned off, then this command will also turn
29013 the error bars on or off.
29015 @cindex @code{LineStyles} variable
29016 @cindex @code{PointStyles} variable
29018 @vindex PointStyles
29019 Another way to specify curve styles is with the @code{LineStyles} and
29020 @code{PointStyles} variables. These variables initially have no stored
29021 values, but if you store a vector of integers in one of these variables,
29022 the @kbd{g a} and @kbd{g f} commands will use those style numbers
29023 instead of the defaults for new curves that are added to the graph.
29024 An entry should be a positive integer for a specific style, or 0 to let
29025 the style be chosen automatically, or @mathit{-1} to turn off lines or points
29026 altogether. If there are more curves than elements in the vector, the
29027 last few curves will continue to have the default styles. Of course,
29028 you can later use @kbd{g s} and @kbd{g S} to change any of these styles.
29030 For example, @kbd{'[2 -1 3] @key{RET} s t LineStyles} causes the first curve
29031 to have lines in style number 2, the second curve to have no connecting
29032 lines, and the third curve to have lines in style 3. Point styles will
29033 still be assigned automatically, but you could store another vector in
29034 @code{PointStyles} to define them, too.
29036 @node Devices, , Graphics Options, Graphics
29037 @section Graphical Devices
29041 @pindex calc-graph-device
29042 The @kbd{g D} (@code{calc-graph-device}) command sets the device name
29043 (or ``terminal name'' in GNUPLOT lingo) to be used by @kbd{g p} commands
29044 on this graph. It does not affect the permanent default device name.
29045 If you enter a blank name, the device name reverts to the default.
29046 Enter @samp{?} to see a list of supported devices.
29048 With a positive numeric prefix argument, @kbd{g D} instead sets
29049 the default device name, used by all plots in the future which do
29050 not override it with a plain @kbd{g D} command. If you enter a
29051 blank line this command shows you the current default. The special
29052 name @code{default} signifies that Calc should choose @code{x11} if
29053 the X window system is in use (as indicated by the presence of a
29054 @code{DISPLAY} environment variable), or otherwise @code{dumb} under
29055 GNUPLOT 3.0 and later, or @code{postscript} under GNUPLOT 2.0.
29056 This is the initial default value.
29058 The @code{dumb} device is an interface to ``dumb terminals,'' i.e.,
29059 terminals with no special graphics facilities. It writes a crude
29060 picture of the graph composed of characters like @code{-} and @code{|}
29061 to a buffer called @samp{*Gnuplot Trail*}, which Calc then displays.
29062 The graph is made the same size as the Emacs screen, which on most
29063 dumb terminals will be
29064 @texline @math{80\times24}
29066 characters. The graph is displayed in
29067 an Emacs ``recursive edit''; type @kbd{q} or @kbd{C-c C-c} to exit
29068 the recursive edit and return to Calc. Note that the @code{dumb}
29069 device is present only in GNUPLOT 3.0 and later versions.
29071 The word @code{dumb} may be followed by two numbers separated by
29072 spaces. These are the desired width and height of the graph in
29073 characters. Also, the device name @code{big} is like @code{dumb}
29074 but creates a graph four times the width and height of the Emacs
29075 screen. You will then have to scroll around to view the entire
29076 graph. In the @samp{*Gnuplot Trail*} buffer, @key{SPC}, @key{DEL},
29077 @kbd{<}, and @kbd{>} are defined to scroll by one screenful in each
29078 of the four directions.
29080 With a negative numeric prefix argument, @kbd{g D} sets or displays
29081 the device name used by @kbd{g P} (@code{calc-graph-print}). This
29082 is initially @code{postscript}. If you don't have a PostScript
29083 printer, you may decide once again to use @code{dumb} to create a
29084 plot on any text-only printer.
29087 @pindex calc-graph-output
29088 The @kbd{g O} (@code{calc-graph-output}) command sets the name of
29089 the output file used by GNUPLOT. For some devices, notably @code{x11},
29090 there is no output file and this information is not used. Many other
29091 ``devices'' are really file formats like @code{postscript}; in these
29092 cases the output in the desired format goes into the file you name
29093 with @kbd{g O}. Type @kbd{g O stdout @key{RET}} to set GNUPLOT to write
29094 to its standard output stream, i.e., to @samp{*Gnuplot Trail*}.
29095 This is the default setting.
29097 Another special output name is @code{tty}, which means that GNUPLOT
29098 is going to write graphics commands directly to its standard output,
29099 which you wish Emacs to pass through to your terminal. Tektronix
29100 graphics terminals, among other devices, operate this way. Calc does
29101 this by telling GNUPLOT to write to a temporary file, then running a
29102 sub-shell executing the command @samp{cat tempfile >/dev/tty}. On
29103 typical Unix systems, this will copy the temporary file directly to
29104 the terminal, bypassing Emacs entirely. You will have to type @kbd{C-l}
29105 to Emacs afterwards to refresh the screen.
29107 Once again, @kbd{g O} with a positive or negative prefix argument
29108 sets the default or printer output file names, respectively. In each
29109 case you can specify @code{auto}, which causes Calc to invent a temporary
29110 file name for each @kbd{g p} (or @kbd{g P}) command. This temporary file
29111 will be deleted once it has been displayed or printed. If the output file
29112 name is not @code{auto}, the file is not automatically deleted.
29114 The default and printer devices and output files can be saved
29115 permanently by the @kbd{m m} (@code{calc-save-modes}) command. The
29116 default number of data points (see @kbd{g N}) and the X geometry
29117 (see @kbd{g X}) are also saved. Other graph information is @emph{not}
29118 saved; you can save a graph's configuration simply by saving the contents
29119 of the @samp{*Gnuplot Commands*} buffer.
29121 @vindex calc-gnuplot-plot-command
29122 @vindex calc-gnuplot-default-device
29123 @vindex calc-gnuplot-default-output
29124 @vindex calc-gnuplot-print-command
29125 @vindex calc-gnuplot-print-device
29126 @vindex calc-gnuplot-print-output
29127 You may wish to configure the default and
29128 printer devices and output files for the whole system. The relevant
29129 Lisp variables are @code{calc-gnuplot-default-device} and @code{-output},
29130 and @code{calc-gnuplot-print-device} and @code{-output}. The output
29131 file names must be either strings as described above, or Lisp
29132 expressions which are evaluated on the fly to get the output file names.
29134 Other important Lisp variables are @code{calc-gnuplot-plot-command} and
29135 @code{calc-gnuplot-print-command}, which give the system commands to
29136 display or print the output of GNUPLOT, respectively. These may be
29137 @code{nil} if no command is necessary, or strings which can include
29138 @samp{%s} to signify the name of the file to be displayed or printed.
29139 Or, these variables may contain Lisp expressions which are evaluated
29140 to display or print the output. These variables are customizable
29141 (@pxref{Customizing Calc}).
29144 @pindex calc-graph-display
29145 The @kbd{g x} (@code{calc-graph-display}) command lets you specify
29146 on which X window system display your graphs should be drawn. Enter
29147 a blank line to see the current display name. This command has no
29148 effect unless the current device is @code{x11}.
29151 @pindex calc-graph-geometry
29152 The @kbd{g X} (@code{calc-graph-geometry}) command is a similar
29153 command for specifying the position and size of the X window.
29154 The normal value is @code{default}, which generally means your
29155 window manager will let you place the window interactively.
29156 Entering @samp{800x500+0+0} would create an 800-by-500 pixel
29157 window in the upper-left corner of the screen.
29159 The buffer called @samp{*Gnuplot Trail*} holds a transcript of the
29160 session with GNUPLOT. This shows the commands Calc has ``typed'' to
29161 GNUPLOT and the responses it has received. Calc tries to notice when an
29162 error message has appeared here and display the buffer for you when
29163 this happens. You can check this buffer yourself if you suspect
29164 something has gone wrong.
29167 @pindex calc-graph-command
29168 The @kbd{g C} (@code{calc-graph-command}) command prompts you to
29169 enter any line of text, then simply sends that line to the current
29170 GNUPLOT process. The @samp{*Gnuplot Trail*} buffer looks deceptively
29171 like a Shell buffer but you can't type commands in it yourself.
29172 Instead, you must use @kbd{g C} for this purpose.
29176 @pindex calc-graph-view-commands
29177 @pindex calc-graph-view-trail
29178 The @kbd{g v} (@code{calc-graph-view-commands}) and @kbd{g V}
29179 (@code{calc-graph-view-trail}) commands display the @samp{*Gnuplot Commands*}
29180 and @samp{*Gnuplot Trail*} buffers, respectively, in another window.
29181 This happens automatically when Calc thinks there is something you
29182 will want to see in either of these buffers. If you type @kbd{g v}
29183 or @kbd{g V} when the relevant buffer is already displayed, the
29184 buffer is hidden again.
29186 One reason to use @kbd{g v} is to add your own commands to the
29187 @samp{*Gnuplot Commands*} buffer. Press @kbd{g v}, then use
29188 @kbd{C-x o} to switch into that window. For example, GNUPLOT has
29189 @samp{set label} and @samp{set arrow} commands that allow you to
29190 annotate your plots. Since Calc doesn't understand these commands,
29191 you have to add them to the @samp{*Gnuplot Commands*} buffer
29192 yourself, then use @w{@kbd{g p}} to replot using these new commands. Note
29193 that your commands must appear @emph{before} the @code{plot} command.
29194 To get help on any GNUPLOT feature, type, e.g., @kbd{g C help set label}.
29195 You may have to type @kbd{g C @key{RET}} a few times to clear the
29196 ``press return for more'' or ``subtopic of @dots{}'' requests.
29197 Note that Calc always sends commands (like @samp{set nolabel}) to
29198 reset all plotting parameters to the defaults before each plot, so
29199 to delete a label all you need to do is delete the @samp{set label}
29200 line you added (or comment it out with @samp{#}) and then replot
29204 @pindex calc-graph-quit
29205 You can use @kbd{g q} (@code{calc-graph-quit}) to kill the GNUPLOT
29206 process that is running. The next graphing command you give will
29207 start a fresh GNUPLOT process. The word @samp{Graph} appears in
29208 the Calc window's mode line whenever a GNUPLOT process is currently
29209 running. The GNUPLOT process is automatically killed when you
29210 exit Emacs if you haven't killed it manually by then.
29213 @pindex calc-graph-kill
29214 The @kbd{g K} (@code{calc-graph-kill}) command is like @kbd{g q}
29215 except that it also views the @samp{*Gnuplot Trail*} buffer so that
29216 you can see the process being killed. This is better if you are
29217 killing GNUPLOT because you think it has gotten stuck.
29219 @node Kill and Yank, Keypad Mode, Graphics, Top
29220 @chapter Kill and Yank Functions
29223 The commands in this chapter move information between the Calculator and
29224 other Emacs editing buffers.
29226 In many cases Embedded mode is an easier and more natural way to
29227 work with Calc from a regular editing buffer. @xref{Embedded Mode}.
29230 * Killing From Stack::
29231 * Yanking Into Stack::
29232 * Saving Into Registers::
29233 * Inserting From Registers::
29234 * Grabbing From Buffers::
29235 * Yanking Into Buffers::
29236 * X Cut and Paste::
29239 @node Killing From Stack, Yanking Into Stack, Kill and Yank, Kill and Yank
29240 @section Killing from the Stack
29246 @pindex calc-copy-as-kill
29248 @pindex calc-kill-region
29250 @pindex calc-copy-region-as-kill
29253 @dfn{Kill} commands are Emacs commands that insert text into the ``kill
29254 ring,'' from which it can later be ``yanked'' by a @kbd{C-y} command.
29255 Three common kill commands in normal Emacs are @kbd{C-k}, which kills
29256 one line, @kbd{C-w}, which kills the region between mark and point, and
29257 @kbd{M-w}, which puts the region into the kill ring without actually
29258 deleting it. All of these commands work in the Calculator, too,
29259 although in the Calculator they operate on whole stack entries, so they
29260 ``round up'' the specified region to encompass full lines. (To copy
29261 only parts of lines, the @kbd{M-C-w} command in the Calculator will copy
29262 the region to the kill ring without any ``rounding up'', just like the
29263 @kbd{M-w} command in normal Emacs.) Also, @kbd{M-k} has been provided
29264 to complete the set; it puts the current line into the kill ring without
29267 The kill commands are unusual in that they pay attention to the location
29268 of the cursor in the Calculator buffer. If the cursor is on or below
29269 the bottom line, the kill commands operate on the top of the stack.
29270 Otherwise, they operate on whatever stack element the cursor is on. The
29271 text is copied into the kill ring exactly as it appears on the screen,
29272 including line numbers if they are enabled.
29274 A numeric prefix argument to @kbd{C-k} or @kbd{M-k} affects the number
29275 of lines killed. A positive argument kills the current line and @expr{n-1}
29276 lines below it. A negative argument kills the @expr{-n} lines above the
29277 current line. Again this mirrors the behavior of the standard Emacs
29278 @kbd{C-k} command. Although a whole line is always deleted, @kbd{C-k}
29279 with no argument copies only the number itself into the kill ring, whereas
29280 @kbd{C-k} with a prefix argument of 1 copies the number with its trailing
29283 @node Yanking Into Stack, Saving Into Registers, Killing From Stack, Kill and Yank
29284 @section Yanking into the Stack
29289 The @kbd{C-y} command yanks the most recently killed text back into the
29290 Calculator. It pushes this value onto the top of the stack regardless of
29291 the cursor position. In general it re-parses the killed text as a number
29292 or formula (or a list of these separated by commas or newlines). However if
29293 the thing being yanked is something that was just killed from the Calculator
29294 itself, its full internal structure is yanked. For example, if you have
29295 set the floating-point display mode to show only four significant digits,
29296 then killing and re-yanking 3.14159 (which displays as 3.142) will yank the
29297 full 3.14159, even though yanking it into any other buffer would yank the
29298 number in its displayed form, 3.142. (Since the default display modes
29299 show all objects to their full precision, this feature normally makes no
29302 @node Saving Into Registers, Inserting From Registers, Yanking Into Stack, Kill and Yank
29303 @section Saving into Registers
29307 @pindex calc-copy-to-register
29308 @pindex calc-prepend-to-register
29309 @pindex calc-append-to-register
29311 An alternative to killing and yanking stack entries is using
29312 registers in Calc. Saving stack entries in registers is like
29313 saving text in normal Emacs registers; although, like Calc's kill
29314 commands, register commands always operate on whole stack
29317 Registers in Calc are places to store stack entries for later use;
29318 each register is indexed by a single character. To store the current
29319 region (rounded up, of course, to include full stack entries) into a
29320 register, use the command @kbd{r s} (@code{calc-copy-to-register}).
29321 You will then be prompted for a register to use, the next character
29322 you type will be the index for the register. To store the region in
29323 register @var{r}, the full command will be @kbd{r s @var{r}}. With an
29324 argument, @kbd{C-u r s @var{r}}, the region being copied to the
29325 register will be deleted from the Calc buffer.
29327 It is possible to add additional stack entries to a register. The
29328 command @kbd{M-x calc-append-to-register} will prompt for a register,
29329 then add the stack entries in the region to the end of the register
29330 contents. The command @kbd{M-x calc-prepend-to-register} will
29331 similarly prompt for a register and add the stack entries in the
29332 region to the beginning of the register contents. Both commands take
29333 @kbd{C-u} arguments, which will cause the region to be deleted after being
29334 added to the register.
29336 @node Inserting From Registers, Grabbing From Buffers, Saving Into Registers, Kill and Yank
29337 @section Inserting from Registers
29340 @pindex calc-insert-register
29341 The command @kbd{r i} (@code{calc-insert-register}) will prompt for a
29342 register, then insert the contents of that register into the
29343 Calculator. If the contents of the register were placed there from
29344 within Calc, then the full internal structure of the contents will be
29345 inserted into the Calculator, otherwise whatever text is in the
29346 register is reparsed and then inserted into the Calculator.
29348 @node Grabbing From Buffers, Yanking Into Buffers, Inserting From Registers, Kill and Yank
29349 @section Grabbing from Other Buffers
29353 @pindex calc-grab-region
29354 The @kbd{C-x * g} (@code{calc-grab-region}) command takes the text between
29355 point and mark in the current buffer and attempts to parse it as a
29356 vector of values. Basically, it wraps the text in vector brackets
29357 @samp{[ ]} unless the text already is enclosed in vector brackets,
29358 then reads the text as if it were an algebraic entry. The contents
29359 of the vector may be numbers, formulas, or any other Calc objects.
29360 If the @kbd{C-x * g} command works successfully, it does an automatic
29361 @kbd{C-x * c} to enter the Calculator buffer.
29363 A numeric prefix argument grabs the specified number of lines around
29364 point, ignoring the mark. A positive prefix grabs from point to the
29365 @expr{n}th following newline (so that @kbd{M-1 C-x * g} grabs from point
29366 to the end of the current line); a negative prefix grabs from point
29367 back to the @expr{n+1}st preceding newline. In these cases the text
29368 that is grabbed is exactly the same as the text that @kbd{C-k} would
29369 delete given that prefix argument.
29371 A prefix of zero grabs the current line; point may be anywhere on the
29374 A plain @kbd{C-u} prefix interprets the region between point and mark
29375 as a single number or formula rather than a vector. For example,
29376 @kbd{C-x * g} on the text @samp{2 a b} produces the vector of three
29377 values @samp{[2, a, b]}, but @kbd{C-u C-x * g} on the same region
29378 reads a formula which is a product of three things: @samp{2 a b}.
29379 (The text @samp{a + b}, on the other hand, will be grabbed as a
29380 vector of one element by plain @kbd{C-x * g} because the interpretation
29381 @samp{[a, +, b]} would be a syntax error.)
29383 If a different language has been specified (@pxref{Language Modes}),
29384 the grabbed text will be interpreted according to that language.
29387 @pindex calc-grab-rectangle
29388 The @kbd{C-x * r} (@code{calc-grab-rectangle}) command takes the text between
29389 point and mark and attempts to parse it as a matrix. If point and mark
29390 are both in the leftmost column, the lines in between are parsed in their
29391 entirety. Otherwise, point and mark define the corners of a rectangle
29392 whose contents are parsed.
29394 Each line of the grabbed area becomes a row of the matrix. The result
29395 will actually be a vector of vectors, which Calc will treat as a matrix
29396 only if every row contains the same number of values.
29398 If a line contains a portion surrounded by square brackets (or curly
29399 braces), that portion is interpreted as a vector which becomes a row
29400 of the matrix. Any text surrounding the bracketed portion on the line
29403 Otherwise, the entire line is interpreted as a row vector as if it
29404 were surrounded by square brackets. Leading line numbers (in the
29405 format used in the Calc stack buffer) are ignored. If you wish to
29406 force this interpretation (even if the line contains bracketed
29407 portions), give a negative numeric prefix argument to the
29408 @kbd{C-x * r} command.
29410 If you give a numeric prefix argument of zero or plain @kbd{C-u}, each
29411 line is instead interpreted as a single formula which is converted into
29412 a one-element vector. Thus the result of @kbd{C-u C-x * r} will be a
29413 one-column matrix. For example, suppose one line of the data is the
29414 expression @samp{2 a}. A plain @w{@kbd{C-x * r}} will interpret this as
29415 @samp{[2 a]}, which in turn is read as a two-element vector that forms
29416 one row of the matrix. But a @kbd{C-u C-x * r} will interpret this row
29419 If you give a positive numeric prefix argument @var{n}, then each line
29420 will be split up into columns of width @var{n}; each column is parsed
29421 separately as a matrix element. If a line contained
29422 @w{@samp{2 +/- 3 4 +/- 5}}, then grabbing with a prefix argument of 8
29423 would correctly split the line into two error forms.
29425 @xref{Matrix Functions}, to see how to pull the matrix apart into its
29426 constituent rows and columns. (If it is a
29427 @texline @math{1\times1}
29429 matrix, just hit @kbd{v u} (@code{calc-unpack}) twice.)
29433 @pindex calc-grab-sum-across
29434 @pindex calc-grab-sum-down
29435 @cindex Summing rows and columns of data
29436 The @kbd{C-x * :} (@code{calc-grab-sum-down}) command is a handy way to
29437 grab a rectangle of data and sum its columns. It is equivalent to
29438 typing @kbd{C-x * r}, followed by @kbd{V R : +} (the vector reduction
29439 command that sums the columns of a matrix; @pxref{Reducing}). The
29440 result of the command will be a vector of numbers, one for each column
29441 in the input data. The @kbd{C-x * _} (@code{calc-grab-sum-across}) command
29442 similarly grabs a rectangle and sums its rows by executing @w{@kbd{V R _ +}}.
29444 As well as being more convenient, @kbd{C-x * :} and @kbd{C-x * _} are also
29445 much faster because they don't actually place the grabbed vector on
29446 the stack. In a @kbd{C-x * r V R : +} sequence, formatting the vector
29447 for display on the stack takes a large fraction of the total time
29448 (unless you have planned ahead and used @kbd{v .} and @kbd{t .} modes).
29450 For example, suppose we have a column of numbers in a file which we
29451 wish to sum. Go to one corner of the column and press @kbd{C-@@} to
29452 set the mark; go to the other corner and type @kbd{C-x * :}. Since there
29453 is only one column, the result will be a vector of one number, the sum.
29454 (You can type @kbd{v u} to unpack this vector into a plain number if
29455 you want to do further arithmetic with it.)
29457 To compute the product of the column of numbers, we would have to do
29458 it ``by hand'' since there's no special grab-and-multiply command.
29459 Use @kbd{C-x * r} to grab the column of numbers into the calculator in
29460 the form of a column matrix. The statistics command @kbd{u *} is a
29461 handy way to find the product of a vector or matrix of numbers.
29462 @xref{Statistical Operations}. Another approach would be to use
29463 an explicit column reduction command, @kbd{V R : *}.
29465 @node Yanking Into Buffers, X Cut and Paste, Grabbing From Buffers, Kill and Yank
29466 @section Yanking into Other Buffers
29470 @pindex calc-copy-to-buffer
29471 The plain @kbd{y} (@code{calc-copy-to-buffer}) command inserts the number
29472 at the top of the stack into the most recently used normal editing buffer.
29473 (More specifically, this is the most recently used buffer which is displayed
29474 in a window and whose name does not begin with @samp{*}. If there is no
29475 such buffer, this is the most recently used buffer except for Calculator
29476 and Calc Trail buffers.) The number is inserted exactly as it appears and
29477 without a newline. (If line-numbering is enabled, the line number is
29478 normally not included.) The number is @emph{not} removed from the stack.
29480 With a prefix argument, @kbd{y} inserts several numbers, one per line.
29481 A positive argument inserts the specified number of values from the top
29482 of the stack. A negative argument inserts the @expr{n}th value from the
29483 top of the stack. An argument of zero inserts the entire stack. Note
29484 that @kbd{y} with an argument of 1 is slightly different from @kbd{y}
29485 with no argument; the former always copies full lines, whereas the
29486 latter strips off the trailing newline.
29488 With a lone @kbd{C-u} as a prefix argument, @kbd{y} @emph{replaces} the
29489 region in the other buffer with the yanked text, then quits the
29490 Calculator, leaving you in that buffer. A typical use would be to use
29491 @kbd{C-x * g} to read a region of data into the Calculator, operate on the
29492 data to produce a new matrix, then type @kbd{C-u y} to replace the
29493 original data with the new data. One might wish to alter the matrix
29494 display style (@pxref{Vector and Matrix Formats}) or change the current
29495 display language (@pxref{Language Modes}) before doing this. Also, note
29496 that this command replaces a linear region of text (as grabbed by
29497 @kbd{C-x * g}), not a rectangle (as grabbed by @kbd{C-x * r}).
29499 If the editing buffer is in overwrite (as opposed to insert) mode,
29500 and the @kbd{C-u} prefix was not used, then the yanked number will
29501 overwrite the characters following point rather than being inserted
29502 before those characters. The usual conventions of overwrite mode
29503 are observed; for example, characters will be inserted at the end of
29504 a line rather than overflowing onto the next line. Yanking a multi-line
29505 object such as a matrix in overwrite mode overwrites the next @var{n}
29506 lines in the buffer, lengthening or shortening each line as necessary.
29507 Finally, if the thing being yanked is a simple integer or floating-point
29508 number (like @samp{-1.2345e-3}) and the characters following point also
29509 make up such a number, then Calc will replace that number with the new
29510 number, lengthening or shortening as necessary. The concept of
29511 ``overwrite mode'' has thus been generalized from overwriting characters
29512 to overwriting one complete number with another.
29515 The @kbd{C-x * y} key sequence is equivalent to @kbd{y} except that
29516 it can be typed anywhere, not just in Calc. This provides an easy
29517 way to guarantee that Calc knows which editing buffer you want to use!
29519 @node X Cut and Paste, , Yanking Into Buffers, Kill and Yank
29520 @section X Cut and Paste
29523 If you are using Emacs with the X window system, there is an easier
29524 way to move small amounts of data into and out of the calculator:
29525 Use the mouse-oriented cut and paste facilities of X.
29527 The default bindings for a three-button mouse cause the left button
29528 to move the Emacs cursor to the given place, the right button to
29529 select the text between the cursor and the clicked location, and
29530 the middle button to yank the selection into the buffer at the
29531 clicked location. So, if you have a Calc window and an editing
29532 window on your Emacs screen, you can use left-click/right-click
29533 to select a number, vector, or formula from one window, then
29534 middle-click to paste that value into the other window. When you
29535 paste text into the Calc window, Calc interprets it as an algebraic
29536 entry. It doesn't matter where you click in the Calc window; the
29537 new value is always pushed onto the top of the stack.
29539 The @code{xterm} program that is typically used for general-purpose
29540 shell windows in X interprets the mouse buttons in the same way.
29541 So you can use the mouse to move data between Calc and any other
29542 Unix program. One nice feature of @code{xterm} is that a double
29543 left-click selects one word, and a triple left-click selects a
29544 whole line. So you can usually transfer a single number into Calc
29545 just by double-clicking on it in the shell, then middle-clicking
29546 in the Calc window.
29548 @node Keypad Mode, Embedded Mode, Kill and Yank, Top
29549 @chapter Keypad Mode
29553 @pindex calc-keypad
29554 The @kbd{C-x * k} (@code{calc-keypad}) command starts the Calculator
29555 and displays a picture of a calculator-style keypad. If you are using
29556 the X window system, you can click on any of the ``keys'' in the
29557 keypad using the left mouse button to operate the calculator.
29558 The original window remains the selected window; in Keypad mode
29559 you can type in your file while simultaneously performing
29560 calculations with the mouse.
29562 @pindex full-calc-keypad
29563 If you have used @kbd{C-x * b} first, @kbd{C-x * k} instead invokes
29564 the @code{full-calc-keypad} command, which takes over the whole
29565 Emacs screen and displays the keypad, the Calc stack, and the Calc
29566 trail all at once. This mode would normally be used when running
29567 Calc standalone (@pxref{Standalone Operation}).
29569 If you aren't using the X window system, you must switch into
29570 the @samp{*Calc Keypad*} window, place the cursor on the desired
29571 ``key,'' and type @key{SPC} or @key{RET}. If you think this
29572 is easier than using Calc normally, go right ahead.
29574 Calc commands are more or less the same in Keypad mode. Certain
29575 keypad keys differ slightly from the corresponding normal Calc
29576 keystrokes; all such deviations are described below.
29578 Keypad mode includes many more commands than will fit on the keypad
29579 at once. Click the right mouse button [@code{calc-keypad-menu}]
29580 to switch to the next menu. The bottom five rows of the keypad
29581 stay the same; the top three rows change to a new set of commands.
29582 To return to earlier menus, click the middle mouse button
29583 [@code{calc-keypad-menu-back}] or simply advance through the menus
29584 until you wrap around. Typing @key{TAB} inside the keypad window
29585 is equivalent to clicking the right mouse button there.
29587 You can always click the @key{EXEC} button and type any normal
29588 Calc key sequence. This is equivalent to switching into the
29589 Calc buffer, typing the keys, then switching back to your
29593 * Keypad Main Menu::
29594 * Keypad Functions Menu::
29595 * Keypad Binary Menu::
29596 * Keypad Vectors Menu::
29597 * Keypad Modes Menu::
29600 @node Keypad Main Menu, Keypad Functions Menu, Keypad Mode, Keypad Mode
29605 |----+----+--Calc---+----+----1
29606 |FLR |CEIL|RND |TRNC|CLN2|FLT |
29607 |----+----+----+----+----+----|
29608 | LN |EXP | |ABS |IDIV|MOD |
29609 |----+----+----+----+----+----|
29610 |SIN |COS |TAN |SQRT|y^x |1/x |
29611 |----+----+----+----+----+----|
29612 | ENTER |+/- |EEX |UNDO| <- |
29613 |-----+---+-+--+--+-+---++----|
29614 | INV | 7 | 8 | 9 | / |
29615 |-----+-----+-----+-----+-----|
29616 | HYP | 4 | 5 | 6 | * |
29617 |-----+-----+-----+-----+-----|
29618 |EXEC | 1 | 2 | 3 | - |
29619 |-----+-----+-----+-----+-----|
29620 | OFF | 0 | . | PI | + |
29621 |-----+-----+-----+-----+-----+
29626 This is the menu that appears the first time you start Keypad mode.
29627 It will show up in a vertical window on the right side of your screen.
29628 Above this menu is the traditional Calc stack display. On a 24-line
29629 screen you will be able to see the top three stack entries.
29631 The ten digit keys, decimal point, and @key{EEX} key are used for
29632 entering numbers in the obvious way. @key{EEX} begins entry of an
29633 exponent in scientific notation. Just as with regular Calc, the
29634 number is pushed onto the stack as soon as you press @key{ENTER}
29635 or any other function key.
29637 The @key{+/-} key corresponds to normal Calc's @kbd{n} key. During
29638 numeric entry it changes the sign of the number or of the exponent.
29639 At other times it changes the sign of the number on the top of the
29642 The @key{INV} and @key{HYP} keys modify other keys. As well as
29643 having the effects described elsewhere in this manual, Keypad mode
29644 defines several other ``inverse'' operations. These are described
29645 below and in the following sections.
29647 The @key{ENTER} key finishes the current numeric entry, or otherwise
29648 duplicates the top entry on the stack.
29650 The @key{UNDO} key undoes the most recent Calc operation.
29651 @kbd{INV UNDO} is the ``redo'' command, and @kbd{HYP UNDO} is
29652 ``last arguments'' (@kbd{M-@key{RET}}).
29654 The @key{<-} key acts as a ``backspace'' during numeric entry.
29655 At other times it removes the top stack entry. @kbd{INV <-}
29656 clears the entire stack. @kbd{HYP <-} takes an integer from
29657 the stack, then removes that many additional stack elements.
29659 The @key{EXEC} key prompts you to enter any keystroke sequence
29660 that would normally work in Calc mode. This can include a
29661 numeric prefix if you wish. It is also possible simply to
29662 switch into the Calc window and type commands in it; there is
29663 nothing ``magic'' about this window when Keypad mode is active.
29665 The other keys in this display perform their obvious calculator
29666 functions. @key{CLN2} rounds the top-of-stack by temporarily
29667 reducing the precision by 2 digits. @key{FLT} converts an
29668 integer or fraction on the top of the stack to floating-point.
29670 The @key{INV} and @key{HYP} keys combined with several of these keys
29671 give you access to some common functions even if the appropriate menu
29672 is not displayed. Obviously you don't need to learn these keys
29673 unless you find yourself wasting time switching among the menus.
29677 is the same as @key{1/x}.
29679 is the same as @key{SQRT}.
29681 is the same as @key{CONJ}.
29683 is the same as @key{y^x}.
29685 is the same as @key{INV y^x} (the @expr{x}th root of @expr{y}).
29687 are the same as @key{SIN} / @kbd{INV SIN}.
29689 are the same as @key{COS} / @kbd{INV COS}.
29691 are the same as @key{TAN} / @kbd{INV TAN}.
29693 are the same as @key{LN} / @kbd{HYP LN}.
29695 are the same as @key{EXP} / @kbd{HYP EXP}.
29697 is the same as @key{ABS}.
29699 is the same as @key{RND} (@code{calc-round}).
29701 is the same as @key{CLN2}.
29703 is the same as @key{FLT} (@code{calc-float}).
29705 is the same as @key{IMAG}.
29707 is the same as @key{PREC}.
29709 is the same as @key{SWAP}.
29711 is the same as @key{RLL3}.
29712 @item INV HYP ENTER
29713 is the same as @key{OVER}.
29715 packs the top two stack entries as an error form.
29717 packs the top two stack entries as a modulo form.
29719 creates an interval form; this removes an integer which is one
29720 of 0 @samp{[]}, 1 @samp{[)}, 2 @samp{(]} or 3 @samp{()}, followed
29721 by the two limits of the interval.
29724 The @kbd{OFF} key turns Calc off; typing @kbd{C-x * k} or @kbd{C-x * *}
29725 again has the same effect. This is analogous to typing @kbd{q} or
29726 hitting @kbd{C-x * c} again in the normal calculator. If Calc is
29727 running standalone (the @code{full-calc-keypad} command appeared in the
29728 command line that started Emacs), then @kbd{OFF} is replaced with
29729 @kbd{EXIT}; clicking on this actually exits Emacs itself.
29731 @node Keypad Functions Menu, Keypad Binary Menu, Keypad Main Menu, Keypad Mode
29732 @section Functions Menu
29736 |----+----+----+----+----+----2
29737 |IGAM|BETA|IBET|ERF |BESJ|BESY|
29738 |----+----+----+----+----+----|
29739 |IMAG|CONJ| RE |ATN2|RAND|RAGN|
29740 |----+----+----+----+----+----|
29741 |GCD |FACT|DFCT|BNOM|PERM|NXTP|
29742 |----+----+----+----+----+----|
29747 This menu provides various operations from the @kbd{f} and @kbd{k}
29750 @key{IMAG} multiplies the number on the stack by the imaginary
29751 number @expr{i = (0, 1)}.
29753 @key{RE} extracts the real part a complex number. @kbd{INV RE}
29754 extracts the imaginary part.
29756 @key{RAND} takes a number from the top of the stack and computes
29757 a random number greater than or equal to zero but less than that
29758 number. (@xref{Random Numbers}.) @key{RAGN} is the ``random
29759 again'' command; it computes another random number using the
29760 same limit as last time.
29762 @key{INV GCD} computes the LCM (least common multiple) function.
29764 @key{INV FACT} is the gamma function.
29765 @texline @math{\Gamma(x) = (x-1)!}.
29766 @infoline @expr{gamma(x) = (x-1)!}.
29768 @key{PERM} is the number-of-permutations function, which is on the
29769 @kbd{H k c} key in normal Calc.
29771 @key{NXTP} finds the next prime after a number. @kbd{INV NXTP}
29772 finds the previous prime.
29774 @node Keypad Binary Menu, Keypad Vectors Menu, Keypad Functions Menu, Keypad Mode
29775 @section Binary Menu
29779 |----+----+----+----+----+----3
29780 |AND | OR |XOR |NOT |LSH |RSH |
29781 |----+----+----+----+----+----|
29782 |DEC |HEX |OCT |BIN |WSIZ|ARSH|
29783 |----+----+----+----+----+----|
29784 | A | B | C | D | E | F |
29785 |----+----+----+----+----+----|
29790 The keys in this menu perform operations on binary integers.
29791 Note that both logical and arithmetic right-shifts are provided.
29792 @key{INV LSH} rotates one bit to the left.
29794 The ``difference'' function (normally on @kbd{b d}) is on @key{INV AND}.
29795 The ``clip'' function (normally on @w{@kbd{b c}}) is on @key{INV NOT}.
29797 The @key{DEC}, @key{HEX}, @key{OCT}, and @key{BIN} keys select the
29798 current radix for display and entry of numbers: Decimal, hexadecimal,
29799 octal, or binary. The six letter keys @key{A} through @key{F} are used
29800 for entering hexadecimal numbers.
29802 The @key{WSIZ} key displays the current word size for binary operations
29803 and allows you to enter a new word size. You can respond to the prompt
29804 using either the keyboard or the digits and @key{ENTER} from the keypad.
29805 The initial word size is 32 bits.
29807 @node Keypad Vectors Menu, Keypad Modes Menu, Keypad Binary Menu, Keypad Mode
29808 @section Vectors Menu
29812 |----+----+----+----+----+----4
29813 |SUM |PROD|MAX |MAP*|MAP^|MAP$|
29814 |----+----+----+----+----+----|
29815 |MINV|MDET|MTRN|IDNT|CRSS|"x" |
29816 |----+----+----+----+----+----|
29817 |PACK|UNPK|INDX|BLD |LEN |... |
29818 |----+----+----+----+----+----|
29823 The keys in this menu operate on vectors and matrices.
29825 @key{PACK} removes an integer @var{n} from the top of the stack;
29826 the next @var{n} stack elements are removed and packed into a vector,
29827 which is replaced onto the stack. Thus the sequence
29828 @kbd{1 ENTER 3 ENTER 5 ENTER 3 PACK} enters the vector
29829 @samp{[1, 3, 5]} onto the stack. To enter a matrix, build each row
29830 on the stack as a vector, then use a final @key{PACK} to collect the
29831 rows into a matrix.
29833 @key{UNPK} unpacks the vector on the stack, pushing each of its
29834 components separately.
29836 @key{INDX} removes an integer @var{n}, then builds a vector of
29837 integers from 1 to @var{n}. @kbd{INV INDX} takes three numbers
29838 from the stack: The vector size @var{n}, the starting number,
29839 and the increment. @kbd{BLD} takes an integer @var{n} and any
29840 value @var{x} and builds a vector of @var{n} copies of @var{x}.
29842 @key{IDNT} removes an integer @var{n}, then builds an @var{n}-by-@var{n}
29845 @key{LEN} replaces a vector by its length, an integer.
29847 @key{...} turns on or off ``abbreviated'' display mode for large vectors.
29849 @key{MINV}, @key{MDET}, @key{MTRN}, and @key{CROSS} are the matrix
29850 inverse, determinant, and transpose, and vector cross product.
29852 @key{SUM} replaces a vector by the sum of its elements. It is
29853 equivalent to @kbd{u +} in normal Calc (@pxref{Statistical Operations}).
29854 @key{PROD} computes the product of the elements of a vector, and
29855 @key{MAX} computes the maximum of all the elements of a vector.
29857 @key{INV SUM} computes the alternating sum of the first element
29858 minus the second, plus the third, minus the fourth, and so on.
29859 @key{INV MAX} computes the minimum of the vector elements.
29861 @key{HYP SUM} computes the mean of the vector elements.
29862 @key{HYP PROD} computes the sample standard deviation.
29863 @key{HYP MAX} computes the median.
29865 @key{MAP*} multiplies two vectors elementwise. It is equivalent
29866 to the @kbd{V M *} command. @key{MAP^} computes powers elementwise.
29867 The arguments must be vectors of equal length, or one must be a vector
29868 and the other must be a plain number. For example, @kbd{2 MAP^} squares
29869 all the elements of a vector.
29871 @key{MAP$} maps the formula on the top of the stack across the
29872 vector in the second-to-top position. If the formula contains
29873 several variables, Calc takes that many vectors starting at the
29874 second-to-top position and matches them to the variables in
29875 alphabetical order. The result is a vector of the same size as
29876 the input vectors, whose elements are the formula evaluated with
29877 the variables set to the various sets of numbers in those vectors.
29878 For example, you could simulate @key{MAP^} using @key{MAP$} with
29879 the formula @samp{x^y}.
29881 The @kbd{"x"} key pushes the variable name @expr{x} onto the
29882 stack. To build the formula @expr{x^2 + 6}, you would use the
29883 key sequence @kbd{"x" 2 y^x 6 +}. This formula would then be
29884 suitable for use with the @key{MAP$} key described above.
29885 With @key{INV}, @key{HYP}, or @key{INV} and @key{HYP}, the
29886 @kbd{"x"} key pushes the variable names @expr{y}, @expr{z}, and
29887 @expr{t}, respectively.
29889 @node Keypad Modes Menu, , Keypad Vectors Menu, Keypad Mode
29890 @section Modes Menu
29894 |----+----+----+----+----+----5
29895 |FLT |FIX |SCI |ENG |GRP | |
29896 |----+----+----+----+----+----|
29897 |RAD |DEG |FRAC|POLR|SYMB|PREC|
29898 |----+----+----+----+----+----|
29899 |SWAP|RLL3|RLL4|OVER|STO |RCL |
29900 |----+----+----+----+----+----|
29905 The keys in this menu manipulate modes, variables, and the stack.
29907 The @key{FLT}, @key{FIX}, @key{SCI}, and @key{ENG} keys select
29908 floating-point, fixed-point, scientific, or engineering notation.
29909 @key{FIX} displays two digits after the decimal by default; the
29910 others display full precision. With the @key{INV} prefix, these
29911 keys pop a number-of-digits argument from the stack.
29913 The @key{GRP} key turns grouping of digits with commas on or off.
29914 @kbd{INV GRP} enables grouping to the right of the decimal point as
29915 well as to the left.
29917 The @key{RAD} and @key{DEG} keys switch between radians and degrees
29918 for trigonometric functions.
29920 The @key{FRAC} key turns Fraction mode on or off. This affects
29921 whether commands like @kbd{/} with integer arguments produce
29922 fractional or floating-point results.
29924 The @key{POLR} key turns Polar mode on or off, determining whether
29925 polar or rectangular complex numbers are used by default.
29927 The @key{SYMB} key turns Symbolic mode on or off, in which
29928 operations that would produce inexact floating-point results
29929 are left unevaluated as algebraic formulas.
29931 The @key{PREC} key selects the current precision. Answer with
29932 the keyboard or with the keypad digit and @key{ENTER} keys.
29934 The @key{SWAP} key exchanges the top two stack elements.
29935 The @key{RLL3} key rotates the top three stack elements upwards.
29936 The @key{RLL4} key rotates the top four stack elements upwards.
29937 The @key{OVER} key duplicates the second-to-top stack element.
29939 The @key{STO} and @key{RCL} keys are analogous to @kbd{s t} and
29940 @kbd{s r} in regular Calc. @xref{Store and Recall}. Click the
29941 @key{STO} or @key{RCL} key, then one of the ten digits. (Named
29942 variables are not available in Keypad mode.) You can also use,
29943 for example, @kbd{STO + 3} to add to register 3.
29945 @node Embedded Mode, Programming, Keypad Mode, Top
29946 @chapter Embedded Mode
29949 Embedded mode in Calc provides an alternative to copying numbers
29950 and formulas back and forth between editing buffers and the Calc
29951 stack. In Embedded mode, your editing buffer becomes temporarily
29952 linked to the stack and this copying is taken care of automatically.
29955 * Basic Embedded Mode::
29956 * More About Embedded Mode::
29957 * Assignments in Embedded Mode::
29958 * Mode Settings in Embedded Mode::
29959 * Customizing Embedded Mode::
29962 @node Basic Embedded Mode, More About Embedded Mode, Embedded Mode, Embedded Mode
29963 @section Basic Embedded Mode
29967 @pindex calc-embedded
29968 To enter Embedded mode, position the Emacs point (cursor) on a
29969 formula in any buffer and press @kbd{C-x * e} (@code{calc-embedded}).
29970 Note that @kbd{C-x * e} is not to be used in the Calc stack buffer
29971 like most Calc commands, but rather in regular editing buffers that
29972 are visiting your own files.
29974 Calc will try to guess an appropriate language based on the major mode
29975 of the editing buffer. (@xref{Language Modes}.) If the current buffer is
29976 in @code{latex-mode}, for example, Calc will set its language to La@TeX{}.
29977 Similarly, Calc will use @TeX{} language for @code{tex-mode},
29978 @code{plain-tex-mode} and @code{context-mode}, C language for
29979 @code{c-mode} and @code{c++-mode}, FORTRAN language for
29980 @code{fortran-mode} and @code{f90-mode}, Pascal for @code{pascal-mode},
29981 and eqn for @code{nroff-mode} (@pxref{Customizing Calc}).
29982 These can be overridden with Calc's mode
29983 changing commands (@pxref{Mode Settings in Embedded Mode}). If no
29984 suitable language is available, Calc will continue with its current language.
29986 Calc normally scans backward and forward in the buffer for the
29987 nearest opening and closing @dfn{formula delimiters}. The simplest
29988 delimiters are blank lines. Other delimiters that Embedded mode
29993 The @TeX{} and La@TeX{} math delimiters @samp{$ $}, @samp{$$ $$},
29994 @samp{\[ \]}, and @samp{\( \)};
29996 Lines beginning with @samp{\begin} and @samp{\end} (except matrix delimiters);
29998 Lines beginning with @samp{@@} (Texinfo delimiters).
30000 Lines beginning with @samp{.EQ} and @samp{.EN} (@dfn{eqn} delimiters);
30002 Lines containing a single @samp{%} or @samp{.\"} symbol and nothing else.
30005 @xref{Customizing Embedded Mode}, to see how to make Calc recognize
30006 your own favorite delimiters. Delimiters like @samp{$ $} can appear
30007 on their own separate lines or in-line with the formula.
30009 If you give a positive or negative numeric prefix argument, Calc
30010 instead uses the current point as one end of the formula, and includes
30011 that many lines forward or backward (respectively, including the current
30012 line). Explicit delimiters are not necessary in this case.
30014 With a prefix argument of zero, Calc uses the current region (delimited
30015 by point and mark) instead of formula delimiters. With a prefix
30016 argument of @kbd{C-u} only, Calc uses the current line as the formula.
30019 @pindex calc-embedded-word
30020 The @kbd{C-x * w} (@code{calc-embedded-word}) command will start Embedded
30021 mode on the current ``word''; in this case Calc will scan for the first
30022 non-numeric character (i.e., the first character that is not a digit,
30023 sign, decimal point, or upper- or lower-case @samp{e}) forward and
30024 backward to delimit the formula.
30026 When you enable Embedded mode for a formula, Calc reads the text
30027 between the delimiters and tries to interpret it as a Calc formula.
30028 Calc can generally identify @TeX{} formulas and
30029 Big-style formulas even if the language mode is wrong. If Calc
30030 can't make sense of the formula, it beeps and refuses to enter
30031 Embedded mode. But if the current language is wrong, Calc can
30032 sometimes parse the formula successfully (but incorrectly);
30033 for example, the C expression @samp{atan(a[1])} can be parsed
30034 in Normal language mode, but the @code{atan} won't correspond to
30035 the built-in @code{arctan} function, and the @samp{a[1]} will be
30036 interpreted as @samp{a} times the vector @samp{[1]}!
30038 If you press @kbd{C-x * e} or @kbd{C-x * w} to activate an embedded
30039 formula which is blank, say with the cursor on the space between
30040 the two delimiters @samp{$ $}, Calc will immediately prompt for
30041 an algebraic entry.
30043 Only one formula in one buffer can be enabled at a time. If you
30044 move to another area of the current buffer and give Calc commands,
30045 Calc turns Embedded mode off for the old formula and then tries
30046 to restart Embedded mode at the new position. Other buffers are
30047 not affected by Embedded mode.
30049 When Embedded mode begins, Calc pushes the current formula onto
30050 the stack. No Calc stack window is created; however, Calc copies
30051 the top-of-stack position into the original buffer at all times.
30052 You can create a Calc window by hand with @kbd{C-x * o} if you
30053 find you need to see the entire stack.
30055 For example, typing @kbd{C-x * e} while somewhere in the formula
30056 @samp{n>2} in the following line enables Embedded mode on that
30060 We define $F_n = F_(n-1)+F_(n-2)$ for all $n>2$.
30064 The formula @expr{n>2} will be pushed onto the Calc stack, and
30065 the top of stack will be copied back into the editing buffer.
30066 This means that spaces will appear around the @samp{>} symbol
30067 to match Calc's usual display style:
30070 We define $F_n = F_(n-1)+F_(n-2)$ for all $n > 2$.
30074 No spaces have appeared around the @samp{+} sign because it's
30075 in a different formula, one which we have not yet touched with
30078 Now that Embedded mode is enabled, keys you type in this buffer
30079 are interpreted as Calc commands. At this point we might use
30080 the ``commute'' command @kbd{j C} to reverse the inequality.
30081 This is a selection-based command for which we first need to
30082 move the cursor onto the operator (@samp{>} in this case) that
30083 needs to be commuted.
30086 We define $F_n = F_(n-1)+F_(n-2)$ for all $2 < n$.
30089 The @kbd{C-x * o} command is a useful way to open a Calc window
30090 without actually selecting that window. Giving this command
30091 verifies that @samp{2 < n} is also on the Calc stack. Typing
30092 @kbd{17 @key{RET}} would produce:
30095 We define $F_n = F_(n-1)+F_(n-2)$ for all $17$.
30099 with @samp{2 < n} and @samp{17} on the stack; typing @key{TAB}
30100 at this point will exchange the two stack values and restore
30101 @samp{2 < n} to the embedded formula. Even though you can't
30102 normally see the stack in Embedded mode, it is still there and
30103 it still operates in the same way. But, as with old-fashioned
30104 RPN calculators, you can only see the value at the top of the
30105 stack at any given time (unless you use @kbd{C-x * o}).
30107 Typing @kbd{C-x * e} again turns Embedded mode off. The Calc
30108 window reveals that the formula @w{@samp{2 < n}} is automatically
30109 removed from the stack, but the @samp{17} is not. Entering
30110 Embedded mode always pushes one thing onto the stack, and
30111 leaving Embedded mode always removes one thing. Anything else
30112 that happens on the stack is entirely your business as far as
30113 Embedded mode is concerned.
30115 If you press @kbd{C-x * e} in the wrong place by accident, it is
30116 possible that Calc will be able to parse the nearby text as a
30117 formula and will mangle that text in an attempt to redisplay it
30118 ``properly'' in the current language mode. If this happens,
30119 press @kbd{C-x * e} again to exit Embedded mode, then give the
30120 regular Emacs ``undo'' command (@kbd{C-_} or @kbd{C-x u}) to put
30121 the text back the way it was before Calc edited it. Note that Calc's
30122 own Undo command (typed before you turn Embedded mode back off)
30123 will not do you any good, because as far as Calc is concerned
30124 you haven't done anything with this formula yet.
30126 @node More About Embedded Mode, Assignments in Embedded Mode, Basic Embedded Mode, Embedded Mode
30127 @section More About Embedded Mode
30130 When Embedded mode ``activates'' a formula, i.e., when it examines
30131 the formula for the first time since the buffer was created or
30132 loaded, Calc tries to sense the language in which the formula was
30133 written. If the formula contains any La@TeX{}-like @samp{\} sequences,
30134 it is parsed (i.e., read) in La@TeX{} mode. If the formula appears to
30135 be written in multi-line Big mode, it is parsed in Big mode. Otherwise,
30136 it is parsed according to the current language mode.
30138 Note that Calc does not change the current language mode according
30139 the formula it reads in. Even though it can read a La@TeX{} formula when
30140 not in La@TeX{} mode, it will immediately rewrite this formula using
30141 whatever language mode is in effect.
30148 @pindex calc-show-plain
30149 Calc's parser is unable to read certain kinds of formulas. For
30150 example, with @kbd{v ]} (@code{calc-matrix-brackets}) you can
30151 specify matrix display styles which the parser is unable to
30152 recognize as matrices. The @kbd{d p} (@code{calc-show-plain})
30153 command turns on a mode in which a ``plain'' version of a
30154 formula is placed in front of the fully-formatted version.
30155 When Calc reads a formula that has such a plain version in
30156 front, it reads the plain version and ignores the formatted
30159 Plain formulas are preceded and followed by @samp{%%%} signs
30160 by default. This notation has the advantage that the @samp{%}
30161 character begins a comment in @TeX{} and La@TeX{}, so if your formula is
30162 embedded in a @TeX{} or La@TeX{} document its plain version will be
30163 invisible in the final printed copy. Certain major modes have different
30164 delimiters to ensure that the ``plain'' version will be
30165 in a comment for those modes, also.
30166 See @ref{Customizing Embedded Mode} to see how to change the ``plain''
30167 formula delimiters.
30169 There are several notations which Calc's parser for ``big''
30170 formatted formulas can't yet recognize. In particular, it can't
30171 read the large symbols for @code{sum}, @code{prod}, and @code{integ},
30172 and it can't handle @samp{=>} with the righthand argument omitted.
30173 Also, Calc won't recognize special formats you have defined with
30174 the @kbd{Z C} command (@pxref{User-Defined Compositions}). In
30175 these cases it is important to use ``plain'' mode to make sure
30176 Calc will be able to read your formula later.
30178 Another example where ``plain'' mode is important is if you have
30179 specified a float mode with few digits of precision. Normally
30180 any digits that are computed but not displayed will simply be
30181 lost when you save and re-load your embedded buffer, but ``plain''
30182 mode allows you to make sure that the complete number is present
30183 in the file as well as the rounded-down number.
30189 Embedded buffers remember active formulas for as long as they
30190 exist in Emacs memory. Suppose you have an embedded formula
30191 which is @cpi{} to the normal 12 decimal places, and then
30192 type @w{@kbd{C-u 5 d n}} to display only five decimal places.
30193 If you then type @kbd{d n}, all 12 places reappear because the
30194 full number is still there on the Calc stack. More surprisingly,
30195 even if you exit Embedded mode and later re-enter it for that
30196 formula, typing @kbd{d n} will restore all 12 places because
30197 each buffer remembers all its active formulas. However, if you
30198 save the buffer in a file and reload it in a new Emacs session,
30199 all non-displayed digits will have been lost unless you used
30206 In some applications of Embedded mode, you will want to have a
30207 sequence of copies of a formula that show its evolution as you
30208 work on it. For example, you might want to have a sequence
30209 like this in your file (elaborating here on the example from
30210 the ``Getting Started'' chapter):
30219 @r{(the derivative of }ln(ln(x))@r{)}
30221 whose value at x = 2 is
30231 @pindex calc-embedded-duplicate
30232 The @kbd{C-x * d} (@code{calc-embedded-duplicate}) command is a
30233 handy way to make sequences like this. If you type @kbd{C-x * d},
30234 the formula under the cursor (which may or may not have Embedded
30235 mode enabled for it at the time) is copied immediately below and
30236 Embedded mode is then enabled for that copy.
30238 For this example, you would start with just
30247 and press @kbd{C-x * d} with the cursor on this formula. The result
30260 with the second copy of the formula enabled in Embedded mode.
30261 You can now press @kbd{a d x @key{RET}} to take the derivative, and
30262 @kbd{C-x * d C-x * d} to make two more copies of the derivative.
30263 To complete the computations, type @kbd{3 s l x @key{RET}} to evaluate
30264 the last formula, then move up to the second-to-last formula
30265 and type @kbd{2 s l x @key{RET}}.
30267 Finally, you would want to press @kbd{C-x * e} to exit Embedded
30268 mode, then go up and insert the necessary text in between the
30269 various formulas and numbers.
30277 @pindex calc-embedded-new-formula
30278 The @kbd{C-x * f} (@code{calc-embedded-new-formula}) command
30279 creates a new embedded formula at the current point. It inserts
30280 some default delimiters, which are usually just blank lines,
30281 and then does an algebraic entry to get the formula (which is
30282 then enabled for Embedded mode). This is just shorthand for
30283 typing the delimiters yourself, positioning the cursor between
30284 the new delimiters, and pressing @kbd{C-x * e}. The key sequence
30285 @kbd{C-x * '} is equivalent to @kbd{C-x * f}.
30289 @pindex calc-embedded-next
30290 @pindex calc-embedded-previous
30291 The @kbd{C-x * n} (@code{calc-embedded-next}) and @kbd{C-x * p}
30292 (@code{calc-embedded-previous}) commands move the cursor to the
30293 next or previous active embedded formula in the buffer. They
30294 can take positive or negative prefix arguments to move by several
30295 formulas. Note that these commands do not actually examine the
30296 text of the buffer looking for formulas; they only see formulas
30297 which have previously been activated in Embedded mode. In fact,
30298 @kbd{C-x * n} and @kbd{C-x * p} are a useful way to tell which
30299 embedded formulas are currently active. Also, note that these
30300 commands do not enable Embedded mode on the next or previous
30301 formula, they just move the cursor.
30304 @pindex calc-embedded-edit
30305 The @kbd{C-x * `} (@code{calc-embedded-edit}) command edits the
30306 embedded formula at the current point as if by @kbd{`} (@code{calc-edit}).
30307 Embedded mode does not have to be enabled for this to work. Press
30308 @kbd{C-c C-c} to finish the edit, or @kbd{C-x k} to cancel.
30310 @node Assignments in Embedded Mode, Mode Settings in Embedded Mode, More About Embedded Mode, Embedded Mode
30311 @section Assignments in Embedded Mode
30314 The @samp{:=} (assignment) and @samp{=>} (``evaluates-to'') operators
30315 are especially useful in Embedded mode. They allow you to make
30316 a definition in one formula, then refer to that definition in
30317 other formulas embedded in the same buffer.
30319 An embedded formula which is an assignment to a variable, as in
30326 records @expr{5} as the stored value of @code{foo} for the
30327 purposes of Embedded mode operations in the current buffer. It
30328 does @emph{not} actually store @expr{5} as the ``global'' value
30329 of @code{foo}, however. Regular Calc operations, and Embedded
30330 formulas in other buffers, will not see this assignment.
30332 One way to use this assigned value is simply to create an
30333 Embedded formula elsewhere that refers to @code{foo}, and to press
30334 @kbd{=} in that formula. However, this permanently replaces the
30335 @code{foo} in the formula with its current value. More interesting
30336 is to use @samp{=>} elsewhere:
30342 @xref{Evaluates-To Operator}, for a general discussion of @samp{=>}.
30344 If you move back and change the assignment to @code{foo}, any
30345 @samp{=>} formulas which refer to it are automatically updated.
30353 The obvious question then is, @emph{how} can one easily change the
30354 assignment to @code{foo}? If you simply select the formula in
30355 Embedded mode and type 17, the assignment itself will be replaced
30356 by the 17. The effect on the other formula will be that the
30357 variable @code{foo} becomes unassigned:
30365 The right thing to do is first to use a selection command (@kbd{j 2}
30366 will do the trick) to select the righthand side of the assignment.
30367 Then, @kbd{17 @key{TAB} @key{DEL}} will swap the 17 into place (@pxref{Selecting
30368 Subformulas}, to see how this works).
30371 @pindex calc-embedded-select
30372 The @kbd{C-x * j} (@code{calc-embedded-select}) command provides an
30373 easy way to operate on assignments. It is just like @kbd{C-x * e},
30374 except that if the enabled formula is an assignment, it uses
30375 @kbd{j 2} to select the righthand side. If the enabled formula
30376 is an evaluates-to, it uses @kbd{j 1} to select the lefthand side.
30377 A formula can also be a combination of both:
30380 bar := foo + 3 => 20
30384 in which case @kbd{C-x * j} will select the middle part (@samp{foo + 3}).
30386 The formula is automatically deselected when you leave Embedded
30390 @pindex calc-embedded-update-formula
30391 Another way to change the assignment to @code{foo} would simply be
30392 to edit the number using regular Emacs editing rather than Embedded
30393 mode. Then, we have to find a way to get Embedded mode to notice
30394 the change. The @kbd{C-x * u} (@code{calc-embedded-update-formula})
30395 command is a convenient way to do this.
30403 Pressing @kbd{C-x * u} is much like pressing @kbd{C-x * e = C-x * e}, that
30404 is, temporarily enabling Embedded mode for the formula under the
30405 cursor and then evaluating it with @kbd{=}. But @kbd{C-x * u} does
30406 not actually use @kbd{C-x * e}, and in fact another formula somewhere
30407 else can be enabled in Embedded mode while you use @kbd{C-x * u} and
30408 that formula will not be disturbed.
30410 With a numeric prefix argument, @kbd{C-x * u} updates all active
30411 @samp{=>} formulas in the buffer. Formulas which have not yet
30412 been activated in Embedded mode, and formulas which do not have
30413 @samp{=>} as their top-level operator, are not affected by this.
30414 (This is useful only if you have used @kbd{m C}; see below.)
30416 With a plain @kbd{C-u} prefix, @kbd{C-u C-x * u} updates only in the
30417 region between mark and point rather than in the whole buffer.
30419 @kbd{C-x * u} is also a handy way to activate a formula, such as an
30420 @samp{=>} formula that has freshly been typed in or loaded from a
30424 @pindex calc-embedded-activate
30425 The @kbd{C-x * a} (@code{calc-embedded-activate}) command scans
30426 through the current buffer and activates all embedded formulas
30427 that contain @samp{:=} or @samp{=>} symbols. This does not mean
30428 that Embedded mode is actually turned on, but only that the
30429 formulas' positions are registered with Embedded mode so that
30430 the @samp{=>} values can be properly updated as assignments are
30433 It is a good idea to type @kbd{C-x * a} right after loading a file
30434 that uses embedded @samp{=>} operators. Emacs includes a nifty
30435 ``buffer-local variables'' feature that you can use to do this
30436 automatically. The idea is to place near the end of your file
30437 a few lines that look like this:
30440 --- Local Variables: ---
30441 --- eval:(calc-embedded-activate) ---
30446 where the leading and trailing @samp{---} can be replaced by
30447 any suitable strings (which must be the same on all three lines)
30448 or omitted altogether; in a @TeX{} or La@TeX{} file, @samp{%} would be a good
30449 leading string and no trailing string would be necessary. In a
30450 C program, @samp{/*} and @samp{*/} would be good leading and
30453 When Emacs loads a file into memory, it checks for a Local Variables
30454 section like this one at the end of the file. If it finds this
30455 section, it does the specified things (in this case, running
30456 @kbd{C-x * a} automatically) before editing of the file begins.
30457 The Local Variables section must be within 3000 characters of the
30458 end of the file for Emacs to find it, and it must be in the last
30459 page of the file if the file has any page separators.
30460 @xref{File Variables, , Local Variables in Files, emacs, the
30463 Note that @kbd{C-x * a} does not update the formulas it finds.
30464 To do this, type, say, @kbd{M-1 C-x * u} after @w{@kbd{C-x * a}}.
30465 Generally this should not be a problem, though, because the
30466 formulas will have been up-to-date already when the file was
30469 Normally, @kbd{C-x * a} activates all the formulas it finds, but
30470 any previous active formulas remain active as well. With a
30471 positive numeric prefix argument, @kbd{C-x * a} first deactivates
30472 all current active formulas, then actives the ones it finds in
30473 its scan of the buffer. With a negative prefix argument,
30474 @kbd{C-x * a} simply deactivates all formulas.
30476 Embedded mode has two symbols, @samp{Active} and @samp{~Active},
30477 which it puts next to the major mode name in a buffer's mode line.
30478 It puts @samp{Active} if it has reason to believe that all
30479 formulas in the buffer are active, because you have typed @kbd{C-x * a}
30480 and Calc has not since had to deactivate any formulas (which can
30481 happen if Calc goes to update an @samp{=>} formula somewhere because
30482 a variable changed, and finds that the formula is no longer there
30483 due to some kind of editing outside of Embedded mode). Calc puts
30484 @samp{~Active} in the mode line if some, but probably not all,
30485 formulas in the buffer are active. This happens if you activate
30486 a few formulas one at a time but never use @kbd{C-x * a}, or if you
30487 used @kbd{C-x * a} but then Calc had to deactivate a formula
30488 because it lost track of it. If neither of these symbols appears
30489 in the mode line, no embedded formulas are active in the buffer
30490 (e.g., before Embedded mode has been used, or after a @kbd{M-- C-x * a}).
30492 Embedded formulas can refer to assignments both before and after them
30493 in the buffer. If there are several assignments to a variable, the
30494 nearest preceding assignment is used if there is one, otherwise the
30495 following assignment is used.
30509 As well as simple variables, you can also assign to subscript
30510 expressions of the form @samp{@var{var}_@var{number}} (as in
30511 @code{x_0}), or @samp{@var{var}_@var{var}} (as in @code{x_max}).
30512 Assignments to other kinds of objects can be represented by Calc,
30513 but the automatic linkage between assignments and references works
30514 only for plain variables and these two kinds of subscript expressions.
30516 If there are no assignments to a given variable, the global
30517 stored value for the variable is used (@pxref{Storing Variables}),
30518 or, if no value is stored, the variable is left in symbolic form.
30519 Note that global stored values will be lost when the file is saved
30520 and loaded in a later Emacs session, unless you have used the
30521 @kbd{s p} (@code{calc-permanent-variable}) command to save them;
30522 @pxref{Operations on Variables}.
30524 The @kbd{m C} (@code{calc-auto-recompute}) command turns automatic
30525 recomputation of @samp{=>} forms on and off. If you turn automatic
30526 recomputation off, you will have to use @kbd{C-x * u} to update these
30527 formulas manually after an assignment has been changed. If you
30528 plan to change several assignments at once, it may be more efficient
30529 to type @kbd{m C}, change all the assignments, then use @kbd{M-1 C-x * u}
30530 to update the entire buffer afterwards. The @kbd{m C} command also
30531 controls @samp{=>} formulas on the stack; @pxref{Evaluates-To
30532 Operator}. When you turn automatic recomputation back on, the
30533 stack will be updated but the Embedded buffer will not; you must
30534 use @kbd{C-x * u} to update the buffer by hand.
30536 @node Mode Settings in Embedded Mode, Customizing Embedded Mode, Assignments in Embedded Mode, Embedded Mode
30537 @section Mode Settings in Embedded Mode
30540 @pindex calc-embedded-preserve-modes
30542 The mode settings can be changed while Calc is in embedded mode, but
30543 by default they will revert to their original values when embedded mode
30544 is ended. However, the modes saved when the mode-recording mode is
30545 @code{Save} (see below) and the modes in effect when the @kbd{m e}
30546 (@code{calc-embedded-preserve-modes}) command is given
30547 will be preserved when embedded mode is ended.
30549 Embedded mode has a rather complicated mechanism for handling mode
30550 settings in Embedded formulas. It is possible to put annotations
30551 in the file that specify mode settings either global to the entire
30552 file or local to a particular formula or formulas. In the latter
30553 case, different modes can be specified for use when a formula
30554 is the enabled Embedded mode formula.
30556 When you give any mode-setting command, like @kbd{m f} (for Fraction
30557 mode) or @kbd{d s} (for scientific notation), Embedded mode adds
30558 a line like the following one to the file just before the opening
30559 delimiter of the formula.
30562 % [calc-mode: fractions: t]
30563 % [calc-mode: float-format: (sci 0)]
30566 When Calc interprets an embedded formula, it scans the text before
30567 the formula for mode-setting annotations like these and sets the
30568 Calc buffer to match these modes. Modes not explicitly described
30569 in the file are not changed. Calc scans all the way to the top of
30570 the file, or up to a line of the form
30577 which you can insert at strategic places in the file if this backward
30578 scan is getting too slow, or just to provide a barrier between one
30579 ``zone'' of mode settings and another.
30581 If the file contains several annotations for the same mode, the
30582 closest one before the formula is used. Annotations after the
30583 formula are never used (except for global annotations, described
30586 The scan does not look for the leading @samp{% }, only for the
30587 square brackets and the text they enclose. In fact, the leading
30588 characters are different for different major modes. You can edit the
30589 mode annotations to a style that works better in context if you wish.
30590 @xref{Customizing Embedded Mode}, to see how to change the style
30591 that Calc uses when it generates the annotations. You can write
30592 mode annotations into the file yourself if you know the syntax;
30593 the easiest way to find the syntax for a given mode is to let
30594 Calc write the annotation for it once and see what it does.
30596 If you give a mode-changing command for a mode that already has
30597 a suitable annotation just above the current formula, Calc will
30598 modify that annotation rather than generating a new, conflicting
30601 Mode annotations have three parts, separated by colons. (Spaces
30602 after the colons are optional.) The first identifies the kind
30603 of mode setting, the second is a name for the mode itself, and
30604 the third is the value in the form of a Lisp symbol, number,
30605 or list. Annotations with unrecognizable text in the first or
30606 second parts are ignored. The third part is not checked to make
30607 sure the value is of a valid type or range; if you write an
30608 annotation by hand, be sure to give a proper value or results
30609 will be unpredictable. Mode-setting annotations are case-sensitive.
30611 While Embedded mode is enabled, the word @code{Local} appears in
30612 the mode line. This is to show that mode setting commands generate
30613 annotations that are ``local'' to the current formula or set of
30614 formulas. The @kbd{m R} (@code{calc-mode-record-mode}) command
30615 causes Calc to generate different kinds of annotations. Pressing
30616 @kbd{m R} repeatedly cycles through the possible modes.
30618 @code{LocEdit} and @code{LocPerm} modes generate annotations
30619 that look like this, respectively:
30622 % [calc-edit-mode: float-format: (sci 0)]
30623 % [calc-perm-mode: float-format: (sci 5)]
30626 The first kind of annotation will be used only while a formula
30627 is enabled in Embedded mode. The second kind will be used only
30628 when the formula is @emph{not} enabled. (Whether the formula
30629 is ``active'' or not, i.e., whether Calc has seen this formula
30630 yet, is not relevant here.)
30632 @code{Global} mode generates an annotation like this at the end
30636 % [calc-global-mode: fractions t]
30639 Global mode annotations affect all formulas throughout the file,
30640 and may appear anywhere in the file. This allows you to tuck your
30641 mode annotations somewhere out of the way, say, on a new page of
30642 the file, as long as those mode settings are suitable for all
30643 formulas in the file.
30645 Enabling a formula with @kbd{C-x * e} causes a fresh scan for local
30646 mode annotations; you will have to use this after adding annotations
30647 above a formula by hand to get the formula to notice them. Updating
30648 a formula with @kbd{C-x * u} will also re-scan the local modes, but
30649 global modes are only re-scanned by @kbd{C-x * a}.
30651 Another way that modes can get out of date is if you add a local
30652 mode annotation to a formula that has another formula after it.
30653 In this example, we have used the @kbd{d s} command while the
30654 first of the two embedded formulas is active. But the second
30655 formula has not changed its style to match, even though by the
30656 rules of reading annotations the @samp{(sci 0)} applies to it, too.
30659 % [calc-mode: float-format: (sci 0)]
30665 We would have to go down to the other formula and press @kbd{C-x * u}
30666 on it in order to get it to notice the new annotation.
30668 Two more mode-recording modes selectable by @kbd{m R} are available
30669 which are also available outside of Embedded mode.
30670 (@pxref{General Mode Commands}.) They are @code{Save}, in which mode
30671 settings are recorded permanently in your Calc init file (the file given
30672 by the variable @code{calc-settings-file}, typically @file{~/.calc.el})
30673 rather than by annotating the current document, and no-recording
30674 mode (where there is no symbol like @code{Save} or @code{Local} in
30675 the mode line), in which mode-changing commands do not leave any
30676 annotations at all.
30678 When Embedded mode is not enabled, mode-recording modes except
30679 for @code{Save} have no effect.
30681 @node Customizing Embedded Mode, , Mode Settings in Embedded Mode, Embedded Mode
30682 @section Customizing Embedded Mode
30685 You can modify Embedded mode's behavior by setting various Lisp
30686 variables described here. These variables are customizable
30687 (@pxref{Customizing Calc}), or you can use @kbd{M-x set-variable}
30688 or @kbd{M-x edit-options} to adjust a variable on the fly.
30689 (Another possibility would be to use a file-local variable annotation at
30690 the end of the file;
30691 @pxref{File Variables, , Local Variables in Files, emacs, the Emacs manual}.)
30692 Many of the variables given mentioned here can be set to depend on the
30693 major mode of the editing buffer (@pxref{Customizing Calc}).
30695 @vindex calc-embedded-open-formula
30696 The @code{calc-embedded-open-formula} variable holds a regular
30697 expression for the opening delimiter of a formula. @xref{Regexp Search,
30698 , Regular Expression Search, emacs, the Emacs manual}, to see
30699 how regular expressions work. Basically, a regular expression is a
30700 pattern that Calc can search for. A regular expression that considers
30701 blank lines, @samp{$}, and @samp{$$} to be opening delimiters is
30702 @code{"\\`\\|^\n\\|\\$\\$?"}. Just in case the meaning of this
30703 regular expression is not completely plain, let's go through it
30706 The surrounding @samp{" "} marks quote the text between them as a
30707 Lisp string. If you left them off, @code{set-variable} or
30708 @code{edit-options} would try to read the regular expression as a
30711 The most obvious property of this regular expression is that it
30712 contains indecently many backslashes. There are actually two levels
30713 of backslash usage going on here. First, when Lisp reads a quoted
30714 string, all pairs of characters beginning with a backslash are
30715 interpreted as special characters. Here, @code{\n} changes to a
30716 new-line character, and @code{\\} changes to a single backslash.
30717 So the actual regular expression seen by Calc is
30718 @samp{\`\|^ @r{(newline)} \|\$\$?}.
30720 Regular expressions also consider pairs beginning with backslash
30721 to have special meanings. Sometimes the backslash is used to quote
30722 a character that otherwise would have a special meaning in a regular
30723 expression, like @samp{$}, which normally means ``end-of-line,''
30724 or @samp{?}, which means that the preceding item is optional. So
30725 @samp{\$\$?} matches either one or two dollar signs.
30727 The other codes in this regular expression are @samp{^}, which matches
30728 ``beginning-of-line,'' @samp{\|}, which means ``or,'' and @samp{\`},
30729 which matches ``beginning-of-buffer.'' So the whole pattern means
30730 that a formula begins at the beginning of the buffer, or on a newline
30731 that occurs at the beginning of a line (i.e., a blank line), or at
30732 one or two dollar signs.
30734 The default value of @code{calc-embedded-open-formula} looks just
30735 like this example, with several more alternatives added on to
30736 recognize various other common kinds of delimiters.
30738 By the way, the reason to use @samp{^\n} rather than @samp{^$}
30739 or @samp{\n\n}, which also would appear to match blank lines,
30740 is that the former expression actually ``consumes'' only one
30741 newline character as @emph{part of} the delimiter, whereas the
30742 latter expressions consume zero or two newlines, respectively.
30743 The former choice gives the most natural behavior when Calc
30744 must operate on a whole formula including its delimiters.
30746 See the Emacs manual for complete details on regular expressions.
30747 But just for your convenience, here is a list of all characters
30748 which must be quoted with backslash (like @samp{\$}) to avoid
30749 some special interpretation: @samp{. * + ? [ ] ^ $ \}. (Note
30750 the backslash in this list; for example, to match @samp{\[} you
30751 must use @code{"\\\\\\["}. An exercise for the reader is to
30752 account for each of these six backslashes!)
30754 @vindex calc-embedded-close-formula
30755 The @code{calc-embedded-close-formula} variable holds a regular
30756 expression for the closing delimiter of a formula. A closing
30757 regular expression to match the above example would be
30758 @code{"\\'\\|\n$\\|\\$\\$?"}. This is almost the same as the
30759 other one, except it now uses @samp{\'} (``end-of-buffer'') and
30760 @samp{\n$} (newline occurring at end of line, yet another way
30761 of describing a blank line that is more appropriate for this
30764 @vindex calc-embedded-word-regexp
30765 The @code{calc-embedded-word-regexp} variable holds a regular expression
30766 used to define an expression to look for (a ``word'') when you type
30767 @kbd{C-x * w} to enable Embedded mode.
30769 @vindex calc-embedded-open-plain
30770 The @code{calc-embedded-open-plain} variable is a string which
30771 begins a ``plain'' formula written in front of the formatted
30772 formula when @kbd{d p} mode is turned on. Note that this is an
30773 actual string, not a regular expression, because Calc must be able
30774 to write this string into a buffer as well as to recognize it.
30775 The default string is @code{"%%% "} (note the trailing space), but may
30776 be different for certain major modes.
30778 @vindex calc-embedded-close-plain
30779 The @code{calc-embedded-close-plain} variable is a string which
30780 ends a ``plain'' formula. The default is @code{" %%%\n"}, but may be
30781 different for different major modes. Without
30782 the trailing newline here, the first line of a Big mode formula
30783 that followed might be shifted over with respect to the other lines.
30785 @vindex calc-embedded-open-new-formula
30786 The @code{calc-embedded-open-new-formula} variable is a string
30787 which is inserted at the front of a new formula when you type
30788 @kbd{C-x * f}. Its default value is @code{"\n\n"}. If this
30789 string begins with a newline character and the @kbd{C-x * f} is
30790 typed at the beginning of a line, @kbd{C-x * f} will skip this
30791 first newline to avoid introducing unnecessary blank lines in
30794 @vindex calc-embedded-close-new-formula
30795 The @code{calc-embedded-close-new-formula} variable is the corresponding
30796 string which is inserted at the end of a new formula. Its default
30797 value is also @code{"\n\n"}. The final newline is omitted by
30798 @w{@kbd{C-x * f}} if typed at the end of a line. (It follows that if
30799 @kbd{C-x * f} is typed on a blank line, both a leading opening
30800 newline and a trailing closing newline are omitted.)
30802 @vindex calc-embedded-announce-formula
30803 The @code{calc-embedded-announce-formula} variable is a regular
30804 expression which is sure to be followed by an embedded formula.
30805 The @kbd{C-x * a} command searches for this pattern as well as for
30806 @samp{=>} and @samp{:=} operators. Note that @kbd{C-x * a} will
30807 not activate just anything surrounded by formula delimiters; after
30808 all, blank lines are considered formula delimiters by default!
30809 But if your language includes a delimiter which can only occur
30810 actually in front of a formula, you can take advantage of it here.
30811 The default pattern is @code{"%Embed\n\\(% .*\n\\)*"}, but may be
30812 different for different major modes.
30813 This pattern will check for @samp{%Embed} followed by any number of
30814 lines beginning with @samp{%} and a space. This last is important to
30815 make Calc consider mode annotations part of the pattern, so that the
30816 formula's opening delimiter really is sure to follow the pattern.
30818 @vindex calc-embedded-open-mode
30819 The @code{calc-embedded-open-mode} variable is a string (not a
30820 regular expression) which should precede a mode annotation.
30821 Calc never scans for this string; Calc always looks for the
30822 annotation itself. But this is the string that is inserted before
30823 the opening bracket when Calc adds an annotation on its own.
30824 The default is @code{"% "}, but may be different for different major
30827 @vindex calc-embedded-close-mode
30828 The @code{calc-embedded-close-mode} variable is a string which
30829 follows a mode annotation written by Calc. Its default value
30830 is simply a newline, @code{"\n"}, but may be different for different
30831 major modes. If you change this, it is a good idea still to end with a
30832 newline so that mode annotations will appear on lines by themselves.
30834 @node Programming, Copying, Embedded Mode, Top
30835 @chapter Programming
30838 There are several ways to ``program'' the Emacs Calculator, depending
30839 on the nature of the problem you need to solve.
30843 @dfn{Keyboard macros} allow you to record a sequence of keystrokes
30844 and play them back at a later time. This is just the standard Emacs
30845 keyboard macro mechanism, dressed up with a few more features such
30846 as loops and conditionals.
30849 @dfn{Algebraic definitions} allow you to use any formula to define a
30850 new function. This function can then be used in algebraic formulas or
30851 as an interactive command.
30854 @dfn{Rewrite rules} are discussed in the section on algebra commands.
30855 @xref{Rewrite Rules}. If you put your rewrite rules in the variable
30856 @code{EvalRules}, they will be applied automatically to all Calc
30857 results in just the same way as an internal ``rule'' is applied to
30858 evaluate @samp{sqrt(9)} to 3 and so on. @xref{Automatic Rewrites}.
30861 @dfn{Lisp} is the programming language that Calc (and most of Emacs)
30862 is written in. If the above techniques aren't powerful enough, you
30863 can write Lisp functions to do anything that built-in Calc commands
30864 can do. Lisp code is also somewhat faster than keyboard macros or
30869 Programming features are available through the @kbd{z} and @kbd{Z}
30870 prefix keys. New commands that you define are two-key sequences
30871 beginning with @kbd{z}. Commands for managing these definitions
30872 use the shift-@kbd{Z} prefix. (The @kbd{Z T} (@code{calc-timing})
30873 command is described elsewhere; @pxref{Troubleshooting Commands}.
30874 The @kbd{Z C} (@code{calc-user-define-composition}) command is also
30875 described elsewhere; @pxref{User-Defined Compositions}.)
30878 * Creating User Keys::
30879 * Keyboard Macros::
30880 * Invocation Macros::
30881 * Algebraic Definitions::
30882 * Lisp Definitions::
30885 @node Creating User Keys, Keyboard Macros, Programming, Programming
30886 @section Creating User Keys
30890 @pindex calc-user-define
30891 Any Calculator command may be bound to a key using the @kbd{Z D}
30892 (@code{calc-user-define}) command. Actually, it is bound to a two-key
30893 sequence beginning with the lower-case @kbd{z} prefix.
30895 The @kbd{Z D} command first prompts for the key to define. For example,
30896 press @kbd{Z D a} to define the new key sequence @kbd{z a}. You are then
30897 prompted for the name of the Calculator command that this key should
30898 run. For example, the @code{calc-sincos} command is not normally
30899 available on a key. Typing @kbd{Z D s sincos @key{RET}} programs the
30900 @kbd{z s} key sequence to run @code{calc-sincos}. This definition will remain
30901 in effect for the rest of this Emacs session, or until you redefine
30902 @kbd{z s} to be something else.
30904 You can actually bind any Emacs command to a @kbd{z} key sequence by
30905 backspacing over the @samp{calc-} when you are prompted for the command name.
30907 As with any other prefix key, you can type @kbd{z ?} to see a list of
30908 all the two-key sequences you have defined that start with @kbd{z}.
30909 Initially, no @kbd{z} sequences (except @kbd{z ?} itself) are defined.
30911 User keys are typically letters, but may in fact be any key.
30912 (@key{META}-keys are not permitted, nor are a terminal's special
30913 function keys which generate multi-character sequences when pressed.)
30914 You can define different commands on the shifted and unshifted versions
30915 of a letter if you wish.
30918 @pindex calc-user-undefine
30919 The @kbd{Z U} (@code{calc-user-undefine}) command unbinds a user key.
30920 For example, the key sequence @kbd{Z U s} will undefine the @code{sincos}
30921 key we defined above.
30924 @pindex calc-user-define-permanent
30925 @cindex Storing user definitions
30926 @cindex Permanent user definitions
30927 @cindex Calc init file, user-defined commands
30928 The @kbd{Z P} (@code{calc-user-define-permanent}) command makes a key
30929 binding permanent so that it will remain in effect even in future Emacs
30930 sessions. (It does this by adding a suitable bit of Lisp code into
30931 your Calc init file; that is, the file given by the variable
30932 @code{calc-settings-file}, typically @file{~/.calc.el}.) For example,
30933 @kbd{Z P s} would register our @code{sincos} command permanently. If
30934 you later wish to unregister this command you must edit your Calc init
30935 file by hand. (@xref{General Mode Commands}, for a way to tell Calc to
30936 use a different file for the Calc init file.)
30938 The @kbd{Z P} command also saves the user definition, if any, for the
30939 command bound to the key. After @kbd{Z F} and @kbd{Z C}, a given user
30940 key could invoke a command, which in turn calls an algebraic function,
30941 which might have one or more special display formats. A single @kbd{Z P}
30942 command will save all of these definitions.
30943 To save an algebraic function, type @kbd{'} (the apostrophe)
30944 when prompted for a key, and type the function name. To save a command
30945 without its key binding, type @kbd{M-x} and enter a function name. (The
30946 @samp{calc-} prefix will automatically be inserted for you.)
30947 (If the command you give implies a function, the function will be saved,
30948 and if the function has any display formats, those will be saved, but
30949 not the other way around: Saving a function will not save any commands
30950 or key bindings associated with the function.)
30953 @pindex calc-user-define-edit
30954 @cindex Editing user definitions
30955 The @kbd{Z E} (@code{calc-user-define-edit}) command edits the definition
30956 of a user key. This works for keys that have been defined by either
30957 keyboard macros or formulas; further details are contained in the relevant
30958 following sections.
30960 @node Keyboard Macros, Invocation Macros, Creating User Keys, Programming
30961 @section Programming with Keyboard Macros
30965 @cindex Programming with keyboard macros
30966 @cindex Keyboard macros
30967 The easiest way to ``program'' the Emacs Calculator is to use standard
30968 keyboard macros. Press @w{@kbd{C-x (}} to begin recording a macro. From
30969 this point on, keystrokes you type will be saved away as well as
30970 performing their usual functions. Press @kbd{C-x )} to end recording.
30971 Press shift-@kbd{X} (or the standard Emacs key sequence @kbd{C-x e}) to
30972 execute your keyboard macro by replaying the recorded keystrokes.
30973 @xref{Keyboard Macros, , , emacs, the Emacs Manual}, for further
30976 When you use @kbd{X} to invoke a keyboard macro, the entire macro is
30977 treated as a single command by the undo and trail features. The stack
30978 display buffer is not updated during macro execution, but is instead
30979 fixed up once the macro completes. Thus, commands defined with keyboard
30980 macros are convenient and efficient. The @kbd{C-x e} command, on the
30981 other hand, invokes the keyboard macro with no special treatment: Each
30982 command in the macro will record its own undo information and trail entry,
30983 and update the stack buffer accordingly. If your macro uses features
30984 outside of Calc's control to operate on the contents of the Calc stack
30985 buffer, or if it includes Undo, Redo, or last-arguments commands, you
30986 must use @kbd{C-x e} to make sure the buffer and undo list are up-to-date
30987 at all times. You could also consider using @kbd{K} (@code{calc-keep-args})
30988 instead of @kbd{M-@key{RET}} (@code{calc-last-args}).
30990 Calc extends the standard Emacs keyboard macros in several ways.
30991 Keyboard macros can be used to create user-defined commands. Keyboard
30992 macros can include conditional and iteration structures, somewhat
30993 analogous to those provided by a traditional programmable calculator.
30996 * Naming Keyboard Macros::
30997 * Conditionals in Macros::
30998 * Loops in Macros::
30999 * Local Values in Macros::
31000 * Queries in Macros::
31003 @node Naming Keyboard Macros, Conditionals in Macros, Keyboard Macros, Keyboard Macros
31004 @subsection Naming Keyboard Macros
31008 @pindex calc-user-define-kbd-macro
31009 Once you have defined a keyboard macro, you can bind it to a @kbd{z}
31010 key sequence with the @kbd{Z K} (@code{calc-user-define-kbd-macro}) command.
31011 This command prompts first for a key, then for a command name. For
31012 example, if you type @kbd{C-x ( n @key{TAB} n @key{TAB} C-x )} you will
31013 define a keyboard macro which negates the top two numbers on the stack
31014 (@key{TAB} swaps the top two stack elements). Now you can type
31015 @kbd{Z K n @key{RET}} to define this keyboard macro onto the @kbd{z n} key
31016 sequence. The default command name (if you answer the second prompt with
31017 just the @key{RET} key as in this example) will be something like
31018 @samp{calc-User-n}. The keyboard macro will now be available as both
31019 @kbd{z n} and @kbd{M-x calc-User-n}. You can backspace and enter a more
31020 descriptive command name if you wish.
31022 Macros defined by @kbd{Z K} act like single commands; they are executed
31023 in the same way as by the @kbd{X} key. If you wish to define the macro
31024 as a standard no-frills Emacs macro (to be executed as if by @kbd{C-x e}),
31025 give a negative prefix argument to @kbd{Z K}.
31027 Once you have bound your keyboard macro to a key, you can use
31028 @kbd{Z P} to register it permanently with Emacs. @xref{Creating User Keys}.
31030 @cindex Keyboard macros, editing
31031 The @kbd{Z E} (@code{calc-user-define-edit}) command on a key that has
31032 been defined by a keyboard macro tries to use the @code{edmacro} package
31033 edit the macro. Type @kbd{C-c C-c} to finish editing and update
31034 the definition stored on the key, or, to cancel the edit, kill the
31035 buffer with @kbd{C-x k}.
31036 The special characters @code{RET}, @code{LFD}, @code{TAB}, @code{SPC},
31037 @code{DEL}, and @code{NUL} must be entered as these three character
31038 sequences, written in all uppercase, as must the prefixes @code{C-} and
31039 @code{M-}. Spaces and line breaks are ignored. Other characters are
31040 copied verbatim into the keyboard macro. Basically, the notation is the
31041 same as is used in all of this manual's examples, except that the manual
31042 takes some liberties with spaces: When we say @kbd{' [1 2 3] @key{RET}},
31043 we take it for granted that it is clear we really mean
31044 @kbd{' [1 @key{SPC} 2 @key{SPC} 3] @key{RET}}.
31047 @pindex read-kbd-macro
31048 The @kbd{C-x * m} (@code{read-kbd-macro}) command reads an Emacs ``region''
31049 of spelled-out keystrokes and defines it as the current keyboard macro.
31050 It is a convenient way to define a keyboard macro that has been stored
31051 in a file, or to define a macro without executing it at the same time.
31053 @node Conditionals in Macros, Loops in Macros, Naming Keyboard Macros, Keyboard Macros
31054 @subsection Conditionals in Keyboard Macros
31059 @pindex calc-kbd-if
31060 @pindex calc-kbd-else
31061 @pindex calc-kbd-else-if
31062 @pindex calc-kbd-end-if
31063 @cindex Conditional structures
31064 The @kbd{Z [} (@code{calc-kbd-if}) and @kbd{Z ]} (@code{calc-kbd-end-if})
31065 commands allow you to put simple tests in a keyboard macro. When Calc
31066 sees the @kbd{Z [}, it pops an object from the stack and, if the object is
31067 a non-zero value, continues executing keystrokes. But if the object is
31068 zero, or if it is not provably nonzero, Calc skips ahead to the matching
31069 @kbd{Z ]} keystroke. @xref{Logical Operations}, for a set of commands for
31070 performing tests which conveniently produce 1 for true and 0 for false.
31072 For example, @kbd{@key{RET} 0 a < Z [ n Z ]} implements an absolute-value
31073 function in the form of a keyboard macro. This macro duplicates the
31074 number on the top of the stack, pushes zero and compares using @kbd{a <}
31075 (@code{calc-less-than}), then, if the number was less than zero,
31076 executes @kbd{n} (@code{calc-change-sign}). Otherwise, the change-sign
31077 command is skipped.
31079 To program this macro, type @kbd{C-x (}, type the above sequence of
31080 keystrokes, then type @kbd{C-x )}. Note that the keystrokes will be
31081 executed while you are making the definition as well as when you later
31082 re-execute the macro by typing @kbd{X}. Thus you should make sure a
31083 suitable number is on the stack before defining the macro so that you
31084 don't get a stack-underflow error during the definition process.
31086 Conditionals can be nested arbitrarily. However, there should be exactly
31087 one @kbd{Z ]} for each @kbd{Z [} in a keyboard macro.
31090 The @kbd{Z :} (@code{calc-kbd-else}) command allows you to choose between
31091 two keystroke sequences. The general format is @kbd{@var{cond} Z [
31092 @var{then-part} Z : @var{else-part} Z ]}. If @var{cond} is true
31093 (i.e., if the top of stack contains a non-zero number after @var{cond}
31094 has been executed), the @var{then-part} will be executed and the
31095 @var{else-part} will be skipped. Otherwise, the @var{then-part} will
31096 be skipped and the @var{else-part} will be executed.
31099 The @kbd{Z |} (@code{calc-kbd-else-if}) command allows you to choose
31100 between any number of alternatives. For example,
31101 @kbd{@var{cond1} Z [ @var{part1} Z : @var{cond2} Z | @var{part2} Z :
31102 @var{part3} Z ]} will execute @var{part1} if @var{cond1} is true,
31103 otherwise it will execute @var{part2} if @var{cond2} is true, otherwise
31104 it will execute @var{part3}.
31106 More precisely, @kbd{Z [} pops a number and conditionally skips to the
31107 next matching @kbd{Z :} or @kbd{Z ]} key. @w{@kbd{Z ]}} has no effect when
31108 actually executed. @kbd{Z :} skips to the next matching @kbd{Z ]}.
31109 @kbd{Z |} pops a number and conditionally skips to the next matching
31110 @kbd{Z :} or @kbd{Z ]}; thus, @kbd{Z [} and @kbd{Z |} are functionally
31111 equivalent except that @kbd{Z [} participates in nesting but @kbd{Z |}
31114 Calc's conditional and looping constructs work by scanning the
31115 keyboard macro for occurrences of character sequences like @samp{Z:}
31116 and @samp{Z]}. One side-effect of this is that if you use these
31117 constructs you must be careful that these character pairs do not
31118 occur by accident in other parts of the macros. Since Calc rarely
31119 uses shift-@kbd{Z} for any purpose except as a prefix character, this
31120 is not likely to be a problem. Another side-effect is that it will
31121 not work to define your own custom key bindings for these commands.
31122 Only the standard shift-@kbd{Z} bindings will work correctly.
31125 If Calc gets stuck while skipping characters during the definition of a
31126 macro, type @kbd{Z C-g} to cancel the definition. (Typing plain @kbd{C-g}
31127 actually adds a @kbd{C-g} keystroke to the macro.)
31129 @node Loops in Macros, Local Values in Macros, Conditionals in Macros, Keyboard Macros
31130 @subsection Loops in Keyboard Macros
31135 @pindex calc-kbd-repeat
31136 @pindex calc-kbd-end-repeat
31137 @cindex Looping structures
31138 @cindex Iterative structures
31139 The @kbd{Z <} (@code{calc-kbd-repeat}) and @kbd{Z >}
31140 (@code{calc-kbd-end-repeat}) commands pop a number from the stack,
31141 which must be an integer, then repeat the keystrokes between the brackets
31142 the specified number of times. If the integer is zero or negative, the
31143 body is skipped altogether. For example, @kbd{1 @key{TAB} Z < 2 * Z >}
31144 computes two to a nonnegative integer power. First, we push 1 on the
31145 stack and then swap the integer argument back to the top. The @kbd{Z <}
31146 pops that argument leaving the 1 back on top of the stack. Then, we
31147 repeat a multiply-by-two step however many times.
31149 Once again, the keyboard macro is executed as it is being entered.
31150 In this case it is especially important to set up reasonable initial
31151 conditions before making the definition: Suppose the integer 1000 just
31152 happened to be sitting on the stack before we typed the above definition!
31153 Another approach is to enter a harmless dummy definition for the macro,
31154 then go back and edit in the real one with a @kbd{Z E} command. Yet
31155 another approach is to type the macro as written-out keystroke names
31156 in a buffer, then use @kbd{C-x * m} (@code{read-kbd-macro}) to read the
31161 The @kbd{Z /} (@code{calc-kbd-break}) command allows you to break out
31162 of a keyboard macro loop prematurely. It pops an object from the stack;
31163 if that object is true (a non-zero number), control jumps out of the
31164 innermost enclosing @kbd{Z <} @dots{} @kbd{Z >} loop and continues
31165 after the @kbd{Z >}. If the object is false, the @kbd{Z /} has no
31166 effect. Thus @kbd{@var{cond} Z /} is similar to @samp{if (@var{cond}) break;}
31171 @pindex calc-kbd-for
31172 @pindex calc-kbd-end-for
31173 The @kbd{Z (} (@code{calc-kbd-for}) and @kbd{Z )} (@code{calc-kbd-end-for})
31174 commands are similar to @kbd{Z <} and @kbd{Z >}, except that they make the
31175 value of the counter available inside the loop. The general layout is
31176 @kbd{@var{init} @var{final} Z ( @var{body} @var{step} Z )}. The @kbd{Z (}
31177 command pops initial and final values from the stack. It then creates
31178 a temporary internal counter and initializes it with the value @var{init}.
31179 The @kbd{Z (} command then repeatedly pushes the counter value onto the
31180 stack and executes @var{body} and @var{step}, adding @var{step} to the
31181 counter each time until the loop finishes.
31183 @cindex Summations (by keyboard macros)
31184 By default, the loop finishes when the counter becomes greater than (or
31185 less than) @var{final}, assuming @var{initial} is less than (greater
31186 than) @var{final}. If @var{initial} is equal to @var{final}, the body
31187 executes exactly once. The body of the loop always executes at least
31188 once. For example, @kbd{0 1 10 Z ( 2 ^ + 1 Z )} computes the sum of the
31189 squares of the integers from 1 to 10, in steps of 1.
31191 If you give a numeric prefix argument of 1 to @kbd{Z (}, the loop is
31192 forced to use upward-counting conventions. In this case, if @var{initial}
31193 is greater than @var{final} the body will not be executed at all.
31194 Note that @var{step} may still be negative in this loop; the prefix
31195 argument merely constrains the loop-finished test. Likewise, a prefix
31196 argument of @mathit{-1} forces downward-counting conventions.
31200 @pindex calc-kbd-loop
31201 @pindex calc-kbd-end-loop
31202 The @kbd{Z @{} (@code{calc-kbd-loop}) and @kbd{Z @}}
31203 (@code{calc-kbd-end-loop}) commands are similar to @kbd{Z <} and
31204 @kbd{Z >}, except that they do not pop a count from the stack---they
31205 effectively create an infinite loop. Every @kbd{Z @{} @dots{} @kbd{Z @}}
31206 loop ought to include at least one @kbd{Z /} to make sure the loop
31207 doesn't run forever. (If any error message occurs which causes Emacs
31208 to beep, the keyboard macro will also be halted; this is a standard
31209 feature of Emacs. You can also generally press @kbd{C-g} to halt a
31210 running keyboard macro, although not all versions of Unix support
31213 The conditional and looping constructs are not actually tied to
31214 keyboard macros, but they are most often used in that context.
31215 For example, the keystrokes @kbd{10 Z < 23 @key{RET} Z >} push
31216 ten copies of 23 onto the stack. This can be typed ``live'' just
31217 as easily as in a macro definition.
31219 @xref{Conditionals in Macros}, for some additional notes about
31220 conditional and looping commands.
31222 @node Local Values in Macros, Queries in Macros, Loops in Macros, Keyboard Macros
31223 @subsection Local Values in Macros
31226 @cindex Local variables
31227 @cindex Restoring saved modes
31228 Keyboard macros sometimes want to operate under known conditions
31229 without affecting surrounding conditions. For example, a keyboard
31230 macro may wish to turn on Fraction mode, or set a particular
31231 precision, independent of the user's normal setting for those
31236 @pindex calc-kbd-push
31237 @pindex calc-kbd-pop
31238 Macros also sometimes need to use local variables. Assignments to
31239 local variables inside the macro should not affect any variables
31240 outside the macro. The @kbd{Z `} (@code{calc-kbd-push}) and @kbd{Z '}
31241 (@code{calc-kbd-pop}) commands give you both of these capabilities.
31243 When you type @kbd{Z `} (with a backquote or accent grave character),
31244 the values of various mode settings are saved away. The ten ``quick''
31245 variables @code{q0} through @code{q9} are also saved. When
31246 you type @w{@kbd{Z '}} (with an apostrophe), these values are restored.
31247 Pairs of @kbd{Z `} and @kbd{Z '} commands may be nested.
31249 If a keyboard macro halts due to an error in between a @kbd{Z `} and
31250 a @kbd{Z '}, the saved values will be restored correctly even though
31251 the macro never reaches the @kbd{Z '} command. Thus you can use
31252 @kbd{Z `} and @kbd{Z '} without having to worry about what happens
31253 in exceptional conditions.
31255 If you type @kbd{Z `} ``live'' (not in a keyboard macro), Calc puts
31256 you into a ``recursive edit.'' You can tell you are in a recursive
31257 edit because there will be extra square brackets in the mode line,
31258 as in @samp{[(Calculator)]}. These brackets will go away when you
31259 type the matching @kbd{Z '} command. The modes and quick variables
31260 will be saved and restored in just the same way as if actual keyboard
31261 macros were involved.
31263 The modes saved by @kbd{Z `} and @kbd{Z '} are the current precision
31264 and binary word size, the angular mode (Deg, Rad, or HMS), the
31265 simplification mode, Algebraic mode, Symbolic mode, Infinite mode,
31266 Matrix or Scalar mode, Fraction mode, and the current complex mode
31267 (Polar or Rectangular). The ten ``quick'' variables' values (or lack
31268 thereof) are also saved.
31270 Most mode-setting commands act as toggles, but with a numeric prefix
31271 they force the mode either on (positive prefix) or off (negative
31272 or zero prefix). Since you don't know what the environment might
31273 be when you invoke your macro, it's best to use prefix arguments
31274 for all mode-setting commands inside the macro.
31276 In fact, @kbd{C-u Z `} is like @kbd{Z `} except that it sets the modes
31277 listed above to their default values. As usual, the matching @kbd{Z '}
31278 will restore the modes to their settings from before the @kbd{C-u Z `}.
31279 Also, @w{@kbd{Z `}} with a negative prefix argument resets the algebraic mode
31280 to its default (off) but leaves the other modes the same as they were
31281 outside the construct.
31283 The contents of the stack and trail, values of non-quick variables, and
31284 other settings such as the language mode and the various display modes,
31285 are @emph{not} affected by @kbd{Z `} and @kbd{Z '}.
31287 @node Queries in Macros, , Local Values in Macros, Keyboard Macros
31288 @subsection Queries in Keyboard Macros
31292 @c @pindex calc-kbd-report
31293 @c The @kbd{Z =} (@code{calc-kbd-report}) command displays an informative
31294 @c message including the value on the top of the stack. You are prompted
31295 @c to enter a string. That string, along with the top-of-stack value,
31296 @c is displayed unless @kbd{m w} (@code{calc-working}) has been used
31297 @c to turn such messages off.
31301 @pindex calc-kbd-query
31302 The @kbd{Z #} (@code{calc-kbd-query}) command prompts for an algebraic
31303 entry which takes its input from the keyboard, even during macro
31304 execution. All the normal conventions of algebraic input, including the
31305 use of @kbd{$} characters, are supported. The prompt message itself is
31306 taken from the top of the stack, and so must be entered (as a string)
31307 before the @kbd{Z #} command. (Recall, as a string it can be entered by
31308 pressing the @kbd{"} key and will appear as a vector when it is put on
31309 the stack. The prompt message is only put on the stack to provide a
31310 prompt for the @kbd{Z #} command; it will not play any role in any
31311 subsequent calculations.) This command allows your keyboard macros to
31312 accept numbers or formulas as interactive input.
31315 @kbd{2 @key{RET} "Power: " @key{RET} Z # 3 @key{RET} ^} will prompt for
31316 input with ``Power: '' in the minibuffer, then return 2 to the provided
31317 power. (The response to the prompt that's given, 3 in this example,
31318 will not be part of the macro.)
31320 @xref{Keyboard Macro Query, , , emacs, the Emacs Manual}, for a description of
31321 @kbd{C-x q} (@code{kbd-macro-query}), the standard Emacs way to accept
31322 keyboard input during a keyboard macro. In particular, you can use
31323 @kbd{C-x q} to enter a recursive edit, which allows the user to perform
31324 any Calculator operations interactively before pressing @kbd{C-M-c} to
31325 return control to the keyboard macro.
31327 @node Invocation Macros, Algebraic Definitions, Keyboard Macros, Programming
31328 @section Invocation Macros
31332 @pindex calc-user-invocation
31333 @pindex calc-user-define-invocation
31334 Calc provides one special keyboard macro, called up by @kbd{C-x * z}
31335 (@code{calc-user-invocation}), that is intended to allow you to define
31336 your own special way of starting Calc. To define this ``invocation
31337 macro,'' create the macro in the usual way with @kbd{C-x (} and
31338 @kbd{C-x )}, then type @kbd{Z I} (@code{calc-user-define-invocation}).
31339 There is only one invocation macro, so you don't need to type any
31340 additional letters after @kbd{Z I}. From now on, you can type
31341 @kbd{C-x * z} at any time to execute your invocation macro.
31343 For example, suppose you find yourself often grabbing rectangles of
31344 numbers into Calc and multiplying their columns. You can do this
31345 by typing @kbd{C-x * r} to grab, and @kbd{V R : *} to multiply columns.
31346 To make this into an invocation macro, just type @kbd{C-x ( C-x * r
31347 V R : * C-x )}, then @kbd{Z I}. Then, to multiply a rectangle of data,
31348 just mark the data in its buffer in the usual way and type @kbd{C-x * z}.
31350 Invocation macros are treated like regular Emacs keyboard macros;
31351 all the special features described above for @kbd{Z K}-style macros
31352 do not apply. @kbd{C-x * z} is just like @kbd{C-x e}, except that it
31353 uses the macro that was last stored by @kbd{Z I}. (In fact, the
31354 macro does not even have to have anything to do with Calc!)
31356 The @kbd{m m} command saves the last invocation macro defined by
31357 @kbd{Z I} along with all the other Calc mode settings.
31358 @xref{General Mode Commands}.
31360 @node Algebraic Definitions, Lisp Definitions, Invocation Macros, Programming
31361 @section Programming with Formulas
31365 @pindex calc-user-define-formula
31366 @cindex Programming with algebraic formulas
31367 Another way to create a new Calculator command uses algebraic formulas.
31368 The @kbd{Z F} (@code{calc-user-define-formula}) command stores the
31369 formula at the top of the stack as the definition for a key. This
31370 command prompts for five things: The key, the command name, the function
31371 name, the argument list, and the behavior of the command when given
31372 non-numeric arguments.
31374 For example, suppose we type @kbd{' a+2b @key{RET}} to push the formula
31375 @samp{a + 2*b} onto the stack. We now type @kbd{Z F m} to define this
31376 formula on the @kbd{z m} key sequence. The next prompt is for a command
31377 name, beginning with @samp{calc-}, which should be the long (@kbd{M-x}) form
31378 for the new command. If you simply press @key{RET}, a default name like
31379 @code{calc-User-m} will be constructed. In our example, suppose we enter
31380 @kbd{spam @key{RET}} to define the new command as @code{calc-spam}.
31382 If you want to give the formula a long-style name only, you can press
31383 @key{SPC} or @key{RET} when asked which single key to use. For example
31384 @kbd{Z F @key{RET} spam @key{RET}} defines the new command as
31385 @kbd{M-x calc-spam}, with no keyboard equivalent.
31387 The third prompt is for an algebraic function name. The default is to
31388 use the same name as the command name but without the @samp{calc-}
31389 prefix. (If this is of the form @samp{User-m}, the hyphen is removed so
31390 it won't be taken for a minus sign in algebraic formulas.)
31391 This is the name you will use if you want to enter your
31392 new function in an algebraic formula. Suppose we enter @kbd{yow @key{RET}}.
31393 Then the new function can be invoked by pushing two numbers on the
31394 stack and typing @kbd{z m} or @kbd{x spam}, or by entering the algebraic
31395 formula @samp{yow(x,y)}.
31397 The fourth prompt is for the function's argument list. This is used to
31398 associate values on the stack with the variables that appear in the formula.
31399 The default is a list of all variables which appear in the formula, sorted
31400 into alphabetical order. In our case, the default would be @samp{(a b)}.
31401 This means that, when the user types @kbd{z m}, the Calculator will remove
31402 two numbers from the stack, substitute these numbers for @samp{a} and
31403 @samp{b} (respectively) in the formula, then simplify the formula and
31404 push the result on the stack. In other words, @kbd{10 @key{RET} 100 z m}
31405 would replace the 10 and 100 on the stack with the number 210, which is
31406 @expr{a + 2 b} with @expr{a=10} and @expr{b=100}. Likewise, the formula
31407 @samp{yow(10, 100)} will be evaluated by substituting @expr{a=10} and
31408 @expr{b=100} in the definition.
31410 You can rearrange the order of the names before pressing @key{RET} to
31411 control which stack positions go to which variables in the formula. If
31412 you remove a variable from the argument list, that variable will be left
31413 in symbolic form by the command. Thus using an argument list of @samp{(b)}
31414 for our function would cause @kbd{10 z m} to replace the 10 on the stack
31415 with the formula @samp{a + 20}. If we had used an argument list of
31416 @samp{(b a)}, the result with inputs 10 and 100 would have been 120.
31418 You can also put a nameless function on the stack instead of just a
31419 formula, as in @samp{<a, b : a + 2 b>}. @xref{Specifying Operators}.
31420 In this example, the command will be defined by the formula @samp{a + 2 b}
31421 using the argument list @samp{(a b)}.
31423 The final prompt is a y-or-n question concerning what to do if symbolic
31424 arguments are given to your function. If you answer @kbd{y}, then
31425 executing @kbd{z m} (using the original argument list @samp{(a b)}) with
31426 arguments @expr{10} and @expr{x} will leave the function in symbolic
31427 form, i.e., @samp{yow(10,x)}. On the other hand, if you answer @kbd{n},
31428 then the formula will always be expanded, even for non-constant
31429 arguments: @samp{10 + 2 x}. If you never plan to feed algebraic
31430 formulas to your new function, it doesn't matter how you answer this
31433 If you answered @kbd{y} to this question you can still cause a function
31434 call to be expanded by typing @kbd{a "} (@code{calc-expand-formula}).
31435 Also, Calc will expand the function if necessary when you take a
31436 derivative or integral or solve an equation involving the function.
31439 @pindex calc-get-user-defn
31440 Once you have defined a formula on a key, you can retrieve this formula
31441 with the @kbd{Z G} (@code{calc-user-define-get-defn}) command. Press a
31442 key, and this command pushes the formula that was used to define that
31443 key onto the stack. Actually, it pushes a nameless function that
31444 specifies both the argument list and the defining formula. You will get
31445 an error message if the key is undefined, or if the key was not defined
31446 by a @kbd{Z F} command.
31448 The @kbd{Z E} (@code{calc-user-define-edit}) command on a key that has
31449 been defined by a formula uses a variant of the @code{calc-edit} command
31450 to edit the defining formula. Press @kbd{C-c C-c} to finish editing and
31451 store the new formula back in the definition, or kill the buffer with
31453 cancel the edit. (The argument list and other properties of the
31454 definition are unchanged; to adjust the argument list, you can use
31455 @kbd{Z G} to grab the function onto the stack, edit with @kbd{`}, and
31456 then re-execute the @kbd{Z F} command.)
31458 As usual, the @kbd{Z P} command records your definition permanently.
31459 In this case it will permanently record all three of the relevant
31460 definitions: the key, the command, and the function.
31462 You may find it useful to turn off the default simplifications with
31463 @kbd{m O} (@code{calc-no-simplify-mode}) when entering a formula to be
31464 used as a function definition. For example, the formula @samp{deriv(a^2,v)}
31465 which might be used to define a new function @samp{dsqr(a,v)} will be
31466 ``simplified'' to 0 immediately upon entry since @code{deriv} considers
31467 @expr{a} to be constant with respect to @expr{v}. Turning off
31468 default simplifications cures this problem: The definition will be stored
31469 in symbolic form without ever activating the @code{deriv} function. Press
31470 @kbd{m D} to turn the default simplifications back on afterwards.
31472 @node Lisp Definitions, , Algebraic Definitions, Programming
31473 @section Programming with Lisp
31476 The Calculator can be programmed quite extensively in Lisp. All you
31477 do is write a normal Lisp function definition, but with @code{defmath}
31478 in place of @code{defun}. This has the same form as @code{defun}, but it
31479 automagically replaces calls to standard Lisp functions like @code{+} and
31480 @code{zerop} with calls to the corresponding functions in Calc's own library.
31481 Thus you can write natural-looking Lisp code which operates on all of the
31482 standard Calculator data types. You can then use @kbd{Z D} if you wish to
31483 bind your new command to a @kbd{z}-prefix key sequence. The @kbd{Z E} command
31484 will not edit a Lisp-based definition.
31486 Emacs Lisp is described in the GNU Emacs Lisp Reference Manual. This section
31487 assumes a familiarity with Lisp programming concepts; if you do not know
31488 Lisp, you may find keyboard macros or rewrite rules to be an easier way
31489 to program the Calculator.
31491 This section first discusses ways to write commands, functions, or
31492 small programs to be executed inside of Calc. Then it discusses how
31493 your own separate programs are able to call Calc from the outside.
31494 Finally, there is a list of internal Calc functions and data structures
31495 for the true Lisp enthusiast.
31498 * Defining Functions::
31499 * Defining Simple Commands::
31500 * Defining Stack Commands::
31501 * Argument Qualifiers::
31502 * Example Definitions::
31504 * Calling Calc from Your Programs::
31508 @node Defining Functions, Defining Simple Commands, Lisp Definitions, Lisp Definitions
31509 @subsection Defining New Functions
31513 The @code{defmath} function (actually a Lisp macro) is like @code{defun}
31514 except that code in the body of the definition can make use of the full
31515 range of Calculator data types. The prefix @samp{calcFunc-} is added
31516 to the specified name to get the actual Lisp function name. As a simple
31520 (defmath myfact (n)
31522 (* n (myfact (1- n)))
31527 This actually expands to the code,
31530 (defun calcFunc-myfact (n)
31532 (math-mul n (calcFunc-myfact (math-add n -1)))
31537 This function can be used in algebraic expressions, e.g., @samp{myfact(5)}.
31539 The @samp{myfact} function as it is defined above has the bug that an
31540 expression @samp{myfact(a+b)} will be simplified to 1 because the
31541 formula @samp{a+b} is not considered to be @code{posp}. A robust
31542 factorial function would be written along the following lines:
31545 (defmath myfact (n)
31547 (* n (myfact (1- n)))
31550 nil))) ; this could be simplified as: (and (= n 0) 1)
31553 If a function returns @code{nil}, it is left unsimplified by the Calculator
31554 (except that its arguments will be simplified). Thus, @samp{myfact(a+1+2)}
31555 will be simplified to @samp{myfact(a+3)} but no further. Beware that every
31556 time the Calculator reexamines this formula it will attempt to resimplify
31557 it, so your function ought to detect the returning-@code{nil} case as
31558 efficiently as possible.
31560 The following standard Lisp functions are treated by @code{defmath}:
31561 @code{+}, @code{-}, @code{*}, @code{/}, @code{%}, @code{^} or
31562 @code{expt}, @code{=}, @code{<}, @code{>}, @code{<=}, @code{>=},
31563 @code{/=}, @code{1+}, @code{1-}, @code{logand}, @code{logior}, @code{logxor},
31564 @code{logandc2}, @code{lognot}. Also, @code{~=} is an abbreviation for
31565 @code{math-nearly-equal}, which is useful in implementing Taylor series.
31567 For other functions @var{func}, if a function by the name
31568 @samp{calcFunc-@var{func}} exists it is used, otherwise if a function by the
31569 name @samp{math-@var{func}} exists it is used, otherwise if @var{func} itself
31570 is defined as a function it is used, otherwise @samp{calcFunc-@var{func}} is
31571 used on the assumption that this is a to-be-defined math function. Also, if
31572 the function name is quoted as in @samp{('integerp a)} the function name is
31573 always used exactly as written (but not quoted).
31575 Variable names have @samp{var-} prepended to them unless they appear in
31576 the function's argument list or in an enclosing @code{let}, @code{let*},
31577 @code{for}, or @code{foreach} form,
31578 or their names already contain a @samp{-} character. Thus a reference to
31579 @samp{foo} is the same as a reference to @samp{var-foo}.
31581 A few other Lisp extensions are available in @code{defmath} definitions:
31585 The @code{elt} function accepts any number of index variables.
31586 Note that Calc vectors are stored as Lisp lists whose first
31587 element is the symbol @code{vec}; thus, @samp{(elt v 2)} yields
31588 the second element of vector @code{v}, and @samp{(elt m i j)}
31589 yields one element of a Calc matrix.
31592 The @code{setq} function has been extended to act like the Common
31593 Lisp @code{setf} function. (The name @code{setf} is recognized as
31594 a synonym of @code{setq}.) Specifically, the first argument of
31595 @code{setq} can be an @code{nth}, @code{elt}, @code{car}, or @code{cdr} form,
31596 in which case the effect is to store into the specified
31597 element of a list. Thus, @samp{(setq (elt m i j) x)} stores @expr{x}
31598 into one element of a matrix.
31601 A @code{for} looping construct is available. For example,
31602 @samp{(for ((i 0 10)) body)} executes @code{body} once for each
31603 binding of @expr{i} from zero to 10. This is like a @code{let}
31604 form in that @expr{i} is temporarily bound to the loop count
31605 without disturbing its value outside the @code{for} construct.
31606 Nested loops, as in @samp{(for ((i 0 10) (j 0 (1- i) 2)) body)},
31607 are also available. For each value of @expr{i} from zero to 10,
31608 @expr{j} counts from 0 to @expr{i-1} in steps of two. Note that
31609 @code{for} has the same general outline as @code{let*}, except
31610 that each element of the header is a list of three or four
31611 things, not just two.
31614 The @code{foreach} construct loops over elements of a list.
31615 For example, @samp{(foreach ((x (cdr v))) body)} executes
31616 @code{body} with @expr{x} bound to each element of Calc vector
31617 @expr{v} in turn. The purpose of @code{cdr} here is to skip over
31618 the initial @code{vec} symbol in the vector.
31621 The @code{break} function breaks out of the innermost enclosing
31622 @code{while}, @code{for}, or @code{foreach} loop. If given a
31623 value, as in @samp{(break x)}, this value is returned by the
31624 loop. (Lisp loops otherwise always return @code{nil}.)
31627 The @code{return} function prematurely returns from the enclosing
31628 function. For example, @samp{(return (+ x y))} returns @expr{x+y}
31629 as the value of a function. You can use @code{return} anywhere
31630 inside the body of the function.
31633 Non-integer numbers (and extremely large integers) cannot be included
31634 directly into a @code{defmath} definition. This is because the Lisp
31635 reader will fail to parse them long before @code{defmath} ever gets control.
31636 Instead, use the notation, @samp{:"3.1415"}. In fact, any algebraic
31637 formula can go between the quotes. For example,
31640 (defmath sqexp (x) ; sqexp(x) == sqrt(exp(x)) == exp(x*0.5)
31648 (defun calcFunc-sqexp (x)
31649 (and (math-numberp x)
31650 (calcFunc-exp (math-mul x '(float 5 -1)))))
31653 Note the use of @code{numberp} as a guard to ensure that the argument is
31654 a number first, returning @code{nil} if not. The exponential function
31655 could itself have been included in the expression, if we had preferred:
31656 @samp{:"exp(x * 0.5)"}. As another example, the multiplication-and-recursion
31657 step of @code{myfact} could have been written
31663 A good place to put your @code{defmath} commands is your Calc init file
31664 (the file given by @code{calc-settings-file}, typically
31665 @file{~/.calc.el}), which will not be loaded until Calc starts.
31666 If a file named @file{.emacs} exists in your home directory, Emacs reads
31667 and executes the Lisp forms in this file as it starts up. While it may
31668 seem reasonable to put your favorite @code{defmath} commands there,
31669 this has the unfortunate side-effect that parts of the Calculator must be
31670 loaded in to process the @code{defmath} commands whether or not you will
31671 actually use the Calculator! If you want to put the @code{defmath}
31672 commands there (for example, if you redefine @code{calc-settings-file}
31673 to be @file{.emacs}), a better effect can be had by writing
31676 (put 'calc-define 'thing '(progn
31683 @vindex calc-define
31684 The @code{put} function adds a @dfn{property} to a symbol. Each Lisp
31685 symbol has a list of properties associated with it. Here we add a
31686 property with a name of @code{thing} and a @samp{(progn ...)} form as
31687 its value. When Calc starts up, and at the start of every Calc command,
31688 the property list for the symbol @code{calc-define} is checked and the
31689 values of any properties found are evaluated as Lisp forms. The
31690 properties are removed as they are evaluated. The property names
31691 (like @code{thing}) are not used; you should choose something like the
31692 name of your project so as not to conflict with other properties.
31694 The net effect is that you can put the above code in your @file{.emacs}
31695 file and it will not be executed until Calc is loaded. Or, you can put
31696 that same code in another file which you load by hand either before or
31697 after Calc itself is loaded.
31699 The properties of @code{calc-define} are evaluated in the same order
31700 that they were added. They can assume that the Calc modules @file{calc.el},
31701 @file{calc-ext.el}, and @file{calc-macs.el} have been fully loaded, and
31702 that the @samp{*Calculator*} buffer will be the current buffer.
31704 If your @code{calc-define} property only defines algebraic functions,
31705 you can be sure that it will have been evaluated before Calc tries to
31706 call your function, even if the file defining the property is loaded
31707 after Calc is loaded. But if the property defines commands or key
31708 sequences, it may not be evaluated soon enough. (Suppose it defines the
31709 new command @code{tweak-calc}; the user can load your file, then type
31710 @kbd{M-x tweak-calc} before Calc has had chance to do anything.) To
31711 protect against this situation, you can put
31714 (run-hooks 'calc-check-defines)
31717 @findex calc-check-defines
31719 at the end of your file. The @code{calc-check-defines} function is what
31720 looks for and evaluates properties on @code{calc-define}; @code{run-hooks}
31721 has the advantage that it is quietly ignored if @code{calc-check-defines}
31722 is not yet defined because Calc has not yet been loaded.
31724 Examples of things that ought to be enclosed in a @code{calc-define}
31725 property are @code{defmath} calls, @code{define-key} calls that modify
31726 the Calc key map, and any calls that redefine things defined inside Calc.
31727 Ordinary @code{defun}s need not be enclosed with @code{calc-define}.
31729 @node Defining Simple Commands, Defining Stack Commands, Defining Functions, Lisp Definitions
31730 @subsection Defining New Simple Commands
31733 @findex interactive
31734 If a @code{defmath} form contains an @code{interactive} clause, it defines
31735 a Calculator command. Actually such a @code{defmath} results in @emph{two}
31736 function definitions: One, a @samp{calcFunc-} function as was just described,
31737 with the @code{interactive} clause removed. Two, a @samp{calc-} function
31738 with a suitable @code{interactive} clause and some sort of wrapper to make
31739 the command work in the Calc environment.
31741 In the simple case, the @code{interactive} clause has the same form as
31742 for normal Emacs Lisp commands:
31745 (defmath increase-precision (delta)
31746 "Increase precision by DELTA." ; This is the "documentation string"
31747 (interactive "p") ; Register this as a M-x-able command
31748 (setq calc-internal-prec (+ calc-internal-prec delta)))
31751 This expands to the pair of definitions,
31754 (defun calc-increase-precision (delta)
31755 "Increase precision by DELTA."
31758 (setq calc-internal-prec (math-add calc-internal-prec delta))))
31760 (defun calcFunc-increase-precision (delta)
31761 "Increase precision by DELTA."
31762 (setq calc-internal-prec (math-add calc-internal-prec delta)))
31766 where in this case the latter function would never really be used! Note
31767 that since the Calculator stores small integers as plain Lisp integers,
31768 the @code{math-add} function will work just as well as the native
31769 @code{+} even when the intent is to operate on native Lisp integers.
31771 @findex calc-wrapper
31772 The @samp{calc-wrapper} call invokes a macro which surrounds the body of
31773 the function with code that looks roughly like this:
31776 (let ((calc-command-flags nil))
31779 (calc-select-buffer)
31780 @emph{body of function}
31781 @emph{renumber stack}
31782 @emph{clear} Working @emph{message})
31783 @emph{realign cursor and window}
31784 @emph{clear Inverse, Hyperbolic, and Keep Args flags}
31785 @emph{update Emacs mode line}))
31788 @findex calc-select-buffer
31789 The @code{calc-select-buffer} function selects the @samp{*Calculator*}
31790 buffer if necessary, say, because the command was invoked from inside
31791 the @samp{*Calc Trail*} window.
31793 @findex calc-set-command-flag
31794 You can call, for example, @code{(calc-set-command-flag 'no-align)} to
31795 set the above-mentioned command flags. Calc routines recognize the
31796 following command flags:
31800 Stack line numbers @samp{1:}, @samp{2:}, and so on must be renumbered
31801 after this command completes. This is set by routines like
31804 @item clear-message
31805 Calc should call @samp{(message "")} if this command completes normally
31806 (to clear a ``Working@dots{}'' message out of the echo area).
31809 Do not move the cursor back to the @samp{.} top-of-stack marker.
31811 @item position-point
31812 Use the variables @code{calc-position-point-line} and
31813 @code{calc-position-point-column} to position the cursor after
31814 this command finishes.
31817 Do not clear @code{calc-inverse-flag}, @code{calc-hyperbolic-flag},
31818 and @code{calc-keep-args-flag} at the end of this command.
31821 Switch to buffer @samp{*Calc Edit*} after this command.
31824 Do not move trail pointer to end of trail when something is recorded
31830 @vindex calc-Y-help-msgs
31831 Calc reserves a special prefix key, shift-@kbd{Y}, for user-written
31832 extensions to Calc. There are no built-in commands that work with
31833 this prefix key; you must call @code{define-key} from Lisp (probably
31834 from inside a @code{calc-define} property) to add to it. Initially only
31835 @kbd{Y ?} is defined; it takes help messages from a list of strings
31836 (initially @code{nil}) in the variable @code{calc-Y-help-msgs}. All
31837 other undefined keys except for @kbd{Y} are reserved for use by
31838 future versions of Calc.
31840 If you are writing a Calc enhancement which you expect to give to
31841 others, it is best to minimize the number of @kbd{Y}-key sequences
31842 you use. In fact, if you have more than one key sequence you should
31843 consider defining three-key sequences with a @kbd{Y}, then a key that
31844 stands for your package, then a third key for the particular command
31845 within your package.
31847 Users may wish to install several Calc enhancements, and it is possible
31848 that several enhancements will choose to use the same key. In the
31849 example below, a variable @code{inc-prec-base-key} has been defined
31850 to contain the key that identifies the @code{inc-prec} package. Its
31851 value is initially @code{"P"}, but a user can change this variable
31852 if necessary without having to modify the file.
31854 Here is a complete file, @file{inc-prec.el}, which makes a @kbd{Y P I}
31855 command that increases the precision, and a @kbd{Y P D} command that
31856 decreases the precision.
31859 ;;; Increase and decrease Calc precision. Dave Gillespie, 5/31/91.
31860 ;; (Include copyright or copyleft stuff here.)
31862 (defvar inc-prec-base-key "P"
31863 "Base key for inc-prec.el commands.")
31865 (put 'calc-define 'inc-prec '(progn
31867 (define-key calc-mode-map (format "Y%sI" inc-prec-base-key)
31868 'increase-precision)
31869 (define-key calc-mode-map (format "Y%sD" inc-prec-base-key)
31870 'decrease-precision)
31872 (setq calc-Y-help-msgs
31873 (cons (format "%s + Inc-prec, Dec-prec" inc-prec-base-key)
31876 (defmath increase-precision (delta)
31877 "Increase precision by DELTA."
31879 (setq calc-internal-prec (+ calc-internal-prec delta)))
31881 (defmath decrease-precision (delta)
31882 "Decrease precision by DELTA."
31884 (setq calc-internal-prec (- calc-internal-prec delta)))
31886 )) ; end of calc-define property
31888 (run-hooks 'calc-check-defines)
31891 @node Defining Stack Commands, Argument Qualifiers, Defining Simple Commands, Lisp Definitions
31892 @subsection Defining New Stack-Based Commands
31895 To define a new computational command which takes and/or leaves arguments
31896 on the stack, a special form of @code{interactive} clause is used.
31899 (interactive @var{num} @var{tag})
31903 where @var{num} is an integer, and @var{tag} is a string. The effect is
31904 to pop @var{num} values off the stack, resimplify them by calling
31905 @code{calc-normalize}, and hand them to your function according to the
31906 function's argument list. Your function may include @code{&optional} and
31907 @code{&rest} parameters, so long as calling the function with @var{num}
31908 parameters is valid.
31910 Your function must return either a number or a formula in a form
31911 acceptable to Calc, or a list of such numbers or formulas. These value(s)
31912 are pushed onto the stack when the function completes. They are also
31913 recorded in the Calc Trail buffer on a line beginning with @var{tag},
31914 a string of (normally) four characters or less. If you omit @var{tag}
31915 or use @code{nil} as a tag, the result is not recorded in the trail.
31917 As an example, the definition
31920 (defmath myfact (n)
31921 "Compute the factorial of the integer at the top of the stack."
31922 (interactive 1 "fact")
31924 (* n (myfact (1- n)))
31929 is a version of the factorial function shown previously which can be used
31930 as a command as well as an algebraic function. It expands to
31933 (defun calc-myfact ()
31934 "Compute the factorial of the integer at the top of the stack."
31937 (calc-enter-result 1 "fact"
31938 (cons 'calcFunc-myfact (calc-top-list-n 1)))))
31940 (defun calcFunc-myfact (n)
31941 "Compute the factorial of the integer at the top of the stack."
31943 (math-mul n (calcFunc-myfact (math-add n -1)))
31944 (and (math-zerop n) 1)))
31947 @findex calc-slow-wrapper
31948 The @code{calc-slow-wrapper} function is a version of @code{calc-wrapper}
31949 that automatically puts up a @samp{Working...} message before the
31950 computation begins. (This message can be turned off by the user
31951 with an @kbd{m w} (@code{calc-working}) command.)
31953 @findex calc-top-list-n
31954 The @code{calc-top-list-n} function returns a list of the specified number
31955 of values from the top of the stack. It resimplifies each value by
31956 calling @code{calc-normalize}. If its argument is zero it returns an
31957 empty list. It does not actually remove these values from the stack.
31959 @findex calc-enter-result
31960 The @code{calc-enter-result} function takes an integer @var{num} and string
31961 @var{tag} as described above, plus a third argument which is either a
31962 Calculator data object or a list of such objects. These objects are
31963 resimplified and pushed onto the stack after popping the specified number
31964 of values from the stack. If @var{tag} is non-@code{nil}, the values
31965 being pushed are also recorded in the trail.
31967 Note that if @code{calcFunc-myfact} returns @code{nil} this represents
31968 ``leave the function in symbolic form.'' To return an actual empty list,
31969 in the sense that @code{calc-enter-result} will push zero elements back
31970 onto the stack, you should return the special value @samp{'(nil)}, a list
31971 containing the single symbol @code{nil}.
31973 The @code{interactive} declaration can actually contain a limited
31974 Emacs-style code string as well which comes just before @var{num} and
31975 @var{tag}. Currently the only Emacs code supported is @samp{"p"}, as in
31978 (defmath foo (a b &optional c)
31979 (interactive "p" 2 "foo")
31983 In this example, the command @code{calc-foo} will evaluate the expression
31984 @samp{foo(a,b)} if executed with no argument, or @samp{foo(a,b,n)} if
31985 executed with a numeric prefix argument of @expr{n}.
31987 The other code string allowed is @samp{"m"} (unrelated to the usual @samp{"m"}
31988 code as used with @code{defun}). It uses the numeric prefix argument as the
31989 number of objects to remove from the stack and pass to the function.
31990 In this case, the integer @var{num} serves as a default number of
31991 arguments to be used when no prefix is supplied.
31993 @node Argument Qualifiers, Example Definitions, Defining Stack Commands, Lisp Definitions
31994 @subsection Argument Qualifiers
31997 Anywhere a parameter name can appear in the parameter list you can also use
31998 an @dfn{argument qualifier}. Thus the general form of a definition is:
32001 (defmath @var{name} (@var{param} @var{param...}
32002 &optional @var{param} @var{param...}
32008 where each @var{param} is either a symbol or a list of the form
32011 (@var{qual} @var{param})
32014 The following qualifiers are recognized:
32019 The argument must not be an incomplete vector, interval, or complex number.
32020 (This is rarely needed since the Calculator itself will never call your
32021 function with an incomplete argument. But there is nothing stopping your
32022 own Lisp code from calling your function with an incomplete argument.)
32026 The argument must be an integer. If it is an integer-valued float
32027 it will be accepted but converted to integer form. Non-integers and
32028 formulas are rejected.
32032 Like @samp{integer}, but the argument must be non-negative.
32036 Like @samp{integer}, but the argument must fit into a native Lisp integer,
32037 which on most systems means less than 2^23 in absolute value. The
32038 argument is converted into Lisp-integer form if necessary.
32042 The argument is converted to floating-point format if it is a number or
32043 vector. If it is a formula it is left alone. (The argument is never
32044 actually rejected by this qualifier.)
32047 The argument must satisfy predicate @var{pred}, which is one of the
32048 standard Calculator predicates. @xref{Predicates}.
32050 @item not-@var{pred}
32051 The argument must @emph{not} satisfy predicate @var{pred}.
32057 (defmath foo (a (constp (not-matrixp b)) &optional (float c)
32066 (defun calcFunc-foo (a b &optional c &rest d)
32067 (and (math-matrixp b)
32068 (math-reject-arg b 'not-matrixp))
32069 (or (math-constp b)
32070 (math-reject-arg b 'constp))
32071 (and c (setq c (math-check-float c)))
32072 (setq d (mapcar 'math-check-integer d))
32077 which performs the necessary checks and conversions before executing the
32078 body of the function.
32080 @node Example Definitions, Calling Calc from Your Programs, Argument Qualifiers, Lisp Definitions
32081 @subsection Example Definitions
32084 This section includes some Lisp programming examples on a larger scale.
32085 These programs make use of some of the Calculator's internal functions;
32089 * Bit Counting Example::
32093 @node Bit Counting Example, Sine Example, Example Definitions, Example Definitions
32094 @subsubsection Bit-Counting
32101 Calc does not include a built-in function for counting the number of
32102 ``one'' bits in a binary integer. It's easy to invent one using @kbd{b u}
32103 to convert the integer to a set, and @kbd{V #} to count the elements of
32104 that set; let's write a function that counts the bits without having to
32105 create an intermediate set.
32108 (defmath bcount ((natnum n))
32109 (interactive 1 "bcnt")
32113 (setq count (1+ count)))
32114 (setq n (lsh n -1)))
32119 When this is expanded by @code{defmath}, it will become the following
32120 Emacs Lisp function:
32123 (defun calcFunc-bcount (n)
32124 (setq n (math-check-natnum n))
32126 (while (math-posp n)
32128 (setq count (math-add count 1)))
32129 (setq n (calcFunc-lsh n -1)))
32133 If the input numbers are large, this function involves a fair amount
32134 of arithmetic. A binary right shift is essentially a division by two;
32135 recall that Calc stores integers in decimal form so bit shifts must
32136 involve actual division.
32138 To gain a bit more efficiency, we could divide the integer into
32139 @var{n}-bit chunks, each of which can be handled quickly because
32140 they fit into Lisp integers. It turns out that Calc's arithmetic
32141 routines are especially fast when dividing by an integer less than
32142 1000, so we can set @var{n = 9} bits and use repeated division by 512:
32145 (defmath bcount ((natnum n))
32146 (interactive 1 "bcnt")
32148 (while (not (fixnump n))
32149 (let ((qr (idivmod n 512)))
32150 (setq count (+ count (bcount-fixnum (cdr qr)))
32152 (+ count (bcount-fixnum n))))
32154 (defun bcount-fixnum (n)
32157 (setq count (+ count (logand n 1))
32163 Note that the second function uses @code{defun}, not @code{defmath}.
32164 Because this function deals only with native Lisp integers (``fixnums''),
32165 it can use the actual Emacs @code{+} and related functions rather
32166 than the slower but more general Calc equivalents which @code{defmath}
32169 The @code{idivmod} function does an integer division, returning both
32170 the quotient and the remainder at once. Again, note that while it
32171 might seem that @samp{(logand n 511)} and @samp{(lsh n -9)} are
32172 more efficient ways to split off the bottom nine bits of @code{n},
32173 actually they are less efficient because each operation is really
32174 a division by 512 in disguise; @code{idivmod} allows us to do the
32175 same thing with a single division by 512.
32177 @node Sine Example, , Bit Counting Example, Example Definitions
32178 @subsubsection The Sine Function
32185 A somewhat limited sine function could be defined as follows, using the
32186 well-known Taylor series expansion for
32187 @texline @math{\sin x}:
32188 @infoline @samp{sin(x)}:
32191 (defmath mysin ((float (anglep x)))
32192 (interactive 1 "mysn")
32193 (setq x (to-radians x)) ; Convert from current angular mode.
32194 (let ((sum x) ; Initial term of Taylor expansion of sin.
32196 (nfact 1) ; "nfact" equals "n" factorial at all times.
32197 (xnegsqr :"-(x^2)")) ; "xnegsqr" equals -x^2.
32198 (for ((n 3 100 2)) ; Upper limit of 100 is a good precaution.
32199 (working "mysin" sum) ; Display "Working" message, if enabled.
32200 (setq nfact (* nfact (1- n) n)
32202 newsum (+ sum (/ x nfact)))
32203 (if (~= newsum sum) ; If newsum is "nearly equal to" sum,
32204 (break)) ; then we are done.
32209 The actual @code{sin} function in Calc works by first reducing the problem
32210 to a sine or cosine of a nonnegative number less than @cpiover{4}. This
32211 ensures that the Taylor series will converge quickly. Also, the calculation
32212 is carried out with two extra digits of precision to guard against cumulative
32213 round-off in @samp{sum}. Finally, complex arguments are allowed and handled
32214 by a separate algorithm.
32217 (defmath mysin ((float (scalarp x)))
32218 (interactive 1 "mysn")
32219 (setq x (to-radians x)) ; Convert from current angular mode.
32220 (with-extra-prec 2 ; Evaluate with extra precision.
32221 (cond ((complexp x)
32224 (- (mysin-raw (- x))) ; Always call mysin-raw with x >= 0.
32225 (t (mysin-raw x))))))
32227 (defmath mysin-raw (x)
32229 (mysin-raw (% x (two-pi)))) ; Now x < 7.
32231 (- (mysin-raw (- x (pi))))) ; Now -pi/2 <= x <= pi/2.
32233 (mycos-raw (- x (pi-over-2)))) ; Now -pi/2 <= x <= pi/4.
32234 ((< x (- (pi-over-4)))
32235 (- (mycos-raw (+ x (pi-over-2))))) ; Now -pi/4 <= x <= pi/4,
32236 (t (mysin-series x)))) ; so the series will be efficient.
32240 where @code{mysin-complex} is an appropriate function to handle complex
32241 numbers, @code{mysin-series} is the routine to compute the sine Taylor
32242 series as before, and @code{mycos-raw} is a function analogous to
32243 @code{mysin-raw} for cosines.
32245 The strategy is to ensure that @expr{x} is nonnegative before calling
32246 @code{mysin-raw}. This function then recursively reduces its argument
32247 to a suitable range, namely, plus-or-minus @cpiover{4}. Note that each
32248 test, and particularly the first comparison against 7, is designed so
32249 that small roundoff errors cannot produce an infinite loop. (Suppose
32250 we compared with @samp{(two-pi)} instead; if due to roundoff problems
32251 the modulo operator ever returned @samp{(two-pi)} exactly, an infinite
32252 recursion could result!) We use modulo only for arguments that will
32253 clearly get reduced, knowing that the next rule will catch any reductions
32254 that this rule misses.
32256 If a program is being written for general use, it is important to code
32257 it carefully as shown in this second example. For quick-and-dirty programs,
32258 when you know that your own use of the sine function will never encounter
32259 a large argument, a simpler program like the first one shown is fine.
32261 @node Calling Calc from Your Programs, Internals, Example Definitions, Lisp Definitions
32262 @subsection Calling Calc from Your Lisp Programs
32265 A later section (@pxref{Internals}) gives a full description of
32266 Calc's internal Lisp functions. It's not hard to call Calc from
32267 inside your programs, but the number of these functions can be daunting.
32268 So Calc provides one special ``programmer-friendly'' function called
32269 @code{calc-eval} that can be made to do just about everything you
32270 need. It's not as fast as the low-level Calc functions, but it's
32271 much simpler to use!
32273 It may seem that @code{calc-eval} itself has a daunting number of
32274 options, but they all stem from one simple operation.
32276 In its simplest manifestation, @samp{(calc-eval "1+2")} parses the
32277 string @code{"1+2"} as if it were a Calc algebraic entry and returns
32278 the result formatted as a string: @code{"3"}.
32280 Since @code{calc-eval} is on the list of recommended @code{autoload}
32281 functions, you don't need to make any special preparations to load
32282 Calc before calling @code{calc-eval} the first time. Calc will be
32283 loaded and initialized for you.
32285 All the Calc modes that are currently in effect will be used when
32286 evaluating the expression and formatting the result.
32293 @subsubsection Additional Arguments to @code{calc-eval}
32296 If the input string parses to a list of expressions, Calc returns
32297 the results separated by @code{", "}. You can specify a different
32298 separator by giving a second string argument to @code{calc-eval}:
32299 @samp{(calc-eval "1+2,3+4" ";")} returns @code{"3;7"}.
32301 The ``separator'' can also be any of several Lisp symbols which
32302 request other behaviors from @code{calc-eval}. These are discussed
32305 You can give additional arguments to be substituted for
32306 @samp{$}, @samp{$$}, and so on in the main expression. For
32307 example, @samp{(calc-eval "$/$$" nil "7" "1+1")} evaluates the
32308 expression @code{"7/(1+1)"} to yield the result @code{"3.5"}
32309 (assuming Fraction mode is not in effect). Note the @code{nil}
32310 used as a placeholder for the item-separator argument.
32317 @subsubsection Error Handling
32320 If @code{calc-eval} encounters an error, it returns a list containing
32321 the character position of the error, plus a suitable message as a
32322 string. Note that @samp{1 / 0} is @emph{not} an error by Calc's
32323 standards; it simply returns the string @code{"1 / 0"} which is the
32324 division left in symbolic form. But @samp{(calc-eval "1/")} will
32325 return the list @samp{(2 "Expected a number")}.
32327 If you bind the variable @code{calc-eval-error} to @code{t}
32328 using a @code{let} form surrounding the call to @code{calc-eval},
32329 errors instead call the Emacs @code{error} function which aborts
32330 to the Emacs command loop with a beep and an error message.
32332 If you bind this variable to the symbol @code{string}, error messages
32333 are returned as strings instead of lists. The character position is
32336 As a courtesy to other Lisp code which may be using Calc, be sure
32337 to bind @code{calc-eval-error} using @code{let} rather than changing
32338 it permanently with @code{setq}.
32345 @subsubsection Numbers Only
32348 Sometimes it is preferable to treat @samp{1 / 0} as an error
32349 rather than returning a symbolic result. If you pass the symbol
32350 @code{num} as the second argument to @code{calc-eval}, results
32351 that are not constants are treated as errors. The error message
32352 reported is the first @code{calc-why} message if there is one,
32353 or otherwise ``Number expected.''
32355 A result is ``constant'' if it is a number, vector, or other
32356 object that does not include variables or function calls. If it
32357 is a vector, the components must themselves be constants.
32364 @subsubsection Default Modes
32367 If the first argument to @code{calc-eval} is a list whose first
32368 element is a formula string, then @code{calc-eval} sets all the
32369 various Calc modes to their default values while the formula is
32370 evaluated and formatted. For example, the precision is set to 12
32371 digits, digit grouping is turned off, and the Normal language
32374 This same principle applies to the other options discussed below.
32375 If the first argument would normally be @var{x}, then it can also
32376 be the list @samp{(@var{x})} to use the default mode settings.
32378 If there are other elements in the list, they are taken as
32379 variable-name/value pairs which override the default mode
32380 settings. Look at the documentation at the front of the
32381 @file{calc.el} file to find the names of the Lisp variables for
32382 the various modes. The mode settings are restored to their
32383 original values when @code{calc-eval} is done.
32385 For example, @samp{(calc-eval '("$+$$" calc-internal-prec 8) 'num a b)}
32386 computes the sum of two numbers, requiring a numeric result, and
32387 using default mode settings except that the precision is 8 instead
32388 of the default of 12.
32390 It's usually best to use this form of @code{calc-eval} unless your
32391 program actually considers the interaction with Calc's mode settings
32392 to be a feature. This will avoid all sorts of potential ``gotchas'';
32393 consider what happens with @samp{(calc-eval "sqrt(2)" 'num)}
32394 when the user has left Calc in Symbolic mode or No-Simplify mode.
32396 As another example, @samp{(equal (calc-eval '("$<$$") nil a b) "1")}
32397 checks if the number in string @expr{a} is less than the one in
32398 string @expr{b}. Without using a list, the integer 1 might
32399 come out in a variety of formats which would be hard to test for
32400 conveniently: @code{"1"}, @code{"8#1"}, @code{"00001"}. (But
32401 see ``Predicates'' mode, below.)
32408 @subsubsection Raw Numbers
32411 Normally all input and output for @code{calc-eval} is done with strings.
32412 You can do arithmetic with, say, @samp{(calc-eval "$+$$" nil a b)}
32413 in place of @samp{(+ a b)}, but this is very inefficient since the
32414 numbers must be converted to and from string format as they are passed
32415 from one @code{calc-eval} to the next.
32417 If the separator is the symbol @code{raw}, the result will be returned
32418 as a raw Calc data structure rather than a string. You can read about
32419 how these objects look in the following sections, but usually you can
32420 treat them as ``black box'' objects with no important internal
32423 There is also a @code{rawnum} symbol, which is a combination of
32424 @code{raw} (returning a raw Calc object) and @code{num} (signaling
32425 an error if that object is not a constant).
32427 You can pass a raw Calc object to @code{calc-eval} in place of a
32428 string, either as the formula itself or as one of the @samp{$}
32429 arguments. Thus @samp{(calc-eval "$+$$" 'raw a b)} is an
32430 addition function that operates on raw Calc objects. Of course
32431 in this case it would be easier to call the low-level @code{math-add}
32432 function in Calc, if you can remember its name.
32434 In particular, note that a plain Lisp integer is acceptable to Calc
32435 as a raw object. (All Lisp integers are accepted on input, but
32436 integers of more than six decimal digits are converted to ``big-integer''
32437 form for output. @xref{Data Type Formats}.)
32439 When it comes time to display the object, just use @samp{(calc-eval a)}
32440 to format it as a string.
32442 It is an error if the input expression evaluates to a list of
32443 values. The separator symbol @code{list} is like @code{raw}
32444 except that it returns a list of one or more raw Calc objects.
32446 Note that a Lisp string is not a valid Calc object, nor is a list
32447 containing a string. Thus you can still safely distinguish all the
32448 various kinds of error returns discussed above.
32455 @subsubsection Predicates
32458 If the separator symbol is @code{pred}, the result of the formula is
32459 treated as a true/false value; @code{calc-eval} returns @code{t} or
32460 @code{nil}, respectively. A value is considered ``true'' if it is a
32461 non-zero number, or false if it is zero or if it is not a number.
32463 For example, @samp{(calc-eval "$<$$" 'pred a b)} tests whether
32464 one value is less than another.
32466 As usual, it is also possible for @code{calc-eval} to return one of
32467 the error indicators described above. Lisp will interpret such an
32468 indicator as ``true'' if you don't check for it explicitly. If you
32469 wish to have an error register as ``false'', use something like
32470 @samp{(eq (calc-eval ...) t)}.
32477 @subsubsection Variable Values
32480 Variables in the formula passed to @code{calc-eval} are not normally
32481 replaced by their values. If you wish this, you can use the
32482 @code{evalv} function (@pxref{Algebraic Manipulation}). For example,
32483 if 4 is stored in Calc variable @code{a} (i.e., in Lisp variable
32484 @code{var-a}), then @samp{(calc-eval "a+pi")} will return the
32485 formula @code{"a + pi"}, but @samp{(calc-eval "evalv(a+pi)")}
32486 will return @code{"7.14159265359"}.
32488 To store in a Calc variable, just use @code{setq} to store in the
32489 corresponding Lisp variable. (This is obtained by prepending
32490 @samp{var-} to the Calc variable name.) Calc routines will
32491 understand either string or raw form values stored in variables,
32492 although raw data objects are much more efficient. For example,
32493 to increment the Calc variable @code{a}:
32496 (setq var-a (calc-eval "evalv(a+1)" 'raw))
32504 @subsubsection Stack Access
32507 If the separator symbol is @code{push}, the formula argument is
32508 evaluated (with possible @samp{$} expansions, as usual). The
32509 result is pushed onto the Calc stack. The return value is @code{nil}
32510 (unless there is an error from evaluating the formula, in which
32511 case the return value depends on @code{calc-eval-error} in the
32514 If the separator symbol is @code{pop}, the first argument to
32515 @code{calc-eval} must be an integer instead of a string. That
32516 many values are popped from the stack and thrown away. A negative
32517 argument deletes the entry at that stack level. The return value
32518 is the number of elements remaining in the stack after popping;
32519 @samp{(calc-eval 0 'pop)} is a good way to measure the size of
32522 If the separator symbol is @code{top}, the first argument to
32523 @code{calc-eval} must again be an integer. The value at that
32524 stack level is formatted as a string and returned. Thus
32525 @samp{(calc-eval 1 'top)} returns the top-of-stack value. If the
32526 integer is out of range, @code{nil} is returned.
32528 The separator symbol @code{rawtop} is just like @code{top} except
32529 that the stack entry is returned as a raw Calc object instead of
32532 In all of these cases the first argument can be made a list in
32533 order to force the default mode settings, as described above.
32534 Thus @samp{(calc-eval '(2 calc-number-radix 16) 'top)} returns the
32535 second-to-top stack entry, formatted as a string using the default
32536 instead of current display modes, except that the radix is
32537 hexadecimal instead of decimal.
32539 It is, of course, polite to put the Calc stack back the way you
32540 found it when you are done, unless the user of your program is
32541 actually expecting it to affect the stack.
32543 Note that you do not actually have to switch into the @samp{*Calculator*}
32544 buffer in order to use @code{calc-eval}; it temporarily switches into
32545 the stack buffer if necessary.
32552 @subsubsection Keyboard Macros
32555 If the separator symbol is @code{macro}, the first argument must be a
32556 string of characters which Calc can execute as a sequence of keystrokes.
32557 This switches into the Calc buffer for the duration of the macro.
32558 For example, @samp{(calc-eval "vx5\rVR+" 'macro)} pushes the
32559 vector @samp{[1,2,3,4,5]} on the stack and then replaces it
32560 with the sum of those numbers. Note that @samp{\r} is the Lisp
32561 notation for the carriage-return, @key{RET}, character.
32563 If your keyboard macro wishes to pop the stack, @samp{\C-d} is
32564 safer than @samp{\177} (the @key{DEL} character) because some
32565 installations may have switched the meanings of @key{DEL} and
32566 @kbd{C-h}. Calc always interprets @kbd{C-d} as a synonym for
32567 ``pop-stack'' regardless of key mapping.
32569 If you provide a third argument to @code{calc-eval}, evaluation
32570 of the keyboard macro will leave a record in the Trail using
32571 that argument as a tag string. Normally the Trail is unaffected.
32573 The return value in this case is always @code{nil}.
32580 @subsubsection Lisp Evaluation
32583 Finally, if the separator symbol is @code{eval}, then the Lisp
32584 @code{eval} function is called on the first argument, which must
32585 be a Lisp expression rather than a Calc formula. Remember to
32586 quote the expression so that it is not evaluated until inside
32589 The difference from plain @code{eval} is that @code{calc-eval}
32590 switches to the Calc buffer before evaluating the expression.
32591 For example, @samp{(calc-eval '(setq calc-internal-prec 17) 'eval)}
32592 will correctly affect the buffer-local Calc precision variable.
32594 An alternative would be @samp{(calc-eval '(calc-precision 17) 'eval)}.
32595 This is evaluating a call to the function that is normally invoked
32596 by the @kbd{p} key, giving it 17 as its ``numeric prefix argument.''
32597 Note that this function will leave a message in the echo area as
32598 a side effect. Also, all Calc functions switch to the Calc buffer
32599 automatically if not invoked from there, so the above call is
32600 also equivalent to @samp{(calc-precision 17)} by itself.
32601 In all cases, Calc uses @code{save-excursion} to switch back to
32602 your original buffer when it is done.
32604 As usual the first argument can be a list that begins with a Lisp
32605 expression to use default instead of current mode settings.
32607 The result of @code{calc-eval} in this usage is just the result
32608 returned by the evaluated Lisp expression.
32615 @subsubsection Example
32618 @findex convert-temp
32619 Here is a sample Emacs command that uses @code{calc-eval}. Suppose
32620 you have a document with lots of references to temperatures on the
32621 Fahrenheit scale, say ``98.6 F'', and you wish to convert these
32622 references to Centigrade. The following command does this conversion.
32623 Place the Emacs cursor right after the letter ``F'' and invoke the
32624 command to change ``98.6 F'' to ``37 C''. Or, if the temperature is
32625 already in Centigrade form, the command changes it back to Fahrenheit.
32628 (defun convert-temp ()
32631 (re-search-backward "[^-.0-9]\\([-.0-9]+\\) *\\([FC]\\)")
32632 (let* ((top1 (match-beginning 1))
32633 (bot1 (match-end 1))
32634 (number (buffer-substring top1 bot1))
32635 (top2 (match-beginning 2))
32636 (bot2 (match-end 2))
32637 (type (buffer-substring top2 bot2)))
32638 (if (equal type "F")
32640 number (calc-eval "($ - 32)*5/9" nil number))
32642 number (calc-eval "$*9/5 + 32" nil number)))
32644 (delete-region top2 bot2)
32645 (insert-before-markers type)
32647 (delete-region top1 bot1)
32648 (if (string-match "\\.$" number) ; change "37." to "37"
32649 (setq number (substring number 0 -1)))
32653 Note the use of @code{insert-before-markers} when changing between
32654 ``F'' and ``C'', so that the character winds up before the cursor
32655 instead of after it.
32657 @node Internals, , Calling Calc from Your Programs, Lisp Definitions
32658 @subsection Calculator Internals
32661 This section describes the Lisp functions defined by the Calculator that
32662 may be of use to user-written Calculator programs (as described in the
32663 rest of this chapter). These functions are shown by their names as they
32664 conventionally appear in @code{defmath}. Their full Lisp names are
32665 generally gotten by prepending @samp{calcFunc-} or @samp{math-} to their
32666 apparent names. (Names that begin with @samp{calc-} are already in
32667 their full Lisp form.) You can use the actual full names instead if you
32668 prefer them, or if you are calling these functions from regular Lisp.
32670 The functions described here are scattered throughout the various
32671 Calc component files. Note that @file{calc.el} includes @code{autoload}s
32672 for only a few component files; when Calc wants to call an advanced
32673 function it calls @samp{(calc-extensions)} first; this function
32674 autoloads @file{calc-ext.el}, which in turn autoloads all the functions
32675 in the remaining component files.
32677 Because @code{defmath} itself uses the extensions, user-written code
32678 generally always executes with the extensions already loaded, so
32679 normally you can use any Calc function and be confident that it will
32680 be autoloaded for you when necessary. If you are doing something
32681 special, check carefully to make sure each function you are using is
32682 from @file{calc.el} or its components, and call @samp{(calc-extensions)}
32683 before using any function based in @file{calc-ext.el} if you can't
32684 prove this file will already be loaded.
32687 * Data Type Formats::
32688 * Interactive Lisp Functions::
32689 * Stack Lisp Functions::
32691 * Computational Lisp Functions::
32692 * Vector Lisp Functions::
32693 * Symbolic Lisp Functions::
32694 * Formatting Lisp Functions::
32698 @node Data Type Formats, Interactive Lisp Functions, Internals, Internals
32699 @subsubsection Data Type Formats
32702 Integers are stored in either of two ways, depending on their magnitude.
32703 Integers less than one million in absolute value are stored as standard
32704 Lisp integers. This is the only storage format for Calc data objects
32705 which is not a Lisp list.
32707 Large integers are stored as lists of the form @samp{(bigpos @var{d0}
32708 @var{d1} @var{d2} @dots{})} for positive integers 1000000 or more, or
32709 @samp{(bigneg @var{d0} @var{d1} @var{d2} @dots{})} for negative integers
32710 @mathit{-1000000} or less. Each @var{d} is a base-1000 ``digit,'' a Lisp integer
32711 from 0 to 999. The least significant digit is @var{d0}; the last digit,
32712 @var{dn}, which is always nonzero, is the most significant digit. For
32713 example, the integer @mathit{-12345678} is stored as @samp{(bigneg 678 345 12)}.
32715 The distinction between small and large integers is entirely hidden from
32716 the user. In @code{defmath} definitions, the Lisp predicate @code{integerp}
32717 returns true for either kind of integer, and in general both big and small
32718 integers are accepted anywhere the word ``integer'' is used in this manual.
32719 If the distinction must be made, native Lisp integers are called @dfn{fixnums}
32720 and large integers are called @dfn{bignums}.
32722 Fractions are stored as a list of the form, @samp{(frac @var{n} @var{d})}
32723 where @var{n} is an integer (big or small) numerator, @var{d} is an
32724 integer denominator greater than one, and @var{n} and @var{d} are relatively
32725 prime. Note that fractions where @var{d} is one are automatically converted
32726 to plain integers by all math routines; fractions where @var{d} is negative
32727 are normalized by negating the numerator and denominator.
32729 Floating-point numbers are stored in the form, @samp{(float @var{mant}
32730 @var{exp})}, where @var{mant} (the ``mantissa'') is an integer less than
32731 @samp{10^@var{p}} in absolute value (@var{p} represents the current
32732 precision), and @var{exp} (the ``exponent'') is a fixnum. The value of
32733 the float is @samp{@var{mant} * 10^@var{exp}}. For example, the number
32734 @mathit{-3.14} is stored as @samp{(float -314 -2) = -314*10^-2}. Other constraints
32735 are that the number 0.0 is always stored as @samp{(float 0 0)}, and,
32736 except for the 0.0 case, the rightmost base-10 digit of @var{mant} is
32737 always nonzero. (If the rightmost digit is zero, the number is
32738 rearranged by dividing @var{mant} by ten and incrementing @var{exp}.)
32740 Rectangular complex numbers are stored in the form @samp{(cplx @var{re}
32741 @var{im})}, where @var{re} and @var{im} are each real numbers, either
32742 integers, fractions, or floats. The value is @samp{@var{re} + @var{im}i}.
32743 The @var{im} part is nonzero; complex numbers with zero imaginary
32744 components are converted to real numbers automatically.
32746 Polar complex numbers are stored in the form @samp{(polar @var{r}
32747 @var{theta})}, where @var{r} is a positive real value and @var{theta}
32748 is a real value or HMS form representing an angle. This angle is
32749 usually normalized to lie in the interval @samp{(-180 ..@: 180)} degrees,
32750 or @samp{(-pi ..@: pi)} radians, according to the current angular mode.
32751 If the angle is 0 the value is converted to a real number automatically.
32752 (If the angle is 180 degrees, the value is usually also converted to a
32753 negative real number.)
32755 Hours-minutes-seconds forms are stored as @samp{(hms @var{h} @var{m}
32756 @var{s})}, where @var{h} is an integer or an integer-valued float (i.e.,
32757 a float with @samp{@var{exp} >= 0}), @var{m} is an integer or integer-valued
32758 float in the range @w{@samp{[0 ..@: 60)}}, and @var{s} is any real number
32759 in the range @samp{[0 ..@: 60)}.
32761 Date forms are stored as @samp{(date @var{n})}, where @var{n} is
32762 a real number that counts days since midnight on the morning of
32763 January 1, 1 AD. If @var{n} is an integer, this is a pure date
32764 form. If @var{n} is a fraction or float, this is a date/time form.
32766 Modulo forms are stored as @samp{(mod @var{n} @var{m})}, where @var{m} is a
32767 positive real number or HMS form, and @var{n} is a real number or HMS
32768 form in the range @samp{[0 ..@: @var{m})}.
32770 Error forms are stored as @samp{(sdev @var{x} @var{sigma})}, where @var{x}
32771 is the mean value and @var{sigma} is the standard deviation. Each
32772 component is either a number, an HMS form, or a symbolic object
32773 (a variable or function call). If @var{sigma} is zero, the value is
32774 converted to a plain real number. If @var{sigma} is negative or
32775 complex, it is automatically normalized to be a positive real.
32777 Interval forms are stored as @samp{(intv @var{mask} @var{lo} @var{hi})},
32778 where @var{mask} is one of the integers 0, 1, 2, or 3, and @var{lo} and
32779 @var{hi} are real numbers, HMS forms, or symbolic objects. The @var{mask}
32780 is a binary integer where 1 represents the fact that the interval is
32781 closed on the high end, and 2 represents the fact that it is closed on
32782 the low end. (Thus 3 represents a fully closed interval.) The interval
32783 @w{@samp{(intv 3 @var{x} @var{x})}} is converted to the plain number @var{x};
32784 intervals @samp{(intv @var{mask} @var{x} @var{x})} for any other @var{mask}
32785 represent empty intervals. If @var{hi} is less than @var{lo}, the interval
32786 is converted to a standard empty interval by replacing @var{hi} with @var{lo}.
32788 Vectors are stored as @samp{(vec @var{v1} @var{v2} @dots{})}, where @var{v1}
32789 is the first element of the vector, @var{v2} is the second, and so on.
32790 An empty vector is stored as @samp{(vec)}. A matrix is simply a vector
32791 where all @var{v}'s are themselves vectors of equal lengths. Note that
32792 Calc vectors are unrelated to the Emacs Lisp ``vector'' type, which is
32793 generally unused by Calc data structures.
32795 Variables are stored as @samp{(var @var{name} @var{sym})}, where
32796 @var{name} is a Lisp symbol whose print name is used as the visible name
32797 of the variable, and @var{sym} is a Lisp symbol in which the variable's
32798 value is actually stored. Thus, @samp{(var pi var-pi)} represents the
32799 special constant @samp{pi}. Almost always, the form is @samp{(var
32800 @var{v} var-@var{v})}. If the variable name was entered with @code{#}
32801 signs (which are converted to hyphens internally), the form is
32802 @samp{(var @var{u} @var{v})}, where @var{u} is a symbol whose name
32803 contains @code{#} characters, and @var{v} is a symbol that contains
32804 @code{-} characters instead. The value of a variable is the Calc
32805 object stored in its @var{sym} symbol's value cell. If the symbol's
32806 value cell is void or if it contains @code{nil}, the variable has no
32807 value. Special constants have the form @samp{(special-const
32808 @var{value})} stored in their value cell, where @var{value} is a formula
32809 which is evaluated when the constant's value is requested. Variables
32810 which represent units are not stored in any special way; they are units
32811 only because their names appear in the units table. If the value
32812 cell contains a string, it is parsed to get the variable's value when
32813 the variable is used.
32815 A Lisp list with any other symbol as the first element is a function call.
32816 The symbols @code{+}, @code{-}, @code{*}, @code{/}, @code{%}, @code{^},
32817 and @code{|} represent special binary operators; these lists are always
32818 of the form @samp{(@var{op} @var{lhs} @var{rhs})} where @var{lhs} is the
32819 sub-formula on the lefthand side and @var{rhs} is the sub-formula on the
32820 right. The symbol @code{neg} represents unary negation; this list is always
32821 of the form @samp{(neg @var{arg})}. Any other symbol @var{func} represents a
32822 function that would be displayed in function-call notation; the symbol
32823 @var{func} is in general always of the form @samp{calcFunc-@var{name}}.
32824 The function cell of the symbol @var{func} should contain a Lisp function
32825 for evaluating a call to @var{func}. This function is passed the remaining
32826 elements of the list (themselves already evaluated) as arguments; such
32827 functions should return @code{nil} or call @code{reject-arg} to signify
32828 that they should be left in symbolic form, or they should return a Calc
32829 object which represents their value, or a list of such objects if they
32830 wish to return multiple values. (The latter case is allowed only for
32831 functions which are the outer-level call in an expression whose value is
32832 about to be pushed on the stack; this feature is considered obsolete
32833 and is not used by any built-in Calc functions.)
32835 @node Interactive Lisp Functions, Stack Lisp Functions, Data Type Formats, Internals
32836 @subsubsection Interactive Functions
32839 The functions described here are used in implementing interactive Calc
32840 commands. Note that this list is not exhaustive! If there is an
32841 existing command that behaves similarly to the one you want to define,
32842 you may find helpful tricks by checking the source code for that command.
32844 @defun calc-set-command-flag flag
32845 Set the command flag @var{flag}. This is generally a Lisp symbol, but
32846 may in fact be anything. The effect is to add @var{flag} to the list
32847 stored in the variable @code{calc-command-flags}, unless it is already
32848 there. @xref{Defining Simple Commands}.
32851 @defun calc-clear-command-flag flag
32852 If @var{flag} appears among the list of currently-set command flags,
32853 remove it from that list.
32856 @defun calc-record-undo rec
32857 Add the ``undo record'' @var{rec} to the list of steps to take if the
32858 current operation should need to be undone. Stack push and pop functions
32859 automatically call @code{calc-record-undo}, so the kinds of undo records
32860 you might need to create take the form @samp{(set @var{sym} @var{value})},
32861 which says that the Lisp variable @var{sym} was changed and had previously
32862 contained @var{value}; @samp{(store @var{var} @var{value})} which says that
32863 the Calc variable @var{var} (a string which is the name of the symbol that
32864 contains the variable's value) was stored and its previous value was
32865 @var{value} (either a Calc data object, or @code{nil} if the variable was
32866 previously void); or @samp{(eval @var{undo} @var{redo} @var{args} @dots{})},
32867 which means that to undo requires calling the function @samp{(@var{undo}
32868 @var{args} @dots{})} and, if the undo is later redone, calling
32869 @samp{(@var{redo} @var{args} @dots{})}.
32872 @defun calc-record-why msg args
32873 Record the error or warning message @var{msg}, which is normally a string.
32874 This message will be replayed if the user types @kbd{w} (@code{calc-why});
32875 if the message string begins with a @samp{*}, it is considered important
32876 enough to display even if the user doesn't type @kbd{w}. If one or more
32877 @var{args} are present, the displayed message will be of the form,
32878 @samp{@var{msg}: @var{arg1}, @var{arg2}, @dots{}}, where the arguments are
32879 formatted on the assumption that they are either strings or Calc objects of
32880 some sort. If @var{msg} is a symbol, it is the name of a Calc predicate
32881 (such as @code{integerp} or @code{numvecp}) which the arguments did not
32882 satisfy; it is expanded to a suitable string such as ``Expected an
32883 integer.'' The @code{reject-arg} function calls @code{calc-record-why}
32884 automatically; @pxref{Predicates}.
32887 @defun calc-is-inverse
32888 This predicate returns true if the current command is inverse,
32889 i.e., if the Inverse (@kbd{I} key) flag was set.
32892 @defun calc-is-hyperbolic
32893 This predicate is the analogous function for the @kbd{H} key.
32896 @node Stack Lisp Functions, Predicates, Interactive Lisp Functions, Internals
32897 @subsubsection Stack-Oriented Functions
32900 The functions described here perform various operations on the Calc
32901 stack and trail. They are to be used in interactive Calc commands.
32903 @defun calc-push-list vals n
32904 Push the Calc objects in list @var{vals} onto the stack at stack level
32905 @var{n}. If @var{n} is omitted it defaults to 1, so that the elements
32906 are pushed at the top of the stack. If @var{n} is greater than 1, the
32907 elements will be inserted into the stack so that the last element will
32908 end up at level @var{n}, the next-to-last at level @var{n}+1, etc.
32909 The elements of @var{vals} are assumed to be valid Calc objects, and
32910 are not evaluated, rounded, or renormalized in any way. If @var{vals}
32911 is an empty list, nothing happens.
32913 The stack elements are pushed without any sub-formula selections.
32914 You can give an optional third argument to this function, which must
32915 be a list the same size as @var{vals} of selections. Each selection
32916 must be @code{eq} to some sub-formula of the corresponding formula
32917 in @var{vals}, or @code{nil} if that formula should have no selection.
32920 @defun calc-top-list n m
32921 Return a list of the @var{n} objects starting at level @var{m} of the
32922 stack. If @var{m} is omitted it defaults to 1, so that the elements are
32923 taken from the top of the stack. If @var{n} is omitted, it also
32924 defaults to 1, so that the top stack element (in the form of a
32925 one-element list) is returned. If @var{m} is greater than 1, the
32926 @var{m}th stack element will be at the end of the list, the @var{m}+1st
32927 element will be next-to-last, etc. If @var{n} or @var{m} are out of
32928 range, the command is aborted with a suitable error message. If @var{n}
32929 is zero, the function returns an empty list. The stack elements are not
32930 evaluated, rounded, or renormalized.
32932 If any stack elements contain selections, and selections have not
32933 been disabled by the @kbd{j e} (@code{calc-enable-selections}) command,
32934 this function returns the selected portions rather than the entire
32935 stack elements. It can be given a third ``selection-mode'' argument
32936 which selects other behaviors. If it is the symbol @code{t}, then
32937 a selection in any of the requested stack elements produces an
32938 ``invalid operation on selections'' error. If it is the symbol @code{full},
32939 the whole stack entry is always returned regardless of selections.
32940 If it is the symbol @code{sel}, the selected portion is always returned,
32941 or @code{nil} if there is no selection. (This mode ignores the @kbd{j e}
32942 command.) If the symbol is @code{entry}, the complete stack entry in
32943 list form is returned; the first element of this list will be the whole
32944 formula, and the third element will be the selection (or @code{nil}).
32947 @defun calc-pop-stack n m
32948 Remove the specified elements from the stack. The parameters @var{n}
32949 and @var{m} are defined the same as for @code{calc-top-list}. The return
32950 value of @code{calc-pop-stack} is uninteresting.
32952 If there are any selected sub-formulas among the popped elements, and
32953 @kbd{j e} has not been used to disable selections, this produces an
32954 error without changing the stack. If you supply an optional third
32955 argument of @code{t}, the stack elements are popped even if they
32956 contain selections.
32959 @defun calc-record-list vals tag
32960 This function records one or more results in the trail. The @var{vals}
32961 are a list of strings or Calc objects. The @var{tag} is the four-character
32962 tag string to identify the values. If @var{tag} is omitted, a blank tag
32966 @defun calc-normalize n
32967 This function takes a Calc object and ``normalizes'' it. At the very
32968 least this involves re-rounding floating-point values according to the
32969 current precision and other similar jobs. Also, unless the user has
32970 selected No-Simplify mode (@pxref{Simplification Modes}), this involves
32971 actually evaluating a formula object by executing the function calls
32972 it contains, and possibly also doing algebraic simplification, etc.
32975 @defun calc-top-list-n n m
32976 This function is identical to @code{calc-top-list}, except that it calls
32977 @code{calc-normalize} on the values that it takes from the stack. They
32978 are also passed through @code{check-complete}, so that incomplete
32979 objects will be rejected with an error message. All computational
32980 commands should use this in preference to @code{calc-top-list}; the only
32981 standard Calc commands that operate on the stack without normalizing
32982 are stack management commands like @code{calc-enter} and @code{calc-roll-up}.
32983 This function accepts the same optional selection-mode argument as
32984 @code{calc-top-list}.
32987 @defun calc-top-n m
32988 This function is a convenient form of @code{calc-top-list-n} in which only
32989 a single element of the stack is taken and returned, rather than a list
32990 of elements. This also accepts an optional selection-mode argument.
32993 @defun calc-enter-result n tag vals
32994 This function is a convenient interface to most of the above functions.
32995 The @var{vals} argument should be either a single Calc object, or a list
32996 of Calc objects; the object or objects are normalized, and the top @var{n}
32997 stack entries are replaced by the normalized objects. If @var{tag} is
32998 non-@code{nil}, the normalized objects are also recorded in the trail.
32999 A typical stack-based computational command would take the form,
33002 (calc-enter-result @var{n} @var{tag} (cons 'calcFunc-@var{func}
33003 (calc-top-list-n @var{n})))
33006 If any of the @var{n} stack elements replaced contain sub-formula
33007 selections, and selections have not been disabled by @kbd{j e},
33008 this function takes one of two courses of action. If @var{n} is
33009 equal to the number of elements in @var{vals}, then each element of
33010 @var{vals} is spliced into the corresponding selection; this is what
33011 happens when you use the @key{TAB} key, or when you use a unary
33012 arithmetic operation like @code{sqrt}. If @var{vals} has only one
33013 element but @var{n} is greater than one, there must be only one
33014 selection among the top @var{n} stack elements; the element from
33015 @var{vals} is spliced into that selection. This is what happens when
33016 you use a binary arithmetic operation like @kbd{+}. Any other
33017 combination of @var{n} and @var{vals} is an error when selections
33021 @defun calc-unary-op tag func arg
33022 This function implements a unary operator that allows a numeric prefix
33023 argument to apply the operator over many stack entries. If the prefix
33024 argument @var{arg} is @code{nil}, this uses @code{calc-enter-result}
33025 as outlined above. Otherwise, it maps the function over several stack
33026 elements; @pxref{Prefix Arguments}. For example,
33029 (defun calc-zeta (arg)
33031 (calc-unary-op "zeta" 'calcFunc-zeta arg))
33035 @defun calc-binary-op tag func arg ident unary
33036 This function implements a binary operator, analogously to
33037 @code{calc-unary-op}. The optional @var{ident} and @var{unary}
33038 arguments specify the behavior when the prefix argument is zero or
33039 one, respectively. If the prefix is zero, the value @var{ident}
33040 is pushed onto the stack, if specified, otherwise an error message
33041 is displayed. If the prefix is one, the unary function @var{unary}
33042 is applied to the top stack element, or, if @var{unary} is not
33043 specified, nothing happens. When the argument is two or more,
33044 the binary function @var{func} is reduced across the top @var{arg}
33045 stack elements; when the argument is negative, the function is
33046 mapped between the next-to-top @mathit{-@var{arg}} stack elements and the
33050 @defun calc-stack-size
33051 Return the number of elements on the stack as an integer. This count
33052 does not include elements that have been temporarily hidden by stack
33053 truncation; @pxref{Truncating the Stack}.
33056 @defun calc-cursor-stack-index n
33057 Move the point to the @var{n}th stack entry. If @var{n} is zero, this
33058 will be the @samp{.} line. If @var{n} is from 1 to the current stack size,
33059 this will be the beginning of the first line of that stack entry's display.
33060 If line numbers are enabled, this will move to the first character of the
33061 line number, not the stack entry itself.
33064 @defun calc-substack-height n
33065 Return the number of lines between the beginning of the @var{n}th stack
33066 entry and the bottom of the buffer. If @var{n} is zero, this
33067 will be one (assuming no stack truncation). If all stack entries are
33068 one line long (i.e., no matrices are displayed), the return value will
33069 be equal @var{n}+1 as long as @var{n} is in range. (Note that in Big
33070 mode, the return value includes the blank lines that separate stack
33074 @defun calc-refresh
33075 Erase the @code{*Calculator*} buffer and reformat its contents from memory.
33076 This must be called after changing any parameter, such as the current
33077 display radix, which might change the appearance of existing stack
33078 entries. (During a keyboard macro invoked by the @kbd{X} key, refreshing
33079 is suppressed, but a flag is set so that the entire stack will be refreshed
33080 rather than just the top few elements when the macro finishes.)
33083 @node Predicates, Computational Lisp Functions, Stack Lisp Functions, Internals
33084 @subsubsection Predicates
33087 The functions described here are predicates, that is, they return a
33088 true/false value where @code{nil} means false and anything else means
33089 true. These predicates are expanded by @code{defmath}, for example,
33090 from @code{zerop} to @code{math-zerop}. In many cases they correspond
33091 to native Lisp functions by the same name, but are extended to cover
33092 the full range of Calc data types.
33095 Returns true if @var{x} is numerically zero, in any of the Calc data
33096 types. (Note that for some types, such as error forms and intervals,
33097 it never makes sense to return true.) In @code{defmath}, the expression
33098 @samp{(= x 0)} will automatically be converted to @samp{(math-zerop x)},
33099 and @samp{(/= x 0)} will be converted to @samp{(not (math-zerop x))}.
33103 Returns true if @var{x} is negative. This accepts negative real numbers
33104 of various types, negative HMS and date forms, and intervals in which
33105 all included values are negative. In @code{defmath}, the expression
33106 @samp{(< x 0)} will automatically be converted to @samp{(math-negp x)},
33107 and @samp{(>= x 0)} will be converted to @samp{(not (math-negp x))}.
33111 Returns true if @var{x} is positive (and non-zero). For complex
33112 numbers, none of these three predicates will return true.
33115 @defun looks-negp x
33116 Returns true if @var{x} is ``negative-looking.'' This returns true if
33117 @var{x} is a negative number, or a formula with a leading minus sign
33118 such as @samp{-a/b}. In other words, this is an object which can be
33119 made simpler by calling @code{(- @var{x})}.
33123 Returns true if @var{x} is an integer of any size.
33127 Returns true if @var{x} is a native Lisp integer.
33131 Returns true if @var{x} is a nonnegative integer of any size.
33134 @defun fixnatnump x
33135 Returns true if @var{x} is a nonnegative Lisp integer.
33138 @defun num-integerp x
33139 Returns true if @var{x} is numerically an integer, i.e., either a
33140 true integer or a float with no significant digits to the right of
33144 @defun messy-integerp x
33145 Returns true if @var{x} is numerically, but not literally, an integer.
33146 A value is @code{num-integerp} if it is @code{integerp} or
33147 @code{messy-integerp} (but it is never both at once).
33150 @defun num-natnump x
33151 Returns true if @var{x} is numerically a nonnegative integer.
33155 Returns true if @var{x} is an even integer.
33158 @defun looks-evenp x
33159 Returns true if @var{x} is an even integer, or a formula with a leading
33160 multiplicative coefficient which is an even integer.
33164 Returns true if @var{x} is an odd integer.
33168 Returns true if @var{x} is a rational number, i.e., an integer or a
33173 Returns true if @var{x} is a real number, i.e., an integer, fraction,
33174 or floating-point number.
33178 Returns true if @var{x} is a real number or HMS form.
33182 Returns true if @var{x} is a float, or a complex number, error form,
33183 interval, date form, or modulo form in which at least one component
33188 Returns true if @var{x} is a rectangular or polar complex number
33189 (but not a real number).
33192 @defun rect-complexp x
33193 Returns true if @var{x} is a rectangular complex number.
33196 @defun polar-complexp x
33197 Returns true if @var{x} is a polar complex number.
33201 Returns true if @var{x} is a real number or a complex number.
33205 Returns true if @var{x} is a real or complex number or an HMS form.
33209 Returns true if @var{x} is a vector (this simply checks if its argument
33210 is a list whose first element is the symbol @code{vec}).
33214 Returns true if @var{x} is a number or vector.
33218 Returns true if @var{x} is a matrix, i.e., a vector of one or more vectors,
33219 all of the same size.
33222 @defun square-matrixp x
33223 Returns true if @var{x} is a square matrix.
33227 Returns true if @var{x} is any numeric Calc object, including real and
33228 complex numbers, HMS forms, date forms, error forms, intervals, and
33229 modulo forms. (Note that error forms and intervals may include formulas
33230 as their components; see @code{constp} below.)
33234 Returns true if @var{x} is an object or a vector. This also accepts
33235 incomplete objects, but it rejects variables and formulas (except as
33236 mentioned above for @code{objectp}).
33240 Returns true if @var{x} is a ``primitive'' or ``atomic'' Calc object,
33241 i.e., one whose components cannot be regarded as sub-formulas. This
33242 includes variables, and all @code{objectp} types except error forms
33247 Returns true if @var{x} is constant, i.e., a real or complex number,
33248 HMS form, date form, or error form, interval, or vector all of whose
33249 components are @code{constp}.
33253 Returns true if @var{x} is numerically less than @var{y}. Returns false
33254 if @var{x} is greater than or equal to @var{y}, or if the order is
33255 undefined or cannot be determined. Generally speaking, this works
33256 by checking whether @samp{@var{x} - @var{y}} is @code{negp}. In
33257 @code{defmath}, the expression @samp{(< x y)} will automatically be
33258 converted to @samp{(lessp x y)}; expressions involving @code{>}, @code{<=},
33259 and @code{>=} are similarly converted in terms of @code{lessp}.
33263 Returns true if @var{x} comes before @var{y} in a canonical ordering
33264 of Calc objects. If @var{x} and @var{y} are both real numbers, this
33265 will be the same as @code{lessp}. But whereas @code{lessp} considers
33266 other types of objects to be unordered, @code{beforep} puts any two
33267 objects into a definite, consistent order. The @code{beforep}
33268 function is used by the @kbd{V S} vector-sorting command, and also
33269 by @kbd{a s} to put the terms of a product into canonical order:
33270 This allows @samp{x y + y x} to be simplified easily to @samp{2 x y}.
33274 This is the standard Lisp @code{equal} predicate; it returns true if
33275 @var{x} and @var{y} are structurally identical. This is the usual way
33276 to compare numbers for equality, but note that @code{equal} will treat
33277 0 and 0.0 as different.
33280 @defun math-equal x y
33281 Returns true if @var{x} and @var{y} are numerically equal, either because
33282 they are @code{equal}, or because their difference is @code{zerop}. In
33283 @code{defmath}, the expression @samp{(= x y)} will automatically be
33284 converted to @samp{(math-equal x y)}.
33287 @defun equal-int x n
33288 Returns true if @var{x} and @var{n} are numerically equal, where @var{n}
33289 is a fixnum which is not a multiple of 10. This will automatically be
33290 used by @code{defmath} in place of the more general @code{math-equal}
33294 @defun nearly-equal x y
33295 Returns true if @var{x} and @var{y}, as floating-point numbers, are
33296 equal except possibly in the last decimal place. For example,
33297 314.159 and 314.166 are considered nearly equal if the current
33298 precision is 6 (since they differ by 7 units), but not if the current
33299 precision is 7 (since they differ by 70 units). Most functions which
33300 use series expansions use @code{with-extra-prec} to evaluate the
33301 series with 2 extra digits of precision, then use @code{nearly-equal}
33302 to decide when the series has converged; this guards against cumulative
33303 error in the series evaluation without doing extra work which would be
33304 lost when the result is rounded back down to the current precision.
33305 In @code{defmath}, this can be written @samp{(~= @var{x} @var{y})}.
33306 The @var{x} and @var{y} can be numbers of any kind, including complex.
33309 @defun nearly-zerop x y
33310 Returns true if @var{x} is nearly zero, compared to @var{y}. This
33311 checks whether @var{x} plus @var{y} would by be @code{nearly-equal}
33312 to @var{y} itself, to within the current precision, in other words,
33313 if adding @var{x} to @var{y} would have a negligible effect on @var{y}
33314 due to roundoff error. @var{X} may be a real or complex number, but
33315 @var{y} must be real.
33319 Return true if the formula @var{x} represents a true value in
33320 Calc, not Lisp, terms. It tests if @var{x} is a non-zero number
33321 or a provably non-zero formula.
33324 @defun reject-arg val pred
33325 Abort the current function evaluation due to unacceptable argument values.
33326 This calls @samp{(calc-record-why @var{pred} @var{val})}, then signals a
33327 Lisp error which @code{normalize} will trap. The net effect is that the
33328 function call which led here will be left in symbolic form.
33331 @defun inexact-value
33332 If Symbolic mode is enabled, this will signal an error that causes
33333 @code{normalize} to leave the formula in symbolic form, with the message
33334 ``Inexact result.'' (This function has no effect when not in Symbolic mode.)
33335 Note that if your function calls @samp{(sin 5)} in Symbolic mode, the
33336 @code{sin} function will call @code{inexact-value}, which will cause your
33337 function to be left unsimplified. You may instead wish to call
33338 @samp{(normalize (list 'calcFunc-sin 5))}, which in Symbolic mode will
33339 return the formula @samp{sin(5)} to your function.
33343 This signals an error that will be reported as a floating-point overflow.
33347 This signals a floating-point underflow.
33350 @node Computational Lisp Functions, Vector Lisp Functions, Predicates, Internals
33351 @subsubsection Computational Functions
33354 The functions described here do the actual computational work of the
33355 Calculator. In addition to these, note that any function described in
33356 the main body of this manual may be called from Lisp; for example, if
33357 the documentation refers to the @code{calc-sqrt} [@code{sqrt}] command,
33358 this means @code{calc-sqrt} is an interactive stack-based square-root
33359 command and @code{sqrt} (which @code{defmath} expands to @code{calcFunc-sqrt})
33360 is the actual Lisp function for taking square roots.
33362 The functions @code{math-add}, @code{math-sub}, @code{math-mul},
33363 @code{math-div}, @code{math-mod}, and @code{math-neg} are not included
33364 in this list, since @code{defmath} allows you to write native Lisp
33365 @code{+}, @code{-}, @code{*}, @code{/}, @code{%}, and unary @code{-},
33366 respectively, instead.
33368 @defun normalize val
33369 (Full form: @code{math-normalize}.)
33370 Reduce the value @var{val} to standard form. For example, if @var{val}
33371 is a fixnum, it will be converted to a bignum if it is too large, and
33372 if @var{val} is a bignum it will be normalized by clipping off trailing
33373 (i.e., most-significant) zero digits and converting to a fixnum if it is
33374 small. All the various data types are similarly converted to their standard
33375 forms. Variables are left alone, but function calls are actually evaluated
33376 in formulas. For example, normalizing @samp{(+ 2 (calcFunc-abs -4))} will
33379 If a function call fails, because the function is void or has the wrong
33380 number of parameters, or because it returns @code{nil} or calls
33381 @code{reject-arg} or @code{inexact-result}, @code{normalize} returns
33382 the formula still in symbolic form.
33384 If the current simplification mode is ``none'' or ``numeric arguments
33385 only,'' @code{normalize} will act appropriately. However, the more
33386 powerful simplification modes (like Algebraic Simplification) are
33387 not handled by @code{normalize}. They are handled by @code{calc-normalize},
33388 which calls @code{normalize} and possibly some other routines, such
33389 as @code{simplify} or @code{simplify-units}. Programs generally will
33390 never call @code{calc-normalize} except when popping or pushing values
33394 @defun evaluate-expr expr
33395 Replace all variables in @var{expr} that have values with their values,
33396 then use @code{normalize} to simplify the result. This is what happens
33397 when you press the @kbd{=} key interactively.
33400 @defmac with-extra-prec n body
33401 Evaluate the Lisp forms in @var{body} with precision increased by @var{n}
33402 digits. This is a macro which expands to
33406 (let ((calc-internal-prec (+ calc-internal-prec @var{n})))
33410 The surrounding call to @code{math-normalize} causes a floating-point
33411 result to be rounded down to the original precision afterwards. This
33412 is important because some arithmetic operations assume a number's
33413 mantissa contains no more digits than the current precision allows.
33416 @defun make-frac n d
33417 Build a fraction @samp{@var{n}:@var{d}}. This is equivalent to calling
33418 @samp{(normalize (list 'frac @var{n} @var{d}))}, but more efficient.
33421 @defun make-float mant exp
33422 Build a floating-point value out of @var{mant} and @var{exp}, both
33423 of which are arbitrary integers. This function will return a
33424 properly normalized float value, or signal an overflow or underflow
33425 if @var{exp} is out of range.
33428 @defun make-sdev x sigma
33429 Build an error form out of @var{x} and the absolute value of @var{sigma}.
33430 If @var{sigma} is zero, the result is the number @var{x} directly.
33431 If @var{sigma} is negative or complex, its absolute value is used.
33432 If @var{x} or @var{sigma} is not a valid type of object for use in
33433 error forms, this calls @code{reject-arg}.
33436 @defun make-intv mask lo hi
33437 Build an interval form out of @var{mask} (which is assumed to be an
33438 integer from 0 to 3), and the limits @var{lo} and @var{hi}. If
33439 @var{lo} is greater than @var{hi}, an empty interval form is returned.
33440 This calls @code{reject-arg} if @var{lo} or @var{hi} is unsuitable.
33443 @defun sort-intv mask lo hi
33444 Build an interval form, similar to @code{make-intv}, except that if
33445 @var{lo} is less than @var{hi} they are simply exchanged, and the
33446 bits of @var{mask} are swapped accordingly.
33449 @defun make-mod n m
33450 Build a modulo form out of @var{n} and the modulus @var{m}. Since modulo
33451 forms do not allow formulas as their components, if @var{n} or @var{m}
33452 is not a real number or HMS form the result will be a formula which
33453 is a call to @code{makemod}, the algebraic version of this function.
33457 Convert @var{x} to floating-point form. Integers and fractions are
33458 converted to numerically equivalent floats; components of complex
33459 numbers, vectors, HMS forms, date forms, error forms, intervals, and
33460 modulo forms are recursively floated. If the argument is a variable
33461 or formula, this calls @code{reject-arg}.
33465 Compare the numbers @var{x} and @var{y}, and return @mathit{-1} if
33466 @samp{(lessp @var{x} @var{y})}, 1 if @samp{(lessp @var{y} @var{x})},
33467 0 if @samp{(math-equal @var{x} @var{y})}, or 2 if the order is
33468 undefined or cannot be determined.
33472 Return the number of digits of integer @var{n}, effectively
33473 @samp{ceil(log10(@var{n}))}, but much more efficient. Zero is
33474 considered to have zero digits.
33477 @defun scale-int x n
33478 Shift integer @var{x} left @var{n} decimal digits, or right @mathit{-@var{n}}
33479 digits with truncation toward zero.
33482 @defun scale-rounding x n
33483 Like @code{scale-int}, except that a right shift rounds to the nearest
33484 integer rather than truncating.
33488 Return the integer @var{n} as a fixnum, i.e., a native Lisp integer.
33489 If @var{n} is outside the permissible range for Lisp integers (usually
33490 24 binary bits) the result is undefined.
33494 Compute the square of @var{x}; short for @samp{(* @var{x} @var{x})}.
33497 @defun quotient x y
33498 Divide integer @var{x} by integer @var{y}; return an integer quotient
33499 and discard the remainder. If @var{x} or @var{y} is negative, the
33500 direction of rounding is undefined.
33504 Perform an integer division; if @var{x} and @var{y} are both nonnegative
33505 integers, this uses the @code{quotient} function, otherwise it computes
33506 @samp{floor(@var{x}/@var{y})}. Thus the result is well-defined but
33507 slower than for @code{quotient}.
33511 Divide integer @var{x} by integer @var{y}; return the integer remainder
33512 and discard the quotient. Like @code{quotient}, this works only for
33513 integer arguments and is not well-defined for negative arguments.
33514 For a more well-defined result, use @samp{(% @var{x} @var{y})}.
33518 Divide integer @var{x} by integer @var{y}; return a cons cell whose
33519 @code{car} is @samp{(quotient @var{x} @var{y})} and whose @code{cdr}
33520 is @samp{(imod @var{x} @var{y})}.
33524 Compute @var{x} to the power @var{y}. In @code{defmath} code, this can
33525 also be written @samp{(^ @var{x} @var{y})} or
33526 @w{@samp{(expt @var{x} @var{y})}}.
33529 @defun abs-approx x
33530 Compute a fast approximation to the absolute value of @var{x}. For
33531 example, for a rectangular complex number the result is the sum of
33532 the absolute values of the components.
33536 @findex gamma-const
33542 @findex pi-over-180
33543 @findex sqrt-two-pi
33547 The function @samp{(pi)} computes @samp{pi} to the current precision.
33548 Other related constant-generating functions are @code{two-pi},
33549 @code{pi-over-2}, @code{pi-over-4}, @code{pi-over-180}, @code{sqrt-two-pi},
33550 @code{e}, @code{sqrt-e}, @code{ln-2}, @code{ln-10}, @code{phi} and
33551 @code{gamma-const}. Each function returns a floating-point value in the
33552 current precision, and each uses caching so that all calls after the
33553 first are essentially free.
33556 @defmac math-defcache @var{func} @var{initial} @var{form}
33557 This macro, usually used as a top-level call like @code{defun} or
33558 @code{defvar}, defines a new cached constant analogous to @code{pi}, etc.
33559 It defines a function @code{func} which returns the requested value;
33560 if @var{initial} is non-@code{nil} it must be a @samp{(float @dots{})}
33561 form which serves as an initial value for the cache. If @var{func}
33562 is called when the cache is empty or does not have enough digits to
33563 satisfy the current precision, the Lisp expression @var{form} is evaluated
33564 with the current precision increased by four, and the result minus its
33565 two least significant digits is stored in the cache. For example,
33566 calling @samp{(pi)} with a precision of 30 computes @samp{pi} to 34
33567 digits, rounds it down to 32 digits for future use, then rounds it
33568 again to 30 digits for use in the present request.
33571 @findex half-circle
33572 @findex quarter-circle
33573 @defun full-circle symb
33574 If the current angular mode is Degrees or HMS, this function returns the
33575 integer 360. In Radians mode, this function returns either the
33576 corresponding value in radians to the current precision, or the formula
33577 @samp{2*pi}, depending on the Symbolic mode. There are also similar
33578 function @code{half-circle} and @code{quarter-circle}.
33581 @defun power-of-2 n
33582 Compute two to the integer power @var{n}, as a (potentially very large)
33583 integer. Powers of two are cached, so only the first call for a
33584 particular @var{n} is expensive.
33587 @defun integer-log2 n
33588 Compute the base-2 logarithm of @var{n}, which must be an integer which
33589 is a power of two. If @var{n} is not a power of two, this function will
33593 @defun div-mod a b m
33594 Divide @var{a} by @var{b}, modulo @var{m}. This returns @code{nil} if
33595 there is no solution, or if any of the arguments are not integers.
33598 @defun pow-mod a b m
33599 Compute @var{a} to the power @var{b}, modulo @var{m}. If @var{a},
33600 @var{b}, and @var{m} are integers, this uses an especially efficient
33601 algorithm. Otherwise, it simply computes @samp{(% (^ a b) m)}.
33605 Compute the integer square root of @var{n}. This is the square root
33606 of @var{n} rounded down toward zero, i.e., @samp{floor(sqrt(@var{n}))}.
33607 If @var{n} is itself an integer, the computation is especially efficient.
33610 @defun to-hms a ang
33611 Convert the argument @var{a} into an HMS form. If @var{ang} is specified,
33612 it is the angular mode in which to interpret @var{a}, either @code{deg}
33613 or @code{rad}. Otherwise, the current angular mode is used. If @var{a}
33614 is already an HMS form it is returned as-is.
33617 @defun from-hms a ang
33618 Convert the HMS form @var{a} into a real number. If @var{ang} is specified,
33619 it is the angular mode in which to express the result, otherwise the
33620 current angular mode is used. If @var{a} is already a real number, it
33624 @defun to-radians a
33625 Convert the number or HMS form @var{a} to radians from the current
33629 @defun from-radians a
33630 Convert the number @var{a} from radians to the current angular mode.
33631 If @var{a} is a formula, this returns the formula @samp{deg(@var{a})}.
33634 @defun to-radians-2 a
33635 Like @code{to-radians}, except that in Symbolic mode a degrees to
33636 radians conversion yields a formula like @samp{@var{a}*pi/180}.
33639 @defun from-radians-2 a
33640 Like @code{from-radians}, except that in Symbolic mode a radians to
33641 degrees conversion yields a formula like @samp{@var{a}*180/pi}.
33644 @defun random-digit
33645 Produce a random base-1000 digit in the range 0 to 999.
33648 @defun random-digits n
33649 Produce a random @var{n}-digit integer; this will be an integer
33650 in the interval @samp{[0, 10^@var{n})}.
33653 @defun random-float
33654 Produce a random float in the interval @samp{[0, 1)}.
33657 @defun prime-test n iters
33658 Determine whether the integer @var{n} is prime. Return a list which has
33659 one of these forms: @samp{(nil @var{f})} means the number is non-prime
33660 because it was found to be divisible by @var{f}; @samp{(nil)} means it
33661 was found to be non-prime by table look-up (so no factors are known);
33662 @samp{(nil unknown)} means it is definitely non-prime but no factors
33663 are known because @var{n} was large enough that Fermat's probabilistic
33664 test had to be used; @samp{(t)} means the number is definitely prime;
33665 and @samp{(maybe @var{i} @var{p})} means that Fermat's test, after @var{i}
33666 iterations, is @var{p} percent sure that the number is prime. The
33667 @var{iters} parameter is the number of Fermat iterations to use, in the
33668 case that this is necessary. If @code{prime-test} returns ``maybe,''
33669 you can call it again with the same @var{n} to get a greater certainty;
33670 @code{prime-test} remembers where it left off.
33673 @defun to-simple-fraction f
33674 If @var{f} is a floating-point number which can be represented exactly
33675 as a small rational number. return that number, else return @var{f}.
33676 For example, 0.75 would be converted to 3:4. This function is very
33680 @defun to-fraction f tol
33681 Find a rational approximation to floating-point number @var{f} to within
33682 a specified tolerance @var{tol}; this corresponds to the algebraic
33683 function @code{frac}, and can be rather slow.
33686 @defun quarter-integer n
33687 If @var{n} is an integer or integer-valued float, this function
33688 returns zero. If @var{n} is a half-integer (i.e., an integer plus
33689 @mathit{1:2} or 0.5), it returns 2. If @var{n} is a quarter-integer,
33690 it returns 1 or 3. If @var{n} is anything else, this function
33691 returns @code{nil}.
33694 @node Vector Lisp Functions, Symbolic Lisp Functions, Computational Lisp Functions, Internals
33695 @subsubsection Vector Functions
33698 The functions described here perform various operations on vectors and
33701 @defun math-concat x y
33702 Do a vector concatenation; this operation is written @samp{@var{x} | @var{y}}
33703 in a symbolic formula. @xref{Building Vectors}.
33706 @defun vec-length v
33707 Return the length of vector @var{v}. If @var{v} is not a vector, the
33708 result is zero. If @var{v} is a matrix, this returns the number of
33709 rows in the matrix.
33712 @defun mat-dimens m
33713 Determine the dimensions of vector or matrix @var{m}. If @var{m} is not
33714 a vector, the result is an empty list. If @var{m} is a plain vector
33715 but not a matrix, the result is a one-element list containing the length
33716 of the vector. If @var{m} is a matrix with @var{r} rows and @var{c} columns,
33717 the result is the list @samp{(@var{r} @var{c})}. Higher-order tensors
33718 produce lists of more than two dimensions. Note that the object
33719 @samp{[[1, 2, 3], [4, 5]]} is a vector of vectors not all the same size,
33720 and is treated by this and other Calc routines as a plain vector of two
33724 @defun dimension-error
33725 Abort the current function with a message of ``Dimension error.''
33726 The Calculator will leave the function being evaluated in symbolic
33727 form; this is really just a special case of @code{reject-arg}.
33730 @defun build-vector args
33731 Return a Calc vector with @var{args} as elements.
33732 For example, @samp{(build-vector 1 2 3)} returns the Calc vector
33733 @samp{[1, 2, 3]}, stored internally as the list @samp{(vec 1 2 3)}.
33736 @defun make-vec obj dims
33737 Return a Calc vector or matrix all of whose elements are equal to
33738 @var{obj}. For example, @samp{(make-vec 27 3 4)} returns a 3x4 matrix
33742 @defun row-matrix v
33743 If @var{v} is a plain vector, convert it into a row matrix, i.e.,
33744 a matrix whose single row is @var{v}. If @var{v} is already a matrix,
33748 @defun col-matrix v
33749 If @var{v} is a plain vector, convert it into a column matrix, i.e., a
33750 matrix with each element of @var{v} as a separate row. If @var{v} is
33751 already a matrix, leave it alone.
33755 Map the Lisp function @var{f} over the Calc vector @var{v}. For example,
33756 @samp{(map-vec 'math-floor v)} returns a vector of the floored components
33760 @defun map-vec-2 f a b
33761 Map the Lisp function @var{f} over the two vectors @var{a} and @var{b}.
33762 If @var{a} and @var{b} are vectors of equal length, the result is a
33763 vector of the results of calling @samp{(@var{f} @var{ai} @var{bi})}
33764 for each pair of elements @var{ai} and @var{bi}. If either @var{a} or
33765 @var{b} is a scalar, it is matched with each value of the other vector.
33766 For example, @samp{(map-vec-2 'math-add v 1)} returns the vector @var{v}
33767 with each element increased by one. Note that using @samp{'+} would not
33768 work here, since @code{defmath} does not expand function names everywhere,
33769 just where they are in the function position of a Lisp expression.
33772 @defun reduce-vec f v
33773 Reduce the function @var{f} over the vector @var{v}. For example, if
33774 @var{v} is @samp{[10, 20, 30, 40]}, this calls @samp{(f (f (f 10 20) 30) 40)}.
33775 If @var{v} is a matrix, this reduces over the rows of @var{v}.
33778 @defun reduce-cols f m
33779 Reduce the function @var{f} over the columns of matrix @var{m}. For
33780 example, if @var{m} is @samp{[[1, 2], [3, 4], [5, 6]]}, the result
33781 is a vector of the two elements @samp{(f (f 1 3) 5)} and @samp{(f (f 2 4) 6)}.
33785 Return the @var{n}th row of matrix @var{m}. This is equivalent to
33786 @samp{(elt m n)}. For a slower but safer version, use @code{mrow}.
33787 (@xref{Extracting Elements}.)
33791 Return the @var{n}th column of matrix @var{m}, in the form of a vector.
33792 The arguments are not checked for correctness.
33795 @defun mat-less-row m n
33796 Return a copy of matrix @var{m} with its @var{n}th row deleted. The
33797 number @var{n} must be in range from 1 to the number of rows in @var{m}.
33800 @defun mat-less-col m n
33801 Return a copy of matrix @var{m} with its @var{n}th column deleted.
33805 Return the transpose of matrix @var{m}.
33808 @defun flatten-vector v
33809 Flatten nested vector @var{v} into a vector of scalars. For example,
33810 if @var{v} is @samp{[[1, 2, 3], [4, 5]]} the result is @samp{[1, 2, 3, 4, 5]}.
33813 @defun copy-matrix m
33814 If @var{m} is a matrix, return a copy of @var{m}. This maps
33815 @code{copy-sequence} over the rows of @var{m}; in Lisp terms, each
33816 element of the result matrix will be @code{eq} to the corresponding
33817 element of @var{m}, but none of the @code{cons} cells that make up
33818 the structure of the matrix will be @code{eq}. If @var{m} is a plain
33819 vector, this is the same as @code{copy-sequence}.
33822 @defun swap-rows m r1 r2
33823 Exchange rows @var{r1} and @var{r2} of matrix @var{m} in-place. In
33824 other words, unlike most of the other functions described here, this
33825 function changes @var{m} itself rather than building up a new result
33826 matrix. The return value is @var{m}, i.e., @samp{(eq (swap-rows m 1 2) m)}
33827 is true, with the side effect of exchanging the first two rows of
33831 @node Symbolic Lisp Functions, Formatting Lisp Functions, Vector Lisp Functions, Internals
33832 @subsubsection Symbolic Functions
33835 The functions described here operate on symbolic formulas in the
33838 @defun calc-prepare-selection num
33839 Prepare a stack entry for selection operations. If @var{num} is
33840 omitted, the stack entry containing the cursor is used; otherwise,
33841 it is the number of the stack entry to use. This function stores
33842 useful information about the current stack entry into a set of
33843 variables. @code{calc-selection-cache-num} contains the number of
33844 the stack entry involved (equal to @var{num} if you specified it);
33845 @code{calc-selection-cache-entry} contains the stack entry as a
33846 list (such as @code{calc-top-list} would return with @code{entry}
33847 as the selection mode); and @code{calc-selection-cache-comp} contains
33848 a special ``tagged'' composition (@pxref{Formatting Lisp Functions})
33849 which allows Calc to relate cursor positions in the buffer with
33850 their corresponding sub-formulas.
33852 A slight complication arises in the selection mechanism because
33853 formulas may contain small integers. For example, in the vector
33854 @samp{[1, 2, 1]} the first and last elements are @code{eq} to each
33855 other; selections are recorded as the actual Lisp object that
33856 appears somewhere in the tree of the whole formula, but storing
33857 @code{1} would falsely select both @code{1}'s in the vector. So
33858 @code{calc-prepare-selection} also checks the stack entry and
33859 replaces any plain integers with ``complex number'' lists of the form
33860 @samp{(cplx @var{n} 0)}. This list will be displayed the same as a
33861 plain @var{n} and the change will be completely invisible to the
33862 user, but it will guarantee that no two sub-formulas of the stack
33863 entry will be @code{eq} to each other. Next time the stack entry
33864 is involved in a computation, @code{calc-normalize} will replace
33865 these lists with plain numbers again, again invisibly to the user.
33868 @defun calc-encase-atoms x
33869 This modifies the formula @var{x} to ensure that each part of the
33870 formula is a unique atom, using the @samp{(cplx @var{n} 0)} trick
33871 described above. This function may use @code{setcar} to modify
33872 the formula in-place.
33875 @defun calc-find-selected-part
33876 Find the smallest sub-formula of the current formula that contains
33877 the cursor. This assumes @code{calc-prepare-selection} has been
33878 called already. If the cursor is not actually on any part of the
33879 formula, this returns @code{nil}.
33882 @defun calc-change-current-selection selection
33883 Change the currently prepared stack element's selection to
33884 @var{selection}, which should be @code{eq} to some sub-formula
33885 of the stack element, or @code{nil} to unselect the formula.
33886 The stack element's appearance in the Calc buffer is adjusted
33887 to reflect the new selection.
33890 @defun calc-find-nth-part expr n
33891 Return the @var{n}th sub-formula of @var{expr}. This function is used
33892 by the selection commands, and (unless @kbd{j b} has been used) treats
33893 sums and products as flat many-element formulas. Thus if @var{expr}
33894 is @samp{((a + b) - c) + d}, calling @code{calc-find-nth-part} with
33895 @var{n} equal to four will return @samp{d}.
33898 @defun calc-find-parent-formula expr part
33899 Return the sub-formula of @var{expr} which immediately contains
33900 @var{part}. If @var{expr} is @samp{a*b + (c+1)*d} and @var{part}
33901 is @code{eq} to the @samp{c+1} term of @var{expr}, then this function
33902 will return @samp{(c+1)*d}. If @var{part} turns out not to be a
33903 sub-formula of @var{expr}, the function returns @code{nil}. If
33904 @var{part} is @code{eq} to @var{expr}, the function returns @code{t}.
33905 This function does not take associativity into account.
33908 @defun calc-find-assoc-parent-formula expr part
33909 This is the same as @code{calc-find-parent-formula}, except that
33910 (unless @kbd{j b} has been used) it continues widening the selection
33911 to contain a complete level of the formula. Given @samp{a} from
33912 @samp{((a + b) - c) + d}, @code{calc-find-parent-formula} will
33913 return @samp{a + b} but @code{calc-find-assoc-parent-formula} will
33914 return the whole expression.
33917 @defun calc-grow-assoc-formula expr part
33918 This expands sub-formula @var{part} of @var{expr} to encompass a
33919 complete level of the formula. If @var{part} and its immediate
33920 parent are not compatible associative operators, or if @kbd{j b}
33921 has been used, this simply returns @var{part}.
33924 @defun calc-find-sub-formula expr part
33925 This finds the immediate sub-formula of @var{expr} which contains
33926 @var{part}. It returns an index @var{n} such that
33927 @samp{(calc-find-nth-part @var{expr} @var{n})} would return @var{part}.
33928 If @var{part} is not a sub-formula of @var{expr}, it returns @code{nil}.
33929 If @var{part} is @code{eq} to @var{expr}, it returns @code{t}. This
33930 function does not take associativity into account.
33933 @defun calc-replace-sub-formula expr old new
33934 This function returns a copy of formula @var{expr}, with the
33935 sub-formula that is @code{eq} to @var{old} replaced by @var{new}.
33938 @defun simplify expr
33939 Simplify the expression @var{expr} by applying various algebraic rules.
33940 This is what the @w{@kbd{a s}} (@code{calc-simplify}) command uses. This
33941 always returns a copy of the expression; the structure @var{expr} points
33942 to remains unchanged in memory.
33944 More precisely, here is what @code{simplify} does: The expression is
33945 first normalized and evaluated by calling @code{normalize}. If any
33946 @code{AlgSimpRules} have been defined, they are then applied. Then
33947 the expression is traversed in a depth-first, bottom-up fashion; at
33948 each level, any simplifications that can be made are made until no
33949 further changes are possible. Once the entire formula has been
33950 traversed in this way, it is compared with the original formula (from
33951 before the call to @code{normalize}) and, if it has changed,
33952 the entire procedure is repeated (starting with @code{normalize})
33953 until no further changes occur. Usually only two iterations are
33954 needed:@: one to simplify the formula, and another to verify that no
33955 further simplifications were possible.
33958 @defun simplify-extended expr
33959 Simplify the expression @var{expr}, with additional rules enabled that
33960 help do a more thorough job, while not being entirely ``safe'' in all
33961 circumstances. (For example, this mode will simplify @samp{sqrt(x^2)}
33962 to @samp{x}, which is only valid when @var{x} is positive.) This is
33963 implemented by temporarily binding the variable @code{math-living-dangerously}
33964 to @code{t} (using a @code{let} form) and calling @code{simplify}.
33965 Dangerous simplification rules are written to check this variable
33966 before taking any action.
33969 @defun simplify-units expr
33970 Simplify the expression @var{expr}, treating variable names as units
33971 whenever possible. This works by binding the variable
33972 @code{math-simplifying-units} to @code{t} while calling @code{simplify}.
33975 @defmac math-defsimplify funcs body
33976 Register a new simplification rule; this is normally called as a top-level
33977 form, like @code{defun} or @code{defmath}. If @var{funcs} is a symbol
33978 (like @code{+} or @code{calcFunc-sqrt}), this simplification rule is
33979 applied to the formulas which are calls to the specified function. Or,
33980 @var{funcs} can be a list of such symbols; the rule applies to all
33981 functions on the list. The @var{body} is written like the body of a
33982 function with a single argument called @code{expr}. The body will be
33983 executed with @code{expr} bound to a formula which is a call to one of
33984 the functions @var{funcs}. If the function body returns @code{nil}, or
33985 if it returns a result @code{equal} to the original @code{expr}, it is
33986 ignored and Calc goes on to try the next simplification rule that applies.
33987 If the function body returns something different, that new formula is
33988 substituted for @var{expr} in the original formula.
33990 At each point in the formula, rules are tried in the order of the
33991 original calls to @code{math-defsimplify}; the search stops after the
33992 first rule that makes a change. Thus later rules for that same
33993 function will not have a chance to trigger until the next iteration
33994 of the main @code{simplify} loop.
33996 Note that, since @code{defmath} is not being used here, @var{body} must
33997 be written in true Lisp code without the conveniences that @code{defmath}
33998 provides. If you prefer, you can have @var{body} simply call another
33999 function (defined with @code{defmath}) which does the real work.
34001 The arguments of a function call will already have been simplified
34002 before any rules for the call itself are invoked. Since a new argument
34003 list is consed up when this happens, this means that the rule's body is
34004 allowed to rearrange the function's arguments destructively if that is
34005 convenient. Here is a typical example of a simplification rule:
34008 (math-defsimplify calcFunc-arcsinh
34009 (or (and (math-looks-negp (nth 1 expr))
34010 (math-neg (list 'calcFunc-arcsinh
34011 (math-neg (nth 1 expr)))))
34012 (and (eq (car-safe (nth 1 expr)) 'calcFunc-sinh)
34013 (or math-living-dangerously
34014 (math-known-realp (nth 1 (nth 1 expr))))
34015 (nth 1 (nth 1 expr)))))
34018 This is really a pair of rules written with one @code{math-defsimplify}
34019 for convenience; the first replaces @samp{arcsinh(-x)} with
34020 @samp{-arcsinh(x)}, and the second, which is safe only for real @samp{x},
34021 replaces @samp{arcsinh(sinh(x))} with @samp{x}.
34024 @defun common-constant-factor expr
34025 Check @var{expr} to see if it is a sum of terms all multiplied by the
34026 same rational value. If so, return this value. If not, return @code{nil}.
34027 For example, if called on @samp{6x + 9y + 12z}, it would return 3, since
34028 3 is a common factor of all the terms.
34031 @defun cancel-common-factor expr factor
34032 Assuming @var{expr} is a sum with @var{factor} as a common factor,
34033 divide each term of the sum by @var{factor}. This is done by
34034 destructively modifying parts of @var{expr}, on the assumption that
34035 it is being used by a simplification rule (where such things are
34036 allowed; see above). For example, consider this built-in rule for
34040 (math-defsimplify calcFunc-sqrt
34041 (let ((fac (math-common-constant-factor (nth 1 expr))))
34042 (and fac (not (eq fac 1))
34043 (math-mul (math-normalize (list 'calcFunc-sqrt fac))
34045 (list 'calcFunc-sqrt
34046 (math-cancel-common-factor
34047 (nth 1 expr) fac)))))))
34051 @defun frac-gcd a b
34052 Compute a ``rational GCD'' of @var{a} and @var{b}, which must both be
34053 rational numbers. This is the fraction composed of the GCD of the
34054 numerators of @var{a} and @var{b}, over the GCD of the denominators.
34055 It is used by @code{common-constant-factor}. Note that the standard
34056 @code{gcd} function uses the LCM to combine the denominators.
34059 @defun map-tree func expr many
34060 Try applying Lisp function @var{func} to various sub-expressions of
34061 @var{expr}. Initially, call @var{func} with @var{expr} itself as an
34062 argument. If this returns an expression which is not @code{equal} to
34063 @var{expr}, apply @var{func} again until eventually it does return
34064 @var{expr} with no changes. Then, if @var{expr} is a function call,
34065 recursively apply @var{func} to each of the arguments. This keeps going
34066 until no changes occur anywhere in the expression; this final expression
34067 is returned by @code{map-tree}. Note that, unlike simplification rules,
34068 @var{func} functions may @emph{not} make destructive changes to
34069 @var{expr}. If a third argument @var{many} is provided, it is an
34070 integer which says how many times @var{func} may be applied; the
34071 default, as described above, is infinitely many times.
34074 @defun compile-rewrites rules
34075 Compile the rewrite rule set specified by @var{rules}, which should
34076 be a formula that is either a vector or a variable name. If the latter,
34077 the compiled rules are saved so that later @code{compile-rules} calls
34078 for that same variable can return immediately. If there are problems
34079 with the rules, this function calls @code{error} with a suitable
34083 @defun apply-rewrites expr crules heads
34084 Apply the compiled rewrite rule set @var{crules} to the expression
34085 @var{expr}. This will make only one rewrite and only checks at the
34086 top level of the expression. The result @code{nil} if no rules
34087 matched, or if the only rules that matched did not actually change
34088 the expression. The @var{heads} argument is optional; if is given,
34089 it should be a list of all function names that (may) appear in
34090 @var{expr}. The rewrite compiler tags each rule with the
34091 rarest-looking function name in the rule; if you specify @var{heads},
34092 @code{apply-rewrites} can use this information to narrow its search
34093 down to just a few rules in the rule set.
34096 @defun rewrite-heads expr
34097 Compute a @var{heads} list for @var{expr} suitable for use with
34098 @code{apply-rewrites}, as discussed above.
34101 @defun rewrite expr rules many
34102 This is an all-in-one rewrite function. It compiles the rule set
34103 specified by @var{rules}, then uses @code{map-tree} to apply the
34104 rules throughout @var{expr} up to @var{many} (default infinity)
34108 @defun match-patterns pat vec not-flag
34109 Given a Calc vector @var{vec} and an uncompiled pattern set or
34110 pattern set variable @var{pat}, this function returns a new vector
34111 of all elements of @var{vec} which do (or don't, if @var{not-flag} is
34112 non-@code{nil}) match any of the patterns in @var{pat}.
34115 @defun deriv expr var value symb
34116 Compute the derivative of @var{expr} with respect to variable @var{var}
34117 (which may actually be any sub-expression). If @var{value} is specified,
34118 the derivative is evaluated at the value of @var{var}; otherwise, the
34119 derivative is left in terms of @var{var}. If the expression contains
34120 functions for which no derivative formula is known, new derivative
34121 functions are invented by adding primes to the names; @pxref{Calculus}.
34122 However, if @var{symb} is non-@code{nil}, the presence of undifferentiable
34123 functions in @var{expr} instead cancels the whole differentiation, and
34124 @code{deriv} returns @code{nil} instead.
34126 Derivatives of an @var{n}-argument function can be defined by
34127 adding a @code{math-derivative-@var{n}} property to the property list
34128 of the symbol for the function's derivative, which will be the
34129 function name followed by an apostrophe. The value of the property
34130 should be a Lisp function; it is called with the same arguments as the
34131 original function call that is being differentiated. It should return
34132 a formula for the derivative. For example, the derivative of @code{ln}
34136 (put 'calcFunc-ln\' 'math-derivative-1
34137 (function (lambda (u) (math-div 1 u))))
34140 The two-argument @code{log} function has two derivatives,
34142 (put 'calcFunc-log\' 'math-derivative-2 ; d(log(x,b)) / dx
34143 (function (lambda (x b) ... )))
34144 (put 'calcFunc-log\'2 'math-derivative-2 ; d(log(x,b)) / db
34145 (function (lambda (x b) ... )))
34149 @defun tderiv expr var value symb
34150 Compute the total derivative of @var{expr}. This is the same as
34151 @code{deriv}, except that variables other than @var{var} are not
34152 assumed to be constant with respect to @var{var}.
34155 @defun integ expr var low high
34156 Compute the integral of @var{expr} with respect to @var{var}.
34157 @xref{Calculus}, for further details.
34160 @defmac math-defintegral funcs body
34161 Define a rule for integrating a function or functions of one argument;
34162 this macro is very similar in format to @code{math-defsimplify}.
34163 The main difference is that here @var{body} is the body of a function
34164 with a single argument @code{u} which is bound to the argument to the
34165 function being integrated, not the function call itself. Also, the
34166 variable of integration is available as @code{math-integ-var}. If
34167 evaluation of the integral requires doing further integrals, the body
34168 should call @samp{(math-integral @var{x})} to find the integral of
34169 @var{x} with respect to @code{math-integ-var}; this function returns
34170 @code{nil} if the integral could not be done. Some examples:
34173 (math-defintegral calcFunc-conj
34174 (let ((int (math-integral u)))
34176 (list 'calcFunc-conj int))))
34178 (math-defintegral calcFunc-cos
34179 (and (equal u math-integ-var)
34180 (math-from-radians-2 (list 'calcFunc-sin u))))
34183 In the @code{cos} example, we define only the integral of @samp{cos(x) dx},
34184 relying on the general integration-by-substitution facility to handle
34185 cosines of more complicated arguments. An integration rule should return
34186 @code{nil} if it can't do the integral; if several rules are defined for
34187 the same function, they are tried in order until one returns a non-@code{nil}
34191 @defmac math-defintegral-2 funcs body
34192 Define a rule for integrating a function or functions of two arguments.
34193 This is exactly analogous to @code{math-defintegral}, except that @var{body}
34194 is written as the body of a function with two arguments, @var{u} and
34198 @defun solve-for lhs rhs var full
34199 Attempt to solve the equation @samp{@var{lhs} = @var{rhs}} by isolating
34200 the variable @var{var} on the lefthand side; return the resulting righthand
34201 side, or @code{nil} if the equation cannot be solved. The variable
34202 @var{var} must appear at least once in @var{lhs} or @var{rhs}. Note that
34203 the return value is a formula which does not contain @var{var}; this is
34204 different from the user-level @code{solve} and @code{finv} functions,
34205 which return a rearranged equation or a functional inverse, respectively.
34206 If @var{full} is non-@code{nil}, a full solution including dummy signs
34207 and dummy integers will be produced. User-defined inverses are provided
34208 as properties in a manner similar to derivatives:
34211 (put 'calcFunc-ln 'math-inverse
34212 (function (lambda (x) (list 'calcFunc-exp x))))
34215 This function can call @samp{(math-solve-get-sign @var{x})} to create
34216 a new arbitrary sign variable, returning @var{x} times that sign, and
34217 @samp{(math-solve-get-int @var{x})} to create a new arbitrary integer
34218 variable multiplied by @var{x}. These functions simply return @var{x}
34219 if the caller requested a non-``full'' solution.
34222 @defun solve-eqn expr var full
34223 This version of @code{solve-for} takes an expression which will
34224 typically be an equation or inequality. (If it is not, it will be
34225 interpreted as the equation @samp{@var{expr} = 0}.) It returns an
34226 equation or inequality, or @code{nil} if no solution could be found.
34229 @defun solve-system exprs vars full
34230 This function solves a system of equations. Generally, @var{exprs}
34231 and @var{vars} will be vectors of equal length.
34232 @xref{Solving Systems of Equations}, for other options.
34235 @defun expr-contains expr var
34236 Returns a non-@code{nil} value if @var{var} occurs as a subexpression
34239 This function might seem at first to be identical to
34240 @code{calc-find-sub-formula}. The key difference is that
34241 @code{expr-contains} uses @code{equal} to test for matches, whereas
34242 @code{calc-find-sub-formula} uses @code{eq}. In the formula
34243 @samp{f(a, a)}, the two @samp{a}s will be @code{equal} but not
34244 @code{eq} to each other.
34247 @defun expr-contains-count expr var
34248 Returns the number of occurrences of @var{var} as a subexpression
34249 of @var{expr}, or @code{nil} if there are no occurrences.
34252 @defun expr-depends expr var
34253 Returns true if @var{expr} refers to any variable the occurs in @var{var}.
34254 In other words, it checks if @var{expr} and @var{var} have any variables
34258 @defun expr-contains-vars expr
34259 Return true if @var{expr} contains any variables, or @code{nil} if @var{expr}
34260 contains only constants and functions with constant arguments.
34263 @defun expr-subst expr old new
34264 Returns a copy of @var{expr}, with all occurrences of @var{old} replaced
34265 by @var{new}. This treats @code{lambda} forms specially with respect
34266 to the dummy argument variables, so that the effect is always to return
34267 @var{expr} evaluated at @var{old} = @var{new}.
34270 @defun multi-subst expr old new
34271 This is like @code{expr-subst}, except that @var{old} and @var{new}
34272 are lists of expressions to be substituted simultaneously. If one
34273 list is shorter than the other, trailing elements of the longer list
34277 @defun expr-weight expr
34278 Returns the ``weight'' of @var{expr}, basically a count of the total
34279 number of objects and function calls that appear in @var{expr}. For
34280 ``primitive'' objects, this will be one.
34283 @defun expr-height expr
34284 Returns the ``height'' of @var{expr}, which is the deepest level to
34285 which function calls are nested. (Note that @samp{@var{a} + @var{b}}
34286 counts as a function call.) For primitive objects, this returns zero.
34289 @defun polynomial-p expr var
34290 Check if @var{expr} is a polynomial in variable (or sub-expression)
34291 @var{var}. If so, return the degree of the polynomial, that is, the
34292 highest power of @var{var} that appears in @var{expr}. For example,
34293 for @samp{(x^2 + 3)^3 + 4} this would return 6. This function returns
34294 @code{nil} unless @var{expr}, when expanded out by @kbd{a x}
34295 (@code{calc-expand}), would consist of a sum of terms in which @var{var}
34296 appears only raised to nonnegative integer powers. Note that if
34297 @var{var} does not occur in @var{expr}, then @var{expr} is considered
34298 a polynomial of degree 0.
34301 @defun is-polynomial expr var degree loose
34302 Check if @var{expr} is a polynomial in variable or sub-expression
34303 @var{var}, and, if so, return a list representation of the polynomial
34304 where the elements of the list are coefficients of successive powers of
34305 @var{var}: @samp{@var{a} + @var{b} x + @var{c} x^3} would produce the
34306 list @samp{(@var{a} @var{b} 0 @var{c})}, and @samp{(x + 1)^2} would
34307 produce the list @samp{(1 2 1)}. The highest element of the list will
34308 be non-zero, with the special exception that if @var{expr} is the
34309 constant zero, the returned value will be @samp{(0)}. Return @code{nil}
34310 if @var{expr} is not a polynomial in @var{var}. If @var{degree} is
34311 specified, this will not consider polynomials of degree higher than that
34312 value. This is a good precaution because otherwise an input of
34313 @samp{(x+1)^1000} will cause a huge coefficient list to be built. If
34314 @var{loose} is non-@code{nil}, then a looser definition of a polynomial
34315 is used in which coefficients are no longer required not to depend on
34316 @var{var}, but are only required not to take the form of polynomials
34317 themselves. For example, @samp{sin(x) x^2 + cos(x)} is a loose
34318 polynomial with coefficients @samp{((calcFunc-cos x) 0 (calcFunc-sin
34319 x))}. The result will never be @code{nil} in loose mode, since any
34320 expression can be interpreted as a ``constant'' loose polynomial.
34323 @defun polynomial-base expr pred
34324 Check if @var{expr} is a polynomial in any variable that occurs in it;
34325 if so, return that variable. (If @var{expr} is a multivariate polynomial,
34326 this chooses one variable arbitrarily.) If @var{pred} is specified, it should
34327 be a Lisp function which is called as @samp{(@var{pred} @var{subexpr})},
34328 and which should return true if @code{mpb-top-expr} (a global name for
34329 the original @var{expr}) is a suitable polynomial in @var{subexpr}.
34330 The default predicate uses @samp{(polynomial-p mpb-top-expr @var{subexpr})};
34331 you can use @var{pred} to specify additional conditions. Or, you could
34332 have @var{pred} build up a list of every suitable @var{subexpr} that
34336 @defun poly-simplify poly
34337 Simplify polynomial coefficient list @var{poly} by (destructively)
34338 clipping off trailing zeros.
34341 @defun poly-mix a ac b bc
34342 Mix two polynomial lists @var{a} and @var{b} (in the form returned by
34343 @code{is-polynomial}) in a linear combination with coefficient expressions
34344 @var{ac} and @var{bc}. The result is a (not necessarily simplified)
34345 polynomial list representing @samp{@var{ac} @var{a} + @var{bc} @var{b}}.
34348 @defun poly-mul a b
34349 Multiply two polynomial coefficient lists @var{a} and @var{b}. The
34350 result will be in simplified form if the inputs were simplified.
34353 @defun build-polynomial-expr poly var
34354 Construct a Calc formula which represents the polynomial coefficient
34355 list @var{poly} applied to variable @var{var}. The @kbd{a c}
34356 (@code{calc-collect}) command uses @code{is-polynomial} to turn an
34357 expression into a coefficient list, then @code{build-polynomial-expr}
34358 to turn the list back into an expression in regular form.
34361 @defun check-unit-name var
34362 Check if @var{var} is a variable which can be interpreted as a unit
34363 name. If so, return the units table entry for that unit. This
34364 will be a list whose first element is the unit name (not counting
34365 prefix characters) as a symbol and whose second element is the
34366 Calc expression which defines the unit. (Refer to the Calc sources
34367 for details on the remaining elements of this list.) If @var{var}
34368 is not a variable or is not a unit name, return @code{nil}.
34371 @defun units-in-expr-p expr sub-exprs
34372 Return true if @var{expr} contains any variables which can be
34373 interpreted as units. If @var{sub-exprs} is @code{t}, the entire
34374 expression is searched. If @var{sub-exprs} is @code{nil}, this
34375 checks whether @var{expr} is directly a units expression.
34378 @defun single-units-in-expr-p expr
34379 Check whether @var{expr} contains exactly one units variable. If so,
34380 return the units table entry for the variable. If @var{expr} does
34381 not contain any units, return @code{nil}. If @var{expr} contains
34382 two or more units, return the symbol @code{wrong}.
34385 @defun to-standard-units expr which
34386 Convert units expression @var{expr} to base units. If @var{which}
34387 is @code{nil}, use Calc's native base units. Otherwise, @var{which}
34388 can specify a units system, which is a list of two-element lists,
34389 where the first element is a Calc base symbol name and the second
34390 is an expression to substitute for it.
34393 @defun remove-units expr
34394 Return a copy of @var{expr} with all units variables replaced by ones.
34395 This expression is generally normalized before use.
34398 @defun extract-units expr
34399 Return a copy of @var{expr} with everything but units variables replaced
34403 @node Formatting Lisp Functions, Hooks, Symbolic Lisp Functions, Internals
34404 @subsubsection I/O and Formatting Functions
34407 The functions described here are responsible for parsing and formatting
34408 Calc numbers and formulas.
34410 @defun calc-eval str sep arg1 arg2 @dots{}
34411 This is the simplest interface to the Calculator from another Lisp program.
34412 @xref{Calling Calc from Your Programs}.
34415 @defun read-number str
34416 If string @var{str} contains a valid Calc number, either integer,
34417 fraction, float, or HMS form, this function parses and returns that
34418 number. Otherwise, it returns @code{nil}.
34421 @defun read-expr str
34422 Read an algebraic expression from string @var{str}. If @var{str} does
34423 not have the form of a valid expression, return a list of the form
34424 @samp{(error @var{pos} @var{msg})} where @var{pos} is an integer index
34425 into @var{str} of the general location of the error, and @var{msg} is
34426 a string describing the problem.
34429 @defun read-exprs str
34430 Read a list of expressions separated by commas, and return it as a
34431 Lisp list. If an error occurs in any expressions, an error list as
34432 shown above is returned instead.
34435 @defun calc-do-alg-entry initial prompt no-norm
34436 Read an algebraic formula or formulas using the minibuffer. All
34437 conventions of regular algebraic entry are observed. The return value
34438 is a list of Calc formulas; there will be more than one if the user
34439 entered a list of values separated by commas. The result is @code{nil}
34440 if the user presses Return with a blank line. If @var{initial} is
34441 given, it is a string which the minibuffer will initially contain.
34442 If @var{prompt} is given, it is the prompt string to use; the default
34443 is ``Algebraic:''. If @var{no-norm} is @code{t}, the formulas will
34444 be returned exactly as parsed; otherwise, they will be passed through
34445 @code{calc-normalize} first.
34447 To support the use of @kbd{$} characters in the algebraic entry, use
34448 @code{let} to bind @code{calc-dollar-values} to a list of the values
34449 to be substituted for @kbd{$}, @kbd{$$}, and so on, and bind
34450 @code{calc-dollar-used} to 0. Upon return, @code{calc-dollar-used}
34451 will have been changed to the highest number of consecutive @kbd{$}s
34452 that actually appeared in the input.
34455 @defun format-number a
34456 Convert the real or complex number or HMS form @var{a} to string form.
34459 @defun format-flat-expr a prec
34460 Convert the arbitrary Calc number or formula @var{a} to string form,
34461 in the style used by the trail buffer and the @code{calc-edit} command.
34462 This is a simple format designed
34463 mostly to guarantee the string is of a form that can be re-parsed by
34464 @code{read-expr}. Most formatting modes, such as digit grouping,
34465 complex number format, and point character, are ignored to ensure the
34466 result will be re-readable. The @var{prec} parameter is normally 0; if
34467 you pass a large integer like 1000 instead, the expression will be
34468 surrounded by parentheses unless it is a plain number or variable name.
34471 @defun format-nice-expr a width
34472 This is like @code{format-flat-expr} (with @var{prec} equal to 0),
34473 except that newlines will be inserted to keep lines down to the
34474 specified @var{width}, and vectors that look like matrices or rewrite
34475 rules are written in a pseudo-matrix format. The @code{calc-edit}
34476 command uses this when only one stack entry is being edited.
34479 @defun format-value a width
34480 Convert the Calc number or formula @var{a} to string form, using the
34481 format seen in the stack buffer. Beware the string returned may
34482 not be re-readable by @code{read-expr}, for example, because of digit
34483 grouping. Multi-line objects like matrices produce strings that
34484 contain newline characters to separate the lines. The @var{w}
34485 parameter, if given, is the target window size for which to format
34486 the expressions. If @var{w} is omitted, the width of the Calculator
34490 @defun compose-expr a prec
34491 Format the Calc number or formula @var{a} according to the current
34492 language mode, returning a ``composition.'' To learn about the
34493 structure of compositions, see the comments in the Calc source code.
34494 You can specify the format of a given type of function call by putting
34495 a @code{math-compose-@var{lang}} property on the function's symbol,
34496 whose value is a Lisp function that takes @var{a} and @var{prec} as
34497 arguments and returns a composition. Here @var{lang} is a language
34498 mode name, one of @code{normal}, @code{big}, @code{c}, @code{pascal},
34499 @code{fortran}, @code{tex}, @code{eqn}, @code{math}, or @code{maple}.
34500 In Big mode, Calc actually tries @code{math-compose-big} first, then
34501 tries @code{math-compose-normal}. If this property does not exist,
34502 or if the function returns @code{nil}, the function is written in the
34503 normal function-call notation for that language.
34506 @defun composition-to-string c w
34507 Convert a composition structure returned by @code{compose-expr} into
34508 a string. Multi-line compositions convert to strings containing
34509 newline characters. The target window size is given by @var{w}.
34510 The @code{format-value} function basically calls @code{compose-expr}
34511 followed by @code{composition-to-string}.
34514 @defun comp-width c
34515 Compute the width in characters of composition @var{c}.
34518 @defun comp-height c
34519 Compute the height in lines of composition @var{c}.
34522 @defun comp-ascent c
34523 Compute the portion of the height of composition @var{c} which is on or
34524 above the baseline. For a one-line composition, this will be one.
34527 @defun comp-descent c
34528 Compute the portion of the height of composition @var{c} which is below
34529 the baseline. For a one-line composition, this will be zero.
34532 @defun comp-first-char c
34533 If composition @var{c} is a ``flat'' composition, return the first
34534 (leftmost) character of the composition as an integer. Otherwise,
34538 @defun comp-last-char c
34539 If composition @var{c} is a ``flat'' composition, return the last
34540 (rightmost) character, otherwise return @code{nil}.
34543 @comment @node Lisp Variables, Hooks, Formatting Lisp Functions, Internals
34544 @comment @subsubsection Lisp Variables
34547 @comment (This section is currently unfinished.)
34549 @node Hooks, , Formatting Lisp Functions, Internals
34550 @subsubsection Hooks
34553 Hooks are variables which contain Lisp functions (or lists of functions)
34554 which are called at various times. Calc defines a number of hooks
34555 that help you to customize it in various ways. Calc uses the Lisp
34556 function @code{run-hooks} to invoke the hooks shown below. Several
34557 other customization-related variables are also described here.
34559 @defvar calc-load-hook
34560 This hook is called at the end of @file{calc.el}, after the file has
34561 been loaded, before any functions in it have been called, but after
34562 @code{calc-mode-map} and similar variables have been set up.
34565 @defvar calc-ext-load-hook
34566 This hook is called at the end of @file{calc-ext.el}.
34569 @defvar calc-start-hook
34570 This hook is called as the last step in a @kbd{M-x calc} command.
34571 At this point, the Calc buffer has been created and initialized if
34572 necessary, the Calc window and trail window have been created,
34573 and the ``Welcome to Calc'' message has been displayed.
34576 @defvar calc-mode-hook
34577 This hook is called when the Calc buffer is being created. Usually
34578 this will only happen once per Emacs session. The hook is called
34579 after Emacs has switched to the new buffer, the mode-settings file
34580 has been read if necessary, and all other buffer-local variables
34581 have been set up. After this hook returns, Calc will perform a
34582 @code{calc-refresh} operation, set up the mode line display, then
34583 evaluate any deferred @code{calc-define} properties that have not
34584 been evaluated yet.
34587 @defvar calc-trail-mode-hook
34588 This hook is called when the Calc Trail buffer is being created.
34589 It is called as the very last step of setting up the Trail buffer.
34590 Like @code{calc-mode-hook}, this will normally happen only once
34594 @defvar calc-end-hook
34595 This hook is called by @code{calc-quit}, generally because the user
34596 presses @kbd{q} or @kbd{C-x * c} while in Calc. The Calc buffer will
34597 be the current buffer. The hook is called as the very first
34598 step, before the Calc window is destroyed.
34601 @defvar calc-window-hook
34602 If this hook is non-@code{nil}, it is called to create the Calc window.
34603 Upon return, this new Calc window should be the current window.
34604 (The Calc buffer will already be the current buffer when the
34605 hook is called.) If the hook is not defined, Calc will
34606 generally use @code{split-window}, @code{set-window-buffer},
34607 and @code{select-window} to create the Calc window.
34610 @defvar calc-trail-window-hook
34611 If this hook is non-@code{nil}, it is called to create the Calc Trail
34612 window. The variable @code{calc-trail-buffer} will contain the buffer
34613 which the window should use. Unlike @code{calc-window-hook}, this hook
34614 must @emph{not} switch into the new window.
34617 @defvar calc-embedded-mode-hook
34618 This hook is called the first time that Embedded mode is entered.
34621 @defvar calc-embedded-new-buffer-hook
34622 This hook is called each time that Embedded mode is entered in a
34626 @defvar calc-embedded-new-formula-hook
34627 This hook is called each time that Embedded mode is enabled for a
34631 @defvar calc-edit-mode-hook
34632 This hook is called by @code{calc-edit} (and the other ``edit''
34633 commands) when the temporary editing buffer is being created.
34634 The buffer will have been selected and set up to be in
34635 @code{calc-edit-mode}, but will not yet have been filled with
34636 text. (In fact it may still have leftover text from a previous
34637 @code{calc-edit} command.)
34640 @defvar calc-mode-save-hook
34641 This hook is called by the @code{calc-save-modes} command,
34642 after Calc's own mode features have been inserted into the
34643 Calc init file and just before the ``End of mode settings''
34644 message is inserted.
34647 @defvar calc-reset-hook
34648 This hook is called after @kbd{C-x * 0} (@code{calc-reset}) has
34649 reset all modes. The Calc buffer will be the current buffer.
34652 @defvar calc-other-modes
34653 This variable contains a list of strings. The strings are
34654 concatenated at the end of the modes portion of the Calc
34655 mode line (after standard modes such as ``Deg'', ``Inv'' and
34656 ``Hyp''). Each string should be a short, single word followed
34657 by a space. The variable is @code{nil} by default.
34660 @defvar calc-mode-map
34661 This is the keymap that is used by Calc mode. The best time
34662 to adjust it is probably in a @code{calc-mode-hook}. If the
34663 Calc extensions package (@file{calc-ext.el}) has not yet been
34664 loaded, many of these keys will be bound to @code{calc-missing-key},
34665 which is a command that loads the extensions package and
34666 ``retypes'' the key. If your @code{calc-mode-hook} rebinds
34667 one of these keys, it will probably be overridden when the
34668 extensions are loaded.
34671 @defvar calc-digit-map
34672 This is the keymap that is used during numeric entry. Numeric
34673 entry uses the minibuffer, but this map binds every non-numeric
34674 key to @code{calcDigit-nondigit} which generally calls
34675 @code{exit-minibuffer} and ``retypes'' the key.
34678 @defvar calc-alg-ent-map
34679 This is the keymap that is used during algebraic entry. This is
34680 mostly a copy of @code{minibuffer-local-map}.
34683 @defvar calc-store-var-map
34684 This is the keymap that is used during entry of variable names for
34685 commands like @code{calc-store} and @code{calc-recall}. This is
34686 mostly a copy of @code{minibuffer-local-completion-map}.
34689 @defvar calc-edit-mode-map
34690 This is the (sparse) keymap used by @code{calc-edit} and other
34691 temporary editing commands. It binds @key{RET}, @key{LFD},
34692 and @kbd{C-c C-c} to @code{calc-edit-finish}.
34695 @defvar calc-mode-var-list
34696 This is a list of variables which are saved by @code{calc-save-modes}.
34697 Each entry is a list of two items, the variable (as a Lisp symbol)
34698 and its default value. When modes are being saved, each variable
34699 is compared with its default value (using @code{equal}) and any
34700 non-default variables are written out.
34703 @defvar calc-local-var-list
34704 This is a list of variables which should be buffer-local to the
34705 Calc buffer. Each entry is a variable name (as a Lisp symbol).
34706 These variables also have their default values manipulated by
34707 the @code{calc} and @code{calc-quit} commands; @pxref{Multiple Calculators}.
34708 Since @code{calc-mode-hook} is called after this list has been
34709 used the first time, your hook should add a variable to the
34710 list and also call @code{make-local-variable} itself.
34713 @node Copying, GNU Free Documentation License, Programming, Top
34714 @appendix GNU GENERAL PUBLIC LICENSE
34717 @node GNU Free Documentation License, Customizing Calc, Copying, Top
34718 @appendix GNU Free Documentation License
34719 @include doclicense.texi
34721 @node Customizing Calc, Reporting Bugs, GNU Free Documentation License, Top
34722 @appendix Customizing Calc
34724 The usual prefix for Calc is the key sequence @kbd{C-x *}. If you wish
34725 to use a different prefix, you can put
34728 (global-set-key "NEWPREFIX" 'calc-dispatch)
34732 in your .emacs file.
34733 (@xref{Key Bindings,,Customizing Key Bindings,emacs,
34734 The GNU Emacs Manual}, for more information on binding keys.)
34735 A convenient way to start Calc is with @kbd{C-x * *}; to make it equally
34736 convenient for users who use a different prefix, the prefix can be
34737 followed by @kbd{=}, @kbd{&}, @kbd{#}, @kbd{\}, @kbd{/}, @kbd{+} or
34738 @kbd{-} as well as @kbd{*} to start Calc, and so in many cases the last
34739 character of the prefix can simply be typed twice.
34741 Calc is controlled by many variables, most of which can be reset
34742 from within Calc. Some variables are less involved with actual
34743 calculation, and can be set outside of Calc using Emacs's
34744 customization facilities. These variables are listed below.
34745 Typing @kbd{M-x customize-variable RET @var{variable-name} RET}
34746 will bring up a buffer in which the variable's value can be redefined.
34747 Typing @kbd{M-x customize-group RET calc RET} will bring up a buffer which
34748 contains all of Calc's customizable variables. (These variables can
34749 also be reset by putting the appropriate lines in your .emacs file;
34750 @xref{Init File, ,Init File, emacs, The GNU Emacs Manual}.)
34752 Some of the customizable variables are regular expressions. A regular
34753 expression is basically a pattern that Calc can search for.
34754 See @ref{Regexp Search,, Regular Expression Search, emacs, The GNU Emacs Manual}
34755 to see how regular expressions work.
34757 @defvar calc-settings-file
34758 The variable @code{calc-settings-file} holds the file name in
34759 which commands like @kbd{m m} and @kbd{Z P} store ``permanent''
34761 If @code{calc-settings-file} is not your user init file (typically
34762 @file{~/.emacs}) and if the variable @code{calc-loaded-settings-file} is
34763 @code{nil}, then Calc will automatically load your settings file (if it
34764 exists) the first time Calc is invoked.
34766 The default value for this variable is @code{"~/.calc.el"}.
34769 @defvar calc-gnuplot-name
34770 See @ref{Graphics}.@*
34771 The variable @code{calc-gnuplot-name} should be the name of the
34772 GNUPLOT program (a string). If you have GNUPLOT installed on your
34773 system but Calc is unable to find it, you may need to set this
34774 variable. You may also need to set some Lisp variables to show Calc how
34775 to run GNUPLOT on your system, see @ref{Devices, ,Graphical Devices} .
34776 The default value of @code{calc-gnuplot-name} is @code{"gnuplot"}.
34779 @defvar calc-gnuplot-plot-command
34780 @defvarx calc-gnuplot-print-command
34781 See @ref{Devices, ,Graphical Devices}.@*
34782 The variables @code{calc-gnuplot-plot-command} and
34783 @code{calc-gnuplot-print-command} represent system commands to
34784 display and print the output of GNUPLOT, respectively. These may be
34785 @code{nil} if no command is necessary, or strings which can include
34786 @samp{%s} to signify the name of the file to be displayed or printed.
34787 Or, these variables may contain Lisp expressions which are evaluated
34788 to display or print the output.
34790 The default value of @code{calc-gnuplot-plot-command} is @code{nil},
34791 and the default value of @code{calc-gnuplot-print-command} is
34795 @defvar calc-language-alist
34796 See @ref{Basic Embedded Mode}.@*
34797 The variable @code{calc-language-alist} controls the languages that
34798 Calc will associate with major modes. When Calc embedded mode is
34799 enabled, it will try to use the current major mode to
34800 determine what language should be used. (This can be overridden using
34801 Calc's mode changing commands, @xref{Mode Settings in Embedded Mode}.)
34802 The variable @code{calc-language-alist} consists of a list of pairs of
34803 the form @code{(@var{MAJOR-MODE} . @var{LANGUAGE})}; for example,
34804 @code{(latex-mode . latex)} is one such pair. If Calc embedded is
34805 activated in a buffer whose major mode is @var{MAJOR-MODE}, it will set itself
34806 to use the language @var{LANGUAGE}.
34808 The default value of @code{calc-language-alist} is
34810 ((latex-mode . latex)
34812 (plain-tex-mode . tex)
34813 (context-mode . tex)
34815 (pascal-mode . pascal)
34818 (fortran-mode . fortran)
34819 (f90-mode . fortran))
34823 @defvar calc-embedded-announce-formula
34824 @defvarx calc-embedded-announce-formula-alist
34825 See @ref{Customizing Embedded Mode}.@*
34826 The variable @code{calc-embedded-announce-formula} helps determine
34827 what formulas @kbd{C-x * a} will activate in a buffer. It is a
34828 regular expression, and when activating embedded formulas with
34829 @kbd{C-x * a}, it will tell Calc that what follows is a formula to be
34830 activated. (Calc also uses other patterns to find formulas, such as
34831 @samp{=>} and @samp{:=}.)
34833 The default pattern is @code{"%Embed\n\\(% .*\n\\)*"}, which checks
34834 for @samp{%Embed} followed by any number of lines beginning with
34835 @samp{%} and a space.
34837 The variable @code{calc-embedded-announce-formula-alist} is used to
34838 set @code{calc-embedded-announce-formula} to different regular
34839 expressions depending on the major mode of the editing buffer.
34840 It consists of a list of pairs of the form @code{(@var{MAJOR-MODE} .
34841 @var{REGEXP})}, and its default value is
34843 ((c++-mode . "//Embed\n\\(// .*\n\\)*")
34844 (c-mode . "/\\*Embed\\*/\n\\(/\\* .*\\*/\n\\)*")
34845 (f90-mode . "!Embed\n\\(! .*\n\\)*")
34846 (fortran-mode . "C Embed\n\\(C .*\n\\)*")
34847 (html-helper-mode . "<!-- Embed -->\n\\(<!-- .* -->\n\\)*")
34848 (html-mode . "<!-- Embed -->\n\\(<!-- .* -->\n\\)*")
34849 (nroff-mode . "\\\\\"Embed\n\\(\\\\\" .*\n\\)*")
34850 (pascal-mode . "@{Embed@}\n\\(@{.*@}\n\\)*")
34851 (sgml-mode . "<!-- Embed -->\n\\(<!-- .* -->\n\\)*")
34852 (xml-mode . "<!-- Embed -->\n\\(<!-- .* -->\n\\)*")
34853 (texinfo-mode . "@@c Embed\n\\(@@c .*\n\\)*"))
34855 Any major modes added to @code{calc-embedded-announce-formula-alist}
34856 should also be added to @code{calc-embedded-open-close-plain-alist}
34857 and @code{calc-embedded-open-close-mode-alist}.
34860 @defvar calc-embedded-open-formula
34861 @defvarx calc-embedded-close-formula
34862 @defvarx calc-embedded-open-close-formula-alist
34863 See @ref{Customizing Embedded Mode}.@*
34864 The variables @code{calc-embedded-open-formula} and
34865 @code{calc-embedded-close-formula} control the region that Calc will
34866 activate as a formula when Embedded mode is entered with @kbd{C-x * e}.
34867 They are regular expressions;
34868 Calc normally scans backward and forward in the buffer for the
34869 nearest text matching these regular expressions to be the ``formula
34872 The simplest delimiters are blank lines. Other delimiters that
34873 Embedded mode understands by default are:
34876 The @TeX{} and La@TeX{} math delimiters @samp{$ $}, @samp{$$ $$},
34877 @samp{\[ \]}, and @samp{\( \)};
34879 Lines beginning with @samp{\begin} and @samp{\end} (except matrix delimiters);
34881 Lines beginning with @samp{@@} (Texinfo delimiters).
34883 Lines beginning with @samp{.EQ} and @samp{.EN} (@dfn{eqn} delimiters);
34885 Lines containing a single @samp{%} or @samp{.\"} symbol and nothing else.
34888 The variable @code{calc-embedded-open-close-formula-alist} is used to
34889 set @code{calc-embedded-open-formula} and
34890 @code{calc-embedded-close-formula} to different regular
34891 expressions depending on the major mode of the editing buffer.
34892 It consists of a list of lists of the form
34893 @code{(@var{MAJOR-MODE} @var{OPEN-FORMULA-REGEXP}
34894 @var{CLOSE-FORMULA-REGEXP})}, and its default value is
34898 @defvar calc-embedded-word-regexp
34899 @defvarx calc-embedded-word-regexp-alist
34900 See @ref{Customizing Embedded Mode}.@*
34901 The variable @code{calc-embedded-word-regexp} determines the expression
34902 that Calc will activate when Embedded mode is entered with @kbd{C-x *
34903 w}. It is a regular expressions.
34905 The default value of @code{calc-embedded-word-regexp} is
34906 @code{"[-+]?[0-9]+\\(\\.[0-9]+\\)?\\([eE][-+]?[0-9]+\\)?"}.
34908 The variable @code{calc-embedded-word-regexp-alist} is used to
34909 set @code{calc-embedded-word-regexp} to a different regular
34910 expression depending on the major mode of the editing buffer.
34911 It consists of a list of lists of the form
34912 @code{(@var{MAJOR-MODE} @var{WORD-REGEXP})}, and its default value is
34916 @defvar calc-embedded-open-plain
34917 @defvarx calc-embedded-close-plain
34918 @defvarx calc-embedded-open-close-plain-alist
34919 See @ref{Customizing Embedded Mode}.@*
34920 The variables @code{calc-embedded-open-plain} and
34921 @code{calc-embedded-open-plain} are used to delimit ``plain''
34922 formulas. Note that these are actual strings, not regular
34923 expressions, because Calc must be able to write these string into a
34924 buffer as well as to recognize them.
34926 The default string for @code{calc-embedded-open-plain} is
34927 @code{"%%% "}, note the trailing space. The default string for
34928 @code{calc-embedded-close-plain} is @code{" %%%\n"}, without
34929 the trailing newline here, the first line of a Big mode formula
34930 that followed might be shifted over with respect to the other lines.
34932 The variable @code{calc-embedded-open-close-plain-alist} is used to
34933 set @code{calc-embedded-open-plain} and
34934 @code{calc-embedded-close-plain} to different strings
34935 depending on the major mode of the editing buffer.
34936 It consists of a list of lists of the form
34937 @code{(@var{MAJOR-MODE} @var{OPEN-PLAIN-STRING}
34938 @var{CLOSE-PLAIN-STRING})}, and its default value is
34940 ((c++-mode "// %% " " %%\n")
34941 (c-mode "/* %% " " %% */\n")
34942 (f90-mode "! %% " " %%\n")
34943 (fortran-mode "C %% " " %%\n")
34944 (html-helper-mode "<!-- %% " " %% -->\n")
34945 (html-mode "<!-- %% " " %% -->\n")
34946 (nroff-mode "\\\" %% " " %%\n")
34947 (pascal-mode "@{%% " " %%@}\n")
34948 (sgml-mode "<!-- %% " " %% -->\n")
34949 (xml-mode "<!-- %% " " %% -->\n")
34950 (texinfo-mode "@@c %% " " %%\n"))
34952 Any major modes added to @code{calc-embedded-open-close-plain-alist}
34953 should also be added to @code{calc-embedded-announce-formula-alist}
34954 and @code{calc-embedded-open-close-mode-alist}.
34957 @defvar calc-embedded-open-new-formula
34958 @defvarx calc-embedded-close-new-formula
34959 @defvarx calc-embedded-open-close-new-formula-alist
34960 See @ref{Customizing Embedded Mode}.@*
34961 The variables @code{calc-embedded-open-new-formula} and
34962 @code{calc-embedded-close-new-formula} are strings which are
34963 inserted before and after a new formula when you type @kbd{C-x * f}.
34965 The default value of @code{calc-embedded-open-new-formula} is
34966 @code{"\n\n"}. If this string begins with a newline character and the
34967 @kbd{C-x * f} is typed at the beginning of a line, @kbd{C-x * f} will skip
34968 this first newline to avoid introducing unnecessary blank lines in the
34969 file. The default value of @code{calc-embedded-close-new-formula} is
34970 also @code{"\n\n"}. The final newline is omitted by @w{@kbd{C-x * f}}
34971 if typed at the end of a line. (It follows that if @kbd{C-x * f} is
34972 typed on a blank line, both a leading opening newline and a trailing
34973 closing newline are omitted.)
34975 The variable @code{calc-embedded-open-close-new-formula-alist} is used to
34976 set @code{calc-embedded-open-new-formula} and
34977 @code{calc-embedded-close-new-formula} to different strings
34978 depending on the major mode of the editing buffer.
34979 It consists of a list of lists of the form
34980 @code{(@var{MAJOR-MODE} @var{OPEN-NEW-FORMULA-STRING}
34981 @var{CLOSE-NEW-FORMULA-STRING})}, and its default value is
34985 @defvar calc-embedded-open-mode
34986 @defvarx calc-embedded-close-mode
34987 @defvarx calc-embedded-open-close-mode-alist
34988 See @ref{Customizing Embedded Mode}.@*
34989 The variables @code{calc-embedded-open-mode} and
34990 @code{calc-embedded-close-mode} are strings which Calc will place before
34991 and after any mode annotations that it inserts. Calc never scans for
34992 these strings; Calc always looks for the annotation itself, so it is not
34993 necessary to add them to user-written annotations.
34995 The default value of @code{calc-embedded-open-mode} is @code{"% "}
34996 and the default value of @code{calc-embedded-close-mode} is
34998 If you change the value of @code{calc-embedded-close-mode}, it is a good
34999 idea still to end with a newline so that mode annotations will appear on
35000 lines by themselves.
35002 The variable @code{calc-embedded-open-close-mode-alist} is used to
35003 set @code{calc-embedded-open-mode} and
35004 @code{calc-embedded-close-mode} to different strings
35005 expressions depending on the major mode of the editing buffer.
35006 It consists of a list of lists of the form
35007 @code{(@var{MAJOR-MODE} @var{OPEN-MODE-STRING}
35008 @var{CLOSE-MODE-STRING})}, and its default value is
35010 ((c++-mode "// " "\n")
35011 (c-mode "/* " " */\n")
35012 (f90-mode "! " "\n")
35013 (fortran-mode "C " "\n")
35014 (html-helper-mode "<!-- " " -->\n")
35015 (html-mode "<!-- " " -->\n")
35016 (nroff-mode "\\\" " "\n")
35017 (pascal-mode "@{ " " @}\n")
35018 (sgml-mode "<!-- " " -->\n")
35019 (xml-mode "<!-- " " -->\n")
35020 (texinfo-mode "@@c " "\n"))
35022 Any major modes added to @code{calc-embedded-open-close-mode-alist}
35023 should also be added to @code{calc-embedded-announce-formula-alist}
35024 and @code{calc-embedded-open-close-plain-alist}.
35027 @defvar calc-multiplication-has-precedence
35028 The variable @code{calc-multiplication-has-precedence} determines
35029 whether multiplication has precedence over division in algebraic
35030 formulas in normal language modes. If
35031 @code{calc-multiplication-has-precedence} is non-@code{nil}, then
35032 multiplication has precedence (and, for certain obscure reasons, is
35033 right associative), and so for example @samp{a/b*c} will be interpreted
35034 as @samp{a/(b*c)}. If @code{calc-multiplication-has-precedence} is
35035 @code{nil}, then multiplication has the same precedence as division
35036 (and, like division, is left associative), and so for example
35037 @samp{a/b*c} will be interpreted as @samp{(a/b)*c}. The default value
35038 of @code{calc-multiplication-has-precedence} is @code{t}.
35041 @node Reporting Bugs, Summary, Customizing Calc, Top
35042 @appendix Reporting Bugs
35045 If you find a bug in Calc, send e-mail to Jay Belanger,
35048 jay.p.belanger@@gmail.com
35052 There is an automatic command @kbd{M-x report-calc-bug} which helps
35053 you to report bugs. This command prompts you for a brief subject
35054 line, then leaves you in a mail editing buffer. Type @kbd{C-c C-c} to
35055 send your mail. Make sure your subject line indicates that you are
35056 reporting a Calc bug; this command sends mail to the maintainer's
35059 If you have suggestions for additional features for Calc, please send
35060 them. Some have dared to suggest that Calc is already top-heavy with
35061 features; this obviously cannot be the case, so if you have ideas, send
35064 At the front of the source file, @file{calc.el}, is a list of ideas for
35065 future work. If any enthusiastic souls wish to take it upon themselves
35066 to work on these, please send a message (using @kbd{M-x report-calc-bug})
35067 so any efforts can be coordinated.
35069 The latest version of Calc is available from Savannah, in the Emacs
35070 CVS tree. See @uref{http://savannah.gnu.org/projects/emacs}.
35073 @node Summary, Key Index, Reporting Bugs, Top
35074 @appendix Calc Summary
35077 This section includes a complete list of Calc keystroke commands.
35078 Each line lists the stack entries used by the command (top-of-stack
35079 last), the keystrokes themselves, the prompts asked by the command,
35080 and the result of the command (also with top-of-stack last).
35081 The result is expressed using the equivalent algebraic function.
35082 Commands which put no results on the stack show the full @kbd{M-x}
35083 command name in that position. Numbers preceding the result or
35084 command name refer to notes at the end.
35086 Algebraic functions and @kbd{M-x} commands that don't have corresponding
35087 keystrokes are not listed in this summary.
35088 @xref{Command Index}. @xref{Function Index}.
35093 \vskip-2\baselineskip \null
35094 \gdef\sumrow#1{\sumrowx#1\relax}%
35095 \gdef\sumrowx#1\:#2\:#3\:#4\:#5\:#6\relax{%
35098 \hbox to5em{\sl\hss#1}%
35099 \hbox to5em{\tt#2\hss}%
35100 \hbox to4em{\sl#3\hss}%
35101 \hbox to5em{\rm\hss#4}%
35106 \gdef\sumlpar{{\rm(}}%
35107 \gdef\sumrpar{{\rm)}}%
35108 \gdef\sumcomma{{\rm,\thinspace}}%
35109 \gdef\sumexcl{{\rm!}}%
35110 \gdef\sumbreak{\vskip-2.5\baselineskip\goodbreak}%
35111 \gdef\minus#1{{\tt-}}%
35115 @catcode`@(=@active @let(=@sumlpar
35116 @catcode`@)=@active @let)=@sumrpar
35117 @catcode`@,=@active @let,=@sumcomma
35118 @catcode`@!=@active @let!=@sumexcl
35122 @advance@baselineskip-2.5pt
35125 @r{ @: C-x * a @: @: 33 @:calc-embedded-activate@:}
35126 @r{ @: C-x * b @: @: @:calc-big-or-small@:}
35127 @r{ @: C-x * c @: @: @:calc@:}
35128 @r{ @: C-x * d @: @: @:calc-embedded-duplicate@:}
35129 @r{ @: C-x * e @: @: 34 @:calc-embedded@:}
35130 @r{ @: C-x * f @:formula @: @:calc-embedded-new-formula@:}
35131 @r{ @: C-x * g @: @: 35 @:calc-grab-region@:}
35132 @r{ @: C-x * i @: @: @:calc-info@:}
35133 @r{ @: C-x * j @: @: @:calc-embedded-select@:}
35134 @r{ @: C-x * k @: @: @:calc-keypad@:}
35135 @r{ @: C-x * l @: @: @:calc-load-everything@:}
35136 @r{ @: C-x * m @: @: @:read-kbd-macro@:}
35137 @r{ @: C-x * n @: @: 4 @:calc-embedded-next@:}
35138 @r{ @: C-x * o @: @: @:calc-other-window@:}
35139 @r{ @: C-x * p @: @: 4 @:calc-embedded-previous@:}
35140 @r{ @: C-x * q @:formula @: @:quick-calc@:}
35141 @r{ @: C-x * r @: @: 36 @:calc-grab-rectangle@:}
35142 @r{ @: C-x * s @: @: @:calc-info-summary@:}
35143 @r{ @: C-x * t @: @: @:calc-tutorial@:}
35144 @r{ @: C-x * u @: @: @:calc-embedded-update-formula@:}
35145 @r{ @: C-x * w @: @: @:calc-embedded-word@:}
35146 @r{ @: C-x * x @: @: @:calc-quit@:}
35147 @r{ @: C-x * y @: @:1,28,49 @:calc-copy-to-buffer@:}
35148 @r{ @: C-x * z @: @: @:calc-user-invocation@:}
35149 @r{ @: C-x * : @: @: 36 @:calc-grab-sum-down@:}
35150 @r{ @: C-x * _ @: @: 36 @:calc-grab-sum-across@:}
35151 @r{ @: C-x * ` @:editing @: 30 @:calc-embedded-edit@:}
35152 @r{ @: C-x * 0 @:(zero) @: @:calc-reset@:}
35155 @r{ @: 0-9 @:number @: @:@:number}
35156 @r{ @: . @:number @: @:@:0.number}
35157 @r{ @: _ @:number @: @:-@:number}
35158 @r{ @: e @:number @: @:@:1e number}
35159 @r{ @: # @:number @: @:@:current-radix@tfn{#}number}
35160 @r{ @: P @:(in number) @: @:+/-@:}
35161 @r{ @: M @:(in number) @: @:mod@:}
35162 @r{ @: @@ ' " @: (in number)@: @:@:HMS form}
35163 @r{ @: h m s @: (in number)@: @:@:HMS form}
35166 @r{ @: ' @:formula @: 37,46 @:@:formula}
35167 @r{ @: $ @:formula @: 37,46 @:$@:formula}
35168 @r{ @: " @:string @: 37,46 @:@:string}
35171 @r{ a b@: + @: @: 2 @:add@:(a,b) a+b}
35172 @r{ a b@: - @: @: 2 @:sub@:(a,b) a@minus{}b}
35173 @r{ a b@: * @: @: 2 @:mul@:(a,b) a b, a*b}
35174 @r{ a b@: / @: @: 2 @:div@:(a,b) a/b}
35175 @r{ a b@: ^ @: @: 2 @:pow@:(a,b) a^b}
35176 @r{ a b@: I ^ @: @: 2 @:nroot@:(a,b) a^(1/b)}
35177 @r{ a b@: % @: @: 2 @:mod@:(a,b) a%b}
35178 @r{ a b@: \ @: @: 2 @:idiv@:(a,b) a\b}
35179 @r{ a b@: : @: @: 2 @:fdiv@:(a,b)}
35180 @r{ a b@: | @: @: 2 @:vconcat@:(a,b) a|b}
35181 @r{ a b@: I | @: @: @:vconcat@:(b,a) b|a}
35182 @r{ a b@: H | @: @: 2 @:append@:(a,b)}
35183 @r{ a b@: I H | @: @: @:append@:(b,a)}
35184 @r{ a@: & @: @: 1 @:inv@:(a) 1/a}
35185 @r{ a@: ! @: @: 1 @:fact@:(a) a!}
35186 @r{ a@: = @: @: 1 @:evalv@:(a)}
35187 @r{ a@: M-% @: @: @:percent@:(a) a%}
35190 @r{ ... a@: @summarykey{RET} @: @: 1 @:@:... a a}
35191 @r{ ... a@: @summarykey{SPC} @: @: 1 @:@:... a a}
35192 @r{... a b@: @summarykey{TAB} @: @: 3 @:@:... b a}
35193 @r{. a b c@: M-@summarykey{TAB} @: @: 3 @:@:... b c a}
35194 @r{... a b@: @summarykey{LFD} @: @: 1 @:@:... a b a}
35195 @r{ ... a@: @summarykey{DEL} @: @: 1 @:@:...}
35196 @r{... a b@: M-@summarykey{DEL} @: @: 1 @:@:... b}
35197 @r{ @: M-@summarykey{RET} @: @: 4 @:calc-last-args@:}
35198 @r{ a@: ` @:editing @: 1,30 @:calc-edit@:}
35201 @r{ ... a@: C-d @: @: 1 @:@:...}
35202 @r{ @: C-k @: @: 27 @:calc-kill@:}
35203 @r{ @: C-w @: @: 27 @:calc-kill-region@:}
35204 @r{ @: C-y @: @: @:calc-yank@:}
35205 @r{ @: C-_ @: @: 4 @:calc-undo@:}
35206 @r{ @: M-k @: @: 27 @:calc-copy-as-kill@:}
35207 @r{ @: M-w @: @: 27 @:calc-copy-region-as-kill@:}
35210 @r{ @: [ @: @: @:@:[...}
35211 @r{[.. a b@: ] @: @: @:@:[a,b]}
35212 @r{ @: ( @: @: @:@:(...}
35213 @r{(.. a b@: ) @: @: @:@:(a,b)}
35214 @r{ @: , @: @: @:@:vector or rect complex}
35215 @r{ @: ; @: @: @:@:matrix or polar complex}
35216 @r{ @: .. @: @: @:@:interval}
35219 @r{ @: ~ @: @: @:calc-num-prefix@:}
35220 @r{ @: < @: @: 4 @:calc-scroll-left@:}
35221 @r{ @: > @: @: 4 @:calc-scroll-right@:}
35222 @r{ @: @{ @: @: 4 @:calc-scroll-down@:}
35223 @r{ @: @} @: @: 4 @:calc-scroll-up@:}
35224 @r{ @: ? @: @: @:calc-help@:}
35227 @r{ a@: n @: @: 1 @:neg@:(a) @minus{}a}
35228 @r{ @: o @: @: 4 @:calc-realign@:}
35229 @r{ @: p @:precision @: 31 @:calc-precision@:}
35230 @r{ @: q @: @: @:calc-quit@:}
35231 @r{ @: w @: @: @:calc-why@:}
35232 @r{ @: x @:command @: @:M-x calc-@:command}
35233 @r{ a@: y @: @:1,28,49 @:calc-copy-to-buffer@:}
35236 @r{ a@: A @: @: 1 @:abs@:(a)}
35237 @r{ a b@: B @: @: 2 @:log@:(a,b)}
35238 @r{ a b@: I B @: @: 2 @:alog@:(a,b) b^a}
35239 @r{ a@: C @: @: 1 @:cos@:(a)}
35240 @r{ a@: I C @: @: 1 @:arccos@:(a)}
35241 @r{ a@: H C @: @: 1 @:cosh@:(a)}
35242 @r{ a@: I H C @: @: 1 @:arccosh@:(a)}
35243 @r{ @: D @: @: 4 @:calc-redo@:}
35244 @r{ a@: E @: @: 1 @:exp@:(a)}
35245 @r{ a@: H E @: @: 1 @:exp10@:(a) 10.^a}
35246 @r{ a@: F @: @: 1,11 @:floor@:(a,d)}
35247 @r{ a@: I F @: @: 1,11 @:ceil@:(a,d)}
35248 @r{ a@: H F @: @: 1,11 @:ffloor@:(a,d)}
35249 @r{ a@: I H F @: @: 1,11 @:fceil@:(a,d)}
35250 @r{ a@: G @: @: 1 @:arg@:(a)}
35251 @r{ @: H @:command @: 32 @:@:Hyperbolic}
35252 @r{ @: I @:command @: 32 @:@:Inverse}
35253 @r{ a@: J @: @: 1 @:conj@:(a)}
35254 @r{ @: K @:command @: 32 @:@:Keep-args}
35255 @r{ a@: L @: @: 1 @:ln@:(a)}
35256 @r{ a@: H L @: @: 1 @:log10@:(a)}
35257 @r{ @: M @: @: @:calc-more-recursion-depth@:}
35258 @r{ @: I M @: @: @:calc-less-recursion-depth@:}
35259 @r{ a@: N @: @: 5 @:evalvn@:(a)}
35260 @r{ @: P @: @: @:@:pi}
35261 @r{ @: I P @: @: @:@:gamma}
35262 @r{ @: H P @: @: @:@:e}
35263 @r{ @: I H P @: @: @:@:phi}
35264 @r{ a@: Q @: @: 1 @:sqrt@:(a)}
35265 @r{ a@: I Q @: @: 1 @:sqr@:(a) a^2}
35266 @r{ a@: R @: @: 1,11 @:round@:(a,d)}
35267 @r{ a@: I R @: @: 1,11 @:trunc@:(a,d)}
35268 @r{ a@: H R @: @: 1,11 @:fround@:(a,d)}
35269 @r{ a@: I H R @: @: 1,11 @:ftrunc@:(a,d)}
35270 @r{ a@: S @: @: 1 @:sin@:(a)}
35271 @r{ a@: I S @: @: 1 @:arcsin@:(a)}
35272 @r{ a@: H S @: @: 1 @:sinh@:(a)}
35273 @r{ a@: I H S @: @: 1 @:arcsinh@:(a)}
35274 @r{ a@: T @: @: 1 @:tan@:(a)}
35275 @r{ a@: I T @: @: 1 @:arctan@:(a)}
35276 @r{ a@: H T @: @: 1 @:tanh@:(a)}
35277 @r{ a@: I H T @: @: 1 @:arctanh@:(a)}
35278 @r{ @: U @: @: 4 @:calc-undo@:}
35279 @r{ @: X @: @: 4 @:calc-call-last-kbd-macro@:}
35282 @r{ a b@: a = @: @: 2 @:eq@:(a,b) a=b}
35283 @r{ a b@: a # @: @: 2 @:neq@:(a,b) a!=b}
35284 @r{ a b@: a < @: @: 2 @:lt@:(a,b) a<b}
35285 @r{ a b@: a > @: @: 2 @:gt@:(a,b) a>b}
35286 @r{ a b@: a [ @: @: 2 @:leq@:(a,b) a<=b}
35287 @r{ a b@: a ] @: @: 2 @:geq@:(a,b) a>=b}
35288 @r{ a b@: a @{ @: @: 2 @:in@:(a,b)}
35289 @r{ a b@: a & @: @: 2,45 @:land@:(a,b) a&&b}
35290 @r{ a b@: a | @: @: 2,45 @:lor@:(a,b) a||b}
35291 @r{ a@: a ! @: @: 1,45 @:lnot@:(a) !a}
35292 @r{ a b c@: a : @: @: 45 @:if@:(a,b,c) a?b:c}
35293 @r{ a@: a . @: @: 1 @:rmeq@:(a)}
35294 @r{ a@: a " @: @: 7,8 @:calc-expand-formula@:}
35297 @r{ a@: a + @:i, l, h @: 6,38 @:sum@:(a,i,l,h)}
35298 @r{ a@: a - @:i, l, h @: 6,38 @:asum@:(a,i,l,h)}
35299 @r{ a@: a * @:i, l, h @: 6,38 @:prod@:(a,i,l,h)}
35300 @r{ a b@: a _ @: @: 2 @:subscr@:(a,b) a_b}
35303 @r{ a b@: a \ @: @: 2 @:pdiv@:(a,b)}
35304 @r{ a b@: a % @: @: 2 @:prem@:(a,b)}
35305 @r{ a b@: a / @: @: 2 @:pdivrem@:(a,b) [q,r]}
35306 @r{ a b@: H a / @: @: 2 @:pdivide@:(a,b) q+r/b}
35309 @r{ a@: a a @: @: 1 @:apart@:(a)}
35310 @r{ a@: a b @:old, new @: 38 @:subst@:(a,old,new)}
35311 @r{ a@: a c @:v @: 38 @:collect@:(a,v)}
35312 @r{ a@: a d @:v @: 4,38 @:deriv@:(a,v)}
35313 @r{ a@: H a d @:v @: 4,38 @:tderiv@:(a,v)}
35314 @r{ a@: a e @: @: @:esimplify@:(a)}
35315 @r{ a@: a f @: @: 1 @:factor@:(a)}
35316 @r{ a@: H a f @: @: 1 @:factors@:(a)}
35317 @r{ a b@: a g @: @: 2 @:pgcd@:(a,b)}
35318 @r{ a@: a i @:v @: 38 @:integ@:(a,v)}
35319 @r{ a@: a m @:pats @: 38 @:match@:(a,pats)}
35320 @r{ a@: I a m @:pats @: 38 @:matchnot@:(a,pats)}
35321 @r{ data x@: a p @: @: 28 @:polint@:(data,x)}
35322 @r{ data x@: H a p @: @: 28 @:ratint@:(data,x)}
35323 @r{ a@: a n @: @: 1 @:nrat@:(a)}
35324 @r{ a@: a r @:rules @:4,8,38 @:rewrite@:(a,rules,n)}
35325 @r{ a@: a s @: @: @:simplify@:(a)}
35326 @r{ a@: a t @:v, n @: 31,39 @:taylor@:(a,v,n)}
35327 @r{ a@: a v @: @: 7,8 @:calc-alg-evaluate@:}
35328 @r{ a@: a x @: @: 4,8 @:expand@:(a)}
35331 @r{ data@: a F @:model, vars @: 48 @:fit@:(m,iv,pv,data)}
35332 @r{ data@: I a F @:model, vars @: 48 @:xfit@:(m,iv,pv,data)}
35333 @r{ data@: H a F @:model, vars @: 48 @:efit@:(m,iv,pv,data)}
35334 @r{ a@: a I @:v, l, h @: 38 @:ninteg@:(a,v,l,h)}
35335 @r{ a b@: a M @:op @: 22 @:mapeq@:(op,a,b)}
35336 @r{ a b@: I a M @:op @: 22 @:mapeqr@:(op,a,b)}
35337 @r{ a b@: H a M @:op @: 22 @:mapeqp@:(op,a,b)}
35338 @r{ a g@: a N @:v @: 38 @:minimize@:(a,v,g)}
35339 @r{ a g@: H a N @:v @: 38 @:wminimize@:(a,v,g)}
35340 @r{ a@: a P @:v @: 38 @:roots@:(a,v)}
35341 @r{ a g@: a R @:v @: 38 @:root@:(a,v,g)}
35342 @r{ a g@: H a R @:v @: 38 @:wroot@:(a,v,g)}
35343 @r{ a@: a S @:v @: 38 @:solve@:(a,v)}
35344 @r{ a@: I a S @:v @: 38 @:finv@:(a,v)}
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35381 @r{ r v@: I b N @: @: 19 @:npvb@:(r,v)}
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35390 @r{ n p a@: H b T @: @: @:ratel@:(n,p,a)}
35391 @r{c s l p@: b Y @: @: @:syd@:(c,s,l,p)}
35393 @r{ r p a@: b # @: @: @:nper@:(r,p,a)}
35394 @r{ r p a@: I b # @: @: @:nperb@:(r,p,a)}
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35401 @r{ a@: H c c @: @: 5 @:clean@:(a,p)}
35402 @r{ a@: H c 0-9 @: @: @:clean@:(a,p)}
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35404 @r{ a@: c f @: @: 1 @:pfloat@:(a)}
35405 @r{ a@: H c f @: @: 1 @:float@:(a)}
35406 @r{ a@: c h @: @: 1 @:hms@:(a)}
35407 @r{ a@: c p @: @: @:polar@:(a)}
35408 @r{ a@: I c p @: @: @:rect@:(a)}
35409 @r{ a@: c r @: @: 1 @:rad@:(a)}
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35413 @r{ a@: H c F @: @: 5 @:frac@:(a,p)}
35416 @r{ a@: c % @: @: @:percent@:(a*100)}
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35428 @r{ @: d " @: @: 12,50 @:calc-display-strings@:}
35429 @r{ @: d @summarykey{SPC} @: @: @:calc-refresh@:}
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35434 @r{ @: d 2 @: @: 50 @:calc-binary-radix@:}
35435 @r{ @: d 6 @: @: 50 @:calc-hex-radix@:}
35436 @r{ @: d 8 @: @: 50 @:calc-octal-radix@:}
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35441 @r{ @: d d @:format @: 50 @:calc-date-notation@:}
35442 @r{ @: d e @: @: 5,50 @:calc-eng-notation@:}
35443 @r{ @: d f @:num @: 31,50 @:calc-fix-notation@:}
35444 @r{ @: d g @: @:12,13,50 @:calc-group-digits@:}
35445 @r{ @: d h @:format @: 50 @:calc-hms-notation@:}
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35447 @r{ @: d j @: @: 50 @:calc-j-notation@:}
35448 @r{ @: d l @: @: 12,50 @:calc-line-numbering@:}
35449 @r{ @: d n @: @: 5,50 @:calc-normal-notation@:}
35450 @r{ @: d o @:format @: 50 @:calc-over-notation@:}
35451 @r{ @: d p @: @: 12,50 @:calc-show-plain@:}
35452 @r{ @: d r @:radix @: 31,50 @:calc-radix@:}
35453 @r{ @: d s @: @: 5,50 @:calc-sci-notation@:}
35454 @r{ @: d t @: @: 27 @:calc-truncate-stack@:}
35455 @r{ @: d w @: @: 12,13 @:calc-auto-why@:}
35456 @r{ @: d z @: @: 12,50 @:calc-leading-zeros@:}
35459 @r{ @: d B @: @: 50 @:calc-big-language@:}
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35461 @r{ @: d E @: @: 50 @:calc-eqn-language@:}
35462 @r{ @: d F @: @: 50 @:calc-fortran-language@:}
35463 @r{ @: d M @: @: 50 @:calc-mathematica-language@:}
35464 @r{ @: d N @: @: 50 @:calc-normal-language@:}
35465 @r{ @: d O @: @: 50 @:calc-flat-language@:}
35466 @r{ @: d P @: @: 50 @:calc-pascal-language@:}
35467 @r{ @: d T @: @: 50 @:calc-tex-language@:}
35468 @r{ @: d L @: @: 50 @:calc-latex-language@:}
35469 @r{ @: d U @: @: 50 @:calc-unformatted-language@:}
35470 @r{ @: d W @: @: 50 @:calc-maple-language@:}
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35479 @r{ a@: I f e @: @: 1 @:erfc@:(a)}
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35493 @r{ x a b@: H f B @: @: @:betaB@:(x,a,b)}
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35496 @r{ a x@: I f G @: @: 2 @:gammaQ@:(a,x)}
35497 @r{ a x@: H f G @: @: 2 @:gammag@:(a,x)}
35498 @r{ a x@: I H f G @: @: 2 @:gammaG@:(a,x)}
35499 @r{ a b@: f I @: @: 2 @:ilog@:(a,b)}
35500 @r{ a b@: I f I @: @: 2 @:alog@:(a,b) b^a}
35501 @r{ a@: f L @: @: 1 @:lnp1@:(a)}
35502 @r{ a@: f M @: @: 1 @:mant@:(a)}
35503 @r{ a@: f Q @: @: 1 @:isqrt@:(a)}
35504 @r{ a@: I f Q @: @: 1 @:sqr@:(a) a^2}
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35525 @r{ @: g t @:title @: @:calc-graph-title-x@:}
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35531 @r{ x y z@: g A @: @: 28,40 @:calc-graph-add-3d@:}
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35543 @r{ @: g T @:title @: @:calc-graph-title-y@:}
35544 @r{ @: g V @: @: @:calc-graph-view-trail@:}
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35550 @r{ @: g C-r @:range @: @:calc-graph-range-z@:}
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35555 @r{ @: h c @:key @: @:calc-describe-key-briefly@:}
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35557 @r{ @: h h @: @: @:calc-full-help@:}
35558 @r{ @: h i @: @: @:calc-info@:}
35559 @r{ @: h k @:key @: @:calc-describe-key@:}
35560 @r{ @: h n @: @: @:calc-view-news@:}
35561 @r{ @: h s @: @: @:calc-info-summary@:}
35562 @r{ @: h t @: @: @:calc-tutorial@:}
35563 @r{ @: h v @:var @: @:calc-describe-variable@:}
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35567 @r{ @: j @summarykey{RET} @: @: 27 @:calc-copy-selection@:}
35568 @r{ @: j @summarykey{DEL} @: @: 27 @:calc-del-selection@:}
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35570 @r{ @: j ` @:editing @: 27,30 @:calc-edit-selection@:}
35571 @r{ @: j " @: @: 7,27 @:calc-sel-expand-formula@:}
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35589 @r{ @: j o @: @: 4,27 @:calc-select-once@:}
35590 @r{ @: j p @: @: 4 @:calc-select-previous@:}
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35594 @r{ @: j v @: @: 7,27 @:calc-sel-evaluate@:}
35597 @r{ @: j C @: @: 27 @:calc-sel-commute@:}
35598 @r{ @: j D @: @: 4,27 @:calc-sel-distribute@:}
35599 @r{ @: j E @: @: 27 @:calc-sel-jump-equals@:}
35600 @r{ @: j I @: @: 27 @:calc-sel-isolate@:}
35601 @r{ @: H j I @: @: 27 @:calc-sel-isolate@: (full)}
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35603 @r{ @: j M @: @: 27 @:calc-sel-merge@:}
35604 @r{ @: j N @: @: 27 @:calc-sel-negate@:}
35605 @r{ @: j O @: @: 4,27 @:calc-select-once-maybe@:}
35606 @r{ @: j R @: @: 4,27 @:calc-commute-right@:}
35607 @r{ @: j S @: @: 4,27 @:calc-select-here-maybe@:}
35608 @r{ @: j U @: @: 27 @:calc-sel-unpack@:}
35611 @r{ @: k a @: @: @:calc-random-again@:}
35612 @r{ n@: k b @: @: 1 @:bern@:(n)}
35613 @r{ n x@: H k b @: @: 2 @:bern@:(n,x)}
35614 @r{ n m@: k c @: @: 2 @:choose@:(n,m)}
35615 @r{ n m@: H k c @: @: 2 @:perm@:(n,m)}
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35617 @r{ n@: k e @: @: 1 @:euler@:(n)}
35618 @r{ n x@: H k e @: @: 2 @:euler@:(n,x)}
35619 @r{ n@: k f @: @: 4 @:prfac@:(n)}
35620 @r{ n m@: k g @: @: 2 @:gcd@:(n,m)}
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35622 @r{ n m@: k l @: @: 2 @:lcm@:(n,m)}
35623 @r{ n@: k m @: @: 1 @:moebius@:(n)}
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35625 @r{ n@: I k n @: @: 4 @:prevprime@:(n)}
35626 @r{ n@: k p @: @: 4,28 @:calc-prime-test@:}
35627 @r{ m@: k r @: @: 14 @:random@:(m)}
35628 @r{ n m@: k s @: @: 2 @:stir1@:(n,m)}
35629 @r{ n m@: H k s @: @: 2 @:stir2@:(n,m)}
35630 @r{ n@: k t @: @: 1 @:totient@:(n)}
35633 @r{ n p x@: k B @: @: @:utpb@:(x,n,p)}
35634 @r{ n p x@: I k B @: @: @:ltpb@:(x,n,p)}
35635 @r{ v x@: k C @: @: @:utpc@:(x,v)}
35636 @r{ v x@: I k C @: @: @:ltpc@:(x,v)}
35637 @r{ n m@: k E @: @: @:egcd@:(n,m)}
35638 @r{v1 v2 x@: k F @: @: @:utpf@:(x,v1,v2)}
35639 @r{v1 v2 x@: I k F @: @: @:ltpf@:(x,v1,v2)}
35640 @r{ m s x@: k N @: @: @:utpn@:(x,m,s)}
35641 @r{ m s x@: I k N @: @: @:ltpn@:(x,m,s)}
35642 @r{ m x@: k P @: @: @:utpp@:(x,m)}
35643 @r{ m x@: I k P @: @: @:ltpp@:(x,m)}
35644 @r{ v x@: k T @: @: @:utpt@:(x,v)}
35645 @r{ v x@: I k T @: @: @:ltpt@:(x,v)}
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35649 @r{ @: m d @: @: @:calc-degrees-mode@:}
35650 @r{ @: m e @: @: @:calc-embedded-preserve-modes@:}
35651 @r{ @: m f @: @: 12 @:calc-frac-mode@:}
35652 @r{ @: m g @: @: 52 @:calc-get-modes@:}
35653 @r{ @: m h @: @: @:calc-hms-mode@:}
35654 @r{ @: m i @: @: 12,13 @:calc-infinite-mode@:}
35655 @r{ @: m m @: @: @:calc-save-modes@:}
35656 @r{ @: m p @: @: 12 @:calc-polar-mode@:}
35657 @r{ @: m r @: @: @:calc-radians-mode@:}
35658 @r{ @: m s @: @: 12 @:calc-symbolic-mode@:}
35659 @r{ @: m t @: @: 12 @:calc-total-algebraic-mode@:}
35660 @r{ @: m v @: @: 12,13 @:calc-matrix-mode@:}
35661 @r{ @: m w @: @: 13 @:calc-working@:}
35662 @r{ @: m x @: @: @:calc-always-load-extensions@:}
35665 @r{ @: m A @: @: 12 @:calc-alg-simplify-mode@:}
35666 @r{ @: m B @: @: 12 @:calc-bin-simplify-mode@:}
35667 @r{ @: m C @: @: 12 @:calc-auto-recompute@:}
35668 @r{ @: m D @: @: @:calc-default-simplify-mode@:}
35669 @r{ @: m E @: @: 12 @:calc-ext-simplify-mode@:}
35670 @r{ @: m F @:filename @: 13 @:calc-settings-file-name@:}
35671 @r{ @: m N @: @: 12 @:calc-num-simplify-mode@:}
35672 @r{ @: m O @: @: 12 @:calc-no-simplify-mode@:}
35673 @r{ @: m R @: @: 12,13 @:calc-mode-record-mode@:}
35674 @r{ @: m S @: @: 12 @:calc-shift-prefix@:}
35675 @r{ @: m U @: @: 12 @:calc-units-simplify-mode@:}
35678 @r{ @: r s @:register @: 27 @:calc-copy-to-register@:}
35679 @r{ @: r i @:register @: @:calc-insert-register@:}
35682 @r{ @: s c @:var1, var2 @: 29 @:calc-copy-variable@:}
35683 @r{ @: s d @:var, decl @: @:calc-declare-variable@:}
35684 @r{ @: s e @:var, editing @: 29,30 @:calc-edit-variable@:}
35685 @r{ @: s i @:buffer @: @:calc-insert-variables@:}
35686 @r{ @: s k @:const, var @: 29 @:calc-copy-special-constant@:}
35687 @r{ a b@: s l @:var @: 29 @:@:a (letting var=b)}
35688 @r{ a ...@: s m @:op, var @: 22,29 @:calc-store-map@:}
35689 @r{ @: s n @:var @: 29,47 @:calc-store-neg@: (v/-1)}
35690 @r{ @: s p @:var @: 29 @:calc-permanent-variable@:}
35691 @r{ @: s r @:var @: 29 @:@:v (recalled value)}
35692 @r{ @: r 0-9 @: @: @:calc-recall-quick@:}
35693 @r{ a@: s s @:var @: 28,29 @:calc-store@:}
35694 @r{ a@: s 0-9 @: @: @:calc-store-quick@:}
35695 @r{ a@: s t @:var @: 29 @:calc-store-into@:}
35696 @r{ a@: t 0-9 @: @: @:calc-store-into-quick@:}
35697 @r{ @: s u @:var @: 29 @:calc-unstore@:}
35698 @r{ a@: s x @:var @: 29 @:calc-store-exchange@:}
35701 @r{ @: s A @:editing @: 30 @:calc-edit-AlgSimpRules@:}
35702 @r{ @: s D @:editing @: 30 @:calc-edit-Decls@:}
35703 @r{ @: s E @:editing @: 30 @:calc-edit-EvalRules@:}
35704 @r{ @: s F @:editing @: 30 @:calc-edit-FitRules@:}
35705 @r{ @: s G @:editing @: 30 @:calc-edit-GenCount@:}
35706 @r{ @: s H @:editing @: 30 @:calc-edit-Holidays@:}
35707 @r{ @: s I @:editing @: 30 @:calc-edit-IntegLimit@:}
35708 @r{ @: s L @:editing @: 30 @:calc-edit-LineStyles@:}
35709 @r{ @: s P @:editing @: 30 @:calc-edit-PointStyles@:}
35710 @r{ @: s R @:editing @: 30 @:calc-edit-PlotRejects@:}
35711 @r{ @: s T @:editing @: 30 @:calc-edit-TimeZone@:}
35712 @r{ @: s U @:editing @: 30 @:calc-edit-Units@:}
35713 @r{ @: s X @:editing @: 30 @:calc-edit-ExtSimpRules@:}
35716 @r{ a@: s + @:var @: 29,47 @:calc-store-plus@: (v+a)}
35717 @r{ a@: s - @:var @: 29,47 @:calc-store-minus@: (v-a)}
35718 @r{ a@: s * @:var @: 29,47 @:calc-store-times@: (v*a)}
35719 @r{ a@: s / @:var @: 29,47 @:calc-store-div@: (v/a)}
35720 @r{ a@: s ^ @:var @: 29,47 @:calc-store-power@: (v^a)}
35721 @r{ a@: s | @:var @: 29,47 @:calc-store-concat@: (v|a)}
35722 @r{ @: s & @:var @: 29,47 @:calc-store-inv@: (v^-1)}
35723 @r{ @: s [ @:var @: 29,47 @:calc-store-decr@: (v-1)}
35724 @r{ @: s ] @:var @: 29,47 @:calc-store-incr@: (v-(-1))}
35725 @r{ a b@: s : @: @: 2 @:assign@:(a,b) a @tfn{:=} b}
35726 @r{ a@: s = @: @: 1 @:evalto@:(a,b) a @tfn{=>}}
35729 @r{ @: t [ @: @: 4 @:calc-trail-first@:}
35730 @r{ @: t ] @: @: 4 @:calc-trail-last@:}
35731 @r{ @: t < @: @: 4 @:calc-trail-scroll-left@:}
35732 @r{ @: t > @: @: 4 @:calc-trail-scroll-right@:}
35733 @r{ @: t . @: @: 12 @:calc-full-trail-vectors@:}
35736 @r{ @: t b @: @: 4 @:calc-trail-backward@:}
35737 @r{ @: t d @: @: 12,50 @:calc-trail-display@:}
35738 @r{ @: t f @: @: 4 @:calc-trail-forward@:}
35739 @r{ @: t h @: @: @:calc-trail-here@:}
35740 @r{ @: t i @: @: @:calc-trail-in@:}
35741 @r{ @: t k @: @: 4 @:calc-trail-kill@:}
35742 @r{ @: t m @:string @: @:calc-trail-marker@:}
35743 @r{ @: t n @: @: 4 @:calc-trail-next@:}
35744 @r{ @: t o @: @: @:calc-trail-out@:}
35745 @r{ @: t p @: @: 4 @:calc-trail-previous@:}
35746 @r{ @: t r @:string @: @:calc-trail-isearch-backward@:}
35747 @r{ @: t s @:string @: @:calc-trail-isearch-forward@:}
35748 @r{ @: t y @: @: 4 @:calc-trail-yank@:}
35751 @r{ d@: t C @:oz, nz @: @:tzconv@:(d,oz,nz)}
35752 @r{d oz nz@: t C @:$ @: @:tzconv@:(d,oz,nz)}
35753 @r{ d@: t D @: @: 15 @:date@:(d)}
35754 @r{ d@: t I @: @: 4 @:incmonth@:(d,n)}
35755 @r{ d@: t J @: @: 16 @:julian@:(d,z)}
35756 @r{ d@: t M @: @: 17 @:newmonth@:(d,n)}
35757 @r{ @: t N @: @: 16 @:now@:(z)}
35758 @r{ d@: t P @:1 @: 31 @:year@:(d)}
35759 @r{ d@: t P @:2 @: 31 @:month@:(d)}
35760 @r{ d@: t P @:3 @: 31 @:day@:(d)}
35761 @r{ d@: t P @:4 @: 31 @:hour@:(d)}
35762 @r{ d@: t P @:5 @: 31 @:minute@:(d)}
35763 @r{ d@: t P @:6 @: 31 @:second@:(d)}
35764 @r{ d@: t P @:7 @: 31 @:weekday@:(d)}
35765 @r{ d@: t P @:8 @: 31 @:yearday@:(d)}
35766 @r{ d@: t P @:9 @: 31 @:time@:(d)}
35767 @r{ d@: t U @: @: 16 @:unixtime@:(d,z)}
35768 @r{ d@: t W @: @: 17 @:newweek@:(d,w)}
35769 @r{ d@: t Y @: @: 17 @:newyear@:(d,n)}
35772 @r{ a b@: t + @: @: 2 @:badd@:(a,b)}
35773 @r{ a b@: t - @: @: 2 @:bsub@:(a,b)}
35776 @r{ @: u a @: @: 12 @:calc-autorange-units@:}
35777 @r{ a@: u b @: @: @:calc-base-units@:}
35778 @r{ a@: u c @:units @: 18 @:calc-convert-units@:}
35779 @r{ defn@: u d @:unit, descr @: @:calc-define-unit@:}
35780 @r{ @: u e @: @: @:calc-explain-units@:}
35781 @r{ @: u g @:unit @: @:calc-get-unit-definition@:}
35782 @r{ @: u p @: @: @:calc-permanent-units@:}
35783 @r{ a@: u r @: @: @:calc-remove-units@:}
35784 @r{ a@: u s @: @: @:usimplify@:(a)}
35785 @r{ a@: u t @:units @: 18 @:calc-convert-temperature@:}
35786 @r{ @: u u @:unit @: @:calc-undefine-unit@:}
35787 @r{ @: u v @: @: @:calc-enter-units-table@:}
35788 @r{ a@: u x @: @: @:calc-extract-units@:}
35789 @r{ a@: u 0-9 @: @: @:calc-quick-units@:}
35792 @r{ v1 v2@: u C @: @: 20 @:vcov@:(v1,v2)}
35793 @r{ v1 v2@: I u C @: @: 20 @:vpcov@:(v1,v2)}
35794 @r{ v1 v2@: H u C @: @: 20 @:vcorr@:(v1,v2)}
35795 @r{ v@: u G @: @: 19 @:vgmean@:(v)}
35796 @r{ a b@: H u G @: @: 2 @:agmean@:(a,b)}
35797 @r{ v@: u M @: @: 19 @:vmean@:(v)}
35798 @r{ v@: I u M @: @: 19 @:vmeane@:(v)}
35799 @r{ v@: H u M @: @: 19 @:vmedian@:(v)}
35800 @r{ v@: I H u M @: @: 19 @:vhmean@:(v)}
35801 @r{ v@: u N @: @: 19 @:vmin@:(v)}
35802 @r{ v@: u S @: @: 19 @:vsdev@:(v)}
35803 @r{ v@: I u S @: @: 19 @:vpsdev@:(v)}
35804 @r{ v@: H u S @: @: 19 @:vvar@:(v)}
35805 @r{ v@: I H u S @: @: 19 @:vpvar@:(v)}
35806 @r{ @: u V @: @: @:calc-view-units-table@:}
35807 @r{ v@: u X @: @: 19 @:vmax@:(v)}
35810 @r{ v@: u + @: @: 19 @:vsum@:(v)}
35811 @r{ v@: u * @: @: 19 @:vprod@:(v)}
35812 @r{ v@: u # @: @: 19 @:vcount@:(v)}
35815 @r{ @: V ( @: @: 50 @:calc-vector-parens@:}
35816 @r{ @: V @{ @: @: 50 @:calc-vector-braces@:}
35817 @r{ @: V [ @: @: 50 @:calc-vector-brackets@:}
35818 @r{ @: V ] @:ROCP @: 50 @:calc-matrix-brackets@:}
35819 @r{ @: V , @: @: 50 @:calc-vector-commas@:}
35820 @r{ @: V < @: @: 50 @:calc-matrix-left-justify@:}
35821 @r{ @: V = @: @: 50 @:calc-matrix-center-justify@:}
35822 @r{ @: V > @: @: 50 @:calc-matrix-right-justify@:}
35823 @r{ @: V / @: @: 12,50 @:calc-break-vectors@:}
35824 @r{ @: V . @: @: 12,50 @:calc-full-vectors@:}
35827 @r{ s t@: V ^ @: @: 2 @:vint@:(s,t)}
35828 @r{ s t@: V - @: @: 2 @:vdiff@:(s,t)}
35829 @r{ s@: V ~ @: @: 1 @:vcompl@:(s)}
35830 @r{ s@: V # @: @: 1 @:vcard@:(s)}
35831 @r{ s@: V : @: @: 1 @:vspan@:(s)}
35832 @r{ s@: V + @: @: 1 @:rdup@:(s)}
35835 @r{ m@: V & @: @: 1 @:inv@:(m) 1/m}
35838 @r{ v@: v a @:n @: @:arrange@:(v,n)}
35839 @r{ a@: v b @:n @: @:cvec@:(a,n)}
35840 @r{ v@: v c @:n >0 @: 21,31 @:mcol@:(v,n)}
35841 @r{ v@: v c @:n <0 @: 31 @:mrcol@:(v,-n)}
35842 @r{ m@: v c @:0 @: 31 @:getdiag@:(m)}
35843 @r{ v@: v d @: @: 25 @:diag@:(v,n)}
35844 @r{ v m@: v e @: @: 2 @:vexp@:(v,m)}
35845 @r{ v m f@: H v e @: @: 2 @:vexp@:(v,m,f)}
35846 @r{ v a@: v f @: @: 26 @:find@:(v,a,n)}
35847 @r{ v@: v h @: @: 1 @:head@:(v)}
35848 @r{ v@: I v h @: @: 1 @:tail@:(v)}
35849 @r{ v@: H v h @: @: 1 @:rhead@:(v)}
35850 @r{ v@: I H v h @: @: 1 @:rtail@:(v)}
35851 @r{ @: v i @:n @: 31 @:idn@:(1,n)}
35852 @r{ @: v i @:0 @: 31 @:idn@:(1)}
35853 @r{ h t@: v k @: @: 2 @:cons@:(h,t)}
35854 @r{ h t@: H v k @: @: 2 @:rcons@:(h,t)}
35855 @r{ v@: v l @: @: 1 @:vlen@:(v)}
35856 @r{ v@: H v l @: @: 1 @:mdims@:(v)}
35857 @r{ v m@: v m @: @: 2 @:vmask@:(v,m)}
35858 @r{ v@: v n @: @: 1 @:rnorm@:(v)}
35859 @r{ a b c@: v p @: @: 24 @:calc-pack@:}
35860 @r{ v@: v r @:n >0 @: 21,31 @:mrow@:(v,n)}
35861 @r{ v@: v r @:n <0 @: 31 @:mrrow@:(v,-n)}
35862 @r{ m@: v r @:0 @: 31 @:getdiag@:(m)}
35863 @r{ v i j@: v s @: @: @:subvec@:(v,i,j)}
35864 @r{ v i j@: I v s @: @: @:rsubvec@:(v,i,j)}
35865 @r{ m@: v t @: @: 1 @:trn@:(m)}
35866 @r{ v@: v u @: @: 24 @:calc-unpack@:}
35867 @r{ v@: v v @: @: 1 @:rev@:(v)}
35868 @r{ @: v x @:n @: 31 @:index@:(n)}
35869 @r{ n s i@: C-u v x @: @: @:index@:(n,s,i)}
35872 @r{ v@: V A @:op @: 22 @:apply@:(op,v)}
35873 @r{ v1 v2@: V C @: @: 2 @:cross@:(v1,v2)}
35874 @r{ m@: V D @: @: 1 @:det@:(m)}
35875 @r{ s@: V E @: @: 1 @:venum@:(s)}
35876 @r{ s@: V F @: @: 1 @:vfloor@:(s)}
35877 @r{ v@: V G @: @: @:grade@:(v)}
35878 @r{ v@: I V G @: @: @:rgrade@:(v)}
35879 @r{ v@: V H @:n @: 31 @:histogram@:(v,n)}
35880 @r{ v w@: H V H @:n @: 31 @:histogram@:(v,w,n)}
35881 @r{ v1 v2@: V I @:mop aop @: 22 @:inner@:(mop,aop,v1,v2)}
35882 @r{ m@: V J @: @: 1 @:ctrn@:(m)}
35883 @r{ m1 m2@: V K @: @: @:kron@:(m1,m2)}
35884 @r{ m@: V L @: @: 1 @:lud@:(m)}
35885 @r{ v@: V M @:op @: 22,23 @:map@:(op,v)}
35886 @r{ v@: V N @: @: 1 @:cnorm@:(v)}
35887 @r{ v1 v2@: V O @:op @: 22 @:outer@:(op,v1,v2)}
35888 @r{ v@: V R @:op @: 22,23 @:reduce@:(op,v)}
35889 @r{ v@: I V R @:op @: 22,23 @:rreduce@:(op,v)}
35890 @r{ a n@: H V R @:op @: 22 @:nest@:(op,a,n)}
35891 @r{ a@: I H V R @:op @: 22 @:fixp@:(op,a)}
35892 @r{ v@: V S @: @: @:sort@:(v)}
35893 @r{ v@: I V S @: @: @:rsort@:(v)}
35894 @r{ m@: V T @: @: 1 @:tr@:(m)}
35895 @r{ v@: V U @:op @: 22 @:accum@:(op,v)}
35896 @r{ v@: I V U @:op @: 22 @:raccum@:(op,v)}
35897 @r{ a n@: H V U @:op @: 22 @:anest@:(op,a,n)}
35898 @r{ a@: I H V U @:op @: 22 @:afixp@:(op,a)}
35899 @r{ s t@: V V @: @: 2 @:vunion@:(s,t)}
35900 @r{ s t@: V X @: @: 2 @:vxor@:(s,t)}
35903 @r{ @: Y @: @: @:@:user commands}
35906 @r{ @: z @: @: @:@:user commands}
35909 @r{ c@: Z [ @: @: 45 @:calc-kbd-if@:}
35910 @r{ c@: Z | @: @: 45 @:calc-kbd-else-if@:}
35911 @r{ @: Z : @: @: @:calc-kbd-else@:}
35912 @r{ @: Z ] @: @: @:calc-kbd-end-if@:}
35915 @r{ @: Z @{ @: @: 4 @:calc-kbd-loop@:}
35916 @r{ c@: Z / @: @: 45 @:calc-kbd-break@:}
35917 @r{ @: Z @} @: @: @:calc-kbd-end-loop@:}
35918 @r{ n@: Z < @: @: @:calc-kbd-repeat@:}
35919 @r{ @: Z > @: @: @:calc-kbd-end-repeat@:}
35920 @r{ n m@: Z ( @: @: @:calc-kbd-for@:}
35921 @r{ s@: Z ) @: @: @:calc-kbd-end-for@:}
35924 @r{ @: Z C-g @: @: @:@:cancel if/loop command}
35927 @r{ @: Z ` @: @: @:calc-kbd-push@:}
35928 @r{ @: Z ' @: @: @:calc-kbd-pop@:}
35929 @r{ @: Z # @: @: @:calc-kbd-query@:}
35932 @r{ comp@: Z C @:func, args @: 50 @:calc-user-define-composition@:}
35933 @r{ @: Z D @:key, command @: @:calc-user-define@:}
35934 @r{ @: Z E @:key, editing @: 30 @:calc-user-define-edit@:}
35935 @r{ defn@: Z F @:k, c, f, a, n@: 28 @:calc-user-define-formula@:}
35936 @r{ @: Z G @:key @: @:calc-get-user-defn@:}
35937 @r{ @: Z I @: @: @:calc-user-define-invocation@:}
35938 @r{ @: Z K @:key, command @: @:calc-user-define-kbd-macro@:}
35939 @r{ @: Z P @:key @: @:calc-user-define-permanent@:}
35940 @r{ @: Z S @: @: 30 @:calc-edit-user-syntax@:}
35941 @r{ @: Z T @: @: 12 @:calc-timing@:}
35942 @r{ @: Z U @:key @: @:calc-user-undefine@:}
35952 Positive prefix arguments apply to @expr{n} stack entries.
35953 Negative prefix arguments apply to the @expr{-n}th stack entry.
35954 A prefix of zero applies to the entire stack. (For @key{LFD} and
35955 @kbd{M-@key{DEL}}, the meaning of the sign is reversed.)
35959 Positive prefix arguments apply to @expr{n} stack entries.
35960 Negative prefix arguments apply to the top stack entry
35961 and the next @expr{-n} stack entries.
35965 Positive prefix arguments rotate top @expr{n} stack entries by one.
35966 Negative prefix arguments rotate the entire stack by @expr{-n}.
35967 A prefix of zero reverses the entire stack.
35971 Prefix argument specifies a repeat count or distance.
35975 Positive prefix arguments specify a precision @expr{p}.
35976 Negative prefix arguments reduce the current precision by @expr{-p}.
35980 A prefix argument is interpreted as an additional step-size parameter.
35981 A plain @kbd{C-u} prefix means to prompt for the step size.
35985 A prefix argument specifies simplification level and depth.
35986 1=Default, 2=like @kbd{a s}, 3=like @kbd{a e}.
35990 A negative prefix operates only on the top level of the input formula.
35994 Positive prefix arguments specify a word size of @expr{w} bits, unsigned.
35995 Negative prefix arguments specify a word size of @expr{w} bits, signed.
35999 Prefix arguments specify the shift amount @expr{n}. The @expr{w} argument
36000 cannot be specified in the keyboard version of this command.
36004 From the keyboard, @expr{d} is omitted and defaults to zero.
36008 Mode is toggled; a positive prefix always sets the mode, and a negative
36009 prefix always clears the mode.
36013 Some prefix argument values provide special variations of the mode.
36017 A prefix argument, if any, is used for @expr{m} instead of taking
36018 @expr{m} from the stack. @expr{M} may take any of these values:
36020 {@advance@tableindent10pt
36024 Random integer in the interval @expr{[0 .. m)}.
36026 Random floating-point number in the interval @expr{[0 .. m)}.
36028 Gaussian with mean 1 and standard deviation 0.
36030 Gaussian with specified mean and standard deviation.
36032 Random integer or floating-point number in that interval.
36034 Random element from the vector.
36042 A prefix argument from 1 to 6 specifies number of date components
36043 to remove from the stack. @xref{Date Conversions}.
36047 A prefix argument specifies a time zone; @kbd{C-u} says to take the
36048 time zone number or name from the top of the stack. @xref{Time Zones}.
36052 A prefix argument specifies a day number (0-6, 0-31, or 0-366).
36056 If the input has no units, you will be prompted for both the old and
36061 With a prefix argument, collect that many stack entries to form the
36062 input data set. Each entry may be a single value or a vector of values.
36066 With a prefix argument of 1, take a single
36067 @texline @var{n}@math{\times2}
36068 @infoline @mathit{@var{N}x2}
36069 matrix from the stack instead of two separate data vectors.
36073 The row or column number @expr{n} may be given as a numeric prefix
36074 argument instead. A plain @kbd{C-u} prefix says to take @expr{n}
36075 from the top of the stack. If @expr{n} is a vector or interval,
36076 a subvector/submatrix of the input is created.
36080 The @expr{op} prompt can be answered with the key sequence for the
36081 desired function, or with @kbd{x} or @kbd{z} followed by a function name,
36082 or with @kbd{$} to take a formula from the top of the stack, or with
36083 @kbd{'} and a typed formula. In the last two cases, the formula may
36084 be a nameless function like @samp{<#1+#2>} or @samp{<x, y : x+y>}, or it
36085 may include @kbd{$}, @kbd{$$}, etc. (where @kbd{$} will correspond to the
36086 last argument of the created function), or otherwise you will be
36087 prompted for an argument list. The number of vectors popped from the
36088 stack by @kbd{V M} depends on the number of arguments of the function.
36092 One of the mapping direction keys @kbd{_} (horizontal, i.e., map
36093 by rows or reduce across), @kbd{:} (vertical, i.e., map by columns or
36094 reduce down), or @kbd{=} (map or reduce by rows) may be used before
36095 entering @expr{op}; these modify the function name by adding the letter
36096 @code{r} for ``rows,'' @code{c} for ``columns,'' @code{a} for ``across,''
36097 or @code{d} for ``down.''
36101 The prefix argument specifies a packing mode. A nonnegative mode
36102 is the number of items (for @kbd{v p}) or the number of levels
36103 (for @kbd{v u}). A negative mode is as described below. With no
36104 prefix argument, the mode is taken from the top of the stack and
36105 may be an integer or a vector of integers.
36107 {@advance@tableindent-20pt
36111 (@var{2}) Rectangular complex number.
36113 (@var{2}) Polar complex number.
36115 (@var{3}) HMS form.
36117 (@var{2}) Error form.
36119 (@var{2}) Modulo form.
36121 (@var{2}) Closed interval.
36123 (@var{2}) Closed .. open interval.
36125 (@var{2}) Open .. closed interval.
36127 (@var{2}) Open interval.
36129 (@var{2}) Fraction.
36131 (@var{2}) Float with integer mantissa.
36133 (@var{2}) Float with mantissa in @expr{[1 .. 10)}.
36135 (@var{1}) Date form (using date numbers).
36137 (@var{3}) Date form (using year, month, day).
36139 (@var{6}) Date form (using year, month, day, hour, minute, second).
36147 A prefix argument specifies the size @expr{n} of the matrix. With no
36148 prefix argument, @expr{n} is omitted and the size is inferred from
36153 The prefix argument specifies the starting position @expr{n} (default 1).
36157 Cursor position within stack buffer affects this command.
36161 Arguments are not actually removed from the stack by this command.
36165 Variable name may be a single digit or a full name.
36169 Editing occurs in a separate buffer. Press @kbd{C-c C-c} (or
36170 @key{LFD}, or in some cases @key{RET}) to finish the edit, or kill the
36171 buffer with @kbd{C-x k} to cancel the edit. The @key{LFD} key prevents evaluation
36172 of the result of the edit.
36176 The number prompted for can also be provided as a prefix argument.
36180 Press this key a second time to cancel the prefix.
36184 With a negative prefix, deactivate all formulas. With a positive
36185 prefix, deactivate and then reactivate from scratch.
36189 Default is to scan for nearest formula delimiter symbols. With a
36190 prefix of zero, formula is delimited by mark and point. With a
36191 non-zero prefix, formula is delimited by scanning forward or
36192 backward by that many lines.
36196 Parse the region between point and mark as a vector. A nonzero prefix
36197 parses @var{n} lines before or after point as a vector. A zero prefix
36198 parses the current line as a vector. A @kbd{C-u} prefix parses the
36199 region between point and mark as a single formula.
36203 Parse the rectangle defined by point and mark as a matrix. A positive
36204 prefix @var{n} divides the rectangle into columns of width @var{n}.
36205 A zero or @kbd{C-u} prefix parses each line as one formula. A negative
36206 prefix suppresses special treatment of bracketed portions of a line.
36210 A numeric prefix causes the current language mode to be ignored.
36214 Responding to a prompt with a blank line answers that and all
36215 later prompts by popping additional stack entries.
36219 Answer for @expr{v} may also be of the form @expr{v = v_0} or
36224 With a positive prefix argument, stack contains many @expr{y}'s and one
36225 common @expr{x}. With a zero prefix, stack contains a vector of
36226 @expr{y}s and a common @expr{x}. With a negative prefix, stack
36227 contains many @expr{[x,y]} vectors. (For 3D plots, substitute
36228 @expr{z} for @expr{y} and @expr{x,y} for @expr{x}.)
36232 With any prefix argument, all curves in the graph are deleted.
36236 With a positive prefix, refines an existing plot with more data points.
36237 With a negative prefix, forces recomputation of the plot data.
36241 With any prefix argument, set the default value instead of the
36242 value for this graph.
36246 With a negative prefix argument, set the value for the printer.
36250 Condition is considered ``true'' if it is a nonzero real or complex
36251 number, or a formula whose value is known to be nonzero; it is ``false''
36256 Several formulas separated by commas are pushed as multiple stack
36257 entries. Trailing @kbd{)}, @kbd{]}, @kbd{@}}, @kbd{>}, and @kbd{"}
36258 delimiters may be omitted. The notation @kbd{$$$} refers to the value
36259 in stack level three, and causes the formula to replace the top three
36260 stack levels. The notation @kbd{$3} refers to stack level three without
36261 causing that value to be removed from the stack. Use @key{LFD} in place
36262 of @key{RET} to prevent evaluation; use @kbd{M-=} in place of @key{RET}
36263 to evaluate variables.
36267 The variable is replaced by the formula shown on the right. The
36268 Inverse flag reverses the order of the operands, e.g., @kbd{I s - x}
36270 @texline @math{x \coloneq a-x}.
36271 @infoline @expr{x := a-x}.
36275 Press @kbd{?} repeatedly to see how to choose a model. Answer the
36276 variables prompt with @expr{iv} or @expr{iv;pv} to specify
36277 independent and parameter variables. A positive prefix argument
36278 takes @mathit{@var{n}+1} vectors from the stack; a zero prefix takes a matrix
36279 and a vector from the stack.
36283 With a plain @kbd{C-u} prefix, replace the current region of the
36284 destination buffer with the yanked text instead of inserting.
36288 All stack entries are reformatted; the @kbd{H} prefix inhibits this.
36289 The @kbd{I} prefix sets the mode temporarily, redraws the top stack
36290 entry, then restores the original setting of the mode.
36294 A negative prefix sets the default 3D resolution instead of the
36295 default 2D resolution.
36299 This grabs a vector of the form [@var{prec}, @var{wsize}, @var{ssize},
36300 @var{radix}, @var{flfmt}, @var{ang}, @var{frac}, @var{symb}, @var{polar},
36301 @var{matrix}, @var{simp}, @var{inf}]. A prefix argument from 1 to 12
36302 grabs the @var{n}th mode value only.
36306 (Space is provided below for you to keep your own written notes.)
36314 @node Key Index, Command Index, Summary, Top
36315 @unnumbered Index of Key Sequences
36319 @node Command Index, Function Index, Key Index, Top
36320 @unnumbered Index of Calculator Commands
36322 Since all Calculator commands begin with the prefix @samp{calc-}, the
36323 @kbd{x} key has been provided as a variant of @kbd{M-x} which automatically
36324 types @samp{calc-} for you. Thus, @kbd{x last-args} is short for
36325 @kbd{M-x calc-last-args}.
36329 @node Function Index, Concept Index, Command Index, Top
36330 @unnumbered Index of Algebraic Functions
36332 This is a list of built-in functions and operators usable in algebraic
36333 expressions. Their full Lisp names are derived by adding the prefix
36334 @samp{calcFunc-}, as in @code{calcFunc-sqrt}.
36336 All functions except those noted with ``*'' have corresponding
36337 Calc keystrokes and can also be found in the Calc Summary.
36342 @node Concept Index, Variable Index, Function Index, Top
36343 @unnumbered Concept Index
36347 @node Variable Index, Lisp Function Index, Concept Index, Top
36348 @unnumbered Index of Variables
36350 The variables in this list that do not contain dashes are accessible
36351 as Calc variables. Add a @samp{var-} prefix to get the name of the
36352 corresponding Lisp variable.
36354 The remaining variables are Lisp variables suitable for @code{setq}ing
36355 in your Calc init file or @file{.emacs} file.
36359 @node Lisp Function Index, , Variable Index, Top
36360 @unnumbered Index of Lisp Math Functions
36362 The following functions are meant to be used with @code{defmath}, not
36363 @code{defun} definitions. For names that do not start with @samp{calc-},
36364 the corresponding full Lisp name is derived by adding a prefix of
36373 arch-tag: 77a71809-fa4d-40be-b2cc-da3e8fb137c0