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 2.1 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
35 @alias texline=comment
36 @macro infoline{stuff}
52 % Suggested by Karl Berry <karl@@freefriends.org>
53 \gdef\!{\mskip-\thinmuskip}
56 @c Fix some other things specifically for this manual.
59 @mathcode`@:=`@: @c Make Calc fractions come out right in math mode
61 \gdef\coloneq{\mathrel{\mathord:\mathord=}}
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66 \gdef\afterdisplayh{\vskip-10pt}
68 @newdimen@kyvpos @kyvpos=0pt
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70 @newcount@calcclubpenalty @calcclubpenalty=1000
73 @newtoks@calcoldeverypar @calcoldeverypar=@everypar
74 @everypar={@calceverypar@the@calcoldeverypar}
75 @ifx@turnoffactive@undefinedzzz@def@turnoffactive{}@fi
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77 @catcode`@\=0 \catcode`\@=11
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84 This file documents Calc, the GNU Emacs calculator.
86 Copyright (C) 1990, 1991, 2001, 2002, 2005 Free Software Foundation, Inc.
89 Permission is granted to copy, distribute and/or modify this document
90 under the terms of the GNU Free Documentation License, Version 1.1 or
91 any later version published by the Free Software Foundation; with the
92 Invariant Sections being just ``GNU GENERAL PUBLIC LICENSE'', with the
93 Front-Cover texts being ``A GNU Manual,'' and with the Back-Cover
94 Texts as in (a) below.
96 (a) The FSF's Back-Cover Text is: ``You have freedom to copy and modify
97 this GNU Manual, like GNU software. Copies published by the Free
98 Software Foundation raise funds for GNU development.''
104 * Calc: (calc). Advanced desk calculator and mathematical tool.
109 @center @titlefont{Calc Manual}
111 @center GNU Emacs Calc Version 2.1
116 @center Dave Gillespie
117 @center daveg@@synaptics.com
120 @vskip 0pt plus 1filll
121 Copyright @copyright{} 1990, 1991, 2001, 2002, 2005
122 Free Software Foundation, Inc.
128 @node Top, , (dir), (dir)
129 @chapter The GNU Emacs Calculator
132 @dfn{Calc} is an advanced desk calculator and mathematical tool
133 that runs as part of the GNU Emacs environment.
135 This manual is divided into three major parts: ``Getting Started,''
136 the ``Calc Tutorial,'' and the ``Calc Reference.'' The Tutorial
137 introduces all the major aspects of Calculator use in an easy,
138 hands-on way. The remainder of the manual is a complete reference to
139 the features of the Calculator.
141 For help in the Emacs Info system (which you are using to read this
142 file), type @kbd{?}. (You can also type @kbd{h} to run through a
143 longer Info tutorial.)
147 * Copying:: How you can copy and share Calc.
149 * Getting Started:: General description and overview.
150 * Interactive Tutorial::
151 * Tutorial:: A step-by-step introduction for beginners.
153 * Introduction:: Introduction to the Calc reference manual.
154 * Data Types:: Types of objects manipulated by Calc.
155 * Stack and Trail:: Manipulating the stack and trail buffers.
156 * Mode Settings:: Adjusting display format and other modes.
157 * Arithmetic:: Basic arithmetic functions.
158 * Scientific Functions:: Transcendentals and other scientific functions.
159 * Matrix Functions:: Operations on vectors and matrices.
160 * Algebra:: Manipulating expressions algebraically.
161 * Units:: Operations on numbers with units.
162 * Store and Recall:: Storing and recalling variables.
163 * Graphics:: Commands for making graphs of data.
164 * Kill and Yank:: Moving data into and out of Calc.
165 * Embedded Mode:: Working with formulas embedded in a file.
166 * Programming:: Calc as a programmable calculator.
168 * Customizable Variables:: Customizable Variables.
169 * Reporting Bugs:: How to report bugs and make suggestions.
171 * Summary:: Summary of Calc commands and functions.
173 * Key Index:: The standard Calc key sequences.
174 * Command Index:: The interactive Calc commands.
175 * Function Index:: Functions (in algebraic formulas).
176 * Concept Index:: General concepts.
177 * Variable Index:: Variables used by Calc (both user and internal).
178 * Lisp Function Index:: Internal Lisp math functions.
181 @node Copying, Getting Started, Top, Top
182 @unnumbered GNU GENERAL PUBLIC LICENSE
183 @center Version 2, June 1991
185 @c This file is intended to be included in another file.
188 Copyright @copyright{} 1989, 1991 Free Software Foundation, Inc.
189 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA
191 Everyone is permitted to copy and distribute verbatim copies
192 of this license document, but changing it is not allowed.
195 @unnumberedsec Preamble
197 The licenses for most software are designed to take away your
198 freedom to share and change it. By contrast, the GNU General Public
199 License is intended to guarantee your freedom to share and change free
200 software---to make sure the software is free for all its users. This
201 General Public License applies to most of the Free Software
202 Foundation's software and to any other program whose authors commit to
203 using it. (Some other Free Software Foundation software is covered by
204 the GNU Library General Public License instead.) You can apply it to
207 When we speak of free software, we are referring to freedom, not
208 price. Our General Public Licenses are designed to make sure that you
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210 this service if you wish), that you receive source code or can get it
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214 To protect your rights, we need to make restrictions that forbid
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216 These restrictions translate to certain responsibilities for you if you
217 distribute copies of the software, or if you modify it.
219 For example, if you distribute copies of such a program, whether
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221 you have. You must make sure that they, too, receive or can get the
222 source code. And you must show them these terms so they know their
225 We protect your rights with two steps: (1) copyright the software, and
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242 The precise terms and conditions for copying, distribution and
246 @unnumberedsec TERMS AND CONDITIONS FOR COPYING, DISTRIBUTION AND MODIFICATION
249 @center TERMS AND CONDITIONS FOR COPYING, DISTRIBUTION AND MODIFICATION
254 This License applies to any program or other work which contains
255 a notice placed by the copyright holder saying it may be distributed
256 under the terms of this General Public License. The ``Program'', below,
257 refers to any such program or work, and a ``work based on the Program''
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272 You may copy and distribute verbatim copies of the Program's
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435 This section is intended to make thoroughly clear what is believed to
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497 POSSIBILITY OF SUCH DAMAGES.
501 @heading END OF TERMS AND CONDITIONS
504 @center END OF TERMS AND CONDITIONS
508 @unnumberedsec Appendix: How to Apply These Terms to Your New Programs
510 If you develop a new program, and you want it to be of the greatest
511 possible use to the public, the best way to achieve this is to make it
512 free software which everyone can redistribute and change under these terms.
514 To do so, attach the following notices to the program. It is safest
515 to attach them to the start of each source file to most effectively
516 convey the exclusion of warranty; and each file should have at least
517 the ``copyright'' line and a pointer to where the full notice is found.
520 @var{one line to give the program's name and a brief idea of what it does.}
521 Copyright (C) @var{yyyy} @var{name of author}
523 This program is free software; you can redistribute it and/or modify
524 it under the terms of the GNU General Public License as published by
525 the Free Software Foundation; either version 2 of the License, or
526 (at your option) any later version.
528 This program is distributed in the hope that it will be useful,
529 but WITHOUT ANY WARRANTY; without even the implied warranty of
530 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
531 GNU General Public License for more details.
533 You should have received a copy of the GNU General Public License
534 along with this program; if not, write to the Free Software
535 Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA.
538 Also add information on how to contact you by electronic and paper mail.
540 If the program is interactive, make it output a short notice like this
541 when it starts in an interactive mode:
544 Gnomovision version 69, Copyright (C) 19@var{yy} @var{name of author}
545 Gnomovision comes with ABSOLUTELY NO WARRANTY; for details type `show w'.
546 This is free software, and you are welcome to redistribute it
547 under certain conditions; type `show c' for details.
550 The hypothetical commands @samp{show w} and @samp{show c} should show
551 the appropriate parts of the General Public License. Of course, the
552 commands you use may be called something other than @samp{show w} and
553 @samp{show c}; they could even be mouse-clicks or menu items---whatever
556 You should also get your employer (if you work as a programmer) or your
557 school, if any, to sign a ``copyright disclaimer'' for the program, if
558 necessary. Here is a sample; alter the names:
561 Yoyodyne, Inc., hereby disclaims all copyright interest in the program
562 `Gnomovision' (which makes passes at compilers) written by James Hacker.
564 @var{signature of Ty Coon}, 1 April 1989
565 Ty Coon, President of Vice
568 This General Public License does not permit incorporating your program into
569 proprietary programs. If your program is a subroutine library, you may
570 consider it more useful to permit linking proprietary applications with the
571 library. If this is what you want to do, use the GNU Library General
572 Public License instead of this License.
574 @node Getting Started, Tutorial, Copying, Top
575 @chapter Getting Started
577 This chapter provides a general overview of Calc, the GNU Emacs
578 Calculator: What it is, how to start it and how to exit from it,
579 and what are the various ways that it can be used.
583 * About This Manual::
584 * Notations Used in This Manual::
586 * Demonstration of Calc::
587 * History and Acknowledgements::
590 @node What is Calc, About This Manual, Getting Started, Getting Started
591 @section What is Calc?
594 @dfn{Calc} is an advanced calculator and mathematical tool that runs as
595 part of the GNU Emacs environment. Very roughly based on the HP-28/48
596 series of calculators, its many features include:
600 Choice of algebraic or RPN (stack-based) entry of calculations.
603 Arbitrary precision integers and floating-point numbers.
606 Arithmetic on rational numbers, complex numbers (rectangular and polar),
607 error forms with standard deviations, open and closed intervals, vectors
608 and matrices, dates and times, infinities, sets, quantities with units,
609 and algebraic formulas.
612 Mathematical operations such as logarithms and trigonometric functions.
615 Programmer's features (bitwise operations, non-decimal numbers).
618 Financial functions such as future value and internal rate of return.
621 Number theoretical features such as prime factorization and arithmetic
622 modulo @var{m} for any @var{m}.
625 Algebraic manipulation features, including symbolic calculus.
628 Moving data to and from regular editing buffers.
631 Embedded mode for manipulating Calc formulas and data directly
632 inside any editing buffer.
635 Graphics using GNUPLOT, a versatile (and free) plotting program.
638 Easy programming using keyboard macros, algebraic formulas,
639 algebraic rewrite rules, or extended Emacs Lisp.
642 Calc tries to include a little something for everyone; as a result it is
643 large and might be intimidating to the first-time user. If you plan to
644 use Calc only as a traditional desk calculator, all you really need to
645 read is the ``Getting Started'' chapter of this manual and possibly the
646 first few sections of the tutorial. As you become more comfortable with
647 the program you can learn its additional features. Calc does not
648 have the scope and depth of a fully-functional symbolic math package,
649 but Calc has the advantages of convenience, portability, and freedom.
651 @node About This Manual, Notations Used in This Manual, What is Calc, Getting Started
652 @section About This Manual
655 This document serves as a complete description of the GNU Emacs
656 Calculator. It works both as an introduction for novices, and as
657 a reference for experienced users. While it helps to have some
658 experience with GNU Emacs in order to get the most out of Calc,
659 this manual ought to be readable even if you don't know or use Emacs
663 The manual is divided into three major parts:@: the ``Getting
664 Started'' chapter you are reading now, the Calc tutorial (chapter 2),
665 and the Calc reference manual (the remaining chapters and appendices).
668 The manual is divided into three major parts:@: the ``Getting
669 Started'' chapter you are reading now, the Calc tutorial (chapter 2),
670 and the Calc reference manual (the remaining chapters and appendices).
672 @c This manual has been printed in two volumes, the @dfn{Tutorial} and the
673 @c @dfn{Reference}. Both volumes include a copy of the ``Getting Started''
677 If you are in a hurry to use Calc, there is a brief ``demonstration''
678 below which illustrates the major features of Calc in just a couple of
679 pages. If you don't have time to go through the full tutorial, this
680 will show you everything you need to know to begin.
681 @xref{Demonstration of Calc}.
683 The tutorial chapter walks you through the various parts of Calc
684 with lots of hands-on examples and explanations. If you are new
685 to Calc and you have some time, try going through at least the
686 beginning of the tutorial. The tutorial includes about 70 exercises
687 with answers. These exercises give you some guided practice with
688 Calc, as well as pointing out some interesting and unusual ways
691 The reference section discusses Calc in complete depth. You can read
692 the reference from start to finish if you want to learn every aspect
693 of Calc. Or, you can look in the table of contents or the Concept
694 Index to find the parts of the manual that discuss the things you
697 @cindex Marginal notes
698 Every Calc keyboard command is listed in the Calc Summary, and also
699 in the Key Index. Algebraic functions, @kbd{M-x} commands, and
700 variables also have their own indices.
702 @infoline In the printed manual, each
703 paragraph that is referenced in the Key or Function Index is marked
704 in the margin with its index entry.
706 @c [fix-ref Help Commands]
707 You can access this manual on-line at any time within Calc by
708 pressing the @kbd{h i} key sequence. Outside of the Calc window,
709 you can press @kbd{M-# i} to read the manual on-line. Also, you
710 can jump directly to the Tutorial by pressing @kbd{h t} or @kbd{M-# t},
711 or to the Summary by pressing @kbd{h s} or @kbd{M-# s}. Within Calc,
712 you can also go to the part of the manual describing any Calc key,
713 function, or variable using @w{@kbd{h k}}, @kbd{h f}, or @kbd{h v},
714 respectively. @xref{Help Commands}.
716 The Calc manual can be printed, but because the manual is so large, you
717 should only make a printed copy if you really need it. To print the
718 manual, you will need the @TeX{} typesetting program (this is a free
719 program by Donald Knuth at Stanford University) as well as the
720 @file{texindex} program and @file{texinfo.tex} file, both of which can
721 be obtained from the FSF as part of the @code{texinfo} package.
722 To print the Calc manual in one huge tome, you will need the
723 source code to this manual, @file{calc.texi}, available as part of the
724 Emacs source. Once you have this file, type @kbd{texi2dvi calc.texi}.
725 Alternatively, change to the @file{man} subdirectory of the Emacs
726 source distribution, and type @kbd{make calc.dvi}. (Don't worry if you
727 get some ``overfull box'' warnings while @TeX{} runs.)
728 The result will be a device-independent output file called
729 @file{calc.dvi}, which you must print in whatever way is right
730 for your system. On many systems, the command is
743 @c Printed copies of this manual are also available from the Free Software
746 @node Notations Used in This Manual, Demonstration of Calc, About This Manual, Getting Started
747 @section Notations Used in This Manual
750 This section describes the various notations that are used
751 throughout the Calc manual.
753 In keystroke sequences, uppercase letters mean you must hold down
754 the shift key while typing the letter. Keys pressed with Control
755 held down are shown as @kbd{C-x}. Keys pressed with Meta held down
756 are shown as @kbd{M-x}. Other notations are @key{RET} for the
757 Return key, @key{SPC} for the space bar, @key{TAB} for the Tab key,
758 @key{DEL} for the Delete key, and @key{LFD} for the Line-Feed key.
759 The @key{DEL} key is called Backspace on some keyboards, it is
760 whatever key you would use to correct a simple typing error when
761 regularly using Emacs.
763 (If you don't have the @key{LFD} or @key{TAB} keys on your keyboard,
764 the @kbd{C-j} and @kbd{C-i} keys are equivalent to them, respectively.
765 If you don't have a Meta key, look for Alt or Extend Char. You can
766 also press @key{ESC} or @key{C-[} first to get the same effect, so
767 that @kbd{M-x}, @kbd{@key{ESC} x}, and @kbd{C-[ x} are all equivalent.)
769 Sometimes the @key{RET} key is not shown when it is ``obvious''
770 that you must press @key{RET} to proceed. For example, the @key{RET}
771 is usually omitted in key sequences like @kbd{M-x calc-keypad @key{RET}}.
773 Commands are generally shown like this: @kbd{p} (@code{calc-precision})
774 or @kbd{M-# k} (@code{calc-keypad}). This means that the command is
775 normally used by pressing the @kbd{p} key or @kbd{M-# k} key sequence,
776 but it also has the full-name equivalent shown, e.g., @kbd{M-x calc-precision}.
778 Commands that correspond to functions in algebraic notation
779 are written: @kbd{C} (@code{calc-cos}) [@code{cos}]. This means
780 the @kbd{C} key is equivalent to @kbd{M-x calc-cos}, and that
781 the corresponding function in an algebraic-style formula would
782 be @samp{cos(@var{x})}.
784 A few commands don't have key equivalents: @code{calc-sincos}
787 @node Demonstration of Calc, Using Calc, Notations Used in This Manual, Getting Started
788 @section A Demonstration of Calc
791 @cindex Demonstration of Calc
792 This section will show some typical small problems being solved with
793 Calc. The focus is more on demonstration than explanation, but
794 everything you see here will be covered more thoroughly in the
797 To begin, start Emacs if necessary (usually the command @code{emacs}
798 does this), and type @kbd{M-# c} (or @kbd{@key{ESC} # c}) to start the
799 Calculator. (@xref{Starting Calc}, if this doesn't work for you.)
801 Be sure to type all the sample input exactly, especially noting the
802 difference between lower-case and upper-case letters. Remember,
803 @key{RET}, @key{TAB}, @key{DEL}, and @key{SPC} are the Return, Tab,
804 Delete, and Space keys.
806 @strong{RPN calculation.} In RPN, you type the input number(s) first,
807 then the command to operate on the numbers.
810 Type @kbd{2 @key{RET} 3 + Q} to compute
811 @texline @math{\sqrt{2+3} = 2.2360679775}.
812 @infoline the square root of 2+3, which is 2.2360679775.
815 Type @kbd{P 2 ^} to compute
816 @texline @math{\pi^2 = 9.86960440109}.
817 @infoline the value of `pi' squared, 9.86960440109.
820 Type @key{TAB} to exchange the order of these two results.
823 Type @kbd{- I H S} to subtract these results and compute the Inverse
824 Hyperbolic sine of the difference, 2.72996136574.
827 Type @key{DEL} to erase this result.
829 @strong{Algebraic calculation.} You can also enter calculations using
830 conventional ``algebraic'' notation. To enter an algebraic formula,
831 use the apostrophe key.
834 Type @kbd{' sqrt(2+3) @key{RET}} to compute
835 @texline @math{\sqrt{2+3}}.
836 @infoline the square root of 2+3.
839 Type @kbd{' pi^2 @key{RET}} to enter
840 @texline @math{\pi^2}.
841 @infoline `pi' squared.
842 To evaluate this symbolic formula as a number, type @kbd{=}.
845 Type @kbd{' arcsinh($ - $$) @key{RET}} to subtract the second-most-recent
846 result from the most-recent and compute the Inverse Hyperbolic sine.
848 @strong{Keypad mode.} If you are using the X window system, press
849 @w{@kbd{M-# k}} to get Keypad mode. (If you don't use X, skip to
853 Click on the @key{2}, @key{ENTER}, @key{3}, @key{+}, and @key{SQRT}
854 ``buttons'' using your left mouse button.
857 Click on @key{PI}, @key{2}, and @tfn{y^x}.
860 Click on @key{INV}, then @key{ENTER} to swap the two results.
863 Click on @key{-}, @key{INV}, @key{HYP}, and @key{SIN}.
866 Click on @key{<-} to erase the result, then click @key{OFF} to turn
867 the Keypad Calculator off.
869 @strong{Grabbing data.} Type @kbd{M-# x} if necessary to exit Calc.
870 Now select the following numbers as an Emacs region: ``Mark'' the
871 front of the list by typing @kbd{C-@key{SPC}} or @kbd{C-@@} there,
872 then move to the other end of the list. (Either get this list from
873 the on-line copy of this manual, accessed by @w{@kbd{M-# i}}, or just
874 type these numbers into a scratch file.) Now type @kbd{M-# g} to
875 ``grab'' these numbers into Calc.
886 The result @samp{[1.23, 1.97, 1.6, 2, 1.19, 1.08]} is a Calc ``vector.''
887 Type @w{@kbd{V R +}} to compute the sum of these numbers.
890 Type @kbd{U} to Undo this command, then type @kbd{V R *} to compute
891 the product of the numbers.
894 You can also grab data as a rectangular matrix. Place the cursor on
895 the upper-leftmost @samp{1} and set the mark, then move to just after
896 the lower-right @samp{8} and press @kbd{M-# r}.
899 Type @kbd{v t} to transpose this
900 @texline @math{3\times2}
903 @texline @math{2\times3}
905 matrix. Type @w{@kbd{v u}} to unpack the rows into two separate
906 vectors. Now type @w{@kbd{V R + @key{TAB} V R +}} to compute the sums
907 of the two original columns. (There is also a special
908 grab-and-sum-columns command, @kbd{M-# :}.)
910 @strong{Units conversion.} Units are entered algebraically.
911 Type @w{@kbd{' 43 mi/hr @key{RET}}} to enter the quantity 43 miles-per-hour.
912 Type @w{@kbd{u c km/hr @key{RET}}}. Type @w{@kbd{u c m/s @key{RET}}}.
914 @strong{Date arithmetic.} Type @kbd{t N} to get the current date and
915 time. Type @kbd{90 +} to find the date 90 days from now. Type
916 @kbd{' <25 dec 87> @key{RET}} to enter a date, then @kbd{- 7 /} to see how
917 many weeks have passed since then.
919 @strong{Algebra.} Algebraic entries can also include formulas
920 or equations involving variables. Type @kbd{@w{' [x + y} = a, x y = 1] @key{RET}}
921 to enter a pair of equations involving three variables.
922 (Note the leading apostrophe in this example; also, note that the space
923 between @samp{x y} is required.) Type @w{@kbd{a S x,y @key{RET}}} to solve
924 these equations for the variables @expr{x} and @expr{y}.
927 Type @kbd{d B} to view the solutions in more readable notation.
928 Type @w{@kbd{d C}} to view them in C language notation, @kbd{d T}
929 to view them in the notation for the @TeX{} typesetting system,
930 and @kbd{d L} to view them in the notation for the La@TeX{} typesetting
931 system. Type @kbd{d N} to return to normal notation.
934 Type @kbd{7.5}, then @kbd{s l a @key{RET}} to let @expr{a = 7.5} in these formulas.
935 (That's a letter @kbd{l}, not a numeral @kbd{1}.)
938 @strong{Help functions.} You can read about any command in the on-line
939 manual. Type @kbd{M-# c} to return to Calc after each of these
940 commands: @kbd{h k t N} to read about the @kbd{t N} command,
941 @kbd{h f sqrt @key{RET}} to read about the @code{sqrt} function, and
942 @kbd{h s} to read the Calc summary.
945 @strong{Help functions.} You can read about any command in the on-line
946 manual. Remember to type the letter @kbd{l}, then @kbd{M-# c}, to
947 return here after each of these commands: @w{@kbd{h k t N}} to read
948 about the @w{@kbd{t N}} command, @kbd{h f sqrt @key{RET}} to read about the
949 @code{sqrt} function, and @kbd{h s} to read the Calc summary.
952 Press @key{DEL} repeatedly to remove any leftover results from the stack.
953 To exit from Calc, press @kbd{q} or @kbd{M-# c} again.
955 @node Using Calc, History and Acknowledgements, Demonstration of Calc, Getting Started
959 Calc has several user interfaces that are specialized for
960 different kinds of tasks. As well as Calc's standard interface,
961 there are Quick mode, Keypad mode, and Embedded mode.
965 * The Standard Interface::
966 * Quick Mode Overview::
967 * Keypad Mode Overview::
968 * Standalone Operation::
969 * Embedded Mode Overview::
970 * Other M-# Commands::
973 @node Starting Calc, The Standard Interface, Using Calc, Using Calc
974 @subsection Starting Calc
977 On most systems, you can type @kbd{M-#} to start the Calculator.
978 The notation @kbd{M-#} is short for Meta-@kbd{#}. On most
979 keyboards this means holding down the Meta (or Alt) and
980 Shift keys while typing @kbd{3}.
983 Once again, if you don't have a Meta key on your keyboard you can type
984 @key{ESC} first, then @kbd{#}, to accomplish the same thing. If you
985 don't even have an @key{ESC} key, you can fake it by holding down
986 Control or @key{CTRL} while typing a left square bracket
987 (that's @kbd{C-[} in Emacs notation).
989 @kbd{M-#} is a @dfn{prefix key}; when you press it, Emacs waits for
990 you to press a second key to complete the command. In this case,
991 you will follow @kbd{M-#} with a letter (upper- or lower-case, it
992 doesn't matter for @kbd{M-#}) that says which Calc interface you
995 To get Calc's standard interface, type @kbd{M-# c}. To get
996 Keypad mode, type @kbd{M-# k}. Type @kbd{M-# ?} to get a brief
997 list of the available options, and type a second @kbd{?} to get
1000 To ease typing, @kbd{M-# M-#} (or @kbd{M-# #} if that's easier)
1001 also works to start Calc. It starts the same interface (either
1002 @kbd{M-# c} or @w{@kbd{M-# k}}) that you last used, selecting the
1003 @kbd{M-# c} interface by default. (If your installation has
1004 a special function key set up to act like @kbd{M-#}, hitting that
1005 function key twice is just like hitting @kbd{M-# M-#}.)
1007 If @kbd{M-#} doesn't work for you, you can always type explicit
1008 commands like @kbd{M-x calc} (for the standard user interface) or
1009 @w{@kbd{M-x calc-keypad}} (for Keypad mode). First type @kbd{M-x}
1010 (that's Meta with the letter @kbd{x}), then, at the prompt,
1011 type the full command (like @kbd{calc-keypad}) and press Return.
1013 The same commands (like @kbd{M-# c} or @kbd{M-# M-#}) that start
1014 the Calculator also turn it off if it is already on.
1016 @node The Standard Interface, Quick Mode Overview, Starting Calc, Using Calc
1017 @subsection The Standard Calc Interface
1020 @cindex Standard user interface
1021 Calc's standard interface acts like a traditional RPN calculator,
1022 operated by the normal Emacs keyboard. When you type @kbd{M-# c}
1023 to start the Calculator, the Emacs screen splits into two windows
1024 with the file you were editing on top and Calc on the bottom.
1030 --**-Emacs: myfile (Fundamental)----All----------------------
1031 --- Emacs Calculator Mode --- |Emacs Calc Mode v2.00...
1039 --%%-Calc: 12 Deg (Calculator)----All----- --%%-Emacs: *Calc Trail*
1043 In this figure, the mode-line for @file{myfile} has moved up and the
1044 ``Calculator'' window has appeared below it. As you can see, Calc
1045 actually makes two windows side-by-side. The lefthand one is
1046 called the @dfn{stack window} and the righthand one is called the
1047 @dfn{trail window.} The stack holds the numbers involved in the
1048 calculation you are currently performing. The trail holds a complete
1049 record of all calculations you have done. In a desk calculator with
1050 a printer, the trail corresponds to the paper tape that records what
1053 In this case, the trail shows that four numbers (17.3, 3, 2, and 4)
1054 were first entered into the Calculator, then the 2 and 4 were
1055 multiplied to get 8, then the 3 and 8 were subtracted to get @mathit{-5}.
1056 (The @samp{>} symbol shows that this was the most recent calculation.)
1057 The net result is the two numbers 17.3 and @mathit{-5} sitting on the stack.
1059 Most Calculator commands deal explicitly with the stack only, but
1060 there is a set of commands that allow you to search back through
1061 the trail and retrieve any previous result.
1063 Calc commands use the digits, letters, and punctuation keys.
1064 Shifted (i.e., upper-case) letters are different from lowercase
1065 letters. Some letters are @dfn{prefix} keys that begin two-letter
1066 commands. For example, @kbd{e} means ``enter exponent'' and shifted
1067 @kbd{E} means @expr{e^x}. With the @kbd{d} (``display modes'') prefix
1068 the letter ``e'' takes on very different meanings: @kbd{d e} means
1069 ``engineering notation'' and @kbd{d E} means ``@dfn{eqn} language mode.''
1071 There is nothing stopping you from switching out of the Calc
1072 window and back into your editing window, say by using the Emacs
1073 @w{@kbd{C-x o}} (@code{other-window}) command. When the cursor is
1074 inside a regular window, Emacs acts just like normal. When the
1075 cursor is in the Calc stack or trail windows, keys are interpreted
1078 When you quit by pressing @kbd{M-# c} a second time, the Calculator
1079 windows go away but the actual Stack and Trail are not gone, just
1080 hidden. When you press @kbd{M-# c} once again you will get the
1081 same stack and trail contents you had when you last used the
1084 The Calculator does not remember its state between Emacs sessions.
1085 Thus if you quit Emacs and start it again, @kbd{M-# c} will give you
1086 a fresh stack and trail. There is a command (@kbd{m m}) that lets
1087 you save your favorite mode settings between sessions, though.
1088 One of the things it saves is which user interface (standard or
1089 Keypad) you last used; otherwise, a freshly started Emacs will
1090 always treat @kbd{M-# M-#} the same as @kbd{M-# c}.
1092 The @kbd{q} key is another equivalent way to turn the Calculator off.
1094 If you type @kbd{M-# b} first and then @kbd{M-# c}, you get a
1095 full-screen version of Calc (@code{full-calc}) in which the stack and
1096 trail windows are still side-by-side but are now as tall as the whole
1097 Emacs screen. When you press @kbd{q} or @kbd{M-# c} again to quit,
1098 the file you were editing before reappears. The @kbd{M-# b} key
1099 switches back and forth between ``big'' full-screen mode and the
1100 normal partial-screen mode.
1102 Finally, @kbd{M-# o} (@code{calc-other-window}) is like @kbd{M-# c}
1103 except that the Calc window is not selected. The buffer you were
1104 editing before remains selected instead. @kbd{M-# o} is a handy
1105 way to switch out of Calc momentarily to edit your file; type
1106 @kbd{M-# c} to switch back into Calc when you are done.
1108 @node Quick Mode Overview, Keypad Mode Overview, The Standard Interface, Using Calc
1109 @subsection Quick Mode (Overview)
1112 @dfn{Quick mode} is a quick way to use Calc when you don't need the
1113 full complexity of the stack and trail. To use it, type @kbd{M-# q}
1114 (@code{quick-calc}) in any regular editing buffer.
1116 Quick mode is very simple: It prompts you to type any formula in
1117 standard algebraic notation (like @samp{4 - 2/3}) and then displays
1118 the result at the bottom of the Emacs screen (@mathit{3.33333333333}
1119 in this case). You are then back in the same editing buffer you
1120 were in before, ready to continue editing or to type @kbd{M-# q}
1121 again to do another quick calculation. The result of the calculation
1122 will also be in the Emacs ``kill ring'' so that a @kbd{C-y} command
1123 at this point will yank the result into your editing buffer.
1125 Calc mode settings affect Quick mode, too, though you will have to
1126 go into regular Calc (with @kbd{M-# c}) to change the mode settings.
1128 @c [fix-ref Quick Calculator mode]
1129 @xref{Quick Calculator}, for further information.
1131 @node Keypad Mode Overview, Standalone Operation, Quick Mode Overview, Using Calc
1132 @subsection Keypad Mode (Overview)
1135 @dfn{Keypad mode} is a mouse-based interface to the Calculator.
1136 It is designed for use with terminals that support a mouse. If you
1137 don't have a mouse, you will have to operate Keypad mode with your
1138 arrow keys (which is probably more trouble than it's worth).
1140 Type @kbd{M-# k} to turn Keypad mode on or off. Once again you
1141 get two new windows, this time on the righthand side of the screen
1142 instead of at the bottom. The upper window is the familiar Calc
1143 Stack; the lower window is a picture of a typical calculator keypad.
1147 \advance \dimen0 by 24\baselineskip%
1148 \ifdim \dimen0>\pagegoal \vfill\eject \fi%
1152 |--- Emacs Calculator Mode ---
1156 |--%%-Calc: 12 Deg (Calcul
1157 |----+-----Calc 2.00-----+----1
1158 |FLR |CEIL|RND |TRNC|CLN2|FLT |
1159 |----+----+----+----+----+----|
1160 | LN |EXP | |ABS |IDIV|MOD |
1161 |----+----+----+----+----+----|
1162 |SIN |COS |TAN |SQRT|y^x |1/x |
1163 |----+----+----+----+----+----|
1164 | ENTER |+/- |EEX |UNDO| <- |
1165 |-----+---+-+--+--+-+---++----|
1166 | INV | 7 | 8 | 9 | / |
1167 |-----+-----+-----+-----+-----|
1168 | HYP | 4 | 5 | 6 | * |
1169 |-----+-----+-----+-----+-----|
1170 |EXEC | 1 | 2 | 3 | - |
1171 |-----+-----+-----+-----+-----|
1172 | OFF | 0 | . | PI | + |
1173 |-----+-----+-----+-----+-----+
1176 Keypad mode is much easier for beginners to learn, because there
1177 is no need to memorize lots of obscure key sequences. But not all
1178 commands in regular Calc are available on the Keypad. You can
1179 always switch the cursor into the Calc stack window to use
1180 standard Calc commands if you need. Serious Calc users, though,
1181 often find they prefer the standard interface over Keypad mode.
1183 To operate the Calculator, just click on the ``buttons'' of the
1184 keypad using your left mouse button. To enter the two numbers
1185 shown here you would click @w{@kbd{1 7 .@: 3 ENTER 5 +/- ENTER}}; to
1186 add them together you would then click @kbd{+} (to get 12.3 on
1189 If you click the right mouse button, the top three rows of the
1190 keypad change to show other sets of commands, such as advanced
1191 math functions, vector operations, and operations on binary
1194 Because Keypad mode doesn't use the regular keyboard, Calc leaves
1195 the cursor in your original editing buffer. You can type in
1196 this buffer in the usual way while also clicking on the Calculator
1197 keypad. One advantage of Keypad mode is that you don't need an
1198 explicit command to switch between editing and calculating.
1200 If you press @kbd{M-# b} first, you get a full-screen Keypad mode
1201 (@code{full-calc-keypad}) with three windows: The keypad in the lower
1202 left, the stack in the lower right, and the trail on top.
1204 @c [fix-ref Keypad Mode]
1205 @xref{Keypad Mode}, for further information.
1207 @node Standalone Operation, Embedded Mode Overview, Keypad Mode Overview, Using Calc
1208 @subsection Standalone Operation
1211 @cindex Standalone Operation
1212 If you are not in Emacs at the moment but you wish to use Calc,
1213 you must start Emacs first. If all you want is to run Calc, you
1214 can give the commands:
1224 emacs -f full-calc-keypad
1228 which run a full-screen Calculator (as if by @kbd{M-# b M-# c}) or
1229 a full-screen X-based Calculator (as if by @kbd{M-# b M-# k}).
1230 In standalone operation, quitting the Calculator (by pressing
1231 @kbd{q} or clicking on the keypad @key{EXIT} button) quits Emacs
1234 @node Embedded Mode Overview, Other M-# Commands, Standalone Operation, Using Calc
1235 @subsection Embedded Mode (Overview)
1238 @dfn{Embedded mode} is a way to use Calc directly from inside an
1239 editing buffer. Suppose you have a formula written as part of a
1253 and you wish to have Calc compute and format the derivative for
1254 you and store this derivative in the buffer automatically. To
1255 do this with Embedded mode, first copy the formula down to where
1256 you want the result to be:
1270 Now, move the cursor onto this new formula and press @kbd{M-# e}.
1271 Calc will read the formula (using the surrounding blank lines to
1272 tell how much text to read), then push this formula (invisibly)
1273 onto the Calc stack. The cursor will stay on the formula in the
1274 editing buffer, but the buffer's mode line will change to look
1275 like the Calc mode line (with mode indicators like @samp{12 Deg}
1276 and so on). Even though you are still in your editing buffer,
1277 the keyboard now acts like the Calc keyboard, and any new result
1278 you get is copied from the stack back into the buffer. To take
1279 the derivative, you would type @kbd{a d x @key{RET}}.
1293 To make this look nicer, you might want to press @kbd{d =} to center
1294 the formula, and even @kbd{d B} to use Big display mode.
1303 % [calc-mode: justify: center]
1304 % [calc-mode: language: big]
1312 Calc has added annotations to the file to help it remember the modes
1313 that were used for this formula. They are formatted like comments
1314 in the @TeX{} typesetting language, just in case you are using @TeX{} or
1315 La@TeX{}. (In this example @TeX{} is not being used, so you might want
1316 to move these comments up to the top of the file or otherwise put them
1319 As an extra flourish, we can add an equation number using a
1320 righthand label: Type @kbd{d @} (1) @key{RET}}.
1324 % [calc-mode: justify: center]
1325 % [calc-mode: language: big]
1326 % [calc-mode: right-label: " (1)"]
1334 To leave Embedded mode, type @kbd{M-# e} again. The mode line
1335 and keyboard will revert to the way they were before. (If you have
1336 actually been trying this as you read along, you'll want to press
1337 @kbd{M-# 0} [with the digit zero] now to reset the modes you changed.)
1339 The related command @kbd{M-# w} operates on a single word, which
1340 generally means a single number, inside text. It uses any
1341 non-numeric characters rather than blank lines to delimit the
1342 formula it reads. Here's an example of its use:
1345 A slope of one-third corresponds to an angle of 1 degrees.
1348 Place the cursor on the @samp{1}, then type @kbd{M-# w} to enable
1349 Embedded mode on that number. Now type @kbd{3 /} (to get one-third),
1350 and @kbd{I T} (the Inverse Tangent converts a slope into an angle),
1351 then @w{@kbd{M-# w}} again to exit Embedded mode.
1354 A slope of one-third corresponds to an angle of 18.4349488229 degrees.
1357 @c [fix-ref Embedded Mode]
1358 @xref{Embedded Mode}, for full details.
1360 @node Other M-# Commands, , Embedded Mode Overview, Using Calc
1361 @subsection Other @kbd{M-#} Commands
1364 Two more Calc-related commands are @kbd{M-# g} and @kbd{M-# r},
1365 which ``grab'' data from a selected region of a buffer into the
1366 Calculator. The region is defined in the usual Emacs way, by
1367 a ``mark'' placed at one end of the region, and the Emacs
1368 cursor or ``point'' placed at the other.
1370 The @kbd{M-# g} command reads the region in the usual left-to-right,
1371 top-to-bottom order. The result is packaged into a Calc vector
1372 of numbers and placed on the stack. Calc (in its standard
1373 user interface) is then started. Type @kbd{v u} if you want
1374 to unpack this vector into separate numbers on the stack. Also,
1375 @kbd{C-u M-# g} interprets the region as a single number or
1378 The @kbd{M-# r} command reads a rectangle, with the point and
1379 mark defining opposite corners of the rectangle. The result
1380 is a matrix of numbers on the Calculator stack.
1382 Complementary to these is @kbd{M-# y}, which ``yanks'' the
1383 value at the top of the Calc stack back into an editing buffer.
1384 If you type @w{@kbd{M-# y}} while in such a buffer, the value is
1385 yanked at the current position. If you type @kbd{M-# y} while
1386 in the Calc buffer, Calc makes an educated guess as to which
1387 editing buffer you want to use. The Calc window does not have
1388 to be visible in order to use this command, as long as there
1389 is something on the Calc stack.
1391 Here, for reference, is the complete list of @kbd{M-#} commands.
1392 The shift, control, and meta keys are ignored for the keystroke
1393 following @kbd{M-#}.
1396 Commands for turning Calc on and off:
1400 Turn Calc on or off, employing the same user interface as last time.
1403 Turn Calc on or off using its standard bottom-of-the-screen
1404 interface. If Calc is already turned on but the cursor is not
1405 in the Calc window, move the cursor into the window.
1408 Same as @kbd{C}, but don't select the new Calc window. If
1409 Calc is already turned on and the cursor is in the Calc window,
1410 move it out of that window.
1413 Control whether @kbd{M-# c} and @kbd{M-# k} use the full screen.
1416 Use Quick mode for a single short calculation.
1419 Turn Calc Keypad mode on or off.
1422 Turn Calc Embedded mode on or off at the current formula.
1425 Turn Calc Embedded mode on or off, select the interesting part.
1428 Turn Calc Embedded mode on or off at the current word (number).
1431 Turn Calc on in a user-defined way, as defined by a @kbd{Z I} command.
1434 Quit Calc; turn off standard, Keypad, or Embedded mode if on.
1435 (This is like @kbd{q} or @key{OFF} inside of Calc.)
1442 Commands for moving data into and out of the Calculator:
1446 Grab the region into the Calculator as a vector.
1449 Grab the rectangular region into the Calculator as a matrix.
1452 Grab the rectangular region and compute the sums of its columns.
1455 Grab the rectangular region and compute the sums of its rows.
1458 Yank a value from the Calculator into the current editing buffer.
1465 Commands for use with Embedded mode:
1469 ``Activate'' the current buffer. Locate all formulas that
1470 contain @samp{:=} or @samp{=>} symbols and record their locations
1471 so that they can be updated automatically as variables are changed.
1474 Duplicate the current formula immediately below and select
1478 Insert a new formula at the current point.
1481 Move the cursor to the next active formula in the buffer.
1484 Move the cursor to the previous active formula in the buffer.
1487 Update (i.e., as if by the @kbd{=} key) the formula at the current point.
1490 Edit (as if by @code{calc-edit}) the formula at the current point.
1497 Miscellaneous commands:
1501 Run the Emacs Info system to read the Calc manual.
1502 (This is the same as @kbd{h i} inside of Calc.)
1505 Run the Emacs Info system to read the Calc Tutorial.
1508 Run the Emacs Info system to read the Calc Summary.
1511 Load Calc entirely into memory. (Normally the various parts
1512 are loaded only as they are needed.)
1515 Read a region of written keystroke names (like @kbd{C-n a b c @key{RET}})
1516 and record them as the current keyboard macro.
1519 (This is the ``zero'' digit key.) Reset the Calculator to
1520 its default state: Empty stack, and default mode settings.
1521 With any prefix argument, reset everything but the stack.
1524 @node History and Acknowledgements, , Using Calc, Getting Started
1525 @section History and Acknowledgements
1528 Calc was originally started as a two-week project to occupy a lull
1529 in the author's schedule. Basically, a friend asked if I remembered
1531 @texline @math{2^{32}}.
1532 @infoline @expr{2^32}.
1533 I didn't offhand, but I said, ``that's easy, just call up an
1534 @code{xcalc}.'' @code{Xcalc} duly reported that the answer to our
1535 question was @samp{4.294967e+09}---with no way to see the full ten
1536 digits even though we knew they were there in the program's memory! I
1537 was so annoyed, I vowed to write a calculator of my own, once and for
1540 I chose Emacs Lisp, a) because I had always been curious about it
1541 and b) because, being only a text editor extension language after
1542 all, Emacs Lisp would surely reach its limits long before the project
1543 got too far out of hand.
1545 To make a long story short, Emacs Lisp turned out to be a distressingly
1546 solid implementation of Lisp, and the humble task of calculating
1547 turned out to be more open-ended than one might have expected.
1549 Emacs Lisp doesn't have built-in floating point math, so it had to be
1550 simulated in software. In fact, Emacs integers will only comfortably
1551 fit six decimal digits or so---not enough for a decent calculator. So
1552 I had to write my own high-precision integer code as well, and once I had
1553 this I figured that arbitrary-size integers were just as easy as large
1554 integers. Arbitrary floating-point precision was the logical next step.
1555 Also, since the large integer arithmetic was there anyway it seemed only
1556 fair to give the user direct access to it, which in turn made it practical
1557 to support fractions as well as floats. All these features inspired me
1558 to look around for other data types that might be worth having.
1560 Around this time, my friend Rick Koshi showed me his nifty new HP-28
1561 calculator. It allowed the user to manipulate formulas as well as
1562 numerical quantities, and it could also operate on matrices. I
1563 decided that these would be good for Calc to have, too. And once
1564 things had gone this far, I figured I might as well take a look at
1565 serious algebra systems for further ideas. Since these systems did
1566 far more than I could ever hope to implement, I decided to focus on
1567 rewrite rules and other programming features so that users could
1568 implement what they needed for themselves.
1570 Rick complained that matrices were hard to read, so I put in code to
1571 format them in a 2D style. Once these routines were in place, Big mode
1572 was obligatory. Gee, what other language modes would be useful?
1574 Scott Hemphill and Allen Knutson, two friends with a strong mathematical
1575 bent, contributed ideas and algorithms for a number of Calc features
1576 including modulo forms, primality testing, and float-to-fraction conversion.
1578 Units were added at the eager insistence of Mass Sivilotti. Later,
1579 Ulrich Mueller at CERN and Przemek Klosowski at NIST provided invaluable
1580 expert assistance with the units table. As far as I can remember, the
1581 idea of using algebraic formulas and variables to represent units dates
1582 back to an ancient article in Byte magazine about muMath, an early
1583 algebra system for microcomputers.
1585 Many people have contributed to Calc by reporting bugs and suggesting
1586 features, large and small. A few deserve special mention: Tim Peters,
1587 who helped develop the ideas that led to the selection commands, rewrite
1588 rules, and many other algebra features;
1589 @texline Fran\c cois
1591 Pinard, who contributed an early prototype of the Calc Summary appendix
1592 as well as providing valuable suggestions in many other areas of Calc;
1593 Carl Witty, whose eagle eyes discovered many typographical and factual
1594 errors in the Calc manual; Tim Kay, who drove the development of
1595 Embedded mode; Ove Ewerlid, who made many suggestions relating to the
1596 algebra commands and contributed some code for polynomial operations;
1597 Randal Schwartz, who suggested the @code{calc-eval} function; Robert
1598 J. Chassell, who suggested the Calc Tutorial and exercises; and Juha
1599 Sarlin, who first worked out how to split Calc into quickly-loading
1600 parts. Bob Weiner helped immensely with the Lucid Emacs port.
1602 @cindex Bibliography
1603 @cindex Knuth, Art of Computer Programming
1604 @cindex Numerical Recipes
1605 @c Should these be expanded into more complete references?
1606 Among the books used in the development of Calc were Knuth's @emph{Art
1607 of Computer Programming} (especially volume II, @emph{Seminumerical
1608 Algorithms}); @emph{Numerical Recipes} by Press, Flannery, Teukolsky,
1609 and Vetterling; Bevington's @emph{Data Reduction and Error Analysis
1610 for the Physical Sciences}; @emph{Concrete Mathematics} by Graham,
1611 Knuth, and Patashnik; Steele's @emph{Common Lisp, the Language}; the
1612 @emph{CRC Standard Math Tables} (William H. Beyer, ed.); and
1613 Abramowitz and Stegun's venerable @emph{Handbook of Mathematical
1614 Functions}. Also, of course, Calc could not have been written without
1615 the excellent @emph{GNU Emacs Lisp Reference Manual}, by Bil Lewis and
1618 Final thanks go to Richard Stallman, without whose fine implementations
1619 of the Emacs editor, language, and environment, Calc would have been
1620 finished in two weeks.
1625 @c This node is accessed by the `M-# t' command.
1626 @node Interactive Tutorial, , , Top
1630 Some brief instructions on using the Emacs Info system for this tutorial:
1632 Press the space bar and Delete keys to go forward and backward in a
1633 section by screenfuls (or use the regular Emacs scrolling commands
1636 Press @kbd{n} or @kbd{p} to go to the Next or Previous section.
1637 If the section has a @dfn{menu}, press a digit key like @kbd{1}
1638 or @kbd{2} to go to a sub-section from the menu. Press @kbd{u} to
1639 go back up from a sub-section to the menu it is part of.
1641 Exercises in the tutorial all have cross-references to the
1642 appropriate page of the ``answers'' section. Press @kbd{f}, then
1643 the exercise number, to see the answer to an exercise. After
1644 you have followed a cross-reference, you can press the letter
1645 @kbd{l} to return to where you were before.
1647 You can press @kbd{?} at any time for a brief summary of Info commands.
1649 Press @kbd{1} now to enter the first section of the Tutorial.
1656 @node Tutorial, Introduction, Getting Started, Top
1660 This chapter explains how to use Calc and its many features, in
1661 a step-by-step, tutorial way. You are encouraged to run Calc and
1662 work along with the examples as you read (@pxref{Starting Calc}).
1663 If you are already familiar with advanced calculators, you may wish
1665 to skip on to the rest of this manual.
1667 @c to skip on to volume II of this manual, the @dfn{Calc Reference}.
1669 @c [fix-ref Embedded Mode]
1670 This tutorial describes the standard user interface of Calc only.
1671 The Quick mode and Keypad mode interfaces are fairly
1672 self-explanatory. @xref{Embedded Mode}, for a description of
1673 the Embedded mode interface.
1676 The easiest way to read this tutorial on-line is to have two windows on
1677 your Emacs screen, one with Calc and one with the Info system. (If you
1678 have a printed copy of the manual you can use that instead.) Press
1679 @kbd{M-# c} to turn Calc on or to switch into the Calc window, and
1680 press @kbd{M-# i} to start the Info system or to switch into its window.
1681 Or, you may prefer to use the tutorial in printed form.
1684 The easiest way to read this tutorial on-line is to have two windows on
1685 your Emacs screen, one with Calc and one with the Info system. (If you
1686 have a printed copy of the manual you can use that instead.) Press
1687 @kbd{M-# c} to turn Calc on or to switch into the Calc window, and
1688 press @kbd{M-# i} to start the Info system or to switch into its window.
1691 This tutorial is designed to be done in sequence. But the rest of this
1692 manual does not assume you have gone through the tutorial. The tutorial
1693 does not cover everything in the Calculator, but it touches on most
1697 You may wish to print out a copy of the Calc Summary and keep notes on
1698 it as you learn Calc. @xref{About This Manual}, to see how to make a
1699 printed summary. @xref{Summary}.
1702 The Calc Summary at the end of the reference manual includes some blank
1703 space for your own use. You may wish to keep notes there as you learn
1709 * Arithmetic Tutorial::
1710 * Vector/Matrix Tutorial::
1712 * Algebra Tutorial::
1713 * Programming Tutorial::
1715 * Answers to Exercises::
1718 @node Basic Tutorial, Arithmetic Tutorial, Tutorial, Tutorial
1719 @section Basic Tutorial
1722 In this section, we learn how RPN and algebraic-style calculations
1723 work, how to undo and redo an operation done by mistake, and how
1724 to control various modes of the Calculator.
1727 * RPN Tutorial:: Basic operations with the stack.
1728 * Algebraic Tutorial:: Algebraic entry; variables.
1729 * Undo Tutorial:: If you make a mistake: Undo and the trail.
1730 * Modes Tutorial:: Common mode-setting commands.
1733 @node RPN Tutorial, Algebraic Tutorial, Basic Tutorial, Basic Tutorial
1734 @subsection RPN Calculations and the Stack
1736 @cindex RPN notation
1739 Calc normally uses RPN notation. You may be familiar with the RPN
1740 system from Hewlett-Packard calculators, FORTH, or PostScript.
1741 (Reverse Polish Notation, RPN, is named after the Polish mathematician
1746 Calc normally uses RPN notation. You may be familiar with the RPN
1747 system from Hewlett-Packard calculators, FORTH, or PostScript.
1748 (Reverse Polish Notation, RPN, is named after the Polish mathematician
1752 The central component of an RPN calculator is the @dfn{stack}. A
1753 calculator stack is like a stack of dishes. New dishes (numbers) are
1754 added at the top of the stack, and numbers are normally only removed
1755 from the top of the stack.
1759 In an operation like @expr{2+3}, the 2 and 3 are called the @dfn{operands}
1760 and the @expr{+} is the @dfn{operator}. In an RPN calculator you always
1761 enter the operands first, then the operator. Each time you type a
1762 number, Calc adds or @dfn{pushes} it onto the top of the Stack.
1763 When you press an operator key like @kbd{+}, Calc @dfn{pops} the appropriate
1764 number of operands from the stack and pushes back the result.
1766 Thus we could add the numbers 2 and 3 in an RPN calculator by typing:
1767 @kbd{2 @key{RET} 3 @key{RET} +}. (The @key{RET} key, Return, corresponds to
1768 the @key{ENTER} key on traditional RPN calculators.) Try this now if
1769 you wish; type @kbd{M-# c} to switch into the Calc window (you can type
1770 @kbd{M-# c} again or @kbd{M-# o} to switch back to the Tutorial window).
1771 The first four keystrokes ``push'' the numbers 2 and 3 onto the stack.
1772 The @kbd{+} key ``pops'' the top two numbers from the stack, adds them,
1773 and pushes the result (5) back onto the stack. Here's how the stack
1774 will look at various points throughout the calculation:
1782 M-# c 2 @key{RET} 3 @key{RET} + @key{DEL}
1786 The @samp{.} symbol is a marker that represents the top of the stack.
1787 Note that the ``top'' of the stack is really shown at the bottom of
1788 the Stack window. This may seem backwards, but it turns out to be
1789 less distracting in regular use.
1791 @cindex Stack levels
1792 @cindex Levels of stack
1793 The numbers @samp{1:} and @samp{2:} on the left are @dfn{stack level
1794 numbers}. Old RPN calculators always had four stack levels called
1795 @expr{x}, @expr{y}, @expr{z}, and @expr{t}. Calc's stack can grow
1796 as large as you like, so it uses numbers instead of letters. Some
1797 stack-manipulation commands accept a numeric argument that says
1798 which stack level to work on. Normal commands like @kbd{+} always
1799 work on the top few levels of the stack.
1801 @c [fix-ref Truncating the Stack]
1802 The Stack buffer is just an Emacs buffer, and you can move around in
1803 it using the regular Emacs motion commands. But no matter where the
1804 cursor is, even if you have scrolled the @samp{.} marker out of
1805 view, most Calc commands always move the cursor back down to level 1
1806 before doing anything. It is possible to move the @samp{.} marker
1807 upwards through the stack, temporarily ``hiding'' some numbers from
1808 commands like @kbd{+}. This is called @dfn{stack truncation} and
1809 we will not cover it in this tutorial; @pxref{Truncating the Stack},
1810 if you are interested.
1812 You don't really need the second @key{RET} in @kbd{2 @key{RET} 3
1813 @key{RET} +}. That's because if you type any operator name or
1814 other non-numeric key when you are entering a number, the Calculator
1815 automatically enters that number and then does the requested command.
1816 Thus @kbd{2 @key{RET} 3 +} will work just as well.
1818 Examples in this tutorial will often omit @key{RET} even when the
1819 stack displays shown would only happen if you did press @key{RET}:
1832 Here, after pressing @kbd{3} the stack would really show @samp{1: 2}
1833 with @samp{Calc:@: 3} in the minibuffer. In these situations, you can
1834 press the optional @key{RET} to see the stack as the figure shows.
1836 (@bullet{}) @strong{Exercise 1.} (This tutorial will include exercises
1837 at various points. Try them if you wish. Answers to all the exercises
1838 are located at the end of the Tutorial chapter. Each exercise will
1839 include a cross-reference to its particular answer. If you are
1840 reading with the Emacs Info system, press @kbd{f} and the
1841 exercise number to go to the answer, then the letter @kbd{l} to
1842 return to where you were.)
1845 Here's the first exercise: What will the keystrokes @kbd{1 @key{RET} 2
1846 @key{RET} 3 @key{RET} 4 + * -} compute? (@samp{*} is the symbol for
1847 multiplication.) Figure it out by hand, then try it with Calc to see
1848 if you're right. @xref{RPN Answer 1, 1}. (@bullet{})
1850 (@bullet{}) @strong{Exercise 2.} Compute
1851 @texline @math{(2\times4) + (7\times9.4) + {5\over4}}
1852 @infoline @expr{2*4 + 7*9.5 + 5/4}
1853 using the stack. @xref{RPN Answer 2, 2}. (@bullet{})
1855 The @key{DEL} key is called Backspace on some keyboards. It is
1856 whatever key you would use to correct a simple typing error when
1857 regularly using Emacs. The @key{DEL} key pops and throws away the
1858 top value on the stack. (You can still get that value back from
1859 the Trail if you should need it later on.) There are many places
1860 in this tutorial where we assume you have used @key{DEL} to erase the
1861 results of the previous example at the beginning of a new example.
1862 In the few places where it is really important to use @key{DEL} to
1863 clear away old results, the text will remind you to do so.
1865 (It won't hurt to let things accumulate on the stack, except that
1866 whenever you give a display-mode-changing command Calc will have to
1867 spend a long time reformatting such a large stack.)
1869 Since the @kbd{-} key is also an operator (it subtracts the top two
1870 stack elements), how does one enter a negative number? Calc uses
1871 the @kbd{_} (underscore) key to act like the minus sign in a number.
1872 So, typing @kbd{-5 @key{RET}} won't work because the @kbd{-} key
1873 will try to do a subtraction, but @kbd{_5 @key{RET}} works just fine.
1875 You can also press @kbd{n}, which means ``change sign.'' It changes
1876 the number at the top of the stack (or the number being entered)
1877 from positive to negative or vice-versa: @kbd{5 n @key{RET}}.
1879 @cindex Duplicating a stack entry
1880 If you press @key{RET} when you're not entering a number, the effect
1881 is to duplicate the top number on the stack. Consider this calculation:
1885 1: 3 2: 3 1: 9 2: 9 1: 81
1889 3 @key{RET} @key{RET} * @key{RET} *
1894 (Of course, an easier way to do this would be @kbd{3 @key{RET} 4 ^},
1895 to raise 3 to the fourth power.)
1897 The space-bar key (denoted @key{SPC} here) performs the same function
1898 as @key{RET}; you could replace all three occurrences of @key{RET} in
1899 the above example with @key{SPC} and the effect would be the same.
1901 @cindex Exchanging stack entries
1902 Another stack manipulation key is @key{TAB}. This exchanges the top
1903 two stack entries. Suppose you have computed @kbd{2 @key{RET} 3 +}
1904 to get 5, and then you realize what you really wanted to compute
1905 was @expr{20 / (2+3)}.
1909 1: 5 2: 5 2: 20 1: 4
1913 2 @key{RET} 3 + 20 @key{TAB} /
1918 Planning ahead, the calculation would have gone like this:
1922 1: 20 2: 20 3: 20 2: 20 1: 4
1927 20 @key{RET} 2 @key{RET} 3 + /
1931 A related stack command is @kbd{M-@key{TAB}} (hold @key{META} and type
1932 @key{TAB}). It rotates the top three elements of the stack upward,
1933 bringing the object in level 3 to the top.
1937 1: 10 2: 10 3: 10 3: 20 3: 30
1938 . 1: 20 2: 20 2: 30 2: 10
1942 10 @key{RET} 20 @key{RET} 30 @key{RET} M-@key{TAB} M-@key{TAB}
1946 (@bullet{}) @strong{Exercise 3.} Suppose the numbers 10, 20, and 30 are
1947 on the stack. Figure out how to add one to the number in level 2
1948 without affecting the rest of the stack. Also figure out how to add
1949 one to the number in level 3. @xref{RPN Answer 3, 3}. (@bullet{})
1951 Operations like @kbd{+}, @kbd{-}, @kbd{*}, @kbd{/}, and @kbd{^} pop two
1952 arguments from the stack and push a result. Operations like @kbd{n} and
1953 @kbd{Q} (square root) pop a single number and push the result. You can
1954 think of them as simply operating on the top element of the stack.
1958 1: 3 1: 9 2: 9 1: 25 1: 5
1962 3 @key{RET} @key{RET} * 4 @key{RET} @key{RET} * + Q
1967 (Note that capital @kbd{Q} means to hold down the Shift key while
1968 typing @kbd{q}. Remember, plain unshifted @kbd{q} is the Quit command.)
1970 @cindex Pythagorean Theorem
1971 Here we've used the Pythagorean Theorem to determine the hypotenuse of a
1972 right triangle. Calc actually has a built-in command for that called
1973 @kbd{f h}, but let's suppose we can't remember the necessary keystrokes.
1974 We can still enter it by its full name using @kbd{M-x} notation:
1982 3 @key{RET} 4 @key{RET} M-x calc-hypot
1986 All Calculator commands begin with the word @samp{calc-}. Since it
1987 gets tiring to type this, Calc provides an @kbd{x} key which is just
1988 like the regular Emacs @kbd{M-x} key except that it types the @samp{calc-}
1997 3 @key{RET} 4 @key{RET} x hypot
2001 What happens if you take the square root of a negative number?
2005 1: 4 1: -4 1: (0, 2)
2013 The notation @expr{(a, b)} represents a complex number.
2014 Complex numbers are more traditionally written @expr{a + b i};
2015 Calc can display in this format, too, but for now we'll stick to the
2016 @expr{(a, b)} notation.
2018 If you don't know how complex numbers work, you can safely ignore this
2019 feature. Complex numbers only arise from operations that would be
2020 errors in a calculator that didn't have complex numbers. (For example,
2021 taking the square root or logarithm of a negative number produces a
2024 Complex numbers are entered in the notation shown. The @kbd{(} and
2025 @kbd{,} and @kbd{)} keys manipulate ``incomplete complex numbers.''
2029 1: ( ... 2: ( ... 1: (2, ... 1: (2, ... 1: (2, 3)
2037 You can perform calculations while entering parts of incomplete objects.
2038 However, an incomplete object cannot actually participate in a calculation:
2042 1: ( ... 2: ( ... 3: ( ... 1: ( ... 1: ( ...
2052 Adding 5 to an incomplete object makes no sense, so the last command
2053 produces an error message and leaves the stack the same.
2055 Incomplete objects can't participate in arithmetic, but they can be
2056 moved around by the regular stack commands.
2060 2: 2 3: 2 3: 3 1: ( ... 1: (2, 3)
2061 1: 3 2: 3 2: ( ... 2 .
2065 2 @key{RET} 3 @key{RET} ( M-@key{TAB} M-@key{TAB} )
2070 Note that the @kbd{,} (comma) key did not have to be used here.
2071 When you press @kbd{)} all the stack entries between the incomplete
2072 entry and the top are collected, so there's never really a reason
2073 to use the comma. It's up to you.
2075 (@bullet{}) @strong{Exercise 4.} To enter the complex number @expr{(2, 3)},
2076 your friend Joe typed @kbd{( 2 , @key{SPC} 3 )}. What happened?
2077 (Joe thought of a clever way to correct his mistake in only two
2078 keystrokes, but it didn't quite work. Try it to find out why.)
2079 @xref{RPN Answer 4, 4}. (@bullet{})
2081 Vectors are entered the same way as complex numbers, but with square
2082 brackets in place of parentheses. We'll meet vectors again later in
2085 Any Emacs command can be given a @dfn{numeric prefix argument} by
2086 typing a series of @key{META}-digits beforehand. If @key{META} is
2087 awkward for you, you can instead type @kbd{C-u} followed by the
2088 necessary digits. Numeric prefix arguments can be negative, as in
2089 @kbd{M-- M-3 M-5} or @w{@kbd{C-u - 3 5}}. Calc commands use numeric
2090 prefix arguments in a variety of ways. For example, a numeric prefix
2091 on the @kbd{+} operator adds any number of stack entries at once:
2095 1: 10 2: 10 3: 10 3: 10 1: 60
2096 . 1: 20 2: 20 2: 20 .
2100 10 @key{RET} 20 @key{RET} 30 @key{RET} C-u 3 +
2104 For stack manipulation commands like @key{RET}, a positive numeric
2105 prefix argument operates on the top @var{n} stack entries at once. A
2106 negative argument operates on the entry in level @var{n} only. An
2107 argument of zero operates on the entire stack. In this example, we copy
2108 the second-to-top element of the stack:
2112 1: 10 2: 10 3: 10 3: 10 4: 10
2113 . 1: 20 2: 20 2: 20 3: 20
2118 10 @key{RET} 20 @key{RET} 30 @key{RET} C-u -2 @key{RET}
2122 @cindex Clearing the stack
2123 @cindex Emptying the stack
2124 Another common idiom is @kbd{M-0 @key{DEL}}, which clears the stack.
2125 (The @kbd{M-0} numeric prefix tells @key{DEL} to operate on the
2128 @node Algebraic Tutorial, Undo Tutorial, RPN Tutorial, Basic Tutorial
2129 @subsection Algebraic-Style Calculations
2132 If you are not used to RPN notation, you may prefer to operate the
2133 Calculator in Algebraic mode, which is closer to the way
2134 non-RPN calculators work. In Algebraic mode, you enter formulas
2135 in traditional @expr{2+3} notation.
2137 You don't really need any special ``mode'' to enter algebraic formulas.
2138 You can enter a formula at any time by pressing the apostrophe (@kbd{'})
2139 key. Answer the prompt with the desired formula, then press @key{RET}.
2140 The formula is evaluated and the result is pushed onto the RPN stack.
2141 If you don't want to think in RPN at all, you can enter your whole
2142 computation as a formula, read the result from the stack, then press
2143 @key{DEL} to delete it from the stack.
2145 Try pressing the apostrophe key, then @kbd{2+3+4}, then @key{RET}.
2146 The result should be the number 9.
2148 Algebraic formulas use the operators @samp{+}, @samp{-}, @samp{*},
2149 @samp{/}, and @samp{^}. You can use parentheses to make the order
2150 of evaluation clear. In the absence of parentheses, @samp{^} is
2151 evaluated first, then @samp{*}, then @samp{/}, then finally
2152 @samp{+} and @samp{-}. For example, the expression
2155 2 + 3*4*5 / 6*7^8 - 9
2162 2 + ((3*4*5) / (6*(7^8)) - 9
2166 or, in large mathematical notation,
2181 $$ 2 + { 3 \times 4 \times 5 \over 6 \times 7^8 } - 9 $$
2186 The result of this expression will be the number @mathit{-6.99999826533}.
2188 Calc's order of evaluation is the same as for most computer languages,
2189 except that @samp{*} binds more strongly than @samp{/}, as the above
2190 example shows. As in normal mathematical notation, the @samp{*} symbol
2191 can often be omitted: @samp{2 a} is the same as @samp{2*a}.
2193 Operators at the same level are evaluated from left to right, except
2194 that @samp{^} is evaluated from right to left. Thus, @samp{2-3-4} is
2195 equivalent to @samp{(2-3)-4} or @mathit{-5}, whereas @samp{2^3^4} is equivalent
2196 to @samp{2^(3^4)} (a very large integer; try it!).
2198 If you tire of typing the apostrophe all the time, there is
2199 Algebraic mode, where Calc automatically senses
2200 when you are about to type an algebraic expression. To enter this
2201 mode, press the two letters @w{@kbd{m a}}. (An @samp{Alg} indicator
2202 should appear in the Calc window's mode line.)
2204 Press @kbd{m a}, then @kbd{2+3+4} with no apostrophe, then @key{RET}.
2206 In Algebraic mode, when you press any key that would normally begin
2207 entering a number (such as a digit, a decimal point, or the @kbd{_}
2208 key), or if you press @kbd{(} or @kbd{[}, Calc automatically begins
2211 Functions which do not have operator symbols like @samp{+} and @samp{*}
2212 must be entered in formulas using function-call notation. For example,
2213 the function name corresponding to the square-root key @kbd{Q} is
2214 @code{sqrt}. To compute a square root in a formula, you would use
2215 the notation @samp{sqrt(@var{x})}.
2217 Press the apostrophe, then type @kbd{sqrt(5*2) - 3}. The result should
2218 be @expr{0.16227766017}.
2220 Note that if the formula begins with a function name, you need to use
2221 the apostrophe even if you are in Algebraic mode. If you type @kbd{arcsin}
2222 out of the blue, the @kbd{a r} will be taken as an Algebraic Rewrite
2223 command, and the @kbd{csin} will be taken as the name of the rewrite
2226 Some people prefer to enter complex numbers and vectors in algebraic
2227 form because they find RPN entry with incomplete objects to be too
2228 distracting, even though they otherwise use Calc as an RPN calculator.
2230 Still in Algebraic mode, type:
2234 1: (2, 3) 2: (2, 3) 1: (8, -1) 2: (8, -1) 1: (9, -1)
2235 . 1: (1, -2) . 1: 1 .
2238 (2,3) @key{RET} (1,-2) @key{RET} * 1 @key{RET} +
2242 Algebraic mode allows us to enter complex numbers without pressing
2243 an apostrophe first, but it also means we need to press @key{RET}
2244 after every entry, even for a simple number like @expr{1}.
2246 (You can type @kbd{C-u m a} to enable a special Incomplete Algebraic
2247 mode in which the @kbd{(} and @kbd{[} keys use algebraic entry even
2248 though regular numeric keys still use RPN numeric entry. There is also
2249 Total Algebraic mode, started by typing @kbd{m t}, in which all
2250 normal keys begin algebraic entry. You must then use the @key{META} key
2251 to type Calc commands: @kbd{M-m t} to get back out of Total Algebraic
2252 mode, @kbd{M-q} to quit, etc.)
2254 If you're still in Algebraic mode, press @kbd{m a} again to turn it off.
2256 Actual non-RPN calculators use a mixture of algebraic and RPN styles.
2257 In general, operators of two numbers (like @kbd{+} and @kbd{*})
2258 use algebraic form, but operators of one number (like @kbd{n} and @kbd{Q})
2259 use RPN form. Also, a non-RPN calculator allows you to see the
2260 intermediate results of a calculation as you go along. You can
2261 accomplish this in Calc by performing your calculation as a series
2262 of algebraic entries, using the @kbd{$} sign to tie them together.
2263 In an algebraic formula, @kbd{$} represents the number on the top
2264 of the stack. Here, we perform the calculation
2265 @texline @math{\sqrt{2\times4+1}},
2266 @infoline @expr{sqrt(2*4+1)},
2267 which on a traditional calculator would be done by pressing
2268 @kbd{2 * 4 + 1 =} and then the square-root key.
2275 ' 2*4 @key{RET} $+1 @key{RET} Q
2280 Notice that we didn't need to press an apostrophe for the @kbd{$+1},
2281 because the dollar sign always begins an algebraic entry.
2283 (@bullet{}) @strong{Exercise 1.} How could you get the same effect as
2284 pressing @kbd{Q} but using an algebraic entry instead? How about
2285 if the @kbd{Q} key on your keyboard were broken?
2286 @xref{Algebraic Answer 1, 1}. (@bullet{})
2288 The notations @kbd{$$}, @kbd{$$$}, and so on stand for higher stack
2289 entries. For example, @kbd{' $$+$ @key{RET}} is just like typing @kbd{+}.
2291 Algebraic formulas can include @dfn{variables}. To store in a
2292 variable, press @kbd{s s}, then type the variable name, then press
2293 @key{RET}. (There are actually two flavors of store command:
2294 @kbd{s s} stores a number in a variable but also leaves the number
2295 on the stack, while @w{@kbd{s t}} removes a number from the stack and
2296 stores it in the variable.) A variable name should consist of one
2297 or more letters or digits, beginning with a letter.
2301 1: 17 . 1: a + a^2 1: 306
2304 17 s t a @key{RET} ' a+a^2 @key{RET} =
2309 The @kbd{=} key @dfn{evaluates} a formula by replacing all its
2310 variables by the values that were stored in them.
2312 For RPN calculations, you can recall a variable's value on the
2313 stack either by entering its name as a formula and pressing @kbd{=},
2314 or by using the @kbd{s r} command.
2318 1: 17 2: 17 3: 17 2: 17 1: 306
2319 . 1: 17 2: 17 1: 289 .
2323 s r a @key{RET} ' a @key{RET} = 2 ^ +
2327 If you press a single digit for a variable name (as in @kbd{s t 3}, you
2328 get one of ten @dfn{quick variables} @code{q0} through @code{q9}.
2329 They are ``quick'' simply because you don't have to type the letter
2330 @code{q} or the @key{RET} after their names. In fact, you can type
2331 simply @kbd{s 3} as a shorthand for @kbd{s s 3}, and likewise for
2332 @kbd{t 3} and @w{@kbd{r 3}}.
2334 Any variables in an algebraic formula for which you have not stored
2335 values are left alone, even when you evaluate the formula.
2339 1: 2 a + 2 b 1: 34 + 2 b
2346 Calls to function names which are undefined in Calc are also left
2347 alone, as are calls for which the value is undefined.
2351 1: 2 + log10(0) + log10(x) + log10(5, 6) + foo(3)
2354 ' log10(100) + log10(0) + log10(x) + log10(5,6) + foo(3) @key{RET}
2359 In this example, the first call to @code{log10} works, but the other
2360 calls are not evaluated. In the second call, the logarithm is
2361 undefined for that value of the argument; in the third, the argument
2362 is symbolic, and in the fourth, there are too many arguments. In the
2363 fifth case, there is no function called @code{foo}. You will see a
2364 ``Wrong number of arguments'' message referring to @samp{log10(5,6)}.
2365 Press the @kbd{w} (``why'') key to see any other messages that may
2366 have arisen from the last calculation. In this case you will get
2367 ``logarithm of zero,'' then ``number expected: @code{x}''. Calc
2368 automatically displays the first message only if the message is
2369 sufficiently important; for example, Calc considers ``wrong number
2370 of arguments'' and ``logarithm of zero'' to be important enough to
2371 report automatically, while a message like ``number expected: @code{x}''
2372 will only show up if you explicitly press the @kbd{w} key.
2374 (@bullet{}) @strong{Exercise 2.} Joe entered the formula @samp{2 x y},
2375 stored 5 in @code{x}, pressed @kbd{=}, and got the expected result,
2376 @samp{10 y}. He then tried the same for the formula @samp{2 x (1+y)},
2377 expecting @samp{10 (1+y)}, but it didn't work. Why not?
2378 @xref{Algebraic Answer 2, 2}. (@bullet{})
2380 (@bullet{}) @strong{Exercise 3.} What result would you expect
2381 @kbd{1 @key{RET} 0 /} to give? What if you then type @kbd{0 *}?
2382 @xref{Algebraic Answer 3, 3}. (@bullet{})
2384 One interesting way to work with variables is to use the
2385 @dfn{evaluates-to} (@samp{=>}) operator. It works like this:
2386 Enter a formula algebraically in the usual way, but follow
2387 the formula with an @samp{=>} symbol. (There is also an @kbd{s =}
2388 command which builds an @samp{=>} formula using the stack.) On
2389 the stack, you will see two copies of the formula with an @samp{=>}
2390 between them. The lefthand formula is exactly like you typed it;
2391 the righthand formula has been evaluated as if by typing @kbd{=}.
2395 2: 2 + 3 => 5 2: 2 + 3 => 5
2396 1: 2 a + 2 b => 34 + 2 b 1: 2 a + 2 b => 20 + 2 b
2399 ' 2+3 => @key{RET} ' 2a+2b @key{RET} s = 10 s t a @key{RET}
2404 Notice that the instant we stored a new value in @code{a}, all
2405 @samp{=>} operators already on the stack that referred to @expr{a}
2406 were updated to use the new value. With @samp{=>}, you can push a
2407 set of formulas on the stack, then change the variables experimentally
2408 to see the effects on the formulas' values.
2410 You can also ``unstore'' a variable when you are through with it:
2415 1: 2 a + 2 b => 2 a + 2 b
2422 We will encounter formulas involving variables and functions again
2423 when we discuss the algebra and calculus features of the Calculator.
2425 @node Undo Tutorial, Modes Tutorial, Algebraic Tutorial, Basic Tutorial
2426 @subsection Undo and Redo
2429 If you make a mistake, you can usually correct it by pressing shift-@kbd{U},
2430 the ``undo'' command. First, clear the stack (@kbd{M-0 @key{DEL}}) and exit
2431 and restart Calc (@kbd{M-# M-# M-# M-#}) to make sure things start off
2432 with a clean slate. Now:
2436 1: 2 2: 2 1: 8 2: 2 1: 6
2444 You can undo any number of times. Calc keeps a complete record of
2445 all you have done since you last opened the Calc window. After the
2446 above example, you could type:
2458 You can also type @kbd{D} to ``redo'' a command that you have undone
2463 . 1: 2 2: 2 1: 6 1: 6
2472 It was not possible to redo past the @expr{6}, since that was placed there
2473 by something other than an undo command.
2476 You can think of undo and redo as a sort of ``time machine.'' Press
2477 @kbd{U} to go backward in time, @kbd{D} to go forward. If you go
2478 backward and do something (like @kbd{*}) then, as any science fiction
2479 reader knows, you have changed your future and you cannot go forward
2480 again. Thus, the inability to redo past the @expr{6} even though there
2481 was an earlier undo command.
2483 You can always recall an earlier result using the Trail. We've ignored
2484 the trail so far, but it has been faithfully recording everything we
2485 did since we loaded the Calculator. If the Trail is not displayed,
2486 press @kbd{t d} now to turn it on.
2488 Let's try grabbing an earlier result. The @expr{8} we computed was
2489 undone by a @kbd{U} command, and was lost even to Redo when we pressed
2490 @kbd{*}, but it's still there in the trail. There should be a little
2491 @samp{>} arrow (the @dfn{trail pointer}) resting on the last trail
2492 entry. If there isn't, press @kbd{t ]} to reset the trail pointer.
2493 Now, press @w{@kbd{t p}} to move the arrow onto the line containing
2494 @expr{8}, and press @w{@kbd{t y}} to ``yank'' that number back onto the
2497 If you press @kbd{t ]} again, you will see that even our Yank command
2498 went into the trail.
2500 Let's go further back in time. Earlier in the tutorial we computed
2501 a huge integer using the formula @samp{2^3^4}. We don't remember
2502 what it was, but the first digits were ``241''. Press @kbd{t r}
2503 (which stands for trail-search-reverse), then type @kbd{241}.
2504 The trail cursor will jump back to the next previous occurrence of
2505 the string ``241'' in the trail. This is just a regular Emacs
2506 incremental search; you can now press @kbd{C-s} or @kbd{C-r} to
2507 continue the search forwards or backwards as you like.
2509 To finish the search, press @key{RET}. This halts the incremental
2510 search and leaves the trail pointer at the thing we found. Now we
2511 can type @kbd{t y} to yank that number onto the stack. If we hadn't
2512 remembered the ``241'', we could simply have searched for @kbd{2^3^4},
2513 then pressed @kbd{@key{RET} t n} to halt and then move to the next item.
2515 You may have noticed that all the trail-related commands begin with
2516 the letter @kbd{t}. (The store-and-recall commands, on the other hand,
2517 all began with @kbd{s}.) Calc has so many commands that there aren't
2518 enough keys for all of them, so various commands are grouped into
2519 two-letter sequences where the first letter is called the @dfn{prefix}
2520 key. If you type a prefix key by accident, you can press @kbd{C-g}
2521 to cancel it. (In fact, you can press @kbd{C-g} to cancel almost
2522 anything in Emacs.) To get help on a prefix key, press that key
2523 followed by @kbd{?}. Some prefixes have several lines of help,
2524 so you need to press @kbd{?} repeatedly to see them all.
2525 You can also type @kbd{h h} to see all the help at once.
2527 Try pressing @kbd{t ?} now. You will see a line of the form,
2530 trail/time: Display; Fwd, Back; Next, Prev, Here, [, ]; Yank: [MORE] t-
2534 The word ``trail'' indicates that the @kbd{t} prefix key contains
2535 trail-related commands. Each entry on the line shows one command,
2536 with a single capital letter showing which letter you press to get
2537 that command. We have used @kbd{t n}, @kbd{t p}, @kbd{t ]}, and
2538 @kbd{t y} so far. The @samp{[MORE]} means you can press @kbd{?}
2539 again to see more @kbd{t}-prefix commands. Notice that the commands
2540 are roughly divided (by semicolons) into related groups.
2542 When you are in the help display for a prefix key, the prefix is
2543 still active. If you press another key, like @kbd{y} for example,
2544 it will be interpreted as a @kbd{t y} command. If all you wanted
2545 was to look at the help messages, press @kbd{C-g} afterwards to cancel
2548 One more way to correct an error is by editing the stack entries.
2549 The actual Stack buffer is marked read-only and must not be edited
2550 directly, but you can press @kbd{`} (the backquote or accent grave)
2551 to edit a stack entry.
2553 Try entering @samp{3.141439} now. If this is supposed to represent
2554 @cpi{}, it's got several errors. Press @kbd{`} to edit this number.
2555 Now use the normal Emacs cursor motion and editing keys to change
2556 the second 4 to a 5, and to transpose the 3 and the 9. When you
2557 press @key{RET}, the number on the stack will be replaced by your
2558 new number. This works for formulas, vectors, and all other types
2559 of values you can put on the stack. The @kbd{`} key also works
2560 during entry of a number or algebraic formula.
2562 @node Modes Tutorial, , Undo Tutorial, Basic Tutorial
2563 @subsection Mode-Setting Commands
2566 Calc has many types of @dfn{modes} that affect the way it interprets
2567 your commands or the way it displays data. We have already seen one
2568 mode, namely Algebraic mode. There are many others, too; we'll
2569 try some of the most common ones here.
2571 Perhaps the most fundamental mode in Calc is the current @dfn{precision}.
2572 Notice the @samp{12} on the Calc window's mode line:
2575 --%%-Calc: 12 Deg (Calculator)----All------
2579 Most of the symbols there are Emacs things you don't need to worry
2580 about, but the @samp{12} and the @samp{Deg} are mode indicators.
2581 The @samp{12} means that calculations should always be carried to
2582 12 significant figures. That is why, when we type @kbd{1 @key{RET} 7 /},
2583 we get @expr{0.142857142857} with exactly 12 digits, not counting
2584 leading and trailing zeros.
2586 You can set the precision to anything you like by pressing @kbd{p},
2587 then entering a suitable number. Try pressing @kbd{p 30 @key{RET}},
2588 then doing @kbd{1 @key{RET} 7 /} again:
2593 2: 0.142857142857142857142857142857
2598 Although the precision can be set arbitrarily high, Calc always
2599 has to have @emph{some} value for the current precision. After
2600 all, the true value @expr{1/7} is an infinitely repeating decimal;
2601 Calc has to stop somewhere.
2603 Of course, calculations are slower the more digits you request.
2604 Press @w{@kbd{p 12}} now to set the precision back down to the default.
2606 Calculations always use the current precision. For example, even
2607 though we have a 30-digit value for @expr{1/7} on the stack, if
2608 we use it in a calculation in 12-digit mode it will be rounded
2609 down to 12 digits before it is used. Try it; press @key{RET} to
2610 duplicate the number, then @w{@kbd{1 +}}. Notice that the @key{RET}
2611 key didn't round the number, because it doesn't do any calculation.
2612 But the instant we pressed @kbd{+}, the number was rounded down.
2617 2: 0.142857142857142857142857142857
2624 In fact, since we added a digit on the left, we had to lose one
2625 digit on the right from even the 12-digit value of @expr{1/7}.
2627 How did we get more than 12 digits when we computed @samp{2^3^4}? The
2628 answer is that Calc makes a distinction between @dfn{integers} and
2629 @dfn{floating-point} numbers, or @dfn{floats}. An integer is a number
2630 that does not contain a decimal point. There is no such thing as an
2631 ``infinitely repeating fraction integer,'' so Calc doesn't have to limit
2632 itself. If you asked for @samp{2^10000} (don't try this!), you would
2633 have to wait a long time but you would eventually get an exact answer.
2634 If you ask for @samp{2.^10000}, you will quickly get an answer which is
2635 correct only to 12 places. The decimal point tells Calc that it should
2636 use floating-point arithmetic to get the answer, not exact integer
2639 You can use the @kbd{F} (@code{calc-floor}) command to convert a
2640 floating-point value to an integer, and @kbd{c f} (@code{calc-float})
2641 to convert an integer to floating-point form.
2643 Let's try entering that last calculation:
2647 1: 2. 2: 2. 1: 1.99506311689e3010
2651 2.0 @key{RET} 10000 @key{RET} ^
2656 @cindex Scientific notation, entry of
2657 Notice the letter @samp{e} in there. It represents ``times ten to the
2658 power of,'' and is used by Calc automatically whenever writing the
2659 number out fully would introduce more extra zeros than you probably
2660 want to see. You can enter numbers in this notation, too.
2664 1: 2. 2: 2. 1: 1.99506311678e3010
2668 2.0 @key{RET} 1e4 @key{RET} ^
2672 @cindex Round-off errors
2674 Hey, the answer is different! Look closely at the middle columns
2675 of the two examples. In the first, the stack contained the
2676 exact integer @expr{10000}, but in the second it contained
2677 a floating-point value with a decimal point. When you raise a
2678 number to an integer power, Calc uses repeated squaring and
2679 multiplication to get the answer. When you use a floating-point
2680 power, Calc uses logarithms and exponentials. As you can see,
2681 a slight error crept in during one of these methods. Which
2682 one should we trust? Let's raise the precision a bit and find
2687 . 1: 2. 2: 2. 1: 1.995063116880828e3010
2691 p 16 @key{RET} 2. @key{RET} 1e4 ^ p 12 @key{RET}
2696 @cindex Guard digits
2697 Presumably, it doesn't matter whether we do this higher-precision
2698 calculation using an integer or floating-point power, since we
2699 have added enough ``guard digits'' to trust the first 12 digits
2700 no matter what. And the verdict is@dots{} Integer powers were more
2701 accurate; in fact, the result was only off by one unit in the
2704 @cindex Guard digits
2705 Calc does many of its internal calculations to a slightly higher
2706 precision, but it doesn't always bump the precision up enough.
2707 In each case, Calc added about two digits of precision during
2708 its calculation and then rounded back down to 12 digits
2709 afterward. In one case, it was enough; in the other, it
2710 wasn't. If you really need @var{x} digits of precision, it
2711 never hurts to do the calculation with a few extra guard digits.
2713 What if we want guard digits but don't want to look at them?
2714 We can set the @dfn{float format}. Calc supports four major
2715 formats for floating-point numbers, called @dfn{normal},
2716 @dfn{fixed-point}, @dfn{scientific notation}, and @dfn{engineering
2717 notation}. You get them by pressing @w{@kbd{d n}}, @kbd{d f},
2718 @kbd{d s}, and @kbd{d e}, respectively. In each case, you can
2719 supply a numeric prefix argument which says how many digits
2720 should be displayed. As an example, let's put a few numbers
2721 onto the stack and try some different display modes. First,
2722 use @kbd{M-0 @key{DEL}} to clear the stack, then enter the four
2727 4: 12345 4: 12345 4: 12345 4: 12345 4: 12345
2728 3: 12345. 3: 12300. 3: 1.2345e4 3: 1.23e4 3: 12345.000
2729 2: 123.45 2: 123. 2: 1.2345e2 2: 1.23e2 2: 123.450
2730 1: 12.345 1: 12.3 1: 1.2345e1 1: 1.23e1 1: 12.345
2733 d n M-3 d n d s M-3 d s M-3 d f
2738 Notice that when we typed @kbd{M-3 d n}, the numbers were rounded down
2739 to three significant digits, but then when we typed @kbd{d s} all
2740 five significant figures reappeared. The float format does not
2741 affect how numbers are stored, it only affects how they are
2742 displayed. Only the current precision governs the actual rounding
2743 of numbers in the Calculator's memory.
2745 Engineering notation, not shown here, is like scientific notation
2746 except the exponent (the power-of-ten part) is always adjusted to be
2747 a multiple of three (as in ``kilo,'' ``micro,'' etc.). As a result
2748 there will be one, two, or three digits before the decimal point.
2750 Whenever you change a display-related mode, Calc redraws everything
2751 in the stack. This may be slow if there are many things on the stack,
2752 so Calc allows you to type shift-@kbd{H} before any mode command to
2753 prevent it from updating the stack. Anything Calc displays after the
2754 mode-changing command will appear in the new format.
2758 4: 12345 4: 12345 4: 12345 4: 12345 4: 12345
2759 3: 12345.000 3: 12345.000 3: 12345.000 3: 1.2345e4 3: 12345.
2760 2: 123.450 2: 123.450 2: 1.2345e1 2: 1.2345e1 2: 123.45
2761 1: 12.345 1: 1.2345e1 1: 1.2345e2 1: 1.2345e2 1: 12.345
2764 H d s @key{DEL} U @key{TAB} d @key{SPC} d n
2769 Here the @kbd{H d s} command changes to scientific notation but without
2770 updating the screen. Deleting the top stack entry and undoing it back
2771 causes it to show up in the new format; swapping the top two stack
2772 entries reformats both entries. The @kbd{d @key{SPC}} command refreshes the
2773 whole stack. The @kbd{d n} command changes back to the normal float
2774 format; since it doesn't have an @kbd{H} prefix, it also updates all
2775 the stack entries to be in @kbd{d n} format.
2777 Notice that the integer @expr{12345} was not affected by any
2778 of the float formats. Integers are integers, and are always
2781 @cindex Large numbers, readability
2782 Large integers have their own problems. Let's look back at
2783 the result of @kbd{2^3^4}.
2786 2417851639229258349412352
2790 Quick---how many digits does this have? Try typing @kbd{d g}:
2793 2,417,851,639,229,258,349,412,352
2797 Now how many digits does this have? It's much easier to tell!
2798 We can actually group digits into clumps of any size. Some
2799 people prefer @kbd{M-5 d g}:
2802 24178,51639,22925,83494,12352
2805 Let's see what happens to floating-point numbers when they are grouped.
2806 First, type @kbd{p 25 @key{RET}} to make sure we have enough precision
2807 to get ourselves into trouble. Now, type @kbd{1e13 /}:
2810 24,17851,63922.9258349412352
2814 The integer part is grouped but the fractional part isn't. Now try
2815 @kbd{M-- M-5 d g} (that's meta-minus-sign, meta-five):
2818 24,17851,63922.92583,49412,352
2821 If you find it hard to tell the decimal point from the commas, try
2822 changing the grouping character to a space with @kbd{d , @key{SPC}}:
2825 24 17851 63922.92583 49412 352
2828 Type @kbd{d , ,} to restore the normal grouping character, then
2829 @kbd{d g} again to turn grouping off. Also, press @kbd{p 12} to
2830 restore the default precision.
2832 Press @kbd{U} enough times to get the original big integer back.
2833 (Notice that @kbd{U} does not undo each mode-setting command; if
2834 you want to undo a mode-setting command, you have to do it yourself.)
2835 Now, type @kbd{d r 16 @key{RET}}:
2838 16#200000000000000000000
2842 The number is now displayed in @dfn{hexadecimal}, or ``base-16'' form.
2843 Suddenly it looks pretty simple; this should be no surprise, since we
2844 got this number by computing a power of two, and 16 is a power of 2.
2845 In fact, we can use @w{@kbd{d r 2 @key{RET}}} to see it in actual binary
2849 2#1000000000000000000000000000000000000000000000000000000 @dots{}
2853 We don't have enough space here to show all the zeros! They won't
2854 fit on a typical screen, either, so you will have to use horizontal
2855 scrolling to see them all. Press @kbd{<} and @kbd{>} to scroll the
2856 stack window left and right by half its width. Another way to view
2857 something large is to press @kbd{`} (back-quote) to edit the top of
2858 stack in a separate window. (Press @kbd{C-c C-c} when you are done.)
2860 You can enter non-decimal numbers using the @kbd{#} symbol, too.
2861 Let's see what the hexadecimal number @samp{5FE} looks like in
2862 binary. Type @kbd{16#5FE} (the letters can be typed in upper or
2863 lower case; they will always appear in upper case). It will also
2864 help to turn grouping on with @kbd{d g}:
2870 Notice that @kbd{d g} groups by fours by default if the display radix
2871 is binary or hexadecimal, but by threes if it is decimal, octal, or any
2874 Now let's see that number in decimal; type @kbd{d r 10}:
2880 Numbers are not @emph{stored} with any particular radix attached. They're
2881 just numbers; they can be entered in any radix, and are always displayed
2882 in whatever radix you've chosen with @kbd{d r}. The current radix applies
2883 to integers, fractions, and floats.
2885 @cindex Roundoff errors, in non-decimal numbers
2886 (@bullet{}) @strong{Exercise 1.} Your friend Joe tried to enter one-third
2887 as @samp{3#0.1} in @kbd{d r 3} mode with a precision of 12. He got
2888 @samp{3#0.0222222...} (with 25 2's) in the display. When he multiplied
2889 that by three, he got @samp{3#0.222222...} instead of the expected
2890 @samp{3#1}. Next, Joe entered @samp{3#0.2} and, to his great relief,
2891 saw @samp{3#0.2} on the screen. But when he typed @kbd{2 /}, he got
2892 @samp{3#0.10000001} (some zeros omitted). What's going on here?
2893 @xref{Modes Answer 1, 1}. (@bullet{})
2895 @cindex Scientific notation, in non-decimal numbers
2896 (@bullet{}) @strong{Exercise 2.} Scientific notation works in non-decimal
2897 modes in the natural way (the exponent is a power of the radix instead of
2898 a power of ten, although the exponent itself is always written in decimal).
2899 Thus @samp{8#1.23e3 = 8#1230.0}. Suppose we have the hexadecimal number
2900 @samp{f.e8f} times 16 to the 15th power: We write @samp{16#f.e8fe15}.
2901 What is wrong with this picture? What could we write instead that would
2902 work better? @xref{Modes Answer 2, 2}. (@bullet{})
2904 The @kbd{m} prefix key has another set of modes, relating to the way
2905 Calc interprets your inputs and does computations. Whereas @kbd{d}-prefix
2906 modes generally affect the way things look, @kbd{m}-prefix modes affect
2907 the way they are actually computed.
2909 The most popular @kbd{m}-prefix mode is the @dfn{angular mode}. Notice
2910 the @samp{Deg} indicator in the mode line. This means that if you use
2911 a command that interprets a number as an angle, it will assume the
2912 angle is measured in degrees. For example,
2916 1: 45 1: 0.707106781187 1: 0.500000000001 1: 0.5
2924 The shift-@kbd{S} command computes the sine of an angle. The sine
2926 @texline @math{\sqrt{2}/2};
2927 @infoline @expr{sqrt(2)/2};
2928 squaring this yields @expr{2/4 = 0.5}. However, there has been a slight
2929 roundoff error because the representation of
2930 @texline @math{\sqrt{2}/2}
2931 @infoline @expr{sqrt(2)/2}
2932 wasn't exact. The @kbd{c 1} command is a handy way to clean up numbers
2933 in this case; it temporarily reduces the precision by one digit while it
2934 re-rounds the number on the top of the stack.
2936 @cindex Roundoff errors, examples
2937 (@bullet{}) @strong{Exercise 3.} Your friend Joe computed the sine
2938 of 45 degrees as shown above, then, hoping to avoid an inexact
2939 result, he increased the precision to 16 digits before squaring.
2940 What happened? @xref{Modes Answer 3, 3}. (@bullet{})
2942 To do this calculation in radians, we would type @kbd{m r} first.
2943 (The indicator changes to @samp{Rad}.) 45 degrees corresponds to
2944 @cpiover{4} radians. To get @cpi{}, press the @kbd{P} key. (Once
2945 again, this is a shifted capital @kbd{P}. Remember, unshifted
2946 @kbd{p} sets the precision.)
2950 1: 3.14159265359 1: 0.785398163398 1: 0.707106781187
2957 Likewise, inverse trigonometric functions generate results in
2958 either radians or degrees, depending on the current angular mode.
2962 1: 0.707106781187 1: 0.785398163398 1: 45.
2965 .5 Q m r I S m d U I S
2970 Here we compute the Inverse Sine of
2971 @texline @math{\sqrt{0.5}},
2972 @infoline @expr{sqrt(0.5)},
2973 first in radians, then in degrees.
2975 Use @kbd{c d} and @kbd{c r} to convert a number from radians to degrees
2980 1: 45 1: 0.785398163397 1: 45.
2987 Another interesting mode is @dfn{Fraction mode}. Normally,
2988 dividing two integers produces a floating-point result if the
2989 quotient can't be expressed as an exact integer. Fraction mode
2990 causes integer division to produce a fraction, i.e., a rational
2995 2: 12 1: 1.33333333333 1: 4:3
2999 12 @key{RET} 9 / m f U / m f
3004 In the first case, we get an approximate floating-point result.
3005 In the second case, we get an exact fractional result (four-thirds).
3007 You can enter a fraction at any time using @kbd{:} notation.
3008 (Calc uses @kbd{:} instead of @kbd{/} as the fraction separator
3009 because @kbd{/} is already used to divide the top two stack
3010 elements.) Calculations involving fractions will always
3011 produce exact fractional results; Fraction mode only says
3012 what to do when dividing two integers.
3014 @cindex Fractions vs. floats
3015 @cindex Floats vs. fractions
3016 (@bullet{}) @strong{Exercise 4.} If fractional arithmetic is exact,
3017 why would you ever use floating-point numbers instead?
3018 @xref{Modes Answer 4, 4}. (@bullet{})
3020 Typing @kbd{m f} doesn't change any existing values in the stack.
3021 In the above example, we had to Undo the division and do it over
3022 again when we changed to Fraction mode. But if you use the
3023 evaluates-to operator you can get commands like @kbd{m f} to
3028 1: 12 / 9 => 1.33333333333 1: 12 / 9 => 1.333 1: 12 / 9 => 4:3
3031 ' 12/9 => @key{RET} p 4 @key{RET} m f
3036 In this example, the righthand side of the @samp{=>} operator
3037 on the stack is recomputed when we change the precision, then
3038 again when we change to Fraction mode. All @samp{=>} expressions
3039 on the stack are recomputed every time you change any mode that
3040 might affect their values.
3042 @node Arithmetic Tutorial, Vector/Matrix Tutorial, Basic Tutorial, Tutorial
3043 @section Arithmetic Tutorial
3046 In this section, we explore the arithmetic and scientific functions
3047 available in the Calculator.
3049 The standard arithmetic commands are @kbd{+}, @kbd{-}, @kbd{*}, @kbd{/},
3050 and @kbd{^}. Each normally takes two numbers from the top of the stack
3051 and pushes back a result. The @kbd{n} and @kbd{&} keys perform
3052 change-sign and reciprocal operations, respectively.
3056 1: 5 1: 0.2 1: 5. 1: -5. 1: 5.
3063 @cindex Binary operators
3064 You can apply a ``binary operator'' like @kbd{+} across any number of
3065 stack entries by giving it a numeric prefix. You can also apply it
3066 pairwise to several stack elements along with the top one if you use
3071 3: 2 1: 9 3: 2 4: 2 3: 12
3072 2: 3 . 2: 3 3: 3 2: 13
3073 1: 4 1: 4 2: 4 1: 14
3077 2 @key{RET} 3 @key{RET} 4 M-3 + U 10 M-- M-3 +
3081 @cindex Unary operators
3082 You can apply a ``unary operator'' like @kbd{&} to the top @var{n}
3083 stack entries with a numeric prefix, too.
3088 2: 3 2: 0.333333333333 2: 3.
3092 2 @key{RET} 3 @key{RET} 4 M-3 & M-2 &
3096 Notice that the results here are left in floating-point form.
3097 We can convert them back to integers by pressing @kbd{F}, the
3098 ``floor'' function. This function rounds down to the next lower
3099 integer. There is also @kbd{R}, which rounds to the nearest
3117 Since dividing-and-flooring (i.e., ``integer quotient'') is such a
3118 common operation, Calc provides a special command for that purpose, the
3119 backslash @kbd{\}. Another common arithmetic operator is @kbd{%}, which
3120 computes the remainder that would arise from a @kbd{\} operation, i.e.,
3121 the ``modulo'' of two numbers. For example,
3125 2: 1234 1: 12 2: 1234 1: 34
3129 1234 @key{RET} 100 \ U %
3133 These commands actually work for any real numbers, not just integers.
3137 2: 3.1415 1: 3 2: 3.1415 1: 0.1415
3141 3.1415 @key{RET} 1 \ U %
3145 (@bullet{}) @strong{Exercise 1.} The @kbd{\} command would appear to be a
3146 frill, since you could always do the same thing with @kbd{/ F}. Think
3147 of a situation where this is not true---@kbd{/ F} would be inadequate.
3148 Now think of a way you could get around the problem if Calc didn't
3149 provide a @kbd{\} command. @xref{Arithmetic Answer 1, 1}. (@bullet{})
3151 We've already seen the @kbd{Q} (square root) and @kbd{S} (sine)
3152 commands. Other commands along those lines are @kbd{C} (cosine),
3153 @kbd{T} (tangent), @kbd{E} (@expr{e^x}) and @kbd{L} (natural
3154 logarithm). These can be modified by the @kbd{I} (inverse) and
3155 @kbd{H} (hyperbolic) prefix keys.
3157 Let's compute the sine and cosine of an angle, and verify the
3159 @texline @math{\sin^2x + \cos^2x = 1}.
3160 @infoline @expr{sin(x)^2 + cos(x)^2 = 1}.
3161 We'll arbitrarily pick @mathit{-64} degrees as a good value for @expr{x}.
3162 With the angular mode set to degrees (type @w{@kbd{m d}}), do:
3166 2: -64 2: -64 2: -0.89879 2: -0.89879 1: 1.
3167 1: -64 1: -0.89879 1: -64 1: 0.43837 .
3170 64 n @key{RET} @key{RET} S @key{TAB} C f h
3175 (For brevity, we're showing only five digits of the results here.
3176 You can of course do these calculations to any precision you like.)
3178 Remember, @kbd{f h} is the @code{calc-hypot}, or square-root of sum
3179 of squares, command.
3182 @texline @math{\displaystyle\tan x = {\sin x \over \cos x}}.
3183 @infoline @expr{tan(x) = sin(x) / cos(x)}.
3187 2: -0.89879 1: -2.0503 1: -64.
3195 A physical interpretation of this calculation is that if you move
3196 @expr{0.89879} units downward and @expr{0.43837} units to the right,
3197 your direction of motion is @mathit{-64} degrees from horizontal. Suppose
3198 we move in the opposite direction, up and to the left:
3202 2: -0.89879 2: 0.89879 1: -2.0503 1: -64.
3203 1: 0.43837 1: -0.43837 . .
3211 How can the angle be the same? The answer is that the @kbd{/} operation
3212 loses information about the signs of its inputs. Because the quotient
3213 is negative, we know exactly one of the inputs was negative, but we
3214 can't tell which one. There is an @kbd{f T} [@code{arctan2}] function which
3215 computes the inverse tangent of the quotient of a pair of numbers.
3216 Since you feed it the two original numbers, it has enough information
3217 to give you a full 360-degree answer.
3221 2: 0.89879 1: 116. 3: 116. 2: 116. 1: 180.
3222 1: -0.43837 . 2: -0.89879 1: -64. .
3226 U U f T M-@key{RET} M-2 n f T -
3231 The resulting angles differ by 180 degrees; in other words, they
3232 point in opposite directions, just as we would expect.
3234 The @key{META}-@key{RET} we used in the third step is the
3235 ``last-arguments'' command. It is sort of like Undo, except that it
3236 restores the arguments of the last command to the stack without removing
3237 the command's result. It is useful in situations like this one,
3238 where we need to do several operations on the same inputs. We could
3239 have accomplished the same thing by using @kbd{M-2 @key{RET}} to duplicate
3240 the top two stack elements right after the @kbd{U U}, then a pair of
3241 @kbd{M-@key{TAB}} commands to cycle the 116 up around the duplicates.
3243 A similar identity is supposed to hold for hyperbolic sines and cosines,
3244 except that it is the @emph{difference}
3245 @texline @math{\cosh^2x - \sinh^2x}
3246 @infoline @expr{cosh(x)^2 - sinh(x)^2}
3247 that always equals one. Let's try to verify this identity.
3251 2: -64 2: -64 2: -64 2: 9.7192e54 2: 9.7192e54
3252 1: -64 1: -3.1175e27 1: 9.7192e54 1: -64 1: 9.7192e54
3255 64 n @key{RET} @key{RET} H C 2 ^ @key{TAB} H S 2 ^
3260 @cindex Roundoff errors, examples
3261 Something's obviously wrong, because when we subtract these numbers
3262 the answer will clearly be zero! But if you think about it, if these
3263 numbers @emph{did} differ by one, it would be in the 55th decimal
3264 place. The difference we seek has been lost entirely to roundoff
3267 We could verify this hypothesis by doing the actual calculation with,
3268 say, 60 decimal places of precision. This will be slow, but not
3269 enormously so. Try it if you wish; sure enough, the answer is
3270 0.99999, reasonably close to 1.
3272 Of course, a more reasonable way to verify the identity is to use
3273 a more reasonable value for @expr{x}!
3275 @cindex Common logarithm
3276 Some Calculator commands use the Hyperbolic prefix for other purposes.
3277 The logarithm and exponential functions, for example, work to the base
3278 @expr{e} normally but use base-10 instead if you use the Hyperbolic
3283 1: 1000 1: 6.9077 1: 1000 1: 3
3291 First, we mistakenly compute a natural logarithm. Then we undo
3292 and compute a common logarithm instead.
3294 The @kbd{B} key computes a general base-@var{b} logarithm for any
3299 2: 1000 1: 3 1: 1000. 2: 1000. 1: 6.9077
3300 1: 10 . . 1: 2.71828 .
3303 1000 @key{RET} 10 B H E H P B
3308 Here we first use @kbd{B} to compute the base-10 logarithm, then use
3309 the ``hyperbolic'' exponential as a cheap hack to recover the number
3310 1000, then use @kbd{B} again to compute the natural logarithm. Note
3311 that @kbd{P} with the hyperbolic prefix pushes the constant @expr{e}
3314 You may have noticed that both times we took the base-10 logarithm
3315 of 1000, we got an exact integer result. Calc always tries to give
3316 an exact rational result for calculations involving rational numbers
3317 where possible. But when we used @kbd{H E}, the result was a
3318 floating-point number for no apparent reason. In fact, if we had
3319 computed @kbd{10 @key{RET} 3 ^} we @emph{would} have gotten an
3320 exact integer 1000. But the @kbd{H E} command is rigged to generate
3321 a floating-point result all of the time so that @kbd{1000 H E} will
3322 not waste time computing a thousand-digit integer when all you
3323 probably wanted was @samp{1e1000}.
3325 (@bullet{}) @strong{Exercise 2.} Find a pair of integer inputs to
3326 the @kbd{B} command for which Calc could find an exact rational
3327 result but doesn't. @xref{Arithmetic Answer 2, 2}. (@bullet{})
3329 The Calculator also has a set of functions relating to combinatorics
3330 and statistics. You may be familiar with the @dfn{factorial} function,
3331 which computes the product of all the integers up to a given number.
3335 1: 100 1: 93326215443... 1: 100. 1: 9.3326e157
3343 Recall, the @kbd{c f} command converts the integer or fraction at the
3344 top of the stack to floating-point format. If you take the factorial
3345 of a floating-point number, you get a floating-point result
3346 accurate to the current precision. But if you give @kbd{!} an
3347 exact integer, you get an exact integer result (158 digits long
3350 If you take the factorial of a non-integer, Calc uses a generalized
3351 factorial function defined in terms of Euler's Gamma function
3352 @texline @math{\Gamma(n)}
3353 @infoline @expr{gamma(n)}
3354 (which is itself available as the @kbd{f g} command).
3358 3: 4. 3: 24. 1: 5.5 1: 52.342777847
3359 2: 4.5 2: 52.3427777847 . .
3363 M-3 ! M-0 @key{DEL} 5.5 f g
3368 Here we verify the identity
3369 @texline @math{n! = \Gamma(n+1)}.
3370 @infoline @expr{@var{n}!@: = gamma(@var{n}+1)}.
3372 The binomial coefficient @var{n}-choose-@var{m}
3373 @texline or @math{\displaystyle {n \choose m}}
3375 @texline @math{\displaystyle {n! \over m! \, (n-m)!}}
3376 @infoline @expr{n!@: / m!@: (n-m)!}
3377 for all reals @expr{n} and @expr{m}. The intermediate results in this
3378 formula can become quite large even if the final result is small; the
3379 @kbd{k c} command computes a binomial coefficient in a way that avoids
3380 large intermediate values.
3382 The @kbd{k} prefix key defines several common functions out of
3383 combinatorics and number theory. Here we compute the binomial
3384 coefficient 30-choose-20, then determine its prime factorization.
3388 2: 30 1: 30045015 1: [3, 3, 5, 7, 11, 13, 23, 29]
3392 30 @key{RET} 20 k c k f
3397 You can verify these prime factors by using @kbd{v u} to ``unpack''
3398 this vector into 8 separate stack entries, then @kbd{M-8 *} to
3399 multiply them back together. The result is the original number,
3403 Suppose a program you are writing needs a hash table with at least
3404 10000 entries. It's best to use a prime number as the actual size
3405 of a hash table. Calc can compute the next prime number after 10000:
3409 1: 10000 1: 10007 1: 9973
3417 Just for kicks we've also computed the next prime @emph{less} than
3420 @c [fix-ref Financial Functions]
3421 @xref{Financial Functions}, for a description of the Calculator
3422 commands that deal with business and financial calculations (functions
3423 like @code{pv}, @code{rate}, and @code{sln}).
3425 @c [fix-ref Binary Number Functions]
3426 @xref{Binary Functions}, to read about the commands for operating
3427 on binary numbers (like @code{and}, @code{xor}, and @code{lsh}).
3429 @node Vector/Matrix Tutorial, Types Tutorial, Arithmetic Tutorial, Tutorial
3430 @section Vector/Matrix Tutorial
3433 A @dfn{vector} is a list of numbers or other Calc data objects.
3434 Calc provides a large set of commands that operate on vectors. Some
3435 are familiar operations from vector analysis. Others simply treat
3436 a vector as a list of objects.
3439 * Vector Analysis Tutorial::
3444 @node Vector Analysis Tutorial, Matrix Tutorial, Vector/Matrix Tutorial, Vector/Matrix Tutorial
3445 @subsection Vector Analysis
3448 If you add two vectors, the result is a vector of the sums of the
3449 elements, taken pairwise.
3453 1: [1, 2, 3] 2: [1, 2, 3] 1: [8, 8, 3]
3457 [1,2,3] s 1 [7 6 0] s 2 +
3462 Note that we can separate the vector elements with either commas or
3463 spaces. This is true whether we are using incomplete vectors or
3464 algebraic entry. The @kbd{s 1} and @kbd{s 2} commands save these
3465 vectors so we can easily reuse them later.
3467 If you multiply two vectors, the result is the sum of the products
3468 of the elements taken pairwise. This is called the @dfn{dot product}
3482 The dot product of two vectors is equal to the product of their
3483 lengths times the cosine of the angle between them. (Here the vector
3484 is interpreted as a line from the origin @expr{(0,0,0)} to the
3485 specified point in three-dimensional space.) The @kbd{A}
3486 (absolute value) command can be used to compute the length of a
3491 3: 19 3: 19 1: 0.550782 1: 56.579
3492 2: [1, 2, 3] 2: 3.741657 . .
3493 1: [7, 6, 0] 1: 9.219544
3496 M-@key{RET} M-2 A * / I C
3501 First we recall the arguments to the dot product command, then
3502 we compute the absolute values of the top two stack entries to
3503 obtain the lengths of the vectors, then we divide the dot product
3504 by the product of the lengths to get the cosine of the angle.
3505 The inverse cosine finds that the angle between the vectors
3506 is about 56 degrees.
3508 @cindex Cross product
3509 @cindex Perpendicular vectors
3510 The @dfn{cross product} of two vectors is a vector whose length
3511 is the product of the lengths of the inputs times the sine of the
3512 angle between them, and whose direction is perpendicular to both
3513 input vectors. Unlike the dot product, the cross product is
3514 defined only for three-dimensional vectors. Let's double-check
3515 our computation of the angle using the cross product.
3519 2: [1, 2, 3] 3: [-18, 21, -8] 1: [-0.52, 0.61, -0.23] 1: 56.579
3520 1: [7, 6, 0] 2: [1, 2, 3] . .
3524 r 1 r 2 V C s 3 M-@key{RET} M-2 A * / A I S
3529 First we recall the original vectors and compute their cross product,
3530 which we also store for later reference. Now we divide the vector
3531 by the product of the lengths of the original vectors. The length of
3532 this vector should be the sine of the angle; sure enough, it is!
3534 @c [fix-ref General Mode Commands]
3535 Vector-related commands generally begin with the @kbd{v} prefix key.
3536 Some are uppercase letters and some are lowercase. To make it easier
3537 to type these commands, the shift-@kbd{V} prefix key acts the same as
3538 the @kbd{v} key. (@xref{General Mode Commands}, for a way to make all
3539 prefix keys have this property.)
3541 If we take the dot product of two perpendicular vectors we expect
3542 to get zero, since the cosine of 90 degrees is zero. Let's check
3543 that the cross product is indeed perpendicular to both inputs:
3547 2: [1, 2, 3] 1: 0 2: [7, 6, 0] 1: 0
3548 1: [-18, 21, -8] . 1: [-18, 21, -8] .
3551 r 1 r 3 * @key{DEL} r 2 r 3 *
3555 @cindex Normalizing a vector
3556 @cindex Unit vectors
3557 (@bullet{}) @strong{Exercise 1.} Given a vector on the top of the
3558 stack, what keystrokes would you use to @dfn{normalize} the
3559 vector, i.e., to reduce its length to one without changing its
3560 direction? @xref{Vector Answer 1, 1}. (@bullet{})
3562 (@bullet{}) @strong{Exercise 2.} Suppose a certain particle can be
3563 at any of several positions along a ruler. You have a list of
3564 those positions in the form of a vector, and another list of the
3565 probabilities for the particle to be at the corresponding positions.
3566 Find the average position of the particle.
3567 @xref{Vector Answer 2, 2}. (@bullet{})
3569 @node Matrix Tutorial, List Tutorial, Vector Analysis Tutorial, Vector/Matrix Tutorial
3570 @subsection Matrices
3573 A @dfn{matrix} is just a vector of vectors, all the same length.
3574 This means you can enter a matrix using nested brackets. You can
3575 also use the semicolon character to enter a matrix. We'll show
3580 1: [ [ 1, 2, 3 ] 1: [ [ 1, 2, 3 ]
3581 [ 4, 5, 6 ] ] [ 4, 5, 6 ] ]
3584 [[1 2 3] [4 5 6]] ' [1 2 3; 4 5 6] @key{RET}
3589 We'll be using this matrix again, so type @kbd{s 4} to save it now.
3591 Note that semicolons work with incomplete vectors, but they work
3592 better in algebraic entry. That's why we use the apostrophe in
3595 When two matrices are multiplied, the lefthand matrix must have
3596 the same number of columns as the righthand matrix has rows.
3597 Row @expr{i}, column @expr{j} of the result is effectively the
3598 dot product of row @expr{i} of the left matrix by column @expr{j}
3599 of the right matrix.
3601 If we try to duplicate this matrix and multiply it by itself,
3602 the dimensions are wrong and the multiplication cannot take place:
3606 1: [ [ 1, 2, 3 ] * [ [ 1, 2, 3 ]
3607 [ 4, 5, 6 ] ] [ 4, 5, 6 ] ]
3615 Though rather hard to read, this is a formula which shows the product
3616 of two matrices. The @samp{*} function, having invalid arguments, has
3617 been left in symbolic form.
3619 We can multiply the matrices if we @dfn{transpose} one of them first.
3623 2: [ [ 1, 2, 3 ] 1: [ [ 14, 32 ] 1: [ [ 17, 22, 27 ]
3624 [ 4, 5, 6 ] ] [ 32, 77 ] ] [ 22, 29, 36 ]
3625 1: [ [ 1, 4 ] . [ 27, 36, 45 ] ]
3630 U v t * U @key{TAB} *
3634 Matrix multiplication is not commutative; indeed, switching the
3635 order of the operands can even change the dimensions of the result
3636 matrix, as happened here!
3638 If you multiply a plain vector by a matrix, it is treated as a
3639 single row or column depending on which side of the matrix it is
3640 on. The result is a plain vector which should also be interpreted
3641 as a row or column as appropriate.
3645 2: [ [ 1, 2, 3 ] 1: [14, 32]
3654 Multiplying in the other order wouldn't work because the number of
3655 rows in the matrix is different from the number of elements in the
3658 (@bullet{}) @strong{Exercise 1.} Use @samp{*} to sum along the rows
3660 @texline @math{2\times3}
3662 matrix to get @expr{[6, 15]}. Now use @samp{*} to sum along the columns
3663 to get @expr{[5, 7, 9]}.
3664 @xref{Matrix Answer 1, 1}. (@bullet{})
3666 @cindex Identity matrix
3667 An @dfn{identity matrix} is a square matrix with ones along the
3668 diagonal and zeros elsewhere. It has the property that multiplication
3669 by an identity matrix, on the left or on the right, always produces
3670 the original matrix.
3674 1: [ [ 1, 2, 3 ] 2: [ [ 1, 2, 3 ] 1: [ [ 1, 2, 3 ]
3675 [ 4, 5, 6 ] ] [ 4, 5, 6 ] ] [ 4, 5, 6 ] ]
3676 . 1: [ [ 1, 0, 0 ] .
3681 r 4 v i 3 @key{RET} *
3685 If a matrix is square, it is often possible to find its @dfn{inverse},
3686 that is, a matrix which, when multiplied by the original matrix, yields
3687 an identity matrix. The @kbd{&} (reciprocal) key also computes the
3688 inverse of a matrix.
3692 1: [ [ 1, 2, 3 ] 1: [ [ -2.4, 1.2, -0.2 ]
3693 [ 4, 5, 6 ] [ 2.8, -1.4, 0.4 ]
3694 [ 7, 6, 0 ] ] [ -0.73333, 0.53333, -0.2 ] ]
3702 The vertical bar @kbd{|} @dfn{concatenates} numbers, vectors, and
3703 matrices together. Here we have used it to add a new row onto
3704 our matrix to make it square.
3706 We can multiply these two matrices in either order to get an identity.
3710 1: [ [ 1., 0., 0. ] 1: [ [ 1., 0., 0. ]
3711 [ 0., 1., 0. ] [ 0., 1., 0. ]
3712 [ 0., 0., 1. ] ] [ 0., 0., 1. ] ]
3715 M-@key{RET} * U @key{TAB} *
3719 @cindex Systems of linear equations
3720 @cindex Linear equations, systems of
3721 Matrix inverses are related to systems of linear equations in algebra.
3722 Suppose we had the following set of equations:
3736 $$ \openup1\jot \tabskip=0pt plus1fil
3737 \halign to\displaywidth{\tabskip=0pt
3738 $\hfil#$&$\hfil{}#{}$&
3739 $\hfil#$&$\hfil{}#{}$&
3740 $\hfil#$&${}#\hfil$\tabskip=0pt plus1fil\cr
3749 This can be cast into the matrix equation,
3754 [ [ 1, 2, 3 ] [ [ a ] [ [ 6 ]
3755 [ 4, 5, 6 ] * [ b ] = [ 2 ]
3756 [ 7, 6, 0 ] ] [ c ] ] [ 3 ] ]
3763 $$ \pmatrix{ 1 & 2 & 3 \cr 4 & 5 & 6 \cr 7 & 6 & 0 }
3765 \pmatrix{ a \cr b \cr c } = \pmatrix{ 6 \cr 2 \cr 3 }
3770 We can solve this system of equations by multiplying both sides by the
3771 inverse of the matrix. Calc can do this all in one step:
3775 2: [6, 2, 3] 1: [-12.6, 15.2, -3.93333]
3786 The result is the @expr{[a, b, c]} vector that solves the equations.
3787 (Dividing by a square matrix is equivalent to multiplying by its
3790 Let's verify this solution:
3794 2: [ [ 1, 2, 3 ] 1: [6., 2., 3.]
3797 1: [-12.6, 15.2, -3.93333]
3805 Note that we had to be careful about the order in which we multiplied
3806 the matrix and vector. If we multiplied in the other order, Calc would
3807 assume the vector was a row vector in order to make the dimensions
3808 come out right, and the answer would be incorrect. If you
3809 don't feel safe letting Calc take either interpretation of your
3810 vectors, use explicit
3811 @texline @math{N\times1}
3814 @texline @math{1\times N}
3816 matrices instead. In this case, you would enter the original column
3817 vector as @samp{[[6], [2], [3]]} or @samp{[6; 2; 3]}.
3819 (@bullet{}) @strong{Exercise 2.} Algebraic entry allows you to make
3820 vectors and matrices that include variables. Solve the following
3821 system of equations to get expressions for @expr{x} and @expr{y}
3822 in terms of @expr{a} and @expr{b}.
3835 $$ \eqalign{ x &+ a y = 6 \cr
3842 @xref{Matrix Answer 2, 2}. (@bullet{})
3844 @cindex Least-squares for over-determined systems
3845 @cindex Over-determined systems of equations
3846 (@bullet{}) @strong{Exercise 3.} A system of equations is ``over-determined''
3847 if it has more equations than variables. It is often the case that
3848 there are no values for the variables that will satisfy all the
3849 equations at once, but it is still useful to find a set of values
3850 which ``nearly'' satisfy all the equations. In terms of matrix equations,
3851 you can't solve @expr{A X = B} directly because the matrix @expr{A}
3852 is not square for an over-determined system. Matrix inversion works
3853 only for square matrices. One common trick is to multiply both sides
3854 on the left by the transpose of @expr{A}:
3856 @samp{trn(A)*A*X = trn(A)*B}.
3860 $A^T A \, X = A^T B$, where $A^T$ is the transpose \samp{trn(A)}.
3863 @texline @math{A^T A}
3864 @infoline @expr{trn(A)*A}
3865 is a square matrix so a solution is possible. It turns out that the
3866 @expr{X} vector you compute in this way will be a ``least-squares''
3867 solution, which can be regarded as the ``closest'' solution to the set
3868 of equations. Use Calc to solve the following over-determined
3884 $$ \openup1\jot \tabskip=0pt plus1fil
3885 \halign to\displaywidth{\tabskip=0pt
3886 $\hfil#$&$\hfil{}#{}$&
3887 $\hfil#$&$\hfil{}#{}$&
3888 $\hfil#$&${}#\hfil$\tabskip=0pt plus1fil\cr
3892 2a&+&4b&+&6c&=11 \cr}
3898 @xref{Matrix Answer 3, 3}. (@bullet{})
3900 @node List Tutorial, , Matrix Tutorial, Vector/Matrix Tutorial
3901 @subsection Vectors as Lists
3905 Although Calc has a number of features for manipulating vectors and
3906 matrices as mathematical objects, you can also treat vectors as
3907 simple lists of values. For example, we saw that the @kbd{k f}
3908 command returns a vector which is a list of the prime factors of a
3911 You can pack and unpack stack entries into vectors:
3915 3: 10 1: [10, 20, 30] 3: 10
3924 You can also build vectors out of consecutive integers, or out
3925 of many copies of a given value:
3929 1: [1, 2, 3, 4] 2: [1, 2, 3, 4] 2: [1, 2, 3, 4]
3930 . 1: 17 1: [17, 17, 17, 17]
3933 v x 4 @key{RET} 17 v b 4 @key{RET}
3937 You can apply an operator to every element of a vector using the
3942 1: [17, 34, 51, 68] 1: [289, 1156, 2601, 4624] 1: [17, 34, 51, 68]
3950 In the first step, we multiply the vector of integers by the vector
3951 of 17's elementwise. In the second step, we raise each element to
3952 the power two. (The general rule is that both operands must be
3953 vectors of the same length, or else one must be a vector and the
3954 other a plain number.) In the final step, we take the square root
3957 (@bullet{}) @strong{Exercise 1.} Compute a vector of powers of two
3959 @texline @math{2^{-4}}
3960 @infoline @expr{2^-4}
3961 to @expr{2^4}. @xref{List Answer 1, 1}. (@bullet{})
3963 You can also @dfn{reduce} a binary operator across a vector.
3964 For example, reducing @samp{*} computes the product of all the
3965 elements in the vector:
3969 1: 123123 1: [3, 7, 11, 13, 41] 1: 123123
3977 In this example, we decompose 123123 into its prime factors, then
3978 multiply those factors together again to yield the original number.
3980 We could compute a dot product ``by hand'' using mapping and
3985 2: [1, 2, 3] 1: [7, 12, 0] 1: 19
3994 Recalling two vectors from the previous section, we compute the
3995 sum of pairwise products of the elements to get the same answer
3996 for the dot product as before.
3998 A slight variant of vector reduction is the @dfn{accumulate} operation,
3999 @kbd{V U}. This produces a vector of the intermediate results from
4000 a corresponding reduction. Here we compute a table of factorials:
4004 1: [1, 2, 3, 4, 5, 6] 1: [1, 2, 6, 24, 120, 720]
4007 v x 6 @key{RET} V U *
4011 Calc allows vectors to grow as large as you like, although it gets
4012 rather slow if vectors have more than about a hundred elements.
4013 Actually, most of the time is spent formatting these large vectors
4014 for display, not calculating on them. Try the following experiment
4015 (if your computer is very fast you may need to substitute a larger
4020 1: [1, 2, 3, 4, ... 1: [2, 3, 4, 5, ...
4023 v x 500 @key{RET} 1 V M +
4027 Now press @kbd{v .} (the letter @kbd{v}, then a period) and try the
4028 experiment again. In @kbd{v .} mode, long vectors are displayed
4029 ``abbreviated'' like this:
4033 1: [1, 2, 3, ..., 500] 1: [2, 3, 4, ..., 501]
4036 v x 500 @key{RET} 1 V M +
4041 (where now the @samp{...} is actually part of the Calc display).
4042 You will find both operations are now much faster. But notice that
4043 even in @w{@kbd{v .}} mode, the full vectors are still shown in the Trail.
4044 Type @w{@kbd{t .}} to cause the trail to abbreviate as well, and try the
4045 experiment one more time. Operations on long vectors are now quite
4046 fast! (But of course if you use @kbd{t .} you will lose the ability
4047 to get old vectors back using the @kbd{t y} command.)
4049 An easy way to view a full vector when @kbd{v .} mode is active is
4050 to press @kbd{`} (back-quote) to edit the vector; editing always works
4051 with the full, unabbreviated value.
4053 @cindex Least-squares for fitting a straight line
4054 @cindex Fitting data to a line
4055 @cindex Line, fitting data to
4056 @cindex Data, extracting from buffers
4057 @cindex Columns of data, extracting
4058 As a larger example, let's try to fit a straight line to some data,
4059 using the method of least squares. (Calc has a built-in command for
4060 least-squares curve fitting, but we'll do it by hand here just to
4061 practice working with vectors.) Suppose we have the following list
4062 of values in a file we have loaded into Emacs:
4089 If you are reading this tutorial in printed form, you will find it
4090 easiest to press @kbd{M-# i} to enter the on-line Info version of
4091 the manual and find this table there. (Press @kbd{g}, then type
4092 @kbd{List Tutorial}, to jump straight to this section.)
4094 Position the cursor at the upper-left corner of this table, just
4095 to the left of the @expr{1.34}. Press @kbd{C-@@} to set the mark.
4096 (On your system this may be @kbd{C-2}, @kbd{C-@key{SPC}}, or @kbd{NUL}.)
4097 Now position the cursor to the lower-right, just after the @expr{1.354}.
4098 You have now defined this region as an Emacs ``rectangle.'' Still
4099 in the Info buffer, type @kbd{M-# r}. This command
4100 (@code{calc-grab-rectangle}) will pop you back into the Calculator, with
4101 the contents of the rectangle you specified in the form of a matrix.
4105 1: [ [ 1.34, 0.234 ]
4112 (You may wish to use @kbd{v .} mode to abbreviate the display of this
4115 We want to treat this as a pair of lists. The first step is to
4116 transpose this matrix into a pair of rows. Remember, a matrix is
4117 just a vector of vectors. So we can unpack the matrix into a pair
4118 of row vectors on the stack.
4122 1: [ [ 1.34, 1.41, 1.49, ... ] 2: [1.34, 1.41, 1.49, ... ]
4123 [ 0.234, 0.298, 0.402, ... ] ] 1: [0.234, 0.298, 0.402, ... ]
4131 Let's store these in quick variables 1 and 2, respectively.
4135 1: [1.34, 1.41, 1.49, ... ] .
4143 (Recall that @kbd{t 2} is a variant of @kbd{s 2} that removes the
4144 stored value from the stack.)
4146 In a least squares fit, the slope @expr{m} is given by the formula
4150 m = (N sum(x y) - sum(x) sum(y)) / (N sum(x^2) - sum(x)^2)
4156 $$ m = {N \sum x y - \sum x \sum y \over
4157 N \sum x^2 - \left( \sum x \right)^2} $$
4163 @texline @math{\sum x}
4164 @infoline @expr{sum(x)}
4165 represents the sum of all the values of @expr{x}. While there is an
4166 actual @code{sum} function in Calc, it's easier to sum a vector using a
4167 simple reduction. First, let's compute the four different sums that
4175 r 1 V R + t 3 r 1 2 V M ^ V R + t 4
4182 1: 13.613 1: 33.36554
4185 r 2 V R + t 5 r 1 r 2 V M * V R + t 6
4191 These are @samp{sum(x)}, @samp{sum(x^2)}, @samp{sum(y)}, and @samp{sum(x y)},
4192 respectively. (We could have used @kbd{*} to compute @samp{sum(x^2)} and
4197 These are $\sum x$, $\sum x^2$, $\sum y$, and $\sum x y$,
4198 respectively. (We could have used \kbd{*} to compute $\sum x^2$ and
4202 Finally, we also need @expr{N}, the number of data points. This is just
4203 the length of either of our lists.
4215 (That's @kbd{v} followed by a lower-case @kbd{l}.)
4217 Now we grind through the formula:
4221 1: 633.94526 2: 633.94526 1: 67.23607
4225 r 7 r 6 * r 3 r 5 * -
4232 2: 67.23607 3: 67.23607 2: 67.23607 1: 0.52141679
4233 1: 1862.0057 2: 1862.0057 1: 128.9488 .
4237 r 7 r 4 * r 3 2 ^ - / t 8
4241 That gives us the slope @expr{m}. The y-intercept @expr{b} can now
4242 be found with the simple formula,
4246 b = (sum(y) - m sum(x)) / N
4252 $$ b = {\sum y - m \sum x \over N} $$
4259 1: 13.613 2: 13.613 1: -8.09358 1: -0.425978
4263 r 5 r 8 r 3 * - r 7 / t 9
4267 Let's ``plot'' this straight line approximation,
4268 @texline @math{y \approx m x + b},
4269 @infoline @expr{m x + b},
4270 and compare it with the original data.
4274 1: [0.699, 0.735, ... ] 1: [0.273, 0.309, ... ]
4282 Notice that multiplying a vector by a constant, and adding a constant
4283 to a vector, can be done without mapping commands since these are
4284 common operations from vector algebra. As far as Calc is concerned,
4285 we've just been doing geometry in 19-dimensional space!
4287 We can subtract this vector from our original @expr{y} vector to get
4288 a feel for the error of our fit. Let's find the maximum error:
4292 1: [0.0387, 0.0112, ... ] 1: [0.0387, 0.0112, ... ] 1: 0.0897
4300 First we compute a vector of differences, then we take the absolute
4301 values of these differences, then we reduce the @code{max} function
4302 across the vector. (The @code{max} function is on the two-key sequence
4303 @kbd{f x}; because it is so common to use @code{max} in a vector
4304 operation, the letters @kbd{X} and @kbd{N} are also accepted for
4305 @code{max} and @code{min} in this context. In general, you answer
4306 the @kbd{V M} or @kbd{V R} prompt with the actual key sequence that
4307 invokes the function you want. You could have typed @kbd{V R f x} or
4308 even @kbd{V R x max @key{RET}} if you had preferred.)
4310 If your system has the GNUPLOT program, you can see graphs of your
4311 data and your straight line to see how well they match. (If you have
4312 GNUPLOT 3.0, the following instructions will work regardless of the
4313 kind of display you have. Some GNUPLOT 2.0, non-X-windows systems
4314 may require additional steps to view the graphs.)
4316 Let's start by plotting the original data. Recall the ``@var{x}'' and ``@var{y}''
4317 vectors onto the stack and press @kbd{g f}. This ``fast'' graphing
4318 command does everything you need to do for simple, straightforward
4323 2: [1.34, 1.41, 1.49, ... ]
4324 1: [0.234, 0.298, 0.402, ... ]
4331 If all goes well, you will shortly get a new window containing a graph
4332 of the data. (If not, contact your GNUPLOT or Calc installer to find
4333 out what went wrong.) In the X window system, this will be a separate
4334 graphics window. For other kinds of displays, the default is to
4335 display the graph in Emacs itself using rough character graphics.
4336 Press @kbd{q} when you are done viewing the character graphics.
4338 Next, let's add the line we got from our least-squares fit.
4340 (If you are reading this tutorial on-line while running Calc, typing
4341 @kbd{g a} may cause the tutorial to disappear from its window and be
4342 replaced by a buffer named @samp{*Gnuplot Commands*}. The tutorial
4343 will reappear when you terminate GNUPLOT by typing @kbd{g q}.)
4348 2: [1.34, 1.41, 1.49, ... ]
4349 1: [0.273, 0.309, 0.351, ... ]
4352 @key{DEL} r 0 g a g p
4356 It's not very useful to get symbols to mark the data points on this
4357 second curve; you can type @kbd{g S g p} to remove them. Type @kbd{g q}
4358 when you are done to remove the X graphics window and terminate GNUPLOT.
4360 (@bullet{}) @strong{Exercise 2.} An earlier exercise showed how to do
4361 least squares fitting to a general system of equations. Our 19 data
4362 points are really 19 equations of the form @expr{y_i = m x_i + b} for
4363 different pairs of @expr{(x_i,y_i)}. Use the matrix-transpose method
4364 to solve for @expr{m} and @expr{b}, duplicating the above result.
4365 @xref{List Answer 2, 2}. (@bullet{})
4367 @cindex Geometric mean
4368 (@bullet{}) @strong{Exercise 3.} If the input data do not form a
4369 rectangle, you can use @w{@kbd{M-# g}} (@code{calc-grab-region})
4370 to grab the data the way Emacs normally works with regions---it reads
4371 left-to-right, top-to-bottom, treating line breaks the same as spaces.
4372 Use this command to find the geometric mean of the following numbers.
4373 (The geometric mean is the @var{n}th root of the product of @var{n} numbers.)
4382 The @kbd{M-# g} command accepts numbers separated by spaces or commas,
4383 with or without surrounding vector brackets.
4384 @xref{List Answer 3, 3}. (@bullet{})
4387 As another example, a theorem about binomial coefficients tells
4388 us that the alternating sum of binomial coefficients
4389 @var{n}-choose-0 minus @var{n}-choose-1 plus @var{n}-choose-2, and so
4390 on up to @var{n}-choose-@var{n},
4391 always comes out to zero. Let's verify this
4395 As another example, a theorem about binomial coefficients tells
4396 us that the alternating sum of binomial coefficients
4397 ${n \choose 0} - {n \choose 1} + {n \choose 2} - \cdots \pm {n \choose n}$
4398 always comes out to zero. Let's verify this
4404 1: [1, 2, 3, 4, 5, 6, 7] 1: [0, 1, 2, 3, 4, 5, 6]
4414 1: [1, -6, 15, -20, 15, -6, 1] 1: 0
4417 V M ' (-1)^$ choose(6,$) @key{RET} V R +
4421 The @kbd{V M '} command prompts you to enter any algebraic expression
4422 to define the function to map over the vector. The symbol @samp{$}
4423 inside this expression represents the argument to the function.
4424 The Calculator applies this formula to each element of the vector,
4425 substituting each element's value for the @samp{$} sign(s) in turn.
4427 To define a two-argument function, use @samp{$$} for the first
4428 argument and @samp{$} for the second: @kbd{V M ' $$-$ @key{RET}} is
4429 equivalent to @kbd{V M -}. This is analogous to regular algebraic
4430 entry, where @samp{$$} would refer to the next-to-top stack entry
4431 and @samp{$} would refer to the top stack entry, and @kbd{' $$-$ @key{RET}}
4432 would act exactly like @kbd{-}.
4434 Notice that the @kbd{V M '} command has recorded two things in the
4435 trail: The result, as usual, and also a funny-looking thing marked
4436 @samp{oper} that represents the operator function you typed in.
4437 The function is enclosed in @samp{< >} brackets, and the argument is
4438 denoted by a @samp{#} sign. If there were several arguments, they
4439 would be shown as @samp{#1}, @samp{#2}, and so on. (For example,
4440 @kbd{V M ' $$-$} will put the function @samp{<#1 - #2>} on the
4441 trail.) This object is a ``nameless function''; you can use nameless
4442 @w{@samp{< >}} notation to answer the @kbd{V M '} prompt if you like.
4443 Nameless function notation has the interesting, occasionally useful
4444 property that a nameless function is not actually evaluated until
4445 it is used. For example, @kbd{V M ' $+random(2.0)} evaluates
4446 @samp{random(2.0)} once and adds that random number to all elements
4447 of the vector, but @kbd{V M ' <#+random(2.0)>} evaluates the
4448 @samp{random(2.0)} separately for each vector element.
4450 Another group of operators that are often useful with @kbd{V M} are
4451 the relational operators: @kbd{a =}, for example, compares two numbers
4452 and gives the result 1 if they are equal, or 0 if not. Similarly,
4453 @w{@kbd{a <}} checks for one number being less than another.
4455 Other useful vector operations include @kbd{v v}, to reverse a
4456 vector end-for-end; @kbd{V S}, to sort the elements of a vector
4457 into increasing order; and @kbd{v r} and @w{@kbd{v c}}, to extract
4458 one row or column of a matrix, or (in both cases) to extract one
4459 element of a plain vector. With a negative argument, @kbd{v r}
4460 and @kbd{v c} instead delete one row, column, or vector element.
4462 @cindex Divisor functions
4463 (@bullet{}) @strong{Exercise 4.} The @expr{k}th @dfn{divisor function}
4467 is the sum of the @expr{k}th powers of all the divisors of an
4468 integer @expr{n}. Figure out a method for computing the divisor
4469 function for reasonably small values of @expr{n}. As a test,
4470 the 0th and 1st divisor functions of 30 are 8 and 72, respectively.
4471 @xref{List Answer 4, 4}. (@bullet{})
4473 @cindex Square-free numbers
4474 @cindex Duplicate values in a list
4475 (@bullet{}) @strong{Exercise 5.} The @kbd{k f} command produces a
4476 list of prime factors for a number. Sometimes it is important to
4477 know that a number is @dfn{square-free}, i.e., that no prime occurs
4478 more than once in its list of prime factors. Find a sequence of
4479 keystrokes to tell if a number is square-free; your method should
4480 leave 1 on the stack if it is, or 0 if it isn't.
4481 @xref{List Answer 5, 5}. (@bullet{})
4483 @cindex Triangular lists
4484 (@bullet{}) @strong{Exercise 6.} Build a list of lists that looks
4485 like the following diagram. (You may wish to use the @kbd{v /}
4486 command to enable multi-line display of vectors.)
4495 [1, 2, 3, 4, 5, 6] ]
4500 @xref{List Answer 6, 6}. (@bullet{})
4502 (@bullet{}) @strong{Exercise 7.} Build the following list of lists.
4510 [10, 11, 12, 13, 14],
4511 [15, 16, 17, 18, 19, 20] ]
4516 @xref{List Answer 7, 7}. (@bullet{})
4518 @cindex Maximizing a function over a list of values
4519 @c [fix-ref Numerical Solutions]
4520 (@bullet{}) @strong{Exercise 8.} Compute a list of values of Bessel's
4521 @texline @math{J_1(x)}
4523 function @samp{besJ(1,x)} for @expr{x} from 0 to 5 in steps of 0.25.
4524 Find the value of @expr{x} (from among the above set of values) for
4525 which @samp{besJ(1,x)} is a maximum. Use an ``automatic'' method,
4526 i.e., just reading along the list by hand to find the largest value
4527 is not allowed! (There is an @kbd{a X} command which does this kind
4528 of thing automatically; @pxref{Numerical Solutions}.)
4529 @xref{List Answer 8, 8}. (@bullet{})
4531 @cindex Digits, vectors of
4532 (@bullet{}) @strong{Exercise 9.} You are given an integer in the range
4533 @texline @math{0 \le N < 10^m}
4534 @infoline @expr{0 <= N < 10^m}
4535 for @expr{m=12} (i.e., an integer of less than
4536 twelve digits). Convert this integer into a vector of @expr{m}
4537 digits, each in the range from 0 to 9. In vector-of-digits notation,
4538 add one to this integer to produce a vector of @expr{m+1} digits
4539 (since there could be a carry out of the most significant digit).
4540 Convert this vector back into a regular integer. A good integer
4541 to try is 25129925999. @xref{List Answer 9, 9}. (@bullet{})
4543 (@bullet{}) @strong{Exercise 10.} Your friend Joe tried to use
4544 @kbd{V R a =} to test if all numbers in a list were equal. What
4545 happened? How would you do this test? @xref{List Answer 10, 10}. (@bullet{})
4547 (@bullet{}) @strong{Exercise 11.} The area of a circle of radius one
4548 is @cpi{}. The area of the
4549 @texline @math{2\times2}
4551 square that encloses that circle is 4. So if we throw @var{n} darts at
4552 random points in the square, about @cpiover{4} of them will land inside
4553 the circle. This gives us an entertaining way to estimate the value of
4554 @cpi{}. The @w{@kbd{k r}}
4555 command picks a random number between zero and the value on the stack.
4556 We could get a random floating-point number between @mathit{-1} and 1 by typing
4557 @w{@kbd{2.0 k r 1 -}}. Build a vector of 100 random @expr{(x,y)} points in
4558 this square, then use vector mapping and reduction to count how many
4559 points lie inside the unit circle. Hint: Use the @kbd{v b} command.
4560 @xref{List Answer 11, 11}. (@bullet{})
4562 @cindex Matchstick problem
4563 (@bullet{}) @strong{Exercise 12.} The @dfn{matchstick problem} provides
4564 another way to calculate @cpi{}. Say you have an infinite field
4565 of vertical lines with a spacing of one inch. Toss a one-inch matchstick
4566 onto the field. The probability that the matchstick will land crossing
4567 a line turns out to be
4568 @texline @math{2/\pi}.
4569 @infoline @expr{2/pi}.
4570 Toss 100 matchsticks to estimate @cpi{}. (If you want still more fun,
4571 the probability that the GCD (@w{@kbd{k g}}) of two large integers is
4573 @texline @math{6/\pi^2}.
4574 @infoline @expr{6/pi^2}.
4575 That provides yet another way to estimate @cpi{}.)
4576 @xref{List Answer 12, 12}. (@bullet{})
4578 (@bullet{}) @strong{Exercise 13.} An algebraic entry of a string in
4579 double-quote marks, @samp{"hello"}, creates a vector of the numerical
4580 (ASCII) codes of the characters (here, @expr{[104, 101, 108, 108, 111]}).
4581 Sometimes it is convenient to compute a @dfn{hash code} of a string,
4582 which is just an integer that represents the value of that string.
4583 Two equal strings have the same hash code; two different strings
4584 @dfn{probably} have different hash codes. (For example, Calc has
4585 over 400 function names, but Emacs can quickly find the definition for
4586 any given name because it has sorted the functions into ``buckets'' by
4587 their hash codes. Sometimes a few names will hash into the same bucket,
4588 but it is easier to search among a few names than among all the names.)
4589 One popular hash function is computed as follows: First set @expr{h = 0}.
4590 Then, for each character from the string in turn, set @expr{h = 3h + c_i}
4591 where @expr{c_i} is the character's ASCII code. If we have 511 buckets,
4592 we then take the hash code modulo 511 to get the bucket number. Develop a
4593 simple command or commands for converting string vectors into hash codes.
4594 The hash code for @samp{"Testing, 1, 2, 3"} is 1960915098, which modulo
4595 511 is 121. @xref{List Answer 13, 13}. (@bullet{})
4597 (@bullet{}) @strong{Exercise 14.} The @kbd{H V R} and @kbd{H V U}
4598 commands do nested function evaluations. @kbd{H V U} takes a starting
4599 value and a number of steps @var{n} from the stack; it then applies the
4600 function you give to the starting value 0, 1, 2, up to @var{n} times
4601 and returns a vector of the results. Use this command to create a
4602 ``random walk'' of 50 steps. Start with the two-dimensional point
4603 @expr{(0,0)}; then take one step a random distance between @mathit{-1} and 1
4604 in both @expr{x} and @expr{y}; then take another step, and so on. Use the
4605 @kbd{g f} command to display this random walk. Now modify your random
4606 walk to walk a unit distance, but in a random direction, at each step.
4607 (Hint: The @code{sincos} function returns a vector of the cosine and
4608 sine of an angle.) @xref{List Answer 14, 14}. (@bullet{})
4610 @node Types Tutorial, Algebra Tutorial, Vector/Matrix Tutorial, Tutorial
4611 @section Types Tutorial
4614 Calc understands a variety of data types as well as simple numbers.
4615 In this section, we'll experiment with each of these types in turn.
4617 The numbers we've been using so far have mainly been either @dfn{integers}
4618 or @dfn{floats}. We saw that floats are usually a good approximation to
4619 the mathematical concept of real numbers, but they are only approximations
4620 and are susceptible to roundoff error. Calc also supports @dfn{fractions},
4621 which can exactly represent any rational number.
4625 1: 3628800 2: 3628800 1: 518400:7 1: 518414:7 1: 7:518414
4629 10 ! 49 @key{RET} : 2 + &
4634 The @kbd{:} command divides two integers to get a fraction; @kbd{/}
4635 would normally divide integers to get a floating-point result.
4636 Notice we had to type @key{RET} between the @kbd{49} and the @kbd{:}
4637 since the @kbd{:} would otherwise be interpreted as part of a
4638 fraction beginning with 49.
4640 You can convert between floating-point and fractional format using
4641 @kbd{c f} and @kbd{c F}:
4645 1: 1.35027217629e-5 1: 7:518414
4652 The @kbd{c F} command replaces a floating-point number with the
4653 ``simplest'' fraction whose floating-point representation is the
4654 same, to within the current precision.
4658 1: 3.14159265359 1: 1146408:364913 1: 3.1416 1: 355:113
4661 P c F @key{DEL} p 5 @key{RET} P c F
4665 (@bullet{}) @strong{Exercise 1.} A calculation has produced the
4666 result 1.26508260337. You suspect it is the square root of the
4667 product of @cpi{} and some rational number. Is it? (Be sure
4668 to allow for roundoff error!) @xref{Types Answer 1, 1}. (@bullet{})
4670 @dfn{Complex numbers} can be stored in both rectangular and polar form.
4674 1: -9 1: (0, 3) 1: (3; 90.) 1: (6; 90.) 1: (2.4495; 45.)
4682 The square root of @mathit{-9} is by default rendered in rectangular form
4683 (@w{@expr{0 + 3i}}), but we can convert it to polar form (3 with a
4684 phase angle of 90 degrees). All the usual arithmetic and scientific
4685 operations are defined on both types of complex numbers.
4687 Another generalized kind of number is @dfn{infinity}. Infinity
4688 isn't really a number, but it can sometimes be treated like one.
4689 Calc uses the symbol @code{inf} to represent positive infinity,
4690 i.e., a value greater than any real number. Naturally, you can
4691 also write @samp{-inf} for minus infinity, a value less than any
4692 real number. The word @code{inf} can only be input using
4697 2: inf 2: -inf 2: -inf 2: -inf 1: nan
4698 1: -17 1: -inf 1: -inf 1: inf .
4701 ' inf @key{RET} 17 n * @key{RET} 72 + A +
4706 Since infinity is infinitely large, multiplying it by any finite
4707 number (like @mathit{-17}) has no effect, except that since @mathit{-17}
4708 is negative, it changes a plus infinity to a minus infinity.
4709 (``A huge positive number, multiplied by @mathit{-17}, yields a huge
4710 negative number.'') Adding any finite number to infinity also
4711 leaves it unchanged. Taking an absolute value gives us plus
4712 infinity again. Finally, we add this plus infinity to the minus
4713 infinity we had earlier. If you work it out, you might expect
4714 the answer to be @mathit{-72} for this. But the 72 has been completely
4715 lost next to the infinities; by the time we compute @w{@samp{inf - inf}}
4716 the finite difference between them, if any, is undetectable.
4717 So we say the result is @dfn{indeterminate}, which Calc writes
4718 with the symbol @code{nan} (for Not A Number).
4720 Dividing by zero is normally treated as an error, but you can get
4721 Calc to write an answer in terms of infinity by pressing @kbd{m i}
4722 to turn on Infinite mode.
4726 3: nan 2: nan 2: nan 2: nan 1: nan
4727 2: 1 1: 1 / 0 1: uinf 1: uinf .
4731 1 @key{RET} 0 / m i U / 17 n * +
4736 Dividing by zero normally is left unevaluated, but after @kbd{m i}
4737 it instead gives an infinite result. The answer is actually
4738 @code{uinf}, ``undirected infinity.'' If you look at a graph of
4739 @expr{1 / x} around @w{@expr{x = 0}}, you'll see that it goes toward
4740 plus infinity as you approach zero from above, but toward minus
4741 infinity as you approach from below. Since we said only @expr{1 / 0},
4742 Calc knows that the answer is infinite but not in which direction.
4743 That's what @code{uinf} means. Notice that multiplying @code{uinf}
4744 by a negative number still leaves plain @code{uinf}; there's no
4745 point in saying @samp{-uinf} because the sign of @code{uinf} is
4746 unknown anyway. Finally, we add @code{uinf} to our @code{nan},
4747 yielding @code{nan} again. It's easy to see that, because
4748 @code{nan} means ``totally unknown'' while @code{uinf} means
4749 ``unknown sign but known to be infinite,'' the more mysterious
4750 @code{nan} wins out when it is combined with @code{uinf}, or, for
4751 that matter, with anything else.
4753 (@bullet{}) @strong{Exercise 2.} Predict what Calc will answer
4754 for each of these formulas: @samp{inf / inf}, @samp{exp(inf)},
4755 @samp{exp(-inf)}, @samp{sqrt(-inf)}, @samp{sqrt(uinf)},
4756 @samp{abs(uinf)}, @samp{ln(0)}.
4757 @xref{Types Answer 2, 2}. (@bullet{})
4759 (@bullet{}) @strong{Exercise 3.} We saw that @samp{inf - inf = nan},
4760 which stands for an unknown value. Can @code{nan} stand for
4761 a complex number? Can it stand for infinity?
4762 @xref{Types Answer 3, 3}. (@bullet{})
4764 @dfn{HMS forms} represent a value in terms of hours, minutes, and
4769 1: 2@@ 30' 0" 1: 3@@ 30' 0" 2: 3@@ 30' 0" 1: 2.
4770 . . 1: 1@@ 45' 0." .
4773 2@@ 30' @key{RET} 1 + @key{RET} 2 / /
4777 HMS forms can also be used to hold angles in degrees, minutes, and
4782 1: 0.5 1: 26.56505 1: 26@@ 33' 54.18" 1: 0.44721
4790 First we convert the inverse tangent of 0.5 to degrees-minutes-seconds
4791 form, then we take the sine of that angle. Note that the trigonometric
4792 functions will accept HMS forms directly as input.
4795 (@bullet{}) @strong{Exercise 4.} The Beatles' @emph{Abbey Road} is
4796 47 minutes and 26 seconds long, and contains 17 songs. What is the
4797 average length of a song on @emph{Abbey Road}? If the Extended Disco
4798 Version of @emph{Abbey Road} added 20 seconds to the length of each
4799 song, how long would the album be? @xref{Types Answer 4, 4}. (@bullet{})
4801 A @dfn{date form} represents a date, or a date and time. Dates must
4802 be entered using algebraic entry. Date forms are surrounded by
4803 @samp{< >} symbols; most standard formats for dates are recognized.
4807 2: <Sun Jan 13, 1991> 1: 2.25
4808 1: <6:00pm Thu Jan 10, 1991> .
4811 ' <13 Jan 1991>, <1/10/91, 6pm> @key{RET} -
4816 In this example, we enter two dates, then subtract to find the
4817 number of days between them. It is also possible to add an
4818 HMS form or a number (of days) to a date form to get another
4823 1: <4:45:59pm Mon Jan 14, 1991> 1: <2:50:59am Thu Jan 17, 1991>
4830 @c [fix-ref Date Arithmetic]
4832 The @kbd{t N} (``now'') command pushes the current date and time on the
4833 stack; then we add two days, ten hours and five minutes to the date and
4834 time. Other date-and-time related commands include @kbd{t J}, which
4835 does Julian day conversions, @kbd{t W}, which finds the beginning of
4836 the week in which a date form lies, and @kbd{t I}, which increments a
4837 date by one or several months. @xref{Date Arithmetic}, for more.
4839 (@bullet{}) @strong{Exercise 5.} How many days until the next
4840 Friday the 13th? @xref{Types Answer 5, 5}. (@bullet{})
4842 (@bullet{}) @strong{Exercise 6.} How many leap years will there be
4843 between now and the year 10001 A.D.? @xref{Types Answer 6, 6}. (@bullet{})
4845 @cindex Slope and angle of a line
4846 @cindex Angle and slope of a line
4847 An @dfn{error form} represents a mean value with an attached standard
4848 deviation, or error estimate. Suppose our measurements indicate that
4849 a certain telephone pole is about 30 meters away, with an estimated
4850 error of 1 meter, and 8 meters tall, with an estimated error of 0.2
4851 meters. What is the slope of a line from here to the top of the
4852 pole, and what is the equivalent angle in degrees?
4856 1: 8 +/- 0.2 2: 8 +/- 0.2 1: 0.266 +/- 0.011 1: 14.93 +/- 0.594
4860 8 p .2 @key{RET} 30 p 1 / I T
4865 This means that the angle is about 15 degrees, and, assuming our
4866 original error estimates were valid standard deviations, there is about
4867 a 60% chance that the result is correct within 0.59 degrees.
4869 @cindex Torus, volume of
4870 (@bullet{}) @strong{Exercise 7.} The volume of a torus (a donut shape) is
4871 @texline @math{2 \pi^2 R r^2}
4872 @infoline @w{@expr{2 pi^2 R r^2}}
4873 where @expr{R} is the radius of the circle that
4874 defines the center of the tube and @expr{r} is the radius of the tube
4875 itself. Suppose @expr{R} is 20 cm and @expr{r} is 4 cm, each known to
4876 within 5 percent. What is the volume and the relative uncertainty of
4877 the volume? @xref{Types Answer 7, 7}. (@bullet{})
4879 An @dfn{interval form} represents a range of values. While an
4880 error form is best for making statistical estimates, intervals give
4881 you exact bounds on an answer. Suppose we additionally know that
4882 our telephone pole is definitely between 28 and 31 meters away,
4883 and that it is between 7.7 and 8.1 meters tall.
4887 1: [7.7 .. 8.1] 2: [7.7 .. 8.1] 1: [0.24 .. 0.28] 1: [13.9 .. 16.1]
4891 [ 7.7 .. 8.1 ] [ 28 .. 31 ] / I T
4896 If our bounds were correct, then the angle to the top of the pole
4897 is sure to lie in the range shown.
4899 The square brackets around these intervals indicate that the endpoints
4900 themselves are allowable values. In other words, the distance to the
4901 telephone pole is between 28 and 31, @emph{inclusive}. You can also
4902 make an interval that is exclusive of its endpoints by writing
4903 parentheses instead of square brackets. You can even make an interval
4904 which is inclusive (``closed'') on one end and exclusive (``open'') on
4909 1: [1 .. 10) 1: (0.1 .. 1] 2: (0.1 .. 1] 1: (0.2 .. 3)
4913 [ 1 .. 10 ) & [ 2 .. 3 ) *
4918 The Calculator automatically keeps track of which end values should
4919 be open and which should be closed. You can also make infinite or
4920 semi-infinite intervals by using @samp{-inf} or @samp{inf} for one
4923 (@bullet{}) @strong{Exercise 8.} What answer would you expect from
4924 @samp{@w{1 /} @w{(0 .. 10)}}? What about @samp{@w{1 /} @w{(-10 .. 0)}}? What
4925 about @samp{@w{1 /} @w{[0 .. 10]}} (where the interval actually includes
4926 zero)? What about @samp{@w{1 /} @w{(-10 .. 10)}}?
4927 @xref{Types Answer 8, 8}. (@bullet{})
4929 (@bullet{}) @strong{Exercise 9.} Two easy ways of squaring a number
4930 are @kbd{@key{RET} *} and @w{@kbd{2 ^}}. Normally these produce the same
4931 answer. Would you expect this still to hold true for interval forms?
4932 If not, which of these will result in a larger interval?
4933 @xref{Types Answer 9, 9}. (@bullet{})
4935 A @dfn{modulo form} is used for performing arithmetic modulo @var{m}.
4936 For example, arithmetic involving time is generally done modulo 12
4941 1: 17 mod 24 1: 3 mod 24 1: 21 mod 24 1: 9 mod 24
4944 17 M 24 @key{RET} 10 + n 5 /
4949 In this last step, Calc has divided by 5 modulo 24; i.e., it has found a
4950 new number which, when multiplied by 5 modulo 24, produces the original
4951 number, 21. If @var{m} is prime and the divisor is not a multiple of
4952 @var{m}, it is always possible to find such a number. For non-prime
4953 @var{m} like 24, it is only sometimes possible.
4957 1: 10 mod 24 1: 16 mod 24 1: 1000000... 1: 16
4960 10 M 24 @key{RET} 100 ^ 10 @key{RET} 100 ^ 24 %
4965 These two calculations get the same answer, but the first one is
4966 much more efficient because it avoids the huge intermediate value
4967 that arises in the second one.
4969 @cindex Fermat, primality test of
4970 (@bullet{}) @strong{Exercise 10.} A theorem of Pierre de Fermat
4972 @texline @w{@math{x^{n-1} \bmod n = 1}}
4973 @infoline @expr{x^(n-1) mod n = 1}
4974 if @expr{n} is a prime number and @expr{x} is an integer less than
4975 @expr{n}. If @expr{n} is @emph{not} a prime number, this will
4976 @emph{not} be true for most values of @expr{x}. Thus we can test
4977 informally if a number is prime by trying this formula for several
4978 values of @expr{x}. Use this test to tell whether the following numbers
4979 are prime: 811749613, 15485863. @xref{Types Answer 10, 10}. (@bullet{})
4981 It is possible to use HMS forms as parts of error forms, intervals,
4982 modulo forms, or as the phase part of a polar complex number.
4983 For example, the @code{calc-time} command pushes the current time
4984 of day on the stack as an HMS/modulo form.
4988 1: 17@@ 34' 45" mod 24@@ 0' 0" 1: 6@@ 22' 15" mod 24@@ 0' 0"
4996 This calculation tells me it is six hours and 22 minutes until midnight.
4998 (@bullet{}) @strong{Exercise 11.} A rule of thumb is that one year
5000 @texline @math{\pi \times 10^7}
5001 @infoline @w{@expr{pi * 10^7}}
5002 seconds. What time will it be that many seconds from right now?
5003 @xref{Types Answer 11, 11}. (@bullet{})
5005 (@bullet{}) @strong{Exercise 12.} You are preparing to order packaging
5006 for the CD release of the Extended Disco Version of @emph{Abbey Road}.
5007 You are told that the songs will actually be anywhere from 20 to 60
5008 seconds longer than the originals. One CD can hold about 75 minutes
5009 of music. Should you order single or double packages?
5010 @xref{Types Answer 12, 12}. (@bullet{})
5012 Another kind of data the Calculator can manipulate is numbers with
5013 @dfn{units}. This isn't strictly a new data type; it's simply an
5014 application of algebraic expressions, where we use variables with
5015 suggestive names like @samp{cm} and @samp{in} to represent units
5016 like centimeters and inches.
5020 1: 2 in 1: 5.08 cm 1: 0.027778 fath 1: 0.0508 m
5023 ' 2in @key{RET} u c cm @key{RET} u c fath @key{RET} u b
5028 We enter the quantity ``2 inches'' (actually an algebraic expression
5029 which means two times the variable @samp{in}), then we convert it
5030 first to centimeters, then to fathoms, then finally to ``base'' units,
5031 which in this case means meters.
5035 1: 9 acre 1: 3 sqrt(acre) 1: 190.84 m 1: 190.84 m + 30 cm
5038 ' 9 acre @key{RET} Q u s ' $+30 cm @key{RET}
5045 1: 191.14 m 1: 36536.3046 m^2 1: 365363046 cm^2
5053 Since units expressions are really just formulas, taking the square
5054 root of @samp{acre} is undefined. After all, @code{acre} might be an
5055 algebraic variable that you will someday assign a value. We use the
5056 ``units-simplify'' command to simplify the expression with variables
5057 being interpreted as unit names.
5059 In the final step, we have converted not to a particular unit, but to a
5060 units system. The ``cgs'' system uses centimeters instead of meters
5061 as its standard unit of length.
5063 There is a wide variety of units defined in the Calculator.
5067 1: 55 mph 1: 88.5139 kph 1: 88.5139 km / hr 1: 8.201407e-8 c
5070 ' 55 mph @key{RET} u c kph @key{RET} u c km/hr @key{RET} u c c @key{RET}
5075 We express a speed first in miles per hour, then in kilometers per
5076 hour, then again using a slightly more explicit notation, then
5077 finally in terms of fractions of the speed of light.
5079 Temperature conversions are a bit more tricky. There are two ways to
5080 interpret ``20 degrees Fahrenheit''---it could mean an actual
5081 temperature, or it could mean a change in temperature. For normal
5082 units there is no difference, but temperature units have an offset
5083 as well as a scale factor and so there must be two explicit commands
5088 1: 20 degF 1: 11.1111 degC 1: -20:3 degC 1: -6.666 degC
5091 ' 20 degF @key{RET} u c degC @key{RET} U u t degC @key{RET} c f
5096 First we convert a change of 20 degrees Fahrenheit into an equivalent
5097 change in degrees Celsius (or Centigrade). Then, we convert the
5098 absolute temperature 20 degrees Fahrenheit into Celsius. Since
5099 this comes out as an exact fraction, we then convert to floating-point
5100 for easier comparison with the other result.
5102 For simple unit conversions, you can put a plain number on the stack.
5103 Then @kbd{u c} and @kbd{u t} will prompt for both old and new units.
5104 When you use this method, you're responsible for remembering which
5105 numbers are in which units:
5109 1: 55 1: 88.5139 1: 8.201407e-8
5112 55 u c mph @key{RET} kph @key{RET} u c km/hr @key{RET} c @key{RET}
5116 To see a complete list of built-in units, type @kbd{u v}. Press
5117 @w{@kbd{M-# c}} again to re-enter the Calculator when you're done looking
5120 (@bullet{}) @strong{Exercise 13.} How many seconds are there really
5121 in a year? @xref{Types Answer 13, 13}. (@bullet{})
5123 @cindex Speed of light
5124 (@bullet{}) @strong{Exercise 14.} Supercomputer designs are limited by
5125 the speed of light (and of electricity, which is nearly as fast).
5126 Suppose a computer has a 4.1 ns (nanosecond) clock cycle, and its
5127 cabinet is one meter across. Is speed of light going to be a
5128 significant factor in its design? @xref{Types Answer 14, 14}. (@bullet{})
5130 (@bullet{}) @strong{Exercise 15.} Sam the Slug normally travels about
5131 five yards in an hour. He has obtained a supply of Power Pills; each
5132 Power Pill he eats doubles his speed. How many Power Pills can he
5133 swallow and still travel legally on most US highways?
5134 @xref{Types Answer 15, 15}. (@bullet{})
5136 @node Algebra Tutorial, Programming Tutorial, Types Tutorial, Tutorial
5137 @section Algebra and Calculus Tutorial
5140 This section shows how to use Calc's algebra facilities to solve
5141 equations, do simple calculus problems, and manipulate algebraic
5145 * Basic Algebra Tutorial::
5146 * Rewrites Tutorial::
5149 @node Basic Algebra Tutorial, Rewrites Tutorial, Algebra Tutorial, Algebra Tutorial
5150 @subsection Basic Algebra
5153 If you enter a formula in Algebraic mode that refers to variables,
5154 the formula itself is pushed onto the stack. You can manipulate
5155 formulas as regular data objects.
5159 1: 2 x^2 - 6 1: 6 - 2 x^2 1: (6 - 2 x^2) (3 x^2 + y)
5162 ' 2x^2-6 @key{RET} n ' 3x^2+y @key{RET} *
5166 (@bullet{}) @strong{Exercise 1.} Do @kbd{' x @key{RET} Q 2 ^} and
5167 @kbd{' x @key{RET} 2 ^ Q} both wind up with the same result (@samp{x})?
5168 Why or why not? @xref{Algebra Answer 1, 1}. (@bullet{})
5170 There are also commands for doing common algebraic operations on
5171 formulas. Continuing with the formula from the last example,
5175 1: 18 x^2 + 6 y - 6 x^4 - 2 x^2 y 1: (18 - 2 y) x^2 - 6 x^4 + 6 y
5183 First we ``expand'' using the distributive law, then we ``collect''
5184 terms involving like powers of @expr{x}.
5186 Let's find the value of this expression when @expr{x} is 2 and @expr{y}
5191 1: 17 x^2 - 6 x^4 + 3 1: -25
5194 1:2 s l y @key{RET} 2 s l x @key{RET}
5199 The @kbd{s l} command means ``let''; it takes a number from the top of
5200 the stack and temporarily assigns it as the value of the variable
5201 you specify. It then evaluates (as if by the @kbd{=} key) the
5202 next expression on the stack. After this command, the variable goes
5203 back to its original value, if any.
5205 (An earlier exercise in this tutorial involved storing a value in the
5206 variable @code{x}; if this value is still there, you will have to
5207 unstore it with @kbd{s u x @key{RET}} before the above example will work
5210 @cindex Maximum of a function using Calculus
5211 Let's find the maximum value of our original expression when @expr{y}
5212 is one-half and @expr{x} ranges over all possible values. We can
5213 do this by taking the derivative with respect to @expr{x} and examining
5214 values of @expr{x} for which the derivative is zero. If the second
5215 derivative of the function at that value of @expr{x} is negative,
5216 the function has a local maximum there.
5220 1: 17 x^2 - 6 x^4 + 3 1: 34 x - 24 x^3
5223 U @key{DEL} s 1 a d x @key{RET} s 2
5228 Well, the derivative is clearly zero when @expr{x} is zero. To find
5229 the other root(s), let's divide through by @expr{x} and then solve:
5233 1: (34 x - 24 x^3) / x 1: 34 x / x - 24 x^3 / x 1: 34 - 24 x^2
5236 ' x @key{RET} / a x a s
5243 1: 34 - 24 x^2 = 0 1: x = 1.19023
5246 0 a = s 3 a S x @key{RET}
5251 Notice the use of @kbd{a s} to ``simplify'' the formula. When the
5252 default algebraic simplifications don't do enough, you can use
5253 @kbd{a s} to tell Calc to spend more time on the job.
5255 Now we compute the second derivative and plug in our values of @expr{x}:
5259 1: 1.19023 2: 1.19023 2: 1.19023
5260 . 1: 34 x - 24 x^3 1: 34 - 72 x^2
5263 a . r 2 a d x @key{RET} s 4
5268 (The @kbd{a .} command extracts just the righthand side of an equation.
5269 Another method would have been to use @kbd{v u} to unpack the equation
5270 @w{@samp{x = 1.19}} to @samp{x} and @samp{1.19}, then use @kbd{M-- M-2 @key{DEL}}
5271 to delete the @samp{x}.)
5275 2: 34 - 72 x^2 1: -68. 2: 34 - 72 x^2 1: 34
5279 @key{TAB} s l x @key{RET} U @key{DEL} 0 s l x @key{RET}
5284 The first of these second derivatives is negative, so we know the function
5285 has a maximum value at @expr{x = 1.19023}. (The function also has a
5286 local @emph{minimum} at @expr{x = 0}.)
5288 When we solved for @expr{x}, we got only one value even though
5289 @expr{34 - 24 x^2 = 0} is a quadratic equation that ought to have
5290 two solutions. The reason is that @w{@kbd{a S}} normally returns a
5291 single ``principal'' solution. If it needs to come up with an
5292 arbitrary sign (as occurs in the quadratic formula) it picks @expr{+}.
5293 If it needs an arbitrary integer, it picks zero. We can get a full
5294 solution by pressing @kbd{H} (the Hyperbolic flag) before @kbd{a S}.
5298 1: 34 - 24 x^2 = 0 1: x = 1.19023 s1 1: x = -1.19023
5301 r 3 H a S x @key{RET} s 5 1 n s l s1 @key{RET}
5306 Calc has invented the variable @samp{s1} to represent an unknown sign;
5307 it is supposed to be either @mathit{+1} or @mathit{-1}. Here we have used
5308 the ``let'' command to evaluate the expression when the sign is negative.
5309 If we plugged this into our second derivative we would get the same,
5310 negative, answer, so @expr{x = -1.19023} is also a maximum.
5312 To find the actual maximum value, we must plug our two values of @expr{x}
5313 into the original formula.
5317 2: 17 x^2 - 6 x^4 + 3 1: 24.08333 s1^2 - 12.04166 s1^4 + 3
5321 r 1 r 5 s l @key{RET}
5326 (Here we see another way to use @kbd{s l}; if its input is an equation
5327 with a variable on the lefthand side, then @kbd{s l} treats the equation
5328 like an assignment to that variable if you don't give a variable name.)
5330 It's clear that this will have the same value for either sign of
5331 @code{s1}, but let's work it out anyway, just for the exercise:
5335 2: [-1, 1] 1: [15.04166, 15.04166]
5336 1: 24.08333 s1^2 ... .
5339 [ 1 n , 1 ] @key{TAB} V M $ @key{RET}
5344 Here we have used a vector mapping operation to evaluate the function
5345 at several values of @samp{s1} at once. @kbd{V M $} is like @kbd{V M '}
5346 except that it takes the formula from the top of the stack. The
5347 formula is interpreted as a function to apply across the vector at the
5348 next-to-top stack level. Since a formula on the stack can't contain
5349 @samp{$} signs, Calc assumes the variables in the formula stand for
5350 different arguments. It prompts you for an @dfn{argument list}, giving
5351 the list of all variables in the formula in alphabetical order as the
5352 default list. In this case the default is @samp{(s1)}, which is just
5353 what we want so we simply press @key{RET} at the prompt.
5355 If there had been several different values, we could have used
5356 @w{@kbd{V R X}} to find the global maximum.
5358 Calc has a built-in @kbd{a P} command that solves an equation using
5359 @w{@kbd{H a S}} and returns a vector of all the solutions. It simply
5360 automates the job we just did by hand. Applied to our original
5361 cubic polynomial, it would produce the vector of solutions
5362 @expr{[1.19023, -1.19023, 0]}. (There is also an @kbd{a X} command
5363 which finds a local maximum of a function. It uses a numerical search
5364 method rather than examining the derivatives, and thus requires you
5365 to provide some kind of initial guess to show it where to look.)
5367 (@bullet{}) @strong{Exercise 2.} Given a vector of the roots of a
5368 polynomial (such as the output of an @kbd{a P} command), what
5369 sequence of commands would you use to reconstruct the original
5370 polynomial? (The answer will be unique to within a constant
5371 multiple; choose the solution where the leading coefficient is one.)
5372 @xref{Algebra Answer 2, 2}. (@bullet{})
5374 The @kbd{m s} command enables Symbolic mode, in which formulas
5375 like @samp{sqrt(5)} that can't be evaluated exactly are left in
5376 symbolic form rather than giving a floating-point approximate answer.
5377 Fraction mode (@kbd{m f}) is also useful when doing algebra.
5381 2: 34 x - 24 x^3 2: 34 x - 24 x^3
5382 1: 34 x - 24 x^3 1: [sqrt(51) / 6, sqrt(51) / -6, 0]
5385 r 2 @key{RET} m s m f a P x @key{RET}
5389 One more mode that makes reading formulas easier is Big mode.
5398 1: [-----, -----, 0]
5407 Here things like powers, square roots, and quotients and fractions
5408 are displayed in a two-dimensional pictorial form. Calc has other
5409 language modes as well, such as C mode, FORTRAN mode, @TeX{} mode
5414 2: 34*x - 24*pow(x, 3) 2: 34*x - 24*x**3
5415 1: @{sqrt(51) / 6, sqrt(51) / -6, 0@} 1: /sqrt(51) / 6, sqrt(51) / -6, 0/
5426 2: [@{\sqrt@{51@} \over 6@}, @{\sqrt@{51@} \over -6@}, 0]
5427 1: @{2 \over 3@} \sqrt@{5@}
5430 d T ' 2 \sqrt@{5@} \over 3 @key{RET}
5435 As you can see, language modes affect both entry and display of
5436 formulas. They affect such things as the names used for built-in
5437 functions, the set of arithmetic operators and their precedences,
5438 and notations for vectors and matrices.
5440 Notice that @samp{sqrt(51)} may cause problems with older
5441 implementations of C and FORTRAN, which would require something more
5442 like @samp{sqrt(51.0)}. It is always wise to check over the formulas
5443 produced by the various language modes to make sure they are fully
5446 Type @kbd{m s}, @kbd{m f}, and @kbd{d N} to reset these modes. (You
5447 may prefer to remain in Big mode, but all the examples in the tutorial
5448 are shown in normal mode.)
5450 @cindex Area under a curve
5451 What is the area under the portion of this curve from @expr{x = 1} to @expr{2}?
5452 This is simply the integral of the function:
5456 1: 17 x^2 - 6 x^4 + 3 1: 5.6666 x^3 - 1.2 x^5 + 3 x
5464 We want to evaluate this at our two values for @expr{x} and subtract.
5465 One way to do it is again with vector mapping and reduction:
5469 2: [2, 1] 1: [12.93333, 7.46666] 1: 5.46666
5470 1: 5.6666 x^3 ... . .
5472 [ 2 , 1 ] @key{TAB} V M $ @key{RET} V R -
5476 (@bullet{}) @strong{Exercise 3.} Find the integral from 1 to @expr{y}
5478 @texline @math{x \sin \pi x}
5479 @infoline @w{@expr{x sin(pi x)}}
5480 (where the sine is calculated in radians). Find the values of the
5481 integral for integers @expr{y} from 1 to 5. @xref{Algebra Answer 3,
5484 Calc's integrator can do many simple integrals symbolically, but many
5485 others are beyond its capabilities. Suppose we wish to find the area
5487 @texline @math{\sin x \ln x}
5488 @infoline @expr{sin(x) ln(x)}
5489 over the same range of @expr{x}. If you entered this formula and typed
5490 @kbd{a i x @key{RET}} (don't bother to try this), Calc would work for a
5491 long time but would be unable to find a solution. In fact, there is no
5492 closed-form solution to this integral. Now what do we do?
5494 @cindex Integration, numerical
5495 @cindex Numerical integration
5496 One approach would be to do the integral numerically. It is not hard
5497 to do this by hand using vector mapping and reduction. It is rather
5498 slow, though, since the sine and logarithm functions take a long time.
5499 We can save some time by reducing the working precision.
5503 3: 10 1: [1, 1.1, 1.2, ... , 1.8, 1.9]
5508 10 @key{RET} 1 @key{RET} .1 @key{RET} C-u v x
5513 (Note that we have used the extended version of @kbd{v x}; we could
5514 also have used plain @kbd{v x} as follows: @kbd{v x 10 @key{RET} 9 + .1 *}.)
5518 2: [1, 1.1, ... ] 1: [0., 0.084941, 0.16993, ... ]
5522 ' sin(x) ln(x) @key{RET} s 1 m r p 5 @key{RET} V M $ @key{RET}
5537 (If you got wildly different results, did you remember to switch
5540 Here we have divided the curve into ten segments of equal width;
5541 approximating these segments as rectangular boxes (i.e., assuming
5542 the curve is nearly flat at that resolution), we compute the areas
5543 of the boxes (height times width), then sum the areas. (It is
5544 faster to sum first, then multiply by the width, since the width
5545 is the same for every box.)
5547 The true value of this integral turns out to be about 0.374, so
5548 we're not doing too well. Let's try another approach.
5552 1: sin(x) ln(x) 1: 0.84147 x - 0.84147 + 0.11957 (x - 1)^2 - ...
5555 r 1 a t x=1 @key{RET} 4 @key{RET}
5560 Here we have computed the Taylor series expansion of the function
5561 about the point @expr{x=1}. We can now integrate this polynomial
5562 approximation, since polynomials are easy to integrate.
5566 1: 0.42074 x^2 + ... 1: [-0.0446, -0.42073] 1: 0.3761
5569 a i x @key{RET} [ 2 , 1 ] @key{TAB} V M $ @key{RET} V R -
5574 Better! By increasing the precision and/or asking for more terms
5575 in the Taylor series, we can get a result as accurate as we like.
5576 (Taylor series converge better away from singularities in the
5577 function such as the one at @code{ln(0)}, so it would also help to
5578 expand the series about the points @expr{x=2} or @expr{x=1.5} instead
5581 @cindex Simpson's rule
5582 @cindex Integration by Simpson's rule
5583 (@bullet{}) @strong{Exercise 4.} Our first method approximated the
5584 curve by stairsteps of width 0.1; the total area was then the sum
5585 of the areas of the rectangles under these stairsteps. Our second
5586 method approximated the function by a polynomial, which turned out
5587 to be a better approximation than stairsteps. A third method is
5588 @dfn{Simpson's rule}, which is like the stairstep method except
5589 that the steps are not required to be flat. Simpson's rule boils
5590 down to the formula,
5594 (h/3) * (f(a) + 4 f(a+h) + 2 f(a+2h) + 4 f(a+3h) + ...
5595 + 2 f(a+(n-2)*h) + 4 f(a+(n-1)*h) + f(a+n*h))
5602 \qquad {h \over 3} (f(a) + 4 f(a+h) + 2 f(a+2h) + 4 f(a+3h) + \cdots
5603 \hfill \cr \hfill {} + 2 f(a+(n-2)h) + 4 f(a+(n-1)h) + f(a+n h)) \qquad
5609 where @expr{n} (which must be even) is the number of slices and @expr{h}
5610 is the width of each slice. These are 10 and 0.1 in our example.
5611 For reference, here is the corresponding formula for the stairstep
5616 h * (f(a) + f(a+h) + f(a+2h) + f(a+3h) + ...
5617 + f(a+(n-2)*h) + f(a+(n-1)*h))
5623 $$ h (f(a) + f(a+h) + f(a+2h) + f(a+3h) + \cdots
5624 + f(a+(n-2)h) + f(a+(n-1)h)) $$
5628 Compute the integral from 1 to 2 of
5629 @texline @math{\sin x \ln x}
5630 @infoline @expr{sin(x) ln(x)}
5631 using Simpson's rule with 10 slices.
5632 @xref{Algebra Answer 4, 4}. (@bullet{})
5634 Calc has a built-in @kbd{a I} command for doing numerical integration.
5635 It uses @dfn{Romberg's method}, which is a more sophisticated cousin
5636 of Simpson's rule. In particular, it knows how to keep refining the
5637 result until the current precision is satisfied.
5639 @c [fix-ref Selecting Sub-Formulas]
5640 Aside from the commands we've seen so far, Calc also provides a
5641 large set of commands for operating on parts of formulas. You
5642 indicate the desired sub-formula by placing the cursor on any part
5643 of the formula before giving a @dfn{selection} command. Selections won't
5644 be covered in the tutorial; @pxref{Selecting Subformulas}, for
5645 details and examples.
5647 @c hard exercise: simplify (2^(n r) - 2^(r*(n - 1))) / (2^r - 1) 2^(n - 1)
5648 @c to 2^((n-1)*(r-1)).
5650 @node Rewrites Tutorial, , Basic Algebra Tutorial, Algebra Tutorial
5651 @subsection Rewrite Rules
5654 No matter how many built-in commands Calc provided for doing algebra,
5655 there would always be something you wanted to do that Calc didn't have
5656 in its repertoire. So Calc also provides a @dfn{rewrite rule} system
5657 that you can use to define your own algebraic manipulations.
5659 Suppose we want to simplify this trigonometric formula:
5663 1: 1 / cos(x) - sin(x) tan(x)
5666 ' 1/cos(x) - sin(x) tan(x) @key{RET} s 1
5671 If we were simplifying this by hand, we'd probably replace the
5672 @samp{tan} with a @samp{sin/cos} first, then combine over a common
5673 denominator. There is no Calc command to do the former; the @kbd{a n}
5674 algebra command will do the latter but we'll do both with rewrite
5675 rules just for practice.
5677 Rewrite rules are written with the @samp{:=} symbol.
5681 1: 1 / cos(x) - sin(x)^2 / cos(x)
5684 a r tan(a) := sin(a)/cos(a) @key{RET}
5689 (The ``assignment operator'' @samp{:=} has several uses in Calc. All
5690 by itself the formula @samp{tan(a) := sin(a)/cos(a)} doesn't do anything,
5691 but when it is given to the @kbd{a r} command, that command interprets
5692 it as a rewrite rule.)
5694 The lefthand side, @samp{tan(a)}, is called the @dfn{pattern} of the
5695 rewrite rule. Calc searches the formula on the stack for parts that
5696 match the pattern. Variables in a rewrite pattern are called
5697 @dfn{meta-variables}, and when matching the pattern each meta-variable
5698 can match any sub-formula. Here, the meta-variable @samp{a} matched
5699 the actual variable @samp{x}.
5701 When the pattern part of a rewrite rule matches a part of the formula,
5702 that part is replaced by the righthand side with all the meta-variables
5703 substituted with the things they matched. So the result is
5704 @samp{sin(x) / cos(x)}. Calc's normal algebraic simplifications then
5705 mix this in with the rest of the original formula.
5707 To merge over a common denominator, we can use another simple rule:
5711 1: (1 - sin(x)^2) / cos(x)
5714 a r a/x + b/x := (a+b)/x @key{RET}
5718 This rule points out several interesting features of rewrite patterns.
5719 First, if a meta-variable appears several times in a pattern, it must
5720 match the same thing everywhere. This rule detects common denominators
5721 because the same meta-variable @samp{x} is used in both of the
5724 Second, meta-variable names are independent from variables in the
5725 target formula. Notice that the meta-variable @samp{x} here matches
5726 the subformula @samp{cos(x)}; Calc never confuses the two meanings of
5729 And third, rewrite patterns know a little bit about the algebraic
5730 properties of formulas. The pattern called for a sum of two quotients;
5731 Calc was able to match a difference of two quotients by matching
5732 @samp{a = 1}, @samp{b = -sin(x)^2}, and @samp{x = cos(x)}.
5734 @c [fix-ref Algebraic Properties of Rewrite Rules]
5735 We could just as easily have written @samp{a/x - b/x := (a-b)/x} for
5736 the rule. It would have worked just the same in all cases. (If we
5737 really wanted the rule to apply only to @samp{+} or only to @samp{-},
5738 we could have used the @code{plain} symbol. @xref{Algebraic Properties
5739 of Rewrite Rules}, for some examples of this.)
5741 One more rewrite will complete the job. We want to use the identity
5742 @samp{sin(x)^2 + cos(x)^2 = 1}, but of course we must first rearrange
5743 the identity in a way that matches our formula. The obvious rule
5744 would be @samp{@w{1 - sin(x)^2} := cos(x)^2}, but a little thought shows
5745 that the rule @samp{sin(x)^2 := 1 - cos(x)^2} will also work. The
5746 latter rule has a more general pattern so it will work in many other
5751 1: (1 + cos(x)^2 - 1) / cos(x) 1: cos(x)
5754 a r sin(x)^2 := 1 - cos(x)^2 @key{RET} a s
5758 You may ask, what's the point of using the most general rule if you
5759 have to type it in every time anyway? The answer is that Calc allows
5760 you to store a rewrite rule in a variable, then give the variable
5761 name in the @kbd{a r} command. In fact, this is the preferred way to
5762 use rewrites. For one, if you need a rule once you'll most likely
5763 need it again later. Also, if the rule doesn't work quite right you
5764 can simply Undo, edit the variable, and run the rule again without
5765 having to retype it.
5769 ' tan(x) := sin(x)/cos(x) @key{RET} s t tsc @key{RET}
5770 ' a/x + b/x := (a+b)/x @key{RET} s t merge @key{RET}
5771 ' sin(x)^2 := 1 - cos(x)^2 @key{RET} s t sinsqr @key{RET}
5773 1: 1 / cos(x) - sin(x) tan(x) 1: cos(x)
5776 r 1 a r tsc @key{RET} a r merge @key{RET} a r sinsqr @key{RET} a s
5780 To edit a variable, type @kbd{s e} and the variable name, use regular
5781 Emacs editing commands as necessary, then type @kbd{C-c C-c} to store
5782 the edited value back into the variable.
5783 You can also use @w{@kbd{s e}} to create a new variable if you wish.
5785 Notice that the first time you use each rule, Calc puts up a ``compiling''
5786 message briefly. The pattern matcher converts rules into a special
5787 optimized pattern-matching language rather than using them directly.
5788 This allows @kbd{a r} to apply even rather complicated rules very
5789 efficiently. If the rule is stored in a variable, Calc compiles it
5790 only once and stores the compiled form along with the variable. That's
5791 another good reason to store your rules in variables rather than
5792 entering them on the fly.
5794 (@bullet{}) @strong{Exercise 1.} Type @kbd{m s} to get Symbolic
5795 mode, then enter the formula @samp{@w{(2 + sqrt(2))} / @w{(1 + sqrt(2))}}.
5796 Using a rewrite rule, simplify this formula by multiplying the top and
5797 bottom by the conjugate @w{@samp{1 - sqrt(2)}}. The result will have
5798 to be expanded by the distributive law; do this with another
5799 rewrite. @xref{Rewrites Answer 1, 1}. (@bullet{})
5801 The @kbd{a r} command can also accept a vector of rewrite rules, or
5802 a variable containing a vector of rules.
5806 1: [tsc, merge, sinsqr] 1: [tan(x) := sin(x) / cos(x), ... ]
5809 ' [tsc,merge,sinsqr] @key{RET} =
5816 1: 1 / cos(x) - sin(x) tan(x) 1: cos(x)
5819 s t trig @key{RET} r 1 a r trig @key{RET} a s
5823 @c [fix-ref Nested Formulas with Rewrite Rules]
5824 Calc tries all the rules you give against all parts of the formula,
5825 repeating until no further change is possible. (The exact order in
5826 which things are tried is rather complex, but for simple rules like
5827 the ones we've used here the order doesn't really matter.
5828 @xref{Nested Formulas with Rewrite Rules}.)
5830 Calc actually repeats only up to 100 times, just in case your rule set
5831 has gotten into an infinite loop. You can give a numeric prefix argument
5832 to @kbd{a r} to specify any limit. In particular, @kbd{M-1 a r} does
5833 only one rewrite at a time.
5837 1: 1 / cos(x) - sin(x)^2 / cos(x) 1: (1 - sin(x)^2) / cos(x)
5840 r 1 M-1 a r trig @key{RET} M-1 a r trig @key{RET}
5844 You can type @kbd{M-0 a r} if you want no limit at all on the number
5845 of rewrites that occur.
5847 Rewrite rules can also be @dfn{conditional}. Simply follow the rule
5848 with a @samp{::} symbol and the desired condition. For example,
5852 1: exp(2 pi i) + exp(3 pi i) + exp(4 pi i)
5855 ' exp(2 pi i) + exp(3 pi i) + exp(4 pi i) @key{RET}
5862 1: 1 + exp(3 pi i) + 1
5865 a r exp(k pi i) := 1 :: k % 2 = 0 @key{RET}
5870 (Recall, @samp{k % 2} is the remainder from dividing @samp{k} by 2,
5871 which will be zero only when @samp{k} is an even integer.)
5873 An interesting point is that the variables @samp{pi} and @samp{i}
5874 were matched literally rather than acting as meta-variables.
5875 This is because they are special-constant variables. The special
5876 constants @samp{e}, @samp{phi}, and so on also match literally.
5877 A common error with rewrite
5878 rules is to write, say, @samp{f(a,b,c,d,e) := g(a+b+c+d+e)}, expecting
5879 to match any @samp{f} with five arguments but in fact matching
5880 only when the fifth argument is literally @samp{e}!
5882 @cindex Fibonacci numbers
5887 Rewrite rules provide an interesting way to define your own functions.
5888 Suppose we want to define @samp{fib(n)} to produce the @var{n}th
5889 Fibonacci number. The first two Fibonacci numbers are each 1;
5890 later numbers are formed by summing the two preceding numbers in
5891 the sequence. This is easy to express in a set of three rules:
5895 ' [fib(1) := 1, fib(2) := 1, fib(n) := fib(n-1) + fib(n-2)] @key{RET} s t fib
5900 ' fib(7) @key{RET} a r fib @key{RET}
5904 One thing that is guaranteed about the order that rewrites are tried
5905 is that, for any given subformula, earlier rules in the rule set will
5906 be tried for that subformula before later ones. So even though the
5907 first and third rules both match @samp{fib(1)}, we know the first will
5908 be used preferentially.
5910 This rule set has one dangerous bug: Suppose we apply it to the
5911 formula @samp{fib(x)}? (Don't actually try this.) The third rule
5912 will match @samp{fib(x)} and replace it with @w{@samp{fib(x-1) + fib(x-2)}}.
5913 Each of these will then be replaced to get @samp{fib(x-2) + 2 fib(x-3) +
5914 fib(x-4)}, and so on, expanding forever. What we really want is to apply
5915 the third rule only when @samp{n} is an integer greater than two. Type
5916 @w{@kbd{s e fib @key{RET}}}, then edit the third rule to:
5919 fib(n) := fib(n-1) + fib(n-2) :: integer(n) :: n > 2
5927 1: fib(6) + fib(x) + fib(0) 1: 8 + fib(x) + fib(0)
5930 ' fib(6)+fib(x)+fib(0) @key{RET} a r fib @key{RET}
5935 We've created a new function, @code{fib}, and a new command,
5936 @w{@kbd{a r fib @key{RET}}}, which means ``evaluate all @code{fib} calls in
5937 this formula.'' To make things easier still, we can tell Calc to
5938 apply these rules automatically by storing them in the special
5939 variable @code{EvalRules}.
5943 1: [fib(1) := ...] . 1: [8, 13]
5946 s r fib @key{RET} s t EvalRules @key{RET} ' [fib(6), fib(7)] @key{RET}
5950 It turns out that this rule set has the problem that it does far
5951 more work than it needs to when @samp{n} is large. Consider the
5952 first few steps of the computation of @samp{fib(6)}:
5958 fib(4) + fib(3) + fib(3) + fib(2) =
5959 fib(3) + fib(2) + fib(2) + fib(1) + fib(2) + fib(1) + 1 = ...
5964 Note that @samp{fib(3)} appears three times here. Unless Calc's
5965 algebraic simplifier notices the multiple @samp{fib(3)}s and combines
5966 them (and, as it happens, it doesn't), this rule set does lots of
5967 needless recomputation. To cure the problem, type @code{s e EvalRules}
5968 to edit the rules (or just @kbd{s E}, a shorthand command for editing
5969 @code{EvalRules}) and add another condition:
5972 fib(n) := fib(n-1) + fib(n-2) :: integer(n) :: n > 2 :: remember
5976 If a @samp{:: remember} condition appears anywhere in a rule, then if
5977 that rule succeeds Calc will add another rule that describes that match
5978 to the front of the rule set. (Remembering works in any rule set, but
5979 for technical reasons it is most effective in @code{EvalRules}.) For
5980 example, if the rule rewrites @samp{fib(7)} to something that evaluates
5981 to 13, then the rule @samp{fib(7) := 13} will be added to the rule set.
5983 Type @kbd{' fib(8) @key{RET}} to compute the eighth Fibonacci number, then
5984 type @kbd{s E} again to see what has happened to the rule set.
5986 With the @code{remember} feature, our rule set can now compute
5987 @samp{fib(@var{n})} in just @var{n} steps. In the process it builds
5988 up a table of all Fibonacci numbers up to @var{n}. After we have
5989 computed the result for a particular @var{n}, we can get it back
5990 (and the results for all smaller @var{n}) later in just one step.
5992 All Calc operations will run somewhat slower whenever @code{EvalRules}
5993 contains any rules. You should type @kbd{s u EvalRules @key{RET}} now to
5994 un-store the variable.
5996 (@bullet{}) @strong{Exercise 2.} Sometimes it is possible to reformulate
5997 a problem to reduce the amount of recursion necessary to solve it.
5998 Create a rule that, in about @var{n} simple steps and without recourse
5999 to the @code{remember} option, replaces @samp{fib(@var{n}, 1, 1)} with
6000 @samp{fib(1, @var{x}, @var{y})} where @var{x} and @var{y} are the
6001 @var{n}th and @var{n+1}st Fibonacci numbers, respectively. This rule is
6002 rather clunky to use, so add a couple more rules to make the ``user
6003 interface'' the same as for our first version: enter @samp{fib(@var{n})},
6004 get back a plain number. @xref{Rewrites Answer 2, 2}. (@bullet{})
6006 There are many more things that rewrites can do. For example, there
6007 are @samp{&&&} and @samp{|||} pattern operators that create ``and''
6008 and ``or'' combinations of rules. As one really simple example, we
6009 could combine our first two Fibonacci rules thusly:
6012 [fib(1 ||| 2) := 1, fib(n) := ... ]
6016 That means ``@code{fib} of something matching either 1 or 2 rewrites
6019 You can also make meta-variables optional by enclosing them in @code{opt}.
6020 For example, the pattern @samp{a + b x} matches @samp{2 + 3 x} but not
6021 @samp{2 + x} or @samp{3 x} or @samp{x}. The pattern @samp{opt(a) + opt(b) x}
6022 matches all of these forms, filling in a default of zero for @samp{a}
6023 and one for @samp{b}.
6025 (@bullet{}) @strong{Exercise 3.} Your friend Joe had @samp{2 + 3 x}
6026 on the stack and tried to use the rule
6027 @samp{opt(a) + opt(b) x := f(a, b, x)}. What happened?
6028 @xref{Rewrites Answer 3, 3}. (@bullet{})
6030 (@bullet{}) @strong{Exercise 4.} Starting with a positive integer @expr{a},
6031 divide @expr{a} by two if it is even, otherwise compute @expr{3 a + 1}.
6032 Now repeat this step over and over. A famous unproved conjecture
6033 is that for any starting @expr{a}, the sequence always eventually
6034 reaches 1. Given the formula @samp{seq(@var{a}, 0)}, write a set of
6035 rules that convert this into @samp{seq(1, @var{n})} where @var{n}
6036 is the number of steps it took the sequence to reach the value 1.
6037 Now enhance the rules to accept @samp{seq(@var{a})} as a starting
6038 configuration, and to stop with just the number @var{n} by itself.
6039 Now make the result be a vector of values in the sequence, from @var{a}
6040 to 1. (The formula @samp{@var{x}|@var{y}} appends the vectors @var{x}
6041 and @var{y}.) For example, rewriting @samp{seq(6)} should yield the
6042 vector @expr{[6, 3, 10, 5, 16, 8, 4, 2, 1]}.
6043 @xref{Rewrites Answer 4, 4}. (@bullet{})
6045 (@bullet{}) @strong{Exercise 5.} Define, using rewrite rules, a function
6046 @samp{nterms(@var{x})} that returns the number of terms in the sum
6047 @var{x}, or 1 if @var{x} is not a sum. (A @dfn{sum} for our purposes
6048 is one or more non-sum terms separated by @samp{+} or @samp{-} signs,
6049 so that @expr{2 - 3 (x + y) + x y} is a sum of three terms.)
6050 @xref{Rewrites Answer 5, 5}. (@bullet{})
6052 (@bullet{}) @strong{Exercise 6.} A Taylor series for a function is an
6053 infinite series that exactly equals the value of that function at
6054 values of @expr{x} near zero.
6058 cos(x) = 1 - x^2 / 2! + x^4 / 4! - x^6 / 6! + ...
6064 $$ \cos x = 1 - {x^2 \over 2!} + {x^4 \over 4!} - {x^6 \over 6!} + \cdots $$
6068 The @kbd{a t} command produces a @dfn{truncated Taylor series} which
6069 is obtained by dropping all the terms higher than, say, @expr{x^2}.
6070 Calc represents the truncated Taylor series as a polynomial in @expr{x}.
6071 Mathematicians often write a truncated series using a ``big-O'' notation
6072 that records what was the lowest term that was truncated.
6076 cos(x) = 1 - x^2 / 2! + O(x^3)
6082 $$ \cos x = 1 - {x^2 \over 2!} + O(x^3) $$
6087 The meaning of @expr{O(x^3)} is ``a quantity which is negligibly small
6088 if @expr{x^3} is considered negligibly small as @expr{x} goes to zero.''
6090 The exercise is to create rewrite rules that simplify sums and products of
6091 power series represented as @samp{@var{polynomial} + O(@var{var}^@var{n})}.
6092 For example, given @samp{1 - x^2 / 2 + O(x^3)} and @samp{x - x^3 / 6 + O(x^4)}
6093 on the stack, we want to be able to type @kbd{*} and get the result
6094 @samp{x - 2:3 x^3 + O(x^4)}. Don't worry if the terms of the sum are
6095 rearranged or if @kbd{a s} needs to be typed after rewriting. (This one
6096 is rather tricky; the solution at the end of this chapter uses 6 rewrite
6097 rules. Hint: The @samp{constant(x)} condition tests whether @samp{x} is
6098 a number.) @xref{Rewrites Answer 6, 6}. (@bullet{})
6100 Just for kicks, try adding the rule @code{2+3 := 6} to @code{EvalRules}.
6101 What happens? (Be sure to remove this rule afterward, or you might get
6102 a nasty surprise when you use Calc to balance your checkbook!)
6104 @xref{Rewrite Rules}, for the whole story on rewrite rules.
6106 @node Programming Tutorial, Answers to Exercises, Algebra Tutorial, Tutorial
6107 @section Programming Tutorial
6110 The Calculator is written entirely in Emacs Lisp, a highly extensible
6111 language. If you know Lisp, you can program the Calculator to do
6112 anything you like. Rewrite rules also work as a powerful programming
6113 system. But Lisp and rewrite rules take a while to master, and often
6114 all you want to do is define a new function or repeat a command a few
6115 times. Calc has features that allow you to do these things easily.
6117 One very limited form of programming is defining your own functions.
6118 Calc's @kbd{Z F} command allows you to define a function name and
6119 key sequence to correspond to any formula. Programming commands use
6120 the shift-@kbd{Z} prefix; the user commands they create use the lower
6121 case @kbd{z} prefix.
6125 1: 1 + x + x^2 / 2 + x^3 / 6 1: 1 + x + x^2 / 2 + x^3 / 6
6128 ' 1 + x + x^2/2! + x^3/3! @key{RET} Z F e myexp @key{RET} @key{RET} @key{RET} y
6132 This polynomial is a Taylor series approximation to @samp{exp(x)}.
6133 The @kbd{Z F} command asks a number of questions. The above answers
6134 say that the key sequence for our function should be @kbd{z e}; the
6135 @kbd{M-x} equivalent should be @code{calc-myexp}; the name of the
6136 function in algebraic formulas should also be @code{myexp}; the
6137 default argument list @samp{(x)} is acceptable; and finally @kbd{y}
6138 answers the question ``leave it in symbolic form for non-constant
6143 1: 1.3495 2: 1.3495 3: 1.3495
6144 . 1: 1.34986 2: 1.34986
6148 .3 z e .3 E ' a+1 @key{RET} z e
6153 First we call our new @code{exp} approximation with 0.3 as an
6154 argument, and compare it with the true @code{exp} function. Then
6155 we note that, as requested, if we try to give @kbd{z e} an
6156 argument that isn't a plain number, it leaves the @code{myexp}
6157 function call in symbolic form. If we had answered @kbd{n} to the
6158 final question, @samp{myexp(a + 1)} would have evaluated by plugging
6159 in @samp{a + 1} for @samp{x} in the defining formula.
6161 @cindex Sine integral Si(x)
6166 (@bullet{}) @strong{Exercise 1.} The ``sine integral'' function
6167 @texline @math{{\rm Si}(x)}
6168 @infoline @expr{Si(x)}
6169 is defined as the integral of @samp{sin(t)/t} for
6170 @expr{t = 0} to @expr{x} in radians. (It was invented because this
6171 integral has no solution in terms of basic functions; if you give it
6172 to Calc's @kbd{a i} command, it will ponder it for a long time and then
6173 give up.) We can use the numerical integration command, however,
6174 which in algebraic notation is written like @samp{ninteg(f(t), t, 0, x)}
6175 with any integrand @samp{f(t)}. Define a @kbd{z s} command and
6176 @code{Si} function that implement this. You will need to edit the
6177 default argument list a bit. As a test, @samp{Si(1)} should return
6178 0.946083. (If you don't get this answer, you might want to check that
6179 Calc is in Radians mode. Also, @code{ninteg} will run a lot faster if
6180 you reduce the precision to, say, six digits beforehand.)
6181 @xref{Programming Answer 1, 1}. (@bullet{})
6183 The simplest way to do real ``programming'' of Emacs is to define a
6184 @dfn{keyboard macro}. A keyboard macro is simply a sequence of
6185 keystrokes which Emacs has stored away and can play back on demand.
6186 For example, if you find yourself typing @kbd{H a S x @key{RET}} often,
6187 you may wish to program a keyboard macro to type this for you.
6191 1: y = sqrt(x) 1: x = y^2
6194 ' y=sqrt(x) @key{RET} C-x ( H a S x @key{RET} C-x )
6196 1: y = cos(x) 1: x = s1 arccos(y) + 2 pi n1
6199 ' y=cos(x) @key{RET} X
6204 When you type @kbd{C-x (}, Emacs begins recording. But it is also
6205 still ready to execute your keystrokes, so you're really ``training''
6206 Emacs by walking it through the procedure once. When you type
6207 @w{@kbd{C-x )}}, the macro is recorded. You can now type @kbd{X} to
6208 re-execute the same keystrokes.
6210 You can give a name to your macro by typing @kbd{Z K}.
6214 1: . 1: y = x^4 1: x = s2 sqrt(s1 sqrt(y))
6217 Z K x @key{RET} ' y=x^4 @key{RET} z x
6222 Notice that we use shift-@kbd{Z} to define the command, and lower-case
6223 @kbd{z} to call it up.
6225 Keyboard macros can call other macros.
6229 1: abs(x) 1: x = s1 y 1: 2 / x 1: x = 2 / y
6232 ' abs(x) @key{RET} C-x ( ' y @key{RET} a = z x C-x ) ' 2/x @key{RET} X
6236 (@bullet{}) @strong{Exercise 2.} Define a keyboard macro to negate
6237 the item in level 3 of the stack, without disturbing the rest of
6238 the stack. @xref{Programming Answer 2, 2}. (@bullet{})
6240 (@bullet{}) @strong{Exercise 3.} Define keyboard macros to compute
6241 the following functions:
6246 @texline @math{\displaystyle{\sin x \over x}},
6247 @infoline @expr{sin(x) / x},
6248 where @expr{x} is the number on the top of the stack.
6251 Compute the base-@expr{b} logarithm, just like the @kbd{B} key except
6252 the arguments are taken in the opposite order.
6255 Produce a vector of integers from 1 to the integer on the top of
6259 @xref{Programming Answer 3, 3}. (@bullet{})
6261 (@bullet{}) @strong{Exercise 4.} Define a keyboard macro to compute
6262 the average (mean) value of a list of numbers.
6263 @xref{Programming Answer 4, 4}. (@bullet{})
6265 In many programs, some of the steps must execute several times.
6266 Calc has @dfn{looping} commands that allow this. Loops are useful
6267 inside keyboard macros, but actually work at any time.
6271 1: x^6 2: x^6 1: 360 x^2
6275 ' x^6 @key{RET} 4 Z < a d x @key{RET} Z >
6280 Here we have computed the fourth derivative of @expr{x^6} by
6281 enclosing a derivative command in a ``repeat loop'' structure.
6282 This structure pops a repeat count from the stack, then
6283 executes the body of the loop that many times.
6285 If you make a mistake while entering the body of the loop,
6286 type @w{@kbd{Z C-g}} to cancel the loop command.
6288 @cindex Fibonacci numbers
6289 Here's another example:
6298 1 @key{RET} @key{RET} 20 Z < @key{TAB} C-j + Z >
6303 The numbers in levels 2 and 1 should be the 21st and 22nd Fibonacci
6304 numbers, respectively. (To see what's going on, try a few repetitions
6305 of the loop body by hand; @kbd{C-j}, also on the Line-Feed or @key{LFD}
6306 key if you have one, makes a copy of the number in level 2.)
6308 @cindex Golden ratio
6309 @cindex Phi, golden ratio
6310 A fascinating property of the Fibonacci numbers is that the @expr{n}th
6311 Fibonacci number can be found directly by computing
6312 @texline @math{\phi^n / \sqrt{5}}
6313 @infoline @expr{phi^n / sqrt(5)}
6314 and then rounding to the nearest integer, where
6315 @texline @math{\phi} (``phi''),
6316 @infoline @expr{phi},
6317 the ``golden ratio,'' is
6318 @texline @math{(1 + \sqrt{5}) / 2}.
6319 @infoline @expr{(1 + sqrt(5)) / 2}.
6320 (For convenience, this constant is available from the @code{phi}
6321 variable, or the @kbd{I H P} command.)
6325 1: 1.61803 1: 24476.0000409 1: 10945.9999817 1: 10946
6332 @cindex Continued fractions
6333 (@bullet{}) @strong{Exercise 5.} The @dfn{continued fraction}
6335 @texline @math{\phi}
6336 @infoline @expr{phi}
6338 @texline @math{1 + 1/(1 + 1/(1 + 1/( \ldots )))}.
6339 @infoline @expr{1 + 1/(1 + 1/(1 + 1/( ...@: )))}.
6340 We can compute an approximate value by carrying this however far
6341 and then replacing the innermost
6342 @texline @math{1/( \ldots )}
6343 @infoline @expr{1/( ...@: )}
6345 @texline @math{\phi}
6346 @infoline @expr{phi}
6347 using a twenty-term continued fraction.
6348 @xref{Programming Answer 5, 5}. (@bullet{})
6350 (@bullet{}) @strong{Exercise 6.} Linear recurrences like the one for
6351 Fibonacci numbers can be expressed in terms of matrices. Given a
6352 vector @w{@expr{[a, b]}} determine a matrix which, when multiplied by this
6353 vector, produces the vector @expr{[b, c]}, where @expr{a}, @expr{b} and
6354 @expr{c} are three successive Fibonacci numbers. Now write a program
6355 that, given an integer @expr{n}, computes the @expr{n}th Fibonacci number
6356 using matrix arithmetic. @xref{Programming Answer 6, 6}. (@bullet{})
6358 @cindex Harmonic numbers
6359 A more sophisticated kind of loop is the @dfn{for} loop. Suppose
6360 we wish to compute the 20th ``harmonic'' number, which is equal to
6361 the sum of the reciprocals of the integers from 1 to 20.
6370 0 @key{RET} 1 @key{RET} 20 Z ( & + 1 Z )
6375 The ``for'' loop pops two numbers, the lower and upper limits, then
6376 repeats the body of the loop as an internal counter increases from
6377 the lower limit to the upper one. Just before executing the loop
6378 body, it pushes the current loop counter. When the loop body
6379 finishes, it pops the ``step,'' i.e., the amount by which to
6380 increment the loop counter. As you can see, our loop always
6383 This harmonic number function uses the stack to hold the running
6384 total as well as for the various loop housekeeping functions. If
6385 you find this disorienting, you can sum in a variable instead:
6389 1: 0 2: 1 . 1: 3.597739
6393 0 t 7 1 @key{RET} 20 Z ( & s + 7 1 Z ) r 7
6398 The @kbd{s +} command adds the top-of-stack into the value in a
6399 variable (and removes that value from the stack).
6401 It's worth noting that many jobs that call for a ``for'' loop can
6402 also be done more easily by Calc's high-level operations. Two
6403 other ways to compute harmonic numbers are to use vector mapping
6404 and reduction (@kbd{v x 20}, then @w{@kbd{V M &}}, then @kbd{V R +}),
6405 or to use the summation command @kbd{a +}. Both of these are
6406 probably easier than using loops. However, there are some
6407 situations where loops really are the way to go:
6409 (@bullet{}) @strong{Exercise 7.} Use a ``for'' loop to find the first
6410 harmonic number which is greater than 4.0.
6411 @xref{Programming Answer 7, 7}. (@bullet{})
6413 Of course, if we're going to be using variables in our programs,
6414 we have to worry about the programs clobbering values that the
6415 caller was keeping in those same variables. This is easy to
6420 . 1: 0.6667 1: 0.6667 3: 0.6667
6425 Z ` p 4 @key{RET} 2 @key{RET} 3 / s 7 s s a @key{RET} Z ' r 7 s r a @key{RET}
6430 When we type @kbd{Z `} (that's a back-quote character), Calc saves
6431 its mode settings and the contents of the ten ``quick variables''
6432 for later reference. When we type @kbd{Z '} (that's an apostrophe
6433 now), Calc restores those saved values. Thus the @kbd{p 4} and
6434 @kbd{s 7} commands have no effect outside this sequence. Wrapping
6435 this around the body of a keyboard macro ensures that it doesn't
6436 interfere with what the user of the macro was doing. Notice that
6437 the contents of the stack, and the values of named variables,
6438 survive past the @kbd{Z '} command.
6440 @cindex Bernoulli numbers, approximate
6441 The @dfn{Bernoulli numbers} are a sequence with the interesting
6442 property that all of the odd Bernoulli numbers are zero, and the
6443 even ones, while difficult to compute, can be roughly approximated
6445 @texline @math{\displaystyle{2 n! \over (2 \pi)^n}}.
6446 @infoline @expr{2 n!@: / (2 pi)^n}.
6447 Let's write a keyboard macro to compute (approximate) Bernoulli numbers.
6448 (Calc has a command, @kbd{k b}, to compute exact Bernoulli numbers, but
6449 this command is very slow for large @expr{n} since the higher Bernoulli
6450 numbers are very large fractions.)
6457 10 C-x ( @key{RET} 2 % Z [ @key{DEL} 0 Z : ' 2 $! / (2 pi)^$ @key{RET} = Z ] C-x )
6462 You can read @kbd{Z [} as ``then,'' @kbd{Z :} as ``else,'' and
6463 @kbd{Z ]} as ``end-if.'' There is no need for an explicit ``if''
6464 command. For the purposes of @w{@kbd{Z [}}, the condition is ``true''
6465 if the value it pops from the stack is a nonzero number, or ``false''
6466 if it pops zero or something that is not a number (like a formula).
6467 Here we take our integer argument modulo 2; this will be nonzero
6468 if we're asking for an odd Bernoulli number.
6470 The actual tenth Bernoulli number is @expr{5/66}.
6474 3: 0.0756823 1: 0 1: 0.25305 1: 0 1: 1.16659
6479 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
6483 Just to exercise loops a bit more, let's compute a table of even
6488 3: [] 1: [0.10132, 0.03079, 0.02340, 0.033197, ...]
6493 [ ] 2 @key{RET} 30 Z ( X | 2 Z )
6498 The vertical-bar @kbd{|} is the vector-concatenation command. When
6499 we execute it, the list we are building will be in stack level 2
6500 (initially this is an empty list), and the next Bernoulli number
6501 will be in level 1. The effect is to append the Bernoulli number
6502 onto the end of the list. (To create a table of exact fractional
6503 Bernoulli numbers, just replace @kbd{X} with @kbd{k b} in the above
6504 sequence of keystrokes.)
6506 With loops and conditionals, you can program essentially anything
6507 in Calc. One other command that makes looping easier is @kbd{Z /},
6508 which takes a condition from the stack and breaks out of the enclosing
6509 loop if the condition is true (non-zero). You can use this to make
6510 ``while'' and ``until'' style loops.
6512 If you make a mistake when entering a keyboard macro, you can edit
6513 it using @kbd{Z E}. First, you must attach it to a key with @kbd{Z K}.
6514 One technique is to enter a throwaway dummy definition for the macro,
6515 then enter the real one in the edit command.
6519 1: 3 1: 3 Calc Macro Edit Mode.
6520 . . Original keys: 1 <return> 2 +
6527 C-x ( 1 @key{RET} 2 + C-x ) Z K h @key{RET} Z E h
6532 A keyboard macro is stored as a pure keystroke sequence. The
6533 @file{edmacro} package (invoked by @kbd{Z E}) scans along the
6534 macro and tries to decode it back into human-readable steps.
6535 Descriptions of the keystrokes are given as comments, which begin with
6536 @samp{;;}, and which are ignored when the edited macro is saved.
6537 Spaces and line breaks are also ignored when the edited macro is saved.
6538 To enter a space into the macro, type @code{SPC}. All the special
6539 characters @code{RET}, @code{LFD}, @code{TAB}, @code{SPC}, @code{DEL},
6540 and @code{NUL} must be written in all uppercase, as must the prefixes
6541 @code{C-} and @code{M-}.
6543 Let's edit in a new definition, for computing harmonic numbers.
6544 First, erase the four lines of the old definition. Then, type
6545 in the new definition (or use Emacs @kbd{M-w} and @kbd{C-y} commands
6546 to copy it from this page of the Info file; you can of course skip
6547 typing the comments, which begin with @samp{;;}).
6550 Z` ;; calc-kbd-push (Save local values)
6551 0 ;; calc digits (Push a zero onto the stack)
6552 st ;; calc-store-into (Store it in the following variable)
6553 1 ;; calc quick variable (Quick variable q1)
6554 1 ;; calc digits (Initial value for the loop)
6555 TAB ;; calc-roll-down (Swap initial and final)
6556 Z( ;; calc-kbd-for (Begin the "for" loop)
6557 & ;; calc-inv (Take the reciprocal)
6558 s+ ;; calc-store-plus (Add to the following variable)
6559 1 ;; calc quick variable (Quick variable q1)
6560 1 ;; calc digits (The loop step is 1)
6561 Z) ;; calc-kbd-end-for (End the "for" loop)
6562 sr ;; calc-recall (Recall the final accumulated value)
6563 1 ;; calc quick variable (Quick variable q1)
6564 Z' ;; calc-kbd-pop (Restore values)
6568 Press @kbd{C-c C-c} to finish editing and return to the Calculator.
6579 The @file{edmacro} package defines a handy @code{read-kbd-macro} command
6580 which reads the current region of the current buffer as a sequence of
6581 keystroke names, and defines that sequence on the @kbd{X}
6582 (and @kbd{C-x e}) key. Because this is so useful, Calc puts this
6583 command on the @kbd{M-# m} key. Try reading in this macro in the
6584 following form: Press @kbd{C-@@} (or @kbd{C-@key{SPC}}) at
6585 one end of the text below, then type @kbd{M-# m} at the other.
6597 (@bullet{}) @strong{Exercise 8.} A general algorithm for solving
6598 equations numerically is @dfn{Newton's Method}. Given the equation
6599 @expr{f(x) = 0} for any function @expr{f}, and an initial guess
6600 @expr{x_0} which is reasonably close to the desired solution, apply
6601 this formula over and over:
6605 new_x = x - f(x)/f'(x)
6610 $$ x_{\rm new} = x - {f(x) \over f'(x)} $$
6615 where @expr{f'(x)} is the derivative of @expr{f}. The @expr{x}
6616 values will quickly converge to a solution, i.e., eventually
6617 @texline @math{x_{\rm new}}
6618 @infoline @expr{new_x}
6619 and @expr{x} will be equal to within the limits
6620 of the current precision. Write a program which takes a formula
6621 involving the variable @expr{x}, and an initial guess @expr{x_0},
6622 on the stack, and produces a value of @expr{x} for which the formula
6623 is zero. Use it to find a solution of
6624 @texline @math{\sin(\cos x) = 0.5}
6625 @infoline @expr{sin(cos(x)) = 0.5}
6626 near @expr{x = 4.5}. (Use angles measured in radians.) Note that
6627 the built-in @w{@kbd{a R}} (@code{calc-find-root}) command uses Newton's
6628 method when it is able. @xref{Programming Answer 8, 8}. (@bullet{})
6630 @cindex Digamma function
6631 @cindex Gamma constant, Euler's
6632 @cindex Euler's gamma constant
6633 (@bullet{}) @strong{Exercise 9.} The @dfn{digamma} function
6634 @texline @math{\psi(z) (``psi'')}
6635 @infoline @expr{psi(z)}
6636 is defined as the derivative of
6637 @texline @math{\ln \Gamma(z)}.
6638 @infoline @expr{ln(gamma(z))}.
6639 For large values of @expr{z}, it can be approximated by the infinite sum
6643 psi(z) ~= ln(z) - 1/2z - sum(bern(2 n) / 2 n z^(2 n), n, 1, inf)
6648 $$ \psi(z) \approx \ln z - {1\over2z} -
6649 \sum_{n=1}^\infty {\code{bern}(2 n) \over 2 n z^{2n}}
6656 @texline @math{\sum}
6657 @infoline @expr{sum}
6658 represents the sum over @expr{n} from 1 to infinity
6659 (or to some limit high enough to give the desired accuracy), and
6660 the @code{bern} function produces (exact) Bernoulli numbers.
6661 While this sum is not guaranteed to converge, in practice it is safe.
6662 An interesting mathematical constant is Euler's gamma, which is equal
6663 to about 0.5772. One way to compute it is by the formula,
6664 @texline @math{\gamma = -\psi(1)}.
6665 @infoline @expr{gamma = -psi(1)}.
6666 Unfortunately, 1 isn't a large enough argument
6667 for the above formula to work (5 is a much safer value for @expr{z}).
6668 Fortunately, we can compute
6669 @texline @math{\psi(1)}
6670 @infoline @expr{psi(1)}
6672 @texline @math{\psi(5)}
6673 @infoline @expr{psi(5)}
6674 using the recurrence
6675 @texline @math{\psi(z+1) = \psi(z) + {1 \over z}}.
6676 @infoline @expr{psi(z+1) = psi(z) + 1/z}.
6677 Your task: Develop a program to compute
6678 @texline @math{\psi(z)};
6679 @infoline @expr{psi(z)};
6680 it should ``pump up'' @expr{z}
6681 if necessary to be greater than 5, then use the above summation
6682 formula. Use looping commands to compute the sum. Use your function
6684 @texline @math{\gamma}
6685 @infoline @expr{gamma}
6686 to twelve decimal places. (Calc has a built-in command
6687 for Euler's constant, @kbd{I P}, which you can use to check your answer.)
6688 @xref{Programming Answer 9, 9}. (@bullet{})
6690 @cindex Polynomial, list of coefficients
6691 (@bullet{}) @strong{Exercise 10.} Given a polynomial in @expr{x} and
6692 a number @expr{m} on the stack, where the polynomial is of degree
6693 @expr{m} or less (i.e., does not have any terms higher than @expr{x^m}),
6694 write a program to convert the polynomial into a list-of-coefficients
6695 notation. For example, @expr{5 x^4 + (x + 1)^2} with @expr{m = 6}
6696 should produce the list @expr{[1, 2, 1, 0, 5, 0, 0]}. Also develop
6697 a way to convert from this form back to the standard algebraic form.
6698 @xref{Programming Answer 10, 10}. (@bullet{})
6701 (@bullet{}) @strong{Exercise 11.} The @dfn{Stirling numbers of the
6702 first kind} are defined by the recurrences,
6706 s(n,n) = 1 for n >= 0,
6707 s(n,0) = 0 for n > 0,
6708 s(n+1,m) = s(n,m-1) - n s(n,m) for n >= m >= 1.
6714 $$ \eqalign{ s(n,n) &= 1 \qquad \hbox{for } n \ge 0, \cr
6715 s(n,0) &= 0 \qquad \hbox{for } n > 0, \cr
6716 s(n+1,m) &= s(n,m-1) - n \, s(n,m) \qquad
6717 \hbox{for } n \ge m \ge 1.}
6721 (These numbers are also sometimes written $\displaystyle{n \brack m}$.)
6724 This can be implemented using a @dfn{recursive} program in Calc; the
6725 program must invoke itself in order to calculate the two righthand
6726 terms in the general formula. Since it always invokes itself with
6727 ``simpler'' arguments, it's easy to see that it must eventually finish
6728 the computation. Recursion is a little difficult with Emacs keyboard
6729 macros since the macro is executed before its definition is complete.
6730 So here's the recommended strategy: Create a ``dummy macro'' and assign
6731 it to a key with, e.g., @kbd{Z K s}. Now enter the true definition,
6732 using the @kbd{z s} command to call itself recursively, then assign it
6733 to the same key with @kbd{Z K s}. Now the @kbd{z s} command will run
6734 the complete recursive program. (Another way is to use @w{@kbd{Z E}}
6735 or @kbd{M-# m} (@code{read-kbd-macro}) to read the whole macro at once,
6736 thus avoiding the ``training'' phase.) The task: Write a program
6737 that computes Stirling numbers of the first kind, given @expr{n} and
6738 @expr{m} on the stack. Test it with @emph{small} inputs like
6739 @expr{s(4,2)}. (There is a built-in command for Stirling numbers,
6740 @kbd{k s}, which you can use to check your answers.)
6741 @xref{Programming Answer 11, 11}. (@bullet{})
6743 The programming commands we've seen in this part of the tutorial
6744 are low-level, general-purpose operations. Often you will find
6745 that a higher-level function, such as vector mapping or rewrite
6746 rules, will do the job much more easily than a detailed, step-by-step
6749 (@bullet{}) @strong{Exercise 12.} Write another program for
6750 computing Stirling numbers of the first kind, this time using
6751 rewrite rules. Once again, @expr{n} and @expr{m} should be taken
6752 from the stack. @xref{Programming Answer 12, 12}. (@bullet{})
6757 This ends the tutorial section of the Calc manual. Now you know enough
6758 about Calc to use it effectively for many kinds of calculations. But
6759 Calc has many features that were not even touched upon in this tutorial.
6761 The rest of this manual tells the whole story.
6763 @c Volume II of this manual, the @dfn{Calc Reference}, tells the whole story.
6766 @node Answers to Exercises, , Programming Tutorial, Tutorial
6767 @section Answers to Exercises
6770 This section includes answers to all the exercises in the Calc tutorial.
6773 * RPN Answer 1:: 1 @key{RET} 2 @key{RET} 3 @key{RET} 4 + * -
6774 * RPN Answer 2:: 2*4 + 7*9.5 + 5/4
6775 * RPN Answer 3:: Operating on levels 2 and 3
6776 * RPN Answer 4:: Joe's complex problems
6777 * Algebraic Answer 1:: Simulating Q command
6778 * Algebraic Answer 2:: Joe's algebraic woes
6779 * Algebraic Answer 3:: 1 / 0
6780 * Modes Answer 1:: 3#0.1 = 3#0.0222222?
6781 * Modes Answer 2:: 16#f.e8fe15
6782 * Modes Answer 3:: Joe's rounding bug
6783 * Modes Answer 4:: Why floating point?
6784 * Arithmetic Answer 1:: Why the \ command?
6785 * Arithmetic Answer 2:: Tripping up the B command
6786 * Vector Answer 1:: Normalizing a vector
6787 * Vector Answer 2:: Average position
6788 * Matrix Answer 1:: Row and column sums
6789 * Matrix Answer 2:: Symbolic system of equations
6790 * Matrix Answer 3:: Over-determined system
6791 * List Answer 1:: Powers of two
6792 * List Answer 2:: Least-squares fit with matrices
6793 * List Answer 3:: Geometric mean
6794 * List Answer 4:: Divisor function
6795 * List Answer 5:: Duplicate factors
6796 * List Answer 6:: Triangular list
6797 * List Answer 7:: Another triangular list
6798 * List Answer 8:: Maximum of Bessel function
6799 * List Answer 9:: Integers the hard way
6800 * List Answer 10:: All elements equal
6801 * List Answer 11:: Estimating pi with darts
6802 * List Answer 12:: Estimating pi with matchsticks
6803 * List Answer 13:: Hash codes
6804 * List Answer 14:: Random walk
6805 * Types Answer 1:: Square root of pi times rational
6806 * Types Answer 2:: Infinities
6807 * Types Answer 3:: What can "nan" be?
6808 * Types Answer 4:: Abbey Road
6809 * Types Answer 5:: Friday the 13th
6810 * Types Answer 6:: Leap years
6811 * Types Answer 7:: Erroneous donut
6812 * Types Answer 8:: Dividing intervals
6813 * Types Answer 9:: Squaring intervals
6814 * Types Answer 10:: Fermat's primality test
6815 * Types Answer 11:: pi * 10^7 seconds
6816 * Types Answer 12:: Abbey Road on CD
6817 * Types Answer 13:: Not quite pi * 10^7 seconds
6818 * Types Answer 14:: Supercomputers and c
6819 * Types Answer 15:: Sam the Slug
6820 * Algebra Answer 1:: Squares and square roots
6821 * Algebra Answer 2:: Building polynomial from roots
6822 * Algebra Answer 3:: Integral of x sin(pi x)
6823 * Algebra Answer 4:: Simpson's rule
6824 * Rewrites Answer 1:: Multiplying by conjugate
6825 * Rewrites Answer 2:: Alternative fib rule
6826 * Rewrites Answer 3:: Rewriting opt(a) + opt(b) x
6827 * Rewrites Answer 4:: Sequence of integers
6828 * Rewrites Answer 5:: Number of terms in sum
6829 * Rewrites Answer 6:: Truncated Taylor series
6830 * Programming Answer 1:: Fresnel's C(x)
6831 * Programming Answer 2:: Negate third stack element
6832 * Programming Answer 3:: Compute sin(x) / x, etc.
6833 * Programming Answer 4:: Average value of a list
6834 * Programming Answer 5:: Continued fraction phi
6835 * Programming Answer 6:: Matrix Fibonacci numbers
6836 * Programming Answer 7:: Harmonic number greater than 4
6837 * Programming Answer 8:: Newton's method
6838 * Programming Answer 9:: Digamma function
6839 * Programming Answer 10:: Unpacking a polynomial
6840 * Programming Answer 11:: Recursive Stirling numbers
6841 * Programming Answer 12:: Stirling numbers with rewrites
6844 @c The following kludgery prevents the individual answers from
6845 @c being entered on the table of contents.
6847 \global\let\oldwrite=\write
6848 \gdef\skipwrite#1#2{\let\write=\oldwrite}
6849 \global\let\oldchapternofonts=\chapternofonts
6850 \gdef\chapternofonts{\let\write=\skipwrite\oldchapternofonts}
6853 @node RPN Answer 1, RPN Answer 2, Answers to Exercises, Answers to Exercises
6854 @subsection RPN Tutorial Exercise 1
6857 @kbd{1 @key{RET} 2 @key{RET} 3 @key{RET} 4 + * -}
6860 @texline @math{1 - (2 \times (3 + 4)) = -13}.
6861 @infoline @expr{1 - (2 * (3 + 4)) = -13}.
6863 @node RPN Answer 2, RPN Answer 3, RPN Answer 1, Answers to Exercises
6864 @subsection RPN Tutorial Exercise 2
6867 @texline @math{2\times4 + 7\times9.5 + {5\over4} = 75.75}
6868 @infoline @expr{2*4 + 7*9.5 + 5/4 = 75.75}
6870 After computing the intermediate term
6871 @texline @math{2\times4 = 8},
6872 @infoline @expr{2*4 = 8},
6873 you can leave that result on the stack while you compute the second
6874 term. With both of these results waiting on the stack you can then
6875 compute the final term, then press @kbd{+ +} to add everything up.
6884 2 @key{RET} 4 * 7 @key{RET} 9.5 *
6891 4: 8 3: 8 2: 8 1: 75.75
6892 3: 66.5 2: 66.5 1: 67.75 .
6901 Alternatively, you could add the first two terms before going on
6902 with the third term.
6906 2: 8 1: 74.5 3: 74.5 2: 74.5 1: 75.75
6907 1: 66.5 . 2: 5 1: 1.25 .
6911 ... + 5 @key{RET} 4 / +
6915 On an old-style RPN calculator this second method would have the
6916 advantage of using only three stack levels. But since Calc's stack
6917 can grow arbitrarily large this isn't really an issue. Which method
6918 you choose is purely a matter of taste.
6920 @node RPN Answer 3, RPN Answer 4, RPN Answer 2, Answers to Exercises
6921 @subsection RPN Tutorial Exercise 3
6924 The @key{TAB} key provides a way to operate on the number in level 2.
6928 3: 10 3: 10 4: 10 3: 10 3: 10
6929 2: 20 2: 30 3: 30 2: 30 2: 21
6930 1: 30 1: 20 2: 20 1: 21 1: 30
6934 @key{TAB} 1 + @key{TAB}
6938 Similarly, @kbd{M-@key{TAB}} gives you access to the number in level 3.
6942 3: 10 3: 21 3: 21 3: 30 3: 11
6943 2: 21 2: 30 2: 30 2: 11 2: 21
6944 1: 30 1: 10 1: 11 1: 21 1: 30
6947 M-@key{TAB} 1 + M-@key{TAB} M-@key{TAB}
6951 @node RPN Answer 4, Algebraic Answer 1, RPN Answer 3, Answers to Exercises
6952 @subsection RPN Tutorial Exercise 4
6955 Either @kbd{( 2 , 3 )} or @kbd{( 2 @key{SPC} 3 )} would have worked,
6956 but using both the comma and the space at once yields:
6960 1: ( ... 2: ( ... 1: (2, ... 2: (2, ... 2: (2, ...
6961 . 1: 2 . 1: (2, ... 1: (2, 3)
6968 Joe probably tried to type @kbd{@key{TAB} @key{DEL}} to swap the
6969 extra incomplete object to the top of the stack and delete it.
6970 But a feature of Calc is that @key{DEL} on an incomplete object
6971 deletes just one component out of that object, so he had to press
6972 @key{DEL} twice to finish the job.
6976 2: (2, ... 2: (2, 3) 2: (2, 3) 1: (2, 3)
6977 1: (2, 3) 1: (2, ... 1: ( ... .
6980 @key{TAB} @key{DEL} @key{DEL}
6984 (As it turns out, deleting the second-to-top stack entry happens often
6985 enough that Calc provides a special key, @kbd{M-@key{DEL}}, to do just that.
6986 @kbd{M-@key{DEL}} is just like @kbd{@key{TAB} @key{DEL}}, except that it doesn't exhibit
6987 the ``feature'' that tripped poor Joe.)
6989 @node Algebraic Answer 1, Algebraic Answer 2, RPN Answer 4, Answers to Exercises
6990 @subsection Algebraic Entry Tutorial Exercise 1
6993 Type @kbd{' sqrt($) @key{RET}}.
6995 If the @kbd{Q} key is broken, you could use @kbd{' $^0.5 @key{RET}}.
6996 Or, RPN style, @kbd{0.5 ^}.
6998 (Actually, @samp{$^1:2}, using the fraction one-half as the power, is
6999 a closer equivalent, since @samp{9^0.5} yields @expr{3.0} whereas
7000 @samp{sqrt(9)} and @samp{9^1:2} yield the exact integer @expr{3}.)
7002 @node Algebraic Answer 2, Algebraic Answer 3, Algebraic Answer 1, Answers to Exercises
7003 @subsection Algebraic Entry Tutorial Exercise 2
7006 In the formula @samp{2 x (1+y)}, @samp{x} was interpreted as a function
7007 name with @samp{1+y} as its argument. Assigning a value to a variable
7008 has no relation to a function by the same name. Joe needed to use an
7009 explicit @samp{*} symbol here: @samp{2 x*(1+y)}.
7011 @node Algebraic Answer 3, Modes Answer 1, Algebraic Answer 2, Answers to Exercises
7012 @subsection Algebraic Entry Tutorial Exercise 3
7015 The result from @kbd{1 @key{RET} 0 /} will be the formula @expr{1 / 0}.
7016 The ``function'' @samp{/} cannot be evaluated when its second argument
7017 is zero, so it is left in symbolic form. When you now type @kbd{0 *},
7018 the result will be zero because Calc uses the general rule that ``zero
7019 times anything is zero.''
7021 @c [fix-ref Infinities]
7022 The @kbd{m i} command enables an @dfn{Infinite mode} in which @expr{1 / 0}
7023 results in a special symbol that represents ``infinity.'' If you
7024 multiply infinity by zero, Calc uses another special new symbol to
7025 show that the answer is ``indeterminate.'' @xref{Infinities}, for
7026 further discussion of infinite and indeterminate values.
7028 @node Modes Answer 1, Modes Answer 2, Algebraic Answer 3, Answers to Exercises
7029 @subsection Modes Tutorial Exercise 1
7032 Calc always stores its numbers in decimal, so even though one-third has
7033 an exact base-3 representation (@samp{3#0.1}), it is still stored as
7034 0.3333333 (chopped off after 12 or however many decimal digits) inside
7035 the calculator's memory. When this inexact number is converted back
7036 to base 3 for display, it may still be slightly inexact. When we
7037 multiply this number by 3, we get 0.999999, also an inexact value.
7039 When Calc displays a number in base 3, it has to decide how many digits
7040 to show. If the current precision is 12 (decimal) digits, that corresponds
7041 to @samp{12 / log10(3) = 25.15} base-3 digits. Because 25.15 is not an
7042 exact integer, Calc shows only 25 digits, with the result that stored
7043 numbers carry a little bit of extra information that may not show up on
7044 the screen. When Joe entered @samp{3#0.2}, the stored number 0.666666
7045 happened to round to a pleasing value when it lost that last 0.15 of a
7046 digit, but it was still inexact in Calc's memory. When he divided by 2,
7047 he still got the dreaded inexact value 0.333333. (Actually, he divided
7048 0.666667 by 2 to get 0.333334, which is why he got something a little
7049 higher than @code{3#0.1} instead of a little lower.)
7051 If Joe didn't want to be bothered with all this, he could have typed
7052 @kbd{M-24 d n} to display with one less digit than the default. (If
7053 you give @kbd{d n} a negative argument, it uses default-minus-that,
7054 so @kbd{M-- d n} would be an easier way to get the same effect.) Those
7055 inexact results would still be lurking there, but they would now be
7056 rounded to nice, natural-looking values for display purposes. (Remember,
7057 @samp{0.022222} in base 3 is like @samp{0.099999} in base 10; rounding
7058 off one digit will round the number up to @samp{0.1}.) Depending on the
7059 nature of your work, this hiding of the inexactness may be a benefit or
7060 a danger. With the @kbd{d n} command, Calc gives you the choice.
7062 Incidentally, another consequence of all this is that if you type
7063 @kbd{M-30 d n} to display more digits than are ``really there,''
7064 you'll see garbage digits at the end of the number. (In decimal
7065 display mode, with decimally-stored numbers, these garbage digits are
7066 always zero so they vanish and you don't notice them.) Because Calc
7067 rounds off that 0.15 digit, there is the danger that two numbers could
7068 be slightly different internally but still look the same. If you feel
7069 uneasy about this, set the @kbd{d n} precision to be a little higher
7070 than normal; you'll get ugly garbage digits, but you'll always be able
7071 to tell two distinct numbers apart.
7073 An interesting side note is that most computers store their
7074 floating-point numbers in binary, and convert to decimal for display.
7075 Thus everyday programs have the same problem: Decimal 0.1 cannot be
7076 represented exactly in binary (try it: @kbd{0.1 d 2}), so @samp{0.1 * 10}
7077 comes out as an inexact approximation to 1 on some machines (though
7078 they generally arrange to hide it from you by rounding off one digit as
7079 we did above). Because Calc works in decimal instead of binary, you can
7080 be sure that numbers that look exact @emph{are} exact as long as you stay
7081 in decimal display mode.
7083 It's not hard to show that any number that can be represented exactly
7084 in binary, octal, or hexadecimal is also exact in decimal, so the kinds
7085 of problems we saw in this exercise are likely to be severe only when
7086 you use a relatively unusual radix like 3.
7088 @node Modes Answer 2, Modes Answer 3, Modes Answer 1, Answers to Exercises
7089 @subsection Modes Tutorial Exercise 2
7091 If the radix is 15 or higher, we can't use the letter @samp{e} to mark
7092 the exponent because @samp{e} is interpreted as a digit. When Calc
7093 needs to display scientific notation in a high radix, it writes
7094 @samp{16#F.E8F*16.^15}. You can enter a number like this as an
7095 algebraic entry. Also, pressing @kbd{e} without any digits before it
7096 normally types @kbd{1e}, but in a high radix it types @kbd{16.^} and
7097 puts you in algebraic entry: @kbd{16#f.e8f @key{RET} e 15 @key{RET} *} is another
7098 way to enter this number.
7100 The reason Calc puts a decimal point in the @samp{16.^} is to prevent
7101 huge integers from being generated if the exponent is large (consider
7102 @samp{16#1.23*16^1000}, where we compute @samp{16^1000} as a giant
7103 exact integer and then throw away most of the digits when we multiply
7104 it by the floating-point @samp{16#1.23}). While this wouldn't normally
7105 matter for display purposes, it could give you a nasty surprise if you
7106 copied that number into a file and later moved it back into Calc.
7108 @node Modes Answer 3, Modes Answer 4, Modes Answer 2, Answers to Exercises
7109 @subsection Modes Tutorial Exercise 3
7112 The answer he got was @expr{0.5000000000006399}.
7114 The problem is not that the square operation is inexact, but that the
7115 sine of 45 that was already on the stack was accurate to only 12 places.
7116 Arbitrary-precision calculations still only give answers as good as
7119 The real problem is that there is no 12-digit number which, when
7120 squared, comes out to 0.5 exactly. The @kbd{f [} and @kbd{f ]}
7121 commands decrease or increase a number by one unit in the last
7122 place (according to the current precision). They are useful for
7123 determining facts like this.
7127 1: 0.707106781187 1: 0.500000000001
7137 1: 0.707106781187 1: 0.707106781186 1: 0.499999999999
7144 A high-precision calculation must be carried out in high precision
7145 all the way. The only number in the original problem which was known
7146 exactly was the quantity 45 degrees, so the precision must be raised
7147 before anything is done after the number 45 has been entered in order
7148 for the higher precision to be meaningful.
7150 @node Modes Answer 4, Arithmetic Answer 1, Modes Answer 3, Answers to Exercises
7151 @subsection Modes Tutorial Exercise 4
7154 Many calculations involve real-world quantities, like the width and
7155 height of a piece of wood or the volume of a jar. Such quantities
7156 can't be measured exactly anyway, and if the data that is input to
7157 a calculation is inexact, doing exact arithmetic on it is a waste
7160 Fractions become unwieldy after too many calculations have been
7161 done with them. For example, the sum of the reciprocals of the
7162 integers from 1 to 10 is 7381:2520. The sum from 1 to 30 is
7163 9304682830147:2329089562800. After a point it will take a long
7164 time to add even one more term to this sum, but a floating-point
7165 calculation of the sum will not have this problem.
7167 Also, rational numbers cannot express the results of all calculations.
7168 There is no fractional form for the square root of two, so if you type
7169 @w{@kbd{2 Q}}, Calc has no choice but to give you a floating-point answer.
7171 @node Arithmetic Answer 1, Arithmetic Answer 2, Modes Answer 4, Answers to Exercises
7172 @subsection Arithmetic Tutorial Exercise 1
7175 Dividing two integers that are larger than the current precision may
7176 give a floating-point result that is inaccurate even when rounded
7177 down to an integer. Consider @expr{123456789 / 2} when the current
7178 precision is 6 digits. The true answer is @expr{61728394.5}, but
7179 with a precision of 6 this will be rounded to
7180 @texline @math{12345700.0/2.0 = 61728500.0}.
7181 @infoline @expr{12345700.@: / 2.@: = 61728500.}.
7182 The result, when converted to an integer, will be off by 106.
7184 Here are two solutions: Raise the precision enough that the
7185 floating-point round-off error is strictly to the right of the
7186 decimal point. Or, convert to Fraction mode so that @expr{123456789 / 2}
7187 produces the exact fraction @expr{123456789:2}, which can be rounded
7188 down by the @kbd{F} command without ever switching to floating-point
7191 @node Arithmetic Answer 2, Vector Answer 1, Arithmetic Answer 1, Answers to Exercises
7192 @subsection Arithmetic Tutorial Exercise 2
7195 @kbd{27 @key{RET} 9 B} could give the exact result @expr{3:2}, but it
7196 does a floating-point calculation instead and produces @expr{1.5}.
7198 Calc will find an exact result for a logarithm if the result is an integer
7199 or (when in Fraction mode) the reciprocal of an integer. But there is
7200 no efficient way to search the space of all possible rational numbers
7201 for an exact answer, so Calc doesn't try.
7203 @node Vector Answer 1, Vector Answer 2, Arithmetic Answer 2, Answers to Exercises
7204 @subsection Vector Tutorial Exercise 1
7207 Duplicate the vector, compute its length, then divide the vector
7208 by its length: @kbd{@key{RET} A /}.
7212 1: [1, 2, 3] 2: [1, 2, 3] 1: [0.27, 0.53, 0.80] 1: 1.
7213 . 1: 3.74165738677 . .
7220 The final @kbd{A} command shows that the normalized vector does
7221 indeed have unit length.
7223 @node Vector Answer 2, Matrix Answer 1, Vector Answer 1, Answers to Exercises
7224 @subsection Vector Tutorial Exercise 2
7227 The average position is equal to the sum of the products of the
7228 positions times their corresponding probabilities. This is the
7229 definition of the dot product operation. So all you need to do
7230 is to put the two vectors on the stack and press @kbd{*}.
7232 @node Matrix Answer 1, Matrix Answer 2, Vector Answer 2, Answers to Exercises
7233 @subsection Matrix Tutorial Exercise 1
7236 The trick is to multiply by a vector of ones. Use @kbd{r 4 [1 1 1] *} to
7237 get the row sum. Similarly, use @kbd{[1 1] r 4 *} to get the column sum.
7239 @node Matrix Answer 2, Matrix Answer 3, Matrix Answer 1, Answers to Exercises
7240 @subsection Matrix Tutorial Exercise 2
7253 $$ \eqalign{ x &+ a y = 6 \cr
7259 Just enter the righthand side vector, then divide by the lefthand side
7264 1: [6, 10] 2: [6, 10] 1: [6 - 4 a / (b - a), 4 / (b - a) ]
7269 ' [6 10] @key{RET} ' [1 a; 1 b] @key{RET} /
7273 This can be made more readable using @kbd{d B} to enable Big display
7279 1: [6 - -----, -----]
7284 Type @kbd{d N} to return to Normal display mode afterwards.
7286 @node Matrix Answer 3, List Answer 1, Matrix Answer 2, Answers to Exercises
7287 @subsection Matrix Tutorial Exercise 3
7291 @texline @math{A^T A \, X = A^T B},
7292 @infoline @expr{trn(A) * A * X = trn(A) * B},
7294 @texline @math{A' = A^T A}
7295 @infoline @expr{A2 = trn(A) * A}
7297 @texline @math{B' = A^T B};
7298 @infoline @expr{B2 = trn(A) * B};
7299 now, we have a system
7300 @texline @math{A' X = B'}
7301 @infoline @expr{A2 * X = B2}
7302 which we can solve using Calc's @samp{/} command.
7317 $$ \openup1\jot \tabskip=0pt plus1fil
7318 \halign to\displaywidth{\tabskip=0pt
7319 $\hfil#$&$\hfil{}#{}$&
7320 $\hfil#$&$\hfil{}#{}$&
7321 $\hfil#$&${}#\hfil$\tabskip=0pt plus1fil\cr
7325 2a&+&4b&+&6c&=11 \cr}
7330 The first step is to enter the coefficient matrix. We'll store it in
7331 quick variable number 7 for later reference. Next, we compute the
7338 1: [ [ 1, 2, 3 ] 2: [ [ 1, 4, 7, 2 ] 1: [57, 84, 96]
7339 [ 4, 5, 6 ] [ 2, 5, 6, 4 ] .
7340 [ 7, 6, 0 ] [ 3, 6, 0, 6 ] ]
7341 [ 2, 4, 6 ] ] 1: [6, 2, 3, 11]
7344 ' [1 2 3; 4 5 6; 7 6 0; 2 4 6] @key{RET} s 7 v t [6 2 3 11] *
7349 Now we compute the matrix
7356 2: [57, 84, 96] 1: [-11.64, 14.08, -3.64]
7357 1: [ [ 70, 72, 39 ] .
7367 (The actual computed answer will be slightly inexact due to
7370 Notice that the answers are similar to those for the
7371 @texline @math{3\times3}
7373 system solved in the text. That's because the fourth equation that was
7374 added to the system is almost identical to the first one multiplied
7375 by two. (If it were identical, we would have gotten the exact same
7377 @texline @math{4\times3}
7379 system would be equivalent to the original
7380 @texline @math{3\times3}
7384 Since the first and fourth equations aren't quite equivalent, they
7385 can't both be satisfied at once. Let's plug our answers back into
7386 the original system of equations to see how well they match.
7390 2: [-11.64, 14.08, -3.64] 1: [5.6, 2., 3., 11.2]
7402 This is reasonably close to our original @expr{B} vector,
7403 @expr{[6, 2, 3, 11]}.
7405 @node List Answer 1, List Answer 2, Matrix Answer 3, Answers to Exercises
7406 @subsection List Tutorial Exercise 1
7409 We can use @kbd{v x} to build a vector of integers. This needs to be
7410 adjusted to get the range of integers we desire. Mapping @samp{-}
7411 across the vector will accomplish this, although it turns out the
7412 plain @samp{-} key will work just as well.
7417 1: [1, 2, 3, 4, 5, 6, 7, 8, 9] 1: [-4, -3, -2, -1, 0, 1, 2, 3, 4]
7420 2 v x 9 @key{RET} 5 V M - or 5 -
7425 Now we use @kbd{V M ^} to map the exponentiation operator across the
7430 1: [0.0625, 0.125, 0.25, 0.5, 1, 2, 4, 8, 16]
7437 @node List Answer 2, List Answer 3, List Answer 1, Answers to Exercises
7438 @subsection List Tutorial Exercise 2
7441 Given @expr{x} and @expr{y} vectors in quick variables 1 and 2 as before,
7442 the first job is to form the matrix that describes the problem.
7452 $$ m \times x + b \times 1 = y $$
7457 @texline @math{19\times2}
7459 matrix with our @expr{x} vector as one column and
7460 ones as the other column. So, first we build the column of ones, then
7461 we combine the two columns to form our @expr{A} matrix.
7465 2: [1.34, 1.41, 1.49, ... ] 1: [ [ 1.34, 1 ]
7466 1: [1, 1, 1, ...] [ 1.41, 1 ]
7470 r 1 1 v b 19 @key{RET} M-2 v p v t s 3
7476 @texline @math{A^T y}
7477 @infoline @expr{trn(A) * y}
7479 @texline @math{A^T A}
7480 @infoline @expr{trn(A) * A}
7485 1: [33.36554, 13.613] 2: [33.36554, 13.613]
7486 . 1: [ [ 98.0003, 41.63 ]
7490 v t r 2 * r 3 v t r 3 *
7495 (Hey, those numbers look familiar!)
7499 1: [0.52141679, -0.425978]
7506 Since we were solving equations of the form
7507 @texline @math{m \times x + b \times 1 = y},
7508 @infoline @expr{m*x + b*1 = y},
7509 these numbers should be @expr{m} and @expr{b}, respectively. Sure
7510 enough, they agree exactly with the result computed using @kbd{V M} and
7513 The moral of this story: @kbd{V M} and @kbd{V R} will probably solve
7514 your problem, but there is often an easier way using the higher-level
7515 arithmetic functions!
7517 @c [fix-ref Curve Fitting]
7518 In fact, there is a built-in @kbd{a F} command that does least-squares
7519 fits. @xref{Curve Fitting}.
7521 @node List Answer 3, List Answer 4, List Answer 2, Answers to Exercises
7522 @subsection List Tutorial Exercise 3
7525 Move to one end of the list and press @kbd{C-@@} (or @kbd{C-@key{SPC}} or
7526 whatever) to set the mark, then move to the other end of the list
7527 and type @w{@kbd{M-# g}}.
7531 1: [2.3, 6, 22, 15.1, 7, 15, 14, 7.5, 2.5]
7536 To make things interesting, let's assume we don't know at a glance
7537 how many numbers are in this list. Then we could type:
7541 2: [2.3, 6, 22, ... ] 2: [2.3, 6, 22, ... ]
7542 1: [2.3, 6, 22, ... ] 1: 126356422.5
7552 2: 126356422.5 2: 126356422.5 1: 7.94652913734
7553 1: [2.3, 6, 22, ... ] 1: 9 .
7561 (The @kbd{I ^} command computes the @var{n}th root of a number.
7562 You could also type @kbd{& ^} to take the reciprocal of 9 and
7563 then raise the number to that power.)
7565 @node List Answer 4, List Answer 5, List Answer 3, Answers to Exercises
7566 @subsection List Tutorial Exercise 4
7569 A number @expr{j} is a divisor of @expr{n} if
7570 @texline @math{n \mathbin{\hbox{\code{\%}}} j = 0}.
7571 @infoline @samp{n % j = 0}.
7572 The first step is to get a vector that identifies the divisors.
7576 2: 30 2: [0, 0, 0, 2, ...] 1: [1, 1, 1, 0, ...]
7577 1: [1, 2, 3, 4, ...] 1: 0 .
7580 30 @key{RET} v x 30 @key{RET} s 1 V M % 0 V M a = s 2
7585 This vector has 1's marking divisors of 30 and 0's marking non-divisors.
7587 The zeroth divisor function is just the total number of divisors.
7588 The first divisor function is the sum of the divisors.
7593 2: [1, 2, 3, 4, ...] 1: [1, 2, 3, 0, ...] 1: 72
7594 1: [1, 1, 1, 0, ...] . .
7597 V R + r 1 r 2 V M * V R +
7602 Once again, the last two steps just compute a dot product for which
7603 a simple @kbd{*} would have worked equally well.
7605 @node List Answer 5, List Answer 6, List Answer 4, Answers to Exercises
7606 @subsection List Tutorial Exercise 5
7609 The obvious first step is to obtain the list of factors with @kbd{k f}.
7610 This list will always be in sorted order, so if there are duplicates
7611 they will be right next to each other. A suitable method is to compare
7612 the list with a copy of itself shifted over by one.
7616 1: [3, 7, 7, 7, 19] 2: [3, 7, 7, 7, 19] 2: [3, 7, 7, 7, 19, 0]
7617 . 1: [3, 7, 7, 7, 19, 0] 1: [0, 3, 7, 7, 7, 19]
7620 19551 k f @key{RET} 0 | @key{TAB} 0 @key{TAB} |
7627 1: [0, 0, 1, 1, 0, 0] 1: 2 1: 0
7635 Note that we have to arrange for both vectors to have the same length
7636 so that the mapping operation works; no prime factor will ever be
7637 zero, so adding zeros on the left and right is safe. From then on
7638 the job is pretty straightforward.
7640 Incidentally, Calc provides the
7641 @texline @dfn{M@"obius} @math{\mu}
7642 @infoline @dfn{Moebius mu}
7643 function which is zero if and only if its argument is square-free. It
7644 would be a much more convenient way to do the above test in practice.
7646 @node List Answer 6, List Answer 7, List Answer 5, Answers to Exercises
7647 @subsection List Tutorial Exercise 6
7650 First use @kbd{v x 6 @key{RET}} to get a list of integers, then @kbd{V M v x}
7651 to get a list of lists of integers!
7653 @node List Answer 7, List Answer 8, List Answer 6, Answers to Exercises
7654 @subsection List Tutorial Exercise 7
7657 Here's one solution. First, compute the triangular list from the previous
7658 exercise and type @kbd{1 -} to subtract one from all the elements.
7671 The numbers down the lefthand edge of the list we desire are called
7672 the ``triangular numbers'' (now you know why!). The @expr{n}th
7673 triangular number is the sum of the integers from 1 to @expr{n}, and
7674 can be computed directly by the formula
7675 @texline @math{n (n+1) \over 2}.
7676 @infoline @expr{n * (n+1) / 2}.
7680 2: [ [0], [0, 1], ... ] 2: [ [0], [0, 1], ... ]
7681 1: [0, 1, 2, 3, 4, 5] 1: [0, 1, 3, 6, 10, 15]
7684 v x 6 @key{RET} 1 - V M ' $ ($+1)/2 @key{RET}
7689 Adding this list to the above list of lists produces the desired
7698 [10, 11, 12, 13, 14],
7699 [15, 16, 17, 18, 19, 20] ]
7706 If we did not know the formula for triangular numbers, we could have
7707 computed them using a @kbd{V U +} command. We could also have
7708 gotten them the hard way by mapping a reduction across the original
7713 2: [ [0], [0, 1], ... ] 2: [ [0], [0, 1], ... ]
7714 1: [ [0], [0, 1], ... ] 1: [0, 1, 3, 6, 10, 15]
7722 (This means ``map a @kbd{V R +} command across the vector,'' and
7723 since each element of the main vector is itself a small vector,
7724 @kbd{V R +} computes the sum of its elements.)
7726 @node List Answer 8, List Answer 9, List Answer 7, Answers to Exercises
7727 @subsection List Tutorial Exercise 8
7730 The first step is to build a list of values of @expr{x}.
7734 1: [1, 2, 3, ..., 21] 1: [0, 1, 2, ..., 20] 1: [0, 0.25, 0.5, ..., 5]
7737 v x 21 @key{RET} 1 - 4 / s 1
7741 Next, we compute the Bessel function values.
7745 1: [0., 0.124, 0.242, ..., -0.328]
7748 V M ' besJ(1,$) @key{RET}
7753 (Another way to do this would be @kbd{1 @key{TAB} V M f j}.)
7755 A way to isolate the maximum value is to compute the maximum using
7756 @kbd{V R X}, then compare all the Bessel values with that maximum.
7760 2: [0., 0.124, 0.242, ... ] 1: [0, 0, 0, ... ] 2: [0, 0, 0, ... ]
7764 @key{RET} V R X V M a = @key{RET} V R + @key{DEL}
7769 It's a good idea to verify, as in the last step above, that only
7770 one value is equal to the maximum. (After all, a plot of
7771 @texline @math{\sin x}
7772 @infoline @expr{sin(x)}
7773 might have many points all equal to the maximum value, 1.)
7775 The vector we have now has a single 1 in the position that indicates
7776 the maximum value of @expr{x}. Now it is a simple matter to convert
7777 this back into the corresponding value itself.
7781 2: [0, 0, 0, ... ] 1: [0, 0., 0., ... ] 1: 1.75
7782 1: [0, 0.25, 0.5, ... ] . .
7789 If @kbd{a =} had produced more than one @expr{1} value, this method
7790 would have given the sum of all maximum @expr{x} values; not very
7791 useful! In this case we could have used @kbd{v m} (@code{calc-mask-vector})
7792 instead. This command deletes all elements of a ``data'' vector that
7793 correspond to zeros in a ``mask'' vector, leaving us with, in this
7794 example, a vector of maximum @expr{x} values.
7796 The built-in @kbd{a X} command maximizes a function using more
7797 efficient methods. Just for illustration, let's use @kbd{a X}
7798 to maximize @samp{besJ(1,x)} over this same interval.
7802 2: besJ(1, x) 1: [1.84115, 0.581865]
7806 ' besJ(1,x), [0..5] @key{RET} a X x @key{RET}
7811 The output from @kbd{a X} is a vector containing the value of @expr{x}
7812 that maximizes the function, and the function's value at that maximum.
7813 As you can see, our simple search got quite close to the right answer.
7815 @node List Answer 9, List Answer 10, List Answer 8, Answers to Exercises
7816 @subsection List Tutorial Exercise 9
7819 Step one is to convert our integer into vector notation.
7823 1: 25129925999 3: 25129925999
7825 1: [11, 10, 9, ..., 1, 0]
7828 25129925999 @key{RET} 10 @key{RET} 12 @key{RET} v x 12 @key{RET} -
7835 1: 25129925999 1: [0, 2, 25, 251, 2512, ... ]
7836 2: [100000000000, ... ] .
7844 (Recall, the @kbd{\} command computes an integer quotient.)
7848 1: [0, 2, 5, 1, 2, 9, 9, 2, 5, 9, 9, 9]
7855 Next we must increment this number. This involves adding one to
7856 the last digit, plus handling carries. There is a carry to the
7857 left out of a digit if that digit is a nine and all the digits to
7858 the right of it are nines.
7862 1: [0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1] 1: [1, 1, 1, 0, 0, 1, ... ]
7872 1: [1, 1, 1, 0, 0, 0, ... ] 1: [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1]
7880 Accumulating @kbd{*} across a vector of ones and zeros will preserve
7881 only the initial run of ones. These are the carries into all digits
7882 except the rightmost digit. Concatenating a one on the right takes
7883 care of aligning the carries properly, and also adding one to the
7888 2: [0, 0, 0, 0, ... ] 1: [0, 0, 2, 5, 1, 2, 9, 9, 2, 6, 0, 0, 0]
7889 1: [0, 0, 2, 5, ... ] .
7892 0 r 2 | V M + 10 V M %
7897 Here we have concatenated 0 to the @emph{left} of the original number;
7898 this takes care of shifting the carries by one with respect to the
7899 digits that generated them.
7901 Finally, we must convert this list back into an integer.
7905 3: [0, 0, 2, 5, ... ] 2: [0, 0, 2, 5, ... ]
7906 2: 1000000000000 1: [1000000000000, 100000000000, ... ]
7907 1: [100000000000, ... ] .
7910 10 @key{RET} 12 ^ r 1 |
7917 1: [0, 0, 20000000000, 5000000000, ... ] 1: 25129926000
7925 Another way to do this final step would be to reduce the formula
7926 @w{@samp{10 $$ + $}} across the vector of digits.
7930 1: [0, 0, 2, 5, ... ] 1: 25129926000
7933 V R ' 10 $$ + $ @key{RET}
7937 @node List Answer 10, List Answer 11, List Answer 9, Answers to Exercises
7938 @subsection List Tutorial Exercise 10
7941 For the list @expr{[a, b, c, d]}, the result is @expr{((a = b) = c) = d},
7942 which will compare @expr{a} and @expr{b} to produce a 1 or 0, which is
7943 then compared with @expr{c} to produce another 1 or 0, which is then
7944 compared with @expr{d}. This is not at all what Joe wanted.
7946 Here's a more correct method:
7950 1: [7, 7, 7, 8, 7] 2: [7, 7, 7, 8, 7]
7954 ' [7,7,7,8,7] @key{RET} @key{RET} v r 1 @key{RET}
7961 1: [1, 1, 1, 0, 1] 1: 0
7968 @node List Answer 11, List Answer 12, List Answer 10, Answers to Exercises
7969 @subsection List Tutorial Exercise 11
7972 The circle of unit radius consists of those points @expr{(x,y)} for which
7973 @expr{x^2 + y^2 < 1}. We start by generating a vector of @expr{x^2}
7974 and a vector of @expr{y^2}.
7976 We can make this go a bit faster by using the @kbd{v .} and @kbd{t .}
7981 2: [2., 2., ..., 2.] 2: [2., 2., ..., 2.]
7982 1: [2., 2., ..., 2.] 1: [1.16, 1.98, ..., 0.81]
7985 v . t . 2. v b 100 @key{RET} @key{RET} V M k r
7992 2: [2., 2., ..., 2.] 1: [0.026, 0.96, ..., 0.036]
7993 1: [0.026, 0.96, ..., 0.036] 2: [0.53, 0.81, ..., 0.094]
7996 1 - 2 V M ^ @key{TAB} V M k r 1 - 2 V M ^
8000 Now we sum the @expr{x^2} and @expr{y^2} values, compare with 1 to
8001 get a vector of 1/0 truth values, then sum the truth values.
8005 1: [0.56, 1.78, ..., 0.13] 1: [1, 0, ..., 1] 1: 84
8013 The ratio @expr{84/100} should approximate the ratio @cpiover{4}.
8017 1: 0.84 1: 3.36 2: 3.36 1: 1.0695
8025 Our estimate, 3.36, is off by about 7%. We could get a better estimate
8026 by taking more points (say, 1000), but it's clear that this method is
8029 (Naturally, since this example uses random numbers your own answer
8030 will be slightly different from the one shown here!)
8032 If you typed @kbd{v .} and @kbd{t .} before, type them again to
8033 return to full-sized display of vectors.
8035 @node List Answer 12, List Answer 13, List Answer 11, Answers to Exercises
8036 @subsection List Tutorial Exercise 12
8039 This problem can be made a lot easier by taking advantage of some
8040 symmetries. First of all, after some thought it's clear that the
8041 @expr{y} axis can be ignored altogether. Just pick a random @expr{x}
8042 component for one end of the match, pick a random direction
8043 @texline @math{\theta},
8044 @infoline @expr{theta},
8045 and see if @expr{x} and
8046 @texline @math{x + \cos \theta}
8047 @infoline @expr{x + cos(theta)}
8048 (which is the @expr{x} coordinate of the other endpoint) cross a line.
8049 The lines are at integer coordinates, so this happens when the two
8050 numbers surround an integer.
8052 Since the two endpoints are equivalent, we may as well choose the leftmost
8053 of the two endpoints as @expr{x}. Then @expr{theta} is an angle pointing
8054 to the right, in the range -90 to 90 degrees. (We could use radians, but
8055 it would feel like cheating to refer to @cpiover{2} radians while trying
8056 to estimate @cpi{}!)
8058 In fact, since the field of lines is infinite we can choose the
8059 coordinates 0 and 1 for the lines on either side of the leftmost
8060 endpoint. The rightmost endpoint will be between 0 and 1 if the
8061 match does not cross a line, or between 1 and 2 if it does. So:
8062 Pick random @expr{x} and
8063 @texline @math{\theta},
8064 @infoline @expr{theta},
8066 @texline @math{x + \cos \theta},
8067 @infoline @expr{x + cos(theta)},
8068 and count how many of the results are greater than one. Simple!
8070 We can make this go a bit faster by using the @kbd{v .} and @kbd{t .}
8075 1: [0.52, 0.71, ..., 0.72] 2: [0.52, 0.71, ..., 0.72]
8076 . 1: [78.4, 64.5, ..., -42.9]
8079 v . t . 1. v b 100 @key{RET} V M k r 180. v b 100 @key{RET} V M k r 90 -
8084 (The next step may be slow, depending on the speed of your computer.)
8088 2: [0.52, 0.71, ..., 0.72] 1: [0.72, 1.14, ..., 1.45]
8089 1: [0.20, 0.43, ..., 0.73] .
8099 1: [0, 1, ..., 1] 1: 0.64 1: 3.125
8102 1 V M a > V R + 100 / 2 @key{TAB} /
8106 Let's try the third method, too. We'll use random integers up to
8107 one million. The @kbd{k r} command with an integer argument picks
8112 2: [1000000, 1000000, ..., 1000000] 2: [78489, 527587, ..., 814975]
8113 1: [1000000, 1000000, ..., 1000000] 1: [324014, 358783, ..., 955450]
8116 1000000 v b 100 @key{RET} @key{RET} V M k r @key{TAB} V M k r
8123 1: [1, 1, ..., 25] 1: [1, 1, ..., 0] 1: 0.56
8126 V M k g 1 V M a = V R + 100 /
8140 For a proof of this property of the GCD function, see section 4.5.2,
8141 exercise 10, of Knuth's @emph{Art of Computer Programming}, volume II.
8143 If you typed @kbd{v .} and @kbd{t .} before, type them again to
8144 return to full-sized display of vectors.
8146 @node List Answer 13, List Answer 14, List Answer 12, Answers to Exercises
8147 @subsection List Tutorial Exercise 13
8150 First, we put the string on the stack as a vector of ASCII codes.
8154 1: [84, 101, 115, ..., 51]
8157 "Testing, 1, 2, 3 @key{RET}
8162 Note that the @kbd{"} key, like @kbd{$}, initiates algebraic entry so
8163 there was no need to type an apostrophe. Also, Calc didn't mind that
8164 we omitted the closing @kbd{"}. (The same goes for all closing delimiters
8165 like @kbd{)} and @kbd{]} at the end of a formula.
8167 We'll show two different approaches here. In the first, we note that
8168 if the input vector is @expr{[a, b, c, d]}, then the hash code is
8169 @expr{3 (3 (3a + b) + c) + d = 27a + 9b + 3c + d}. In other words,
8170 it's a sum of descending powers of three times the ASCII codes.
8174 2: [84, 101, 115, ..., 51] 2: [84, 101, 115, ..., 51]
8175 1: 16 1: [15, 14, 13, ..., 0]
8178 @key{RET} v l v x 16 @key{RET} -
8185 2: [84, 101, 115, ..., 51] 1: 1960915098 1: 121
8186 1: [14348907, ..., 1] . .
8189 3 @key{TAB} V M ^ * 511 %
8194 Once again, @kbd{*} elegantly summarizes most of the computation.
8195 But there's an even more elegant approach: Reduce the formula
8196 @kbd{3 $$ + $} across the vector. Recall that this represents a
8197 function of two arguments that computes its first argument times three
8198 plus its second argument.
8202 1: [84, 101, 115, ..., 51] 1: 1960915098
8205 "Testing, 1, 2, 3 @key{RET} V R ' 3$$+$ @key{RET}
8210 If you did the decimal arithmetic exercise, this will be familiar.
8211 Basically, we're turning a base-3 vector of digits into an integer,
8212 except that our ``digits'' are much larger than real digits.
8214 Instead of typing @kbd{511 %} again to reduce the result, we can be
8215 cleverer still and notice that rather than computing a huge integer
8216 and taking the modulo at the end, we can take the modulo at each step
8217 without affecting the result. While this means there are more
8218 arithmetic operations, the numbers we operate on remain small so
8219 the operations are faster.
8223 1: [84, 101, 115, ..., 51] 1: 121
8226 "Testing, 1, 2, 3 @key{RET} V R ' (3$$+$)%511 @key{RET}
8230 Why does this work? Think about a two-step computation:
8231 @w{@expr{3 (3a + b) + c}}. Taking a result modulo 511 basically means
8232 subtracting off enough 511's to put the result in the desired range.
8233 So the result when we take the modulo after every step is,
8237 3 (3 a + b - 511 m) + c - 511 n
8243 $$ 3 (3 a + b - 511 m) + c - 511 n $$
8248 for some suitable integers @expr{m} and @expr{n}. Expanding out by
8249 the distributive law yields
8253 9 a + 3 b + c - 511*3 m - 511 n
8259 $$ 9 a + 3 b + c - 511\times3 m - 511 n $$
8264 The @expr{m} term in the latter formula is redundant because any
8265 contribution it makes could just as easily be made by the @expr{n}
8266 term. So we can take it out to get an equivalent formula with
8271 9 a + 3 b + c - 511 n'
8277 $$ 9 a + 3 b + c - 511 n' $$
8282 which is just the formula for taking the modulo only at the end of
8283 the calculation. Therefore the two methods are essentially the same.
8285 Later in the tutorial we will encounter @dfn{modulo forms}, which
8286 basically automate the idea of reducing every intermediate result
8287 modulo some value @var{m}.
8289 @node List Answer 14, Types Answer 1, List Answer 13, Answers to Exercises
8290 @subsection List Tutorial Exercise 14
8292 We want to use @kbd{H V U} to nest a function which adds a random
8293 step to an @expr{(x,y)} coordinate. The function is a bit long, but
8294 otherwise the problem is quite straightforward.
8298 2: [0, 0] 1: [ [ 0, 0 ]
8299 1: 50 [ 0.4288, -0.1695 ]
8300 . [ -0.4787, -0.9027 ]
8303 [0,0] 50 H V U ' <# + [random(2.0)-1, random(2.0)-1]> @key{RET}
8307 Just as the text recommended, we used @samp{< >} nameless function
8308 notation to keep the two @code{random} calls from being evaluated
8309 before nesting even begins.
8311 We now have a vector of @expr{[x, y]} sub-vectors, which by Calc's
8312 rules acts like a matrix. We can transpose this matrix and unpack
8313 to get a pair of vectors, @expr{x} and @expr{y}, suitable for graphing.
8317 2: [ 0, 0.4288, -0.4787, ... ]
8318 1: [ 0, -0.1696, -0.9027, ... ]
8325 Incidentally, because the @expr{x} and @expr{y} are completely
8326 independent in this case, we could have done two separate commands
8327 to create our @expr{x} and @expr{y} vectors of numbers directly.
8329 To make a random walk of unit steps, we note that @code{sincos} of
8330 a random direction exactly gives us an @expr{[x, y]} step of unit
8331 length; in fact, the new nesting function is even briefer, though
8332 we might want to lower the precision a bit for it.
8336 2: [0, 0] 1: [ [ 0, 0 ]
8337 1: 50 [ 0.1318, 0.9912 ]
8338 . [ -0.5965, 0.3061 ]
8341 [0,0] 50 m d p 6 @key{RET} H V U ' <# + sincos(random(360.0))> @key{RET}
8345 Another @kbd{v t v u g f} sequence will graph this new random walk.
8347 An interesting twist on these random walk functions would be to use
8348 complex numbers instead of 2-vectors to represent points on the plane.
8349 In the first example, we'd use something like @samp{random + random*(0,1)},
8350 and in the second we could use polar complex numbers with random phase
8351 angles. (This exercise was first suggested in this form by Randal
8354 @node Types Answer 1, Types Answer 2, List Answer 14, Answers to Exercises
8355 @subsection Types Tutorial Exercise 1
8358 If the number is the square root of @cpi{} times a rational number,
8359 then its square, divided by @cpi{}, should be a rational number.
8363 1: 1.26508260337 1: 0.509433962268 1: 2486645810:4881193627
8371 Technically speaking this is a rational number, but not one that is
8372 likely to have arisen in the original problem. More likely, it just
8373 happens to be the fraction which most closely represents some
8374 irrational number to within 12 digits.
8376 But perhaps our result was not quite exact. Let's reduce the
8377 precision slightly and try again:
8381 1: 0.509433962268 1: 27:53
8384 U p 10 @key{RET} c F
8389 Aha! It's unlikely that an irrational number would equal a fraction
8390 this simple to within ten digits, so our original number was probably
8391 @texline @math{\sqrt{27 \pi / 53}}.
8392 @infoline @expr{sqrt(27 pi / 53)}.
8394 Notice that we didn't need to re-round the number when we reduced the
8395 precision. Remember, arithmetic operations always round their inputs
8396 to the current precision before they begin.
8398 @node Types Answer 2, Types Answer 3, Types Answer 1, Answers to Exercises
8399 @subsection Types Tutorial Exercise 2
8402 @samp{inf / inf = nan}. Perhaps @samp{1} is the ``obvious'' answer.
8403 But if @w{@samp{17 inf = inf}}, then @samp{17 inf / inf = inf / inf = 17}, too.
8405 @samp{exp(inf) = inf}. It's tempting to say that the exponential
8406 of infinity must be ``bigger'' than ``regular'' infinity, but as
8407 far as Calc is concerned all infinities are as just as big.
8408 In other words, as @expr{x} goes to infinity, @expr{e^x} also goes
8409 to infinity, but the fact the @expr{e^x} grows much faster than
8410 @expr{x} is not relevant here.
8412 @samp{exp(-inf) = 0}. Here we have a finite answer even though
8413 the input is infinite.
8415 @samp{sqrt(-inf) = (0, 1) inf}. Remember that @expr{(0, 1)}
8416 represents the imaginary number @expr{i}. Here's a derivation:
8417 @samp{sqrt(-inf) = @w{sqrt((-1) * inf)} = sqrt(-1) * sqrt(inf)}.
8418 The first part is, by definition, @expr{i}; the second is @code{inf}
8419 because, once again, all infinities are the same size.
8421 @samp{sqrt(uinf) = uinf}. In fact, we do know something about the
8422 direction because @code{sqrt} is defined to return a value in the
8423 right half of the complex plane. But Calc has no notation for this,
8424 so it settles for the conservative answer @code{uinf}.
8426 @samp{abs(uinf) = inf}. No matter which direction @expr{x} points,
8427 @samp{abs(x)} always points along the positive real axis.
8429 @samp{ln(0) = -inf}. Here we have an infinite answer to a finite
8430 input. As in the @expr{1 / 0} case, Calc will only use infinities
8431 here if you have turned on Infinite mode. Otherwise, it will
8432 treat @samp{ln(0)} as an error.
8434 @node Types Answer 3, Types Answer 4, Types Answer 2, Answers to Exercises
8435 @subsection Types Tutorial Exercise 3
8438 We can make @samp{inf - inf} be any real number we like, say,
8439 @expr{a}, just by claiming that we added @expr{a} to the first
8440 infinity but not to the second. This is just as true for complex
8441 values of @expr{a}, so @code{nan} can stand for a complex number.
8442 (And, similarly, @code{uinf} can stand for an infinity that points
8443 in any direction in the complex plane, such as @samp{(0, 1) inf}).
8445 In fact, we can multiply the first @code{inf} by two. Surely
8446 @w{@samp{2 inf - inf = inf}}, but also @samp{2 inf - inf = inf - inf = nan}.
8447 So @code{nan} can even stand for infinity. Obviously it's just
8448 as easy to make it stand for minus infinity as for plus infinity.
8450 The moral of this story is that ``infinity'' is a slippery fish
8451 indeed, and Calc tries to handle it by having a very simple model
8452 for infinities (only the direction counts, not the ``size''); but
8453 Calc is careful to write @code{nan} any time this simple model is
8454 unable to tell what the true answer is.
8456 @node Types Answer 4, Types Answer 5, Types Answer 3, Answers to Exercises
8457 @subsection Types Tutorial Exercise 4
8461 2: 0@@ 47' 26" 1: 0@@ 2' 47.411765"
8465 0@@ 47' 26" @key{RET} 17 /
8470 The average song length is two minutes and 47.4 seconds.
8474 2: 0@@ 2' 47.411765" 1: 0@@ 3' 7.411765" 1: 0@@ 53' 6.000005"
8483 The album would be 53 minutes and 6 seconds long.
8485 @node Types Answer 5, Types Answer 6, Types Answer 4, Answers to Exercises
8486 @subsection Types Tutorial Exercise 5
8489 Let's suppose it's January 14, 1991. The easiest thing to do is
8490 to keep trying 13ths of months until Calc reports a Friday.
8491 We can do this by manually entering dates, or by using @kbd{t I}:
8495 1: <Wed Feb 13, 1991> 1: <Wed Mar 13, 1991> 1: <Sat Apr 13, 1991>
8498 ' <2/13> @key{RET} @key{DEL} ' <3/13> @key{RET} t I
8503 (Calc assumes the current year if you don't say otherwise.)
8505 This is getting tedious---we can keep advancing the date by typing
8506 @kbd{t I} over and over again, but let's automate the job by using
8507 vector mapping. The @kbd{t I} command actually takes a second
8508 ``how-many-months'' argument, which defaults to one. This
8509 argument is exactly what we want to map over:
8513 2: <Sat Apr 13, 1991> 1: [<Mon May 13, 1991>, <Thu Jun 13, 1991>,
8514 1: [1, 2, 3, 4, 5, 6] <Sat Jul 13, 1991>, <Tue Aug 13, 1991>,
8515 . <Fri Sep 13, 1991>, <Sun Oct 13, 1991>]
8518 v x 6 @key{RET} V M t I
8523 Et voil@`a, September 13, 1991 is a Friday.
8530 ' <sep 13> - <jan 14> @key{RET}
8535 And the answer to our original question: 242 days to go.
8537 @node Types Answer 6, Types Answer 7, Types Answer 5, Answers to Exercises
8538 @subsection Types Tutorial Exercise 6
8541 The full rule for leap years is that they occur in every year divisible
8542 by four, except that they don't occur in years divisible by 100, except
8543 that they @emph{do} in years divisible by 400. We could work out the
8544 answer by carefully counting the years divisible by four and the
8545 exceptions, but there is a much simpler way that works even if we
8546 don't know the leap year rule.
8548 Let's assume the present year is 1991. Years have 365 days, except
8549 that leap years (whenever they occur) have 366 days. So let's count
8550 the number of days between now and then, and compare that to the
8551 number of years times 365. The number of extra days we find must be
8552 equal to the number of leap years there were.
8556 1: <Mon Jan 1, 10001> 2: <Mon Jan 1, 10001> 1: 2925593
8557 . 1: <Tue Jan 1, 1991> .
8560 ' <jan 1 10001> @key{RET} ' <jan 1 1991> @key{RET} -
8567 3: 2925593 2: 2925593 2: 2925593 1: 1943
8568 2: 10001 1: 8010 1: 2923650 .
8572 10001 @key{RET} 1991 - 365 * -
8576 @c [fix-ref Date Forms]
8578 There will be 1943 leap years before the year 10001. (Assuming,
8579 of course, that the algorithm for computing leap years remains
8580 unchanged for that long. @xref{Date Forms}, for some interesting
8581 background information in that regard.)
8583 @node Types Answer 7, Types Answer 8, Types Answer 6, Answers to Exercises
8584 @subsection Types Tutorial Exercise 7
8587 The relative errors must be converted to absolute errors so that
8588 @samp{+/-} notation may be used.
8596 20 @key{RET} .05 * 4 @key{RET} .05 *
8600 Now we simply chug through the formula.
8604 1: 19.7392088022 1: 394.78 +/- 19.739 1: 6316.5 +/- 706.21
8607 2 P 2 ^ * 20 p 1 * 4 p .2 @key{RET} 2 ^ *
8611 It turns out the @kbd{v u} command will unpack an error form as
8612 well as a vector. This saves us some retyping of numbers.
8616 3: 6316.5 +/- 706.21 2: 6316.5 +/- 706.21
8621 @key{RET} v u @key{TAB} /
8626 Thus the volume is 6316 cubic centimeters, within about 11 percent.
8628 @node Types Answer 8, Types Answer 9, Types Answer 7, Answers to Exercises
8629 @subsection Types Tutorial Exercise 8
8632 The first answer is pretty simple: @samp{1 / (0 .. 10) = (0.1 .. inf)}.
8633 Since a number in the interval @samp{(0 .. 10)} can get arbitrarily
8634 close to zero, its reciprocal can get arbitrarily large, so the answer
8635 is an interval that effectively means, ``any number greater than 0.1''
8636 but with no upper bound.
8638 The second answer, similarly, is @samp{1 / (-10 .. 0) = (-inf .. -0.1)}.
8640 Calc normally treats division by zero as an error, so that the formula
8641 @w{@samp{1 / 0}} is left unsimplified. Our third problem,
8642 @w{@samp{1 / [0 .. 10]}}, also (potentially) divides by zero because zero
8643 is now a member of the interval. So Calc leaves this one unevaluated, too.
8645 If you turn on Infinite mode by pressing @kbd{m i}, you will
8646 instead get the answer @samp{[0.1 .. inf]}, which includes infinity
8647 as a possible value.
8649 The fourth calculation, @samp{1 / (-10 .. 10)}, has the same problem.
8650 Zero is buried inside the interval, but it's still a possible value.
8651 It's not hard to see that the actual result of @samp{1 / (-10 .. 10)}
8652 will be either greater than @mathit{0.1}, or less than @mathit{-0.1}. Thus
8653 the interval goes from minus infinity to plus infinity, with a ``hole''
8654 in it from @mathit{-0.1} to @mathit{0.1}. Calc doesn't have any way to
8655 represent this, so it just reports @samp{[-inf .. inf]} as the answer.
8656 It may be disappointing to hear ``the answer lies somewhere between
8657 minus infinity and plus infinity, inclusive,'' but that's the best
8658 that interval arithmetic can do in this case.
8660 @node Types Answer 9, Types Answer 10, Types Answer 8, Answers to Exercises
8661 @subsection Types Tutorial Exercise 9
8665 1: [-3 .. 3] 2: [-3 .. 3] 2: [0 .. 9]
8666 . 1: [0 .. 9] 1: [-9 .. 9]
8669 [ 3 n .. 3 ] @key{RET} 2 ^ @key{TAB} @key{RET} *
8674 In the first case the result says, ``if a number is between @mathit{-3} and
8675 3, its square is between 0 and 9.'' The second case says, ``the product
8676 of two numbers each between @mathit{-3} and 3 is between @mathit{-9} and 9.''
8678 An interval form is not a number; it is a symbol that can stand for
8679 many different numbers. Two identical-looking interval forms can stand
8680 for different numbers.
8682 The same issue arises when you try to square an error form.
8684 @node Types Answer 10, Types Answer 11, Types Answer 9, Answers to Exercises
8685 @subsection Types Tutorial Exercise 10
8688 Testing the first number, we might arbitrarily choose 17 for @expr{x}.
8692 1: 17 mod 811749613 2: 17 mod 811749613 1: 533694123 mod 811749613
8696 17 M 811749613 @key{RET} 811749612 ^
8701 Since 533694123 is (considerably) different from 1, the number 811749613
8704 It's awkward to type the number in twice as we did above. There are
8705 various ways to avoid this, and algebraic entry is one. In fact, using
8706 a vector mapping operation we can perform several tests at once. Let's
8707 use this method to test the second number.
8711 2: [17, 42, 100000] 1: [1 mod 15485863, 1 mod ... ]
8715 [17 42 100000] 15485863 @key{RET} V M ' ($$ mod $)^($-1) @key{RET}
8720 The result is three ones (modulo @expr{n}), so it's very probable that
8721 15485863 is prime. (In fact, this number is the millionth prime.)
8723 Note that the functions @samp{($$^($-1)) mod $} or @samp{$$^($-1) % $}
8724 would have been hopelessly inefficient, since they would have calculated
8725 the power using full integer arithmetic.
8727 Calc has a @kbd{k p} command that does primality testing. For small
8728 numbers it does an exact test; for large numbers it uses a variant
8729 of the Fermat test we used here. You can use @kbd{k p} repeatedly
8730 to prove that a large integer is prime with any desired probability.
8732 @node Types Answer 11, Types Answer 12, Types Answer 10, Answers to Exercises
8733 @subsection Types Tutorial Exercise 11
8736 There are several ways to insert a calculated number into an HMS form.
8737 One way to convert a number of seconds to an HMS form is simply to
8738 multiply the number by an HMS form representing one second:
8742 1: 31415926.5359 2: 31415926.5359 1: 8726@@ 38' 46.5359"
8753 2: 8726@@ 38' 46.5359" 1: 6@@ 6' 2.5359" mod 24@@ 0' 0"
8754 1: 15@@ 27' 16" mod 24@@ 0' 0" .
8762 It will be just after six in the morning.
8764 The algebraic @code{hms} function can also be used to build an
8769 1: hms(0, 0, 10000000. pi) 1: 8726@@ 38' 46.5359"
8772 ' hms(0, 0, 1e7 pi) @key{RET} =
8777 The @kbd{=} key is necessary to evaluate the symbol @samp{pi} to
8778 the actual number 3.14159...
8780 @node Types Answer 12, Types Answer 13, Types Answer 11, Answers to Exercises
8781 @subsection Types Tutorial Exercise 12
8784 As we recall, there are 17 songs of about 2 minutes and 47 seconds
8789 2: 0@@ 2' 47" 1: [0@@ 3' 7" .. 0@@ 3' 47"]
8790 1: [0@@ 0' 20" .. 0@@ 1' 0"] .
8793 [ 0@@ 20" .. 0@@ 1' ] +
8800 1: [0@@ 52' 59." .. 1@@ 4' 19."]
8808 No matter how long it is, the album will fit nicely on one CD.
8810 @node Types Answer 13, Types Answer 14, Types Answer 12, Answers to Exercises
8811 @subsection Types Tutorial Exercise 13
8814 Type @kbd{' 1 yr @key{RET} u c s @key{RET}}. The answer is 31557600 seconds.
8816 @node Types Answer 14, Types Answer 15, Types Answer 13, Answers to Exercises
8817 @subsection Types Tutorial Exercise 14
8820 How long will it take for a signal to get from one end of the computer
8825 1: m / c 1: 3.3356 ns
8828 ' 1 m / c @key{RET} u c ns @key{RET}
8833 (Recall, @samp{c} is a ``unit'' corresponding to the speed of light.)
8837 1: 3.3356 ns 1: 0.81356 ns / ns 1: 0.81356
8841 ' 4.1 ns @key{RET} / u s
8846 Thus a signal could take up to 81 percent of a clock cycle just to
8847 go from one place to another inside the computer, assuming the signal
8848 could actually attain the full speed of light. Pretty tight!
8850 @node Types Answer 15, Algebra Answer 1, Types Answer 14, Answers to Exercises
8851 @subsection Types Tutorial Exercise 15
8854 The speed limit is 55 miles per hour on most highways. We want to
8855 find the ratio of Sam's speed to the US speed limit.
8859 1: 55 mph 2: 55 mph 3: 11 hr mph / yd
8863 ' 55 mph @key{RET} ' 5 yd/hr @key{RET} /
8867 The @kbd{u s} command cancels out these units to get a plain
8868 number. Now we take the logarithm base two to find the final
8869 answer, assuming that each successive pill doubles his speed.
8873 1: 19360. 2: 19360. 1: 14.24
8882 Thus Sam can take up to 14 pills without a worry.
8884 @node Algebra Answer 1, Algebra Answer 2, Types Answer 15, Answers to Exercises
8885 @subsection Algebra Tutorial Exercise 1
8888 @c [fix-ref Declarations]
8889 The result @samp{sqrt(x)^2} is simplified back to @expr{x} by the
8890 Calculator, but @samp{sqrt(x^2)} is not. (Consider what happens
8891 if @w{@expr{x = -4}}.) If @expr{x} is real, this formula could be
8892 simplified to @samp{abs(x)}, but for general complex arguments even
8893 that is not safe. (@xref{Declarations}, for a way to tell Calc
8894 that @expr{x} is known to be real.)
8896 @node Algebra Answer 2, Algebra Answer 3, Algebra Answer 1, Answers to Exercises
8897 @subsection Algebra Tutorial Exercise 2
8900 Suppose our roots are @expr{[a, b, c]}. We want a polynomial which
8901 is zero when @expr{x} is any of these values. The trivial polynomial
8902 @expr{x-a} is zero when @expr{x=a}, so the product @expr{(x-a)(x-b)(x-c)}
8903 will do the job. We can use @kbd{a c x} to write this in a more
8908 1: 34 x - 24 x^3 1: [1.19023, -1.19023, 0]
8918 1: [x - 1.19023, x + 1.19023, x] 1: (x - 1.19023) (x + 1.19023) x
8921 V M ' x-$ @key{RET} V R *
8928 1: x^3 - 1.41666 x 1: 34 x - 24 x^3
8931 a c x @key{RET} 24 n * a x
8936 Sure enough, our answer (multiplied by a suitable constant) is the
8937 same as the original polynomial.
8939 @node Algebra Answer 3, Algebra Answer 4, Algebra Answer 2, Answers to Exercises
8940 @subsection Algebra Tutorial Exercise 3
8944 1: x sin(pi x) 1: (sin(pi x) - pi x cos(pi x)) / pi^2
8947 ' x sin(pi x) @key{RET} m r a i x @key{RET}
8955 2: (sin(pi x) - pi x cos(pi x)) / pi^2
8958 ' [y,1] @key{RET} @key{TAB}
8965 1: [(sin(pi y) - pi y cos(pi y)) / pi^2, (sin(pi) - pi cos(pi)) / pi^2]
8975 1: (sin(pi y) - pi y cos(pi y)) / pi^2 + (pi cos(pi) - sin(pi)) / pi^2
8985 1: (sin(3.14159 y) - 3.14159 y cos(3.14159 y)) / 9.8696 - 0.3183
8995 1: [0., -0.95493, 0.63662, -1.5915, 1.2732]
8998 v x 5 @key{RET} @key{TAB} V M $ @key{RET}
9002 @node Algebra Answer 4, Rewrites Answer 1, Algebra Answer 3, Answers to Exercises
9003 @subsection Algebra Tutorial Exercise 4
9006 The hard part is that @kbd{V R +} is no longer sufficient to add up all
9007 the contributions from the slices, since the slices have varying
9008 coefficients. So first we must come up with a vector of these
9009 coefficients. Here's one way:
9013 2: -1 2: 3 1: [4, 2, ..., 4]
9014 1: [1, 2, ..., 9] 1: [-1, 1, ..., -1] .
9017 1 n v x 9 @key{RET} V M ^ 3 @key{TAB} -
9024 1: [4, 2, ..., 4, 1] 1: [1, 4, 2, ..., 4, 1]
9032 Now we compute the function values. Note that for this method we need
9033 eleven values, including both endpoints of the desired interval.
9037 2: [1, 4, 2, ..., 4, 1]
9038 1: [1, 1.1, 1.2, ... , 1.8, 1.9, 2.]
9041 11 @key{RET} 1 @key{RET} .1 @key{RET} C-u v x
9048 2: [1, 4, 2, ..., 4, 1]
9049 1: [0., 0.084941, 0.16993, ... ]
9052 ' sin(x) ln(x) @key{RET} m r p 5 @key{RET} V M $ @key{RET}
9057 Once again this calls for @kbd{V M * V R +}; a simple @kbd{*} does the
9062 1: 11.22 1: 1.122 1: 0.374
9070 Wow! That's even better than the result from the Taylor series method.
9072 @node Rewrites Answer 1, Rewrites Answer 2, Algebra Answer 4, Answers to Exercises
9073 @subsection Rewrites Tutorial Exercise 1
9076 We'll use Big mode to make the formulas more readable.
9082 1: (2 + sqrt(2)) / (1 + sqrt(2)) 1: --------
9088 ' (2+sqrt(2)) / (1+sqrt(2)) @key{RET} d B
9093 Multiplying by the conjugate helps because @expr{(a+b) (a-b) = a^2 - b^2}.
9098 1: (2 + V 2 ) (V 2 - 1)
9101 a r a/(b+c) := a*(b-c) / (b^2-c^2) @key{RET}
9109 1: 2 + V 2 - 2 1: V 2
9112 a r a*(b+c) := a*b + a*c a s
9117 (We could have used @kbd{a x} instead of a rewrite rule for the
9120 The multiply-by-conjugate rule turns out to be useful in many
9121 different circumstances, such as when the denominator involves
9122 sines and cosines or the imaginary constant @code{i}.
9124 @node Rewrites Answer 2, Rewrites Answer 3, Rewrites Answer 1, Answers to Exercises
9125 @subsection Rewrites Tutorial Exercise 2
9128 Here is the rule set:
9132 [ fib(n) := fib(n, 1, 1) :: integer(n) :: n >= 1,
9134 fib(n, x, y) := fib(n-1, y, x+y) ]
9139 The first rule turns a one-argument @code{fib} that people like to write
9140 into a three-argument @code{fib} that makes computation easier. The
9141 second rule converts back from three-argument form once the computation
9142 is done. The third rule does the computation itself. It basically
9143 says that if @expr{x} and @expr{y} are two consecutive Fibonacci numbers,
9144 then @expr{y} and @expr{x+y} are the next (overlapping) pair of Fibonacci
9147 Notice that because the number @expr{n} was ``validated'' by the
9148 conditions on the first rule, there is no need to put conditions on
9149 the other rules because the rule set would never get that far unless
9150 the input were valid. That further speeds computation, since no
9151 extra conditions need to be checked at every step.
9153 Actually, a user with a nasty sense of humor could enter a bad
9154 three-argument @code{fib} call directly, say, @samp{fib(0, 1, 1)},
9155 which would get the rules into an infinite loop. One thing that would
9156 help keep this from happening by accident would be to use something like
9157 @samp{ZzFib} instead of @code{fib} as the name of the three-argument
9160 @node Rewrites Answer 3, Rewrites Answer 4, Rewrites Answer 2, Answers to Exercises
9161 @subsection Rewrites Tutorial Exercise 3
9164 He got an infinite loop. First, Calc did as expected and rewrote
9165 @w{@samp{2 + 3 x}} to @samp{f(2, 3, x)}. Then it looked for ways to
9166 apply the rule again, and found that @samp{f(2, 3, x)} looks like
9167 @samp{a + b x} with @w{@samp{a = 0}} and @samp{b = 1}, so it rewrote to
9168 @samp{f(0, 1, f(2, 3, x))}. It then wrapped another @samp{f(0, 1, ...)}
9169 around that, and so on, ad infinitum. Joe should have used @kbd{M-1 a r}
9170 to make sure the rule applied only once.
9172 (Actually, even the first step didn't work as he expected. What Calc
9173 really gives for @kbd{M-1 a r} in this situation is @samp{f(3 x, 1, 2)},
9174 treating 2 as the ``variable,'' and @samp{3 x} as a constant being added
9175 to it. While this may seem odd, it's just as valid a solution as the
9176 ``obvious'' one. One way to fix this would be to add the condition
9177 @samp{:: variable(x)} to the rule, to make sure the thing that matches
9178 @samp{x} is indeed a variable, or to change @samp{x} to @samp{quote(x)}
9179 on the lefthand side, so that the rule matches the actual variable
9180 @samp{x} rather than letting @samp{x} stand for something else.)
9182 @node Rewrites Answer 4, Rewrites Answer 5, Rewrites Answer 3, Answers to Exercises
9183 @subsection Rewrites Tutorial Exercise 4
9190 Here is a suitable set of rules to solve the first part of the problem:
9194 [ seq(n, c) := seq(n/2, c+1) :: n%2 = 0,
9195 seq(n, c) := seq(3n+1, c+1) :: n%2 = 1 :: n > 1 ]
9199 Given the initial formula @samp{seq(6, 0)}, application of these
9200 rules produces the following sequence of formulas:
9214 whereupon neither of the rules match, and rewriting stops.
9216 We can pretty this up a bit with a couple more rules:
9220 [ seq(n) := seq(n, 0),
9227 Now, given @samp{seq(6)} as the starting configuration, we get 8
9230 The change to return a vector is quite simple:
9234 [ seq(n) := seq(n, []) :: integer(n) :: n > 0,
9236 seq(n, v) := seq(n/2, v | n) :: n%2 = 0,
9237 seq(n, v) := seq(3n+1, v | n) :: n%2 = 1 ]
9242 Given @samp{seq(6)}, the result is @samp{[6, 3, 10, 5, 16, 8, 4, 2, 1]}.
9244 Notice that the @expr{n > 1} guard is no longer necessary on the last
9245 rule since the @expr{n = 1} case is now detected by another rule.
9246 But a guard has been added to the initial rule to make sure the
9247 initial value is suitable before the computation begins.
9249 While still a good idea, this guard is not as vitally important as it
9250 was for the @code{fib} function, since calling, say, @samp{seq(x, [])}
9251 will not get into an infinite loop. Calc will not be able to prove
9252 the symbol @samp{x} is either even or odd, so none of the rules will
9253 apply and the rewrites will stop right away.
9255 @node Rewrites Answer 5, Rewrites Answer 6, Rewrites Answer 4, Answers to Exercises
9256 @subsection Rewrites Tutorial Exercise 5
9263 If @expr{x} is the sum @expr{a + b}, then `@tfn{nterms(}@var{x}@tfn{)}' must
9264 be `@tfn{nterms(}@var{a}@tfn{)}' plus `@tfn{nterms(}@var{b}@tfn{)}'. If @expr{x}
9265 is not a sum, then `@tfn{nterms(}@var{x}@tfn{)}' = 1.
9269 [ nterms(a + b) := nterms(a) + nterms(b),
9275 Here we have taken advantage of the fact that earlier rules always
9276 match before later rules; @samp{nterms(x)} will only be tried if we
9277 already know that @samp{x} is not a sum.
9279 @node Rewrites Answer 6, Programming Answer 1, Rewrites Answer 5, Answers to Exercises
9280 @subsection Rewrites Tutorial Exercise 6
9283 Here is a rule set that will do the job:
9287 [ a*(b + c) := a*b + a*c,
9288 opt(a) O(x^n) + opt(b) O(x^m) := O(x^n) :: n <= m
9289 :: constant(a) :: constant(b),
9290 opt(a) O(x^n) + opt(b) x^m := O(x^n) :: n <= m
9291 :: constant(a) :: constant(b),
9292 a O(x^n) := O(x^n) :: constant(a),
9293 x^opt(m) O(x^n) := O(x^(n+m)),
9294 O(x^n) O(x^m) := O(x^(n+m)) ]
9298 If we really want the @kbd{+} and @kbd{*} keys to operate naturally
9299 on power series, we should put these rules in @code{EvalRules}. For
9300 testing purposes, it is better to put them in a different variable,
9301 say, @code{O}, first.
9303 The first rule just expands products of sums so that the rest of the
9304 rules can assume they have an expanded-out polynomial to work with.
9305 Note that this rule does not mention @samp{O} at all, so it will
9306 apply to any product-of-sum it encounters---this rule may surprise
9307 you if you put it into @code{EvalRules}!
9309 In the second rule, the sum of two O's is changed to the smaller O.
9310 The optional constant coefficients are there mostly so that
9311 @samp{O(x^2) - O(x^3)} and @samp{O(x^3) - O(x^2)} are handled
9312 as well as @samp{O(x^2) + O(x^3)}.
9314 The third rule absorbs higher powers of @samp{x} into O's.
9316 The fourth rule says that a constant times a negligible quantity
9317 is still negligible. (This rule will also match @samp{O(x^3) / 4},
9318 with @samp{a = 1/4}.)
9320 The fifth rule rewrites, for example, @samp{x^2 O(x^3)} to @samp{O(x^5)}.
9321 (It is easy to see that if one of these forms is negligible, the other
9322 is, too.) Notice the @samp{x^opt(m)} to pick up terms like
9323 @w{@samp{x O(x^3)}}. Optional powers will match @samp{x} as @samp{x^1}
9324 but not 1 as @samp{x^0}. This turns out to be exactly what we want here.
9326 The sixth rule is the corresponding rule for products of two O's.
9328 Another way to solve this problem would be to create a new ``data type''
9329 that represents truncated power series. We might represent these as
9330 function calls @samp{series(@var{coefs}, @var{x})} where @var{coefs} is
9331 a vector of coefficients for @expr{x^0}, @expr{x^1}, @expr{x^2}, and so
9332 on. Rules would exist for sums and products of such @code{series}
9333 objects, and as an optional convenience could also know how to combine a
9334 @code{series} object with a normal polynomial. (With this, and with a
9335 rule that rewrites @samp{O(x^n)} to the equivalent @code{series} form,
9336 you could still enter power series in exactly the same notation as
9337 before.) Operations on such objects would probably be more efficient,
9338 although the objects would be a bit harder to read.
9340 @c [fix-ref Compositions]
9341 Some other symbolic math programs provide a power series data type
9342 similar to this. Mathematica, for example, has an object that looks
9343 like @samp{PowerSeries[@var{x}, @var{x0}, @var{coefs}, @var{nmin},
9344 @var{nmax}, @var{den}]}, where @var{x0} is the point about which the
9345 power series is taken (we've been assuming this was always zero),
9346 and @var{nmin}, @var{nmax}, and @var{den} allow pseudo-power-series
9347 with fractional or negative powers. Also, the @code{PowerSeries}
9348 objects have a special display format that makes them look like
9349 @samp{2 x^2 + O(x^4)} when they are printed out. (@xref{Compositions},
9350 for a way to do this in Calc, although for something as involved as
9351 this it would probably be better to write the formatting routine
9354 @node Programming Answer 1, Programming Answer 2, Rewrites Answer 6, Answers to Exercises
9355 @subsection Programming Tutorial Exercise 1
9358 Just enter the formula @samp{ninteg(sin(t)/t, t, 0, x)}, type
9359 @kbd{Z F}, and answer the questions. Since this formula contains two
9360 variables, the default argument list will be @samp{(t x)}. We want to
9361 change this to @samp{(x)} since @expr{t} is really a dummy variable
9362 to be used within @code{ninteg}.
9364 The exact keystrokes are @kbd{Z F s Si @key{RET} @key{RET} C-b C-b @key{DEL} @key{DEL} @key{RET} y}.
9365 (The @kbd{C-b C-b @key{DEL} @key{DEL}} are what fix the argument list.)
9367 @node Programming Answer 2, Programming Answer 3, Programming Answer 1, Answers to Exercises
9368 @subsection Programming Tutorial Exercise 2
9371 One way is to move the number to the top of the stack, operate on
9372 it, then move it back: @kbd{C-x ( M-@key{TAB} n M-@key{TAB} M-@key{TAB} C-x )}.
9374 Another way is to negate the top three stack entries, then negate
9375 again the top two stack entries: @kbd{C-x ( M-3 n M-2 n C-x )}.
9377 Finally, it turns out that a negative prefix argument causes a
9378 command like @kbd{n} to operate on the specified stack entry only,
9379 which is just what we want: @kbd{C-x ( M-- 3 n C-x )}.
9381 Just for kicks, let's also do it algebraically:
9382 @w{@kbd{C-x ( ' -$$$, $$, $ @key{RET} C-x )}}.
9384 @node Programming Answer 3, Programming Answer 4, Programming Answer 2, Answers to Exercises
9385 @subsection Programming Tutorial Exercise 3
9388 Each of these functions can be computed using the stack, or using
9389 algebraic entry, whichever way you prefer:
9393 @texline @math{\displaystyle{\sin x \over x}}:
9394 @infoline @expr{sin(x) / x}:
9396 Using the stack: @kbd{C-x ( @key{RET} S @key{TAB} / C-x )}.
9398 Using algebraic entry: @kbd{C-x ( ' sin($)/$ @key{RET} C-x )}.
9401 Computing the logarithm:
9403 Using the stack: @kbd{C-x ( @key{TAB} B C-x )}
9405 Using algebraic entry: @kbd{C-x ( ' log($,$$) @key{RET} C-x )}.
9408 Computing the vector of integers:
9410 Using the stack: @kbd{C-x ( 1 @key{RET} 1 C-u v x C-x )}. (Recall that
9411 @kbd{C-u v x} takes the vector size, starting value, and increment
9414 Alternatively: @kbd{C-x ( ~ v x C-x )}. (The @kbd{~} key pops a
9415 number from the stack and uses it as the prefix argument for the
9418 Using algebraic entry: @kbd{C-x ( ' index($) @key{RET} C-x )}.
9420 @node Programming Answer 4, Programming Answer 5, Programming Answer 3, Answers to Exercises
9421 @subsection Programming Tutorial Exercise 4
9424 Here's one way: @kbd{C-x ( @key{RET} V R + @key{TAB} v l / C-x )}.
9426 @node Programming Answer 5, Programming Answer 6, Programming Answer 4, Answers to Exercises
9427 @subsection Programming Tutorial Exercise 5
9431 2: 1 1: 1.61803398502 2: 1.61803398502
9432 1: 20 . 1: 1.61803398875
9435 1 @key{RET} 20 Z < & 1 + Z > I H P
9440 This answer is quite accurate.
9442 @node Programming Answer 6, Programming Answer 7, Programming Answer 5, Answers to Exercises
9443 @subsection Programming Tutorial Exercise 6
9449 [ [ 0, 1 ] * [a, b] = [b, a + b]
9454 Thus @samp{[0, 1; 1, 1]^n * [1, 1]} computes Fibonacci numbers @expr{n+1}
9455 and @expr{n+2}. Here's one program that does the job:
9458 C-x ( ' [0, 1; 1, 1] ^ ($-1) * [1, 1] @key{RET} v u @key{DEL} C-x )
9462 This program is quite efficient because Calc knows how to raise a
9463 matrix (or other value) to the power @expr{n} in only
9464 @texline @math{\log_2 n}
9465 @infoline @expr{log(n,2)}
9466 steps. For example, this program can compute the 1000th Fibonacci
9467 number (a 209-digit integer!) in about 10 steps; even though the
9468 @kbd{Z < ... Z >} solution had much simpler steps, it would have
9469 required so many steps that it would not have been practical.
9471 @node Programming Answer 7, Programming Answer 8, Programming Answer 6, Answers to Exercises
9472 @subsection Programming Tutorial Exercise 7
9475 The trick here is to compute the harmonic numbers differently, so that
9476 the loop counter itself accumulates the sum of reciprocals. We use
9477 a separate variable to hold the integer counter.
9485 1 t 1 1 @key{RET} 4 Z ( t 2 r 1 1 + s 1 & Z )
9490 The body of the loop goes as follows: First save the harmonic sum
9491 so far in variable 2. Then delete it from the stack; the for loop
9492 itself will take care of remembering it for us. Next, recall the
9493 count from variable 1, add one to it, and feed its reciprocal to
9494 the for loop to use as the step value. The for loop will increase
9495 the ``loop counter'' by that amount and keep going until the
9496 loop counter exceeds 4.
9501 1: 3.99498713092 2: 3.99498713092
9505 r 1 r 2 @key{RET} 31 & +
9509 Thus we find that the 30th harmonic number is 3.99, and the 31st
9510 harmonic number is 4.02.
9512 @node Programming Answer 8, Programming Answer 9, Programming Answer 7, Answers to Exercises
9513 @subsection Programming Tutorial Exercise 8
9516 The first step is to compute the derivative @expr{f'(x)} and thus
9518 @texline @math{\displaystyle{x - {f(x) \over f'(x)}}}.
9519 @infoline @expr{x - f(x)/f'(x)}.
9521 (Because this definition is long, it will be repeated in concise form
9522 below. You can use @w{@kbd{M-# m}} to load it from there. While you are
9523 entering a @kbd{Z ` Z '} body in a macro, Calc simply collects
9524 keystrokes without executing them. In the following diagrams we'll
9525 pretend Calc actually executed the keystrokes as you typed them,
9526 just for purposes of illustration.)
9530 2: sin(cos(x)) - 0.5 3: 4.5
9531 1: 4.5 2: sin(cos(x)) - 0.5
9532 . 1: -(sin(x) cos(cos(x)))
9535 ' sin(cos(x))-0.5 @key{RET} 4.5 m r C-x ( Z ` @key{TAB} @key{RET} a d x @key{RET}
9543 1: x + (sin(cos(x)) - 0.5) / sin(x) cos(cos(x))
9546 / ' x @key{RET} @key{TAB} - t 1
9550 Now, we enter the loop. We'll use a repeat loop with a 20-repetition
9551 limit just in case the method fails to converge for some reason.
9552 (Normally, the @w{@kbd{Z /}} command will stop the loop before all 20
9553 repetitions are done.)
9557 1: 4.5 3: 4.5 2: 4.5
9558 . 2: x + (sin(cos(x)) ... 1: 5.24196456928
9562 20 Z < @key{RET} r 1 @key{TAB} s l x @key{RET}
9566 This is the new guess for @expr{x}. Now we compare it with the
9567 old one to see if we've converged.
9571 3: 5.24196 2: 5.24196 1: 5.24196 1: 5.26345856348
9576 @key{RET} M-@key{TAB} a = Z / Z > Z ' C-x )
9580 The loop converges in just a few steps to this value. To check
9581 the result, we can simply substitute it back into the equation.
9589 @key{RET} ' sin(cos($)) @key{RET}
9593 Let's test the new definition again:
9601 ' x^2-9 @key{RET} 1 X
9605 Once again, here's the full Newton's Method definition:
9609 C-x ( Z ` @key{TAB} @key{RET} a d x @key{RET} / ' x @key{RET} @key{TAB} - t 1
9610 20 Z < @key{RET} r 1 @key{TAB} s l x @key{RET}
9611 @key{RET} M-@key{TAB} a = Z /
9618 @c [fix-ref Nesting and Fixed Points]
9619 It turns out that Calc has a built-in command for applying a formula
9620 repeatedly until it converges to a number. @xref{Nesting and Fixed Points},
9621 to see how to use it.
9623 @c [fix-ref Root Finding]
9624 Also, of course, @kbd{a R} is a built-in command that uses Newton's
9625 method (among others) to look for numerical solutions to any equation.
9626 @xref{Root Finding}.
9628 @node Programming Answer 9, Programming Answer 10, Programming Answer 8, Answers to Exercises
9629 @subsection Programming Tutorial Exercise 9
9632 The first step is to adjust @expr{z} to be greater than 5. A simple
9633 ``for'' loop will do the job here. If @expr{z} is less than 5, we
9634 reduce the problem using
9635 @texline @math{\psi(z) = \psi(z+1) - 1/z}.
9636 @infoline @expr{psi(z) = psi(z+1) - 1/z}. We go
9638 @texline @math{\psi(z+1)},
9639 @infoline @expr{psi(z+1)},
9640 and remember to add back a factor of @expr{-1/z} when we're done. This
9641 step is repeated until @expr{z > 5}.
9643 (Because this definition is long, it will be repeated in concise form
9644 below. You can use @w{@kbd{M-# m}} to load it from there. While you are
9645 entering a @kbd{Z ` Z '} body in a macro, Calc simply collects
9646 keystrokes without executing them. In the following diagrams we'll
9647 pretend Calc actually executed the keystrokes as you typed them,
9648 just for purposes of illustration.)
9655 1.0 @key{RET} C-x ( Z ` s 1 0 t 2
9659 Here, variable 1 holds @expr{z} and variable 2 holds the adjustment
9660 factor. If @expr{z < 5}, we use a loop to increase it.
9662 (By the way, we started with @samp{1.0} instead of the integer 1 because
9663 otherwise the calculation below will try to do exact fractional arithmetic,
9664 and will never converge because fractions compare equal only if they
9665 are exactly equal, not just equal to within the current precision.)
9674 @key{RET} 5 a < Z [ 5 Z ( & s + 2 1 s + 1 1 Z ) r 1 Z ]
9678 Now we compute the initial part of the sum:
9679 @texline @math{\ln z - {1 \over 2z}}
9680 @infoline @expr{ln(z) - 1/2z}
9681 minus the adjustment factor.
9685 2: 1.79175946923 2: 1.7084261359 1: -0.57490719743
9686 1: 0.0833333333333 1: 2.28333333333 .
9693 Now we evaluate the series. We'll use another ``for'' loop counting
9694 up the value of @expr{2 n}. (Calc does have a summation command,
9695 @kbd{a +}, but we'll use loops just to get more practice with them.)
9699 3: -0.5749 3: -0.5749 4: -0.5749 2: -0.5749
9700 2: 2 2: 1:6 3: 1:6 1: 2.3148e-3
9705 2 @key{RET} 40 Z ( @key{RET} k b @key{TAB} @key{RET} r 1 @key{TAB} ^ * /
9712 3: -0.5749 3: -0.5772 2: -0.5772 1: -0.577215664892
9713 2: -0.5749 2: -0.5772 1: 0 .
9714 1: 2.3148e-3 1: -0.5749 .
9717 @key{TAB} @key{RET} M-@key{TAB} - @key{RET} M-@key{TAB} a = Z / 2 Z ) Z ' C-x )
9721 This is the value of
9722 @texline @math{-\gamma},
9723 @infoline @expr{- gamma},
9724 with a slight bit of roundoff error. To get a full 12 digits, let's use
9729 2: -0.577215664892 2: -0.577215664892
9730 1: 1. 1: -0.577215664901532
9732 1. @key{RET} p 16 @key{RET} X
9736 Here's the complete sequence of keystrokes:
9741 @key{RET} 5 a < Z [ 5 Z ( & s + 2 1 s + 1 1 Z ) r 1 Z ]
9743 2 @key{RET} 40 Z ( @key{RET} k b @key{TAB} @key{RET} r 1 @key{TAB} ^ * /
9744 @key{TAB} @key{RET} M-@key{TAB} - @key{RET} M-@key{TAB} a = Z /
9751 @node Programming Answer 10, Programming Answer 11, Programming Answer 9, Answers to Exercises
9752 @subsection Programming Tutorial Exercise 10
9755 Taking the derivative of a term of the form @expr{x^n} will produce
9757 @texline @math{n x^{n-1}}.
9758 @infoline @expr{n x^(n-1)}.
9759 Taking the derivative of a constant
9760 produces zero. From this it is easy to see that the @expr{n}th
9761 derivative of a polynomial, evaluated at @expr{x = 0}, will equal the
9762 coefficient on the @expr{x^n} term times @expr{n!}.
9764 (Because this definition is long, it will be repeated in concise form
9765 below. You can use @w{@kbd{M-# m}} to load it from there. While you are
9766 entering a @kbd{Z ` Z '} body in a macro, Calc simply collects
9767 keystrokes without executing them. In the following diagrams we'll
9768 pretend Calc actually executed the keystrokes as you typed them,
9769 just for purposes of illustration.)
9773 2: 5 x^4 + (x + 1)^2 3: 5 x^4 + (x + 1)^2
9778 ' 5 x^4 + (x+1)^2 @key{RET} 6 C-x ( Z ` [ ] t 1 0 @key{TAB}
9783 Variable 1 will accumulate the vector of coefficients.
9787 2: 0 3: 0 2: 5 x^4 + ...
9788 1: 5 x^4 + ... 2: 5 x^4 + ... 1: 1
9792 Z ( @key{TAB} @key{RET} 0 s l x @key{RET} M-@key{TAB} ! / s | 1
9797 Note that @kbd{s | 1} appends the top-of-stack value to the vector
9798 in a variable; it is completely analogous to @kbd{s + 1}. We could
9799 have written instead, @kbd{r 1 @key{TAB} | t 1}.
9803 1: 20 x^3 + 2 x + 2 1: 0 1: [1, 2, 1, 0, 5, 0, 0]
9806 a d x @key{RET} 1 Z ) @key{DEL} r 1 Z ' C-x )
9810 To convert back, a simple method is just to map the coefficients
9811 against a table of powers of @expr{x}.
9815 2: [1, 2, 1, 0, 5, 0, 0] 2: [1, 2, 1, 0, 5, 0, 0]
9816 1: 6 1: [0, 1, 2, 3, 4, 5, 6]
9819 6 @key{RET} 1 + 0 @key{RET} 1 C-u v x
9826 2: [1, 2, 1, 0, 5, 0, 0] 2: 1 + 2 x + x^2 + 5 x^4
9827 1: [1, x, x^2, x^3, ... ] .
9830 ' x @key{RET} @key{TAB} V M ^ *
9834 Once again, here are the whole polynomial to/from vector programs:
9838 C-x ( Z ` [ ] t 1 0 @key{TAB}
9839 Z ( @key{TAB} @key{RET} 0 s l x @key{RET} M-@key{TAB} ! / s | 1
9845 C-x ( 1 + 0 @key{RET} 1 C-u v x ' x @key{RET} @key{TAB} V M ^ * C-x )
9849 @node Programming Answer 11, Programming Answer 12, Programming Answer 10, Answers to Exercises
9850 @subsection Programming Tutorial Exercise 11
9853 First we define a dummy program to go on the @kbd{z s} key. The true
9854 @w{@kbd{z s}} key is supposed to take two numbers from the stack and
9855 return one number, so @key{DEL} as a dummy definition will make
9856 sure the stack comes out right.
9864 4 @key{RET} 2 C-x ( @key{DEL} C-x ) Z K s @key{RET} 2
9868 The last step replaces the 2 that was eaten during the creation
9869 of the dummy @kbd{z s} command. Now we move on to the real
9870 definition. The recurrence needs to be rewritten slightly,
9871 to the form @expr{s(n,m) = s(n-1,m-1) - (n-1) s(n-1,m)}.
9873 (Because this definition is long, it will be repeated in concise form
9874 below. You can use @kbd{M-# m} to load it from there.)
9884 C-x ( M-2 @key{RET} a = Z [ @key{DEL} @key{DEL} 1 Z :
9891 4: 4 2: 4 2: 3 4: 3 4: 3 3: 3
9892 3: 2 1: 2 1: 2 3: 2 3: 2 2: 2
9893 2: 2 . . 2: 3 2: 3 1: 3
9897 @key{RET} 0 a = Z [ @key{DEL} @key{DEL} 0 Z : @key{TAB} 1 - @key{TAB} M-2 @key{RET} 1 - z s
9902 (Note that the value 3 that our dummy @kbd{z s} produces is not correct;
9903 it is merely a placeholder that will do just as well for now.)
9907 3: 3 4: 3 3: 3 2: 3 1: -6
9908 2: 3 3: 3 2: 3 1: 9 .
9913 M-@key{TAB} M-@key{TAB} @key{TAB} @key{RET} M-@key{TAB} z s * -
9920 1: -6 2: 4 1: 11 2: 11
9924 Z ] Z ] C-x ) Z K s @key{RET} @key{DEL} 4 @key{RET} 2 z s M-@key{RET} k s
9928 Even though the result that we got during the definition was highly
9929 bogus, once the definition is complete the @kbd{z s} command gets
9932 Here's the full program once again:
9936 C-x ( M-2 @key{RET} a =
9937 Z [ @key{DEL} @key{DEL} 1
9939 Z [ @key{DEL} @key{DEL} 0
9940 Z : @key{TAB} 1 - @key{TAB} M-2 @key{RET} 1 - z s
9941 M-@key{TAB} M-@key{TAB} @key{TAB} @key{RET} M-@key{TAB} z s * -
9948 You can read this definition using @kbd{M-# m} (@code{read-kbd-macro})
9949 followed by @kbd{Z K s}, without having to make a dummy definition
9950 first, because @code{read-kbd-macro} doesn't need to execute the
9951 definition as it reads it in. For this reason, @code{M-# m} is often
9952 the easiest way to create recursive programs in Calc.
9954 @node Programming Answer 12, , Programming Answer 11, Answers to Exercises
9955 @subsection Programming Tutorial Exercise 12
9958 This turns out to be a much easier way to solve the problem. Let's
9959 denote Stirling numbers as calls of the function @samp{s}.
9961 First, we store the rewrite rules corresponding to the definition of
9962 Stirling numbers in a convenient variable:
9965 s e StirlingRules @key{RET}
9966 [ s(n,n) := 1 :: n >= 0,
9967 s(n,0) := 0 :: n > 0,
9968 s(n,m) := s(n-1,m-1) - (n-1) s(n-1,m) :: n >= m :: m >= 1 ]
9972 Now, it's just a matter of applying the rules:
9976 2: 4 1: s(4, 2) 1: 11
9980 4 @key{RET} 2 C-x ( ' s($$,$) @key{RET} a r StirlingRules @key{RET} C-x )
9984 As in the case of the @code{fib} rules, it would be useful to put these
9985 rules in @code{EvalRules} and to add a @samp{:: remember} condition to
9988 @c This ends the table-of-contents kludge from above:
9990 \global\let\chapternofonts=\oldchapternofonts
9995 @node Introduction, Data Types, Tutorial, Top
9996 @chapter Introduction
9999 This chapter is the beginning of the Calc reference manual.
10000 It covers basic concepts such as the stack, algebraic and
10001 numeric entry, undo, numeric prefix arguments, etc.
10004 @c (Chapter 2, the Tutorial, has been printed in a separate volume.)
10011 * Algebraic Entry::
10012 * Quick Calculator::
10014 * Prefix Arguments::
10017 * Multiple Calculators::
10018 * Troubleshooting Commands::
10021 @node Basic Commands, Help Commands, Introduction, Introduction
10022 @section Basic Commands
10027 @cindex Starting the Calculator
10028 @cindex Running the Calculator
10029 To start the Calculator in its standard interface, type @kbd{M-x calc}.
10030 By default this creates a pair of small windows, @samp{*Calculator*}
10031 and @samp{*Calc Trail*}. The former displays the contents of the
10032 Calculator stack and is manipulated exclusively through Calc commands.
10033 It is possible (though not usually necessary) to create several Calc
10034 mode buffers each of which has an independent stack, undo list, and
10035 mode settings. There is exactly one Calc Trail buffer; it records a
10036 list of the results of all calculations that have been done. The
10037 Calc Trail buffer uses a variant of Calc mode, so Calculator commands
10038 still work when the trail buffer's window is selected. It is possible
10039 to turn the trail window off, but the @samp{*Calc Trail*} buffer itself
10040 still exists and is updated silently. @xref{Trail Commands}.
10048 In most installations, the @kbd{M-# c} key sequence is a more
10049 convenient way to start the Calculator. Also, @kbd{M-# M-#} and
10050 @kbd{M-# #} are synonyms for @kbd{M-# c} unless you last used Calc
10051 in its Keypad mode.
10055 @pindex calc-execute-extended-command
10056 Most Calc commands use one or two keystrokes. Lower- and upper-case
10057 letters are distinct. Commands may also be entered in full @kbd{M-x} form;
10058 for some commands this is the only form. As a convenience, the @kbd{x}
10059 key (@code{calc-execute-extended-command})
10060 is like @kbd{M-x} except that it enters the initial string @samp{calc-}
10061 for you. For example, the following key sequences are equivalent:
10062 @kbd{S}, @kbd{M-x calc-sin @key{RET}}, @kbd{x sin @key{RET}}.
10064 @cindex Extensions module
10065 @cindex @file{calc-ext} module
10066 The Calculator exists in many parts. When you type @kbd{M-# c}, the
10067 Emacs ``auto-load'' mechanism will bring in only the first part, which
10068 contains the basic arithmetic functions. The other parts will be
10069 auto-loaded the first time you use the more advanced commands like trig
10070 functions or matrix operations. This is done to improve the response time
10071 of the Calculator in the common case when all you need to do is a
10072 little arithmetic. If for some reason the Calculator fails to load an
10073 extension module automatically, you can force it to load all the
10074 extensions by using the @kbd{M-# L} (@code{calc-load-everything})
10075 command. @xref{Mode Settings}.
10077 If you type @kbd{M-x calc} or @kbd{M-# c} with any numeric prefix argument,
10078 the Calculator is loaded if necessary, but it is not actually started.
10079 If the argument is positive, the @file{calc-ext} extensions are also
10080 loaded if necessary. User-written Lisp code that wishes to make use
10081 of Calc's arithmetic routines can use @samp{(calc 0)} or @samp{(calc 1)}
10082 to auto-load the Calculator.
10086 If you type @kbd{M-# b}, then next time you use @kbd{M-# c} you
10087 will get a Calculator that uses the full height of the Emacs screen.
10088 When full-screen mode is on, @kbd{M-# c} runs the @code{full-calc}
10089 command instead of @code{calc}. From the Unix shell you can type
10090 @samp{emacs -f full-calc} to start a new Emacs specifically for use
10091 as a calculator. When Calc is started from the Emacs command line
10092 like this, Calc's normal ``quit'' commands actually quit Emacs itself.
10095 @pindex calc-other-window
10096 The @kbd{M-# o} command is like @kbd{M-# c} except that the Calc
10097 window is not actually selected. If you are already in the Calc
10098 window, @kbd{M-# o} switches you out of it. (The regular Emacs
10099 @kbd{C-x o} command would also work for this, but it has a
10100 tendency to drop you into the Calc Trail window instead, which
10101 @kbd{M-# o} takes care not to do.)
10106 For one quick calculation, you can type @kbd{M-# q} (@code{quick-calc})
10107 which prompts you for a formula (like @samp{2+3/4}). The result is
10108 displayed at the bottom of the Emacs screen without ever creating
10109 any special Calculator windows. @xref{Quick Calculator}.
10114 Finally, if you are using the X window system you may want to try
10115 @kbd{M-# k} (@code{calc-keypad}) which runs Calc with a
10116 ``calculator keypad'' picture as well as a stack display. Click on
10117 the keys with the mouse to operate the calculator. @xref{Keypad Mode}.
10121 @cindex Quitting the Calculator
10122 @cindex Exiting the Calculator
10123 The @kbd{q} key (@code{calc-quit}) exits Calc mode and closes the
10124 Calculator's window(s). It does not delete the Calculator buffers.
10125 If you type @kbd{M-x calc} again, the Calculator will reappear with the
10126 contents of the stack intact. Typing @kbd{M-# c} or @kbd{M-# M-#}
10127 again from inside the Calculator buffer is equivalent to executing
10128 @code{calc-quit}; you can think of @kbd{M-# M-#} as toggling the
10129 Calculator on and off.
10132 The @kbd{M-# x} command also turns the Calculator off, no matter which
10133 user interface (standard, Keypad, or Embedded) is currently active.
10134 It also cancels @code{calc-edit} mode if used from there.
10136 @kindex d @key{SPC}
10137 @pindex calc-refresh
10138 @cindex Refreshing a garbled display
10139 @cindex Garbled displays, refreshing
10140 The @kbd{d @key{SPC}} key sequence (@code{calc-refresh}) redraws the contents
10141 of the Calculator buffer from memory. Use this if the contents of the
10142 buffer have been damaged somehow.
10147 The @kbd{o} key (@code{calc-realign}) moves the cursor back to its
10148 ``home'' position at the bottom of the Calculator buffer.
10152 @pindex calc-scroll-left
10153 @pindex calc-scroll-right
10154 @cindex Horizontal scrolling
10156 @cindex Wide text, scrolling
10157 The @kbd{<} and @kbd{>} keys are bound to @code{calc-scroll-left} and
10158 @code{calc-scroll-right}. These are just like the normal horizontal
10159 scrolling commands except that they scroll one half-screen at a time by
10160 default. (Calc formats its output to fit within the bounds of the
10161 window whenever it can.)
10165 @pindex calc-scroll-down
10166 @pindex calc-scroll-up
10167 @cindex Vertical scrolling
10168 The @kbd{@{} and @kbd{@}} keys are bound to @code{calc-scroll-down}
10169 and @code{calc-scroll-up}. They scroll up or down by one-half the
10170 height of the Calc window.
10174 The @kbd{M-# 0} command (@code{calc-reset}; that's @kbd{M-#} followed
10175 by a zero) resets the Calculator to its initial state. This clears
10176 the stack, resets all the modes to their initial values (the values
10177 that were saved with @kbd{m m} (@code{calc-save-modes})), clears the
10178 caches (@pxref{Caches}), and so on. (It does @emph{not} erase the
10179 values of any variables.) With an argument of 0, Calc will be reset to
10180 its default state; namely, the modes will be given their default values.
10181 With a positive prefix argument, @kbd{M-# 0} preserves the contents of
10182 the stack but resets everything else to its initial state; with a
10183 negative prefix argument, @kbd{M-# 0} preserves the contents of the
10184 stack but resets everything else to its default state.
10186 @pindex calc-version
10187 The @kbd{M-x calc-version} command displays the current version number
10188 of Calc and the name of the person who installed it on your system.
10189 (This information is also present in the @samp{*Calc Trail*} buffer,
10190 and in the output of the @kbd{h h} command.)
10192 @node Help Commands, Stack Basics, Basic Commands, Introduction
10193 @section Help Commands
10196 @cindex Help commands
10199 The @kbd{?} key (@code{calc-help}) displays a series of brief help messages.
10200 Some keys (such as @kbd{b} and @kbd{d}) are prefix keys, like Emacs'
10201 @key{ESC} and @kbd{C-x} prefixes. You can type
10202 @kbd{?} after a prefix to see a list of commands beginning with that
10203 prefix. (If the message includes @samp{[MORE]}, press @kbd{?} again
10204 to see additional commands for that prefix.)
10207 @pindex calc-full-help
10208 The @kbd{h h} (@code{calc-full-help}) command displays all the @kbd{?}
10209 responses at once. When printed, this makes a nice, compact (three pages)
10210 summary of Calc keystrokes.
10212 In general, the @kbd{h} key prefix introduces various commands that
10213 provide help within Calc. Many of the @kbd{h} key functions are
10214 Calc-specific analogues to the @kbd{C-h} functions for Emacs help.
10220 The @kbd{h i} (@code{calc-info}) command runs the Emacs Info system
10221 to read this manual on-line. This is basically the same as typing
10222 @kbd{C-h i} (the regular way to run the Info system), then, if Info
10223 is not already in the Calc manual, selecting the beginning of the
10224 manual. The @kbd{M-# i} command is another way to read the Calc
10225 manual; it is different from @kbd{h i} in that it works any time,
10226 not just inside Calc. The plain @kbd{i} key is also equivalent to
10227 @kbd{h i}, though this key is obsolete and may be replaced with a
10228 different command in a future version of Calc.
10232 @pindex calc-tutorial
10233 The @kbd{h t} (@code{calc-tutorial}) command runs the Info system on
10234 the Tutorial section of the Calc manual. It is like @kbd{h i},
10235 except that it selects the starting node of the tutorial rather
10236 than the beginning of the whole manual. (It actually selects the
10237 node ``Interactive Tutorial'' which tells a few things about
10238 using the Info system before going on to the actual tutorial.)
10239 The @kbd{M-# t} key is equivalent to @kbd{h t} (but it works at
10244 @pindex calc-info-summary
10245 The @kbd{h s} (@code{calc-info-summary}) command runs the Info system
10246 on the Summary node of the Calc manual. @xref{Summary}. The @kbd{M-# s}
10247 key is equivalent to @kbd{h s}.
10250 @pindex calc-describe-key
10251 The @kbd{h k} (@code{calc-describe-key}) command looks up a key
10252 sequence in the Calc manual. For example, @kbd{h k H a S} looks
10253 up the documentation on the @kbd{H a S} (@code{calc-solve-for})
10254 command. This works by looking up the textual description of
10255 the key(s) in the Key Index of the manual, then jumping to the
10256 node indicated by the index.
10258 Most Calc commands do not have traditional Emacs documentation
10259 strings, since the @kbd{h k} command is both more convenient and
10260 more instructive. This means the regular Emacs @kbd{C-h k}
10261 (@code{describe-key}) command will not be useful for Calc keystrokes.
10264 @pindex calc-describe-key-briefly
10265 The @kbd{h c} (@code{calc-describe-key-briefly}) command reads a
10266 key sequence and displays a brief one-line description of it at
10267 the bottom of the screen. It looks for the key sequence in the
10268 Summary node of the Calc manual; if it doesn't find the sequence
10269 there, it acts just like its regular Emacs counterpart @kbd{C-h c}
10270 (@code{describe-key-briefly}). For example, @kbd{h c H a S}
10271 gives the description:
10274 H a S runs calc-solve-for: a `H a S' v => fsolve(a,v) (?=notes)
10278 which means the command @kbd{H a S} or @kbd{H M-x calc-solve-for}
10279 takes a value @expr{a} from the stack, prompts for a value @expr{v},
10280 then applies the algebraic function @code{fsolve} to these values.
10281 The @samp{?=notes} message means you can now type @kbd{?} to see
10282 additional notes from the summary that apply to this command.
10285 @pindex calc-describe-function
10286 The @kbd{h f} (@code{calc-describe-function}) command looks up an
10287 algebraic function or a command name in the Calc manual. Enter an
10288 algebraic function name to look up that function in the Function
10289 Index or enter a command name beginning with @samp{calc-} to look it
10290 up in the Command Index. This command will also look up operator
10291 symbols that can appear in algebraic formulas, like @samp{%} and
10295 @pindex calc-describe-variable
10296 The @kbd{h v} (@code{calc-describe-variable}) command looks up a
10297 variable in the Calc manual. Enter a variable name like @code{pi} or
10298 @code{PlotRejects}.
10301 @pindex describe-bindings
10302 The @kbd{h b} (@code{calc-describe-bindings}) command is just like
10303 @kbd{C-h b}, except that only local (Calc-related) key bindings are
10307 The @kbd{h n} or @kbd{h C-n} (@code{calc-view-news}) command displays
10308 the ``news'' or change history of Calc. This is kept in the file
10309 @file{README}, which Calc looks for in the same directory as the Calc
10315 The @kbd{h C-c}, @kbd{h C-d}, and @kbd{h C-w} keys display copying,
10316 distribution, and warranty information about Calc. These work by
10317 pulling up the appropriate parts of the ``Copying'' or ``Reporting
10318 Bugs'' sections of the manual.
10320 @node Stack Basics, Numeric Entry, Help Commands, Introduction
10321 @section Stack Basics
10324 @cindex Stack basics
10325 @c [fix-tut RPN Calculations and the Stack]
10326 Calc uses RPN notation. If you are not familiar with RPN, @pxref{RPN
10329 To add the numbers 1 and 2 in Calc you would type the keys:
10330 @kbd{1 @key{RET} 2 +}.
10331 (@key{RET} corresponds to the @key{ENTER} key on most calculators.)
10332 The first three keystrokes ``push'' the numbers 1 and 2 onto the stack. The
10333 @kbd{+} key always ``pops'' the top two numbers from the stack, adds them,
10334 and pushes the result (3) back onto the stack. This number is ready for
10335 further calculations: @kbd{5 -} pushes 5 onto the stack, then pops the
10336 3 and 5, subtracts them, and pushes the result (@mathit{-2}).
10338 Note that the ``top'' of the stack actually appears at the @emph{bottom}
10339 of the buffer. A line containing a single @samp{.} character signifies
10340 the end of the buffer; Calculator commands operate on the number(s)
10341 directly above this line. The @kbd{d t} (@code{calc-truncate-stack})
10342 command allows you to move the @samp{.} marker up and down in the stack;
10343 @pxref{Truncating the Stack}.
10346 @pindex calc-line-numbering
10347 Stack elements are numbered consecutively, with number 1 being the top of
10348 the stack. These line numbers are ordinarily displayed on the lefthand side
10349 of the window. The @kbd{d l} (@code{calc-line-numbering}) command controls
10350 whether these numbers appear. (Line numbers may be turned off since they
10351 slow the Calculator down a bit and also clutter the display.)
10354 @pindex calc-realign
10355 The unshifted letter @kbd{o} (@code{calc-realign}) command repositions
10356 the cursor to its top-of-stack ``home'' position. It also undoes any
10357 horizontal scrolling in the window. If you give it a numeric prefix
10358 argument, it instead moves the cursor to the specified stack element.
10360 The @key{RET} (or equivalent @key{SPC}) key is only required to separate
10361 two consecutive numbers.
10362 (After all, if you typed @kbd{1 2} by themselves the Calculator
10363 would enter the number 12.) If you press @key{RET} or @key{SPC} @emph{not}
10364 right after typing a number, the key duplicates the number on the top of
10365 the stack. @kbd{@key{RET} *} is thus a handy way to square a number.
10367 The @key{DEL} key pops and throws away the top number on the stack.
10368 The @key{TAB} key swaps the top two objects on the stack.
10369 @xref{Stack and Trail}, for descriptions of these and other stack-related
10372 @node Numeric Entry, Algebraic Entry, Stack Basics, Introduction
10373 @section Numeric Entry
10379 @cindex Numeric entry
10380 @cindex Entering numbers
10381 Pressing a digit or other numeric key begins numeric entry using the
10382 minibuffer. The number is pushed on the stack when you press the @key{RET}
10383 or @key{SPC} keys. If you press any other non-numeric key, the number is
10384 pushed onto the stack and the appropriate operation is performed. If
10385 you press a numeric key which is not valid, the key is ignored.
10387 @cindex Minus signs
10388 @cindex Negative numbers, entering
10390 There are three different concepts corresponding to the word ``minus,''
10391 typified by @expr{a-b} (subtraction), @expr{-x}
10392 (change-sign), and @expr{-5} (negative number). Calc uses three
10393 different keys for these operations, respectively:
10394 @kbd{-}, @kbd{n}, and @kbd{_} (the underscore). The @kbd{-} key subtracts
10395 the two numbers on the top of the stack. The @kbd{n} key changes the sign
10396 of the number on the top of the stack or the number currently being entered.
10397 The @kbd{_} key begins entry of a negative number or changes the sign of
10398 the number currently being entered. The following sequences all enter the
10399 number @mathit{-5} onto the stack: @kbd{0 @key{RET} 5 -}, @kbd{5 n @key{RET}},
10400 @kbd{5 @key{RET} n}, @kbd{_ 5 @key{RET}}, @kbd{5 _ @key{RET}}.
10402 Some other keys are active during numeric entry, such as @kbd{#} for
10403 non-decimal numbers, @kbd{:} for fractions, and @kbd{@@} for HMS forms.
10404 These notations are described later in this manual with the corresponding
10405 data types. @xref{Data Types}.
10407 During numeric entry, the only editing key available is @key{DEL}.
10409 @node Algebraic Entry, Quick Calculator, Numeric Entry, Introduction
10410 @section Algebraic Entry
10414 @pindex calc-algebraic-entry
10415 @cindex Algebraic notation
10416 @cindex Formulas, entering
10417 Calculations can also be entered in algebraic form. This is accomplished
10418 by typing the apostrophe key, @kbd{'}, followed by the expression in
10419 standard format: @kbd{@key{'} 2+3*4 @key{RET}} computes
10420 @texline @math{2+(3\times4) = 14}
10421 @infoline @expr{2+(3*4) = 14}
10422 and pushes that on the stack. If you wish you can
10423 ignore the RPN aspect of Calc altogether and simply enter algebraic
10424 expressions in this way. You may want to use @key{DEL} every so often to
10425 clear previous results off the stack.
10427 You can press the apostrophe key during normal numeric entry to switch
10428 the half-entered number into Algebraic entry mode. One reason to do this
10429 would be to use the full Emacs cursor motion and editing keys, which are
10430 available during algebraic entry but not during numeric entry.
10432 In the same vein, during either numeric or algebraic entry you can
10433 press @kbd{`} (backquote) to switch to @code{calc-edit} mode, where
10434 you complete your half-finished entry in a separate buffer.
10435 @xref{Editing Stack Entries}.
10438 @pindex calc-algebraic-mode
10439 @cindex Algebraic Mode
10440 If you prefer algebraic entry, you can use the command @kbd{m a}
10441 (@code{calc-algebraic-mode}) to set Algebraic mode. In this mode,
10442 digits and other keys that would normally start numeric entry instead
10443 start full algebraic entry; as long as your formula begins with a digit
10444 you can omit the apostrophe. Open parentheses and square brackets also
10445 begin algebraic entry. You can still do RPN calculations in this mode,
10446 but you will have to press @key{RET} to terminate every number:
10447 @kbd{2 @key{RET} 3 @key{RET} * 4 @key{RET} +} would accomplish the same
10448 thing as @kbd{2*3+4 @key{RET}}.
10450 @cindex Incomplete Algebraic Mode
10451 If you give a numeric prefix argument like @kbd{C-u} to the @kbd{m a}
10452 command, it enables Incomplete Algebraic mode; this is like regular
10453 Algebraic mode except that it applies to the @kbd{(} and @kbd{[} keys
10454 only. Numeric keys still begin a numeric entry in this mode.
10457 @pindex calc-total-algebraic-mode
10458 @cindex Total Algebraic Mode
10459 The @kbd{m t} (@code{calc-total-algebraic-mode}) gives you an even
10460 stronger algebraic-entry mode, in which @emph{all} regular letter and
10461 punctuation keys begin algebraic entry. Use this if you prefer typing
10462 @w{@kbd{sqrt( )}} instead of @kbd{Q}, @w{@kbd{factor( )}} instead of
10463 @kbd{a f}, and so on. To type regular Calc commands when you are in
10464 Total Algebraic mode, hold down the @key{META} key. Thus @kbd{M-q}
10465 is the command to quit Calc, @kbd{M-p} sets the precision, and
10466 @kbd{M-m t} (or @kbd{M-m M-t}, if you prefer) turns Total Algebraic
10467 mode back off again. Meta keys also terminate algebraic entry, so
10468 that @kbd{2+3 M-S} is equivalent to @kbd{2+3 @key{RET} M-S}. The symbol
10469 @samp{Alg*} will appear in the mode line whenever you are in this mode.
10471 Pressing @kbd{'} (the apostrophe) a second time re-enters the previous
10472 algebraic formula. You can then use the normal Emacs editing keys to
10473 modify this formula to your liking before pressing @key{RET}.
10476 @cindex Formulas, referring to stack
10477 Within a formula entered from the keyboard, the symbol @kbd{$}
10478 represents the number on the top of the stack. If an entered formula
10479 contains any @kbd{$} characters, the Calculator replaces the top of
10480 stack with that formula rather than simply pushing the formula onto the
10481 stack. Thus, @kbd{' 1+2 @key{RET}} pushes 3 on the stack, and @kbd{$*2
10482 @key{RET}} replaces it with 6. Note that the @kbd{$} key always
10483 initiates algebraic entry; the @kbd{'} is unnecessary if @kbd{$} is the
10484 first character in the new formula.
10486 Higher stack elements can be accessed from an entered formula with the
10487 symbols @kbd{$$}, @kbd{$$$}, and so on. The number of stack elements
10488 removed (to be replaced by the entered values) equals the number of dollar
10489 signs in the longest such symbol in the formula. For example, @samp{$$+$$$}
10490 adds the second and third stack elements, replacing the top three elements
10491 with the answer. (All information about the top stack element is thus lost
10492 since no single @samp{$} appears in this formula.)
10494 A slightly different way to refer to stack elements is with a dollar
10495 sign followed by a number: @samp{$1}, @samp{$2}, and so on are much
10496 like @samp{$}, @samp{$$}, etc., except that stack entries referred
10497 to numerically are not replaced by the algebraic entry. That is, while
10498 @samp{$+1} replaces 5 on the stack with 6, @samp{$1+1} leaves the 5
10499 on the stack and pushes an additional 6.
10501 If a sequence of formulas are entered separated by commas, each formula
10502 is pushed onto the stack in turn. For example, @samp{1,2,3} pushes
10503 those three numbers onto the stack (leaving the 3 at the top), and
10504 @samp{$+1,$-1} replaces a 5 on the stack with 4 followed by 6. Also,
10505 @samp{$,$$} exchanges the top two elements of the stack, just like the
10508 You can finish an algebraic entry with @kbd{M-=} or @kbd{M-@key{RET}} instead
10509 of @key{RET}. This uses @kbd{=} to evaluate the variables in each
10510 formula that goes onto the stack. (Thus @kbd{' pi @key{RET}} pushes
10511 the variable @samp{pi}, but @kbd{' pi M-@key{RET}} pushes 3.1415.)
10513 If you finish your algebraic entry by pressing @key{LFD} (or @kbd{C-j})
10514 instead of @key{RET}, Calc disables the default simplifications
10515 (as if by @kbd{m O}; @pxref{Simplification Modes}) while the entry
10516 is being pushed on the stack. Thus @kbd{' 1+2 @key{RET}} pushes 3
10517 on the stack, but @kbd{' 1+2 @key{LFD}} pushes the formula @expr{1+2};
10518 you might then press @kbd{=} when it is time to evaluate this formula.
10520 @node Quick Calculator, Prefix Arguments, Algebraic Entry, Introduction
10521 @section ``Quick Calculator'' Mode
10526 @cindex Quick Calculator
10527 There is another way to invoke the Calculator if all you need to do
10528 is make one or two quick calculations. Type @kbd{M-# q} (or
10529 @kbd{M-x quick-calc}), then type any formula as an algebraic entry.
10530 The Calculator will compute the result and display it in the echo
10531 area, without ever actually putting up a Calc window.
10533 You can use the @kbd{$} character in a Quick Calculator formula to
10534 refer to the previous Quick Calculator result. Older results are
10535 not retained; the Quick Calculator has no effect on the full
10536 Calculator's stack or trail. If you compute a result and then
10537 forget what it was, just run @code{M-# q} again and enter
10538 @samp{$} as the formula.
10540 If this is the first time you have used the Calculator in this Emacs
10541 session, the @kbd{M-# q} command will create the @code{*Calculator*}
10542 buffer and perform all the usual initializations; it simply will
10543 refrain from putting that buffer up in a new window. The Quick
10544 Calculator refers to the @code{*Calculator*} buffer for all mode
10545 settings. Thus, for example, to set the precision that the Quick
10546 Calculator uses, simply run the full Calculator momentarily and use
10547 the regular @kbd{p} command.
10549 If you use @code{M-# q} from inside the Calculator buffer, the
10550 effect is the same as pressing the apostrophe key (algebraic entry).
10552 The result of a Quick calculation is placed in the Emacs ``kill ring''
10553 as well as being displayed. A subsequent @kbd{C-y} command will
10554 yank the result into the editing buffer. You can also use this
10555 to yank the result into the next @kbd{M-# q} input line as a more
10556 explicit alternative to @kbd{$} notation, or to yank the result
10557 into the Calculator stack after typing @kbd{M-# c}.
10559 If you finish your formula by typing @key{LFD} (or @kbd{C-j}) instead
10560 of @key{RET}, the result is inserted immediately into the current
10561 buffer rather than going into the kill ring.
10563 Quick Calculator results are actually evaluated as if by the @kbd{=}
10564 key (which replaces variable names by their stored values, if any).
10565 If the formula you enter is an assignment to a variable using the
10566 @samp{:=} operator, say, @samp{foo := 2 + 3} or @samp{foo := foo + 1},
10567 then the result of the evaluation is stored in that Calc variable.
10568 @xref{Store and Recall}.
10570 If the result is an integer and the current display radix is decimal,
10571 the number will also be displayed in hex and octal formats. If the
10572 integer is in the range from 1 to 126, it will also be displayed as
10573 an ASCII character.
10575 For example, the quoted character @samp{"x"} produces the vector
10576 result @samp{[120]} (because 120 is the ASCII code of the lower-case
10577 `x'; @pxref{Strings}). Since this is a vector, not an integer, it
10578 is displayed only according to the current mode settings. But
10579 running Quick Calc again and entering @samp{120} will produce the
10580 result @samp{120 (16#78, 8#170, x)} which shows the number in its
10581 decimal, hexadecimal, octal, and ASCII forms.
10583 Please note that the Quick Calculator is not any faster at loading
10584 or computing the answer than the full Calculator; the name ``quick''
10585 merely refers to the fact that it's much less hassle to use for
10586 small calculations.
10588 @node Prefix Arguments, Undo, Quick Calculator, Introduction
10589 @section Numeric Prefix Arguments
10592 Many Calculator commands use numeric prefix arguments. Some, such as
10593 @kbd{d s} (@code{calc-sci-notation}), set a parameter to the value of
10594 the prefix argument or use a default if you don't use a prefix.
10595 Others (like @kbd{d f} (@code{calc-fix-notation})) require an argument
10596 and prompt for a number if you don't give one as a prefix.
10598 As a rule, stack-manipulation commands accept a numeric prefix argument
10599 which is interpreted as an index into the stack. A positive argument
10600 operates on the top @var{n} stack entries; a negative argument operates
10601 on the @var{n}th stack entry in isolation; and a zero argument operates
10602 on the entire stack.
10604 Most commands that perform computations (such as the arithmetic and
10605 scientific functions) accept a numeric prefix argument that allows the
10606 operation to be applied across many stack elements. For unary operations
10607 (that is, functions of one argument like absolute value or complex
10608 conjugate), a positive prefix argument applies that function to the top
10609 @var{n} stack entries simultaneously, and a negative argument applies it
10610 to the @var{n}th stack entry only. For binary operations (functions of
10611 two arguments like addition, GCD, and vector concatenation), a positive
10612 prefix argument ``reduces'' the function across the top @var{n}
10613 stack elements (for example, @kbd{C-u 5 +} sums the top 5 stack entries;
10614 @pxref{Reducing and Mapping}), and a negative argument maps the next-to-top
10615 @var{n} stack elements with the top stack element as a second argument
10616 (for example, @kbd{7 c-u -5 +} adds 7 to the top 5 stack elements).
10617 This feature is not available for operations which use the numeric prefix
10618 argument for some other purpose.
10620 Numeric prefixes are specified the same way as always in Emacs: Press
10621 a sequence of @key{META}-digits, or press @key{ESC} followed by digits,
10622 or press @kbd{C-u} followed by digits. Some commands treat plain
10623 @kbd{C-u} (without any actual digits) specially.
10626 @pindex calc-num-prefix
10627 You can type @kbd{~} (@code{calc-num-prefix}) to pop an integer from the
10628 top of the stack and enter it as the numeric prefix for the next command.
10629 For example, @kbd{C-u 16 p} sets the precision to 16 digits; an alternate
10630 (silly) way to do this would be @kbd{2 @key{RET} 4 ^ ~ p}, i.e., compute 2
10631 to the fourth power and set the precision to that value.
10633 Conversely, if you have typed a numeric prefix argument the @kbd{~} key
10634 pushes it onto the stack in the form of an integer.
10636 @node Undo, Error Messages, Prefix Arguments, Introduction
10637 @section Undoing Mistakes
10643 @cindex Mistakes, undoing
10644 @cindex Undoing mistakes
10645 @cindex Errors, undoing
10646 The shift-@kbd{U} key (@code{calc-undo}) undoes the most recent operation.
10647 If that operation added or dropped objects from the stack, those objects
10648 are removed or restored. If it was a ``store'' operation, you are
10649 queried whether or not to restore the variable to its original value.
10650 The @kbd{U} key may be pressed any number of times to undo successively
10651 farther back in time; with a numeric prefix argument it undoes a
10652 specified number of operations. The undo history is cleared only by the
10653 @kbd{q} (@code{calc-quit}) command. (Recall that @kbd{M-# c} is
10654 synonymous with @code{calc-quit} while inside the Calculator; this
10655 also clears the undo history.)
10657 Currently the mode-setting commands (like @code{calc-precision}) are not
10658 undoable. You can undo past a point where you changed a mode, but you
10659 will need to reset the mode yourself.
10663 @cindex Redoing after an Undo
10664 The shift-@kbd{D} key (@code{calc-redo}) redoes an operation that was
10665 mistakenly undone. Pressing @kbd{U} with a negative prefix argument is
10666 equivalent to executing @code{calc-redo}. You can redo any number of
10667 times, up to the number of recent consecutive undo commands. Redo
10668 information is cleared whenever you give any command that adds new undo
10669 information, i.e., if you undo, then enter a number on the stack or make
10670 any other change, then it will be too late to redo.
10672 @kindex M-@key{RET}
10673 @pindex calc-last-args
10674 @cindex Last-arguments feature
10675 @cindex Arguments, restoring
10676 The @kbd{M-@key{RET}} key (@code{calc-last-args}) is like undo in that
10677 it restores the arguments of the most recent command onto the stack;
10678 however, it does not remove the result of that command. Given a numeric
10679 prefix argument, this command applies to the @expr{n}th most recent
10680 command which removed items from the stack; it pushes those items back
10683 The @kbd{K} (@code{calc-keep-args}) command provides a related function
10684 to @kbd{M-@key{RET}}. @xref{Stack and Trail}.
10686 It is also possible to recall previous results or inputs using the trail.
10687 @xref{Trail Commands}.
10689 The standard Emacs @kbd{C-_} undo key is recognized as a synonym for @kbd{U}.
10691 @node Error Messages, Multiple Calculators, Undo, Introduction
10692 @section Error Messages
10697 @cindex Errors, messages
10698 @cindex Why did an error occur?
10699 Many situations that would produce an error message in other calculators
10700 simply create unsimplified formulas in the Emacs Calculator. For example,
10701 @kbd{1 @key{RET} 0 /} pushes the formula @expr{1 / 0}; @w{@kbd{0 L}} pushes
10702 the formula @samp{ln(0)}. Floating-point overflow and underflow are also
10703 reasons for this to happen.
10705 When a function call must be left in symbolic form, Calc usually
10706 produces a message explaining why. Messages that are probably
10707 surprising or indicative of user errors are displayed automatically.
10708 Other messages are simply kept in Calc's memory and are displayed only
10709 if you type @kbd{w} (@code{calc-why}). You can also press @kbd{w} if
10710 the same computation results in several messages. (The first message
10711 will end with @samp{[w=more]} in this case.)
10714 @pindex calc-auto-why
10715 The @kbd{d w} (@code{calc-auto-why}) command controls when error messages
10716 are displayed automatically. (Calc effectively presses @kbd{w} for you
10717 after your computation finishes.) By default, this occurs only for
10718 ``important'' messages. The other possible modes are to report
10719 @emph{all} messages automatically, or to report none automatically (so
10720 that you must always press @kbd{w} yourself to see the messages).
10722 @node Multiple Calculators, Troubleshooting Commands, Error Messages, Introduction
10723 @section Multiple Calculators
10726 @pindex another-calc
10727 It is possible to have any number of Calc mode buffers at once.
10728 Usually this is done by executing @kbd{M-x another-calc}, which
10729 is similar to @kbd{M-# c} except that if a @samp{*Calculator*}
10730 buffer already exists, a new, independent one with a name of the
10731 form @samp{*Calculator*<@var{n}>} is created. You can also use the
10732 command @code{calc-mode} to put any buffer into Calculator mode, but
10733 this would ordinarily never be done.
10735 The @kbd{q} (@code{calc-quit}) command does not destroy a Calculator buffer;
10736 it only closes its window. Use @kbd{M-x kill-buffer} to destroy a
10739 Each Calculator buffer keeps its own stack, undo list, and mode settings
10740 such as precision, angular mode, and display formats. In Emacs terms,
10741 variables such as @code{calc-stack} are buffer-local variables. The
10742 global default values of these variables are used only when a new
10743 Calculator buffer is created. The @code{calc-quit} command saves
10744 the stack and mode settings of the buffer being quit as the new defaults.
10746 There is only one trail buffer, @samp{*Calc Trail*}, used by all
10747 Calculator buffers.
10749 @node Troubleshooting Commands, , Multiple Calculators, Introduction
10750 @section Troubleshooting Commands
10753 This section describes commands you can use in case a computation
10754 incorrectly fails or gives the wrong answer.
10756 @xref{Reporting Bugs}, if you find a problem that appears to be due
10757 to a bug or deficiency in Calc.
10760 * Autoloading Problems::
10761 * Recursion Depth::
10766 @node Autoloading Problems, Recursion Depth, Troubleshooting Commands, Troubleshooting Commands
10767 @subsection Autoloading Problems
10770 The Calc program is split into many component files; components are
10771 loaded automatically as you use various commands that require them.
10772 Occasionally Calc may lose track of when a certain component is
10773 necessary; typically this means you will type a command and it won't
10774 work because some function you've never heard of was undefined.
10777 @pindex calc-load-everything
10778 If this happens, the easiest workaround is to type @kbd{M-# L}
10779 (@code{calc-load-everything}) to force all the parts of Calc to be
10780 loaded right away. This will cause Emacs to take up a lot more
10781 memory than it would otherwise, but it's guaranteed to fix the problem.
10783 @node Recursion Depth, Caches, Autoloading Problems, Troubleshooting Commands
10784 @subsection Recursion Depth
10789 @pindex calc-more-recursion-depth
10790 @pindex calc-less-recursion-depth
10791 @cindex Recursion depth
10792 @cindex ``Computation got stuck'' message
10793 @cindex @code{max-lisp-eval-depth}
10794 @cindex @code{max-specpdl-size}
10795 Calc uses recursion in many of its calculations. Emacs Lisp keeps a
10796 variable @code{max-lisp-eval-depth} which limits the amount of recursion
10797 possible in an attempt to recover from program bugs. If a calculation
10798 ever halts incorrectly with the message ``Computation got stuck or
10799 ran too long,'' use the @kbd{M} command (@code{calc-more-recursion-depth})
10800 to increase this limit. (Of course, this will not help if the
10801 calculation really did get stuck due to some problem inside Calc.)
10803 The limit is always increased (multiplied) by a factor of two. There
10804 is also an @kbd{I M} (@code{calc-less-recursion-depth}) command which
10805 decreases this limit by a factor of two, down to a minimum value of 200.
10806 The default value is 1000.
10808 These commands also double or halve @code{max-specpdl-size}, another
10809 internal Lisp recursion limit. The minimum value for this limit is 600.
10811 @node Caches, Debugging Calc, Recursion Depth, Troubleshooting Commands
10816 @cindex Flushing caches
10817 Calc saves certain values after they have been computed once. For
10818 example, the @kbd{P} (@code{calc-pi}) command initially ``knows'' the
10819 constant @cpi{} to about 20 decimal places; if the current precision
10820 is greater than this, it will recompute @cpi{} using a series
10821 approximation. This value will not need to be recomputed ever again
10822 unless you raise the precision still further. Many operations such as
10823 logarithms and sines make use of similarly cached values such as
10825 @texline @math{\ln 2}.
10826 @infoline @expr{ln(2)}.
10827 The visible effect of caching is that
10828 high-precision computations may seem to do extra work the first time.
10829 Other things cached include powers of two (for the binary arithmetic
10830 functions), matrix inverses and determinants, symbolic integrals, and
10831 data points computed by the graphing commands.
10833 @pindex calc-flush-caches
10834 If you suspect a Calculator cache has become corrupt, you can use the
10835 @code{calc-flush-caches} command to reset all caches to the empty state.
10836 (This should only be necessary in the event of bugs in the Calculator.)
10837 The @kbd{M-# 0} (with the zero key) command also resets caches along
10838 with all other aspects of the Calculator's state.
10840 @node Debugging Calc, , Caches, Troubleshooting Commands
10841 @subsection Debugging Calc
10844 A few commands exist to help in the debugging of Calc commands.
10845 @xref{Programming}, to see the various ways that you can write
10846 your own Calc commands.
10849 @pindex calc-timing
10850 The @kbd{Z T} (@code{calc-timing}) command turns on and off a mode
10851 in which the timing of slow commands is reported in the Trail.
10852 Any Calc command that takes two seconds or longer writes a line
10853 to the Trail showing how many seconds it took. This value is
10854 accurate only to within one second.
10856 All steps of executing a command are included; in particular, time
10857 taken to format the result for display in the stack and trail is
10858 counted. Some prompts also count time taken waiting for them to
10859 be answered, while others do not; this depends on the exact
10860 implementation of the command. For best results, if you are timing
10861 a sequence that includes prompts or multiple commands, define a
10862 keyboard macro to run the whole sequence at once. Calc's @kbd{X}
10863 command (@pxref{Keyboard Macros}) will then report the time taken
10864 to execute the whole macro.
10866 Another advantage of the @kbd{X} command is that while it is
10867 executing, the stack and trail are not updated from step to step.
10868 So if you expect the output of your test sequence to leave a result
10869 that may take a long time to format and you don't wish to count
10870 this formatting time, end your sequence with a @key{DEL} keystroke
10871 to clear the result from the stack. When you run the sequence with
10872 @kbd{X}, Calc will never bother to format the large result.
10874 Another thing @kbd{Z T} does is to increase the Emacs variable
10875 @code{gc-cons-threshold} to a much higher value (two million; the
10876 usual default in Calc is 250,000) for the duration of each command.
10877 This generally prevents garbage collection during the timing of
10878 the command, though it may cause your Emacs process to grow
10879 abnormally large. (Garbage collection time is a major unpredictable
10880 factor in the timing of Emacs operations.)
10882 Another command that is useful when debugging your own Lisp
10883 extensions to Calc is @kbd{M-x calc-pass-errors}, which disables
10884 the error handler that changes the ``@code{max-lisp-eval-depth}
10885 exceeded'' message to the much more friendly ``Computation got
10886 stuck or ran too long.'' This handler interferes with the Emacs
10887 Lisp debugger's @code{debug-on-error} mode. Errors are reported
10888 in the handler itself rather than at the true location of the
10889 error. After you have executed @code{calc-pass-errors}, Lisp
10890 errors will be reported correctly but the user-friendly message
10893 @node Data Types, Stack and Trail, Introduction, Top
10894 @chapter Data Types
10897 This chapter discusses the various types of objects that can be placed
10898 on the Calculator stack, how they are displayed, and how they are
10899 entered. (@xref{Data Type Formats}, for information on how these data
10900 types are represented as underlying Lisp objects.)
10902 Integers, fractions, and floats are various ways of describing real
10903 numbers. HMS forms also for many purposes act as real numbers. These
10904 types can be combined to form complex numbers, modulo forms, error forms,
10905 or interval forms. (But these last four types cannot be combined
10906 arbitrarily:@: error forms may not contain modulo forms, for example.)
10907 Finally, all these types of numbers may be combined into vectors,
10908 matrices, or algebraic formulas.
10911 * Integers:: The most basic data type.
10912 * Fractions:: This and above are called @dfn{rationals}.
10913 * Floats:: This and above are called @dfn{reals}.
10914 * Complex Numbers:: This and above are called @dfn{numbers}.
10916 * Vectors and Matrices::
10923 * Incomplete Objects::
10928 @node Integers, Fractions, Data Types, Data Types
10933 The Calculator stores integers to arbitrary precision. Addition,
10934 subtraction, and multiplication of integers always yields an exact
10935 integer result. (If the result of a division or exponentiation of
10936 integers is not an integer, it is expressed in fractional or
10937 floating-point form according to the current Fraction mode.
10938 @xref{Fraction Mode}.)
10940 A decimal integer is represented as an optional sign followed by a
10941 sequence of digits. Grouping (@pxref{Grouping Digits}) can be used to
10942 insert a comma at every third digit for display purposes, but you
10943 must not type commas during the entry of numbers.
10946 A non-decimal integer is represented as an optional sign, a radix
10947 between 2 and 36, a @samp{#} symbol, and one or more digits. For radix 11
10948 and above, the letters A through Z (upper- or lower-case) count as
10949 digits and do not terminate numeric entry mode. @xref{Radix Modes}, for how
10950 to set the default radix for display of integers. Numbers of any radix
10951 may be entered at any time. If you press @kbd{#} at the beginning of a
10952 number, the current display radix is used.
10954 @node Fractions, Floats, Integers, Data Types
10959 A @dfn{fraction} is a ratio of two integers. Fractions are traditionally
10960 written ``2/3'' but Calc uses the notation @samp{2:3}. (The @kbd{/} key
10961 performs RPN division; the following two sequences push the number
10962 @samp{2:3} on the stack: @kbd{2 :@: 3 @key{RET}}, or @kbd{2 @key{RET} 3 /}
10963 assuming Fraction mode has been enabled.)
10964 When the Calculator produces a fractional result it always reduces it to
10965 simplest form, which may in fact be an integer.
10967 Fractions may also be entered in a three-part form, where @samp{2:3:4}
10968 represents two-and-three-quarters. @xref{Fraction Formats}, for fraction
10971 Non-decimal fractions are entered and displayed as
10972 @samp{@var{radix}#@var{num}:@var{denom}} (or in the analogous three-part
10973 form). The numerator and denominator always use the same radix.
10975 @node Floats, Complex Numbers, Fractions, Data Types
10979 @cindex Floating-point numbers
10980 A floating-point number or @dfn{float} is a number stored in scientific
10981 notation. The number of significant digits in the fractional part is
10982 governed by the current floating precision (@pxref{Precision}). The
10983 range of acceptable values is from
10984 @texline @math{10^{-3999999}}
10985 @infoline @expr{10^-3999999}
10987 @texline @math{10^{4000000}}
10988 @infoline @expr{10^4000000}
10989 (exclusive), plus the corresponding negative values and zero.
10991 Calculations that would exceed the allowable range of values (such
10992 as @samp{exp(exp(20))}) are left in symbolic form by Calc. The
10993 messages ``floating-point overflow'' or ``floating-point underflow''
10994 indicate that during the calculation a number would have been produced
10995 that was too large or too close to zero, respectively, to be represented
10996 by Calc. This does not necessarily mean the final result would have
10997 overflowed, just that an overflow occurred while computing the result.
10998 (In fact, it could report an underflow even though the final result
10999 would have overflowed!)
11001 If a rational number and a float are mixed in a calculation, the result
11002 will in general be expressed as a float. Commands that require an integer
11003 value (such as @kbd{k g} [@code{gcd}]) will also accept integer-valued
11004 floats, i.e., floating-point numbers with nothing after the decimal point.
11006 Floats are identified by the presence of a decimal point and/or an
11007 exponent. In general a float consists of an optional sign, digits
11008 including an optional decimal point, and an optional exponent consisting
11009 of an @samp{e}, an optional sign, and up to seven exponent digits.
11010 For example, @samp{23.5e-2} is 23.5 times ten to the minus-second power,
11013 Floating-point numbers are normally displayed in decimal notation with
11014 all significant figures shown. Exceedingly large or small numbers are
11015 displayed in scientific notation. Various other display options are
11016 available. @xref{Float Formats}.
11018 @cindex Accuracy of calculations
11019 Floating-point numbers are stored in decimal, not binary. The result
11020 of each operation is rounded to the nearest value representable in the
11021 number of significant digits specified by the current precision,
11022 rounding away from zero in the case of a tie. Thus (in the default
11023 display mode) what you see is exactly what you get. Some operations such
11024 as square roots and transcendental functions are performed with several
11025 digits of extra precision and then rounded down, in an effort to make the
11026 final result accurate to the full requested precision. However,
11027 accuracy is not rigorously guaranteed. If you suspect the validity of a
11028 result, try doing the same calculation in a higher precision. The
11029 Calculator's arithmetic is not intended to be IEEE-conformant in any
11032 While floats are always @emph{stored} in decimal, they can be entered
11033 and displayed in any radix just like integers and fractions. The
11034 notation @samp{@var{radix}#@var{ddd}.@var{ddd}} is a floating-point
11035 number whose digits are in the specified radix. Note that the @samp{.}
11036 is more aptly referred to as a ``radix point'' than as a decimal
11037 point in this case. The number @samp{8#123.4567} is defined as
11038 @samp{8#1234567 * 8^-4}. If the radix is 14 or less, you can use
11039 @samp{e} notation to write a non-decimal number in scientific notation.
11040 The exponent is written in decimal, and is considered to be a power
11041 of the radix: @samp{8#1234567e-4}. If the radix is 15 or above, the
11042 letter @samp{e} is a digit, so scientific notation must be written
11043 out, e.g., @samp{16#123.4567*16^2}. The first two exercises of the
11044 Modes Tutorial explore some of the properties of non-decimal floats.
11046 @node Complex Numbers, Infinities, Floats, Data Types
11047 @section Complex Numbers
11050 @cindex Complex numbers
11051 There are two supported formats for complex numbers: rectangular and
11052 polar. The default format is rectangular, displayed in the form
11053 @samp{(@var{real},@var{imag})} where @var{real} is the real part and
11054 @var{imag} is the imaginary part, each of which may be any real number.
11055 Rectangular complex numbers can also be displayed in @samp{@var{a}+@var{b}i}
11056 notation; @pxref{Complex Formats}.
11058 Polar complex numbers are displayed in the form
11059 @texline `@tfn{(}@var{r}@tfn{;}@math{\theta}@tfn{)}'
11060 @infoline `@tfn{(}@var{r}@tfn{;}@var{theta}@tfn{)}'
11061 where @var{r} is the nonnegative magnitude and
11062 @texline @math{\theta}
11063 @infoline @var{theta}
11064 is the argument or phase angle. The range of
11065 @texline @math{\theta}
11066 @infoline @var{theta}
11067 depends on the current angular mode (@pxref{Angular Modes}); it is
11068 generally between @mathit{-180} and @mathit{+180} degrees or the equivalent range
11071 Complex numbers are entered in stages using incomplete objects.
11072 @xref{Incomplete Objects}.
11074 Operations on rectangular complex numbers yield rectangular complex
11075 results, and similarly for polar complex numbers. Where the two types
11076 are mixed, or where new complex numbers arise (as for the square root of
11077 a negative real), the current @dfn{Polar mode} is used to determine the
11078 type. @xref{Polar Mode}.
11080 A complex result in which the imaginary part is zero (or the phase angle
11081 is 0 or 180 degrees or @cpi{} radians) is automatically converted to a real
11084 @node Infinities, Vectors and Matrices, Complex Numbers, Data Types
11085 @section Infinities
11089 @cindex @code{inf} variable
11090 @cindex @code{uinf} variable
11091 @cindex @code{nan} variable
11095 The word @code{inf} represents the mathematical concept of @dfn{infinity}.
11096 Calc actually has three slightly different infinity-like values:
11097 @code{inf}, @code{uinf}, and @code{nan}. These are just regular
11098 variable names (@pxref{Variables}); you should avoid using these
11099 names for your own variables because Calc gives them special
11100 treatment. Infinities, like all variable names, are normally
11101 entered using algebraic entry.
11103 Mathematically speaking, it is not rigorously correct to treat
11104 ``infinity'' as if it were a number, but mathematicians often do
11105 so informally. When they say that @samp{1 / inf = 0}, what they
11106 really mean is that @expr{1 / x}, as @expr{x} becomes larger and
11107 larger, becomes arbitrarily close to zero. So you can imagine
11108 that if @expr{x} got ``all the way to infinity,'' then @expr{1 / x}
11109 would go all the way to zero. Similarly, when they say that
11110 @samp{exp(inf) = inf}, they mean that
11111 @texline @math{e^x}
11112 @infoline @expr{exp(x)}
11113 grows without bound as @expr{x} grows. The symbol @samp{-inf} likewise
11114 stands for an infinitely negative real value; for example, we say that
11115 @samp{exp(-inf) = 0}. You can have an infinity pointing in any
11116 direction on the complex plane: @samp{sqrt(-inf) = i inf}.
11118 The same concept of limits can be used to define @expr{1 / 0}. We
11119 really want the value that @expr{1 / x} approaches as @expr{x}
11120 approaches zero. But if all we have is @expr{1 / 0}, we can't
11121 tell which direction @expr{x} was coming from. If @expr{x} was
11122 positive and decreasing toward zero, then we should say that
11123 @samp{1 / 0 = inf}. But if @expr{x} was negative and increasing
11124 toward zero, the answer is @samp{1 / 0 = -inf}. In fact, @expr{x}
11125 could be an imaginary number, giving the answer @samp{i inf} or
11126 @samp{-i inf}. Calc uses the special symbol @samp{uinf} to mean
11127 @dfn{undirected infinity}, i.e., a value which is infinitely
11128 large but with an unknown sign (or direction on the complex plane).
11130 Calc actually has three modes that say how infinities are handled.
11131 Normally, infinities never arise from calculations that didn't
11132 already have them. Thus, @expr{1 / 0} is treated simply as an
11133 error and left unevaluated. The @kbd{m i} (@code{calc-infinite-mode})
11134 command (@pxref{Infinite Mode}) enables a mode in which
11135 @expr{1 / 0} evaluates to @code{uinf} instead. There is also
11136 an alternative type of infinite mode which says to treat zeros
11137 as if they were positive, so that @samp{1 / 0 = inf}. While this
11138 is less mathematically correct, it may be the answer you want in
11141 Since all infinities are ``as large'' as all others, Calc simplifies,
11142 e.g., @samp{5 inf} to @samp{inf}. Another example is
11143 @samp{5 - inf = -inf}, where the @samp{-inf} is so large that
11144 adding a finite number like five to it does not affect it.
11145 Note that @samp{a - inf} also results in @samp{-inf}; Calc assumes
11146 that variables like @code{a} always stand for finite quantities.
11147 Just to show that infinities really are all the same size,
11148 note that @samp{sqrt(inf) = inf^2 = exp(inf) = inf} in Calc's
11151 It's not so easy to define certain formulas like @samp{0 * inf} and
11152 @samp{inf / inf}. Depending on where these zeros and infinities
11153 came from, the answer could be literally anything. The latter
11154 formula could be the limit of @expr{x / x} (giving a result of one),
11155 or @expr{2 x / x} (giving two), or @expr{x^2 / x} (giving @code{inf}),
11156 or @expr{x / x^2} (giving zero). Calc uses the symbol @code{nan}
11157 to represent such an @dfn{indeterminate} value. (The name ``nan''
11158 comes from analogy with the ``NAN'' concept of IEEE standard
11159 arithmetic; it stands for ``Not A Number.'' This is somewhat of a
11160 misnomer, since @code{nan} @emph{does} stand for some number or
11161 infinity, it's just that @emph{which} number it stands for
11162 cannot be determined.) In Calc's notation, @samp{0 * inf = nan}
11163 and @samp{inf / inf = nan}. A few other common indeterminate
11164 expressions are @samp{inf - inf} and @samp{inf ^ 0}. Also,
11165 @samp{0 / 0 = nan} if you have turned on Infinite mode
11166 (as described above).
11168 Infinities are especially useful as parts of @dfn{intervals}.
11169 @xref{Interval Forms}.
11171 @node Vectors and Matrices, Strings, Infinities, Data Types
11172 @section Vectors and Matrices
11176 @cindex Plain vectors
11178 The @dfn{vector} data type is flexible and general. A vector is simply a
11179 list of zero or more data objects. When these objects are numbers, the
11180 whole is a vector in the mathematical sense. When these objects are
11181 themselves vectors of equal (nonzero) length, the whole is a @dfn{matrix}.
11182 A vector which is not a matrix is referred to here as a @dfn{plain vector}.
11184 A vector is displayed as a list of values separated by commas and enclosed
11185 in square brackets: @samp{[1, 2, 3]}. Thus the following is a 2 row by
11186 3 column matrix: @samp{[[1, 2, 3], [4, 5, 6]]}. Vectors, like complex
11187 numbers, are entered as incomplete objects. @xref{Incomplete Objects}.
11188 During algebraic entry, vectors are entered all at once in the usual
11189 brackets-and-commas form. Matrices may be entered algebraically as nested
11190 vectors, or using the shortcut notation @w{@samp{[1, 2, 3; 4, 5, 6]}},
11191 with rows separated by semicolons. The commas may usually be omitted
11192 when entering vectors: @samp{[1 2 3]}. Curly braces may be used in
11193 place of brackets: @samp{@{1, 2, 3@}}, but the commas are required in
11196 Traditional vector and matrix arithmetic is also supported;
11197 @pxref{Basic Arithmetic} and @pxref{Matrix Functions}.
11198 Many other operations are applied to vectors element-wise. For example,
11199 the complex conjugate of a vector is a vector of the complex conjugates
11206 Algebraic functions for building vectors include @samp{vec(a, b, c)}
11207 to build @samp{[a, b, c]}, @samp{cvec(a, n, m)} to build an
11208 @texline @math{n\times m}
11209 @infoline @var{n}x@var{m}
11210 matrix of @samp{a}s, and @samp{index(n)} to build a vector of integers
11211 from 1 to @samp{n}.
11213 @node Strings, HMS Forms, Vectors and Matrices, Data Types
11219 @cindex Character strings
11220 Character strings are not a special data type in the Calculator.
11221 Rather, a string is represented simply as a vector all of whose
11222 elements are integers in the range 0 to 255 (ASCII codes). You can
11223 enter a string at any time by pressing the @kbd{"} key. Quotation
11224 marks and backslashes are written @samp{\"} and @samp{\\}, respectively,
11225 inside strings. Other notations introduced by backslashes are:
11241 Finally, a backslash followed by three octal digits produces any
11242 character from its ASCII code.
11245 @pindex calc-display-strings
11246 Strings are normally displayed in vector-of-integers form. The
11247 @w{@kbd{d "}} (@code{calc-display-strings}) command toggles a mode in
11248 which any vectors of small integers are displayed as quoted strings
11251 The backslash notations shown above are also used for displaying
11252 strings. Characters 128 and above are not translated by Calc; unless
11253 you have an Emacs modified for 8-bit fonts, these will show up in
11254 backslash-octal-digits notation. For characters below 32, and
11255 for character 127, Calc uses the backslash-letter combination if
11256 there is one, or otherwise uses a @samp{\^} sequence.
11258 The only Calc feature that uses strings is @dfn{compositions};
11259 @pxref{Compositions}. Strings also provide a convenient
11260 way to do conversions between ASCII characters and integers.
11266 There is a @code{string} function which provides a different display
11267 format for strings. Basically, @samp{string(@var{s})}, where @var{s}
11268 is a vector of integers in the proper range, is displayed as the
11269 corresponding string of characters with no surrounding quotation
11270 marks or other modifications. Thus @samp{string("ABC")} (or
11271 @samp{string([65 66 67])}) will look like @samp{ABC} on the stack.
11272 This happens regardless of whether @w{@kbd{d "}} has been used. The
11273 only way to turn it off is to use @kbd{d U} (unformatted language
11274 mode) which will display @samp{string("ABC")} instead.
11276 Control characters are displayed somewhat differently by @code{string}.
11277 Characters below 32, and character 127, are shown using @samp{^} notation
11278 (same as shown above, but without the backslash). The quote and
11279 backslash characters are left alone, as are characters 128 and above.
11285 The @code{bstring} function is just like @code{string} except that
11286 the resulting string is breakable across multiple lines if it doesn't
11287 fit all on one line. Potential break points occur at every space
11288 character in the string.
11290 @node HMS Forms, Date Forms, Strings, Data Types
11294 @cindex Hours-minutes-seconds forms
11295 @cindex Degrees-minutes-seconds forms
11296 @dfn{HMS} stands for Hours-Minutes-Seconds; when used as an angular
11297 argument, the interpretation is Degrees-Minutes-Seconds. All functions
11298 that operate on angles accept HMS forms. These are interpreted as
11299 degrees regardless of the current angular mode. It is also possible to
11300 use HMS as the angular mode so that calculated angles are expressed in
11301 degrees, minutes, and seconds.
11307 @kindex ' (HMS forms)
11311 @kindex " (HMS forms)
11315 @kindex h (HMS forms)
11319 @kindex o (HMS forms)
11323 @kindex m (HMS forms)
11327 @kindex s (HMS forms)
11328 The default format for HMS values is
11329 @samp{@var{hours}@@ @var{mins}' @var{secs}"}. During entry, the letters
11330 @samp{h} (for ``hours'') or
11331 @samp{o} (approximating the ``degrees'' symbol) are accepted as well as
11332 @samp{@@}, @samp{m} is accepted in place of @samp{'}, and @samp{s} is
11333 accepted in place of @samp{"}.
11334 The @var{hours} value is an integer (or integer-valued float).
11335 The @var{mins} value is an integer or integer-valued float between 0 and 59.
11336 The @var{secs} value is a real number between 0 (inclusive) and 60
11337 (exclusive). A positive HMS form is interpreted as @var{hours} +
11338 @var{mins}/60 + @var{secs}/3600. A negative HMS form is interpreted
11339 as @mathit{- @var{hours}} @mathit{-} @var{mins}/60 @mathit{-} @var{secs}/3600.
11340 Display format for HMS forms is quite flexible. @xref{HMS Formats}.
11342 HMS forms can be added and subtracted. When they are added to numbers,
11343 the numbers are interpreted according to the current angular mode. HMS
11344 forms can also be multiplied and divided by real numbers. Dividing
11345 two HMS forms produces a real-valued ratio of the two angles.
11348 @cindex Time of day
11349 Just for kicks, @kbd{M-x calc-time} pushes the current time of day on
11350 the stack as an HMS form.
11352 @node Date Forms, Modulo Forms, HMS Forms, Data Types
11353 @section Date Forms
11357 A @dfn{date form} represents a date and possibly an associated time.
11358 Simple date arithmetic is supported: Adding a number to a date
11359 produces a new date shifted by that many days; adding an HMS form to
11360 a date shifts it by that many hours. Subtracting two date forms
11361 computes the number of days between them (represented as a simple
11362 number). Many other operations, such as multiplying two date forms,
11363 are nonsensical and are not allowed by Calc.
11365 Date forms are entered and displayed enclosed in @samp{< >} brackets.
11366 The default format is, e.g., @samp{<Wed Jan 9, 1991>} for dates,
11367 or @samp{<3:32:20pm Wed Jan 9, 1991>} for dates with times.
11368 Input is flexible; date forms can be entered in any of the usual
11369 notations for dates and times. @xref{Date Formats}.
11371 Date forms are stored internally as numbers, specifically the number
11372 of days since midnight on the morning of January 1 of the year 1 AD.
11373 If the internal number is an integer, the form represents a date only;
11374 if the internal number is a fraction or float, the form represents
11375 a date and time. For example, @samp{<6:00am Wed Jan 9, 1991>}
11376 is represented by the number 726842.25. The standard precision of
11377 12 decimal digits is enough to ensure that a (reasonable) date and
11378 time can be stored without roundoff error.
11380 If the current precision is greater than 12, date forms will keep
11381 additional digits in the seconds position. For example, if the
11382 precision is 15, the seconds will keep three digits after the
11383 decimal point. Decreasing the precision below 12 may cause the
11384 time part of a date form to become inaccurate. This can also happen
11385 if astronomically high years are used, though this will not be an
11386 issue in everyday (or even everymillennium) use. Note that date
11387 forms without times are stored as exact integers, so roundoff is
11388 never an issue for them.
11390 You can use the @kbd{v p} (@code{calc-pack}) and @kbd{v u}
11391 (@code{calc-unpack}) commands to get at the numerical representation
11392 of a date form. @xref{Packing and Unpacking}.
11394 Date forms can go arbitrarily far into the future or past. Negative
11395 year numbers represent years BC. Calc uses a combination of the
11396 Gregorian and Julian calendars, following the history of Great
11397 Britain and the British colonies. This is the same calendar that
11398 is used by the @code{cal} program in most Unix implementations.
11400 @cindex Julian calendar
11401 @cindex Gregorian calendar
11402 Some historical background: The Julian calendar was created by
11403 Julius Caesar in the year 46 BC as an attempt to fix the gradual
11404 drift caused by the lack of leap years in the calendar used
11405 until that time. The Julian calendar introduced an extra day in
11406 all years divisible by four. After some initial confusion, the
11407 calendar was adopted around the year we call 8 AD. Some centuries
11408 later it became apparent that the Julian year of 365.25 days was
11409 itself not quite right. In 1582 Pope Gregory XIII introduced the
11410 Gregorian calendar, which added the new rule that years divisible
11411 by 100, but not by 400, were not to be considered leap years
11412 despite being divisible by four. Many countries delayed adoption
11413 of the Gregorian calendar because of religious differences;
11414 in Britain it was put off until the year 1752, by which time
11415 the Julian calendar had fallen eleven days behind the true
11416 seasons. So the switch to the Gregorian calendar in early
11417 September 1752 introduced a discontinuity: The day after
11418 Sep 2, 1752 is Sep 14, 1752. Calc follows this convention.
11419 To take another example, Russia waited until 1918 before
11420 adopting the new calendar, and thus needed to remove thirteen
11421 days (between Feb 1, 1918 and Feb 14, 1918). This means that
11422 Calc's reckoning will be inconsistent with Russian history between
11423 1752 and 1918, and similarly for various other countries.
11425 Today's timekeepers introduce an occasional ``leap second'' as
11426 well, but Calc does not take these minor effects into account.
11427 (If it did, it would have to report a non-integer number of days
11428 between, say, @samp{<12:00am Mon Jan 1, 1900>} and
11429 @samp{<12:00am Sat Jan 1, 2000>}.)
11431 Calc uses the Julian calendar for all dates before the year 1752,
11432 including dates BC when the Julian calendar technically had not
11433 yet been invented. Thus the claim that day number @mathit{-10000} is
11434 called ``August 16, 28 BC'' should be taken with a grain of salt.
11436 Please note that there is no ``year 0''; the day before
11437 @samp{<Sat Jan 1, +1>} is @samp{<Fri Dec 31, -1>}. These are
11438 days 0 and @mathit{-1} respectively in Calc's internal numbering scheme.
11440 @cindex Julian day counting
11441 Another day counting system in common use is, confusingly, also
11442 called ``Julian.'' It was invented in 1583 by Joseph Justus
11443 Scaliger, who named it in honor of his father Julius Caesar
11444 Scaliger. For obscure reasons he chose to start his day
11445 numbering on Jan 1, 4713 BC at noon, which in Calc's scheme
11446 is @mathit{-1721423.5} (recall that Calc starts at midnight instead
11447 of noon). Thus to convert a Calc date code obtained by
11448 unpacking a date form into a Julian day number, simply add
11449 1721423.5. The Julian code for @samp{6:00am Jan 9, 1991}
11450 is 2448265.75. The built-in @kbd{t J} command performs
11451 this conversion for you.
11453 @cindex Unix time format
11454 The Unix operating system measures time as an integer number of
11455 seconds since midnight, Jan 1, 1970. To convert a Calc date
11456 value into a Unix time stamp, first subtract 719164 (the code
11457 for @samp{<Jan 1, 1970>}), then multiply by 86400 (the number of
11458 seconds in a day) and press @kbd{R} to round to the nearest
11459 integer. If you have a date form, you can simply subtract the
11460 day @samp{<Jan 1, 1970>} instead of unpacking and subtracting
11461 719164. Likewise, divide by 86400 and add @samp{<Jan 1, 1970>}
11462 to convert from Unix time to a Calc date form. (Note that
11463 Unix normally maintains the time in the GMT time zone; you may
11464 need to subtract five hours to get New York time, or eight hours
11465 for California time. The same is usually true of Julian day
11466 counts.) The built-in @kbd{t U} command performs these
11469 @node Modulo Forms, Error Forms, Date Forms, Data Types
11470 @section Modulo Forms
11473 @cindex Modulo forms
11474 A @dfn{modulo form} is a real number which is taken modulo (i.e., within
11475 an integer multiple of) some value @var{M}. Arithmetic modulo @var{M}
11476 often arises in number theory. Modulo forms are written
11477 `@var{a} @tfn{mod} @var{M}',
11478 where @var{a} and @var{M} are real numbers or HMS forms, and
11479 @texline @math{0 \le a < M}.
11480 @infoline @expr{0 <= a < @var{M}}.
11481 In many applications @expr{a} and @expr{M} will be
11482 integers but this is not required.
11487 @kindex M (modulo forms)
11491 @tindex mod (operator)
11492 To create a modulo form during numeric entry, press the shift-@kbd{M}
11493 key to enter the word @samp{mod}. As a special convenience, pressing
11494 shift-@kbd{M} a second time automatically enters the value of @expr{M}
11495 that was most recently used before. During algebraic entry, either
11496 type @samp{mod} by hand or press @kbd{M-m} (that's @kbd{@key{META}-m}).
11497 Once again, pressing this a second time enters the current modulo.
11499 Modulo forms are not to be confused with the modulo operator @samp{%}.
11500 The expression @samp{27 % 10} means to compute 27 modulo 10 to produce
11501 the result 7. Further computations treat this 7 as just a regular integer.
11502 The expression @samp{27 mod 10} produces the result @samp{7 mod 10};
11503 further computations with this value are again reduced modulo 10 so that
11504 the result always lies in the desired range.
11506 When two modulo forms with identical @expr{M}'s are added or multiplied,
11507 the Calculator simply adds or multiplies the values, then reduces modulo
11508 @expr{M}. If one argument is a modulo form and the other a plain number,
11509 the plain number is treated like a compatible modulo form. It is also
11510 possible to raise modulo forms to powers; the result is the value raised
11511 to the power, then reduced modulo @expr{M}. (When all values involved
11512 are integers, this calculation is done much more efficiently than
11513 actually computing the power and then reducing.)
11515 @cindex Modulo division
11516 Two modulo forms `@var{a} @tfn{mod} @var{M}' and `@var{b} @tfn{mod} @var{M}'
11517 can be divided if @expr{a}, @expr{b}, and @expr{M} are all
11518 integers. The result is the modulo form which, when multiplied by
11519 `@var{b} @tfn{mod} @var{M}', produces `@var{a} @tfn{mod} @var{M}'. If
11520 there is no solution to this equation (which can happen only when
11521 @expr{M} is non-prime), or if any of the arguments are non-integers, the
11522 division is left in symbolic form. Other operations, such as square
11523 roots, are not yet supported for modulo forms. (Note that, although
11524 @w{`@tfn{(}@var{a} @tfn{mod} @var{M}@tfn{)^.5}'} will compute a ``modulo square root''
11525 in the sense of reducing
11526 @texline @math{\sqrt a}
11527 @infoline @expr{sqrt(a)}
11528 modulo @expr{M}, this is not a useful definition from the
11529 number-theoretical point of view.)
11531 It is possible to mix HMS forms and modulo forms. For example, an
11532 HMS form modulo 24 could be used to manipulate clock times; an HMS
11533 form modulo 360 would be suitable for angles. Making the modulo @expr{M}
11534 also be an HMS form eliminates troubles that would arise if the angular
11535 mode were inadvertently set to Radians, in which case
11536 @w{@samp{2@@ 0' 0" mod 24}} would be interpreted as two degrees modulo
11539 Modulo forms cannot have variables or formulas for components. If you
11540 enter the formula @samp{(x + 2) mod 5}, Calc propagates the modulus
11541 to each of the coefficients: @samp{(1 mod 5) x + (2 mod 5)}.
11543 You can use @kbd{v p} and @kbd{%} to modify modulo forms.
11544 @xref{Packing and Unpacking}. @xref{Basic Arithmetic}.
11550 The algebraic function @samp{makemod(a, m)} builds the modulo form
11551 @w{@samp{a mod m}}.
11553 @node Error Forms, Interval Forms, Modulo Forms, Data Types
11554 @section Error Forms
11557 @cindex Error forms
11558 @cindex Standard deviations
11559 An @dfn{error form} is a number with an associated standard
11560 deviation, as in @samp{2.3 +/- 0.12}. The notation
11561 @texline `@var{x} @tfn{+/-} @math{\sigma}'
11562 @infoline `@var{x} @tfn{+/-} sigma'
11563 stands for an uncertain value which follows
11564 a normal or Gaussian distribution of mean @expr{x} and standard
11565 deviation or ``error''
11566 @texline @math{\sigma}.
11567 @infoline @expr{sigma}.
11568 Both the mean and the error can be either numbers or
11569 formulas. Generally these are real numbers but the mean may also be
11570 complex. If the error is negative or complex, it is changed to its
11571 absolute value. An error form with zero error is converted to a
11572 regular number by the Calculator.
11574 All arithmetic and transcendental functions accept error forms as input.
11575 Operations on the mean-value part work just like operations on regular
11576 numbers. The error part for any function @expr{f(x)} (such as
11577 @texline @math{\sin x}
11578 @infoline @expr{sin(x)})
11579 is defined by the error of @expr{x} times the derivative of @expr{f}
11580 evaluated at the mean value of @expr{x}. For a two-argument function
11581 @expr{f(x,y)} (such as addition) the error is the square root of the sum
11582 of the squares of the errors due to @expr{x} and @expr{y}.
11585 f(x \hbox{\code{ +/- }} \sigma)
11586 &= f(x) \hbox{\code{ +/- }} \sigma \left| {df(x) \over dx} \right| \cr
11587 f(x \hbox{\code{ +/- }} \sigma_x, y \hbox{\code{ +/- }} \sigma_y)
11588 &= f(x,y) \hbox{\code{ +/- }}
11589 \sqrt{\left(\sigma_x \left| {\partial f(x,y) \over \partial x}
11591 +\left(\sigma_y \left| {\partial f(x,y) \over \partial y}
11592 \right| \right)^2 } \cr
11596 definition assumes the errors in @expr{x} and @expr{y} are uncorrelated.
11597 A side effect of this definition is that @samp{(2 +/- 1) * (2 +/- 1)}
11598 is not the same as @samp{(2 +/- 1)^2}; the former represents the product
11599 of two independent values which happen to have the same probability
11600 distributions, and the latter is the product of one random value with itself.
11601 The former will produce an answer with less error, since on the average
11602 the two independent errors can be expected to cancel out.
11604 Consult a good text on error analysis for a discussion of the proper use
11605 of standard deviations. Actual errors often are neither Gaussian-distributed
11606 nor uncorrelated, and the above formulas are valid only when errors
11607 are small. As an example, the error arising from
11608 @texline `@tfn{sin(}@var{x} @tfn{+/-} @math{\sigma}@tfn{)}'
11609 @infoline `@tfn{sin(}@var{x} @tfn{+/-} @var{sigma}@tfn{)}'
11611 @texline `@math{\sigma} @tfn{abs(cos(}@var{x}@tfn{))}'.
11612 @infoline `@var{sigma} @tfn{abs(cos(}@var{x}@tfn{))}'.
11613 When @expr{x} is close to zero,
11614 @texline @math{\cos x}
11615 @infoline @expr{cos(x)}
11616 is close to one so the error in the sine is close to
11617 @texline @math{\sigma};
11618 @infoline @expr{sigma};
11619 this makes sense, since
11620 @texline @math{\sin x}
11621 @infoline @expr{sin(x)}
11622 is approximately @expr{x} near zero, so a given error in @expr{x} will
11623 produce about the same error in the sine. Likewise, near 90 degrees
11624 @texline @math{\cos x}
11625 @infoline @expr{cos(x)}
11626 is nearly zero and so the computed error is
11627 small: The sine curve is nearly flat in that region, so an error in @expr{x}
11628 has relatively little effect on the value of
11629 @texline @math{\sin x}.
11630 @infoline @expr{sin(x)}.
11631 However, consider @samp{sin(90 +/- 1000)}. The cosine of 90 is zero, so
11632 Calc will report zero error! We get an obviously wrong result because
11633 we have violated the small-error approximation underlying the error
11634 analysis. If the error in @expr{x} had been small, the error in
11635 @texline @math{\sin x}
11636 @infoline @expr{sin(x)}
11637 would indeed have been negligible.
11642 @kindex p (error forms)
11644 To enter an error form during regular numeric entry, use the @kbd{p}
11645 (``plus-or-minus'') key to type the @samp{+/-} symbol. (If you try actually
11646 typing @samp{+/-} the @kbd{+} key will be interpreted as the Calculator's
11647 @kbd{+} command!) Within an algebraic formula, you can press @kbd{M-p} to
11648 type the @samp{+/-} symbol, or type it out by hand.
11650 Error forms and complex numbers can be mixed; the formulas shown above
11651 are used for complex numbers, too; note that if the error part evaluates
11652 to a complex number its absolute value (or the square root of the sum of
11653 the squares of the absolute values of the two error contributions) is
11654 used. Mathematically, this corresponds to a radially symmetric Gaussian
11655 distribution of numbers on the complex plane. However, note that Calc
11656 considers an error form with real components to represent a real number,
11657 not a complex distribution around a real mean.
11659 Error forms may also be composed of HMS forms. For best results, both
11660 the mean and the error should be HMS forms if either one is.
11666 The algebraic function @samp{sdev(a, b)} builds the error form @samp{a +/- b}.
11668 @node Interval Forms, Incomplete Objects, Error Forms, Data Types
11669 @section Interval Forms
11672 @cindex Interval forms
11673 An @dfn{interval} is a subset of consecutive real numbers. For example,
11674 the interval @samp{[2 ..@: 4]} represents all the numbers from 2 to 4,
11675 inclusive. If you multiply it by the interval @samp{[0.5 ..@: 2]} you
11676 obtain @samp{[1 ..@: 8]}. This calculation represents the fact that if
11677 you multiply some number in the range @samp{[2 ..@: 4]} by some other
11678 number in the range @samp{[0.5 ..@: 2]}, your result will lie in the range
11679 from 1 to 8. Interval arithmetic is used to get a worst-case estimate
11680 of the possible range of values a computation will produce, given the
11681 set of possible values of the input.
11684 Calc supports several varieties of intervals, including @dfn{closed}
11685 intervals of the type shown above, @dfn{open} intervals such as
11686 @samp{(2 ..@: 4)}, which represents the range of numbers from 2 to 4
11687 @emph{exclusive}, and @dfn{semi-open} intervals in which one end
11688 uses a round parenthesis and the other a square bracket. In mathematical
11690 @samp{[2 ..@: 4]} means @expr{2 <= x <= 4}, whereas
11691 @samp{[2 ..@: 4)} represents @expr{2 <= x < 4},
11692 @samp{(2 ..@: 4]} represents @expr{2 < x <= 4}, and
11693 @samp{(2 ..@: 4)} represents @expr{2 < x < 4}.
11696 Calc supports several varieties of intervals, including \dfn{closed}
11697 intervals of the type shown above, \dfn{open} intervals such as
11698 \samp{(2 ..\: 4)}, which represents the range of numbers from 2 to 4
11699 \emph{exclusive}, and \dfn{semi-open} intervals in which one end
11700 uses a round parenthesis and the other a square bracket. In mathematical
11703 [2 \hbox{\cite{..}} 4] &\quad\hbox{means}\quad 2 \le x \le 4 \cr
11704 [2 \hbox{\cite{..}} 4) &\quad\hbox{means}\quad 2 \le x < 4 \cr
11705 (2 \hbox{\cite{..}} 4] &\quad\hbox{means}\quad 2 < x \le 4 \cr
11706 (2 \hbox{\cite{..}} 4) &\quad\hbox{means}\quad 2 < x < 4 \cr
11710 The lower and upper limits of an interval must be either real numbers
11711 (or HMS or date forms), or symbolic expressions which are assumed to be
11712 real-valued, or @samp{-inf} and @samp{inf}. In general the lower limit
11713 must be less than the upper limit. A closed interval containing only
11714 one value, @samp{[3 ..@: 3]}, is converted to a plain number (3)
11715 automatically. An interval containing no values at all (such as
11716 @samp{[3 ..@: 2]} or @samp{[2 ..@: 2)}) can be represented but is not
11717 guaranteed to behave well when used in arithmetic. Note that the
11718 interval @samp{[3 .. inf)} represents all real numbers greater than
11719 or equal to 3, and @samp{(-inf .. inf)} represents all real numbers.
11720 In fact, @samp{[-inf .. inf]} represents all real numbers including
11721 the real infinities.
11723 Intervals are entered in the notation shown here, either as algebraic
11724 formulas, or using incomplete forms. (@xref{Incomplete Objects}.)
11725 In algebraic formulas, multiple periods in a row are collected from
11726 left to right, so that @samp{1...1e2} is interpreted as @samp{1.0 ..@: 1e2}
11727 rather than @samp{1 ..@: 0.1e2}. Add spaces or zeros if you want to
11728 get the other interpretation. If you omit the lower or upper limit,
11729 a default of @samp{-inf} or @samp{inf} (respectively) is furnished.
11731 Infinite mode also affects operations on intervals
11732 (@pxref{Infinities}). Calc will always introduce an open infinity,
11733 as in @samp{1 / (0 .. 2] = [0.5 .. inf)}. But closed infinities,
11734 @w{@samp{1 / [0 .. 2] = [0.5 .. inf]}}, arise only in Infinite mode;
11735 otherwise they are left unevaluated. Note that the ``direction'' of
11736 a zero is not an issue in this case since the zero is always assumed
11737 to be continuous with the rest of the interval. For intervals that
11738 contain zero inside them Calc is forced to give the result,
11739 @samp{1 / (-2 .. 2) = [-inf .. inf]}.
11741 While it may seem that intervals and error forms are similar, they are
11742 based on entirely different concepts of inexact quantities. An error
11744 @texline `@var{x} @tfn{+/-} @math{\sigma}'
11745 @infoline `@var{x} @tfn{+/-} @var{sigma}'
11746 means a variable is random, and its value could
11747 be anything but is ``probably'' within one
11748 @texline @math{\sigma}
11749 @infoline @var{sigma}
11750 of the mean value @expr{x}. An interval
11751 `@tfn{[}@var{a} @tfn{..@:} @var{b}@tfn{]}' means a
11752 variable's value is unknown, but guaranteed to lie in the specified
11753 range. Error forms are statistical or ``average case'' approximations;
11754 interval arithmetic tends to produce ``worst case'' bounds on an
11757 Intervals may not contain complex numbers, but they may contain
11758 HMS forms or date forms.
11760 @xref{Set Operations}, for commands that interpret interval forms
11761 as subsets of the set of real numbers.
11767 The algebraic function @samp{intv(n, a, b)} builds an interval form
11768 from @samp{a} to @samp{b}; @samp{n} is an integer code which must
11769 be 0 for @samp{(..)}, 1 for @samp{(..]}, 2 for @samp{[..)}, or
11772 Please note that in fully rigorous interval arithmetic, care would be
11773 taken to make sure that the computation of the lower bound rounds toward
11774 minus infinity, while upper bound computations round toward plus
11775 infinity. Calc's arithmetic always uses a round-to-nearest mode,
11776 which means that roundoff errors could creep into an interval
11777 calculation to produce intervals slightly smaller than they ought to
11778 be. For example, entering @samp{[1..2]} and pressing @kbd{Q 2 ^}
11779 should yield the interval @samp{[1..2]} again, but in fact it yields the
11780 (slightly too small) interval @samp{[1..1.9999999]} due to roundoff
11783 @node Incomplete Objects, Variables, Interval Forms, Data Types
11784 @section Incomplete Objects
11804 @cindex Incomplete vectors
11805 @cindex Incomplete complex numbers
11806 @cindex Incomplete interval forms
11807 When @kbd{(} or @kbd{[} is typed to begin entering a complex number or
11808 vector, respectively, the effect is to push an @dfn{incomplete} complex
11809 number or vector onto the stack. The @kbd{,} key adds the value(s) at
11810 the top of the stack onto the current incomplete object. The @kbd{)}
11811 and @kbd{]} keys ``close'' the incomplete object after adding any values
11812 on the top of the stack in front of the incomplete object.
11814 As a result, the sequence of keystrokes @kbd{[ 2 , 3 @key{RET} 2 * , 9 ]}
11815 pushes the vector @samp{[2, 6, 9]} onto the stack. Likewise, @kbd{( 1 , 2 Q )}
11816 pushes the complex number @samp{(1, 1.414)} (approximately).
11818 If several values lie on the stack in front of the incomplete object,
11819 all are collected and appended to the object. Thus the @kbd{,} key
11820 is redundant: @kbd{[ 2 @key{RET} 3 @key{RET} 2 * 9 ]}. Some people
11821 prefer the equivalent @key{SPC} key to @key{RET}.
11823 As a special case, typing @kbd{,} immediately after @kbd{(}, @kbd{[}, or
11824 @kbd{,} adds a zero or duplicates the preceding value in the list being
11825 formed. Typing @key{DEL} during incomplete entry removes the last item
11829 The @kbd{;} key is used in the same way as @kbd{,} to create polar complex
11830 numbers: @kbd{( 1 ; 2 )}. When entering a vector, @kbd{;} is useful for
11831 creating a matrix. In particular, @kbd{[ [ 1 , 2 ; 3 , 4 ; 5 , 6 ] ]} is
11832 equivalent to @kbd{[ [ 1 , 2 ] , [ 3 , 4 ] , [ 5 , 6 ] ]}.
11836 Incomplete entry is also used to enter intervals. For example,
11837 @kbd{[ 2 ..@: 4 )} enters a semi-open interval. Note that when you type
11838 the first period, it will be interpreted as a decimal point, but when
11839 you type a second period immediately afterward, it is re-interpreted as
11840 part of the interval symbol. Typing @kbd{..} corresponds to executing
11841 the @code{calc-dots} command.
11843 If you find incomplete entry distracting, you may wish to enter vectors
11844 and complex numbers as algebraic formulas by pressing the apostrophe key.
11846 @node Variables, Formulas, Incomplete Objects, Data Types
11850 @cindex Variables, in formulas
11851 A @dfn{variable} is somewhere between a storage register on a conventional
11852 calculator, and a variable in a programming language. (In fact, a Calc
11853 variable is really just an Emacs Lisp variable that contains a Calc number
11854 or formula.) A variable's name is normally composed of letters and digits.
11855 Calc also allows apostrophes and @code{#} signs in variable names.
11856 (The Calc variable @code{foo} corresponds to the Emacs Lisp variable
11857 @code{var-foo}, but unless you access the variable from within Emacs
11858 Lisp, you don't need to worry about it. Variable names in algebraic
11859 formulas implicitly have @samp{var-} prefixed to their names. The
11860 @samp{#} character in variable names used in algebraic formulas
11861 corresponds to a dash @samp{-} in the Lisp variable name. If the name
11862 contains any dashes, the prefix @samp{var-} is @emph{not} automatically
11863 added. Thus the two formulas @samp{foo + 1} and @samp{var#foo + 1} both
11864 refer to the same variable.)
11866 In a command that takes a variable name, you can either type the full
11867 name of a variable, or type a single digit to use one of the special
11868 convenience variables @code{q0} through @code{q9}. For example,
11869 @kbd{3 s s 2} stores the number 3 in variable @code{q2}, and
11870 @w{@kbd{3 s s foo @key{RET}}} stores that number in variable
11873 To push a variable itself (as opposed to the variable's value) on the
11874 stack, enter its name as an algebraic expression using the apostrophe
11878 @pindex calc-evaluate
11879 @cindex Evaluation of variables in a formula
11880 @cindex Variables, evaluation
11881 @cindex Formulas, evaluation
11882 The @kbd{=} (@code{calc-evaluate}) key ``evaluates'' a formula by
11883 replacing all variables in the formula which have been given values by a
11884 @code{calc-store} or @code{calc-let} command by their stored values.
11885 Other variables are left alone. Thus a variable that has not been
11886 stored acts like an abstract variable in algebra; a variable that has
11887 been stored acts more like a register in a traditional calculator.
11888 With a positive numeric prefix argument, @kbd{=} evaluates the top
11889 @var{n} stack entries; with a negative argument, @kbd{=} evaluates
11890 the @var{n}th stack entry.
11892 @cindex @code{e} variable
11893 @cindex @code{pi} variable
11894 @cindex @code{i} variable
11895 @cindex @code{phi} variable
11896 @cindex @code{gamma} variable
11902 A few variables are called @dfn{special constants}. Their names are
11903 @samp{e}, @samp{pi}, @samp{i}, @samp{phi}, and @samp{gamma}.
11904 (@xref{Scientific Functions}.) When they are evaluated with @kbd{=},
11905 their values are calculated if necessary according to the current precision
11906 or complex polar mode. If you wish to use these symbols for other purposes,
11907 simply undefine or redefine them using @code{calc-store}.
11909 The variables @samp{inf}, @samp{uinf}, and @samp{nan} stand for
11910 infinite or indeterminate values. It's best not to use them as
11911 regular variables, since Calc uses special algebraic rules when
11912 it manipulates them. Calc displays a warning message if you store
11913 a value into any of these special variables.
11915 @xref{Store and Recall}, for a discussion of commands dealing with variables.
11917 @node Formulas, , Variables, Data Types
11922 @cindex Expressions
11923 @cindex Operators in formulas
11924 @cindex Precedence of operators
11925 When you press the apostrophe key you may enter any expression or formula
11926 in algebraic form. (Calc uses the terms ``expression'' and ``formula''
11927 interchangeably.) An expression is built up of numbers, variable names,
11928 and function calls, combined with various arithmetic operators.
11930 be used to indicate grouping. Spaces are ignored within formulas, except
11931 that spaces are not permitted within variable names or numbers.
11932 Arithmetic operators, in order from highest to lowest precedence, and
11933 with their equivalent function names, are:
11935 @samp{_} [@code{subscr}] (subscripts);
11937 postfix @samp{%} [@code{percent}] (as in @samp{25% = 0.25});
11939 prefix @samp{+} and @samp{-} [@code{neg}] (as in @samp{-x})
11940 and prefix @samp{!} [@code{lnot}] (logical ``not,'' as in @samp{!x});
11942 @samp{+/-} [@code{sdev}] (the standard deviation symbol) and
11943 @samp{mod} [@code{makemod}] (the symbol for modulo forms);
11945 postfix @samp{!} [@code{fact}] (factorial, as in @samp{n!})
11946 and postfix @samp{!!} [@code{dfact}] (double factorial);
11948 @samp{^} [@code{pow}] (raised-to-the-power-of);
11950 @samp{*} [@code{mul}];
11952 @samp{/} [@code{div}], @samp{%} [@code{mod}] (modulo), and
11953 @samp{\} [@code{idiv}] (integer division);
11955 infix @samp{+} [@code{add}] and @samp{-} [@code{sub}] (as in @samp{x-y});
11957 @samp{|} [@code{vconcat}] (vector concatenation);
11959 relations @samp{=} [@code{eq}], @samp{!=} [@code{neq}], @samp{<} [@code{lt}],
11960 @samp{>} [@code{gt}], @samp{<=} [@code{leq}], and @samp{>=} [@code{geq}];
11962 @samp{&&} [@code{land}] (logical ``and'');
11964 @samp{||} [@code{lor}] (logical ``or'');
11966 the C-style ``if'' operator @samp{a?b:c} [@code{if}];
11968 @samp{!!!} [@code{pnot}] (rewrite pattern ``not'');
11970 @samp{&&&} [@code{pand}] (rewrite pattern ``and'');
11972 @samp{|||} [@code{por}] (rewrite pattern ``or'');
11974 @samp{:=} [@code{assign}] (for assignments and rewrite rules);
11976 @samp{::} [@code{condition}] (rewrite pattern condition);
11978 @samp{=>} [@code{evalto}].
11980 Note that, unlike in usual computer notation, multiplication binds more
11981 strongly than division: @samp{a*b/c*d} is equivalent to
11982 @texline @math{a b \over c d}.
11983 @infoline @expr{(a*b)/(c*d)}.
11985 @cindex Multiplication, implicit
11986 @cindex Implicit multiplication
11987 The multiplication sign @samp{*} may be omitted in many cases. In particular,
11988 if the righthand side is a number, variable name, or parenthesized
11989 expression, the @samp{*} may be omitted. Implicit multiplication has the
11990 same precedence as the explicit @samp{*} operator. The one exception to
11991 the rule is that a variable name followed by a parenthesized expression,
11993 is interpreted as a function call, not an implicit @samp{*}. In many
11994 cases you must use a space if you omit the @samp{*}: @samp{2a} is the
11995 same as @samp{2*a}, and @samp{a b} is the same as @samp{a*b}, but @samp{ab}
11996 is a variable called @code{ab}, @emph{not} the product of @samp{a} and
11997 @samp{b}! Also note that @samp{f (x)} is still a function call.
11999 @cindex Implicit comma in vectors
12000 The rules are slightly different for vectors written with square brackets.
12001 In vectors, the space character is interpreted (like the comma) as a
12002 separator of elements of the vector. Thus @w{@samp{[ 2a b+c d ]}} is
12003 equivalent to @samp{[2*a, b+c, d]}, whereas @samp{2a b+c d} is equivalent
12004 to @samp{2*a*b + c*d}.
12005 Note that spaces around the brackets, and around explicit commas, are
12006 ignored. To force spaces to be interpreted as multiplication you can
12007 enclose a formula in parentheses as in @samp{[(a b) 2(c d)]}, which is
12008 interpreted as @samp{[a*b, 2*c*d]}. An implicit comma is also inserted
12009 between @samp{][}, as in the matrix @samp{[[1 2][3 4]]}.
12011 Vectors that contain commas (not embedded within nested parentheses or
12012 brackets) do not treat spaces specially: @samp{[a b, 2 c d]} is a vector
12013 of two elements. Also, if it would be an error to treat spaces as
12014 separators, but not otherwise, then Calc will ignore spaces:
12015 @w{@samp{[a - b]}} is a vector of one element, but @w{@samp{[a -b]}} is
12016 a vector of two elements. Finally, vectors entered with curly braces
12017 instead of square brackets do not give spaces any special treatment.
12018 When Calc displays a vector that does not contain any commas, it will
12019 insert parentheses if necessary to make the meaning clear:
12020 @w{@samp{[(a b)]}}.
12022 The expression @samp{5%-2} is ambiguous; is this five-percent minus two,
12023 or five modulo minus-two? Calc always interprets the leftmost symbol as
12024 an infix operator preferentially (modulo, in this case), so you would
12025 need to write @samp{(5%)-2} to get the former interpretation.
12027 @cindex Function call notation
12028 A function call is, e.g., @samp{sin(1+x)}. (The Calc algebraic function
12029 @code{foo} corresponds to the Emacs Lisp function @code{calcFunc-foo},
12030 but unless you access the function from within Emacs Lisp, you don't
12031 need to worry about it.) Most mathematical Calculator commands like
12032 @code{calc-sin} have function equivalents like @code{sin}.
12033 If no Lisp function is defined for a function called by a formula, the
12034 call is left as it is during algebraic manipulation: @samp{f(x+y)} is
12035 left alone. Beware that many innocent-looking short names like @code{in}
12036 and @code{re} have predefined meanings which could surprise you; however,
12037 single letters or single letters followed by digits are always safe to
12038 use for your own function names. @xref{Function Index}.
12040 In the documentation for particular commands, the notation @kbd{H S}
12041 (@code{calc-sinh}) [@code{sinh}] means that the key sequence @kbd{H S}, the
12042 command @kbd{M-x calc-sinh}, and the algebraic function @code{sinh(x)} all
12043 represent the same operation.
12045 Commands that interpret (``parse'') text as algebraic formulas include
12046 algebraic entry (@kbd{'}), editing commands like @kbd{`} which parse
12047 the contents of the editing buffer when you finish, the @kbd{M-# g}
12048 and @w{@kbd{M-# r}} commands, the @kbd{C-y} command, the X window system
12049 ``paste'' mouse operation, and Embedded mode. All of these operations
12050 use the same rules for parsing formulas; in particular, language modes
12051 (@pxref{Language Modes}) affect them all in the same way.
12053 When you read a large amount of text into the Calculator (say a vector
12054 which represents a big set of rewrite rules; @pxref{Rewrite Rules}),
12055 you may wish to include comments in the text. Calc's formula parser
12056 ignores the symbol @samp{%%} and anything following it on a line:
12059 [ a + b, %% the sum of "a" and "b"
12061 %% last line is coming up:
12066 This is parsed exactly the same as @samp{[ a + b, c + d, e + f ]}.
12068 @xref{Syntax Tables}, for a way to create your own operators and other
12069 input notations. @xref{Compositions}, for a way to create new display
12072 @xref{Algebra}, for commands for manipulating formulas symbolically.
12074 @node Stack and Trail, Mode Settings, Data Types, Top
12075 @chapter Stack and Trail Commands
12078 This chapter describes the Calc commands for manipulating objects on the
12079 stack and in the trail buffer. (These commands operate on objects of any
12080 type, such as numbers, vectors, formulas, and incomplete objects.)
12083 * Stack Manipulation::
12084 * Editing Stack Entries::
12089 @node Stack Manipulation, Editing Stack Entries, Stack and Trail, Stack and Trail
12090 @section Stack Manipulation Commands
12096 @cindex Duplicating stack entries
12097 To duplicate the top object on the stack, press @key{RET} or @key{SPC}
12098 (two equivalent keys for the @code{calc-enter} command).
12099 Given a positive numeric prefix argument, these commands duplicate
12100 several elements at the top of the stack.
12101 Given a negative argument,
12102 these commands duplicate the specified element of the stack.
12103 Given an argument of zero, they duplicate the entire stack.
12104 For example, with @samp{10 20 30} on the stack,
12105 @key{RET} creates @samp{10 20 30 30},
12106 @kbd{C-u 2 @key{RET}} creates @samp{10 20 30 20 30},
12107 @kbd{C-u - 2 @key{RET}} creates @samp{10 20 30 20}, and
12108 @kbd{C-u 0 @key{RET}} creates @samp{10 20 30 10 20 30}.
12112 The @key{LFD} (@code{calc-over}) command (on a key marked Line-Feed if you
12113 have it, else on @kbd{C-j}) is like @code{calc-enter}
12114 except that the sign of the numeric prefix argument is interpreted
12115 oppositely. Also, with no prefix argument the default argument is 2.
12116 Thus with @samp{10 20 30} on the stack, @key{LFD} and @kbd{C-u 2 @key{LFD}}
12117 are both equivalent to @kbd{C-u - 2 @key{RET}}, producing
12118 @samp{10 20 30 20}.
12123 @cindex Removing stack entries
12124 @cindex Deleting stack entries
12125 To remove the top element from the stack, press @key{DEL} (@code{calc-pop}).
12126 The @kbd{C-d} key is a synonym for @key{DEL}.
12127 (If the top element is an incomplete object with at least one element, the
12128 last element is removed from it.) Given a positive numeric prefix argument,
12129 several elements are removed. Given a negative argument, the specified
12130 element of the stack is deleted. Given an argument of zero, the entire
12132 For example, with @samp{10 20 30} on the stack,
12133 @key{DEL} leaves @samp{10 20},
12134 @kbd{C-u 2 @key{DEL}} leaves @samp{10},
12135 @kbd{C-u - 2 @key{DEL}} leaves @samp{10 30}, and
12136 @kbd{C-u 0 @key{DEL}} leaves an empty stack.
12138 @kindex M-@key{DEL}
12139 @pindex calc-pop-above
12140 The @kbd{M-@key{DEL}} (@code{calc-pop-above}) command is to @key{DEL} what
12141 @key{LFD} is to @key{RET}: It interprets the sign of the numeric
12142 prefix argument in the opposite way, and the default argument is 2.
12143 Thus @kbd{M-@key{DEL}} by itself removes the second-from-top stack element,
12144 leaving the first, third, fourth, and so on; @kbd{M-3 M-@key{DEL}} deletes
12145 the third stack element.
12148 @pindex calc-roll-down
12149 To exchange the top two elements of the stack, press @key{TAB}
12150 (@code{calc-roll-down}). Given a positive numeric prefix argument, the
12151 specified number of elements at the top of the stack are rotated downward.
12152 Given a negative argument, the entire stack is rotated downward the specified
12153 number of times. Given an argument of zero, the entire stack is reversed
12155 For example, with @samp{10 20 30 40 50} on the stack,
12156 @key{TAB} creates @samp{10 20 30 50 40},
12157 @kbd{C-u 3 @key{TAB}} creates @samp{10 20 50 30 40},
12158 @kbd{C-u - 2 @key{TAB}} creates @samp{40 50 10 20 30}, and
12159 @kbd{C-u 0 @key{TAB}} creates @samp{50 40 30 20 10}.
12161 @kindex M-@key{TAB}
12162 @pindex calc-roll-up
12163 The command @kbd{M-@key{TAB}} (@code{calc-roll-up}) is analogous to @key{TAB}
12164 except that it rotates upward instead of downward. Also, the default
12165 with no prefix argument is to rotate the top 3 elements.
12166 For example, with @samp{10 20 30 40 50} on the stack,
12167 @kbd{M-@key{TAB}} creates @samp{10 20 40 50 30},
12168 @kbd{C-u 4 M-@key{TAB}} creates @samp{10 30 40 50 20},
12169 @kbd{C-u - 2 M-@key{TAB}} creates @samp{30 40 50 10 20}, and
12170 @kbd{C-u 0 M-@key{TAB}} creates @samp{50 40 30 20 10}.
12172 A good way to view the operation of @key{TAB} and @kbd{M-@key{TAB}} is in
12173 terms of moving a particular element to a new position in the stack.
12174 With a positive argument @var{n}, @key{TAB} moves the top stack
12175 element down to level @var{n}, making room for it by pulling all the
12176 intervening stack elements toward the top. @kbd{M-@key{TAB}} moves the
12177 element at level @var{n} up to the top. (Compare with @key{LFD},
12178 which copies instead of moving the element in level @var{n}.)
12180 With a negative argument @mathit{-@var{n}}, @key{TAB} rotates the stack
12181 to move the object in level @var{n} to the deepest place in the
12182 stack, and the object in level @mathit{@var{n}+1} to the top. @kbd{M-@key{TAB}}
12183 rotates the deepest stack element to be in level @mathit{n}, also
12184 putting the top stack element in level @mathit{@var{n}+1}.
12186 @xref{Selecting Subformulas}, for a way to apply these commands to
12187 any portion of a vector or formula on the stack.
12189 @node Editing Stack Entries, Trail Commands, Stack Manipulation, Stack and Trail
12190 @section Editing Stack Entries
12195 @pindex calc-edit-finish
12196 @cindex Editing the stack with Emacs
12197 The backquote, @kbd{`} (@code{calc-edit}) command creates a temporary
12198 buffer (@samp{*Calc Edit*}) for editing the top-of-stack value using
12199 regular Emacs commands. With a numeric prefix argument, it edits the
12200 specified number of stack entries at once. (An argument of zero edits
12201 the entire stack; a negative argument edits one specific stack entry.)
12203 When you are done editing, press @kbd{C-c C-c} to finish and return
12204 to Calc. The @key{RET} and @key{LFD} keys also work to finish most
12205 sorts of editing, though in some cases Calc leaves @key{RET} with its
12206 usual meaning (``insert a newline'') if it's a situation where you
12207 might want to insert new lines into the editing buffer.
12209 When you finish editing, the Calculator parses the lines of text in
12210 the @samp{*Calc Edit*} buffer as numbers or formulas, replaces the
12211 original stack elements in the original buffer with these new values,
12212 then kills the @samp{*Calc Edit*} buffer. The original Calculator buffer
12213 continues to exist during editing, but for best results you should be
12214 careful not to change it until you have finished the edit. You can
12215 also cancel the edit by killing the buffer with @kbd{C-x k}.
12217 The formula is normally reevaluated as it is put onto the stack.
12218 For example, editing @samp{a + 2} to @samp{3 + 2} and pressing
12219 @kbd{C-c C-c} will push 5 on the stack. If you use @key{LFD} to
12220 finish, Calc will put the result on the stack without evaluating it.
12222 If you give a prefix argument to @kbd{C-c C-c},
12223 Calc will not kill the @samp{*Calc Edit*} buffer. You can switch
12224 back to that buffer and continue editing if you wish. However, you
12225 should understand that if you initiated the edit with @kbd{`}, the
12226 @kbd{C-c C-c} operation will be programmed to replace the top of the
12227 stack with the new edited value, and it will do this even if you have
12228 rearranged the stack in the meanwhile. This is not so much of a problem
12229 with other editing commands, though, such as @kbd{s e}
12230 (@code{calc-edit-variable}; @pxref{Operations on Variables}).
12232 If the @code{calc-edit} command involves more than one stack entry,
12233 each line of the @samp{*Calc Edit*} buffer is interpreted as a
12234 separate formula. Otherwise, the entire buffer is interpreted as
12235 one formula, with line breaks ignored. (You can use @kbd{C-o} or
12236 @kbd{C-q C-j} to insert a newline in the buffer without pressing @key{RET}.)
12238 The @kbd{`} key also works during numeric or algebraic entry. The
12239 text entered so far is moved to the @code{*Calc Edit*} buffer for
12240 more extensive editing than is convenient in the minibuffer.
12242 @node Trail Commands, Keep Arguments, Editing Stack Entries, Stack and Trail
12243 @section Trail Commands
12246 @cindex Trail buffer
12247 The commands for manipulating the Calc Trail buffer are two-key sequences
12248 beginning with the @kbd{t} prefix.
12251 @pindex calc-trail-display
12252 The @kbd{t d} (@code{calc-trail-display}) command turns display of the
12253 trail on and off. Normally the trail display is toggled on if it was off,
12254 off if it was on. With a numeric prefix of zero, this command always
12255 turns the trail off; with a prefix of one, it always turns the trail on.
12256 The other trail-manipulation commands described here automatically turn
12257 the trail on. Note that when the trail is off values are still recorded
12258 there; they are simply not displayed. To set Emacs to turn the trail
12259 off by default, type @kbd{t d} and then save the mode settings with
12260 @kbd{m m} (@code{calc-save-modes}).
12263 @pindex calc-trail-in
12265 @pindex calc-trail-out
12266 The @kbd{t i} (@code{calc-trail-in}) and @kbd{t o}
12267 (@code{calc-trail-out}) commands switch the cursor into and out of the
12268 Calc Trail window. In practice they are rarely used, since the commands
12269 shown below are a more convenient way to move around in the
12270 trail, and they work ``by remote control'' when the cursor is still
12271 in the Calculator window.
12273 @cindex Trail pointer
12274 There is a @dfn{trail pointer} which selects some entry of the trail at
12275 any given time. The trail pointer looks like a @samp{>} symbol right
12276 before the selected number. The following commands operate on the
12277 trail pointer in various ways.
12280 @pindex calc-trail-yank
12281 @cindex Retrieving previous results
12282 The @kbd{t y} (@code{calc-trail-yank}) command reads the selected value in
12283 the trail and pushes it onto the Calculator stack. It allows you to
12284 re-use any previously computed value without retyping. With a numeric
12285 prefix argument @var{n}, it yanks the value @var{n} lines above the current
12289 @pindex calc-trail-scroll-left
12291 @pindex calc-trail-scroll-right
12292 The @kbd{t <} (@code{calc-trail-scroll-left}) and @kbd{t >}
12293 (@code{calc-trail-scroll-right}) commands horizontally scroll the trail
12294 window left or right by one half of its width.
12297 @pindex calc-trail-next
12299 @pindex calc-trail-previous
12301 @pindex calc-trail-forward
12303 @pindex calc-trail-backward
12304 The @kbd{t n} (@code{calc-trail-next}) and @kbd{t p}
12305 (@code{calc-trail-previous)} commands move the trail pointer down or up
12306 one line. The @kbd{t f} (@code{calc-trail-forward}) and @kbd{t b}
12307 (@code{calc-trail-backward}) commands move the trail pointer down or up
12308 one screenful at a time. All of these commands accept numeric prefix
12309 arguments to move several lines or screenfuls at a time.
12312 @pindex calc-trail-first
12314 @pindex calc-trail-last
12316 @pindex calc-trail-here
12317 The @kbd{t [} (@code{calc-trail-first}) and @kbd{t ]}
12318 (@code{calc-trail-last}) commands move the trail pointer to the first or
12319 last line of the trail. The @kbd{t h} (@code{calc-trail-here}) command
12320 moves the trail pointer to the cursor position; unlike the other trail
12321 commands, @kbd{t h} works only when Calc Trail is the selected window.
12324 @pindex calc-trail-isearch-forward
12326 @pindex calc-trail-isearch-backward
12328 The @kbd{t s} (@code{calc-trail-isearch-forward}) and @kbd{t r}
12329 (@code{calc-trail-isearch-backward}) commands perform an incremental
12330 search forward or backward through the trail. You can press @key{RET}
12331 to terminate the search; the trail pointer moves to the current line.
12332 If you cancel the search with @kbd{C-g}, the trail pointer stays where
12333 it was when the search began.
12336 The @kbd{t s} (@code{calc-trail-isearch-forward}) and @kbd{t r}
12337 (@code{calc-trail-isearch-backward}) com\-mands perform an incremental
12338 search forward or backward through the trail. You can press @key{RET}
12339 to terminate the search; the trail pointer moves to the current line.
12340 If you cancel the search with @kbd{C-g}, the trail pointer stays where
12341 it was when the search began.
12345 @pindex calc-trail-marker
12346 The @kbd{t m} (@code{calc-trail-marker}) command allows you to enter a
12347 line of text of your own choosing into the trail. The text is inserted
12348 after the line containing the trail pointer; this usually means it is
12349 added to the end of the trail. Trail markers are useful mainly as the
12350 targets for later incremental searches in the trail.
12353 @pindex calc-trail-kill
12354 The @kbd{t k} (@code{calc-trail-kill}) command removes the selected line
12355 from the trail. The line is saved in the Emacs kill ring suitable for
12356 yanking into another buffer, but it is not easy to yank the text back
12357 into the trail buffer. With a numeric prefix argument, this command
12358 kills the @var{n} lines below or above the selected one.
12360 The @kbd{t .} (@code{calc-full-trail-vectors}) command is described
12361 elsewhere; @pxref{Vector and Matrix Formats}.
12363 @node Keep Arguments, , Trail Commands, Stack and Trail
12364 @section Keep Arguments
12368 @pindex calc-keep-args
12369 The @kbd{K} (@code{calc-keep-args}) command acts like a prefix for
12370 the following command. It prevents that command from removing its
12371 arguments from the stack. For example, after @kbd{2 @key{RET} 3 +},
12372 the stack contains the sole number 5, but after @kbd{2 @key{RET} 3 K +},
12373 the stack contains the arguments and the result: @samp{2 3 5}.
12375 With the exception of keyboard macros, this works for all commands that
12376 take arguments off the stack. (To avoid potentially unpleasant behavior,
12377 a @kbd{K} prefix before a keyboard macro will be ignored. A @kbd{K}
12378 prefix called @emph{within} the keyboard macro will still take effect.)
12379 As another example, @kbd{K a s} simplifies a formula, pushing the
12380 simplified version of the formula onto the stack after the original
12381 formula (rather than replacing the original formula). Note that you
12382 could get the same effect by typing @kbd{@key{RET} a s}, copying the
12383 formula and then simplifying the copy. One difference is that for a very
12384 large formula the time taken to format the intermediate copy in
12385 @kbd{@key{RET} a s} could be noticeable; @kbd{K a s} would avoid this
12388 Even stack manipulation commands are affected. @key{TAB} works by
12389 popping two values and pushing them back in the opposite order,
12390 so @kbd{2 @key{RET} 3 K @key{TAB}} produces @samp{2 3 3 2}.
12392 A few Calc commands provide other ways of doing the same thing.
12393 For example, @kbd{' sin($)} replaces the number on the stack with
12394 its sine using algebraic entry; to push the sine and keep the
12395 original argument you could use either @kbd{' sin($1)} or
12396 @kbd{K ' sin($)}. @xref{Algebraic Entry}. Also, the @kbd{s s}
12397 command is effectively the same as @kbd{K s t}. @xref{Storing Variables}.
12399 If you execute a command and then decide you really wanted to keep
12400 the argument, you can press @kbd{M-@key{RET}} (@code{calc-last-args}).
12401 This command pushes the last arguments that were popped by any command
12402 onto the stack. Note that the order of things on the stack will be
12403 different than with @kbd{K}: @kbd{2 @key{RET} 3 + M-@key{RET}} leaves
12404 @samp{5 2 3} on the stack instead of @samp{2 3 5}. @xref{Undo}.
12406 @node Mode Settings, Arithmetic, Stack and Trail, Top
12407 @chapter Mode Settings
12410 This chapter describes commands that set modes in the Calculator.
12411 They do not affect the contents of the stack, although they may change
12412 the @emph{appearance} or @emph{interpretation} of the stack's contents.
12415 * General Mode Commands::
12417 * Inverse and Hyperbolic::
12418 * Calculation Modes::
12419 * Simplification Modes::
12427 @node General Mode Commands, Precision, Mode Settings, Mode Settings
12428 @section General Mode Commands
12432 @pindex calc-save-modes
12433 @cindex Continuous memory
12434 @cindex Saving mode settings
12435 @cindex Permanent mode settings
12436 @cindex Calc init file, mode settings
12437 You can save all of the current mode settings in your Calc init file
12438 (the file given by the variable @code{calc-settings-file}, typically
12439 @file{~/.calc.el}) with the @kbd{m m} (@code{calc-save-modes}) command.
12440 This will cause Emacs to reestablish these modes each time it starts up.
12441 The modes saved in the file include everything controlled by the @kbd{m}
12442 and @kbd{d} prefix keys, the current precision and binary word size,
12443 whether or not the trail is displayed, the current height of the Calc
12444 window, and more. The current interface (used when you type @kbd{M-#
12445 M-#}) is also saved. If there were already saved mode settings in the
12446 file, they are replaced. Otherwise, the new mode information is
12447 appended to the end of the file.
12450 @pindex calc-mode-record-mode
12451 The @kbd{m R} (@code{calc-mode-record-mode}) command tells Calc to
12452 record all the mode settings (as if by pressing @kbd{m m}) every
12453 time a mode setting changes. If the modes are saved this way, then this
12454 ``automatic mode recording'' mode is also saved.
12455 Type @kbd{m R} again to disable this method of recording the mode
12456 settings. To turn it off permanently, the @kbd{m m} command will also be
12457 necessary. (If Embedded mode is enabled, other options for recording
12458 the modes are available; @pxref{Mode Settings in Embedded Mode}.)
12461 @pindex calc-settings-file-name
12462 The @kbd{m F} (@code{calc-settings-file-name}) command allows you to
12463 choose a different file than the current value of @code{calc-settings-file}
12464 for @kbd{m m}, @kbd{Z P}, and similar commands to save permanent information.
12465 You are prompted for a file name. All Calc modes are then reset to
12466 their default values, then settings from the file you named are loaded
12467 if this file exists, and this file becomes the one that Calc will
12468 use in the future for commands like @kbd{m m}. The default settings
12469 file name is @file{~/.calc.el}. You can see the current file name by
12470 giving a blank response to the @kbd{m F} prompt. See also the
12471 discussion of the @code{calc-settings-file} variable; @pxref{Customizable Variables}.
12473 If the file name you give is your user init file (typically
12474 @file{~/.emacs}), @kbd{m F} will not automatically load the new file. This
12475 is because your user init file may contain other things you don't want
12476 to reread. You can give
12477 a numeric prefix argument of 1 to @kbd{m F} to force it to read the
12478 file no matter what. Conversely, an argument of @mathit{-1} tells
12479 @kbd{m F} @emph{not} to read the new file. An argument of 2 or @mathit{-2}
12480 tells @kbd{m F} not to reset the modes to their defaults beforehand,
12481 which is useful if you intend your new file to have a variant of the
12482 modes present in the file you were using before.
12485 @pindex calc-always-load-extensions
12486 The @kbd{m x} (@code{calc-always-load-extensions}) command enables a mode
12487 in which the first use of Calc loads the entire program, including all
12488 extensions modules. Otherwise, the extensions modules will not be loaded
12489 until the various advanced Calc features are used. Since this mode only
12490 has effect when Calc is first loaded, @kbd{m x} is usually followed by
12491 @kbd{m m} to make the mode-setting permanent. To load all of Calc just
12492 once, rather than always in the future, you can press @kbd{M-# L}.
12495 @pindex calc-shift-prefix
12496 The @kbd{m S} (@code{calc-shift-prefix}) command enables a mode in which
12497 all of Calc's letter prefix keys may be typed shifted as well as unshifted.
12498 If you are typing, say, @kbd{a S} (@code{calc-solve-for}) quite often
12499 you might find it easier to turn this mode on so that you can type
12500 @kbd{A S} instead. When this mode is enabled, the commands that used to
12501 be on those single shifted letters (e.g., @kbd{A} (@code{calc-abs})) can
12502 now be invoked by pressing the shifted letter twice: @kbd{A A}. Note
12503 that the @kbd{v} prefix key always works both shifted and unshifted, and
12504 the @kbd{z} and @kbd{Z} prefix keys are always distinct. Also, the @kbd{h}
12505 prefix is not affected by this mode. Press @kbd{m S} again to disable
12506 shifted-prefix mode.
12508 @node Precision, Inverse and Hyperbolic, General Mode Commands, Mode Settings
12513 @pindex calc-precision
12514 @cindex Precision of calculations
12515 The @kbd{p} (@code{calc-precision}) command controls the precision to
12516 which floating-point calculations are carried. The precision must be
12517 at least 3 digits and may be arbitrarily high, within the limits of
12518 memory and time. This affects only floats: Integer and rational
12519 calculations are always carried out with as many digits as necessary.
12521 The @kbd{p} key prompts for the current precision. If you wish you
12522 can instead give the precision as a numeric prefix argument.
12524 Many internal calculations are carried to one or two digits higher
12525 precision than normal. Results are rounded down afterward to the
12526 current precision. Unless a special display mode has been selected,
12527 floats are always displayed with their full stored precision, i.e.,
12528 what you see is what you get. Reducing the current precision does not
12529 round values already on the stack, but those values will be rounded
12530 down before being used in any calculation. The @kbd{c 0} through
12531 @kbd{c 9} commands (@pxref{Conversions}) can be used to round an
12532 existing value to a new precision.
12534 @cindex Accuracy of calculations
12535 It is important to distinguish the concepts of @dfn{precision} and
12536 @dfn{accuracy}. In the normal usage of these words, the number
12537 123.4567 has a precision of 7 digits but an accuracy of 4 digits.
12538 The precision is the total number of digits not counting leading
12539 or trailing zeros (regardless of the position of the decimal point).
12540 The accuracy is simply the number of digits after the decimal point
12541 (again not counting trailing zeros). In Calc you control the precision,
12542 not the accuracy of computations. If you were to set the accuracy
12543 instead, then calculations like @samp{exp(100)} would generate many
12544 more digits than you would typically need, while @samp{exp(-100)} would
12545 probably round to zero! In Calc, both these computations give you
12546 exactly 12 (or the requested number of) significant digits.
12548 The only Calc features that deal with accuracy instead of precision
12549 are fixed-point display mode for floats (@kbd{d f}; @pxref{Float Formats}),
12550 and the rounding functions like @code{floor} and @code{round}
12551 (@pxref{Integer Truncation}). Also, @kbd{c 0} through @kbd{c 9}
12552 deal with both precision and accuracy depending on the magnitudes
12553 of the numbers involved.
12555 If you need to work with a particular fixed accuracy (say, dollars and
12556 cents with two digits after the decimal point), one solution is to work
12557 with integers and an ``implied'' decimal point. For example, $8.99
12558 divided by 6 would be entered @kbd{899 @key{RET} 6 /}, yielding 149.833
12559 (actually $1.49833 with our implied decimal point); pressing @kbd{R}
12560 would round this to 150 cents, i.e., $1.50.
12562 @xref{Floats}, for still more on floating-point precision and related
12565 @node Inverse and Hyperbolic, Calculation Modes, Precision, Mode Settings
12566 @section Inverse and Hyperbolic Flags
12570 @pindex calc-inverse
12571 There is no single-key equivalent to the @code{calc-arcsin} function.
12572 Instead, you must first press @kbd{I} (@code{calc-inverse}) to set
12573 the @dfn{Inverse Flag}, then press @kbd{S} (@code{calc-sin}).
12574 The @kbd{I} key actually toggles the Inverse Flag. When this flag
12575 is set, the word @samp{Inv} appears in the mode line.
12578 @pindex calc-hyperbolic
12579 Likewise, the @kbd{H} key (@code{calc-hyperbolic}) sets or clears the
12580 Hyperbolic Flag, which transforms @code{calc-sin} into @code{calc-sinh}.
12581 If both of these flags are set at once, the effect will be
12582 @code{calc-arcsinh}. (The Hyperbolic flag is also used by some
12583 non-trigonometric commands; for example @kbd{H L} computes a base-10,
12584 instead of base-@mathit{e}, logarithm.)
12586 Command names like @code{calc-arcsin} are provided for completeness, and
12587 may be executed with @kbd{x} or @kbd{M-x}. Their effect is simply to
12588 toggle the Inverse and/or Hyperbolic flags and then execute the
12589 corresponding base command (@code{calc-sin} in this case).
12591 The Inverse and Hyperbolic flags apply only to the next Calculator
12592 command, after which they are automatically cleared. (They are also
12593 cleared if the next keystroke is not a Calc command.) Digits you
12594 type after @kbd{I} or @kbd{H} (or @kbd{K}) are treated as prefix
12595 arguments for the next command, not as numeric entries. The same
12596 is true of @kbd{C-u}, but not of the minus sign (@kbd{K -} means to
12597 subtract and keep arguments).
12599 The third Calc prefix flag, @kbd{K} (keep-arguments), is discussed
12600 elsewhere. @xref{Keep Arguments}.
12602 @node Calculation Modes, Simplification Modes, Inverse and Hyperbolic, Mode Settings
12603 @section Calculation Modes
12606 The commands in this section are two-key sequences beginning with
12607 the @kbd{m} prefix. (That's the letter @kbd{m}, not the @key{META} key.)
12608 The @samp{m a} (@code{calc-algebraic-mode}) command is described elsewhere
12609 (@pxref{Algebraic Entry}).
12618 * Automatic Recomputation::
12619 * Working Message::
12622 @node Angular Modes, Polar Mode, Calculation Modes, Calculation Modes
12623 @subsection Angular Modes
12626 @cindex Angular mode
12627 The Calculator supports three notations for angles: radians, degrees,
12628 and degrees-minutes-seconds. When a number is presented to a function
12629 like @code{sin} that requires an angle, the current angular mode is
12630 used to interpret the number as either radians or degrees. If an HMS
12631 form is presented to @code{sin}, it is always interpreted as
12632 degrees-minutes-seconds.
12634 Functions that compute angles produce a number in radians, a number in
12635 degrees, or an HMS form depending on the current angular mode. If the
12636 result is a complex number and the current mode is HMS, the number is
12637 instead expressed in degrees. (Complex-number calculations would
12638 normally be done in Radians mode, though. Complex numbers are converted
12639 to degrees by calculating the complex result in radians and then
12640 multiplying by 180 over @cpi{}.)
12643 @pindex calc-radians-mode
12645 @pindex calc-degrees-mode
12647 @pindex calc-hms-mode
12648 The @kbd{m r} (@code{calc-radians-mode}), @kbd{m d} (@code{calc-degrees-mode}),
12649 and @kbd{m h} (@code{calc-hms-mode}) commands control the angular mode.
12650 The current angular mode is displayed on the Emacs mode line.
12651 The default angular mode is Degrees.
12653 @node Polar Mode, Fraction Mode, Angular Modes, Calculation Modes
12654 @subsection Polar Mode
12658 The Calculator normally ``prefers'' rectangular complex numbers in the
12659 sense that rectangular form is used when the proper form can not be
12660 decided from the input. This might happen by multiplying a rectangular
12661 number by a polar one, by taking the square root of a negative real
12662 number, or by entering @kbd{( 2 @key{SPC} 3 )}.
12665 @pindex calc-polar-mode
12666 The @kbd{m p} (@code{calc-polar-mode}) command toggles complex-number
12667 preference between rectangular and polar forms. In Polar mode, all
12668 of the above example situations would produce polar complex numbers.
12670 @node Fraction Mode, Infinite Mode, Polar Mode, Calculation Modes
12671 @subsection Fraction Mode
12674 @cindex Fraction mode
12675 @cindex Division of integers
12676 Division of two integers normally yields a floating-point number if the
12677 result cannot be expressed as an integer. In some cases you would
12678 rather get an exact fractional answer. One way to accomplish this is
12679 to use the @kbd{:} (@code{calc-fdiv}) [@code{fdiv}] command, which
12680 divides the two integers on the top of the stack to produce a fraction:
12681 @kbd{6 @key{RET} 4 :} produces @expr{3:2} even though
12682 @kbd{6 @key{RET} 4 /} produces @expr{1.5}.
12685 @pindex calc-frac-mode
12686 To set the Calculator to produce fractional results for normal integer
12687 divisions, use the @kbd{m f} (@code{calc-frac-mode}) command.
12688 For example, @expr{8/4} produces @expr{2} in either mode,
12689 but @expr{6/4} produces @expr{3:2} in Fraction mode, @expr{1.5} in
12692 At any time you can use @kbd{c f} (@code{calc-float}) to convert a
12693 fraction to a float, or @kbd{c F} (@code{calc-fraction}) to convert a
12694 float to a fraction. @xref{Conversions}.
12696 @node Infinite Mode, Symbolic Mode, Fraction Mode, Calculation Modes
12697 @subsection Infinite Mode
12700 @cindex Infinite mode
12701 The Calculator normally treats results like @expr{1 / 0} as errors;
12702 formulas like this are left in unsimplified form. But Calc can be
12703 put into a mode where such calculations instead produce ``infinite''
12707 @pindex calc-infinite-mode
12708 The @kbd{m i} (@code{calc-infinite-mode}) command turns this mode
12709 on and off. When the mode is off, infinities do not arise except
12710 in calculations that already had infinities as inputs. (One exception
12711 is that infinite open intervals like @samp{[0 .. inf)} can be
12712 generated; however, intervals closed at infinity (@samp{[0 .. inf]})
12713 will not be generated when Infinite mode is off.)
12715 With Infinite mode turned on, @samp{1 / 0} will generate @code{uinf},
12716 an undirected infinity. @xref{Infinities}, for a discussion of the
12717 difference between @code{inf} and @code{uinf}. Also, @expr{0 / 0}
12718 evaluates to @code{nan}, the ``indeterminate'' symbol. Various other
12719 functions can also return infinities in this mode; for example,
12720 @samp{ln(0) = -inf}, and @samp{gamma(-7) = uinf}. Once again,
12721 note that @samp{exp(inf) = inf} regardless of Infinite mode because
12722 this calculation has infinity as an input.
12724 @cindex Positive Infinite mode
12725 The @kbd{m i} command with a numeric prefix argument of zero,
12726 i.e., @kbd{C-u 0 m i}, turns on a Positive Infinite mode in
12727 which zero is treated as positive instead of being directionless.
12728 Thus, @samp{1 / 0 = inf} and @samp{-1 / 0 = -inf} in this mode.
12729 Note that zero never actually has a sign in Calc; there are no
12730 separate representations for @mathit{+0} and @mathit{-0}. Positive
12731 Infinite mode merely changes the interpretation given to the
12732 single symbol, @samp{0}. One consequence of this is that, while
12733 you might expect @samp{1 / -0 = -inf}, actually @samp{1 / -0}
12734 is equivalent to @samp{1 / 0}, which is equal to positive @code{inf}.
12736 @node Symbolic Mode, Matrix Mode, Infinite Mode, Calculation Modes
12737 @subsection Symbolic Mode
12740 @cindex Symbolic mode
12741 @cindex Inexact results
12742 Calculations are normally performed numerically wherever possible.
12743 For example, the @code{calc-sqrt} command, or @code{sqrt} function in an
12744 algebraic expression, produces a numeric answer if the argument is a
12745 number or a symbolic expression if the argument is an expression:
12746 @kbd{2 Q} pushes 1.4142 but @kbd{@key{'} x+1 @key{RET} Q} pushes @samp{sqrt(x+1)}.
12749 @pindex calc-symbolic-mode
12750 In @dfn{Symbolic mode}, controlled by the @kbd{m s} (@code{calc-symbolic-mode})
12751 command, functions which would produce inexact, irrational results are
12752 left in symbolic form. Thus @kbd{16 Q} pushes 4, but @kbd{2 Q} pushes
12756 @pindex calc-eval-num
12757 The shift-@kbd{N} (@code{calc-eval-num}) command evaluates numerically
12758 the expression at the top of the stack, by temporarily disabling
12759 @code{calc-symbolic-mode} and executing @kbd{=} (@code{calc-evaluate}).
12760 Given a numeric prefix argument, it also
12761 sets the floating-point precision to the specified value for the duration
12764 To evaluate a formula numerically without expanding the variables it
12765 contains, you can use the key sequence @kbd{m s a v m s} (this uses
12766 @code{calc-alg-evaluate}, which resimplifies but doesn't evaluate
12769 @node Matrix Mode, Automatic Recomputation, Symbolic Mode, Calculation Modes
12770 @subsection Matrix and Scalar Modes
12773 @cindex Matrix mode
12774 @cindex Scalar mode
12775 Calc sometimes makes assumptions during algebraic manipulation that
12776 are awkward or incorrect when vectors and matrices are involved.
12777 Calc has two modes, @dfn{Matrix mode} and @dfn{Scalar mode}, which
12778 modify its behavior around vectors in useful ways.
12781 @pindex calc-matrix-mode
12782 Press @kbd{m v} (@code{calc-matrix-mode}) once to enter Matrix mode.
12783 In this mode, all objects are assumed to be matrices unless provably
12784 otherwise. One major effect is that Calc will no longer consider
12785 multiplication to be commutative. (Recall that in matrix arithmetic,
12786 @samp{A*B} is not the same as @samp{B*A}.) This assumption affects
12787 rewrite rules and algebraic simplification. Another effect of this
12788 mode is that calculations that would normally produce constants like
12789 0 and 1 (e.g., @expr{a - a} and @expr{a / a}, respectively) will now
12790 produce function calls that represent ``generic'' zero or identity
12791 matrices: @samp{idn(0)}, @samp{idn(1)}. The @code{idn} function
12792 @samp{idn(@var{a},@var{n})} returns @var{a} times an @var{n}x@var{n}
12793 identity matrix; if @var{n} is omitted, it doesn't know what
12794 dimension to use and so the @code{idn} call remains in symbolic
12795 form. However, if this generic identity matrix is later combined
12796 with a matrix whose size is known, it will be converted into
12797 a true identity matrix of the appropriate size. On the other hand,
12798 if it is combined with a scalar (as in @samp{idn(1) + 2}), Calc
12799 will assume it really was a scalar after all and produce, e.g., 3.
12801 Press @kbd{m v} a second time to get Scalar mode. Here, objects are
12802 assumed @emph{not} to be vectors or matrices unless provably so.
12803 For example, normally adding a variable to a vector, as in
12804 @samp{[x, y, z] + a}, will leave the sum in symbolic form because
12805 as far as Calc knows, @samp{a} could represent either a number or
12806 another 3-vector. In Scalar mode, @samp{a} is assumed to be a
12807 non-vector, and the addition is evaluated to @samp{[x+a, y+a, z+a]}.
12809 Press @kbd{m v} a third time to return to the normal mode of operation.
12811 If you press @kbd{m v} with a numeric prefix argument @var{n}, you
12812 get a special ``dimensioned'' Matrix mode in which matrices of
12813 unknown size are assumed to be @var{n}x@var{n} square matrices.
12814 Then, the function call @samp{idn(1)} will expand into an actual
12815 matrix rather than representing a ``generic'' matrix.
12817 @cindex Declaring scalar variables
12818 Of course these modes are approximations to the true state of
12819 affairs, which is probably that some quantities will be matrices
12820 and others will be scalars. One solution is to ``declare''
12821 certain variables or functions to be scalar-valued.
12822 @xref{Declarations}, to see how to make declarations in Calc.
12824 There is nothing stopping you from declaring a variable to be
12825 scalar and then storing a matrix in it; however, if you do, the
12826 results you get from Calc may not be valid. Suppose you let Calc
12827 get the result @samp{[x+a, y+a, z+a]} shown above, and then stored
12828 @samp{[1, 2, 3]} in @samp{a}. The result would not be the same as
12829 for @samp{[x, y, z] + [1, 2, 3]}, but that's because you have broken
12830 your earlier promise to Calc that @samp{a} would be scalar.
12832 Another way to mix scalars and matrices is to use selections
12833 (@pxref{Selecting Subformulas}). Use Matrix mode when operating on
12834 your formula normally; then, to apply Scalar mode to a certain part
12835 of the formula without affecting the rest just select that part,
12836 change into Scalar mode and press @kbd{=} to resimplify the part
12837 under this mode, then change back to Matrix mode before deselecting.
12839 @node Automatic Recomputation, Working Message, Matrix Mode, Calculation Modes
12840 @subsection Automatic Recomputation
12843 The @dfn{evaluates-to} operator, @samp{=>}, has the special
12844 property that any @samp{=>} formulas on the stack are recomputed
12845 whenever variable values or mode settings that might affect them
12846 are changed. @xref{Evaluates-To Operator}.
12849 @pindex calc-auto-recompute
12850 The @kbd{m C} (@code{calc-auto-recompute}) command turns this
12851 automatic recomputation on and off. If you turn it off, Calc will
12852 not update @samp{=>} operators on the stack (nor those in the
12853 attached Embedded mode buffer, if there is one). They will not
12854 be updated unless you explicitly do so by pressing @kbd{=} or until
12855 you press @kbd{m C} to turn recomputation back on. (While automatic
12856 recomputation is off, you can think of @kbd{m C m C} as a command
12857 to update all @samp{=>} operators while leaving recomputation off.)
12859 To update @samp{=>} operators in an Embedded buffer while
12860 automatic recomputation is off, use @w{@kbd{M-# u}}.
12861 @xref{Embedded Mode}.
12863 @node Working Message, , Automatic Recomputation, Calculation Modes
12864 @subsection Working Messages
12867 @cindex Performance
12868 @cindex Working messages
12869 Since the Calculator is written entirely in Emacs Lisp, which is not
12870 designed for heavy numerical work, many operations are quite slow.
12871 The Calculator normally displays the message @samp{Working...} in the
12872 echo area during any command that may be slow. In addition, iterative
12873 operations such as square roots and trigonometric functions display the
12874 intermediate result at each step. Both of these types of messages can
12875 be disabled if you find them distracting.
12878 @pindex calc-working
12879 Type @kbd{m w} (@code{calc-working}) with a numeric prefix of 0 to
12880 disable all ``working'' messages. Use a numeric prefix of 1 to enable
12881 only the plain @samp{Working...} message. Use a numeric prefix of 2 to
12882 see intermediate results as well. With no numeric prefix this displays
12885 While it may seem that the ``working'' messages will slow Calc down
12886 considerably, experiments have shown that their impact is actually
12887 quite small. But if your terminal is slow you may find that it helps
12888 to turn the messages off.
12890 @node Simplification Modes, Declarations, Calculation Modes, Mode Settings
12891 @section Simplification Modes
12894 The current @dfn{simplification mode} controls how numbers and formulas
12895 are ``normalized'' when being taken from or pushed onto the stack.
12896 Some normalizations are unavoidable, such as rounding floating-point
12897 results to the current precision, and reducing fractions to simplest
12898 form. Others, such as simplifying a formula like @expr{a+a} (or @expr{2+3}),
12899 are done by default but can be turned off when necessary.
12901 When you press a key like @kbd{+} when @expr{2} and @expr{3} are on the
12902 stack, Calc pops these numbers, normalizes them, creates the formula
12903 @expr{2+3}, normalizes it, and pushes the result. Of course the standard
12904 rules for normalizing @expr{2+3} will produce the result @expr{5}.
12906 Simplification mode commands consist of the lower-case @kbd{m} prefix key
12907 followed by a shifted letter.
12910 @pindex calc-no-simplify-mode
12911 The @kbd{m O} (@code{calc-no-simplify-mode}) command turns off all optional
12912 simplifications. These would leave a formula like @expr{2+3} alone. In
12913 fact, nothing except simple numbers are ever affected by normalization
12917 @pindex calc-num-simplify-mode
12918 The @kbd{m N} (@code{calc-num-simplify-mode}) command turns off simplification
12919 of any formulas except those for which all arguments are constants. For
12920 example, @expr{1+2} is simplified to @expr{3}, and @expr{a+(2-2)} is
12921 simplified to @expr{a+0} but no further, since one argument of the sum
12922 is not a constant. Unfortunately, @expr{(a+2)-2} is @emph{not} simplified
12923 because the top-level @samp{-} operator's arguments are not both
12924 constant numbers (one of them is the formula @expr{a+2}).
12925 A constant is a number or other numeric object (such as a constant
12926 error form or modulo form), or a vector all of whose
12927 elements are constant.
12930 @pindex calc-default-simplify-mode
12931 The @kbd{m D} (@code{calc-default-simplify-mode}) command restores the
12932 default simplifications for all formulas. This includes many easy and
12933 fast algebraic simplifications such as @expr{a+0} to @expr{a}, and
12934 @expr{a + 2 a} to @expr{3 a}, as well as evaluating functions like
12935 @expr{@tfn{deriv}(x^2, x)} to @expr{2 x}.
12938 @pindex calc-bin-simplify-mode
12939 The @kbd{m B} (@code{calc-bin-simplify-mode}) mode applies the default
12940 simplifications to a result and then, if the result is an integer,
12941 uses the @kbd{b c} (@code{calc-clip}) command to clip the integer according
12942 to the current binary word size. @xref{Binary Functions}. Real numbers
12943 are rounded to the nearest integer and then clipped; other kinds of
12944 results (after the default simplifications) are left alone.
12947 @pindex calc-alg-simplify-mode
12948 The @kbd{m A} (@code{calc-alg-simplify-mode}) mode does algebraic
12949 simplification; it applies all the default simplifications, and also
12950 the more powerful (and slower) simplifications made by @kbd{a s}
12951 (@code{calc-simplify}). @xref{Algebraic Simplifications}.
12954 @pindex calc-ext-simplify-mode
12955 The @kbd{m E} (@code{calc-ext-simplify-mode}) mode does ``extended''
12956 algebraic simplification, as by the @kbd{a e} (@code{calc-simplify-extended})
12957 command. @xref{Unsafe Simplifications}.
12960 @pindex calc-units-simplify-mode
12961 The @kbd{m U} (@code{calc-units-simplify-mode}) mode does units
12962 simplification; it applies the command @kbd{u s}
12963 (@code{calc-simplify-units}), which in turn
12964 is a superset of @kbd{a s}. In this mode, variable names which
12965 are identifiable as unit names (like @samp{mm} for ``millimeters'')
12966 are simplified with their unit definitions in mind.
12968 A common technique is to set the simplification mode down to the lowest
12969 amount of simplification you will allow to be applied automatically, then
12970 use manual commands like @kbd{a s} and @kbd{c c} (@code{calc-clean}) to
12971 perform higher types of simplifications on demand. @xref{Algebraic
12972 Definitions}, for another sample use of No-Simplification mode.
12974 @node Declarations, Display Modes, Simplification Modes, Mode Settings
12975 @section Declarations
12978 A @dfn{declaration} is a statement you make that promises you will
12979 use a certain variable or function in a restricted way. This may
12980 give Calc the freedom to do things that it couldn't do if it had to
12981 take the fully general situation into account.
12984 * Declaration Basics::
12985 * Kinds of Declarations::
12986 * Functions for Declarations::
12989 @node Declaration Basics, Kinds of Declarations, Declarations, Declarations
12990 @subsection Declaration Basics
12994 @pindex calc-declare-variable
12995 The @kbd{s d} (@code{calc-declare-variable}) command is the easiest
12996 way to make a declaration for a variable. This command prompts for
12997 the variable name, then prompts for the declaration. The default
12998 at the declaration prompt is the previous declaration, if any.
12999 You can edit this declaration, or press @kbd{C-k} to erase it and
13000 type a new declaration. (Or, erase it and press @key{RET} to clear
13001 the declaration, effectively ``undeclaring'' the variable.)
13003 A declaration is in general a vector of @dfn{type symbols} and
13004 @dfn{range} values. If there is only one type symbol or range value,
13005 you can write it directly rather than enclosing it in a vector.
13006 For example, @kbd{s d foo @key{RET} real @key{RET}} declares @code{foo} to
13007 be a real number, and @kbd{s d bar @key{RET} [int, const, [1..6]] @key{RET}}
13008 declares @code{bar} to be a constant integer between 1 and 6.
13009 (Actually, you can omit the outermost brackets and Calc will
13010 provide them for you: @kbd{s d bar @key{RET} int, const, [1..6] @key{RET}}.)
13012 @cindex @code{Decls} variable
13014 Declarations in Calc are kept in a special variable called @code{Decls}.
13015 This variable encodes the set of all outstanding declarations in
13016 the form of a matrix. Each row has two elements: A variable or
13017 vector of variables declared by that row, and the declaration
13018 specifier as described above. You can use the @kbd{s D} command to
13019 edit this variable if you wish to see all the declarations at once.
13020 @xref{Operations on Variables}, for a description of this command
13021 and the @kbd{s p} command that allows you to save your declarations
13022 permanently if you wish.
13024 Items being declared can also be function calls. The arguments in
13025 the call are ignored; the effect is to say that this function returns
13026 values of the declared type for any valid arguments. The @kbd{s d}
13027 command declares only variables, so if you wish to make a function
13028 declaration you will have to edit the @code{Decls} matrix yourself.
13030 For example, the declaration matrix
13036 [ f(1,2,3), [0 .. inf) ] ]
13041 declares that @code{foo} represents a real number, @code{j}, @code{k}
13042 and @code{n} represent integers, and the function @code{f} always
13043 returns a real number in the interval shown.
13046 If there is a declaration for the variable @code{All}, then that
13047 declaration applies to all variables that are not otherwise declared.
13048 It does not apply to function names. For example, using the row
13049 @samp{[All, real]} says that all your variables are real unless they
13050 are explicitly declared without @code{real} in some other row.
13051 The @kbd{s d} command declares @code{All} if you give a blank
13052 response to the variable-name prompt.
13054 @node Kinds of Declarations, Functions for Declarations, Declaration Basics, Declarations
13055 @subsection Kinds of Declarations
13058 The type-specifier part of a declaration (that is, the second prompt
13059 in the @kbd{s d} command) can be a type symbol, an interval, or a
13060 vector consisting of zero or more type symbols followed by zero or
13061 more intervals or numbers that represent the set of possible values
13066 [ [ a, [1, 2, 3, 4, 5] ]
13068 [ c, [int, 1 .. 5] ] ]
13072 Here @code{a} is declared to contain one of the five integers shown;
13073 @code{b} is any number in the interval from 1 to 5 (any real number
13074 since we haven't specified), and @code{c} is any integer in that
13075 interval. Thus the declarations for @code{a} and @code{c} are
13076 nearly equivalent (see below).
13078 The type-specifier can be the empty vector @samp{[]} to say that
13079 nothing is known about a given variable's value. This is the same
13080 as not declaring the variable at all except that it overrides any
13081 @code{All} declaration which would otherwise apply.
13083 The initial value of @code{Decls} is the empty vector @samp{[]}.
13084 If @code{Decls} has no stored value or if the value stored in it
13085 is not valid, it is ignored and there are no declarations as far
13086 as Calc is concerned. (The @kbd{s d} command will replace such a
13087 malformed value with a fresh empty matrix, @samp{[]}, before recording
13088 the new declaration.) Unrecognized type symbols are ignored.
13090 The following type symbols describe what sorts of numbers will be
13091 stored in a variable:
13097 Numerical integers. (Integers or integer-valued floats.)
13099 Fractions. (Rational numbers which are not integers.)
13101 Rational numbers. (Either integers or fractions.)
13103 Floating-point numbers.
13105 Real numbers. (Integers, fractions, or floats. Actually,
13106 intervals and error forms with real components also count as
13109 Positive real numbers. (Strictly greater than zero.)
13111 Nonnegative real numbers. (Greater than or equal to zero.)
13113 Numbers. (Real or complex.)
13116 Calc uses this information to determine when certain simplifications
13117 of formulas are safe. For example, @samp{(x^y)^z} cannot be
13118 simplified to @samp{x^(y z)} in general; for example,
13119 @samp{((-3)^2)^1:2} is 3, but @samp{(-3)^(2*1:2) = (-3)^1} is @mathit{-3}.
13120 However, this simplification @emph{is} safe if @code{z} is known
13121 to be an integer, or if @code{x} is known to be a nonnegative
13122 real number. If you have given declarations that allow Calc to
13123 deduce either of these facts, Calc will perform this simplification
13126 Calc can apply a certain amount of logic when using declarations.
13127 For example, @samp{(x^y)^(2n+1)} will be simplified if @code{n}
13128 has been declared @code{int}; Calc knows that an integer times an
13129 integer, plus an integer, must always be an integer. (In fact,
13130 Calc would simplify @samp{(-x)^(2n+1)} to @samp{-(x^(2n+1))} since
13131 it is able to determine that @samp{2n+1} must be an odd integer.)
13133 Similarly, @samp{(abs(x)^y)^z} will be simplified to @samp{abs(x)^(y z)}
13134 because Calc knows that the @code{abs} function always returns a
13135 nonnegative real. If you had a @code{myabs} function that also had
13136 this property, you could get Calc to recognize it by adding the row
13137 @samp{[myabs(), nonneg]} to the @code{Decls} matrix.
13139 One instance of this simplification is @samp{sqrt(x^2)} (since the
13140 @code{sqrt} function is effectively a one-half power). Normally
13141 Calc leaves this formula alone. After the command
13142 @kbd{s d x @key{RET} real @key{RET}}, however, it can simplify the formula to
13143 @samp{abs(x)}. And after @kbd{s d x @key{RET} nonneg @key{RET}}, Calc can
13144 simplify this formula all the way to @samp{x}.
13146 If there are any intervals or real numbers in the type specifier,
13147 they comprise the set of possible values that the variable or
13148 function being declared can have. In particular, the type symbol
13149 @code{real} is effectively the same as the range @samp{[-inf .. inf]}
13150 (note that infinity is included in the range of possible values);
13151 @code{pos} is the same as @samp{(0 .. inf]}, and @code{nonneg} is
13152 the same as @samp{[0 .. inf]}. Saying @samp{[real, [-5 .. 5]]} is
13153 redundant because the fact that the variable is real can be
13154 deduced just from the interval, but @samp{[int, [-5 .. 5]]} and
13155 @samp{[rat, [-5 .. 5]]} are useful combinations.
13157 Note that the vector of intervals or numbers is in the same format
13158 used by Calc's set-manipulation commands. @xref{Set Operations}.
13160 The type specifier @samp{[1, 2, 3]} is equivalent to
13161 @samp{[numint, 1, 2, 3]}, @emph{not} to @samp{[int, 1, 2, 3]}.
13162 In other words, the range of possible values means only that
13163 the variable's value must be numerically equal to a number in
13164 that range, but not that it must be equal in type as well.
13165 Calc's set operations act the same way; @samp{in(2, [1., 2., 3.])}
13166 and @samp{in(1.5, [1:2, 3:2, 5:2])} both report ``true.''
13168 If you use a conflicting combination of type specifiers, the
13169 results are unpredictable. An example is @samp{[pos, [0 .. 5]]},
13170 where the interval does not lie in the range described by the
13173 ``Real'' declarations mostly affect simplifications involving powers
13174 like the one described above. Another case where they are used
13175 is in the @kbd{a P} command which returns a list of all roots of a
13176 polynomial; if the variable has been declared real, only the real
13177 roots (if any) will be included in the list.
13179 ``Integer'' declarations are used for simplifications which are valid
13180 only when certain values are integers (such as @samp{(x^y)^z}
13183 Another command that makes use of declarations is @kbd{a s}, when
13184 simplifying equations and inequalities. It will cancel @code{x}
13185 from both sides of @samp{a x = b x} only if it is sure @code{x}
13186 is non-zero, say, because it has a @code{pos} declaration.
13187 To declare specifically that @code{x} is real and non-zero,
13188 use @samp{[[-inf .. 0), (0 .. inf]]}. (There is no way in the
13189 current notation to say that @code{x} is nonzero but not necessarily
13190 real.) The @kbd{a e} command does ``unsafe'' simplifications,
13191 including cancelling @samp{x} from the equation when @samp{x} is
13192 not known to be nonzero.
13194 Another set of type symbols distinguish between scalars and vectors.
13198 The value is not a vector.
13200 The value is a vector.
13202 The value is a matrix (a rectangular vector of vectors).
13205 These type symbols can be combined with the other type symbols
13206 described above; @samp{[int, matrix]} describes an object which
13207 is a matrix of integers.
13209 Scalar/vector declarations are used to determine whether certain
13210 algebraic operations are safe. For example, @samp{[a, b, c] + x}
13211 is normally not simplified to @samp{[a + x, b + x, c + x]}, but
13212 it will be if @code{x} has been declared @code{scalar}. On the
13213 other hand, multiplication is usually assumed to be commutative,
13214 but the terms in @samp{x y} will never be exchanged if both @code{x}
13215 and @code{y} are known to be vectors or matrices. (Calc currently
13216 never distinguishes between @code{vector} and @code{matrix}
13219 @xref{Matrix Mode}, for a discussion of Matrix mode and
13220 Scalar mode, which are similar to declaring @samp{[All, matrix]}
13221 or @samp{[All, scalar]} but much more convenient.
13223 One more type symbol that is recognized is used with the @kbd{H a d}
13224 command for taking total derivatives of a formula. @xref{Calculus}.
13228 The value is a constant with respect to other variables.
13231 Calc does not check the declarations for a variable when you store
13232 a value in it. However, storing @mathit{-3.5} in a variable that has
13233 been declared @code{pos}, @code{int}, or @code{matrix} may have
13234 unexpected effects; Calc may evaluate @samp{sqrt(x^2)} to @expr{3.5}
13235 if it substitutes the value first, or to @expr{-3.5} if @code{x}
13236 was declared @code{pos} and the formula @samp{sqrt(x^2)} is
13237 simplified to @samp{x} before the value is substituted. Before
13238 using a variable for a new purpose, it is best to use @kbd{s d}
13239 or @kbd{s D} to check to make sure you don't still have an old
13240 declaration for the variable that will conflict with its new meaning.
13242 @node Functions for Declarations, , Kinds of Declarations, Declarations
13243 @subsection Functions for Declarations
13246 Calc has a set of functions for accessing the current declarations
13247 in a convenient manner. These functions return 1 if the argument
13248 can be shown to have the specified property, or 0 if the argument
13249 can be shown @emph{not} to have that property; otherwise they are
13250 left unevaluated. These functions are suitable for use with rewrite
13251 rules (@pxref{Conditional Rewrite Rules}) or programming constructs
13252 (@pxref{Conditionals in Macros}). They can be entered only using
13253 algebraic notation. @xref{Logical Operations}, for functions
13254 that perform other tests not related to declarations.
13256 For example, @samp{dint(17)} returns 1 because 17 is an integer, as
13257 do @samp{dint(n)} and @samp{dint(2 n - 3)} if @code{n} has been declared
13258 @code{int}, but @samp{dint(2.5)} and @samp{dint(n + 0.5)} return 0.
13259 Calc consults knowledge of its own built-in functions as well as your
13260 own declarations: @samp{dint(floor(x))} returns 1.
13274 The @code{dint} function checks if its argument is an integer.
13275 The @code{dnatnum} function checks if its argument is a natural
13276 number, i.e., a nonnegative integer. The @code{dnumint} function
13277 checks if its argument is numerically an integer, i.e., either an
13278 integer or an integer-valued float. Note that these and the other
13279 data type functions also accept vectors or matrices composed of
13280 suitable elements, and that real infinities @samp{inf} and @samp{-inf}
13281 are considered to be integers for the purposes of these functions.
13287 The @code{drat} function checks if its argument is rational, i.e.,
13288 an integer or fraction. Infinities count as rational, but intervals
13289 and error forms do not.
13295 The @code{dreal} function checks if its argument is real. This
13296 includes integers, fractions, floats, real error forms, and intervals.
13302 The @code{dimag} function checks if its argument is imaginary,
13303 i.e., is mathematically equal to a real number times @expr{i}.
13317 The @code{dpos} function checks for positive (but nonzero) reals.
13318 The @code{dneg} function checks for negative reals. The @code{dnonneg}
13319 function checks for nonnegative reals, i.e., reals greater than or
13320 equal to zero. Note that the @kbd{a s} command can simplify an
13321 expression like @expr{x > 0} to 1 or 0 using @code{dpos}, and that
13322 @kbd{a s} is effectively applied to all conditions in rewrite rules,
13323 so the actual functions @code{dpos}, @code{dneg}, and @code{dnonneg}
13324 are rarely necessary.
13330 The @code{dnonzero} function checks that its argument is nonzero.
13331 This includes all nonzero real or complex numbers, all intervals that
13332 do not include zero, all nonzero modulo forms, vectors all of whose
13333 elements are nonzero, and variables or formulas whose values can be
13334 deduced to be nonzero. It does not include error forms, since they
13335 represent values which could be anything including zero. (This is
13336 also the set of objects considered ``true'' in conditional contexts.)
13346 The @code{deven} function returns 1 if its argument is known to be
13347 an even integer (or integer-valued float); it returns 0 if its argument
13348 is known not to be even (because it is known to be odd or a non-integer).
13349 The @kbd{a s} command uses this to simplify a test of the form
13350 @samp{x % 2 = 0}. There is also an analogous @code{dodd} function.
13356 The @code{drange} function returns a set (an interval or a vector
13357 of intervals and/or numbers; @pxref{Set Operations}) that describes
13358 the set of possible values of its argument. If the argument is
13359 a variable or a function with a declaration, the range is copied
13360 from the declaration. Otherwise, the possible signs of the
13361 expression are determined using a method similar to @code{dpos},
13362 etc., and a suitable set like @samp{[0 .. inf]} is returned. If
13363 the expression is not provably real, the @code{drange} function
13364 remains unevaluated.
13370 The @code{dscalar} function returns 1 if its argument is provably
13371 scalar, or 0 if its argument is provably non-scalar. It is left
13372 unevaluated if this cannot be determined. (If Matrix mode or Scalar
13373 mode is in effect, this function returns 1 or 0, respectively,
13374 if it has no other information.) When Calc interprets a condition
13375 (say, in a rewrite rule) it considers an unevaluated formula to be
13376 ``false.'' Thus, @samp{dscalar(a)} is ``true'' only if @code{a} is
13377 provably scalar, and @samp{!dscalar(a)} is ``true'' only if @code{a}
13378 is provably non-scalar; both are ``false'' if there is insufficient
13379 information to tell.
13381 @node Display Modes, Language Modes, Declarations, Mode Settings
13382 @section Display Modes
13385 The commands in this section are two-key sequences beginning with the
13386 @kbd{d} prefix. The @kbd{d l} (@code{calc-line-numbering}) and @kbd{d b}
13387 (@code{calc-line-breaking}) commands are described elsewhere;
13388 @pxref{Stack Basics} and @pxref{Normal Language Modes}, respectively.
13389 Display formats for vectors and matrices are also covered elsewhere;
13390 @pxref{Vector and Matrix Formats}.
13392 One thing all display modes have in common is their treatment of the
13393 @kbd{H} prefix. This prefix causes any mode command that would normally
13394 refresh the stack to leave the stack display alone. The word ``Dirty''
13395 will appear in the mode line when Calc thinks the stack display may not
13396 reflect the latest mode settings.
13398 @kindex d @key{RET}
13399 @pindex calc-refresh-top
13400 The @kbd{d @key{RET}} (@code{calc-refresh-top}) command reformats the
13401 top stack entry according to all the current modes. Positive prefix
13402 arguments reformat the top @var{n} entries; negative prefix arguments
13403 reformat the specified entry, and a prefix of zero is equivalent to
13404 @kbd{d @key{SPC}} (@code{calc-refresh}), which reformats the entire stack.
13405 For example, @kbd{H d s M-2 d @key{RET}} changes to scientific notation
13406 but reformats only the top two stack entries in the new mode.
13408 The @kbd{I} prefix has another effect on the display modes. The mode
13409 is set only temporarily; the top stack entry is reformatted according
13410 to that mode, then the original mode setting is restored. In other
13411 words, @kbd{I d s} is equivalent to @kbd{H d s d @key{RET} H d (@var{old mode})}.
13415 * Grouping Digits::
13417 * Complex Formats::
13418 * Fraction Formats::
13421 * Truncating the Stack::
13426 @node Radix Modes, Grouping Digits, Display Modes, Display Modes
13427 @subsection Radix Modes
13430 @cindex Radix display
13431 @cindex Non-decimal numbers
13432 @cindex Decimal and non-decimal numbers
13433 Calc normally displays numbers in decimal (@dfn{base-10} or @dfn{radix-10})
13434 notation. Calc can actually display in any radix from two (binary) to 36.
13435 When the radix is above 10, the letters @code{A} to @code{Z} are used as
13436 digits. When entering such a number, letter keys are interpreted as
13437 potential digits rather than terminating numeric entry mode.
13443 @cindex Hexadecimal integers
13444 @cindex Octal integers
13445 The key sequences @kbd{d 2}, @kbd{d 8}, @kbd{d 6}, and @kbd{d 0} select
13446 binary, octal, hexadecimal, and decimal as the current display radix,
13447 respectively. Numbers can always be entered in any radix, though the
13448 current radix is used as a default if you press @kbd{#} without any initial
13449 digits. A number entered without a @kbd{#} is @emph{always} interpreted
13454 To set the radix generally, use @kbd{d r} (@code{calc-radix}) and enter
13455 an integer from 2 to 36. You can specify the radix as a numeric prefix
13456 argument; otherwise you will be prompted for it.
13459 @pindex calc-leading-zeros
13460 @cindex Leading zeros
13461 Integers normally are displayed with however many digits are necessary to
13462 represent the integer and no more. The @kbd{d z} (@code{calc-leading-zeros})
13463 command causes integers to be padded out with leading zeros according to the
13464 current binary word size. (@xref{Binary Functions}, for a discussion of
13465 word size.) If the absolute value of the word size is @expr{w}, all integers
13466 are displayed with at least enough digits to represent
13467 @texline @math{2^w-1}
13468 @infoline @expr{(2^w)-1}
13469 in the current radix. (Larger integers will still be displayed in their
13472 @node Grouping Digits, Float Formats, Radix Modes, Display Modes
13473 @subsection Grouping Digits
13477 @pindex calc-group-digits
13478 @cindex Grouping digits
13479 @cindex Digit grouping
13480 Long numbers can be hard to read if they have too many digits. For
13481 example, the factorial of 30 is 33 digits long! Press @kbd{d g}
13482 (@code{calc-group-digits}) to enable @dfn{Grouping} mode, in which digits
13483 are displayed in clumps of 3 or 4 (depending on the current radix)
13484 separated by commas.
13486 The @kbd{d g} command toggles grouping on and off.
13487 With a numeric prefix of 0, this command displays the current state of
13488 the grouping flag; with an argument of minus one it disables grouping;
13489 with a positive argument @expr{N} it enables grouping on every @expr{N}
13490 digits. For floating-point numbers, grouping normally occurs only
13491 before the decimal point. A negative prefix argument @expr{-N} enables
13492 grouping every @expr{N} digits both before and after the decimal point.
13495 @pindex calc-group-char
13496 The @kbd{d ,} (@code{calc-group-char}) command allows you to choose any
13497 character as the grouping separator. The default is the comma character.
13498 If you find it difficult to read vectors of large integers grouped with
13499 commas, you may wish to use spaces or some other character instead.
13500 This command takes the next character you type, whatever it is, and
13501 uses it as the digit separator. As a special case, @kbd{d , \} selects
13502 @samp{\,} (@TeX{}'s thin-space symbol) as the digit separator.
13504 Please note that grouped numbers will not generally be parsed correctly
13505 if re-read in textual form, say by the use of @kbd{M-# y} and @kbd{M-# g}.
13506 (@xref{Kill and Yank}, for details on these commands.) One exception is
13507 the @samp{\,} separator, which doesn't interfere with parsing because it
13508 is ignored by @TeX{} language mode.
13510 @node Float Formats, Complex Formats, Grouping Digits, Display Modes
13511 @subsection Float Formats
13514 Floating-point quantities are normally displayed in standard decimal
13515 form, with scientific notation used if the exponent is especially high
13516 or low. All significant digits are normally displayed. The commands
13517 in this section allow you to choose among several alternative display
13518 formats for floats.
13521 @pindex calc-normal-notation
13522 The @kbd{d n} (@code{calc-normal-notation}) command selects the normal
13523 display format. All significant figures in a number are displayed.
13524 With a positive numeric prefix, numbers are rounded if necessary to
13525 that number of significant digits. With a negative numerix prefix,
13526 the specified number of significant digits less than the current
13527 precision is used. (Thus @kbd{C-u -2 d n} displays 10 digits if the
13528 current precision is 12.)
13531 @pindex calc-fix-notation
13532 The @kbd{d f} (@code{calc-fix-notation}) command selects fixed-point
13533 notation. The numeric argument is the number of digits after the
13534 decimal point, zero or more. This format will relax into scientific
13535 notation if a nonzero number would otherwise have been rounded all the
13536 way to zero. Specifying a negative number of digits is the same as
13537 for a positive number, except that small nonzero numbers will be rounded
13538 to zero rather than switching to scientific notation.
13541 @pindex calc-sci-notation
13542 @cindex Scientific notation, display of
13543 The @kbd{d s} (@code{calc-sci-notation}) command selects scientific
13544 notation. A positive argument sets the number of significant figures
13545 displayed, of which one will be before and the rest after the decimal
13546 point. A negative argument works the same as for @kbd{d n} format.
13547 The default is to display all significant digits.
13550 @pindex calc-eng-notation
13551 @cindex Engineering notation, display of
13552 The @kbd{d e} (@code{calc-eng-notation}) command selects engineering
13553 notation. This is similar to scientific notation except that the
13554 exponent is rounded down to a multiple of three, with from one to three
13555 digits before the decimal point. An optional numeric prefix sets the
13556 number of significant digits to display, as for @kbd{d s}.
13558 It is important to distinguish between the current @emph{precision} and
13559 the current @emph{display format}. After the commands @kbd{C-u 10 p}
13560 and @kbd{C-u 6 d n} the Calculator computes all results to ten
13561 significant figures but displays only six. (In fact, intermediate
13562 calculations are often carried to one or two more significant figures,
13563 but values placed on the stack will be rounded down to ten figures.)
13564 Numbers are never actually rounded to the display precision for storage,
13565 except by commands like @kbd{C-k} and @kbd{M-# y} which operate on the
13566 actual displayed text in the Calculator buffer.
13569 @pindex calc-point-char
13570 The @kbd{d .} (@code{calc-point-char}) command selects the character used
13571 as a decimal point. Normally this is a period; users in some countries
13572 may wish to change this to a comma. Note that this is only a display
13573 style; on entry, periods must always be used to denote floating-point
13574 numbers, and commas to separate elements in a list.
13576 @node Complex Formats, Fraction Formats, Float Formats, Display Modes
13577 @subsection Complex Formats
13581 @pindex calc-complex-notation
13582 There are three supported notations for complex numbers in rectangular
13583 form. The default is as a pair of real numbers enclosed in parentheses
13584 and separated by a comma: @samp{(a,b)}. The @kbd{d c}
13585 (@code{calc-complex-notation}) command selects this style.
13588 @pindex calc-i-notation
13590 @pindex calc-j-notation
13591 The other notations are @kbd{d i} (@code{calc-i-notation}), in which
13592 numbers are displayed in @samp{a+bi} form, and @kbd{d j}
13593 (@code{calc-j-notation}) which displays the form @samp{a+bj} preferred
13594 in some disciplines.
13596 @cindex @code{i} variable
13598 Complex numbers are normally entered in @samp{(a,b)} format.
13599 If you enter @samp{2+3i} as an algebraic formula, it will be stored as
13600 the formula @samp{2 + 3 * i}. However, if you use @kbd{=} to evaluate
13601 this formula and you have not changed the variable @samp{i}, the @samp{i}
13602 will be interpreted as @samp{(0,1)} and the formula will be simplified
13603 to @samp{(2,3)}. Other commands (like @code{calc-sin}) will @emph{not}
13604 interpret the formula @samp{2 + 3 * i} as a complex number.
13605 @xref{Variables}, under ``special constants.''
13607 @node Fraction Formats, HMS Formats, Complex Formats, Display Modes
13608 @subsection Fraction Formats
13612 @pindex calc-over-notation
13613 Display of fractional numbers is controlled by the @kbd{d o}
13614 (@code{calc-over-notation}) command. By default, a number like
13615 eight thirds is displayed in the form @samp{8:3}. The @kbd{d o} command
13616 prompts for a one- or two-character format. If you give one character,
13617 that character is used as the fraction separator. Common separators are
13618 @samp{:} and @samp{/}. (During input of numbers, the @kbd{:} key must be
13619 used regardless of the display format; in particular, the @kbd{/} is used
13620 for RPN-style division, @emph{not} for entering fractions.)
13622 If you give two characters, fractions use ``integer-plus-fractional-part''
13623 notation. For example, the format @samp{+/} would display eight thirds
13624 as @samp{2+2/3}. If two colons are present in a number being entered,
13625 the number is interpreted in this form (so that the entries @kbd{2:2:3}
13626 and @kbd{8:3} are equivalent).
13628 It is also possible to follow the one- or two-character format with
13629 a number. For example: @samp{:10} or @samp{+/3}. In this case,
13630 Calc adjusts all fractions that are displayed to have the specified
13631 denominator, if possible. Otherwise it adjusts the denominator to
13632 be a multiple of the specified value. For example, in @samp{:6} mode
13633 the fraction @expr{1:6} will be unaffected, but @expr{2:3} will be
13634 displayed as @expr{4:6}, @expr{1:2} will be displayed as @expr{3:6},
13635 and @expr{1:8} will be displayed as @expr{3:24}. Integers are also
13636 affected by this mode: 3 is displayed as @expr{18:6}. Note that the
13637 format @samp{:1} writes fractions the same as @samp{:}, but it writes
13638 integers as @expr{n:1}.
13640 The fraction format does not affect the way fractions or integers are
13641 stored, only the way they appear on the screen. The fraction format
13642 never affects floats.
13644 @node HMS Formats, Date Formats, Fraction Formats, Display Modes
13645 @subsection HMS Formats
13649 @pindex calc-hms-notation
13650 The @kbd{d h} (@code{calc-hms-notation}) command controls the display of
13651 HMS (hours-minutes-seconds) forms. It prompts for a string which
13652 consists basically of an ``hours'' marker, optional punctuation, a
13653 ``minutes'' marker, more optional punctuation, and a ``seconds'' marker.
13654 Punctuation is zero or more spaces, commas, or semicolons. The hours
13655 marker is one or more non-punctuation characters. The minutes and
13656 seconds markers must be single non-punctuation characters.
13658 The default HMS format is @samp{@@ ' "}, producing HMS values of the form
13659 @samp{23@@ 30' 15.75"}. The format @samp{deg, ms} would display this same
13660 value as @samp{23deg, 30m15.75s}. During numeric entry, the @kbd{h} or @kbd{o}
13661 keys are recognized as synonyms for @kbd{@@} regardless of display format.
13662 The @kbd{m} and @kbd{s} keys are recognized as synonyms for @kbd{'} and
13663 @kbd{"}, respectively, but only if an @kbd{@@} (or @kbd{h} or @kbd{o}) has
13664 already been typed; otherwise, they have their usual meanings
13665 (@kbd{m-} prefix and @kbd{s-} prefix). Thus, @kbd{5 "}, @kbd{0 @@ 5 "}, and
13666 @kbd{0 h 5 s} are some of the ways to enter the quantity ``five seconds.''
13667 The @kbd{'} key is recognized as ``minutes'' only if @kbd{@@} (or @kbd{h} or
13668 @kbd{o}) has already been pressed; otherwise it means to switch to algebraic
13671 @node Date Formats, Truncating the Stack, HMS Formats, Display Modes
13672 @subsection Date Formats
13676 @pindex calc-date-notation
13677 The @kbd{d d} (@code{calc-date-notation}) command controls the display
13678 of date forms (@pxref{Date Forms}). It prompts for a string which
13679 contains letters that represent the various parts of a date and time.
13680 To show which parts should be omitted when the form represents a pure
13681 date with no time, parts of the string can be enclosed in @samp{< >}
13682 marks. If you don't include @samp{< >} markers in the format, Calc
13683 guesses at which parts, if any, should be omitted when formatting
13686 The default format is: @samp{<H:mm:SSpp >Www Mmm D, YYYY}.
13687 An example string in this format is @samp{3:32pm Wed Jan 9, 1991}.
13688 If you enter a blank format string, this default format is
13691 Calc uses @samp{< >} notation for nameless functions as well as for
13692 dates. @xref{Specifying Operators}. To avoid confusion with nameless
13693 functions, your date formats should avoid using the @samp{#} character.
13696 * Date Formatting Codes::
13697 * Free-Form Dates::
13698 * Standard Date Formats::
13701 @node Date Formatting Codes, Free-Form Dates, Date Formats, Date Formats
13702 @subsubsection Date Formatting Codes
13705 When displaying a date, the current date format is used. All
13706 characters except for letters and @samp{<} and @samp{>} are
13707 copied literally when dates are formatted. The portion between
13708 @samp{< >} markers is omitted for pure dates, or included for
13709 date/time forms. Letters are interpreted according to the table
13712 When dates are read in during algebraic entry, Calc first tries to
13713 match the input string to the current format either with or without
13714 the time part. The punctuation characters (including spaces) must
13715 match exactly; letter fields must correspond to suitable text in
13716 the input. If this doesn't work, Calc checks if the input is a
13717 simple number; if so, the number is interpreted as a number of days
13718 since Jan 1, 1 AD. Otherwise, Calc tries a much more relaxed and
13719 flexible algorithm which is described in the next section.
13721 Weekday names are ignored during reading.
13723 Two-digit year numbers are interpreted as lying in the range
13724 from 1941 to 2039. Years outside that range are always
13725 entered and displayed in full. Year numbers with a leading
13726 @samp{+} sign are always interpreted exactly, allowing the
13727 entry and display of the years 1 through 99 AD.
13729 Here is a complete list of the formatting codes for dates:
13733 Year: ``91'' for 1991, ``7'' for 2007, ``+23'' for 23 AD.
13735 Year: ``91'' for 1991, ``07'' for 2007, ``+23'' for 23 AD.
13737 Year: ``91'' for 1991, `` 7'' for 2007, ``+23'' for 23 AD.
13739 Year: ``1991'' for 1991, ``23'' for 23 AD.
13741 Year: ``1991'' for 1991, ``+23'' for 23 AD.
13743 Year: ``ad'' or blank.
13745 Year: ``AD'' or blank.
13747 Year: ``ad '' or blank. (Note trailing space.)
13749 Year: ``AD '' or blank.
13751 Year: ``a.d.'' or blank.
13753 Year: ``A.D.'' or blank.
13755 Year: ``bc'' or blank.
13757 Year: ``BC'' or blank.
13759 Year: `` bc'' or blank. (Note leading space.)
13761 Year: `` BC'' or blank.
13763 Year: ``b.c.'' or blank.
13765 Year: ``B.C.'' or blank.
13767 Month: ``8'' for August.
13769 Month: ``08'' for August.
13771 Month: `` 8'' for August.
13773 Month: ``AUG'' for August.
13775 Month: ``Aug'' for August.
13777 Month: ``aug'' for August.
13779 Month: ``AUGUST'' for August.
13781 Month: ``August'' for August.
13783 Day: ``7'' for 7th day of month.
13785 Day: ``07'' for 7th day of month.
13787 Day: `` 7'' for 7th day of month.
13789 Weekday: ``0'' for Sunday, ``6'' for Saturday.
13791 Weekday: ``SUN'' for Sunday.
13793 Weekday: ``Sun'' for Sunday.
13795 Weekday: ``sun'' for Sunday.
13797 Weekday: ``SUNDAY'' for Sunday.
13799 Weekday: ``Sunday'' for Sunday.
13801 Day of year: ``34'' for Feb. 3.
13803 Day of year: ``034'' for Feb. 3.
13805 Day of year: `` 34'' for Feb. 3.
13807 Hour: ``5'' for 5 AM; ``17'' for 5 PM.
13809 Hour: ``05'' for 5 AM; ``17'' for 5 PM.
13811 Hour: `` 5'' for 5 AM; ``17'' for 5 PM.
13813 Hour: ``5'' for 5 AM and 5 PM.
13815 Hour: ``05'' for 5 AM and 5 PM.
13817 Hour: `` 5'' for 5 AM and 5 PM.
13819 AM/PM: ``a'' or ``p''.
13821 AM/PM: ``A'' or ``P''.
13823 AM/PM: ``am'' or ``pm''.
13825 AM/PM: ``AM'' or ``PM''.
13827 AM/PM: ``a.m.'' or ``p.m.''.
13829 AM/PM: ``A.M.'' or ``P.M.''.
13831 Minutes: ``7'' for 7.
13833 Minutes: ``07'' for 7.
13835 Minutes: `` 7'' for 7.
13837 Seconds: ``7'' for 7; ``7.23'' for 7.23.
13839 Seconds: ``07'' for 7; ``07.23'' for 7.23.
13841 Seconds: `` 7'' for 7; `` 7.23'' for 7.23.
13843 Optional seconds: ``07'' for 7; blank for 0.
13845 Optional seconds: `` 7'' for 7; blank for 0.
13847 Numeric date/time: ``726842.25'' for 6:00am Wed Jan 9, 1991.
13849 Numeric date: ``726842'' for any time on Wed Jan 9, 1991.
13851 Julian date/time: ``2448265.75'' for 6:00am Wed Jan 9, 1991.
13853 Julian date: ``2448266'' for any time on Wed Jan 9, 1991.
13855 Unix time: ``663400800'' for 6:00am Wed Jan 9, 1991.
13857 Brackets suppression. An ``X'' at the front of the format
13858 causes the surrounding @w{@samp{< >}} delimiters to be omitted
13859 when formatting dates. Note that the brackets are still
13860 required for algebraic entry.
13863 If ``SS'' or ``BS'' (optional seconds) is preceded by a colon, the
13864 colon is also omitted if the seconds part is zero.
13866 If ``bb,'' ``bbb'' or ``bbbb'' or their upper-case equivalents
13867 appear in the format, then negative year numbers are displayed
13868 without a minus sign. Note that ``aa'' and ``bb'' are mutually
13869 exclusive. Some typical usages would be @samp{YYYY AABB};
13870 @samp{AAAYYYYBBB}; @samp{YYYYBBB}.
13872 The formats ``YY,'' ``YYYY,'' ``MM,'' ``DD,'' ``ddd,'' ``hh,'' ``HH,''
13873 ``mm,'' ``ss,'' and ``SS'' actually match any number of digits during
13874 reading unless several of these codes are strung together with no
13875 punctuation in between, in which case the input must have exactly as
13876 many digits as there are letters in the format.
13878 The ``j,'' ``J,'' and ``U'' formats do not make any time zone
13879 adjustment. They effectively use @samp{julian(x,0)} and
13880 @samp{unixtime(x,0)} to make the conversion; @pxref{Date Arithmetic}.
13882 @node Free-Form Dates, Standard Date Formats, Date Formatting Codes, Date Formats
13883 @subsubsection Free-Form Dates
13886 When reading a date form during algebraic entry, Calc falls back
13887 on the algorithm described here if the input does not exactly
13888 match the current date format. This algorithm generally
13889 ``does the right thing'' and you don't have to worry about it,
13890 but it is described here in full detail for the curious.
13892 Calc does not distinguish between upper- and lower-case letters
13893 while interpreting dates.
13895 First, the time portion, if present, is located somewhere in the
13896 text and then removed. The remaining text is then interpreted as
13899 A time is of the form @samp{hh:mm:ss}, possibly with the seconds
13900 part omitted and possibly with an AM/PM indicator added to indicate
13901 12-hour time. If the AM/PM is present, the minutes may also be
13902 omitted. The AM/PM part may be any of the words @samp{am},
13903 @samp{pm}, @samp{noon}, or @samp{midnight}; each of these may be
13904 abbreviated to one letter, and the alternate forms @samp{a.m.},
13905 @samp{p.m.}, and @samp{mid} are also understood. Obviously
13906 @samp{noon} and @samp{midnight} are allowed only on 12:00:00.
13907 The words @samp{noon}, @samp{mid}, and @samp{midnight} are also
13908 recognized with no number attached.
13910 If there is no AM/PM indicator, the time is interpreted in 24-hour
13913 To read the date portion, all words and numbers are isolated
13914 from the string; other characters are ignored. All words must
13915 be either month names or day-of-week names (the latter of which
13916 are ignored). Names can be written in full or as three-letter
13919 Large numbers, or numbers with @samp{+} or @samp{-} signs,
13920 are interpreted as years. If one of the other numbers is
13921 greater than 12, then that must be the day and the remaining
13922 number in the input is therefore the month. Otherwise, Calc
13923 assumes the month, day and year are in the same order that they
13924 appear in the current date format. If the year is omitted, the
13925 current year is taken from the system clock.
13927 If there are too many or too few numbers, or any unrecognizable
13928 words, then the input is rejected.
13930 If there are any large numbers (of five digits or more) other than
13931 the year, they are ignored on the assumption that they are something
13932 like Julian dates that were included along with the traditional
13933 date components when the date was formatted.
13935 One of the words @samp{ad}, @samp{a.d.}, @samp{bc}, or @samp{b.c.}
13936 may optionally be used; the latter two are equivalent to a
13937 minus sign on the year value.
13939 If you always enter a four-digit year, and use a name instead
13940 of a number for the month, there is no danger of ambiguity.
13942 @node Standard Date Formats, , Free-Form Dates, Date Formats
13943 @subsubsection Standard Date Formats
13946 There are actually ten standard date formats, numbered 0 through 9.
13947 Entering a blank line at the @kbd{d d} command's prompt gives
13948 you format number 1, Calc's usual format. You can enter any digit
13949 to select the other formats.
13951 To create your own standard date formats, give a numeric prefix
13952 argument from 0 to 9 to the @w{@kbd{d d}} command. The format you
13953 enter will be recorded as the new standard format of that
13954 number, as well as becoming the new current date format.
13955 You can save your formats permanently with the @w{@kbd{m m}}
13956 command (@pxref{Mode Settings}).
13960 @samp{N} (Numerical format)
13962 @samp{<H:mm:SSpp >Www Mmm D, YYYY} (American format)
13964 @samp{D Mmm YYYY<, h:mm:SS>} (European format)
13966 @samp{Www Mmm BD< hh:mm:ss> YYYY} (Unix written date format)
13968 @samp{M/D/Y< H:mm:SSpp>} (American slashed format)
13970 @samp{D.M.Y< h:mm:SS>} (European dotted format)
13972 @samp{M-D-Y< H:mm:SSpp>} (American dashed format)
13974 @samp{D-M-Y< h:mm:SS>} (European dashed format)
13976 @samp{j<, h:mm:ss>} (Julian day plus time)
13978 @samp{YYddd< hh:mm:ss>} (Year-day format)
13981 @node Truncating the Stack, Justification, Date Formats, Display Modes
13982 @subsection Truncating the Stack
13986 @pindex calc-truncate-stack
13987 @cindex Truncating the stack
13988 @cindex Narrowing the stack
13989 The @kbd{d t} (@code{calc-truncate-stack}) command moves the @samp{.}@:
13990 line that marks the top-of-stack up or down in the Calculator buffer.
13991 The number right above that line is considered to the be at the top of
13992 the stack. Any numbers below that line are ``hidden'' from all stack
13993 operations (although still visible to the user). This is similar to the
13994 Emacs ``narrowing'' feature, except that the values below the @samp{.}
13995 are @emph{visible}, just temporarily frozen. This feature allows you to
13996 keep several independent calculations running at once in different parts
13997 of the stack, or to apply a certain command to an element buried deep in
14000 Pressing @kbd{d t} by itself moves the @samp{.} to the line the cursor
14001 is on. Thus, this line and all those below it become hidden. To un-hide
14002 these lines, move down to the end of the buffer and press @w{@kbd{d t}}.
14003 With a positive numeric prefix argument @expr{n}, @kbd{d t} hides the
14004 bottom @expr{n} values in the buffer. With a negative argument, it hides
14005 all but the top @expr{n} values. With an argument of zero, it hides zero
14006 values, i.e., moves the @samp{.} all the way down to the bottom.
14009 @pindex calc-truncate-up
14011 @pindex calc-truncate-down
14012 The @kbd{d [} (@code{calc-truncate-up}) and @kbd{d ]}
14013 (@code{calc-truncate-down}) commands move the @samp{.} up or down one
14014 line at a time (or several lines with a prefix argument).
14016 @node Justification, Labels, Truncating the Stack, Display Modes
14017 @subsection Justification
14021 @pindex calc-left-justify
14023 @pindex calc-center-justify
14025 @pindex calc-right-justify
14026 Values on the stack are normally left-justified in the window. You can
14027 control this arrangement by typing @kbd{d <} (@code{calc-left-justify}),
14028 @kbd{d >} (@code{calc-right-justify}), or @kbd{d =}
14029 (@code{calc-center-justify}). For example, in Right-Justification mode,
14030 stack entries are displayed flush-right against the right edge of the
14033 If you change the width of the Calculator window you may have to type
14034 @kbd{d @key{SPC}} (@code{calc-refresh}) to re-align right-justified or centered
14037 Right-justification is especially useful together with fixed-point
14038 notation (see @code{d f}; @code{calc-fix-notation}). With these modes
14039 together, the decimal points on numbers will always line up.
14041 With a numeric prefix argument, the justification commands give you
14042 a little extra control over the display. The argument specifies the
14043 horizontal ``origin'' of a display line. It is also possible to
14044 specify a maximum line width using the @kbd{d b} command (@pxref{Normal
14045 Language Modes}). For reference, the precise rules for formatting and
14046 breaking lines are given below. Notice that the interaction between
14047 origin and line width is slightly different in each justification
14050 In Left-Justified mode, the line is indented by a number of spaces
14051 given by the origin (default zero). If the result is longer than the
14052 maximum line width, if given, or too wide to fit in the Calc window
14053 otherwise, then it is broken into lines which will fit; each broken
14054 line is indented to the origin.
14056 In Right-Justified mode, lines are shifted right so that the rightmost
14057 character is just before the origin, or just before the current
14058 window width if no origin was specified. If the line is too long
14059 for this, then it is broken; the current line width is used, if
14060 specified, or else the origin is used as a width if that is
14061 specified, or else the line is broken to fit in the window.
14063 In Centering mode, the origin is the column number of the center of
14064 each stack entry. If a line width is specified, lines will not be
14065 allowed to go past that width; Calc will either indent less or
14066 break the lines if necessary. If no origin is specified, half the
14067 line width or Calc window width is used.
14069 Note that, in each case, if line numbering is enabled the display
14070 is indented an additional four spaces to make room for the line
14071 number. The width of the line number is taken into account when
14072 positioning according to the current Calc window width, but not
14073 when positioning by explicit origins and widths. In the latter
14074 case, the display is formatted as specified, and then uniformly
14075 shifted over four spaces to fit the line numbers.
14077 @node Labels, , Justification, Display Modes
14082 @pindex calc-left-label
14083 The @kbd{d @{} (@code{calc-left-label}) command prompts for a string,
14084 then displays that string to the left of every stack entry. If the
14085 entries are left-justified (@pxref{Justification}), then they will
14086 appear immediately after the label (unless you specified an origin
14087 greater than the length of the label). If the entries are centered
14088 or right-justified, the label appears on the far left and does not
14089 affect the horizontal position of the stack entry.
14091 Give a blank string (with @kbd{d @{ @key{RET}}) to turn the label off.
14094 @pindex calc-right-label
14095 The @kbd{d @}} (@code{calc-right-label}) command similarly adds a
14096 label on the righthand side. It does not affect positioning of
14097 the stack entries unless they are right-justified. Also, if both
14098 a line width and an origin are given in Right-Justified mode, the
14099 stack entry is justified to the origin and the righthand label is
14100 justified to the line width.
14102 One application of labels would be to add equation numbers to
14103 formulas you are manipulating in Calc and then copying into a
14104 document (possibly using Embedded mode). The equations would
14105 typically be centered, and the equation numbers would be on the
14106 left or right as you prefer.
14108 @node Language Modes, Modes Variable, Display Modes, Mode Settings
14109 @section Language Modes
14112 The commands in this section change Calc to use a different notation for
14113 entry and display of formulas, corresponding to the conventions of some
14114 other common language such as Pascal or La@TeX{}. Objects displayed on the
14115 stack or yanked from the Calculator to an editing buffer will be formatted
14116 in the current language; objects entered in algebraic entry or yanked from
14117 another buffer will be interpreted according to the current language.
14119 The current language has no effect on things written to or read from the
14120 trail buffer, nor does it affect numeric entry. Only algebraic entry is
14121 affected. You can make even algebraic entry ignore the current language
14122 and use the standard notation by giving a numeric prefix, e.g., @kbd{C-u '}.
14124 For example, suppose the formula @samp{2*a[1] + atan(a[2])} occurs in a C
14125 program; elsewhere in the program you need the derivatives of this formula
14126 with respect to @samp{a[1]} and @samp{a[2]}. First, type @kbd{d C}
14127 to switch to C notation. Now use @code{C-u M-# g} to grab the formula
14128 into the Calculator, @kbd{a d a[1] @key{RET}} to differentiate with respect
14129 to the first variable, and @kbd{M-# y} to yank the formula for the derivative
14130 back into your C program. Press @kbd{U} to undo the differentiation and
14131 repeat with @kbd{a d a[2] @key{RET}} for the other derivative.
14133 Without being switched into C mode first, Calc would have misinterpreted
14134 the brackets in @samp{a[1]} and @samp{a[2]}, would not have known that
14135 @code{atan} was equivalent to Calc's built-in @code{arctan} function,
14136 and would have written the formula back with notations (like implicit
14137 multiplication) which would not have been valid for a C program.
14139 As another example, suppose you are maintaining a C program and a La@TeX{}
14140 document, each of which needs a copy of the same formula. You can grab the
14141 formula from the program in C mode, switch to La@TeX{} mode, and yank the
14142 formula into the document in La@TeX{} math-mode format.
14144 Language modes are selected by typing the letter @kbd{d} followed by a
14145 shifted letter key.
14148 * Normal Language Modes::
14149 * C FORTRAN Pascal::
14150 * TeX and LaTeX Language Modes::
14151 * Eqn Language Mode::
14152 * Mathematica Language Mode::
14153 * Maple Language Mode::
14158 @node Normal Language Modes, C FORTRAN Pascal, Language Modes, Language Modes
14159 @subsection Normal Language Modes
14163 @pindex calc-normal-language
14164 The @kbd{d N} (@code{calc-normal-language}) command selects the usual
14165 notation for Calc formulas, as described in the rest of this manual.
14166 Matrices are displayed in a multi-line tabular format, but all other
14167 objects are written in linear form, as they would be typed from the
14171 @pindex calc-flat-language
14172 @cindex Matrix display
14173 The @kbd{d O} (@code{calc-flat-language}) command selects a language
14174 identical with the normal one, except that matrices are written in
14175 one-line form along with everything else. In some applications this
14176 form may be more suitable for yanking data into other buffers.
14179 @pindex calc-line-breaking
14180 @cindex Line breaking
14181 @cindex Breaking up long lines
14182 Even in one-line mode, long formulas or vectors will still be split
14183 across multiple lines if they exceed the width of the Calculator window.
14184 The @kbd{d b} (@code{calc-line-breaking}) command turns this line-breaking
14185 feature on and off. (It works independently of the current language.)
14186 If you give a numeric prefix argument of five or greater to the @kbd{d b}
14187 command, that argument will specify the line width used when breaking
14191 @pindex calc-big-language
14192 The @kbd{d B} (@code{calc-big-language}) command selects a language
14193 which uses textual approximations to various mathematical notations,
14194 such as powers, quotients, and square roots:
14204 in place of @samp{sqrt((a+1)/b + c^2)}.
14206 Subscripts like @samp{a_i} are displayed as actual subscripts in Big
14207 mode. Double subscripts, @samp{a_i_j} (@samp{subscr(subscr(a, i), j)})
14208 are displayed as @samp{a} with subscripts separated by commas:
14209 @samp{i, j}. They must still be entered in the usual underscore
14212 One slight ambiguity of Big notation is that
14221 can represent either the negative rational number @expr{-3:4}, or the
14222 actual expression @samp{-(3/4)}; but the latter formula would normally
14223 never be displayed because it would immediately be evaluated to
14224 @expr{-3:4} or @expr{-0.75}, so this ambiguity is not a problem in
14227 Non-decimal numbers are displayed with subscripts. Thus there is no
14228 way to tell the difference between @samp{16#C2} and @samp{C2_16},
14229 though generally you will know which interpretation is correct.
14230 Logarithms @samp{log(x,b)} and @samp{log10(x)} also use subscripts
14233 In Big mode, stack entries often take up several lines. To aid
14234 readability, stack entries are separated by a blank line in this mode.
14235 You may find it useful to expand the Calc window's height using
14236 @kbd{C-x ^} (@code{enlarge-window}) or to make the Calc window the only
14237 one on the screen with @kbd{C-x 1} (@code{delete-other-windows}).
14239 Long lines are currently not rearranged to fit the window width in
14240 Big mode, so you may need to use the @kbd{<} and @kbd{>} keys
14241 to scroll across a wide formula. For really big formulas, you may
14242 even need to use @kbd{@{} and @kbd{@}} to scroll up and down.
14245 @pindex calc-unformatted-language
14246 The @kbd{d U} (@code{calc-unformatted-language}) command altogether disables
14247 the use of operator notation in formulas. In this mode, the formula
14248 shown above would be displayed:
14251 sqrt(add(div(add(a, 1), b), pow(c, 2)))
14254 These four modes differ only in display format, not in the format
14255 expected for algebraic entry. The standard Calc operators work in
14256 all four modes, and unformatted notation works in any language mode
14257 (except that Mathematica mode expects square brackets instead of
14260 @node C FORTRAN Pascal, TeX and LaTeX Language Modes, Normal Language Modes, Language Modes
14261 @subsection C, FORTRAN, and Pascal Modes
14265 @pindex calc-c-language
14267 The @kbd{d C} (@code{calc-c-language}) command selects the conventions
14268 of the C language for display and entry of formulas. This differs from
14269 the normal language mode in a variety of (mostly minor) ways. In
14270 particular, C language operators and operator precedences are used in
14271 place of Calc's usual ones. For example, @samp{a^b} means @samp{xor(a,b)}
14272 in C mode; a value raised to a power is written as a function call,
14275 In C mode, vectors and matrices use curly braces instead of brackets.
14276 Octal and hexadecimal values are written with leading @samp{0} or @samp{0x}
14277 rather than using the @samp{#} symbol. Array subscripting is
14278 translated into @code{subscr} calls, so that @samp{a[i]} in C
14279 mode is the same as @samp{a_i} in Normal mode. Assignments
14280 turn into the @code{assign} function, which Calc normally displays
14281 using the @samp{:=} symbol.
14283 The variables @code{pi} and @code{e} would be displayed @samp{pi}
14284 and @samp{e} in Normal mode, but in C mode they are displayed as
14285 @samp{M_PI} and @samp{M_E}, corresponding to the names of constants
14286 typically provided in the @file{<math.h>} header. Functions whose
14287 names are different in C are translated automatically for entry and
14288 display purposes. For example, entering @samp{asin(x)} will push the
14289 formula @samp{arcsin(x)} onto the stack; this formula will be displayed
14290 as @samp{asin(x)} as long as C mode is in effect.
14293 @pindex calc-pascal-language
14294 @cindex Pascal language
14295 The @kbd{d P} (@code{calc-pascal-language}) command selects Pascal
14296 conventions. Like C mode, Pascal mode interprets array brackets and uses
14297 a different table of operators. Hexadecimal numbers are entered and
14298 displayed with a preceding dollar sign. (Thus the regular meaning of
14299 @kbd{$2} during algebraic entry does not work in Pascal mode, though
14300 @kbd{$} (and @kbd{$$}, etc.) not followed by digits works the same as
14301 always.) No special provisions are made for other non-decimal numbers,
14302 vectors, and so on, since there is no universally accepted standard way
14303 of handling these in Pascal.
14306 @pindex calc-fortran-language
14307 @cindex FORTRAN language
14308 The @kbd{d F} (@code{calc-fortran-language}) command selects FORTRAN
14309 conventions. Various function names are transformed into FORTRAN
14310 equivalents. Vectors are written as @samp{/1, 2, 3/}, and may be
14311 entered this way or using square brackets. Since FORTRAN uses round
14312 parentheses for both function calls and array subscripts, Calc displays
14313 both in the same way; @samp{a(i)} is interpreted as a function call
14314 upon reading, and subscripts must be entered as @samp{subscr(a, i)}.
14315 Also, if the variable @code{a} has been declared to have type
14316 @code{vector} or @code{matrix} then @samp{a(i)} will be parsed as a
14317 subscript. (@xref{Declarations}.) Usually it doesn't matter, though;
14318 if you enter the subscript expression @samp{a(i)} and Calc interprets
14319 it as a function call, you'll never know the difference unless you
14320 switch to another language mode or replace @code{a} with an actual
14321 vector (or unless @code{a} happens to be the name of a built-in
14324 Underscores are allowed in variable and function names in all of these
14325 language modes. The underscore here is equivalent to the @samp{#} in
14326 Normal mode, or to hyphens in the underlying Emacs Lisp variable names.
14328 FORTRAN and Pascal modes normally do not adjust the case of letters in
14329 formulas. Most built-in Calc names use lower-case letters. If you use a
14330 positive numeric prefix argument with @kbd{d P} or @kbd{d F}, these
14331 modes will use upper-case letters exclusively for display, and will
14332 convert to lower-case on input. With a negative prefix, these modes
14333 convert to lower-case for display and input.
14335 @node TeX and LaTeX Language Modes, Eqn Language Mode, C FORTRAN Pascal, Language Modes
14336 @subsection @TeX{} and La@TeX{} Language Modes
14340 @pindex calc-tex-language
14341 @cindex TeX language
14343 @pindex calc-latex-language
14344 @cindex LaTeX language
14345 The @kbd{d T} (@code{calc-tex-language}) command selects the conventions
14346 of ``math mode'' in Donald Knuth's @TeX{} typesetting language,
14347 and the @kbd{d L} (@code{calc-latex-language}) command selects the
14348 conventions of ``math mode'' in La@TeX{}, a typesetting language that
14349 uses @TeX{} as its formatting engine. Calc's La@TeX{} language mode can
14350 read any formula that the @TeX{} language mode can, although La@TeX{}
14351 mode may display it differently.
14353 Formulas are entered and displayed in the appropriate notation;
14354 @texline @math{\sin(a/b)}
14355 @infoline @expr{sin(a/b)}
14356 will appear as @samp{\sin\left( a \over b \right)} in @TeX{} mode and
14357 @samp{\sin\left(\frac@{a@}@{b@}\right)} in La@TeX{} mode.
14358 Math formulas are often enclosed by @samp{$ $} signs in @TeX{} and
14359 La@TeX{}; these should be omitted when interfacing with Calc. To Calc,
14360 the @samp{$} sign has the same meaning it always does in algebraic
14361 formulas (a reference to an existing entry on the stack).
14363 Complex numbers are displayed as in @samp{3 + 4i}. Fractions and
14364 quotients are written using @code{\over} in @TeX{} mode (as in
14365 @code{@{a \over b@}}) and @code{\frac} in La@TeX{} mode (as in
14366 @code{\frac@{a@}@{b@}}); binomial coefficients are written with
14367 @code{\choose} in @TeX{} mode (as in @code{@{a \choose b@}}) and
14368 @code{\binom} in La@TeX{} mode (as in @code{\binom@{a@}@{b@}}).
14369 Interval forms are written with @code{\ldots}, and error forms are
14370 written with @code{\pm}. Absolute values are written as in
14371 @samp{|x + 1|}, and the floor and ceiling functions are written with
14372 @code{\lfloor}, @code{\rfloor}, etc. The words @code{\left} and
14373 @code{\right} are ignored when reading formulas in @TeX{} and La@TeX{}
14374 modes. Both @code{inf} and @code{uinf} are written as @code{\infty};
14375 when read, @code{\infty} always translates to @code{inf}.
14377 Function calls are written the usual way, with the function name followed
14378 by the arguments in parentheses. However, functions for which @TeX{}
14379 and La@TeX{} have special names (like @code{\sin}) will use curly braces
14380 instead of parentheses for very simple arguments. During input, curly
14381 braces and parentheses work equally well for grouping, but when the
14382 document is formatted the curly braces will be invisible. Thus the
14384 @texline @math{\sin{2 x}}
14385 @infoline @expr{sin 2x}
14387 @texline @math{\sin(2 + x)}.
14388 @infoline @expr{sin(2 + x)}.
14390 Function and variable names not treated specially by @TeX{} and La@TeX{}
14391 are simply written out as-is, which will cause them to come out in
14392 italic letters in the printed document. If you invoke @kbd{d T} or
14393 @kbd{d L} with a positive numeric prefix argument, names of more than
14394 one character will instead be enclosed in a protective commands that
14395 will prevent them from being typeset in the math italics; they will be
14396 written @samp{\hbox@{@var{name}@}} in @TeX{} mode and
14397 @samp{\text@{@var{name}@}} in La@TeX{} mode. The
14398 @samp{\hbox@{ @}} and @samp{\text@{ @}} notations are ignored during
14399 reading. If you use a negative prefix argument, such function names are
14400 written @samp{\@var{name}}, and function names that begin with @code{\} during
14401 reading have the @code{\} removed. (Note that in this mode, long
14402 variable names are still written with @code{\hbox} or @code{\text}.
14403 However, you can always make an actual variable name like @code{\bar} in
14406 During reading, text of the form @samp{\matrix@{ ...@: @}} is replaced
14407 by @samp{[ ...@: ]}. The same also applies to @code{\pmatrix} and
14408 @code{\bmatrix}. In La@TeX{} mode this also applies to
14409 @samp{\begin@{matrix@} ... \end@{matrix@}},
14410 @samp{\begin@{bmatrix@} ... \end@{bmatrix@}},
14411 @samp{\begin@{pmatrix@} ... \end@{pmatrix@}}, as well as
14412 @samp{\begin@{smallmatrix@} ... \end@{smallmatrix@}}.
14413 The symbol @samp{&} is interpreted as a comma,
14414 and the symbols @samp{\cr} and @samp{\\} are interpreted as semicolons.
14415 During output, matrices are displayed in @samp{\matrix@{ a & b \\ c & d@}}
14416 format in @TeX{} mode and in
14417 @samp{\begin@{pmatrix@} a & b \\ c & d \end@{pmatrix@}} format in
14418 La@TeX{} mode; you may need to edit this afterwards to change to your
14419 preferred matrix form. If you invoke @kbd{d T} or @kbd{d L} with an
14420 argument of 2 or -2, then matrices will be displayed in two-dimensional
14431 This may be convenient for isolated matrices, but could lead to
14432 expressions being displayed like
14435 \begin@{pmatrix@} \times x
14442 While this wouldn't bother Calc, it is incorrect La@TeX{}.
14443 (Similarly for @TeX{}.)
14445 Accents like @code{\tilde} and @code{\bar} translate into function
14446 calls internally (@samp{tilde(x)}, @samp{bar(x)}). The @code{\underline}
14447 sequence is treated as an accent. The @code{\vec} accent corresponds
14448 to the function name @code{Vec}, because @code{vec} is the name of
14449 a built-in Calc function. The following table shows the accents
14450 in Calc, @TeX{}, La@TeX{} and @dfn{eqn} (described in the next section):
14454 @let@calcindexershow=@calcindexernoshow @c Suppress marginal notes
14455 @let@calcindexersh=@calcindexernoshow
14563 acute \acute \acute
14567 breve \breve \breve
14569 check \check \check
14575 dotdot \ddot \ddot dotdot
14578 grave \grave \grave
14583 tilde \tilde \tilde tilde
14585 under \underline \underline under
14590 The @samp{=>} (evaluates-to) operator appears as a @code{\to} symbol:
14591 @samp{@{@var{a} \to @var{b}@}}. @TeX{} defines @code{\to} as an
14592 alias for @code{\rightarrow}. However, if the @samp{=>} is the
14593 top-level expression being formatted, a slightly different notation
14594 is used: @samp{\evalto @var{a} \to @var{b}}. The @code{\evalto}
14595 word is ignored by Calc's input routines, and is undefined in @TeX{}.
14596 You will typically want to include one of the following definitions
14597 at the top of a @TeX{} file that uses @code{\evalto}:
14601 \def\evalto#1\to@{@}
14604 The first definition formats evaluates-to operators in the usual
14605 way. The second causes only the @var{b} part to appear in the
14606 printed document; the @var{a} part and the arrow are hidden.
14607 Another definition you may wish to use is @samp{\let\to=\Rightarrow}
14608 which causes @code{\to} to appear more like Calc's @samp{=>} symbol.
14609 @xref{Evaluates-To Operator}, for a discussion of @code{evalto}.
14611 The complete set of @TeX{} control sequences that are ignored during
14615 \hbox \mbox \text \left \right
14616 \, \> \: \; \! \quad \qquad \hfil \hfill
14617 \displaystyle \textstyle \dsize \tsize
14618 \scriptstyle \scriptscriptstyle \ssize \ssize
14619 \rm \bf \it \sl \roman \bold \italic \slanted
14620 \cal \mit \Cal \Bbb \frak \goth
14624 Note that, because these symbols are ignored, reading a @TeX{} or
14625 La@TeX{} formula into Calc and writing it back out may lose spacing and
14628 Also, the ``discretionary multiplication sign'' @samp{\*} is read
14629 the same as @samp{*}.
14632 The @TeX{} version of this manual includes some printed examples at the
14633 end of this section.
14636 Here are some examples of how various Calc formulas are formatted in @TeX{}:
14641 \sin\left( {a^2 \over b_i} \right)
14645 $$ \sin\left( a^2 \over b_i \right) $$
14651 [(3, 4), 3:4, 3 +/- 4, [3 .. inf)]
14652 [3 + 4i, @{3 \over 4@}, 3 \pm 4, [3 \ldots \infty)]
14657 $$ [3 + 4i, {3 \over 4}, 3 \pm 4, [ 3 \ldots \infty)] $$
14663 [abs(a), abs(a / b), floor(a), ceil(a / b)]
14664 [|a|, \left| a \over b \right|,
14665 \lfloor a \rfloor, \left\lceil a \over b \right\rceil]
14669 $$ [|a|, \left| a \over b \right|,
14670 \lfloor a \rfloor, \left\lceil a \over b \right\rceil] $$
14676 [sin(a), sin(2 a), sin(2 + a), sin(a / b)]
14677 [\sin@{a@}, \sin@{2 a@}, \sin(2 + a),
14678 \sin\left( @{a \over b@} \right)]
14683 $$ [\sin{a}, \sin{2 a}, \sin(2 + a), \sin\left( {a \over b} \right)] $$
14687 First with plain @kbd{d T}, then with @kbd{C-u d T}, then finally with
14688 @kbd{C-u - d T} (using the example definition
14689 @samp{\def\foo#1@{\tilde F(#1)@}}:
14693 [f(a), foo(bar), sin(pi)]
14694 [f(a), foo(bar), \sin{\pi}]
14695 [f(a), \hbox@{foo@}(\hbox@{bar@}), \sin@{\pi@}]
14696 [f(a), \foo@{\hbox@{bar@}@}, \sin@{\pi@}]
14700 $$ [f(a), foo(bar), \sin{\pi}] $$
14701 $$ [f(a), \hbox{foo}(\hbox{bar}), \sin{\pi}] $$
14702 $$ [f(a), \tilde F(\hbox{bar}), \sin{\pi}] $$
14706 First with @samp{\def\evalto@{@}}, then with @samp{\def\evalto#1\to@{@}}:
14711 \evalto 2 + 3 \to 5
14721 First with standard @code{\to}, then with @samp{\let\to\Rightarrow}:
14725 [2 + 3 => 5, a / 2 => (b + c) / 2]
14726 [@{2 + 3 \to 5@}, @{@{a \over 2@} \to @{b + c \over 2@}@}]
14731 $$ [{2 + 3 \to 5}, {{a \over 2} \to {b + c \over 2}}] $$
14732 {\let\to\Rightarrow
14733 $$ [{2 + 3 \to 5}, {{a \over 2} \to {b + c \over 2}}] $$}
14737 Matrices normally, then changing @code{\matrix} to @code{\pmatrix}:
14741 [ [ a / b, 0 ], [ 0, 2^(x + 1) ] ]
14742 \matrix@{ @{a \over b@} & 0 \\ 0 & 2^@{(x + 1)@} @}
14743 \pmatrix@{ @{a \over b@} & 0 \\ 0 & 2^@{(x + 1)@} @}
14748 $$ \matrix{ {a \over b} & 0 \cr 0 & 2^{(x + 1)} } $$
14749 $$ \pmatrix{ {a \over b} & 0 \cr 0 & 2^{(x + 1)} } $$
14754 @node Eqn Language Mode, Mathematica Language Mode, TeX and LaTeX Language Modes, Language Modes
14755 @subsection Eqn Language Mode
14759 @pindex calc-eqn-language
14760 @dfn{Eqn} is another popular formatter for math formulas. It is
14761 designed for use with the TROFF text formatter, and comes standard
14762 with many versions of Unix. The @kbd{d E} (@code{calc-eqn-language})
14763 command selects @dfn{eqn} notation.
14765 The @dfn{eqn} language's main idiosyncrasy is that whitespace plays
14766 a significant part in the parsing of the language. For example,
14767 @samp{sqrt x+1 + y} treats @samp{x+1} as the argument of the
14768 @code{sqrt} operator. @dfn{Eqn} also understands more conventional
14769 grouping using curly braces: @samp{sqrt@{x+1@} + y}. Braces are
14770 required only when the argument contains spaces.
14772 In Calc's @dfn{eqn} mode, however, curly braces are required to
14773 delimit arguments of operators like @code{sqrt}. The first of the
14774 above examples would treat only the @samp{x} as the argument of
14775 @code{sqrt}, and in fact @samp{sin x+1} would be interpreted as
14776 @samp{sin * x + 1}, because @code{sin} is not a special operator
14777 in the @dfn{eqn} language. If you always surround the argument
14778 with curly braces, Calc will never misunderstand.
14780 Calc also understands parentheses as grouping characters. Another
14781 peculiarity of @dfn{eqn}'s syntax makes it advisable to separate
14782 words with spaces from any surrounding characters that aren't curly
14783 braces, so Calc writes @samp{sin ( x + y )} in @dfn{eqn} mode.
14784 (The spaces around @code{sin} are important to make @dfn{eqn}
14785 recognize that @code{sin} should be typeset in a roman font, and
14786 the spaces around @code{x} and @code{y} are a good idea just in
14787 case the @dfn{eqn} document has defined special meanings for these
14790 Powers and subscripts are written with the @code{sub} and @code{sup}
14791 operators, respectively. Note that the caret symbol @samp{^} is
14792 treated the same as a space in @dfn{eqn} mode, as is the @samp{~}
14793 symbol (these are used to introduce spaces of various widths into
14794 the typeset output of @dfn{eqn}).
14796 As in La@TeX{} mode, Calc's formatter omits parentheses around the
14797 arguments of functions like @code{ln} and @code{sin} if they are
14798 ``simple-looking''; in this case Calc surrounds the argument with
14799 braces, separated by a @samp{~} from the function name: @samp{sin~@{x@}}.
14801 Font change codes (like @samp{roman @var{x}}) and positioning codes
14802 (like @samp{~} and @samp{down @var{n} @var{x}}) are ignored by the
14803 @dfn{eqn} reader. Also ignored are the words @code{left}, @code{right},
14804 @code{mark}, and @code{lineup}. Quotation marks in @dfn{eqn} mode input
14805 are treated the same as curly braces: @samp{sqrt "1+x"} is equivalent to
14806 @samp{sqrt @{1+x@}}; this is only an approximation to the true meaning
14807 of quotes in @dfn{eqn}, but it is good enough for most uses.
14809 Accent codes (@samp{@var{x} dot}) are handled by treating them as
14810 function calls (@samp{dot(@var{x})}) internally.
14811 @xref{TeX and LaTeX Language Modes}, for a table of these accent
14812 functions. The @code{prime} accent is treated specially if it occurs on
14813 a variable or function name: @samp{f prime prime @w{( x prime )}} is
14814 stored internally as @samp{f'@w{'}(x')}. For example, taking the
14815 derivative of @samp{f(2 x)} with @kbd{a d x} will produce @samp{2 f'(2
14816 x)}, which @dfn{eqn} mode will display as @samp{2 f prime ( 2 x )}.
14818 Assignments are written with the @samp{<-} (left-arrow) symbol,
14819 and @code{evalto} operators are written with @samp{->} or
14820 @samp{evalto ... ->} (@pxref{TeX and LaTeX Language Modes}, for a discussion
14821 of this). The regular Calc symbols @samp{:=} and @samp{=>} are also
14822 recognized for these operators during reading.
14824 Vectors in @dfn{eqn} mode use regular Calc square brackets, but
14825 matrices are formatted as @samp{matrix @{ ccol @{ a above b @} ... @}}.
14826 The words @code{lcol} and @code{rcol} are recognized as synonyms
14827 for @code{ccol} during input, and are generated instead of @code{ccol}
14828 if the matrix justification mode so specifies.
14830 @node Mathematica Language Mode, Maple Language Mode, Eqn Language Mode, Language Modes
14831 @subsection Mathematica Language Mode
14835 @pindex calc-mathematica-language
14836 @cindex Mathematica language
14837 The @kbd{d M} (@code{calc-mathematica-language}) command selects the
14838 conventions of Mathematica. Notable differences in Mathematica mode
14839 are that the names of built-in functions are capitalized, and function
14840 calls use square brackets instead of parentheses. Thus the Calc
14841 formula @samp{sin(2 x)} is entered and displayed @w{@samp{Sin[2 x]}} in
14844 Vectors and matrices use curly braces in Mathematica. Complex numbers
14845 are written @samp{3 + 4 I}. The standard special constants in Calc are
14846 written @code{Pi}, @code{E}, @code{I}, @code{GoldenRatio}, @code{EulerGamma},
14847 @code{Infinity}, @code{ComplexInfinity}, and @code{Indeterminate} in
14849 Non-decimal numbers are written, e.g., @samp{16^^7fff}. Floating-point
14850 numbers in scientific notation are written @samp{1.23*10.^3}.
14851 Subscripts use double square brackets: @samp{a[[i]]}.
14853 @node Maple Language Mode, Compositions, Mathematica Language Mode, Language Modes
14854 @subsection Maple Language Mode
14858 @pindex calc-maple-language
14859 @cindex Maple language
14860 The @kbd{d W} (@code{calc-maple-language}) command selects the
14861 conventions of Maple.
14863 Maple's language is much like C. Underscores are allowed in symbol
14864 names; square brackets are used for subscripts; explicit @samp{*}s for
14865 multiplications are required. Use either @samp{^} or @samp{**} to
14868 Maple uses square brackets for lists and curly braces for sets. Calc
14869 interprets both notations as vectors, and displays vectors with square
14870 brackets. This means Maple sets will be converted to lists when they
14871 pass through Calc. As a special case, matrices are written as calls
14872 to the function @code{matrix}, given a list of lists as the argument,
14873 and can be read in this form or with all-capitals @code{MATRIX}.
14875 The Maple interval notation @samp{2 .. 3} has no surrounding brackets;
14876 Calc reads @samp{2 .. 3} as the closed interval @samp{[2 .. 3]}, and
14877 writes any kind of interval as @samp{2 .. 3}. This means you cannot
14878 see the difference between an open and a closed interval while in
14879 Maple display mode.
14881 Maple writes complex numbers as @samp{3 + 4*I}. Its special constants
14882 are @code{Pi}, @code{E}, @code{I}, and @code{infinity} (all three of
14883 @code{inf}, @code{uinf}, and @code{nan} display as @code{infinity}).
14884 Floating-point numbers are written @samp{1.23*10.^3}.
14886 Among things not currently handled by Calc's Maple mode are the
14887 various quote symbols, procedures and functional operators, and
14888 inert (@samp{&}) operators.
14890 @node Compositions, Syntax Tables, Maple Language Mode, Language Modes
14891 @subsection Compositions
14894 @cindex Compositions
14895 There are several @dfn{composition functions} which allow you to get
14896 displays in a variety of formats similar to those in Big language
14897 mode. Most of these functions do not evaluate to anything; they are
14898 placeholders which are left in symbolic form by Calc's evaluator but
14899 are recognized by Calc's display formatting routines.
14901 Two of these, @code{string} and @code{bstring}, are described elsewhere.
14902 @xref{Strings}. For example, @samp{string("ABC")} is displayed as
14903 @samp{ABC}. When viewed on the stack it will be indistinguishable from
14904 the variable @code{ABC}, but internally it will be stored as
14905 @samp{string([65, 66, 67])} and can still be manipulated this way; for
14906 example, the selection and vector commands @kbd{j 1 v v j u} would
14907 select the vector portion of this object and reverse the elements, then
14908 deselect to reveal a string whose characters had been reversed.
14910 The composition functions do the same thing in all language modes
14911 (although their components will of course be formatted in the current
14912 language mode). The one exception is Unformatted mode (@kbd{d U}),
14913 which does not give the composition functions any special treatment.
14914 The functions are discussed here because of their relationship to
14915 the language modes.
14918 * Composition Basics::
14919 * Horizontal Compositions::
14920 * Vertical Compositions::
14921 * Other Compositions::
14922 * Information about Compositions::
14923 * User-Defined Compositions::
14926 @node Composition Basics, Horizontal Compositions, Compositions, Compositions
14927 @subsubsection Composition Basics
14930 Compositions are generally formed by stacking formulas together
14931 horizontally or vertically in various ways. Those formulas are
14932 themselves compositions. @TeX{} users will find this analogous
14933 to @TeX{}'s ``boxes.'' Each multi-line composition has a
14934 @dfn{baseline}; horizontal compositions use the baselines to
14935 decide how formulas should be positioned relative to one another.
14936 For example, in the Big mode formula
14948 the second term of the sum is four lines tall and has line three as
14949 its baseline. Thus when the term is combined with 17, line three
14950 is placed on the same level as the baseline of 17.
14956 Another important composition concept is @dfn{precedence}. This is
14957 an integer that represents the binding strength of various operators.
14958 For example, @samp{*} has higher precedence (195) than @samp{+} (180),
14959 which means that @samp{(a * b) + c} will be formatted without the
14960 parentheses, but @samp{a * (b + c)} will keep the parentheses.
14962 The operator table used by normal and Big language modes has the
14963 following precedences:
14966 _ 1200 @r{(subscripts)}
14967 % 1100 @r{(as in n}%@r{)}
14968 - 1000 @r{(as in }-@r{n)}
14969 ! 1000 @r{(as in }!@r{n)}
14972 !! 210 @r{(as in n}!!@r{)}
14973 ! 210 @r{(as in n}!@r{)}
14975 * 195 @r{(or implicit multiplication)}
14977 + - 180 @r{(as in a}+@r{b)}
14979 < = 160 @r{(and other relations)}
14991 The general rule is that if an operator with precedence @expr{n}
14992 occurs as an argument to an operator with precedence @expr{m}, then
14993 the argument is enclosed in parentheses if @expr{n < m}. Top-level
14994 expressions and expressions which are function arguments, vector
14995 components, etc., are formatted with precedence zero (so that they
14996 normally never get additional parentheses).
14998 For binary left-associative operators like @samp{+}, the righthand
14999 argument is actually formatted with one-higher precedence than shown
15000 in the table. This makes sure @samp{(a + b) + c} omits the parentheses,
15001 but the unnatural form @samp{a + (b + c)} keeps its parentheses.
15002 Right-associative operators like @samp{^} format the lefthand argument
15003 with one-higher precedence.
15009 The @code{cprec} function formats an expression with an arbitrary
15010 precedence. For example, @samp{cprec(abc, 185)} will combine into
15011 sums and products as follows: @samp{7 + abc}, @samp{7 (abc)} (because
15012 this @code{cprec} form has higher precedence than addition, but lower
15013 precedence than multiplication).
15019 A final composition issue is @dfn{line breaking}. Calc uses two
15020 different strategies for ``flat'' and ``non-flat'' compositions.
15021 A non-flat composition is anything that appears on multiple lines
15022 (not counting line breaking). Examples would be matrices and Big
15023 mode powers and quotients. Non-flat compositions are displayed
15024 exactly as specified. If they come out wider than the current
15025 window, you must use horizontal scrolling (@kbd{<} and @kbd{>}) to
15028 Flat compositions, on the other hand, will be broken across several
15029 lines if they are too wide to fit the window. Certain points in a
15030 composition are noted internally as @dfn{break points}. Calc's
15031 general strategy is to fill each line as much as possible, then to
15032 move down to the next line starting at the first break point that
15033 didn't fit. However, the line breaker understands the hierarchical
15034 structure of formulas. It will not break an ``inner'' formula if
15035 it can use an earlier break point from an ``outer'' formula instead.
15036 For example, a vector of sums might be formatted as:
15040 [ a + b + c, d + e + f,
15041 g + h + i, j + k + l, m ]
15046 If the @samp{m} can fit, then so, it seems, could the @samp{g}.
15047 But Calc prefers to break at the comma since the comma is part
15048 of a ``more outer'' formula. Calc would break at a plus sign
15049 only if it had to, say, if the very first sum in the vector had
15050 itself been too large to fit.
15052 Of the composition functions described below, only @code{choriz}
15053 generates break points. The @code{bstring} function (@pxref{Strings})
15054 also generates breakable items: A break point is added after every
15055 space (or group of spaces) except for spaces at the very beginning or
15058 Composition functions themselves count as levels in the formula
15059 hierarchy, so a @code{choriz} that is a component of a larger
15060 @code{choriz} will be less likely to be broken. As a special case,
15061 if a @code{bstring} occurs as a component of a @code{choriz} or
15062 @code{choriz}-like object (such as a vector or a list of arguments
15063 in a function call), then the break points in that @code{bstring}
15064 will be on the same level as the break points of the surrounding
15067 @node Horizontal Compositions, Vertical Compositions, Composition Basics, Compositions
15068 @subsubsection Horizontal Compositions
15075 The @code{choriz} function takes a vector of objects and composes
15076 them horizontally. For example, @samp{choriz([17, a b/c, d])} formats
15077 as @w{@samp{17a b / cd}} in Normal language mode, or as
15088 in Big language mode. This is actually one case of the general
15089 function @samp{choriz(@var{vec}, @var{sep}, @var{prec})}, where
15090 either or both of @var{sep} and @var{prec} may be omitted.
15091 @var{Prec} gives the @dfn{precedence} to use when formatting
15092 each of the components of @var{vec}. The default precedence is
15093 the precedence from the surrounding environment.
15095 @var{Sep} is a string (i.e., a vector of character codes as might
15096 be entered with @code{" "} notation) which should separate components
15097 of the composition. Also, if @var{sep} is given, the line breaker
15098 will allow lines to be broken after each occurrence of @var{sep}.
15099 If @var{sep} is omitted, the composition will not be breakable
15100 (unless any of its component compositions are breakable).
15102 For example, @samp{2 choriz([a, b c, d = e], " + ", 180)} is
15103 formatted as @samp{2 a + b c + (d = e)}. To get the @code{choriz}
15104 to have precedence 180 ``outwards'' as well as ``inwards,''
15105 enclose it in a @code{cprec} form: @samp{2 cprec(choriz(...), 180)}
15106 formats as @samp{2 (a + b c + (d = e))}.
15108 The baseline of a horizontal composition is the same as the
15109 baselines of the component compositions, which are all aligned.
15111 @node Vertical Compositions, Other Compositions, Horizontal Compositions, Compositions
15112 @subsubsection Vertical Compositions
15119 The @code{cvert} function makes a vertical composition. Each
15120 component of the vector is centered in a column. The baseline of
15121 the result is by default the top line of the resulting composition.
15122 For example, @samp{f(cvert([a, bb, ccc]), cvert([a^2 + 1, b^2]))}
15123 formats in Big mode as
15138 There are several special composition functions that work only as
15139 components of a vertical composition. The @code{cbase} function
15140 controls the baseline of the vertical composition; the baseline
15141 will be the same as the baseline of whatever component is enclosed
15142 in @code{cbase}. Thus @samp{f(cvert([a, cbase(bb), ccc]),
15143 cvert([a^2 + 1, cbase(b^2)]))} displays as
15163 There are also @code{ctbase} and @code{cbbase} functions which
15164 make the baseline of the vertical composition equal to the top
15165 or bottom line (rather than the baseline) of that component.
15166 Thus @samp{cvert([cbase(a / b)]) + cvert([ctbase(a / b)]) +
15167 cvert([cbbase(a / b)])} gives
15179 There should be only one @code{cbase}, @code{ctbase}, or @code{cbbase}
15180 function in a given vertical composition. These functions can also
15181 be written with no arguments: @samp{ctbase()} is a zero-height object
15182 which means the baseline is the top line of the following item, and
15183 @samp{cbbase()} means the baseline is the bottom line of the preceding
15190 The @code{crule} function builds a ``rule,'' or horizontal line,
15191 across a vertical composition. By itself @samp{crule()} uses @samp{-}
15192 characters to build the rule. You can specify any other character,
15193 e.g., @samp{crule("=")}. The argument must be a character code or
15194 vector of exactly one character code. It is repeated to match the
15195 width of the widest item in the stack. For example, a quotient
15196 with a thick line is @samp{cvert([a + 1, cbase(crule("=")), b^2])}:
15215 Finally, the functions @code{clvert} and @code{crvert} act exactly
15216 like @code{cvert} except that the items are left- or right-justified
15217 in the stack. Thus @samp{clvert([a, bb, ccc]) + crvert([a, bb, ccc])}
15228 Like @code{choriz}, the vertical compositions accept a second argument
15229 which gives the precedence to use when formatting the components.
15230 Vertical compositions do not support separator strings.
15232 @node Other Compositions, Information about Compositions, Vertical Compositions, Compositions
15233 @subsubsection Other Compositions
15240 The @code{csup} function builds a superscripted expression. For
15241 example, @samp{csup(a, b)} looks the same as @samp{a^b} does in Big
15242 language mode. This is essentially a horizontal composition of
15243 @samp{a} and @samp{b}, where @samp{b} is shifted up so that its
15244 bottom line is one above the baseline.
15250 Likewise, the @code{csub} function builds a subscripted expression.
15251 This shifts @samp{b} down so that its top line is one below the
15252 bottom line of @samp{a} (note that this is not quite analogous to
15253 @code{csup}). Other arrangements can be obtained by using
15254 @code{choriz} and @code{cvert} directly.
15260 The @code{cflat} function formats its argument in ``flat'' mode,
15261 as obtained by @samp{d O}, if the current language mode is normal
15262 or Big. It has no effect in other language modes. For example,
15263 @samp{a^(b/c)} is formatted by Big mode like @samp{csup(a, cflat(b/c))}
15264 to improve its readability.
15270 The @code{cspace} function creates horizontal space. For example,
15271 @samp{cspace(4)} is effectively the same as @samp{string(" ")}.
15272 A second string (i.e., vector of characters) argument is repeated
15273 instead of the space character. For example, @samp{cspace(4, "ab")}
15274 looks like @samp{abababab}. If the second argument is not a string,
15275 it is formatted in the normal way and then several copies of that
15276 are composed together: @samp{cspace(4, a^2)} yields
15286 If the number argument is zero, this is a zero-width object.
15292 The @code{cvspace} function creates vertical space, or a vertical
15293 stack of copies of a certain string or formatted object. The
15294 baseline is the center line of the resulting stack. A numerical
15295 argument of zero will produce an object which contributes zero
15296 height if used in a vertical composition.
15306 There are also @code{ctspace} and @code{cbspace} functions which
15307 create vertical space with the baseline the same as the baseline
15308 of the top or bottom copy, respectively, of the second argument.
15309 Thus @samp{cvspace(2, a/b) + ctspace(2, a/b) + cbspace(2, a/b)}
15326 @node Information about Compositions, User-Defined Compositions, Other Compositions, Compositions
15327 @subsubsection Information about Compositions
15330 The functions in this section are actual functions; they compose their
15331 arguments according to the current language and other display modes,
15332 then return a certain measurement of the composition as an integer.
15338 The @code{cwidth} function measures the width, in characters, of a
15339 composition. For example, @samp{cwidth(a + b)} is 5, and
15340 @samp{cwidth(a / b)} is 5 in Normal mode, 1 in Big mode, and 11 in
15341 @TeX{} mode (for @samp{@{a \over b@}}). The argument may involve
15342 the composition functions described in this section.
15348 The @code{cheight} function measures the height of a composition.
15349 This is the total number of lines in the argument's printed form.
15359 The functions @code{cascent} and @code{cdescent} measure the amount
15360 of the height that is above (and including) the baseline, or below
15361 the baseline, respectively. Thus @samp{cascent(@var{x}) + cdescent(@var{x})}
15362 always equals @samp{cheight(@var{x})}. For a one-line formula like
15363 @samp{a + b}, @code{cascent} returns 1 and @code{cdescent} returns 0.
15364 For @samp{a / b} in Big mode, @code{cascent} returns 2 and @code{cdescent}
15365 returns 1. The only formula for which @code{cascent} will return zero
15366 is @samp{cvspace(0)} or equivalents.
15368 @node User-Defined Compositions, , Information about Compositions, Compositions
15369 @subsubsection User-Defined Compositions
15373 @pindex calc-user-define-composition
15374 The @kbd{Z C} (@code{calc-user-define-composition}) command lets you
15375 define the display format for any algebraic function. You provide a
15376 formula containing a certain number of argument variables on the stack.
15377 Any time Calc formats a call to the specified function in the current
15378 language mode and with that number of arguments, Calc effectively
15379 replaces the function call with that formula with the arguments
15382 Calc builds the default argument list by sorting all the variable names
15383 that appear in the formula into alphabetical order. You can edit this
15384 argument list before pressing @key{RET} if you wish. Any variables in
15385 the formula that do not appear in the argument list will be displayed
15386 literally; any arguments that do not appear in the formula will not
15387 affect the display at all.
15389 You can define formats for built-in functions, for functions you have
15390 defined with @kbd{Z F} (@pxref{Algebraic Definitions}), or for functions
15391 which have no definitions but are being used as purely syntactic objects.
15392 You can define different formats for each language mode, and for each
15393 number of arguments, using a succession of @kbd{Z C} commands. When
15394 Calc formats a function call, it first searches for a format defined
15395 for the current language mode (and number of arguments); if there is
15396 none, it uses the format defined for the Normal language mode. If
15397 neither format exists, Calc uses its built-in standard format for that
15398 function (usually just @samp{@var{func}(@var{args})}).
15400 If you execute @kbd{Z C} with the number 0 on the stack instead of a
15401 formula, any defined formats for the function in the current language
15402 mode will be removed. The function will revert to its standard format.
15404 For example, the default format for the binomial coefficient function
15405 @samp{choose(n, m)} in the Big language mode is
15416 You might prefer the notation,
15426 To define this notation, first make sure you are in Big mode,
15427 then put the formula
15430 choriz([cvert([cvspace(1), n]), C, cvert([cvspace(1), m])])
15434 on the stack and type @kbd{Z C}. Answer the first prompt with
15435 @code{choose}. The second prompt will be the default argument list
15436 of @samp{(C m n)}. Edit this list to be @samp{(n m)} and press
15437 @key{RET}. Now, try it out: For example, turn simplification
15438 off with @kbd{m O} and enter @samp{choose(a,b) + choose(7,3)}
15439 as an algebraic entry.
15448 As another example, let's define the usual notation for Stirling
15449 numbers of the first kind, @samp{stir1(n, m)}. This is just like
15450 the regular format for binomial coefficients but with square brackets
15451 instead of parentheses.
15454 choriz([string("["), cvert([n, cbase(cvspace(1)), m]), string("]")])
15457 Now type @kbd{Z C stir1 @key{RET}}, edit the argument list to
15458 @samp{(n m)}, and type @key{RET}.
15460 The formula provided to @kbd{Z C} usually will involve composition
15461 functions, but it doesn't have to. Putting the formula @samp{a + b + c}
15462 onto the stack and typing @kbd{Z C foo @key{RET} @key{RET}} would define
15463 the function @samp{foo(x,y,z)} to display like @samp{x + y + z}.
15464 This ``sum'' will act exactly like a real sum for all formatting
15465 purposes (it will be parenthesized the same, and so on). However
15466 it will be computationally unrelated to a sum. For example, the
15467 formula @samp{2 * foo(1, 2, 3)} will display as @samp{2 (1 + 2 + 3)}.
15468 Operator precedences have caused the ``sum'' to be written in
15469 parentheses, but the arguments have not actually been summed.
15470 (Generally a display format like this would be undesirable, since
15471 it can easily be confused with a real sum.)
15473 The special function @code{eval} can be used inside a @kbd{Z C}
15474 composition formula to cause all or part of the formula to be
15475 evaluated at display time. For example, if the formula is
15476 @samp{a + eval(b + c)}, then @samp{foo(1, 2, 3)} will be displayed
15477 as @samp{1 + 5}. Evaluation will use the default simplifications,
15478 regardless of the current simplification mode. There are also
15479 @code{evalsimp} and @code{evalextsimp} which simplify as if by
15480 @kbd{a s} and @kbd{a e} (respectively). Note that these ``functions''
15481 operate only in the context of composition formulas (and also in
15482 rewrite rules, where they serve a similar purpose; @pxref{Rewrite
15483 Rules}). On the stack, a call to @code{eval} will be left in
15486 It is not a good idea to use @code{eval} except as a last resort.
15487 It can cause the display of formulas to be extremely slow. For
15488 example, while @samp{eval(a + b)} might seem quite fast and simple,
15489 there are several situations where it could be slow. For example,
15490 @samp{a} and/or @samp{b} could be polar complex numbers, in which
15491 case doing the sum requires trigonometry. Or, @samp{a} could be
15492 the factorial @samp{fact(100)} which is unevaluated because you
15493 have typed @kbd{m O}; @code{eval} will evaluate it anyway to
15494 produce a large, unwieldy integer.
15496 You can save your display formats permanently using the @kbd{Z P}
15497 command (@pxref{Creating User Keys}).
15499 @node Syntax Tables, , Compositions, Language Modes
15500 @subsection Syntax Tables
15503 @cindex Syntax tables
15504 @cindex Parsing formulas, customized
15505 Syntax tables do for input what compositions do for output: They
15506 allow you to teach custom notations to Calc's formula parser.
15507 Calc keeps a separate syntax table for each language mode.
15509 (Note that the Calc ``syntax tables'' discussed here are completely
15510 unrelated to the syntax tables described in the Emacs manual.)
15513 @pindex calc-edit-user-syntax
15514 The @kbd{Z S} (@code{calc-edit-user-syntax}) command edits the
15515 syntax table for the current language mode. If you want your
15516 syntax to work in any language, define it in the Normal language
15517 mode. Type @kbd{C-c C-c} to finish editing the syntax table, or
15518 @kbd{C-x k} to cancel the edit. The @kbd{m m} command saves all
15519 the syntax tables along with the other mode settings;
15520 @pxref{General Mode Commands}.
15523 * Syntax Table Basics::
15524 * Precedence in Syntax Tables::
15525 * Advanced Syntax Patterns::
15526 * Conditional Syntax Rules::
15529 @node Syntax Table Basics, Precedence in Syntax Tables, Syntax Tables, Syntax Tables
15530 @subsubsection Syntax Table Basics
15533 @dfn{Parsing} is the process of converting a raw string of characters,
15534 such as you would type in during algebraic entry, into a Calc formula.
15535 Calc's parser works in two stages. First, the input is broken down
15536 into @dfn{tokens}, such as words, numbers, and punctuation symbols
15537 like @samp{+}, @samp{:=}, and @samp{+/-}. Space between tokens is
15538 ignored (except when it serves to separate adjacent words). Next,
15539 the parser matches this string of tokens against various built-in
15540 syntactic patterns, such as ``an expression followed by @samp{+}
15541 followed by another expression'' or ``a name followed by @samp{(},
15542 zero or more expressions separated by commas, and @samp{)}.''
15544 A @dfn{syntax table} is a list of user-defined @dfn{syntax rules},
15545 which allow you to specify new patterns to define your own
15546 favorite input notations. Calc's parser always checks the syntax
15547 table for the current language mode, then the table for the Normal
15548 language mode, before it uses its built-in rules to parse an
15549 algebraic formula you have entered. Each syntax rule should go on
15550 its own line; it consists of a @dfn{pattern}, a @samp{:=} symbol,
15551 and a Calc formula with an optional @dfn{condition}. (Syntax rules
15552 resemble algebraic rewrite rules, but the notation for patterns is
15553 completely different.)
15555 A syntax pattern is a list of tokens, separated by spaces.
15556 Except for a few special symbols, tokens in syntax patterns are
15557 matched literally, from left to right. For example, the rule,
15564 would cause Calc to parse the formula @samp{4+foo()*5} as if it
15565 were @samp{4+(2+3)*5}. Notice that the parentheses were written
15566 as two separate tokens in the rule. As a result, the rule works
15567 for both @samp{foo()} and @w{@samp{foo ( )}}. If we had written
15568 the rule as @samp{foo () := 2+3}, then Calc would treat @samp{()}
15569 as a single, indivisible token, so that @w{@samp{foo( )}} would
15570 not be recognized by the rule. (It would be parsed as a regular
15571 zero-argument function call instead.) In fact, this rule would
15572 also make trouble for the rest of Calc's parser: An unrelated
15573 formula like @samp{bar()} would now be tokenized into @samp{bar ()}
15574 instead of @samp{bar ( )}, so that the standard parser for function
15575 calls would no longer recognize it!
15577 While it is possible to make a token with a mixture of letters
15578 and punctuation symbols, this is not recommended. It is better to
15579 break it into several tokens, as we did with @samp{foo()} above.
15581 The symbol @samp{#} in a syntax pattern matches any Calc expression.
15582 On the righthand side, the things that matched the @samp{#}s can
15583 be referred to as @samp{#1}, @samp{#2}, and so on (where @samp{#1}
15584 matches the leftmost @samp{#} in the pattern). For example, these
15585 rules match a user-defined function, prefix operator, infix operator,
15586 and postfix operator, respectively:
15589 foo ( # ) := myfunc(#1)
15590 foo # := myprefix(#1)
15591 # foo # := myinfix(#1,#2)
15592 # foo := mypostfix(#1)
15595 Thus @samp{foo(3)} will parse as @samp{myfunc(3)}, and @samp{2+3 foo}
15596 will parse as @samp{mypostfix(2+3)}.
15598 It is important to write the first two rules in the order shown,
15599 because Calc tries rules in order from first to last. If the
15600 pattern @samp{foo #} came first, it would match anything that could
15601 match the @samp{foo ( # )} rule, since an expression in parentheses
15602 is itself a valid expression. Thus the @w{@samp{foo ( # )}} rule would
15603 never get to match anything. Likewise, the last two rules must be
15604 written in the order shown or else @samp{3 foo 4} will be parsed as
15605 @samp{mypostfix(3) * 4}. (Of course, the best way to avoid these
15606 ambiguities is not to use the same symbol in more than one way at
15607 the same time! In case you're not convinced, try the following
15608 exercise: How will the above rules parse the input @samp{foo(3,4)},
15609 if at all? Work it out for yourself, then try it in Calc and see.)
15611 Calc is quite flexible about what sorts of patterns are allowed.
15612 The only rule is that every pattern must begin with a literal
15613 token (like @samp{foo} in the first two patterns above), or with
15614 a @samp{#} followed by a literal token (as in the last two
15615 patterns). After that, any mixture is allowed, although putting
15616 two @samp{#}s in a row will not be very useful since two
15617 expressions with nothing between them will be parsed as one
15618 expression that uses implicit multiplication.
15620 As a more practical example, Maple uses the notation
15621 @samp{sum(a(i), i=1..10)} for sums, which Calc's Maple mode doesn't
15622 recognize at present. To handle this syntax, we simply add the
15626 sum ( # , # = # .. # ) := sum(#1,#2,#3,#4)
15630 to the Maple mode syntax table. As another example, C mode can't
15631 read assignment operators like @samp{++} and @samp{*=}. We can
15632 define these operators quite easily:
15635 # *= # := muleq(#1,#2)
15636 # ++ := postinc(#1)
15641 To complete the job, we would use corresponding composition functions
15642 and @kbd{Z C} to cause these functions to display in their respective
15643 Maple and C notations. (Note that the C example ignores issues of
15644 operator precedence, which are discussed in the next section.)
15646 You can enclose any token in quotes to prevent its usual
15647 interpretation in syntax patterns:
15650 # ":=" # := becomes(#1,#2)
15653 Quotes also allow you to include spaces in a token, although once
15654 again it is generally better to use two tokens than one token with
15655 an embedded space. To include an actual quotation mark in a quoted
15656 token, precede it with a backslash. (This also works to include
15657 backslashes in tokens.)
15660 # "bad token" # "/\"\\" # := silly(#1,#2,#3)
15664 This will parse @samp{3 bad token 4 /"\ 5} to @samp{silly(3,4,5)}.
15666 The token @kbd{#} has a predefined meaning in Calc's formula parser;
15667 it is not valid to use @samp{"#"} in a syntax rule. However, longer
15668 tokens that include the @samp{#} character are allowed. Also, while
15669 @samp{"$"} and @samp{"\""} are allowed as tokens, their presence in
15670 the syntax table will prevent those characters from working in their
15671 usual ways (referring to stack entries and quoting strings,
15674 Finally, the notation @samp{%%} anywhere in a syntax table causes
15675 the rest of the line to be ignored as a comment.
15677 @node Precedence in Syntax Tables, Advanced Syntax Patterns, Syntax Table Basics, Syntax Tables
15678 @subsubsection Precedence
15681 Different operators are generally assigned different @dfn{precedences}.
15682 By default, an operator defined by a rule like
15685 # foo # := foo(#1,#2)
15689 will have an extremely low precedence, so that @samp{2*3+4 foo 5 == 6}
15690 will be parsed as @samp{(2*3+4) foo (5 == 6)}. To change the
15691 precedence of an operator, use the notation @samp{#/@var{p}} in
15692 place of @samp{#}, where @var{p} is an integer precedence level.
15693 For example, 185 lies between the precedences for @samp{+} and
15694 @samp{*}, so if we change this rule to
15697 #/185 foo #/186 := foo(#1,#2)
15701 then @samp{2+3 foo 4*5} will be parsed as @samp{2+(3 foo (4*5))}.
15702 Also, because we've given the righthand expression slightly higher
15703 precedence, our new operator will be left-associative:
15704 @samp{1 foo 2 foo 3} will be parsed as @samp{(1 foo 2) foo 3}.
15705 By raising the precedence of the lefthand expression instead, we
15706 can create a right-associative operator.
15708 @xref{Composition Basics}, for a table of precedences of the
15709 standard Calc operators. For the precedences of operators in other
15710 language modes, look in the Calc source file @file{calc-lang.el}.
15712 @node Advanced Syntax Patterns, Conditional Syntax Rules, Precedence in Syntax Tables, Syntax Tables
15713 @subsubsection Advanced Syntax Patterns
15716 To match a function with a variable number of arguments, you could
15720 foo ( # ) := myfunc(#1)
15721 foo ( # , # ) := myfunc(#1,#2)
15722 foo ( # , # , # ) := myfunc(#1,#2,#3)
15726 but this isn't very elegant. To match variable numbers of items,
15727 Calc uses some notations inspired regular expressions and the
15728 ``extended BNF'' style used by some language designers.
15731 foo ( @{ # @}*, ) := apply(myfunc,#1)
15734 The token @samp{@{} introduces a repeated or optional portion.
15735 One of the three tokens @samp{@}*}, @samp{@}+}, or @samp{@}?}
15736 ends the portion. These will match zero or more, one or more,
15737 or zero or one copies of the enclosed pattern, respectively.
15738 In addition, @samp{@}*} and @samp{@}+} can be followed by a
15739 separator token (with no space in between, as shown above).
15740 Thus @samp{@{ # @}*,} matches nothing, or one expression, or
15741 several expressions separated by commas.
15743 A complete @samp{@{ ... @}} item matches as a vector of the
15744 items that matched inside it. For example, the above rule will
15745 match @samp{foo(1,2,3)} to get @samp{apply(myfunc,[1,2,3])}.
15746 The Calc @code{apply} function takes a function name and a vector
15747 of arguments and builds a call to the function with those
15748 arguments, so the net result is the formula @samp{myfunc(1,2,3)}.
15750 If the body of a @samp{@{ ... @}} contains several @samp{#}s
15751 (or nested @samp{@{ ... @}} constructs), then the items will be
15752 strung together into the resulting vector. If the body
15753 does not contain anything but literal tokens, the result will
15754 always be an empty vector.
15757 foo ( @{ # , # @}+, ) := bar(#1)
15758 foo ( @{ @{ # @}*, @}*; ) := matrix(#1)
15762 will parse @samp{foo(1, 2, 3, 4)} as @samp{bar([1, 2, 3, 4])}, and
15763 @samp{foo(1, 2; 3, 4)} as @samp{matrix([[1, 2], [3, 4]])}. Also, after
15764 some thought it's easy to see how this pair of rules will parse
15765 @samp{foo(1, 2, 3)} as @samp{matrix([[1, 2, 3]])}, since the first
15766 rule will only match an even number of arguments. The rule
15769 foo ( # @{ , # , # @}? ) := bar(#1,#2)
15773 will parse @samp{foo(2,3,4)} as @samp{bar(2,[3,4])}, and
15774 @samp{foo(2)} as @samp{bar(2,[])}.
15776 The notation @samp{@{ ... @}?.} (note the trailing period) works
15777 just the same as regular @samp{@{ ... @}?}, except that it does not
15778 count as an argument; the following two rules are equivalent:
15781 foo ( # , @{ also @}? # ) := bar(#1,#3)
15782 foo ( # , @{ also @}?. # ) := bar(#1,#2)
15786 Note that in the first case the optional text counts as @samp{#2},
15787 which will always be an empty vector, but in the second case no
15788 empty vector is produced.
15790 Another variant is @samp{@{ ... @}?$}, which means the body is
15791 optional only at the end of the input formula. All built-in syntax
15792 rules in Calc use this for closing delimiters, so that during
15793 algebraic entry you can type @kbd{[sqrt(2), sqrt(3 @key{RET}}, omitting
15794 the closing parenthesis and bracket. Calc does this automatically
15795 for trailing @samp{)}, @samp{]}, and @samp{>} tokens in syntax
15796 rules, but you can use @samp{@{ ... @}?$} explicitly to get
15797 this effect with any token (such as @samp{"@}"} or @samp{end}).
15798 Like @samp{@{ ... @}?.}, this notation does not count as an
15799 argument. Conversely, you can use quotes, as in @samp{")"}, to
15800 prevent a closing-delimiter token from being automatically treated
15803 Calc's parser does not have full backtracking, which means some
15804 patterns will not work as you might expect:
15807 foo ( @{ # , @}? # , # ) := bar(#1,#2,#3)
15811 Here we are trying to make the first argument optional, so that
15812 @samp{foo(2,3)} parses as @samp{bar([],2,3)}. Unfortunately, Calc
15813 first tries to match @samp{2,} against the optional part of the
15814 pattern, finds a match, and so goes ahead to match the rest of the
15815 pattern. Later on it will fail to match the second comma, but it
15816 doesn't know how to go back and try the other alternative at that
15817 point. One way to get around this would be to use two rules:
15820 foo ( # , # , # ) := bar([#1],#2,#3)
15821 foo ( # , # ) := bar([],#1,#2)
15824 More precisely, when Calc wants to match an optional or repeated
15825 part of a pattern, it scans forward attempting to match that part.
15826 If it reaches the end of the optional part without failing, it
15827 ``finalizes'' its choice and proceeds. If it fails, though, it
15828 backs up and tries the other alternative. Thus Calc has ``partial''
15829 backtracking. A fully backtracking parser would go on to make sure
15830 the rest of the pattern matched before finalizing the choice.
15832 @node Conditional Syntax Rules, , Advanced Syntax Patterns, Syntax Tables
15833 @subsubsection Conditional Syntax Rules
15836 It is possible to attach a @dfn{condition} to a syntax rule. For
15840 foo ( # ) := ifoo(#1) :: integer(#1)
15841 foo ( # ) := gfoo(#1)
15845 will parse @samp{foo(3)} as @samp{ifoo(3)}, but will parse
15846 @samp{foo(3.5)} and @samp{foo(x)} as calls to @code{gfoo}. Any
15847 number of conditions may be attached; all must be true for the
15848 rule to succeed. A condition is ``true'' if it evaluates to a
15849 nonzero number. @xref{Logical Operations}, for a list of Calc
15850 functions like @code{integer} that perform logical tests.
15852 The exact sequence of events is as follows: When Calc tries a
15853 rule, it first matches the pattern as usual. It then substitutes
15854 @samp{#1}, @samp{#2}, etc., in the conditions, if any. Next, the
15855 conditions are simplified and evaluated in order from left to right,
15856 as if by the @w{@kbd{a s}} algebra command (@pxref{Simplifying Formulas}).
15857 Each result is true if it is a nonzero number, or an expression
15858 that can be proven to be nonzero (@pxref{Declarations}). If the
15859 results of all conditions are true, the expression (such as
15860 @samp{ifoo(#1)}) has its @samp{#}s substituted, and that is the
15861 result of the parse. If the result of any condition is false, Calc
15862 goes on to try the next rule in the syntax table.
15864 Syntax rules also support @code{let} conditions, which operate in
15865 exactly the same way as they do in algebraic rewrite rules.
15866 @xref{Other Features of Rewrite Rules}, for details. A @code{let}
15867 condition is always true, but as a side effect it defines a
15868 variable which can be used in later conditions, and also in the
15869 expression after the @samp{:=} sign:
15872 foo ( # ) := hifoo(x) :: let(x := #1 + 0.5) :: dnumint(x)
15876 The @code{dnumint} function tests if a value is numerically an
15877 integer, i.e., either a true integer or an integer-valued float.
15878 This rule will parse @code{foo} with a half-integer argument,
15879 like @samp{foo(3.5)}, to a call like @samp{hifoo(4.)}.
15881 The lefthand side of a syntax rule @code{let} must be a simple
15882 variable, not the arbitrary pattern that is allowed in rewrite
15885 The @code{matches} function is also treated specially in syntax
15886 rule conditions (again, in the same way as in rewrite rules).
15887 @xref{Matching Commands}. If the matching pattern contains
15888 meta-variables, then those meta-variables may be used in later
15889 conditions and in the result expression. The arguments to
15890 @code{matches} are not evaluated in this situation.
15893 sum ( # , # ) := sum(#1,a,b,c) :: matches(#2, a=[b..c])
15897 This is another way to implement the Maple mode @code{sum} notation.
15898 In this approach, we allow @samp{#2} to equal the whole expression
15899 @samp{i=1..10}. Then, we use @code{matches} to break it apart into
15900 its components. If the expression turns out not to match the pattern,
15901 the syntax rule will fail. Note that @kbd{Z S} always uses Calc's
15902 Normal language mode for editing expressions in syntax rules, so we
15903 must use regular Calc notation for the interval @samp{[b..c]} that
15904 will correspond to the Maple mode interval @samp{1..10}.
15906 @node Modes Variable, Calc Mode Line, Language Modes, Mode Settings
15907 @section The @code{Modes} Variable
15911 @pindex calc-get-modes
15912 The @kbd{m g} (@code{calc-get-modes}) command pushes onto the stack
15913 a vector of numbers that describes the various mode settings that
15914 are in effect. With a numeric prefix argument, it pushes only the
15915 @var{n}th mode, i.e., the @var{n}th element of this vector. Keyboard
15916 macros can use the @kbd{m g} command to modify their behavior based
15917 on the current mode settings.
15919 @cindex @code{Modes} variable
15921 The modes vector is also available in the special variable
15922 @code{Modes}. In other words, @kbd{m g} is like @kbd{s r Modes @key{RET}}.
15923 It will not work to store into this variable; in fact, if you do,
15924 @code{Modes} will cease to track the current modes. (The @kbd{m g}
15925 command will continue to work, however.)
15927 In general, each number in this vector is suitable as a numeric
15928 prefix argument to the associated mode-setting command. (Recall
15929 that the @kbd{~} key takes a number from the stack and gives it as
15930 a numeric prefix to the next command.)
15932 The elements of the modes vector are as follows:
15936 Current precision. Default is 12; associated command is @kbd{p}.
15939 Binary word size. Default is 32; associated command is @kbd{b w}.
15942 Stack size (not counting the value about to be pushed by @kbd{m g}).
15943 This is zero if @kbd{m g} is executed with an empty stack.
15946 Number radix. Default is 10; command is @kbd{d r}.
15949 Floating-point format. This is the number of digits, plus the
15950 constant 0 for normal notation, 10000 for scientific notation,
15951 20000 for engineering notation, or 30000 for fixed-point notation.
15952 These codes are acceptable as prefix arguments to the @kbd{d n}
15953 command, but note that this may lose information: For example,
15954 @kbd{d s} and @kbd{C-u 12 d s} have similar (but not quite
15955 identical) effects if the current precision is 12, but they both
15956 produce a code of 10012, which will be treated by @kbd{d n} as
15957 @kbd{C-u 12 d s}. If the precision then changes, the float format
15958 will still be frozen at 12 significant figures.
15961 Angular mode. Default is 1 (degrees). Other values are 2 (radians)
15962 and 3 (HMS). The @kbd{m d} command accepts these prefixes.
15965 Symbolic mode. Value is 0 or 1; default is 0. Command is @kbd{m s}.
15968 Fraction mode. Value is 0 or 1; default is 0. Command is @kbd{m f}.
15971 Polar mode. Value is 0 (rectangular) or 1 (polar); default is 0.
15972 Command is @kbd{m p}.
15975 Matrix/Scalar mode. Default value is @mathit{-1}. Value is 0 for Scalar
15976 mode, @mathit{-2} for Matrix mode, or @var{N} for
15977 @texline @math{N\times N}
15978 @infoline @var{N}x@var{N}
15979 Matrix mode. Command is @kbd{m v}.
15982 Simplification mode. Default is 1. Value is @mathit{-1} for off (@kbd{m O}),
15983 0 for @kbd{m N}, 2 for @kbd{m B}, 3 for @kbd{m A}, 4 for @kbd{m E},
15984 or 5 for @w{@kbd{m U}}. The @kbd{m D} command accepts these prefixes.
15987 Infinite mode. Default is @mathit{-1} (off). Value is 1 if the mode is on,
15988 or 0 if the mode is on with positive zeros. Command is @kbd{m i}.
15991 For example, the sequence @kbd{M-1 m g @key{RET} 2 + ~ p} increases the
15992 precision by two, leaving a copy of the old precision on the stack.
15993 Later, @kbd{~ p} will restore the original precision using that
15994 stack value. (This sequence might be especially useful inside a
15997 As another example, @kbd{M-3 m g 1 - ~ @key{DEL}} deletes all but the
15998 oldest (bottommost) stack entry.
16000 Yet another example: The HP-48 ``round'' command rounds a number
16001 to the current displayed precision. You could roughly emulate this
16002 in Calc with the sequence @kbd{M-5 m g 10000 % ~ c c}. (This
16003 would not work for fixed-point mode, but it wouldn't be hard to
16004 do a full emulation with the help of the @kbd{Z [} and @kbd{Z ]}
16005 programming commands. @xref{Conditionals in Macros}.)
16007 @node Calc Mode Line, , Modes Variable, Mode Settings
16008 @section The Calc Mode Line
16011 @cindex Mode line indicators
16012 This section is a summary of all symbols that can appear on the
16013 Calc mode line, the highlighted bar that appears under the Calc
16014 stack window (or under an editing window in Embedded mode).
16016 The basic mode line format is:
16019 --%%-Calc: 12 Deg @var{other modes} (Calculator)
16022 The @samp{%%} is the Emacs symbol for ``read-only''; it shows that
16023 regular Emacs commands are not allowed to edit the stack buffer
16024 as if it were text.
16026 The word @samp{Calc:} changes to @samp{CalcEmbed:} if Embedded mode
16027 is enabled. The words after this describe the various Calc modes
16028 that are in effect.
16030 The first mode is always the current precision, an integer.
16031 The second mode is always the angular mode, either @code{Deg},
16032 @code{Rad}, or @code{Hms}.
16034 Here is a complete list of the remaining symbols that can appear
16039 Algebraic mode (@kbd{m a}; @pxref{Algebraic Entry}).
16042 Incomplete algebraic mode (@kbd{C-u m a}).
16045 Total algebraic mode (@kbd{m t}).
16048 Symbolic mode (@kbd{m s}; @pxref{Symbolic Mode}).
16051 Matrix mode (@kbd{m v}; @pxref{Matrix Mode}).
16053 @item Matrix@var{n}
16054 Dimensioned Matrix mode (@kbd{C-u @var{n} m v}).
16057 Scalar mode (@kbd{m v}; @pxref{Matrix Mode}).
16060 Polar complex mode (@kbd{m p}; @pxref{Polar Mode}).
16063 Fraction mode (@kbd{m f}; @pxref{Fraction Mode}).
16066 Infinite mode (@kbd{m i}; @pxref{Infinite Mode}).
16069 Positive Infinite mode (@kbd{C-u 0 m i}).
16072 Default simplifications off (@kbd{m O}; @pxref{Simplification Modes}).
16075 Default simplifications for numeric arguments only (@kbd{m N}).
16077 @item BinSimp@var{w}
16078 Binary-integer simplification mode; word size @var{w} (@kbd{m B}, @kbd{b w}).
16081 Algebraic simplification mode (@kbd{m A}).
16084 Extended algebraic simplification mode (@kbd{m E}).
16087 Units simplification mode (@kbd{m U}).
16090 Current radix is 2 (@kbd{d 2}; @pxref{Radix Modes}).
16093 Current radix is 8 (@kbd{d 8}).
16096 Current radix is 16 (@kbd{d 6}).
16099 Current radix is @var{n} (@kbd{d r}).
16102 Leading zeros (@kbd{d z}; @pxref{Radix Modes}).
16105 Big language mode (@kbd{d B}; @pxref{Normal Language Modes}).
16108 One-line normal language mode (@kbd{d O}).
16111 Unformatted language mode (@kbd{d U}).
16114 C language mode (@kbd{d C}; @pxref{C FORTRAN Pascal}).
16117 Pascal language mode (@kbd{d P}).
16120 FORTRAN language mode (@kbd{d F}).
16123 @TeX{} language mode (@kbd{d T}; @pxref{TeX and LaTeX Language Modes}).
16126 La@TeX{} language mode (@kbd{d L}; @pxref{TeX and LaTeX Language Modes}).
16129 @dfn{Eqn} language mode (@kbd{d E}; @pxref{Eqn Language Mode}).
16132 Mathematica language mode (@kbd{d M}; @pxref{Mathematica Language Mode}).
16135 Maple language mode (@kbd{d W}; @pxref{Maple Language Mode}).
16138 Normal float mode with @var{n} digits (@kbd{d n}; @pxref{Float Formats}).
16141 Fixed point mode with @var{n} digits after the point (@kbd{d f}).
16144 Scientific notation mode (@kbd{d s}).
16147 Scientific notation with @var{n} digits (@kbd{d s}).
16150 Engineering notation mode (@kbd{d e}).
16153 Engineering notation with @var{n} digits (@kbd{d e}).
16156 Left-justified display indented by @var{n} (@kbd{d <}; @pxref{Justification}).
16159 Right-justified display (@kbd{d >}).
16162 Right-justified display with width @var{n} (@kbd{d >}).
16165 Centered display (@kbd{d =}).
16167 @item Center@var{n}
16168 Centered display with center column @var{n} (@kbd{d =}).
16171 Line breaking with width @var{n} (@kbd{d b}; @pxref{Normal Language Modes}).
16174 No line breaking (@kbd{d b}).
16177 Selections show deep structure (@kbd{j b}; @pxref{Making Selections}).
16180 Record modes in @file{~/.calc.el} (@kbd{m R}; @pxref{General Mode Commands}).
16183 Record modes in Embedded buffer (@kbd{m R}).
16186 Record modes as editing-only in Embedded buffer (@kbd{m R}).
16189 Record modes as permanent-only in Embedded buffer (@kbd{m R}).
16192 Record modes as global in Embedded buffer (@kbd{m R}).
16195 Automatic recomputation turned off (@kbd{m C}; @pxref{Automatic
16199 GNUPLOT process is alive in background (@pxref{Graphics}).
16202 Top-of-stack has a selection (Embedded only; @pxref{Making Selections}).
16205 The stack display may not be up-to-date (@pxref{Display Modes}).
16208 ``Inverse'' prefix was pressed (@kbd{I}; @pxref{Inverse and Hyperbolic}).
16211 ``Hyperbolic'' prefix was pressed (@kbd{H}).
16214 ``Keep-arguments'' prefix was pressed (@kbd{K}).
16217 Stack is truncated (@kbd{d t}; @pxref{Truncating the Stack}).
16220 In addition, the symbols @code{Active} and @code{~Active} can appear
16221 as minor modes on an Embedded buffer's mode line. @xref{Embedded Mode}.
16223 @node Arithmetic, Scientific Functions, Mode Settings, Top
16224 @chapter Arithmetic Functions
16227 This chapter describes the Calc commands for doing simple calculations
16228 on numbers, such as addition, absolute value, and square roots. These
16229 commands work by removing the top one or two values from the stack,
16230 performing the desired operation, and pushing the result back onto the
16231 stack. If the operation cannot be performed, the result pushed is a
16232 formula instead of a number, such as @samp{2/0} (because division by zero
16233 is invalid) or @samp{sqrt(x)} (because the argument @samp{x} is a formula).
16235 Most of the commands described here can be invoked by a single keystroke.
16236 Some of the more obscure ones are two-letter sequences beginning with
16237 the @kbd{f} (``functions'') prefix key.
16239 @xref{Prefix Arguments}, for a discussion of the effect of numeric
16240 prefix arguments on commands in this chapter which do not otherwise
16241 interpret a prefix argument.
16244 * Basic Arithmetic::
16245 * Integer Truncation::
16246 * Complex Number Functions::
16248 * Date Arithmetic::
16249 * Financial Functions::
16250 * Binary Functions::
16253 @node Basic Arithmetic, Integer Truncation, Arithmetic, Arithmetic
16254 @section Basic Arithmetic
16263 The @kbd{+} (@code{calc-plus}) command adds two numbers. The numbers may
16264 be any of the standard Calc data types. The resulting sum is pushed back
16267 If both arguments of @kbd{+} are vectors or matrices (of matching dimensions),
16268 the result is a vector or matrix sum. If one argument is a vector and the
16269 other a scalar (i.e., a non-vector), the scalar is added to each of the
16270 elements of the vector to form a new vector. If the scalar is not a
16271 number, the operation is left in symbolic form: Suppose you added @samp{x}
16272 to the vector @samp{[1,2]}. You may want the result @samp{[1+x,2+x]}, or
16273 you may plan to substitute a 2-vector for @samp{x} in the future. Since
16274 the Calculator can't tell which interpretation you want, it makes the
16275 safest assumption. @xref{Reducing and Mapping}, for a way to add @samp{x}
16276 to every element of a vector.
16278 If either argument of @kbd{+} is a complex number, the result will in general
16279 be complex. If one argument is in rectangular form and the other polar,
16280 the current Polar mode determines the form of the result. If Symbolic
16281 mode is enabled, the sum may be left as a formula if the necessary
16282 conversions for polar addition are non-trivial.
16284 If both arguments of @kbd{+} are HMS forms, the forms are added according to
16285 the usual conventions of hours-minutes-seconds notation. If one argument
16286 is an HMS form and the other is a number, that number is converted from
16287 degrees or radians (depending on the current Angular mode) to HMS format
16288 and then the two HMS forms are added.
16290 If one argument of @kbd{+} is a date form, the other can be either a
16291 real number, which advances the date by a certain number of days, or
16292 an HMS form, which advances the date by a certain amount of time.
16293 Subtracting two date forms yields the number of days between them.
16294 Adding two date forms is meaningless, but Calc interprets it as the
16295 subtraction of one date form and the negative of the other. (The
16296 negative of a date form can be understood by remembering that dates
16297 are stored as the number of days before or after Jan 1, 1 AD.)
16299 If both arguments of @kbd{+} are error forms, the result is an error form
16300 with an appropriately computed standard deviation. If one argument is an
16301 error form and the other is a number, the number is taken to have zero error.
16302 Error forms may have symbolic formulas as their mean and/or error parts;
16303 adding these will produce a symbolic error form result. However, adding an
16304 error form to a plain symbolic formula (as in @samp{(a +/- b) + c}) will not
16305 work, for the same reasons just mentioned for vectors. Instead you must
16306 write @samp{(a +/- b) + (c +/- 0)}.
16308 If both arguments of @kbd{+} are modulo forms with equal values of @expr{M},
16309 or if one argument is a modulo form and the other a plain number, the
16310 result is a modulo form which represents the sum, modulo @expr{M}, of
16313 If both arguments of @kbd{+} are intervals, the result is an interval
16314 which describes all possible sums of the possible input values. If
16315 one argument is a plain number, it is treated as the interval
16316 @w{@samp{[x ..@: x]}}.
16318 If one argument of @kbd{+} is an infinity and the other is not, the
16319 result is that same infinity. If both arguments are infinite and in
16320 the same direction, the result is the same infinity, but if they are
16321 infinite in different directions the result is @code{nan}.
16329 The @kbd{-} (@code{calc-minus}) command subtracts two values. The top
16330 number on the stack is subtracted from the one behind it, so that the
16331 computation @kbd{5 @key{RET} 2 -} produces 3, not @mathit{-3}. All options
16332 available for @kbd{+} are available for @kbd{-} as well.
16340 The @kbd{*} (@code{calc-times}) command multiplies two numbers. If one
16341 argument is a vector and the other a scalar, the scalar is multiplied by
16342 the elements of the vector to produce a new vector. If both arguments
16343 are vectors, the interpretation depends on the dimensions of the
16344 vectors: If both arguments are matrices, a matrix multiplication is
16345 done. If one argument is a matrix and the other a plain vector, the
16346 vector is interpreted as a row vector or column vector, whichever is
16347 dimensionally correct. If both arguments are plain vectors, the result
16348 is a single scalar number which is the dot product of the two vectors.
16350 If one argument of @kbd{*} is an HMS form and the other a number, the
16351 HMS form is multiplied by that amount. It is an error to multiply two
16352 HMS forms together, or to attempt any multiplication involving date
16353 forms. Error forms, modulo forms, and intervals can be multiplied;
16354 see the comments for addition of those forms. When two error forms
16355 or intervals are multiplied they are considered to be statistically
16356 independent; thus, @samp{[-2 ..@: 3] * [-2 ..@: 3]} is @samp{[-6 ..@: 9]},
16357 whereas @w{@samp{[-2 ..@: 3] ^ 2}} is @samp{[0 ..@: 9]}.
16360 @pindex calc-divide
16365 The @kbd{/} (@code{calc-divide}) command divides two numbers. When
16366 dividing a scalar @expr{B} by a square matrix @expr{A}, the computation
16367 performed is @expr{B} times the inverse of @expr{A}. This also occurs
16368 if @expr{B} is itself a vector or matrix, in which case the effect is
16369 to solve the set of linear equations represented by @expr{B}. If @expr{B}
16370 is a matrix with the same number of rows as @expr{A}, or a plain vector
16371 (which is interpreted here as a column vector), then the equation
16372 @expr{A X = B} is solved for the vector or matrix @expr{X}. Otherwise,
16373 if @expr{B} is a non-square matrix with the same number of @emph{columns}
16374 as @expr{A}, the equation @expr{X A = B} is solved. If you wish a vector
16375 @expr{B} to be interpreted as a row vector to be solved as @expr{X A = B},
16376 make it into a one-row matrix with @kbd{C-u 1 v p} first. To force a
16377 left-handed solution with a square matrix @expr{B}, transpose @expr{A} and
16378 @expr{B} before dividing, then transpose the result.
16380 HMS forms can be divided by real numbers or by other HMS forms. Error
16381 forms can be divided in any combination of ways. Modulo forms where both
16382 values and the modulo are integers can be divided to get an integer modulo
16383 form result. Intervals can be divided; dividing by an interval that
16384 encompasses zero or has zero as a limit will result in an infinite
16393 The @kbd{^} (@code{calc-power}) command raises a number to a power. If
16394 the power is an integer, an exact result is computed using repeated
16395 multiplications. For non-integer powers, Calc uses Newton's method or
16396 logarithms and exponentials. Square matrices can be raised to integer
16397 powers. If either argument is an error (or interval or modulo) form,
16398 the result is also an error (or interval or modulo) form.
16402 If you press the @kbd{I} (inverse) key first, the @kbd{I ^} command
16403 computes an Nth root: @kbd{125 @key{RET} 3 I ^} computes the number 5.
16404 (This is entirely equivalent to @kbd{125 @key{RET} 1:3 ^}.)
16413 The @kbd{\} (@code{calc-idiv}) command divides two numbers on the stack
16414 to produce an integer result. It is equivalent to dividing with
16415 @key{/}, then rounding down with @kbd{F} (@code{calc-floor}), only a bit
16416 more convenient and efficient. Also, since it is an all-integer
16417 operation when the arguments are integers, it avoids problems that
16418 @kbd{/ F} would have with floating-point roundoff.
16426 The @kbd{%} (@code{calc-mod}) command performs a ``modulo'' (or ``remainder'')
16427 operation. Mathematically, @samp{a%b = a - (a\b)*b}, and is defined
16428 for all real numbers @expr{a} and @expr{b} (except @expr{b=0}). For
16429 positive @expr{b}, the result will always be between 0 (inclusive) and
16430 @expr{b} (exclusive). Modulo does not work for HMS forms and error forms.
16431 If @expr{a} is a modulo form, its modulo is changed to @expr{b}, which
16432 must be positive real number.
16437 The @kbd{:} (@code{calc-fdiv}) [@code{fdiv}] command
16438 divides the two integers on the top of the stack to produce a fractional
16439 result. This is a convenient shorthand for enabling Fraction mode (with
16440 @kbd{m f}) temporarily and using @samp{/}. Note that during numeric entry
16441 the @kbd{:} key is interpreted as a fraction separator, so to divide 8 by 6
16442 you would have to type @kbd{8 @key{RET} 6 @key{RET} :}. (Of course, in
16443 this case, it would be much easier simply to enter the fraction directly
16444 as @kbd{8:6 @key{RET}}!)
16447 @pindex calc-change-sign
16448 The @kbd{n} (@code{calc-change-sign}) command negates the number on the top
16449 of the stack. It works on numbers, vectors and matrices, HMS forms, date
16450 forms, error forms, intervals, and modulo forms.
16455 The @kbd{A} (@code{calc-abs}) [@code{abs}] command computes the absolute
16456 value of a number. The result of @code{abs} is always a nonnegative
16457 real number: With a complex argument, it computes the complex magnitude.
16458 With a vector or matrix argument, it computes the Frobenius norm, i.e.,
16459 the square root of the sum of the squares of the absolute values of the
16460 elements. The absolute value of an error form is defined by replacing
16461 the mean part with its absolute value and leaving the error part the same.
16462 The absolute value of a modulo form is undefined. The absolute value of
16463 an interval is defined in the obvious way.
16466 @pindex calc-abssqr
16468 The @kbd{f A} (@code{calc-abssqr}) [@code{abssqr}] command computes the
16469 absolute value squared of a number, vector or matrix, or error form.
16474 The @kbd{f s} (@code{calc-sign}) [@code{sign}] command returns 1 if its
16475 argument is positive, @mathit{-1} if its argument is negative, or 0 if its
16476 argument is zero. In algebraic form, you can also write @samp{sign(a,x)}
16477 which evaluates to @samp{x * sign(a)}, i.e., either @samp{x}, @samp{-x}, or
16478 zero depending on the sign of @samp{a}.
16484 The @kbd{&} (@code{calc-inv}) [@code{inv}] command computes the
16485 reciprocal of a number, i.e., @expr{1 / x}. Operating on a square
16486 matrix, it computes the inverse of that matrix.
16491 The @kbd{Q} (@code{calc-sqrt}) [@code{sqrt}] command computes the square
16492 root of a number. For a negative real argument, the result will be a
16493 complex number whose form is determined by the current Polar mode.
16498 The @kbd{f h} (@code{calc-hypot}) [@code{hypot}] command computes the square
16499 root of the sum of the squares of two numbers. That is, @samp{hypot(a,b)}
16500 is the length of the hypotenuse of a right triangle with sides @expr{a}
16501 and @expr{b}. If the arguments are complex numbers, their squared
16502 magnitudes are used.
16507 The @kbd{f Q} (@code{calc-isqrt}) [@code{isqrt}] command computes the
16508 integer square root of an integer. This is the true square root of the
16509 number, rounded down to an integer. For example, @samp{isqrt(10)}
16510 produces 3. Note that, like @kbd{\} [@code{idiv}], this uses exact
16511 integer arithmetic throughout to avoid roundoff problems. If the input
16512 is a floating-point number or other non-integer value, this is exactly
16513 the same as @samp{floor(sqrt(x))}.
16521 The @kbd{f n} (@code{calc-min}) [@code{min}] and @kbd{f x} (@code{calc-max})
16522 [@code{max}] commands take the minimum or maximum of two real numbers,
16523 respectively. These commands also work on HMS forms, date forms,
16524 intervals, and infinities. (In algebraic expressions, these functions
16525 take any number of arguments and return the maximum or minimum among
16526 all the arguments.)
16530 @pindex calc-mant-part
16532 @pindex calc-xpon-part
16534 The @kbd{f M} (@code{calc-mant-part}) [@code{mant}] function extracts
16535 the ``mantissa'' part @expr{m} of its floating-point argument; @kbd{f X}
16536 (@code{calc-xpon-part}) [@code{xpon}] extracts the ``exponent'' part
16537 @expr{e}. The original number is equal to
16538 @texline @math{m \times 10^e},
16539 @infoline @expr{m * 10^e},
16540 where @expr{m} is in the interval @samp{[1.0 ..@: 10.0)} except that
16541 @expr{m=e=0} if the original number is zero. For integers
16542 and fractions, @code{mant} returns the number unchanged and @code{xpon}
16543 returns zero. The @kbd{v u} (@code{calc-unpack}) command can also be
16544 used to ``unpack'' a floating-point number; this produces an integer
16545 mantissa and exponent, with the constraint that the mantissa is not
16546 a multiple of ten (again except for the @expr{m=e=0} case).
16549 @pindex calc-scale-float
16551 The @kbd{f S} (@code{calc-scale-float}) [@code{scf}] function scales a number
16552 by a given power of ten. Thus, @samp{scf(mant(x), xpon(x)) = x} for any
16553 real @samp{x}. The second argument must be an integer, but the first
16554 may actually be any numeric value. For example, @samp{scf(5,-2) = 0.05}
16555 or @samp{1:20} depending on the current Fraction mode.
16559 @pindex calc-decrement
16560 @pindex calc-increment
16563 The @kbd{f [} (@code{calc-decrement}) [@code{decr}] and @kbd{f ]}
16564 (@code{calc-increment}) [@code{incr}] functions decrease or increase
16565 a number by one unit. For integers, the effect is obvious. For
16566 floating-point numbers, the change is by one unit in the last place.
16567 For example, incrementing @samp{12.3456} when the current precision
16568 is 6 digits yields @samp{12.3457}. If the current precision had been
16569 8 digits, the result would have been @samp{12.345601}. Incrementing
16570 @samp{0.0} produces
16571 @texline @math{10^{-p}},
16572 @infoline @expr{10^-p},
16573 where @expr{p} is the current
16574 precision. These operations are defined only on integers and floats.
16575 With numeric prefix arguments, they change the number by @expr{n} units.
16577 Note that incrementing followed by decrementing, or vice-versa, will
16578 almost but not quite always cancel out. Suppose the precision is
16579 6 digits and the number @samp{9.99999} is on the stack. Incrementing
16580 will produce @samp{10.0000}; decrementing will produce @samp{9.9999}.
16581 One digit has been dropped. This is an unavoidable consequence of the
16582 way floating-point numbers work.
16584 Incrementing a date/time form adjusts it by a certain number of seconds.
16585 Incrementing a pure date form adjusts it by a certain number of days.
16587 @node Integer Truncation, Complex Number Functions, Basic Arithmetic, Arithmetic
16588 @section Integer Truncation
16591 There are four commands for truncating a real number to an integer,
16592 differing mainly in their treatment of negative numbers. All of these
16593 commands have the property that if the argument is an integer, the result
16594 is the same integer. An integer-valued floating-point argument is converted
16597 If you press @kbd{H} (@code{calc-hyperbolic}) first, the result will be
16598 expressed as an integer-valued floating-point number.
16600 @cindex Integer part of a number
16609 The @kbd{F} (@code{calc-floor}) [@code{floor} or @code{ffloor}] command
16610 truncates a real number to the next lower integer, i.e., toward minus
16611 infinity. Thus @kbd{3.6 F} produces 3, but @kbd{_3.6 F} produces
16615 @pindex calc-ceiling
16622 The @kbd{I F} (@code{calc-ceiling}) [@code{ceil} or @code{fceil}]
16623 command truncates toward positive infinity. Thus @kbd{3.6 I F} produces
16624 4, and @kbd{_3.6 I F} produces @mathit{-3}.
16634 The @kbd{R} (@code{calc-round}) [@code{round} or @code{fround}] command
16635 rounds to the nearest integer. When the fractional part is .5 exactly,
16636 this command rounds away from zero. (All other rounding in the
16637 Calculator uses this convention as well.) Thus @kbd{3.5 R} produces 4
16638 but @kbd{3.4 R} produces 3; @kbd{_3.5 R} produces @mathit{-4}.
16648 The @kbd{I R} (@code{calc-trunc}) [@code{trunc} or @code{ftrunc}]
16649 command truncates toward zero. In other words, it ``chops off''
16650 everything after the decimal point. Thus @kbd{3.6 I R} produces 3 and
16651 @kbd{_3.6 I R} produces @mathit{-3}.
16653 These functions may not be applied meaningfully to error forms, but they
16654 do work for intervals. As a convenience, applying @code{floor} to a
16655 modulo form floors the value part of the form. Applied to a vector,
16656 these functions operate on all elements of the vector one by one.
16657 Applied to a date form, they operate on the internal numerical
16658 representation of dates, converting a date/time form into a pure date.
16676 There are two more rounding functions which can only be entered in
16677 algebraic notation. The @code{roundu} function is like @code{round}
16678 except that it rounds up, toward plus infinity, when the fractional
16679 part is .5. This distinction matters only for negative arguments.
16680 Also, @code{rounde} rounds to an even number in the case of a tie,
16681 rounding up or down as necessary. For example, @samp{rounde(3.5)} and
16682 @samp{rounde(4.5)} both return 4, but @samp{rounde(5.5)} returns 6.
16683 The advantage of round-to-even is that the net error due to rounding
16684 after a long calculation tends to cancel out to zero. An important
16685 subtle point here is that the number being fed to @code{rounde} will
16686 already have been rounded to the current precision before @code{rounde}
16687 begins. For example, @samp{rounde(2.500001)} with a current precision
16688 of 6 will incorrectly, or at least surprisingly, yield 2 because the
16689 argument will first have been rounded down to @expr{2.5} (which
16690 @code{rounde} sees as an exact tie between 2 and 3).
16692 Each of these functions, when written in algebraic formulas, allows
16693 a second argument which specifies the number of digits after the
16694 decimal point to keep. For example, @samp{round(123.4567, 2)} will
16695 produce the answer 123.46, and @samp{round(123.4567, -1)} will
16696 produce 120 (i.e., the cutoff is one digit to the @emph{left} of
16697 the decimal point). A second argument of zero is equivalent to
16698 no second argument at all.
16700 @cindex Fractional part of a number
16701 To compute the fractional part of a number (i.e., the amount which, when
16702 added to `@tfn{floor(}@var{n}@tfn{)}', will produce @var{n}) just take @var{n}
16703 modulo 1 using the @code{%} command.
16705 Note also the @kbd{\} (integer quotient), @kbd{f I} (integer logarithm),
16706 and @kbd{f Q} (integer square root) commands, which are analogous to
16707 @kbd{/}, @kbd{B}, and @kbd{Q}, respectively, except that they take integer
16708 arguments and return the result rounded down to an integer.
16710 @node Complex Number Functions, Conversions, Integer Truncation, Arithmetic
16711 @section Complex Number Functions
16717 The @kbd{J} (@code{calc-conj}) [@code{conj}] command computes the
16718 complex conjugate of a number. For complex number @expr{a+bi}, the
16719 complex conjugate is @expr{a-bi}. If the argument is a real number,
16720 this command leaves it the same. If the argument is a vector or matrix,
16721 this command replaces each element by its complex conjugate.
16724 @pindex calc-argument
16726 The @kbd{G} (@code{calc-argument}) [@code{arg}] command computes the
16727 ``argument'' or polar angle of a complex number. For a number in polar
16728 notation, this is simply the second component of the pair
16729 @texline `@tfn{(}@var{r}@tfn{;}@math{\theta}@tfn{)}'.
16730 @infoline `@tfn{(}@var{r}@tfn{;}@var{theta}@tfn{)}'.
16731 The result is expressed according to the current angular mode and will
16732 be in the range @mathit{-180} degrees (exclusive) to @mathit{+180} degrees
16733 (inclusive), or the equivalent range in radians.
16735 @pindex calc-imaginary
16736 The @code{calc-imaginary} command multiplies the number on the
16737 top of the stack by the imaginary number @expr{i = (0,1)}. This
16738 command is not normally bound to a key in Calc, but it is available
16739 on the @key{IMAG} button in Keypad mode.
16744 The @kbd{f r} (@code{calc-re}) [@code{re}] command replaces a complex number
16745 by its real part. This command has no effect on real numbers. (As an
16746 added convenience, @code{re} applied to a modulo form extracts
16752 The @kbd{f i} (@code{calc-im}) [@code{im}] command replaces a complex number
16753 by its imaginary part; real numbers are converted to zero. With a vector
16754 or matrix argument, these functions operate element-wise.
16759 @kindex v p (complex)
16761 The @kbd{v p} (@code{calc-pack}) command can pack the top two numbers on
16762 the stack into a composite object such as a complex number. With
16763 a prefix argument of @mathit{-1}, it produces a rectangular complex number;
16764 with an argument of @mathit{-2}, it produces a polar complex number.
16765 (Also, @pxref{Building Vectors}.)
16770 @kindex v u (complex)
16771 @pindex calc-unpack
16772 The @kbd{v u} (@code{calc-unpack}) command takes the complex number
16773 (or other composite object) on the top of the stack and unpacks it
16774 into its separate components.
16776 @node Conversions, Date Arithmetic, Complex Number Functions, Arithmetic
16777 @section Conversions
16780 The commands described in this section convert numbers from one form
16781 to another; they are two-key sequences beginning with the letter @kbd{c}.
16786 The @kbd{c f} (@code{calc-float}) [@code{pfloat}] command converts the
16787 number on the top of the stack to floating-point form. For example,
16788 @expr{23} is converted to @expr{23.0}, @expr{3:2} is converted to
16789 @expr{1.5}, and @expr{2.3} is left the same. If the value is a composite
16790 object such as a complex number or vector, each of the components is
16791 converted to floating-point. If the value is a formula, all numbers
16792 in the formula are converted to floating-point. Note that depending
16793 on the current floating-point precision, conversion to floating-point
16794 format may lose information.
16796 As a special exception, integers which appear as powers or subscripts
16797 are not floated by @kbd{c f}. If you really want to float a power,
16798 you can use a @kbd{j s} command to select the power followed by @kbd{c f}.
16799 Because @kbd{c f} cannot examine the formula outside of the selection,
16800 it does not notice that the thing being floated is a power.
16801 @xref{Selecting Subformulas}.
16803 The normal @kbd{c f} command is ``pervasive'' in the sense that it
16804 applies to all numbers throughout the formula. The @code{pfloat}
16805 algebraic function never stays around in a formula; @samp{pfloat(a + 1)}
16806 changes to @samp{a + 1.0} as soon as it is evaluated.
16810 With the Hyperbolic flag, @kbd{H c f} [@code{float}] operates
16811 only on the number or vector of numbers at the top level of its
16812 argument. Thus, @samp{float(1)} is 1.0, but @samp{float(a + 1)}
16813 is left unevaluated because its argument is not a number.
16815 You should use @kbd{H c f} if you wish to guarantee that the final
16816 value, once all the variables have been assigned, is a float; you
16817 would use @kbd{c f} if you wish to do the conversion on the numbers
16818 that appear right now.
16821 @pindex calc-fraction
16823 The @kbd{c F} (@code{calc-fraction}) [@code{pfrac}] command converts a
16824 floating-point number into a fractional approximation. By default, it
16825 produces a fraction whose decimal representation is the same as the
16826 input number, to within the current precision. You can also give a
16827 numeric prefix argument to specify a tolerance, either directly, or,
16828 if the prefix argument is zero, by using the number on top of the stack
16829 as the tolerance. If the tolerance is a positive integer, the fraction
16830 is correct to within that many significant figures. If the tolerance is
16831 a non-positive integer, it specifies how many digits fewer than the current
16832 precision to use. If the tolerance is a floating-point number, the
16833 fraction is correct to within that absolute amount.
16837 The @code{pfrac} function is pervasive, like @code{pfloat}.
16838 There is also a non-pervasive version, @kbd{H c F} [@code{frac}],
16839 which is analogous to @kbd{H c f} discussed above.
16842 @pindex calc-to-degrees
16844 The @kbd{c d} (@code{calc-to-degrees}) [@code{deg}] command converts a
16845 number into degrees form. The value on the top of the stack may be an
16846 HMS form (interpreted as degrees-minutes-seconds), or a real number which
16847 will be interpreted in radians regardless of the current angular mode.
16850 @pindex calc-to-radians
16852 The @kbd{c r} (@code{calc-to-radians}) [@code{rad}] command converts an
16853 HMS form or angle in degrees into an angle in radians.
16856 @pindex calc-to-hms
16858 The @kbd{c h} (@code{calc-to-hms}) [@code{hms}] command converts a real
16859 number, interpreted according to the current angular mode, to an HMS
16860 form describing the same angle. In algebraic notation, the @code{hms}
16861 function also accepts three arguments: @samp{hms(@var{h}, @var{m}, @var{s})}.
16862 (The three-argument version is independent of the current angular mode.)
16864 @pindex calc-from-hms
16865 The @code{calc-from-hms} command converts the HMS form on the top of the
16866 stack into a real number according to the current angular mode.
16873 The @kbd{c p} (@code{calc-polar}) command converts the complex number on
16874 the top of the stack from polar to rectangular form, or from rectangular
16875 to polar form, whichever is appropriate. Real numbers are left the same.
16876 This command is equivalent to the @code{rect} or @code{polar}
16877 functions in algebraic formulas, depending on the direction of
16878 conversion. (It uses @code{polar}, except that if the argument is
16879 already a polar complex number, it uses @code{rect} instead. The
16880 @kbd{I c p} command always uses @code{rect}.)
16885 The @kbd{c c} (@code{calc-clean}) [@code{pclean}] command ``cleans'' the
16886 number on the top of the stack. Floating point numbers are re-rounded
16887 according to the current precision. Polar numbers whose angular
16888 components have strayed from the @mathit{-180} to @mathit{+180} degree range
16889 are normalized. (Note that results will be undesirable if the current
16890 angular mode is different from the one under which the number was
16891 produced!) Integers and fractions are generally unaffected by this
16892 operation. Vectors and formulas are cleaned by cleaning each component
16893 number (i.e., pervasively).
16895 If the simplification mode is set below the default level, it is raised
16896 to the default level for the purposes of this command. Thus, @kbd{c c}
16897 applies the default simplifications even if their automatic application
16898 is disabled. @xref{Simplification Modes}.
16900 @cindex Roundoff errors, correcting
16901 A numeric prefix argument to @kbd{c c} sets the floating-point precision
16902 to that value for the duration of the command. A positive prefix (of at
16903 least 3) sets the precision to the specified value; a negative or zero
16904 prefix decreases the precision by the specified amount.
16907 @pindex calc-clean-num
16908 The keystroke sequences @kbd{c 0} through @kbd{c 9} are equivalent
16909 to @kbd{c c} with the corresponding negative prefix argument. If roundoff
16910 errors have changed 2.0 into 1.999999, typing @kbd{c 1} to clip off one
16911 decimal place often conveniently does the trick.
16913 The @kbd{c c} command with a numeric prefix argument, and the @kbd{c 0}
16914 through @kbd{c 9} commands, also ``clip'' very small floating-point
16915 numbers to zero. If the exponent is less than or equal to the negative
16916 of the specified precision, the number is changed to 0.0. For example,
16917 if the current precision is 12, then @kbd{c 2} changes the vector
16918 @samp{[1e-8, 1e-9, 1e-10, 1e-11]} to @samp{[1e-8, 1e-9, 0, 0]}.
16919 Numbers this small generally arise from roundoff noise.
16921 If the numbers you are using really are legitimately this small,
16922 you should avoid using the @kbd{c 0} through @kbd{c 9} commands.
16923 (The plain @kbd{c c} command rounds to the current precision but
16924 does not clip small numbers.)
16926 One more property of @kbd{c 0} through @kbd{c 9}, and of @kbd{c c} with
16927 a prefix argument, is that integer-valued floats are converted to
16928 plain integers, so that @kbd{c 1} on @samp{[1., 1.5, 2., 2.5, 3.]}
16929 produces @samp{[1, 1.5, 2, 2.5, 3]}. This is not done for huge
16930 numbers (@samp{1e100} is technically an integer-valued float, but
16931 you wouldn't want it automatically converted to a 100-digit integer).
16936 With the Hyperbolic flag, @kbd{H c c} and @kbd{H c 0} through @kbd{H c 9}
16937 operate non-pervasively [@code{clean}].
16939 @node Date Arithmetic, Financial Functions, Conversions, Arithmetic
16940 @section Date Arithmetic
16943 @cindex Date arithmetic, additional functions
16944 The commands described in this section perform various conversions
16945 and calculations involving date forms (@pxref{Date Forms}). They
16946 use the @kbd{t} (for time/date) prefix key followed by shifted
16949 The simplest date arithmetic is done using the regular @kbd{+} and @kbd{-}
16950 commands. In particular, adding a number to a date form advances the
16951 date form by a certain number of days; adding an HMS form to a date
16952 form advances the date by a certain amount of time; and subtracting two
16953 date forms produces a difference measured in days. The commands
16954 described here provide additional, more specialized operations on dates.
16956 Many of these commands accept a numeric prefix argument; if you give
16957 plain @kbd{C-u} as the prefix, these commands will instead take the
16958 additional argument from the top of the stack.
16961 * Date Conversions::
16967 @node Date Conversions, Date Functions, Date Arithmetic, Date Arithmetic
16968 @subsection Date Conversions
16974 The @kbd{t D} (@code{calc-date}) [@code{date}] command converts a
16975 date form into a number, measured in days since Jan 1, 1 AD. The
16976 result will be an integer if @var{date} is a pure date form, or a
16977 fraction or float if @var{date} is a date/time form. Or, if its
16978 argument is a number, it converts this number into a date form.
16980 With a numeric prefix argument, @kbd{t D} takes that many objects
16981 (up to six) from the top of the stack and interprets them in one
16982 of the following ways:
16984 The @samp{date(@var{year}, @var{month}, @var{day})} function
16985 builds a pure date form out of the specified year, month, and
16986 day, which must all be integers. @var{Year} is a year number,
16987 such as 1991 (@emph{not} the same as 91!). @var{Month} must be
16988 an integer in the range 1 to 12; @var{day} must be in the range
16989 1 to 31. If the specified month has fewer than 31 days and
16990 @var{day} is too large, the equivalent day in the following
16991 month will be used.
16993 The @samp{date(@var{month}, @var{day})} function builds a
16994 pure date form using the current year, as determined by the
16997 The @samp{date(@var{year}, @var{month}, @var{day}, @var{hms})}
16998 function builds a date/time form using an @var{hms} form.
17000 The @samp{date(@var{year}, @var{month}, @var{day}, @var{hour},
17001 @var{minute}, @var{second})} function builds a date/time form.
17002 @var{hour} should be an integer in the range 0 to 23;
17003 @var{minute} should be an integer in the range 0 to 59;
17004 @var{second} should be any real number in the range @samp{[0 .. 60)}.
17005 The last two arguments default to zero if omitted.
17008 @pindex calc-julian
17010 @cindex Julian day counts, conversions
17011 The @kbd{t J} (@code{calc-julian}) [@code{julian}] command converts
17012 a date form into a Julian day count, which is the number of days
17013 since noon on Jan 1, 4713 BC. A pure date is converted to an integer
17014 Julian count representing noon of that day. A date/time form is
17015 converted to an exact floating-point Julian count, adjusted to
17016 interpret the date form in the current time zone but the Julian
17017 day count in Greenwich Mean Time. A numeric prefix argument allows
17018 you to specify the time zone; @pxref{Time Zones}. Use a prefix of
17019 zero to suppress the time zone adjustment. Note that pure date forms
17020 are never time-zone adjusted.
17022 This command can also do the opposite conversion, from a Julian day
17023 count (either an integer day, or a floating-point day and time in
17024 the GMT zone), into a pure date form or a date/time form in the
17025 current or specified time zone.
17028 @pindex calc-unix-time
17030 @cindex Unix time format, conversions
17031 The @kbd{t U} (@code{calc-unix-time}) [@code{unixtime}] command
17032 converts a date form into a Unix time value, which is the number of
17033 seconds since midnight on Jan 1, 1970, or vice-versa. The numeric result
17034 will be an integer if the current precision is 12 or less; for higher
17035 precisions, the result may be a float with (@var{precision}@minus{}12)
17036 digits after the decimal. Just as for @kbd{t J}, the numeric time
17037 is interpreted in the GMT time zone and the date form is interpreted
17038 in the current or specified zone. Some systems use Unix-like
17039 numbering but with the local time zone; give a prefix of zero to
17040 suppress the adjustment if so.
17043 @pindex calc-convert-time-zones
17045 @cindex Time Zones, converting between
17046 The @kbd{t C} (@code{calc-convert-time-zones}) [@code{tzconv}]
17047 command converts a date form from one time zone to another. You
17048 are prompted for each time zone name in turn; you can answer with
17049 any suitable Calc time zone expression (@pxref{Time Zones}).
17050 If you answer either prompt with a blank line, the local time
17051 zone is used for that prompt. You can also answer the first
17052 prompt with @kbd{$} to take the two time zone names from the
17053 stack (and the date to be converted from the third stack level).
17055 @node Date Functions, Business Days, Date Conversions, Date Arithmetic
17056 @subsection Date Functions
17062 The @kbd{t N} (@code{calc-now}) [@code{now}] command pushes the
17063 current date and time on the stack as a date form. The time is
17064 reported in terms of the specified time zone; with no numeric prefix
17065 argument, @kbd{t N} reports for the current time zone.
17068 @pindex calc-date-part
17069 The @kbd{t P} (@code{calc-date-part}) command extracts one part
17070 of a date form. The prefix argument specifies the part; with no
17071 argument, this command prompts for a part code from 1 to 9.
17072 The various part codes are described in the following paragraphs.
17075 The @kbd{M-1 t P} [@code{year}] function extracts the year number
17076 from a date form as an integer, e.g., 1991. This and the
17077 following functions will also accept a real number for an
17078 argument, which is interpreted as a standard Calc day number.
17079 Note that this function will never return zero, since the year
17080 1 BC immediately precedes the year 1 AD.
17083 The @kbd{M-2 t P} [@code{month}] function extracts the month number
17084 from a date form as an integer in the range 1 to 12.
17087 The @kbd{M-3 t P} [@code{day}] function extracts the day number
17088 from a date form as an integer in the range 1 to 31.
17091 The @kbd{M-4 t P} [@code{hour}] function extracts the hour from
17092 a date form as an integer in the range 0 (midnight) to 23. Note
17093 that 24-hour time is always used. This returns zero for a pure
17094 date form. This function (and the following two) also accept
17095 HMS forms as input.
17098 The @kbd{M-5 t P} [@code{minute}] function extracts the minute
17099 from a date form as an integer in the range 0 to 59.
17102 The @kbd{M-6 t P} [@code{second}] function extracts the second
17103 from a date form. If the current precision is 12 or less,
17104 the result is an integer in the range 0 to 59. For higher
17105 precisions, the result may instead be a floating-point number.
17108 The @kbd{M-7 t P} [@code{weekday}] function extracts the weekday
17109 number from a date form as an integer in the range 0 (Sunday)
17113 The @kbd{M-8 t P} [@code{yearday}] function extracts the day-of-year
17114 number from a date form as an integer in the range 1 (January 1)
17115 to 366 (December 31 of a leap year).
17118 The @kbd{M-9 t P} [@code{time}] function extracts the time portion
17119 of a date form as an HMS form. This returns @samp{0@@ 0' 0"}
17120 for a pure date form.
17123 @pindex calc-new-month
17125 The @kbd{t M} (@code{calc-new-month}) [@code{newmonth}] command
17126 computes a new date form that represents the first day of the month
17127 specified by the input date. The result is always a pure date
17128 form; only the year and month numbers of the input are retained.
17129 With a numeric prefix argument @var{n} in the range from 1 to 31,
17130 @kbd{t M} computes the @var{n}th day of the month. (If @var{n}
17131 is greater than the actual number of days in the month, or if
17132 @var{n} is zero, the last day of the month is used.)
17135 @pindex calc-new-year
17137 The @kbd{t Y} (@code{calc-new-year}) [@code{newyear}] command
17138 computes a new pure date form that represents the first day of
17139 the year specified by the input. The month, day, and time
17140 of the input date form are lost. With a numeric prefix argument
17141 @var{n} in the range from 1 to 366, @kbd{t Y} computes the
17142 @var{n}th day of the year (366 is treated as 365 in non-leap
17143 years). A prefix argument of 0 computes the last day of the
17144 year (December 31). A negative prefix argument from @mathit{-1} to
17145 @mathit{-12} computes the first day of the @var{n}th month of the year.
17148 @pindex calc-new-week
17150 The @kbd{t W} (@code{calc-new-week}) [@code{newweek}] command
17151 computes a new pure date form that represents the Sunday on or before
17152 the input date. With a numeric prefix argument, it can be made to
17153 use any day of the week as the starting day; the argument must be in
17154 the range from 0 (Sunday) to 6 (Saturday). This function always
17155 subtracts between 0 and 6 days from the input date.
17157 Here's an example use of @code{newweek}: Find the date of the next
17158 Wednesday after a given date. Using @kbd{M-3 t W} or @samp{newweek(d, 3)}
17159 will give you the @emph{preceding} Wednesday, so @samp{newweek(d+7, 3)}
17160 will give you the following Wednesday. A further look at the definition
17161 of @code{newweek} shows that if the input date is itself a Wednesday,
17162 this formula will return the Wednesday one week in the future. An
17163 exercise for the reader is to modify this formula to yield the same day
17164 if the input is already a Wednesday. Another interesting exercise is
17165 to preserve the time-of-day portion of the input (@code{newweek} resets
17166 the time to midnight; hint:@: how can @code{newweek} be defined in terms
17167 of the @code{weekday} function?).
17173 The @samp{pwday(@var{date})} function (not on any key) computes the
17174 day-of-month number of the Sunday on or before @var{date}. With
17175 two arguments, @samp{pwday(@var{date}, @var{day})} computes the day
17176 number of the Sunday on or before day number @var{day} of the month
17177 specified by @var{date}. The @var{day} must be in the range from
17178 7 to 31; if the day number is greater than the actual number of days
17179 in the month, the true number of days is used instead. Thus
17180 @samp{pwday(@var{date}, 7)} finds the first Sunday of the month, and
17181 @samp{pwday(@var{date}, 31)} finds the last Sunday of the month.
17182 With a third @var{weekday} argument, @code{pwday} can be made to look
17183 for any day of the week instead of Sunday.
17186 @pindex calc-inc-month
17188 The @kbd{t I} (@code{calc-inc-month}) [@code{incmonth}] command
17189 increases a date form by one month, or by an arbitrary number of
17190 months specified by a numeric prefix argument. The time portion,
17191 if any, of the date form stays the same. The day also stays the
17192 same, except that if the new month has fewer days the day
17193 number may be reduced to lie in the valid range. For example,
17194 @samp{incmonth(<Jan 31, 1991>)} produces @samp{<Feb 28, 1991>}.
17195 Because of this, @kbd{t I t I} and @kbd{M-2 t I} do not always give
17196 the same results (@samp{<Mar 28, 1991>} versus @samp{<Mar 31, 1991>}
17203 The @samp{incyear(@var{date}, @var{step})} function increases
17204 a date form by the specified number of years, which may be
17205 any positive or negative integer. Note that @samp{incyear(d, n)}
17206 is equivalent to @w{@samp{incmonth(d, 12*n)}}, but these do not have
17207 simple equivalents in terms of day arithmetic because
17208 months and years have varying lengths. If the @var{step}
17209 argument is omitted, 1 year is assumed. There is no keyboard
17210 command for this function; use @kbd{C-u 12 t I} instead.
17212 There is no @code{newday} function at all because @kbd{F} [@code{floor}]
17213 serves this purpose. Similarly, instead of @code{incday} and
17214 @code{incweek} simply use @expr{d + n} or @expr{d + 7 n}.
17216 @xref{Basic Arithmetic}, for the @kbd{f ]} [@code{incr}] command
17217 which can adjust a date/time form by a certain number of seconds.
17219 @node Business Days, Time Zones, Date Functions, Date Arithmetic
17220 @subsection Business Days
17223 Often time is measured in ``business days'' or ``working days,''
17224 where weekends and holidays are skipped. Calc's normal date
17225 arithmetic functions use calendar days, so that subtracting two
17226 consecutive Mondays will yield a difference of 7 days. By contrast,
17227 subtracting two consecutive Mondays would yield 5 business days
17228 (assuming two-day weekends and the absence of holidays).
17234 @pindex calc-business-days-plus
17235 @pindex calc-business-days-minus
17236 The @kbd{t +} (@code{calc-business-days-plus}) [@code{badd}]
17237 and @kbd{t -} (@code{calc-business-days-minus}) [@code{bsub}]
17238 commands perform arithmetic using business days. For @kbd{t +},
17239 one argument must be a date form and the other must be a real
17240 number (positive or negative). If the number is not an integer,
17241 then a certain amount of time is added as well as a number of
17242 days; for example, adding 0.5 business days to a time in Friday
17243 evening will produce a time in Monday morning. It is also
17244 possible to add an HMS form; adding @samp{12@@ 0' 0"} also adds
17245 half a business day. For @kbd{t -}, the arguments are either a
17246 date form and a number or HMS form, or two date forms, in which
17247 case the result is the number of business days between the two
17250 @cindex @code{Holidays} variable
17252 By default, Calc considers any day that is not a Saturday or
17253 Sunday to be a business day. You can define any number of
17254 additional holidays by editing the variable @code{Holidays}.
17255 (There is an @w{@kbd{s H}} convenience command for editing this
17256 variable.) Initially, @code{Holidays} contains the vector
17257 @samp{[sat, sun]}. Entries in the @code{Holidays} vector may
17258 be any of the following kinds of objects:
17262 Date forms (pure dates, not date/time forms). These specify
17263 particular days which are to be treated as holidays.
17266 Intervals of date forms. These specify a range of days, all of
17267 which are holidays (e.g., Christmas week). @xref{Interval Forms}.
17270 Nested vectors of date forms. Each date form in the vector is
17271 considered to be a holiday.
17274 Any Calc formula which evaluates to one of the above three things.
17275 If the formula involves the variable @expr{y}, it stands for a
17276 yearly repeating holiday; @expr{y} will take on various year
17277 numbers like 1992. For example, @samp{date(y, 12, 25)} specifies
17278 Christmas day, and @samp{newweek(date(y, 11, 7), 4) + 21} specifies
17279 Thanksgiving (which is held on the fourth Thursday of November).
17280 If the formula involves the variable @expr{m}, that variable
17281 takes on month numbers from 1 to 12: @samp{date(y, m, 15)} is
17282 a holiday that takes place on the 15th of every month.
17285 A weekday name, such as @code{sat} or @code{sun}. This is really
17286 a variable whose name is a three-letter, lower-case day name.
17289 An interval of year numbers (integers). This specifies the span of
17290 years over which this holiday list is to be considered valid. Any
17291 business-day arithmetic that goes outside this range will result
17292 in an error message. Use this if you are including an explicit
17293 list of holidays, rather than a formula to generate them, and you
17294 want to make sure you don't accidentally go beyond the last point
17295 where the holidays you entered are complete. If there is no
17296 limiting interval in the @code{Holidays} vector, the default
17297 @samp{[1 .. 2737]} is used. (This is the absolute range of years
17298 for which Calc's business-day algorithms will operate.)
17301 An interval of HMS forms. This specifies the span of hours that
17302 are to be considered one business day. For example, if this
17303 range is @samp{[9@@ 0' 0" .. 17@@ 0' 0"]} (i.e., 9am to 5pm), then
17304 the business day is only eight hours long, so that @kbd{1.5 t +}
17305 on @samp{<4:00pm Fri Dec 13, 1991>} will add one business day and
17306 four business hours to produce @samp{<12:00pm Tue Dec 17, 1991>}.
17307 Likewise, @kbd{t -} will now express differences in time as
17308 fractions of an eight-hour day. Times before 9am will be treated
17309 as 9am by business date arithmetic, and times at or after 5pm will
17310 be treated as 4:59:59pm. If there is no HMS interval in @code{Holidays},
17311 the full 24-hour day @samp{[0@ 0' 0" .. 24@ 0' 0"]} is assumed.
17312 (Regardless of the type of bounds you specify, the interval is
17313 treated as inclusive on the low end and exclusive on the high end,
17314 so that the work day goes from 9am up to, but not including, 5pm.)
17317 If the @code{Holidays} vector is empty, then @kbd{t +} and
17318 @kbd{t -} will act just like @kbd{+} and @kbd{-} because there will
17319 then be no difference between business days and calendar days.
17321 Calc expands the intervals and formulas you give into a complete
17322 list of holidays for internal use. This is done mainly to make
17323 sure it can detect multiple holidays. (For example,
17324 @samp{<Jan 1, 1989>} is both New Year's Day and a Sunday, but
17325 Calc's algorithms take care to count it only once when figuring
17326 the number of holidays between two dates.)
17328 Since the complete list of holidays for all the years from 1 to
17329 2737 would be huge, Calc actually computes only the part of the
17330 list between the smallest and largest years that have been involved
17331 in business-day calculations so far. Normally, you won't have to
17332 worry about this. Keep in mind, however, that if you do one
17333 calculation for 1992, and another for 1792, even if both involve
17334 only a small range of years, Calc will still work out all the
17335 holidays that fall in that 200-year span.
17337 If you add a (positive) number of days to a date form that falls on a
17338 weekend or holiday, the date form is treated as if it were the most
17339 recent business day. (Thus adding one business day to a Friday,
17340 Saturday, or Sunday will all yield the following Monday.) If you
17341 subtract a number of days from a weekend or holiday, the date is
17342 effectively on the following business day. (So subtracting one business
17343 day from Saturday, Sunday, or Monday yields the preceding Friday.) The
17344 difference between two dates one or both of which fall on holidays
17345 equals the number of actual business days between them. These
17346 conventions are consistent in the sense that, if you add @var{n}
17347 business days to any date, the difference between the result and the
17348 original date will come out to @var{n} business days. (It can't be
17349 completely consistent though; a subtraction followed by an addition
17350 might come out a bit differently, since @kbd{t +} is incapable of
17351 producing a date that falls on a weekend or holiday.)
17357 There is a @code{holiday} function, not on any keys, that takes
17358 any date form and returns 1 if that date falls on a weekend or
17359 holiday, as defined in @code{Holidays}, or 0 if the date is a
17362 @node Time Zones, , Business Days, Date Arithmetic
17363 @subsection Time Zones
17367 @cindex Daylight savings time
17368 Time zones and daylight savings time are a complicated business.
17369 The conversions to and from Julian and Unix-style dates automatically
17370 compute the correct time zone and daylight savings adjustment to use,
17371 provided they can figure out this information. This section describes
17372 Calc's time zone adjustment algorithm in detail, in case you want to
17373 do conversions in different time zones or in case Calc's algorithms
17374 can't determine the right correction to use.
17376 Adjustments for time zones and daylight savings time are done by
17377 @kbd{t U}, @kbd{t J}, @kbd{t N}, and @kbd{t C}, but not by any other
17378 commands. In particular, @samp{<may 1 1991> - <apr 1 1991>} evaluates
17379 to exactly 30 days even though there is a daylight-savings
17380 transition in between. This is also true for Julian pure dates:
17381 @samp{julian(<may 1 1991>) - julian(<apr 1 1991>)}. But Julian
17382 and Unix date/times will adjust for daylight savings time:
17383 @samp{julian(<12am may 1 1991>) - julian(<12am apr 1 1991>)}
17384 evaluates to @samp{29.95834} (that's 29 days and 23 hours)
17385 because one hour was lost when daylight savings commenced on
17388 In brief, the idiom @samp{julian(@var{date1}) - julian(@var{date2})}
17389 computes the actual number of 24-hour periods between two dates, whereas
17390 @samp{@var{date1} - @var{date2}} computes the number of calendar
17391 days between two dates without taking daylight savings into account.
17393 @pindex calc-time-zone
17398 The @code{calc-time-zone} [@code{tzone}] command converts the time
17399 zone specified by its numeric prefix argument into a number of
17400 seconds difference from Greenwich mean time (GMT). If the argument
17401 is a number, the result is simply that value multiplied by 3600.
17402 Typical arguments for North America are 5 (Eastern) or 8 (Pacific). If
17403 Daylight Savings time is in effect, one hour should be subtracted from
17404 the normal difference.
17406 If you give a prefix of plain @kbd{C-u}, @code{calc-time-zone} (like other
17407 date arithmetic commands that include a time zone argument) takes the
17408 zone argument from the top of the stack. (In the case of @kbd{t J}
17409 and @kbd{t U}, the normal argument is then taken from the second-to-top
17410 stack position.) This allows you to give a non-integer time zone
17411 adjustment. The time-zone argument can also be an HMS form, or
17412 it can be a variable which is a time zone name in upper- or lower-case.
17413 For example @samp{tzone(PST) = tzone(8)} and @samp{tzone(pdt) = tzone(7)}
17414 (for Pacific standard and daylight savings times, respectively).
17416 North American and European time zone names are defined as follows;
17417 note that for each time zone there is one name for standard time,
17418 another for daylight savings time, and a third for ``generalized'' time
17419 in which the daylight savings adjustment is computed from context.
17423 YST PST MST CST EST AST NST GMT WET MET MEZ
17424 9 8 7 6 5 4 3.5 0 -1 -2 -2
17426 YDT PDT MDT CDT EDT ADT NDT BST WETDST METDST MESZ
17427 8 7 6 5 4 3 2.5 -1 -2 -3 -3
17429 YGT PGT MGT CGT EGT AGT NGT BGT WEGT MEGT MEGZ
17430 9/8 8/7 7/6 6/5 5/4 4/3 3.5/2.5 0/-1 -1/-2 -2/-3 -2/-3
17434 @vindex math-tzone-names
17435 To define time zone names that do not appear in the above table,
17436 you must modify the Lisp variable @code{math-tzone-names}. This
17437 is a list of lists describing the different time zone names; its
17438 structure is best explained by an example. The three entries for
17439 Pacific Time look like this:
17443 ( ( "PST" 8 0 ) ; Name as an upper-case string, then standard
17444 ( "PDT" 8 -1 ) ; adjustment, then daylight savings adjustment.
17445 ( "PGT" 8 "PST" "PDT" ) ) ; Generalized time zone.
17449 @cindex @code{TimeZone} variable
17451 With no arguments, @code{calc-time-zone} or @samp{tzone()} obtains an
17452 argument from the Calc variable @code{TimeZone} if a value has been
17453 stored for that variable. If not, Calc runs the Unix @samp{date}
17454 command and looks for one of the above time zone names in the output;
17455 if this does not succeed, @samp{tzone()} leaves itself unevaluated.
17456 The time zone name in the @samp{date} output may be followed by a signed
17457 adjustment, e.g., @samp{GMT+5} or @samp{GMT+0500} which specifies a
17458 number of hours and minutes to be added to the base time zone.
17459 Calc stores the time zone it finds into @code{TimeZone} to speed
17460 later calls to @samp{tzone()}.
17462 The special time zone name @code{local} is equivalent to no argument,
17463 i.e., it uses the local time zone as obtained from the @code{date}
17466 If the time zone name found is one of the standard or daylight
17467 savings zone names from the above table, and Calc's internal
17468 daylight savings algorithm says that time and zone are consistent
17469 (e.g., @code{PDT} accompanies a date that Calc's algorithm would also
17470 consider to be daylight savings, or @code{PST} accompanies a date
17471 that Calc would consider to be standard time), then Calc substitutes
17472 the corresponding generalized time zone (like @code{PGT}).
17474 If your system does not have a suitable @samp{date} command, you
17475 may wish to put a @samp{(setq var-TimeZone ...)} in your Emacs
17476 initialization file to set the time zone. (Since you are interacting
17477 with the variable @code{TimeZone} directly from Emacs Lisp, the
17478 @code{var-} prefix needs to be present.) The easiest way to do
17479 this is to edit the @code{TimeZone} variable using Calc's @kbd{s T}
17480 command, then use the @kbd{s p} (@code{calc-permanent-variable})
17481 command to save the value of @code{TimeZone} permanently.
17483 The @kbd{t J} and @code{t U} commands with no numeric prefix
17484 arguments do the same thing as @samp{tzone()}. If the current
17485 time zone is a generalized time zone, e.g., @code{EGT}, Calc
17486 examines the date being converted to tell whether to use standard
17487 or daylight savings time. But if the current time zone is explicit,
17488 e.g., @code{EST} or @code{EDT}, then that adjustment is used exactly
17489 and Calc's daylight savings algorithm is not consulted.
17491 Some places don't follow the usual rules for daylight savings time.
17492 The state of Arizona, for example, does not observe daylight savings
17493 time. If you run Calc during the winter season in Arizona, the
17494 Unix @code{date} command will report @code{MST} time zone, which
17495 Calc will change to @code{MGT}. If you then convert a time that
17496 lies in the summer months, Calc will apply an incorrect daylight
17497 savings time adjustment. To avoid this, set your @code{TimeZone}
17498 variable explicitly to @code{MST} to force the use of standard,
17499 non-daylight-savings time.
17501 @vindex math-daylight-savings-hook
17502 @findex math-std-daylight-savings
17503 By default Calc always considers daylight savings time to begin at
17504 2 a.m.@: on the first Sunday of April, and to end at 2 a.m.@: on the
17505 last Sunday of October. This is the rule that has been in effect
17506 in North America since 1987. If you are in a country that uses
17507 different rules for computing daylight savings time, you have two
17508 choices: Write your own daylight savings hook, or control time
17509 zones explicitly by setting the @code{TimeZone} variable and/or
17510 always giving a time-zone argument for the conversion functions.
17512 The Lisp variable @code{math-daylight-savings-hook} holds the
17513 name of a function that is used to compute the daylight savings
17514 adjustment for a given date. The default is
17515 @code{math-std-daylight-savings}, which computes an adjustment
17516 (either 0 or @mathit{-1}) using the North American rules given above.
17518 The daylight savings hook function is called with four arguments:
17519 The date, as a floating-point number in standard Calc format;
17520 a six-element list of the date decomposed into year, month, day,
17521 hour, minute, and second, respectively; a string which contains
17522 the generalized time zone name in upper-case, e.g., @code{"WEGT"};
17523 and a special adjustment to be applied to the hour value when
17524 converting into a generalized time zone (see below).
17526 @findex math-prev-weekday-in-month
17527 The Lisp function @code{math-prev-weekday-in-month} is useful for
17528 daylight savings computations. This is an internal version of
17529 the user-level @code{pwday} function described in the previous
17530 section. It takes four arguments: The floating-point date value,
17531 the corresponding six-element date list, the day-of-month number,
17532 and the weekday number (0-6).
17534 The default daylight savings hook ignores the time zone name, but a
17535 more sophisticated hook could use different algorithms for different
17536 time zones. It would also be possible to use different algorithms
17537 depending on the year number, but the default hook always uses the
17538 algorithm for 1987 and later. Here is a listing of the default
17539 daylight savings hook:
17542 (defun math-std-daylight-savings (date dt zone bump)
17543 (cond ((< (nth 1 dt) 4) 0)
17545 (let ((sunday (math-prev-weekday-in-month date dt 7 0)))
17546 (cond ((< (nth 2 dt) sunday) 0)
17547 ((= (nth 2 dt) sunday)
17548 (if (>= (nth 3 dt) (+ 3 bump)) -1 0))
17550 ((< (nth 1 dt) 10) -1)
17552 (let ((sunday (math-prev-weekday-in-month date dt 31 0)))
17553 (cond ((< (nth 2 dt) sunday) -1)
17554 ((= (nth 2 dt) sunday)
17555 (if (>= (nth 3 dt) (+ 2 bump)) 0 -1))
17562 The @code{bump} parameter is equal to zero when Calc is converting
17563 from a date form in a generalized time zone into a GMT date value.
17564 It is @mathit{-1} when Calc is converting in the other direction. The
17565 adjustments shown above ensure that the conversion behaves correctly
17566 and reasonably around the 2 a.m.@: transition in each direction.
17568 There is a ``missing'' hour between 2 a.m.@: and 3 a.m.@: at the
17569 beginning of daylight savings time; converting a date/time form that
17570 falls in this hour results in a time value for the following hour,
17571 from 3 a.m.@: to 4 a.m. At the end of daylight savings time, the
17572 hour from 1 a.m.@: to 2 a.m.@: repeats itself; converting a date/time
17573 form that falls in in this hour results in a time value for the first
17574 manifestation of that time (@emph{not} the one that occurs one hour later).
17576 If @code{math-daylight-savings-hook} is @code{nil}, then the
17577 daylight savings adjustment is always taken to be zero.
17579 In algebraic formulas, @samp{tzone(@var{zone}, @var{date})}
17580 computes the time zone adjustment for a given zone name at a
17581 given date. The @var{date} is ignored unless @var{zone} is a
17582 generalized time zone. If @var{date} is a date form, the
17583 daylight savings computation is applied to it as it appears.
17584 If @var{date} is a numeric date value, it is adjusted for the
17585 daylight-savings version of @var{zone} before being given to
17586 the daylight savings hook. This odd-sounding rule ensures
17587 that the daylight-savings computation is always done in
17588 local time, not in the GMT time that a numeric @var{date}
17589 is typically represented in.
17595 The @samp{dsadj(@var{date}, @var{zone})} function computes the
17596 daylight savings adjustment that is appropriate for @var{date} in
17597 time zone @var{zone}. If @var{zone} is explicitly in or not in
17598 daylight savings time (e.g., @code{PDT} or @code{PST}) the
17599 @var{date} is ignored. If @var{zone} is a generalized time zone,
17600 the algorithms described above are used. If @var{zone} is omitted,
17601 the computation is done for the current time zone.
17603 @xref{Reporting Bugs}, for the address of Calc's author, if you
17604 should wish to contribute your improved versions of
17605 @code{math-tzone-names} and @code{math-daylight-savings-hook}
17606 to the Calc distribution.
17608 @node Financial Functions, Binary Functions, Date Arithmetic, Arithmetic
17609 @section Financial Functions
17612 Calc's financial or business functions use the @kbd{b} prefix
17613 key followed by a shifted letter. (The @kbd{b} prefix followed by
17614 a lower-case letter is used for operations on binary numbers.)
17616 Note that the rate and the number of intervals given to these
17617 functions must be on the same time scale, e.g., both months or
17618 both years. Mixing an annual interest rate with a time expressed
17619 in months will give you very wrong answers!
17621 It is wise to compute these functions to a higher precision than
17622 you really need, just to make sure your answer is correct to the
17623 last penny; also, you may wish to check the definitions at the end
17624 of this section to make sure the functions have the meaning you expect.
17630 * Related Financial Functions::
17631 * Depreciation Functions::
17632 * Definitions of Financial Functions::
17635 @node Percentages, Future Value, Financial Functions, Financial Functions
17636 @subsection Percentages
17639 @pindex calc-percent
17642 The @kbd{M-%} (@code{calc-percent}) command takes a percentage value,
17643 say 5.4, and converts it to an equivalent actual number. For example,
17644 @kbd{5.4 M-%} enters 0.054 on the stack. (That's the @key{META} or
17645 @key{ESC} key combined with @kbd{%}.)
17647 Actually, @kbd{M-%} creates a formula of the form @samp{5.4%}.
17648 You can enter @samp{5.4%} yourself during algebraic entry. The
17649 @samp{%} operator simply means, ``the preceding value divided by
17650 100.'' The @samp{%} operator has very high precedence, so that
17651 @samp{1+8%} is interpreted as @samp{1+(8%)}, not as @samp{(1+8)%}.
17652 (The @samp{%} operator is just a postfix notation for the
17653 @code{percent} function, just like @samp{20!} is the notation for
17654 @samp{fact(20)}, or twenty-factorial.)
17656 The formula @samp{5.4%} would normally evaluate immediately to
17657 0.054, but the @kbd{M-%} command suppresses evaluation as it puts
17658 the formula onto the stack. However, the next Calc command that
17659 uses the formula @samp{5.4%} will evaluate it as its first step.
17660 The net effect is that you get to look at @samp{5.4%} on the stack,
17661 but Calc commands see it as @samp{0.054}, which is what they expect.
17663 In particular, @samp{5.4%} and @samp{0.054} are suitable values
17664 for the @var{rate} arguments of the various financial functions,
17665 but the number @samp{5.4} is probably @emph{not} suitable---it
17666 represents a rate of 540 percent!
17668 The key sequence @kbd{M-% *} effectively means ``percent-of.''
17669 For example, @kbd{68 @key{RET} 25 M-% *} computes 17, which is 25% of
17670 68 (and also 68% of 25, which comes out to the same thing).
17673 @pindex calc-convert-percent
17674 The @kbd{c %} (@code{calc-convert-percent}) command converts the
17675 value on the top of the stack from numeric to percentage form.
17676 For example, if 0.08 is on the stack, @kbd{c %} converts it to
17677 @samp{8%}. The quantity is the same, it's just represented
17678 differently. (Contrast this with @kbd{M-%}, which would convert
17679 this number to @samp{0.08%}.) The @kbd{=} key is a convenient way
17680 to convert a formula like @samp{8%} back to numeric form, 0.08.
17682 To compute what percentage one quantity is of another quantity,
17683 use @kbd{/ c %}. For example, @w{@kbd{17 @key{RET} 68 / c %}} displays
17687 @pindex calc-percent-change
17689 The @kbd{b %} (@code{calc-percent-change}) [@code{relch}] command
17690 calculates the percentage change from one number to another.
17691 For example, @kbd{40 @key{RET} 50 b %} produces the answer @samp{25%},
17692 since 50 is 25% larger than 40. A negative result represents a
17693 decrease: @kbd{50 @key{RET} 40 b %} produces @samp{-20%}, since 40 is
17694 20% smaller than 50. (The answers are different in magnitude
17695 because, in the first case, we're increasing by 25% of 40, but
17696 in the second case, we're decreasing by 20% of 50.) The effect
17697 of @kbd{40 @key{RET} 50 b %} is to compute @expr{(50-40)/40}, converting
17698 the answer to percentage form as if by @kbd{c %}.
17700 @node Future Value, Present Value, Percentages, Financial Functions
17701 @subsection Future Value
17705 @pindex calc-fin-fv
17707 The @kbd{b F} (@code{calc-fin-fv}) [@code{fv}] command computes
17708 the future value of an investment. It takes three arguments
17709 from the stack: @samp{fv(@var{rate}, @var{n}, @var{payment})}.
17710 If you give payments of @var{payment} every year for @var{n}
17711 years, and the money you have paid earns interest at @var{rate} per
17712 year, then this function tells you what your investment would be
17713 worth at the end of the period. (The actual interval doesn't
17714 have to be years, as long as @var{n} and @var{rate} are expressed
17715 in terms of the same intervals.) This function assumes payments
17716 occur at the @emph{end} of each interval.
17720 The @kbd{I b F} [@code{fvb}] command does the same computation,
17721 but assuming your payments are at the beginning of each interval.
17722 Suppose you plan to deposit $1000 per year in a savings account
17723 earning 5.4% interest, starting right now. How much will be
17724 in the account after five years? @code{fvb(5.4%, 5, 1000) = 5870.73}.
17725 Thus you will have earned $870 worth of interest over the years.
17726 Using the stack, this calculation would have been
17727 @kbd{5.4 M-% 5 @key{RET} 1000 I b F}. Note that the rate is expressed
17728 as a number between 0 and 1, @emph{not} as a percentage.
17732 The @kbd{H b F} [@code{fvl}] command computes the future value
17733 of an initial lump sum investment. Suppose you could deposit
17734 those five thousand dollars in the bank right now; how much would
17735 they be worth in five years? @code{fvl(5.4%, 5, 5000) = 6503.89}.
17737 The algebraic functions @code{fv} and @code{fvb} accept an optional
17738 fourth argument, which is used as an initial lump sum in the sense
17739 of @code{fvl}. In other words, @code{fv(@var{rate}, @var{n},
17740 @var{payment}, @var{initial}) = fv(@var{rate}, @var{n}, @var{payment})
17741 + fvl(@var{rate}, @var{n}, @var{initial})}.
17743 To illustrate the relationships between these functions, we could
17744 do the @code{fvb} calculation ``by hand'' using @code{fvl}. The
17745 final balance will be the sum of the contributions of our five
17746 deposits at various times. The first deposit earns interest for
17747 five years: @code{fvl(5.4%, 5, 1000) = 1300.78}. The second
17748 deposit only earns interest for four years: @code{fvl(5.4%, 4, 1000) =
17749 1234.13}. And so on down to the last deposit, which earns one
17750 year's interest: @code{fvl(5.4%, 1, 1000) = 1054.00}. The sum of
17751 these five values is, sure enough, $5870.73, just as was computed
17752 by @code{fvb} directly.
17754 What does @code{fv(5.4%, 5, 1000) = 5569.96} mean? The payments
17755 are now at the ends of the periods. The end of one year is the same
17756 as the beginning of the next, so what this really means is that we've
17757 lost the payment at year zero (which contributed $1300.78), but we're
17758 now counting the payment at year five (which, since it didn't have
17759 a chance to earn interest, counts as $1000). Indeed, @expr{5569.96 =
17760 5870.73 - 1300.78 + 1000} (give or take a bit of roundoff error).
17762 @node Present Value, Related Financial Functions, Future Value, Financial Functions
17763 @subsection Present Value
17767 @pindex calc-fin-pv
17769 The @kbd{b P} (@code{calc-fin-pv}) [@code{pv}] command computes
17770 the present value of an investment. Like @code{fv}, it takes
17771 three arguments: @code{pv(@var{rate}, @var{n}, @var{payment})}.
17772 It computes the present value of a series of regular payments.
17773 Suppose you have the chance to make an investment that will
17774 pay $2000 per year over the next four years; as you receive
17775 these payments you can put them in the bank at 9% interest.
17776 You want to know whether it is better to make the investment, or
17777 to keep the money in the bank where it earns 9% interest right
17778 from the start. The calculation @code{pv(9%, 4, 2000)} gives the
17779 result 6479.44. If your initial investment must be less than this,
17780 say, $6000, then the investment is worthwhile. But if you had to
17781 put up $7000, then it would be better just to leave it in the bank.
17783 Here is the interpretation of the result of @code{pv}: You are
17784 trying to compare the return from the investment you are
17785 considering, which is @code{fv(9%, 4, 2000) = 9146.26}, with
17786 the return from leaving the money in the bank, which is
17787 @code{fvl(9%, 4, @var{x})} where @var{x} is the amount of money
17788 you would have to put up in advance. The @code{pv} function
17789 finds the break-even point, @expr{x = 6479.44}, at which
17790 @code{fvl(9%, 4, 6479.44)} is also equal to 9146.26. This is
17791 the largest amount you should be willing to invest.
17795 The @kbd{I b P} [@code{pvb}] command solves the same problem,
17796 but with payments occurring at the beginning of each interval.
17797 It has the same relationship to @code{fvb} as @code{pv} has
17798 to @code{fv}. For example @code{pvb(9%, 4, 2000) = 7062.59},
17799 a larger number than @code{pv} produced because we get to start
17800 earning interest on the return from our investment sooner.
17804 The @kbd{H b P} [@code{pvl}] command computes the present value of
17805 an investment that will pay off in one lump sum at the end of the
17806 period. For example, if we get our $8000 all at the end of the
17807 four years, @code{pvl(9%, 4, 8000) = 5667.40}. This is much
17808 less than @code{pv} reported, because we don't earn any interest
17809 on the return from this investment. Note that @code{pvl} and
17810 @code{fvl} are simple inverses: @code{fvl(9%, 4, 5667.40) = 8000}.
17812 You can give an optional fourth lump-sum argument to @code{pv}
17813 and @code{pvb}; this is handled in exactly the same way as the
17814 fourth argument for @code{fv} and @code{fvb}.
17817 @pindex calc-fin-npv
17819 The @kbd{b N} (@code{calc-fin-npv}) [@code{npv}] command computes
17820 the net present value of a series of irregular investments.
17821 The first argument is the interest rate. The second argument is
17822 a vector which represents the expected return from the investment
17823 at the end of each interval. For example, if the rate represents
17824 a yearly interest rate, then the vector elements are the return
17825 from the first year, second year, and so on.
17827 Thus, @code{npv(9%, [2000,2000,2000,2000]) = pv(9%, 4, 2000) = 6479.44}.
17828 Obviously this function is more interesting when the payments are
17831 The @code{npv} function can actually have two or more arguments.
17832 Multiple arguments are interpreted in the same way as for the
17833 vector statistical functions like @code{vsum}.
17834 @xref{Single-Variable Statistics}. Basically, if there are several
17835 payment arguments, each either a vector or a plain number, all these
17836 values are collected left-to-right into the complete list of payments.
17837 A numeric prefix argument on the @kbd{b N} command says how many
17838 payment values or vectors to take from the stack.
17842 The @kbd{I b N} [@code{npvb}] command computes the net present
17843 value where payments occur at the beginning of each interval
17844 rather than at the end.
17846 @node Related Financial Functions, Depreciation Functions, Present Value, Financial Functions
17847 @subsection Related Financial Functions
17850 The functions in this section are basically inverses of the
17851 present value functions with respect to the various arguments.
17854 @pindex calc-fin-pmt
17856 The @kbd{b M} (@code{calc-fin-pmt}) [@code{pmt}] command computes
17857 the amount of periodic payment necessary to amortize a loan.
17858 Thus @code{pmt(@var{rate}, @var{n}, @var{amount})} equals the
17859 value of @var{payment} such that @code{pv(@var{rate}, @var{n},
17860 @var{payment}) = @var{amount}}.
17864 The @kbd{I b M} [@code{pmtb}] command does the same computation
17865 but using @code{pvb} instead of @code{pv}. Like @code{pv} and
17866 @code{pvb}, these functions can also take a fourth argument which
17867 represents an initial lump-sum investment.
17870 The @kbd{H b M} key just invokes the @code{fvl} function, which is
17871 the inverse of @code{pvl}. There is no explicit @code{pmtl} function.
17874 @pindex calc-fin-nper
17876 The @kbd{b #} (@code{calc-fin-nper}) [@code{nper}] command computes
17877 the number of regular payments necessary to amortize a loan.
17878 Thus @code{nper(@var{rate}, @var{payment}, @var{amount})} equals
17879 the value of @var{n} such that @code{pv(@var{rate}, @var{n},
17880 @var{payment}) = @var{amount}}. If @var{payment} is too small
17881 ever to amortize a loan for @var{amount} at interest rate @var{rate},
17882 the @code{nper} function is left in symbolic form.
17886 The @kbd{I b #} [@code{nperb}] command does the same computation
17887 but using @code{pvb} instead of @code{pv}. You can give a fourth
17888 lump-sum argument to these functions, but the computation will be
17889 rather slow in the four-argument case.
17893 The @kbd{H b #} [@code{nperl}] command does the same computation
17894 using @code{pvl}. By exchanging @var{payment} and @var{amount} you
17895 can also get the solution for @code{fvl}. For example,
17896 @code{nperl(8%, 2000, 1000) = 9.006}, so if you place $1000 in a
17897 bank account earning 8%, it will take nine years to grow to $2000.
17900 @pindex calc-fin-rate
17902 The @kbd{b T} (@code{calc-fin-rate}) [@code{rate}] command computes
17903 the rate of return on an investment. This is also an inverse of @code{pv}:
17904 @code{rate(@var{n}, @var{payment}, @var{amount})} computes the value of
17905 @var{rate} such that @code{pv(@var{rate}, @var{n}, @var{payment}) =
17906 @var{amount}}. The result is expressed as a formula like @samp{6.3%}.
17912 The @kbd{I b T} [@code{rateb}] and @kbd{H b T} [@code{ratel}]
17913 commands solve the analogous equations with @code{pvb} or @code{pvl}
17914 in place of @code{pv}. Also, @code{rate} and @code{rateb} can
17915 accept an optional fourth argument just like @code{pv} and @code{pvb}.
17916 To redo the above example from a different perspective,
17917 @code{ratel(9, 2000, 1000) = 8.00597%}, which says you will need an
17918 interest rate of 8% in order to double your account in nine years.
17921 @pindex calc-fin-irr
17923 The @kbd{b I} (@code{calc-fin-irr}) [@code{irr}] command is the
17924 analogous function to @code{rate} but for net present value.
17925 Its argument is a vector of payments. Thus @code{irr(@var{payments})}
17926 computes the @var{rate} such that @code{npv(@var{rate}, @var{payments}) = 0};
17927 this rate is known as the @dfn{internal rate of return}.
17931 The @kbd{I b I} [@code{irrb}] command computes the internal rate of
17932 return assuming payments occur at the beginning of each period.
17934 @node Depreciation Functions, Definitions of Financial Functions, Related Financial Functions, Financial Functions
17935 @subsection Depreciation Functions
17938 The functions in this section calculate @dfn{depreciation}, which is
17939 the amount of value that a possession loses over time. These functions
17940 are characterized by three parameters: @var{cost}, the original cost
17941 of the asset; @var{salvage}, the value the asset will have at the end
17942 of its expected ``useful life''; and @var{life}, the number of years
17943 (or other periods) of the expected useful life.
17945 There are several methods for calculating depreciation that differ in
17946 the way they spread the depreciation over the lifetime of the asset.
17949 @pindex calc-fin-sln
17951 The @kbd{b S} (@code{calc-fin-sln}) [@code{sln}] command computes the
17952 ``straight-line'' depreciation. In this method, the asset depreciates
17953 by the same amount every year (or period). For example,
17954 @samp{sln(12000, 2000, 5)} returns 2000. The asset costs $12000
17955 initially and will be worth $2000 after five years; it loses $2000
17959 @pindex calc-fin-syd
17961 The @kbd{b Y} (@code{calc-fin-syd}) [@code{syd}] command computes the
17962 accelerated ``sum-of-years'-digits'' depreciation. Here the depreciation
17963 is higher during the early years of the asset's life. Since the
17964 depreciation is different each year, @kbd{b Y} takes a fourth @var{period}
17965 parameter which specifies which year is requested, from 1 to @var{life}.
17966 If @var{period} is outside this range, the @code{syd} function will
17970 @pindex calc-fin-ddb
17972 The @kbd{b D} (@code{calc-fin-ddb}) [@code{ddb}] command computes an
17973 accelerated depreciation using the double-declining balance method.
17974 It also takes a fourth @var{period} parameter.
17976 For symmetry, the @code{sln} function will accept a @var{period}
17977 parameter as well, although it will ignore its value except that the
17978 return value will as usual be zero if @var{period} is out of range.
17980 For example, pushing the vector @expr{[1,2,3,4,5]} (perhaps with @kbd{v x 5})
17981 and then mapping @kbd{V M ' [sln(12000,2000,5,$), syd(12000,2000,5,$),
17982 ddb(12000,2000,5,$)] @key{RET}} produces a matrix that allows us to compare
17983 the three depreciation methods:
17987 [ [ 2000, 3333, 4800 ]
17988 [ 2000, 2667, 2880 ]
17989 [ 2000, 2000, 1728 ]
17990 [ 2000, 1333, 592 ]
17996 (Values have been rounded to nearest integers in this figure.)
17997 We see that @code{sln} depreciates by the same amount each year,
17998 @kbd{syd} depreciates more at the beginning and less at the end,
17999 and @kbd{ddb} weights the depreciation even more toward the beginning.
18001 Summing columns with @kbd{V R : +} yields @expr{[10000, 10000, 10000]};
18002 the total depreciation in any method is (by definition) the
18003 difference between the cost and the salvage value.
18005 @node Definitions of Financial Functions, , Depreciation Functions, Financial Functions
18006 @subsection Definitions
18009 For your reference, here are the actual formulas used to compute
18010 Calc's financial functions.
18012 Calc will not evaluate a financial function unless the @var{rate} or
18013 @var{n} argument is known. However, @var{payment} or @var{amount} can
18014 be a variable. Calc expands these functions according to the
18015 formulas below for symbolic arguments only when you use the @kbd{a "}
18016 (@code{calc-expand-formula}) command, or when taking derivatives or
18017 integrals or solving equations involving the functions.
18020 These formulas are shown using the conventions of Big display
18021 mode (@kbd{d B}); for example, the formula for @code{fv} written
18022 linearly is @samp{pmt * ((1 + rate)^n) - 1) / rate}.
18027 fv(rate, n, pmt) = pmt * ---------------
18031 ((1 + rate) - 1) (1 + rate)
18032 fvb(rate, n, pmt) = pmt * ----------------------------
18036 fvl(rate, n, pmt) = pmt * (1 + rate)
18040 pv(rate, n, pmt) = pmt * ----------------
18044 (1 - (1 + rate) ) (1 + rate)
18045 pvb(rate, n, pmt) = pmt * -----------------------------
18049 pvl(rate, n, pmt) = pmt * (1 + rate)
18052 npv(rate, [a, b, c]) = a*(1 + rate) + b*(1 + rate) + c*(1 + rate)
18055 npvb(rate, [a, b, c]) = a + b*(1 + rate) + c*(1 + rate)
18058 (amt - x * (1 + rate) ) * rate
18059 pmt(rate, n, amt, x) = -------------------------------
18064 (amt - x * (1 + rate) ) * rate
18065 pmtb(rate, n, amt, x) = -------------------------------
18067 (1 - (1 + rate) ) (1 + rate)
18070 nper(rate, pmt, amt) = - log(1 - ------------, 1 + rate)
18074 nperb(rate, pmt, amt) = - log(1 - ---------------, 1 + rate)
18078 nperl(rate, pmt, amt) = - log(---, 1 + rate)
18083 ratel(n, pmt, amt) = ------ - 1
18088 sln(cost, salv, life) = -----------
18091 (cost - salv) * (life - per + 1)
18092 syd(cost, salv, life, per) = --------------------------------
18093 life * (life + 1) / 2
18096 ddb(cost, salv, life, per) = --------, book = cost - depreciation so far
18102 $$ \code{fv}(r, n, p) = p { (1 + r)^n - 1 \over r } $$
18103 $$ \code{fvb}(r, n, p) = p { ((1 + r)^n - 1) (1 + r) \over r } $$
18104 $$ \code{fvl}(r, n, p) = p (1 + r)^n $$
18105 $$ \code{pv}(r, n, p) = p { 1 - (1 + r)^{-n} \over r } $$
18106 $$ \code{pvb}(r, n, p) = p { (1 - (1 + r)^{-n}) (1 + r) \over r } $$
18107 $$ \code{pvl}(r, n, p) = p (1 + r)^{-n} $$
18108 $$ \code{npv}(r, [a,b,c]) = a (1 + r)^{-1} + b (1 + r)^{-2} + c (1 + r)^{-3} $$
18109 $$ \code{npvb}(r, [a,b,c]) = a + b (1 + r)^{-1} + c (1 + r)^{-2} $$
18110 $$ \code{pmt}(r, n, a, x) = { (a - x (1 + r)^{-n}) r \over 1 - (1 + r)^{-n} }$$
18111 $$ \code{pmtb}(r, n, a, x) = { (a - x (1 + r)^{-n}) r \over
18112 (1 - (1 + r)^{-n}) (1 + r) } $$
18113 $$ \code{nper}(r, p, a) = -\code{log}(1 - { a r \over p }, 1 + r) $$
18114 $$ \code{nperb}(r, p, a) = -\code{log}(1 - { a r \over p (1 + r) }, 1 + r) $$
18115 $$ \code{nperl}(r, p, a) = -\code{log}({a \over p}, 1 + r) $$
18116 $$ \code{ratel}(n, p, a) = { p^{1/n} \over a^{1/n} } - 1 $$
18117 $$ \code{sln}(c, s, l) = { c - s \over l } $$
18118 $$ \code{syd}(c, s, l, p) = { (c - s) (l - p + 1) \over l (l+1) / 2 } $$
18119 $$ \code{ddb}(c, s, l, p) = { 2 (c - \hbox{depreciation so far}) \over l } $$
18123 In @code{pmt} and @code{pmtb}, @expr{x=0} if omitted.
18125 These functions accept any numeric objects, including error forms,
18126 intervals, and even (though not very usefully) complex numbers. The
18127 above formulas specify exactly the behavior of these functions with
18128 all sorts of inputs.
18130 Note that if the first argument to the @code{log} in @code{nper} is
18131 negative, @code{nper} leaves itself in symbolic form rather than
18132 returning a (financially meaningless) complex number.
18134 @samp{rate(num, pmt, amt)} solves the equation
18135 @samp{pv(rate, num, pmt) = amt} for @samp{rate} using @kbd{H a R}
18136 (@code{calc-find-root}), with the interval @samp{[.01% .. 100%]}
18137 for an initial guess. The @code{rateb} function is the same except
18138 that it uses @code{pvb}. Note that @code{ratel} can be solved
18139 directly; its formula is shown in the above list.
18141 Similarly, @samp{irr(pmts)} solves the equation @samp{npv(rate, pmts) = 0}
18144 If you give a fourth argument to @code{nper} or @code{nperb}, Calc
18145 will also use @kbd{H a R} to solve the equation using an initial
18146 guess interval of @samp{[0 .. 100]}.
18148 A fourth argument to @code{fv} simply sums the two components
18149 calculated from the above formulas for @code{fv} and @code{fvl}.
18150 The same is true of @code{fvb}, @code{pv}, and @code{pvb}.
18152 The @kbd{ddb} function is computed iteratively; the ``book'' value
18153 starts out equal to @var{cost}, and decreases according to the above
18154 formula for the specified number of periods. If the book value
18155 would decrease below @var{salvage}, it only decreases to @var{salvage}
18156 and the depreciation is zero for all subsequent periods. The @code{ddb}
18157 function returns the amount the book value decreased in the specified
18160 @node Binary Functions, , Financial Functions, Arithmetic
18161 @section Binary Number Functions
18164 The commands in this chapter all use two-letter sequences beginning with
18165 the @kbd{b} prefix.
18167 @cindex Binary numbers
18168 The ``binary'' operations actually work regardless of the currently
18169 displayed radix, although their results make the most sense in a radix
18170 like 2, 8, or 16 (as obtained by the @kbd{d 2}, @kbd{d 8}, or @w{@kbd{d 6}}
18171 commands, respectively). You may also wish to enable display of leading
18172 zeros with @kbd{d z}. @xref{Radix Modes}.
18174 @cindex Word size for binary operations
18175 The Calculator maintains a current @dfn{word size} @expr{w}, an
18176 arbitrary positive or negative integer. For a positive word size, all
18177 of the binary operations described here operate modulo @expr{2^w}. In
18178 particular, negative arguments are converted to positive integers modulo
18179 @expr{2^w} by all binary functions.
18181 If the word size is negative, binary operations produce 2's complement
18183 @texline @math{-2^{-w-1}}
18184 @infoline @expr{-(2^(-w-1))}
18186 @texline @math{2^{-w-1}-1}
18187 @infoline @expr{2^(-w-1)-1}
18188 inclusive. Either mode accepts inputs in any range; the sign of
18189 @expr{w} affects only the results produced.
18194 The @kbd{b c} (@code{calc-clip})
18195 [@code{clip}] command can be used to clip a number by reducing it modulo
18196 @expr{2^w}. The commands described in this chapter automatically clip
18197 their results to the current word size. Note that other operations like
18198 addition do not use the current word size, since integer addition
18199 generally is not ``binary.'' (However, @pxref{Simplification Modes},
18200 @code{calc-bin-simplify-mode}.) For example, with a word size of 8
18201 bits @kbd{b c} converts a number to the range 0 to 255; with a word
18202 size of @mathit{-8} @kbd{b c} converts to the range @mathit{-128} to 127.
18205 @pindex calc-word-size
18206 The default word size is 32 bits. All operations except the shifts and
18207 rotates allow you to specify a different word size for that one
18208 operation by giving a numeric prefix argument: @kbd{C-u 8 b c} clips the
18209 top of stack to the range 0 to 255 regardless of the current word size.
18210 To set the word size permanently, use @kbd{b w} (@code{calc-word-size}).
18211 This command displays a prompt with the current word size; press @key{RET}
18212 immediately to keep this word size, or type a new word size at the prompt.
18214 When the binary operations are written in symbolic form, they take an
18215 optional second (or third) word-size parameter. When a formula like
18216 @samp{and(a,b)} is finally evaluated, the word size current at that time
18217 will be used, but when @samp{and(a,b,-8)} is evaluated, a word size of
18218 @mathit{-8} will always be used. A symbolic binary function will be left
18219 in symbolic form unless the all of its argument(s) are integers or
18220 integer-valued floats.
18222 If either or both arguments are modulo forms for which @expr{M} is a
18223 power of two, that power of two is taken as the word size unless a
18224 numeric prefix argument overrides it. The current word size is never
18225 consulted when modulo-power-of-two forms are involved.
18230 The @kbd{b a} (@code{calc-and}) [@code{and}] command computes the bitwise
18231 AND of the two numbers on the top of the stack. In other words, for each
18232 of the @expr{w} binary digits of the two numbers (pairwise), the corresponding
18233 bit of the result is 1 if and only if both input bits are 1:
18234 @samp{and(2#1100, 2#1010) = 2#1000}.
18239 The @kbd{b o} (@code{calc-or}) [@code{or}] command computes the bitwise
18240 inclusive OR of two numbers. A bit is 1 if either of the input bits, or
18241 both, are 1: @samp{or(2#1100, 2#1010) = 2#1110}.
18246 The @kbd{b x} (@code{calc-xor}) [@code{xor}] command computes the bitwise
18247 exclusive OR of two numbers. A bit is 1 if exactly one of the input bits
18248 is 1: @samp{xor(2#1100, 2#1010) = 2#0110}.
18253 The @kbd{b d} (@code{calc-diff}) [@code{diff}] command computes the bitwise
18254 difference of two numbers; this is defined by @samp{diff(a,b) = and(a,not(b))},
18255 so that @samp{diff(2#1100, 2#1010) = 2#0100}.
18260 The @kbd{b n} (@code{calc-not}) [@code{not}] command computes the bitwise
18261 NOT of a number. A bit is 1 if the input bit is 0 and vice-versa.
18264 @pindex calc-lshift-binary
18266 The @kbd{b l} (@code{calc-lshift-binary}) [@code{lsh}] command shifts a
18267 number left by one bit, or by the number of bits specified in the numeric
18268 prefix argument. A negative prefix argument performs a logical right shift,
18269 in which zeros are shifted in on the left. In symbolic form, @samp{lsh(a)}
18270 is short for @samp{lsh(a,1)}, which in turn is short for @samp{lsh(a,n,w)}.
18271 Bits shifted ``off the end,'' according to the current word size, are lost.
18287 The @kbd{H b l} command also does a left shift, but it takes two arguments
18288 from the stack (the value to shift, and, at top-of-stack, the number of
18289 bits to shift). This version interprets the prefix argument just like
18290 the regular binary operations, i.e., as a word size. The Hyperbolic flag
18291 has a similar effect on the rest of the binary shift and rotate commands.
18294 @pindex calc-rshift-binary
18296 The @kbd{b r} (@code{calc-rshift-binary}) [@code{rsh}] command shifts a
18297 number right by one bit, or by the number of bits specified in the numeric
18298 prefix argument: @samp{rsh(a,n) = lsh(a,-n)}.
18301 @pindex calc-lshift-arith
18303 The @kbd{b L} (@code{calc-lshift-arith}) [@code{ash}] command shifts a
18304 number left. It is analogous to @code{lsh}, except that if the shift
18305 is rightward (the prefix argument is negative), an arithmetic shift
18306 is performed as described below.
18309 @pindex calc-rshift-arith
18311 The @kbd{b R} (@code{calc-rshift-arith}) [@code{rash}] command performs
18312 an ``arithmetic'' shift to the right, in which the leftmost bit (according
18313 to the current word size) is duplicated rather than shifting in zeros.
18314 This corresponds to dividing by a power of two where the input is interpreted
18315 as a signed, twos-complement number. (The distinction between the @samp{rsh}
18316 and @samp{rash} operations is totally independent from whether the word
18317 size is positive or negative.) With a negative prefix argument, this
18318 performs a standard left shift.
18321 @pindex calc-rotate-binary
18323 The @kbd{b t} (@code{calc-rotate-binary}) [@code{rot}] command rotates a
18324 number one bit to the left. The leftmost bit (according to the current
18325 word size) is dropped off the left and shifted in on the right. With a
18326 numeric prefix argument, the number is rotated that many bits to the left
18329 @xref{Set Operations}, for the @kbd{b p} and @kbd{b u} commands that
18330 pack and unpack binary integers into sets. (For example, @kbd{b u}
18331 unpacks the number @samp{2#11001} to the set of bit-numbers
18332 @samp{[0, 3, 4]}.) Type @kbd{b u V #} to count the number of ``1''
18333 bits in a binary integer.
18335 Another interesting use of the set representation of binary integers
18336 is to reverse the bits in, say, a 32-bit integer. Type @kbd{b u} to
18337 unpack; type @kbd{31 @key{TAB} -} to replace each bit-number in the set
18338 with 31 minus that bit-number; type @kbd{b p} to pack the set back
18339 into a binary integer.
18341 @node Scientific Functions, Matrix Functions, Arithmetic, Top
18342 @chapter Scientific Functions
18345 The functions described here perform trigonometric and other transcendental
18346 calculations. They generally produce floating-point answers correct to the
18347 full current precision. The @kbd{H} (Hyperbolic) and @kbd{I} (Inverse)
18348 flag keys must be used to get some of these functions from the keyboard.
18352 @cindex @code{pi} variable
18355 @cindex @code{e} variable
18358 @cindex @code{gamma} variable
18360 @cindex Gamma constant, Euler's
18361 @cindex Euler's gamma constant
18363 @cindex @code{phi} variable
18364 @cindex Phi, golden ratio
18365 @cindex Golden ratio
18366 One miscellaneous command is shift-@kbd{P} (@code{calc-pi}), which pushes
18367 the value of @cpi{} (at the current precision) onto the stack. With the
18368 Hyperbolic flag, it pushes the value @expr{e}, the base of natural logarithms.
18369 With the Inverse flag, it pushes Euler's constant
18370 @texline @math{\gamma}
18371 @infoline @expr{gamma}
18372 (about 0.5772). With both Inverse and Hyperbolic, it
18373 pushes the ``golden ratio''
18374 @texline @math{\phi}
18375 @infoline @expr{phi}
18376 (about 1.618). (At present, Euler's constant is not available
18377 to unlimited precision; Calc knows only the first 100 digits.)
18378 In Symbolic mode, these commands push the
18379 actual variables @samp{pi}, @samp{e}, @samp{gamma}, and @samp{phi},
18380 respectively, instead of their values; @pxref{Symbolic Mode}.
18390 The @kbd{Q} (@code{calc-sqrt}) [@code{sqrt}] function is described elsewhere;
18391 @pxref{Basic Arithmetic}. With the Inverse flag [@code{sqr}], this command
18392 computes the square of the argument.
18394 @xref{Prefix Arguments}, for a discussion of the effect of numeric
18395 prefix arguments on commands in this chapter which do not otherwise
18396 interpret a prefix argument.
18399 * Logarithmic Functions::
18400 * Trigonometric and Hyperbolic Functions::
18401 * Advanced Math Functions::
18404 * Combinatorial Functions::
18405 * Probability Distribution Functions::
18408 @node Logarithmic Functions, Trigonometric and Hyperbolic Functions, Scientific Functions, Scientific Functions
18409 @section Logarithmic Functions
18419 The shift-@kbd{L} (@code{calc-ln}) [@code{ln}] command computes the natural
18420 logarithm of the real or complex number on the top of the stack. With
18421 the Inverse flag it computes the exponential function instead, although
18422 this is redundant with the @kbd{E} command.
18431 The shift-@kbd{E} (@code{calc-exp}) [@code{exp}] command computes the
18432 exponential, i.e., @expr{e} raised to the power of the number on the stack.
18433 The meanings of the Inverse and Hyperbolic flags follow from those for
18434 the @code{calc-ln} command.
18449 The @kbd{H L} (@code{calc-log10}) [@code{log10}] command computes the common
18450 (base-10) logarithm of a number. (With the Inverse flag [@code{exp10}],
18451 it raises ten to a given power.) Note that the common logarithm of a
18452 complex number is computed by taking the natural logarithm and dividing
18454 @texline @math{\ln10}.
18455 @infoline @expr{ln(10)}.
18462 The @kbd{B} (@code{calc-log}) [@code{log}] command computes a logarithm
18463 to any base. For example, @kbd{1024 @key{RET} 2 B} produces 10, since
18464 @texline @math{2^{10} = 1024}.
18465 @infoline @expr{2^10 = 1024}.
18466 In certain cases like @samp{log(3,9)}, the result
18467 will be either @expr{1:2} or @expr{0.5} depending on the current Fraction
18468 mode setting. With the Inverse flag [@code{alog}], this command is
18469 similar to @kbd{^} except that the order of the arguments is reversed.
18474 The @kbd{f I} (@code{calc-ilog}) [@code{ilog}] command computes the
18475 integer logarithm of a number to any base. The number and the base must
18476 themselves be positive integers. This is the true logarithm, rounded
18477 down to an integer. Thus @kbd{ilog(x,10)} is 3 for all @expr{x} in the
18478 range from 1000 to 9999. If both arguments are positive integers, exact
18479 integer arithmetic is used; otherwise, this is equivalent to
18480 @samp{floor(log(x,b))}.
18485 The @kbd{f E} (@code{calc-expm1}) [@code{expm1}] command computes
18486 @texline @math{e^x - 1},
18487 @infoline @expr{exp(x)-1},
18488 but using an algorithm that produces a more accurate
18489 answer when the result is close to zero, i.e., when
18490 @texline @math{e^x}
18491 @infoline @expr{exp(x)}
18497 The @kbd{f L} (@code{calc-lnp1}) [@code{lnp1}] command computes
18498 @texline @math{\ln(x+1)},
18499 @infoline @expr{ln(x+1)},
18500 producing a more accurate answer when @expr{x} is close to zero.
18502 @node Trigonometric and Hyperbolic Functions, Advanced Math Functions, Logarithmic Functions, Scientific Functions
18503 @section Trigonometric/Hyperbolic Functions
18509 The shift-@kbd{S} (@code{calc-sin}) [@code{sin}] command computes the sine
18510 of an angle or complex number. If the input is an HMS form, it is interpreted
18511 as degrees-minutes-seconds; otherwise, the input is interpreted according
18512 to the current angular mode. It is best to use Radians mode when operating
18513 on complex numbers.
18515 Calc's ``units'' mechanism includes angular units like @code{deg},
18516 @code{rad}, and @code{grad}. While @samp{sin(45 deg)} is not evaluated
18517 all the time, the @kbd{u s} (@code{calc-simplify-units}) command will
18518 simplify @samp{sin(45 deg)} by taking the sine of 45 degrees, regardless
18519 of the current angular mode. @xref{Basic Operations on Units}.
18521 Also, the symbolic variable @code{pi} is not ordinarily recognized in
18522 arguments to trigonometric functions, as in @samp{sin(3 pi / 4)}, but
18523 the @kbd{a s} (@code{calc-simplify}) command recognizes many such
18524 formulas when the current angular mode is Radians @emph{and} Symbolic
18525 mode is enabled; this example would be replaced by @samp{sqrt(2) / 2}.
18526 @xref{Symbolic Mode}. Beware, this simplification occurs even if you
18527 have stored a different value in the variable @samp{pi}; this is one
18528 reason why changing built-in variables is a bad idea. Arguments of
18529 the form @expr{x} plus a multiple of @cpiover{2} are also simplified.
18530 Calc includes similar formulas for @code{cos} and @code{tan}.
18532 The @kbd{a s} command knows all angles which are integer multiples of
18533 @cpiover{12}, @cpiover{10}, or @cpiover{8} radians. In Degrees mode,
18534 analogous simplifications occur for integer multiples of 15 or 18
18535 degrees, and for arguments plus multiples of 90 degrees.
18538 @pindex calc-arcsin
18540 With the Inverse flag, @code{calc-sin} computes an arcsine. This is also
18541 available as the @code{calc-arcsin} command or @code{arcsin} algebraic
18542 function. The returned argument is converted to degrees, radians, or HMS
18543 notation depending on the current angular mode.
18549 @pindex calc-arcsinh
18551 With the Hyperbolic flag, @code{calc-sin} computes the hyperbolic
18552 sine, also available as @code{calc-sinh} [@code{sinh}]. With the
18553 Hyperbolic and Inverse flags, it computes the hyperbolic arcsine
18554 (@code{calc-arcsinh}) [@code{arcsinh}].
18563 @pindex calc-arccos
18581 @pindex calc-arccosh
18599 @pindex calc-arctan
18617 @pindex calc-arctanh
18622 The shift-@kbd{C} (@code{calc-cos}) [@code{cos}] command computes the cosine
18623 of an angle or complex number, and shift-@kbd{T} (@code{calc-tan}) [@code{tan}]
18624 computes the tangent, along with all the various inverse and hyperbolic
18625 variants of these functions.
18628 @pindex calc-arctan2
18630 The @kbd{f T} (@code{calc-arctan2}) [@code{arctan2}] command takes two
18631 numbers from the stack and computes the arc tangent of their ratio. The
18632 result is in the full range from @mathit{-180} (exclusive) to @mathit{+180}
18633 (inclusive) degrees, or the analogous range in radians. A similar
18634 result would be obtained with @kbd{/} followed by @kbd{I T}, but the
18635 value would only be in the range from @mathit{-90} to @mathit{+90} degrees
18636 since the division loses information about the signs of the two
18637 components, and an error might result from an explicit division by zero
18638 which @code{arctan2} would avoid. By (arbitrary) definition,
18639 @samp{arctan2(0,0)=0}.
18641 @pindex calc-sincos
18653 The @code{calc-sincos} [@code{sincos}] command computes the sine and
18654 cosine of a number, returning them as a vector of the form
18655 @samp{[@var{cos}, @var{sin}]}.
18656 With the Inverse flag [@code{arcsincos}], this command takes a two-element
18657 vector as an argument and computes @code{arctan2} of the elements.
18658 (This command does not accept the Hyperbolic flag.)
18672 The remaining trigonometric functions, @code{calc-sec} [@code{sec}],
18673 @code{calc-csc} [@code{csc}] and @code{calc-sec} [@code{sec}], are also
18674 available. With the Hyperbolic flag, these compute their hyperbolic
18675 counterparts, which are also available separately as @code{calc-sech}
18676 [@code{sech}], @code{calc-csch} [@code{csch}] and @code{calc-sech}
18677 [@code{sech}]. (These commmands do not accept the Inverse flag.)
18679 @node Advanced Math Functions, Branch Cuts, Trigonometric and Hyperbolic Functions, Scientific Functions
18680 @section Advanced Mathematical Functions
18683 Calc can compute a variety of less common functions that arise in
18684 various branches of mathematics. All of the functions described in
18685 this section allow arbitrary complex arguments and, except as noted,
18686 will work to arbitrarily large precisions. They can not at present
18687 handle error forms or intervals as arguments.
18689 NOTE: These functions are still experimental. In particular, their
18690 accuracy is not guaranteed in all domains. It is advisable to set the
18691 current precision comfortably higher than you actually need when
18692 using these functions. Also, these functions may be impractically
18693 slow for some values of the arguments.
18698 The @kbd{f g} (@code{calc-gamma}) [@code{gamma}] command computes the Euler
18699 gamma function. For positive integer arguments, this is related to the
18700 factorial function: @samp{gamma(n+1) = fact(n)}. For general complex
18701 arguments the gamma function can be defined by the following definite
18703 @texline @math{\Gamma(a) = \int_0^\infty t^{a-1} e^t dt}.
18704 @infoline @expr{gamma(a) = integ(t^(a-1) exp(t), t, 0, inf)}.
18705 (The actual implementation uses far more efficient computational methods.)
18721 @pindex calc-inc-gamma
18734 The @kbd{f G} (@code{calc-inc-gamma}) [@code{gammaP}] command computes
18735 the incomplete gamma function, denoted @samp{P(a,x)}. This is defined by
18737 @texline @math{P(a,x) = \left( \int_0^x t^{a-1} e^t dt \right) / \Gamma(a)}.
18738 @infoline @expr{gammaP(a,x) = integ(t^(a-1) exp(t), t, 0, x) / gamma(a)}.
18739 This implies that @samp{gammaP(a,inf) = 1} for any @expr{a} (see the
18740 definition of the normal gamma function).
18742 Several other varieties of incomplete gamma function are defined.
18743 The complement of @expr{P(a,x)}, called @expr{Q(a,x) = 1-P(a,x)} by
18744 some authors, is computed by the @kbd{I f G} [@code{gammaQ}] command.
18745 You can think of this as taking the other half of the integral, from
18746 @expr{x} to infinity.
18749 The functions corresponding to the integrals that define @expr{P(a,x)}
18750 and @expr{Q(a,x)} but without the normalizing @expr{1/gamma(a)}
18751 factor are called @expr{g(a,x)} and @expr{G(a,x)}, respectively
18752 (where @expr{g} and @expr{G} represent the lower- and upper-case Greek
18753 letter gamma). You can obtain these using the @kbd{H f G} [@code{gammag}]
18754 and @kbd{H I f G} [@code{gammaG}] commands.
18758 The functions corresponding to the integrals that define $P(a,x)$
18759 and $Q(a,x)$ but without the normalizing $1/\Gamma(a)$
18760 factor are called $\gamma(a,x)$ and $\Gamma(a,x)$, respectively.
18761 You can obtain these using the \kbd{H f G} [\code{gammag}] and
18762 \kbd{I H f G} [\code{gammaG}] commands.
18768 The @kbd{f b} (@code{calc-beta}) [@code{beta}] command computes the
18769 Euler beta function, which is defined in terms of the gamma function as
18770 @texline @math{B(a,b) = \Gamma(a) \Gamma(b) / \Gamma(a+b)},
18771 @infoline @expr{beta(a,b) = gamma(a) gamma(b) / gamma(a+b)},
18773 @texline @math{B(a,b) = \int_0^1 t^{a-1} (1-t)^{b-1} dt}.
18774 @infoline @expr{beta(a,b) = integ(t^(a-1) (1-t)^(b-1), t, 0, 1)}.
18778 @pindex calc-inc-beta
18781 The @kbd{f B} (@code{calc-inc-beta}) [@code{betaI}] command computes
18782 the incomplete beta function @expr{I(x,a,b)}. It is defined by
18783 @texline @math{I(x,a,b) = \left( \int_0^x t^{a-1} (1-t)^{b-1} dt \right) / B(a,b)}.
18784 @infoline @expr{betaI(x,a,b) = integ(t^(a-1) (1-t)^(b-1), t, 0, x) / beta(a,b)}.
18785 Once again, the @kbd{H} (hyperbolic) prefix gives the corresponding
18786 un-normalized version [@code{betaB}].
18793 The @kbd{f e} (@code{calc-erf}) [@code{erf}] command computes the
18795 @texline @math{\hbox{erf}(x) = {2 \over \sqrt{\pi}} \int_0^x e^{-t^2} dt}.
18796 @infoline @expr{erf(x) = 2 integ(exp(-(t^2)), t, 0, x) / sqrt(pi)}.
18797 The complementary error function @kbd{I f e} (@code{calc-erfc}) [@code{erfc}]
18798 is the corresponding integral from @samp{x} to infinity; the sum
18799 @texline @math{\hbox{erf}(x) + \hbox{erfc}(x) = 1}.
18800 @infoline @expr{erf(x) + erfc(x) = 1}.
18804 @pindex calc-bessel-J
18805 @pindex calc-bessel-Y
18808 The @kbd{f j} (@code{calc-bessel-J}) [@code{besJ}] and @kbd{f y}
18809 (@code{calc-bessel-Y}) [@code{besY}] commands compute the Bessel
18810 functions of the first and second kinds, respectively.
18811 In @samp{besJ(n,x)} and @samp{besY(n,x)} the ``order'' parameter
18812 @expr{n} is often an integer, but is not required to be one.
18813 Calc's implementation of the Bessel functions currently limits the
18814 precision to 8 digits, and may not be exact even to that precision.
18817 @node Branch Cuts, Random Numbers, Advanced Math Functions, Scientific Functions
18818 @section Branch Cuts and Principal Values
18821 @cindex Branch cuts
18822 @cindex Principal values
18823 All of the logarithmic, trigonometric, and other scientific functions are
18824 defined for complex numbers as well as for reals.
18825 This section describes the values
18826 returned in cases where the general result is a family of possible values.
18827 Calc follows section 12.5.3 of Steele's @dfn{Common Lisp, the Language},
18828 second edition, in these matters. This section will describe each
18829 function briefly; for a more detailed discussion (including some nifty
18830 diagrams), consult Steele's book.
18832 Note that the branch cuts for @code{arctan} and @code{arctanh} were
18833 changed between the first and second editions of Steele. Versions of
18834 Calc starting with 2.00 follow the second edition.
18836 The new branch cuts exactly match those of the HP-28/48 calculators.
18837 They also match those of Mathematica 1.2, except that Mathematica's
18838 @code{arctan} cut is always in the right half of the complex plane,
18839 and its @code{arctanh} cut is always in the top half of the plane.
18840 Calc's cuts are continuous with quadrants I and III for @code{arctan},
18841 or II and IV for @code{arctanh}.
18843 Note: The current implementations of these functions with complex arguments
18844 are designed with proper behavior around the branch cuts in mind, @emph{not}
18845 efficiency or accuracy. You may need to increase the floating precision
18846 and wait a while to get suitable answers from them.
18848 For @samp{sqrt(a+bi)}: When @expr{a<0} and @expr{b} is small but positive
18849 or zero, the result is close to the @expr{+i} axis. For @expr{b} small and
18850 negative, the result is close to the @expr{-i} axis. The result always lies
18851 in the right half of the complex plane.
18853 For @samp{ln(a+bi)}: The real part is defined as @samp{ln(abs(a+bi))}.
18854 The imaginary part is defined as @samp{arg(a+bi) = arctan2(b,a)}.
18855 Thus the branch cuts for @code{sqrt} and @code{ln} both lie on the
18856 negative real axis.
18858 The following table describes these branch cuts in another way.
18859 If the real and imaginary parts of @expr{z} are as shown, then
18860 the real and imaginary parts of @expr{f(z)} will be as shown.
18861 Here @code{eps} stands for a small positive value; each
18862 occurrence of @code{eps} may stand for a different small value.
18866 ----------------------------------------
18869 -, +eps +eps, + +eps, +
18870 -, -eps +eps, - +eps, -
18873 For @samp{z1^z2}: This is defined by @samp{exp(ln(z1)*z2)}.
18874 One interesting consequence of this is that @samp{(-8)^1:3} does
18875 not evaluate to @mathit{-2} as you might expect, but to the complex
18876 number @expr{(1., 1.732)}. Both of these are valid cube roots
18877 of @mathit{-8} (as is @expr{(1., -1.732)}); Calc chooses a perhaps
18878 less-obvious root for the sake of mathematical consistency.
18880 For @samp{arcsin(z)}: This is defined by @samp{-i*ln(i*z + sqrt(1-z^2))}.
18881 The branch cuts are on the real axis, less than @mathit{-1} and greater than 1.
18883 For @samp{arccos(z)}: This is defined by @samp{-i*ln(z + i*sqrt(1-z^2))},
18884 or equivalently by @samp{pi/2 - arcsin(z)}. The branch cuts are on
18885 the real axis, less than @mathit{-1} and greater than 1.
18887 For @samp{arctan(z)}: This is defined by
18888 @samp{(ln(1+i*z) - ln(1-i*z)) / (2*i)}. The branch cuts are on the
18889 imaginary axis, below @expr{-i} and above @expr{i}.
18891 For @samp{arcsinh(z)}: This is defined by @samp{ln(z + sqrt(1+z^2))}.
18892 The branch cuts are on the imaginary axis, below @expr{-i} and
18895 For @samp{arccosh(z)}: This is defined by
18896 @samp{ln(z + (z+1)*sqrt((z-1)/(z+1)))}. The branch cut is on the
18897 real axis less than 1.
18899 For @samp{arctanh(z)}: This is defined by @samp{(ln(1+z) - ln(1-z)) / 2}.
18900 The branch cuts are on the real axis, less than @mathit{-1} and greater than 1.
18902 The following tables for @code{arcsin}, @code{arccos}, and
18903 @code{arctan} assume the current angular mode is Radians. The
18904 hyperbolic functions operate independently of the angular mode.
18907 z arcsin(z) arccos(z)
18908 -------------------------------------------------------
18909 (-1..1), 0 (-pi/2..pi/2), 0 (0..pi), 0
18910 (-1..1), +eps (-pi/2..pi/2), +eps (0..pi), -eps
18911 (-1..1), -eps (-pi/2..pi/2), -eps (0..pi), +eps
18912 <-1, 0 -pi/2, + pi, -
18913 <-1, +eps -pi/2 + eps, + pi - eps, -
18914 <-1, -eps -pi/2 + eps, - pi - eps, +
18916 >1, +eps pi/2 - eps, + +eps, -
18917 >1, -eps pi/2 - eps, - +eps, +
18921 z arccosh(z) arctanh(z)
18922 -----------------------------------------------------
18923 (-1..1), 0 0, (0..pi) any, 0
18924 (-1..1), +eps +eps, (0..pi) any, +eps
18925 (-1..1), -eps +eps, (-pi..0) any, -eps
18926 <-1, 0 +, pi -, pi/2
18927 <-1, +eps +, pi - eps -, pi/2 - eps
18928 <-1, -eps +, -pi + eps -, -pi/2 + eps
18929 >1, 0 +, 0 +, -pi/2
18930 >1, +eps +, +eps +, pi/2 - eps
18931 >1, -eps +, -eps +, -pi/2 + eps
18935 z arcsinh(z) arctan(z)
18936 -----------------------------------------------------
18937 0, (-1..1) 0, (-pi/2..pi/2) 0, any
18938 0, <-1 -, -pi/2 -pi/2, -
18939 +eps, <-1 +, -pi/2 + eps pi/2 - eps, -
18940 -eps, <-1 -, -pi/2 + eps -pi/2 + eps, -
18941 0, >1 +, pi/2 pi/2, +
18942 +eps, >1 +, pi/2 - eps pi/2 - eps, +
18943 -eps, >1 -, pi/2 - eps -pi/2 + eps, +
18946 Finally, the following identities help to illustrate the relationship
18947 between the complex trigonometric and hyperbolic functions. They
18948 are valid everywhere, including on the branch cuts.
18951 sin(i*z) = i*sinh(z) arcsin(i*z) = i*arcsinh(z)
18952 cos(i*z) = cosh(z) arcsinh(i*z) = i*arcsin(z)
18953 tan(i*z) = i*tanh(z) arctan(i*z) = i*arctanh(z)
18954 sinh(i*z) = i*sin(z) cosh(i*z) = cos(z)
18957 The ``advanced math'' functions (gamma, Bessel, etc.@:) are also defined
18958 for general complex arguments, but their branch cuts and principal values
18959 are not rigorously specified at present.
18961 @node Random Numbers, Combinatorial Functions, Branch Cuts, Scientific Functions
18962 @section Random Numbers
18966 @pindex calc-random
18968 The @kbd{k r} (@code{calc-random}) [@code{random}] command produces
18969 random numbers of various sorts.
18971 Given a positive numeric prefix argument @expr{M}, it produces a random
18972 integer @expr{N} in the range
18973 @texline @math{0 \le N < M}.
18974 @infoline @expr{0 <= N < M}.
18975 Each of the @expr{M} values appears with equal probability.
18977 With no numeric prefix argument, the @kbd{k r} command takes its argument
18978 from the stack instead. Once again, if this is a positive integer @expr{M}
18979 the result is a random integer less than @expr{M}. However, note that
18980 while numeric prefix arguments are limited to six digits or so, an @expr{M}
18981 taken from the stack can be arbitrarily large. If @expr{M} is negative,
18982 the result is a random integer in the range
18983 @texline @math{M < N \le 0}.
18984 @infoline @expr{M < N <= 0}.
18986 If the value on the stack is a floating-point number @expr{M}, the result
18987 is a random floating-point number @expr{N} in the range
18988 @texline @math{0 \le N < M}
18989 @infoline @expr{0 <= N < M}
18991 @texline @math{M < N \le 0},
18992 @infoline @expr{M < N <= 0},
18993 according to the sign of @expr{M}.
18995 If @expr{M} is zero, the result is a Gaussian-distributed random real
18996 number; the distribution has a mean of zero and a standard deviation
18997 of one. The algorithm used generates random numbers in pairs; thus,
18998 every other call to this function will be especially fast.
19000 If @expr{M} is an error form
19001 @texline @math{m} @code{+/-} @math{\sigma}
19002 @infoline @samp{m +/- s}
19004 @texline @math{\sigma}
19006 are both real numbers, the result uses a Gaussian distribution with mean
19007 @var{m} and standard deviation
19008 @texline @math{\sigma}.
19011 If @expr{M} is an interval form, the lower and upper bounds specify the
19012 acceptable limits of the random numbers. If both bounds are integers,
19013 the result is a random integer in the specified range. If either bound
19014 is floating-point, the result is a random real number in the specified
19015 range. If the interval is open at either end, the result will be sure
19016 not to equal that end value. (This makes a big difference for integer
19017 intervals, but for floating-point intervals it's relatively minor:
19018 with a precision of 6, @samp{random([1.0..2.0))} will return any of one
19019 million numbers from 1.00000 to 1.99999; @samp{random([1.0..2.0])} may
19020 additionally return 2.00000, but the probability of this happening is
19023 If @expr{M} is a vector, the result is one element taken at random from
19024 the vector. All elements of the vector are given equal probabilities.
19027 The sequence of numbers produced by @kbd{k r} is completely random by
19028 default, i.e., the sequence is seeded each time you start Calc using
19029 the current time and other information. You can get a reproducible
19030 sequence by storing a particular ``seed value'' in the Calc variable
19031 @code{RandSeed}. Any integer will do for a seed; integers of from 1
19032 to 12 digits are good. If you later store a different integer into
19033 @code{RandSeed}, Calc will switch to a different pseudo-random
19034 sequence. If you ``unstore'' @code{RandSeed}, Calc will re-seed itself
19035 from the current time. If you store the same integer that you used
19036 before back into @code{RandSeed}, you will get the exact same sequence
19037 of random numbers as before.
19039 @pindex calc-rrandom
19040 The @code{calc-rrandom} command (not on any key) produces a random real
19041 number between zero and one. It is equivalent to @samp{random(1.0)}.
19044 @pindex calc-random-again
19045 The @kbd{k a} (@code{calc-random-again}) command produces another random
19046 number, re-using the most recent value of @expr{M}. With a numeric
19047 prefix argument @var{n}, it produces @var{n} more random numbers using
19048 that value of @expr{M}.
19051 @pindex calc-shuffle
19053 The @kbd{k h} (@code{calc-shuffle}) command produces a vector of several
19054 random values with no duplicates. The value on the top of the stack
19055 specifies the set from which the random values are drawn, and may be any
19056 of the @expr{M} formats described above. The numeric prefix argument
19057 gives the length of the desired list. (If you do not provide a numeric
19058 prefix argument, the length of the list is taken from the top of the
19059 stack, and @expr{M} from second-to-top.)
19061 If @expr{M} is a floating-point number, zero, or an error form (so
19062 that the random values are being drawn from the set of real numbers)
19063 there is little practical difference between using @kbd{k h} and using
19064 @kbd{k r} several times. But if the set of possible values consists
19065 of just a few integers, or the elements of a vector, then there is
19066 a very real chance that multiple @kbd{k r}'s will produce the same
19067 number more than once. The @kbd{k h} command produces a vector whose
19068 elements are always distinct. (Actually, there is a slight exception:
19069 If @expr{M} is a vector, no given vector element will be drawn more
19070 than once, but if several elements of @expr{M} are equal, they may
19071 each make it into the result vector.)
19073 One use of @kbd{k h} is to rearrange a list at random. This happens
19074 if the prefix argument is equal to the number of values in the list:
19075 @kbd{[1, 1.5, 2, 2.5, 3] 5 k h} might produce the permuted list
19076 @samp{[2.5, 1, 1.5, 3, 2]}. As a convenient feature, if the argument
19077 @var{n} is negative it is replaced by the size of the set represented
19078 by @expr{M}. Naturally, this is allowed only when @expr{M} specifies
19079 a small discrete set of possibilities.
19081 To do the equivalent of @kbd{k h} but with duplications allowed,
19082 given @expr{M} on the stack and with @var{n} just entered as a numeric
19083 prefix, use @kbd{v b} to build a vector of copies of @expr{M}, then use
19084 @kbd{V M k r} to ``map'' the normal @kbd{k r} function over the
19085 elements of this vector. @xref{Matrix Functions}.
19088 * Random Number Generator:: (Complete description of Calc's algorithm)
19091 @node Random Number Generator, , Random Numbers, Random Numbers
19092 @subsection Random Number Generator
19094 Calc's random number generator uses several methods to ensure that
19095 the numbers it produces are highly random. Knuth's @emph{Art of
19096 Computer Programming}, Volume II, contains a thorough description
19097 of the theory of random number generators and their measurement and
19100 If @code{RandSeed} has no stored value, Calc calls Emacs' built-in
19101 @code{random} function to get a stream of random numbers, which it
19102 then treats in various ways to avoid problems inherent in the simple
19103 random number generators that many systems use to implement @code{random}.
19105 When Calc's random number generator is first invoked, it ``seeds''
19106 the low-level random sequence using the time of day, so that the
19107 random number sequence will be different every time you use Calc.
19109 Since Emacs Lisp doesn't specify the range of values that will be
19110 returned by its @code{random} function, Calc exercises the function
19111 several times to estimate the range. When Calc subsequently uses
19112 the @code{random} function, it takes only 10 bits of the result
19113 near the most-significant end. (It avoids at least the bottom
19114 four bits, preferably more, and also tries to avoid the top two
19115 bits.) This strategy works well with the linear congruential
19116 generators that are typically used to implement @code{random}.
19118 If @code{RandSeed} contains an integer, Calc uses this integer to
19119 seed an ``additive congruential'' method (Knuth's algorithm 3.2.2A,
19121 @texline @math{X_{n-55} - X_{n-24}}.
19122 @infoline @expr{X_n-55 - X_n-24}).
19123 This method expands the seed
19124 value into a large table which is maintained internally; the variable
19125 @code{RandSeed} is changed from, e.g., 42 to the vector @expr{[42]}
19126 to indicate that the seed has been absorbed into this table. When
19127 @code{RandSeed} contains a vector, @kbd{k r} and related commands
19128 continue to use the same internal table as last time. There is no
19129 way to extract the complete state of the random number generator
19130 so that you can restart it from any point; you can only restart it
19131 from the same initial seed value. A simple way to restart from the
19132 same seed is to type @kbd{s r RandSeed} to get the seed vector,
19133 @kbd{v u} to unpack it back into a number, then @kbd{s t RandSeed}
19134 to reseed the generator with that number.
19136 Calc uses a ``shuffling'' method as described in algorithm 3.2.2B
19137 of Knuth. It fills a table with 13 random 10-bit numbers. Then,
19138 to generate a new random number, it uses the previous number to
19139 index into the table, picks the value it finds there as the new
19140 random number, then replaces that table entry with a new value
19141 obtained from a call to the base random number generator (either
19142 the additive congruential generator or the @code{random} function
19143 supplied by the system). If there are any flaws in the base
19144 generator, shuffling will tend to even them out. But if the system
19145 provides an excellent @code{random} function, shuffling will not
19146 damage its randomness.
19148 To create a random integer of a certain number of digits, Calc
19149 builds the integer three decimal digits at a time. For each group
19150 of three digits, Calc calls its 10-bit shuffling random number generator
19151 (which returns a value from 0 to 1023); if the random value is 1000
19152 or more, Calc throws it out and tries again until it gets a suitable
19155 To create a random floating-point number with precision @var{p}, Calc
19156 simply creates a random @var{p}-digit integer and multiplies by
19157 @texline @math{10^{-p}}.
19158 @infoline @expr{10^-p}.
19159 The resulting random numbers should be very clean, but note
19160 that relatively small numbers will have few significant random digits.
19161 In other words, with a precision of 12, you will occasionally get
19162 numbers on the order of
19163 @texline @math{10^{-9}}
19164 @infoline @expr{10^-9}
19166 @texline @math{10^{-10}},
19167 @infoline @expr{10^-10},
19168 but those numbers will only have two or three random digits since they
19169 correspond to small integers times
19170 @texline @math{10^{-12}}.
19171 @infoline @expr{10^-12}.
19173 To create a random integer in the interval @samp{[0 .. @var{m})}, Calc
19174 counts the digits in @var{m}, creates a random integer with three
19175 additional digits, then reduces modulo @var{m}. Unless @var{m} is a
19176 power of ten the resulting values will be very slightly biased toward
19177 the lower numbers, but this bias will be less than 0.1%. (For example,
19178 if @var{m} is 42, Calc will reduce a random integer less than 100000
19179 modulo 42 to get a result less than 42. It is easy to show that the
19180 numbers 40 and 41 will be only 2380/2381 as likely to result from this
19181 modulo operation as numbers 39 and below.) If @var{m} is a power of
19182 ten, however, the numbers should be completely unbiased.
19184 The Gaussian random numbers generated by @samp{random(0.0)} use the
19185 ``polar'' method described in Knuth section 3.4.1C. This method
19186 generates a pair of Gaussian random numbers at a time, so only every
19187 other call to @samp{random(0.0)} will require significant calculations.
19189 @node Combinatorial Functions, Probability Distribution Functions, Random Numbers, Scientific Functions
19190 @section Combinatorial Functions
19193 Commands relating to combinatorics and number theory begin with the
19194 @kbd{k} key prefix.
19199 The @kbd{k g} (@code{calc-gcd}) [@code{gcd}] command computes the
19200 Greatest Common Divisor of two integers. It also accepts fractions;
19201 the GCD of two fractions is defined by taking the GCD of the
19202 numerators, and the LCM of the denominators. This definition is
19203 consistent with the idea that @samp{a / gcd(a,x)} should yield an
19204 integer for any @samp{a} and @samp{x}. For other types of arguments,
19205 the operation is left in symbolic form.
19210 The @kbd{k l} (@code{calc-lcm}) [@code{lcm}] command computes the
19211 Least Common Multiple of two integers or fractions. The product of
19212 the LCM and GCD of two numbers is equal to the product of the
19216 @pindex calc-extended-gcd
19218 The @kbd{k E} (@code{calc-extended-gcd}) [@code{egcd}] command computes
19219 the GCD of two integers @expr{x} and @expr{y} and returns a vector
19220 @expr{[g, a, b]} where
19221 @texline @math{g = \gcd(x,y) = a x + b y}.
19222 @infoline @expr{g = gcd(x,y) = a x + b y}.
19225 @pindex calc-factorial
19231 The @kbd{!} (@code{calc-factorial}) [@code{fact}] command computes the
19232 factorial of the number at the top of the stack. If the number is an
19233 integer, the result is an exact integer. If the number is an
19234 integer-valued float, the result is a floating-point approximation. If
19235 the number is a non-integral real number, the generalized factorial is used,
19236 as defined by the Euler Gamma function. Please note that computation of
19237 large factorials can be slow; using floating-point format will help
19238 since fewer digits must be maintained. The same is true of many of
19239 the commands in this section.
19242 @pindex calc-double-factorial
19248 The @kbd{k d} (@code{calc-double-factorial}) [@code{dfact}] command
19249 computes the ``double factorial'' of an integer. For an even integer,
19250 this is the product of even integers from 2 to @expr{N}. For an odd
19251 integer, this is the product of odd integers from 3 to @expr{N}. If
19252 the argument is an integer-valued float, the result is a floating-point
19253 approximation. This function is undefined for negative even integers.
19254 The notation @expr{N!!} is also recognized for double factorials.
19257 @pindex calc-choose
19259 The @kbd{k c} (@code{calc-choose}) [@code{choose}] command computes the
19260 binomial coefficient @expr{N}-choose-@expr{M}, where @expr{M} is the number
19261 on the top of the stack and @expr{N} is second-to-top. If both arguments
19262 are integers, the result is an exact integer. Otherwise, the result is a
19263 floating-point approximation. The binomial coefficient is defined for all
19265 @texline @math{N! \over M! (N-M)!\,}.
19266 @infoline @expr{N! / M! (N-M)!}.
19272 The @kbd{H k c} (@code{calc-perm}) [@code{perm}] command computes the
19273 number-of-permutations function @expr{N! / (N-M)!}.
19276 The \kbd{H k c} (\code{calc-perm}) [\code{perm}] command computes the
19277 number-of-perm\-utations function $N! \over (N-M)!\,$.
19282 @pindex calc-bernoulli-number
19284 The @kbd{k b} (@code{calc-bernoulli-number}) [@code{bern}] command
19285 computes a given Bernoulli number. The value at the top of the stack
19286 is a nonnegative integer @expr{n} that specifies which Bernoulli number
19287 is desired. The @kbd{H k b} command computes a Bernoulli polynomial,
19288 taking @expr{n} from the second-to-top position and @expr{x} from the
19289 top of the stack. If @expr{x} is a variable or formula the result is
19290 a polynomial in @expr{x}; if @expr{x} is a number the result is a number.
19294 @pindex calc-euler-number
19296 The @kbd{k e} (@code{calc-euler-number}) [@code{euler}] command similarly
19297 computes an Euler number, and @w{@kbd{H k e}} computes an Euler polynomial.
19298 Bernoulli and Euler numbers occur in the Taylor expansions of several
19303 @pindex calc-stirling-number
19306 The @kbd{k s} (@code{calc-stirling-number}) [@code{stir1}] command
19307 computes a Stirling number of the first
19308 @texline kind@tie{}@math{n \brack m},
19310 given two integers @expr{n} and @expr{m} on the stack. The @kbd{H k s}
19311 [@code{stir2}] command computes a Stirling number of the second
19312 @texline kind@tie{}@math{n \brace m}.
19314 These are the number of @expr{m}-cycle permutations of @expr{n} objects,
19315 and the number of ways to partition @expr{n} objects into @expr{m}
19316 non-empty sets, respectively.
19319 @pindex calc-prime-test
19321 The @kbd{k p} (@code{calc-prime-test}) command checks if the integer on
19322 the top of the stack is prime. For integers less than eight million, the
19323 answer is always exact and reasonably fast. For larger integers, a
19324 probabilistic method is used (see Knuth vol. II, section 4.5.4, algorithm P).
19325 The number is first checked against small prime factors (up to 13). Then,
19326 any number of iterations of the algorithm are performed. Each step either
19327 discovers that the number is non-prime, or substantially increases the
19328 certainty that the number is prime. After a few steps, the chance that
19329 a number was mistakenly described as prime will be less than one percent.
19330 (Indeed, this is a worst-case estimate of the probability; in practice
19331 even a single iteration is quite reliable.) After the @kbd{k p} command,
19332 the number will be reported as definitely prime or non-prime if possible,
19333 or otherwise ``probably'' prime with a certain probability of error.
19339 The normal @kbd{k p} command performs one iteration of the primality
19340 test. Pressing @kbd{k p} repeatedly for the same integer will perform
19341 additional iterations. Also, @kbd{k p} with a numeric prefix performs
19342 the specified number of iterations. There is also an algebraic function
19343 @samp{prime(n)} or @samp{prime(n,iters)} which returns 1 if @expr{n}
19344 is (probably) prime and 0 if not.
19347 @pindex calc-prime-factors
19349 The @kbd{k f} (@code{calc-prime-factors}) [@code{prfac}] command
19350 attempts to decompose an integer into its prime factors. For numbers up
19351 to 25 million, the answer is exact although it may take some time. The
19352 result is a vector of the prime factors in increasing order. For larger
19353 inputs, prime factors above 5000 may not be found, in which case the
19354 last number in the vector will be an unfactored integer greater than 25
19355 million (with a warning message). For negative integers, the first
19356 element of the list will be @mathit{-1}. For inputs @mathit{-1}, @mathit{0}, and
19357 @mathit{1}, the result is a list of the same number.
19360 @pindex calc-next-prime
19362 @mindex nextpr@idots
19365 The @kbd{k n} (@code{calc-next-prime}) [@code{nextprime}] command finds
19366 the next prime above a given number. Essentially, it searches by calling
19367 @code{calc-prime-test} on successive integers until it finds one that
19368 passes the test. This is quite fast for integers less than eight million,
19369 but once the probabilistic test comes into play the search may be rather
19370 slow. Ordinarily this command stops for any prime that passes one iteration
19371 of the primality test. With a numeric prefix argument, a number must pass
19372 the specified number of iterations before the search stops. (This only
19373 matters when searching above eight million.) You can always use additional
19374 @kbd{k p} commands to increase your certainty that the number is indeed
19378 @pindex calc-prev-prime
19380 @mindex prevpr@idots
19383 The @kbd{I k n} (@code{calc-prev-prime}) [@code{prevprime}] command
19384 analogously finds the next prime less than a given number.
19387 @pindex calc-totient
19389 The @kbd{k t} (@code{calc-totient}) [@code{totient}] command computes the
19391 @texline function@tie{}@math{\phi(n)},
19392 @infoline function,
19393 the number of integers less than @expr{n} which
19394 are relatively prime to @expr{n}.
19397 @pindex calc-moebius
19399 The @kbd{k m} (@code{calc-moebius}) [@code{moebius}] command computes the
19400 @texline M@"obius @math{\mu}
19401 @infoline Moebius ``mu''
19402 function. If the input number is a product of @expr{k}
19403 distinct factors, this is @expr{(-1)^k}. If the input number has any
19404 duplicate factors (i.e., can be divided by the same prime more than once),
19405 the result is zero.
19407 @node Probability Distribution Functions, , Combinatorial Functions, Scientific Functions
19408 @section Probability Distribution Functions
19411 The functions in this section compute various probability distributions.
19412 For continuous distributions, this is the integral of the probability
19413 density function from @expr{x} to infinity. (These are the ``upper
19414 tail'' distribution functions; there are also corresponding ``lower
19415 tail'' functions which integrate from minus infinity to @expr{x}.)
19416 For discrete distributions, the upper tail function gives the sum
19417 from @expr{x} to infinity; the lower tail function gives the sum
19418 from minus infinity up to, but not including,@w{ }@expr{x}.
19420 To integrate from @expr{x} to @expr{y}, just use the distribution
19421 function twice and subtract. For example, the probability that a
19422 Gaussian random variable with mean 2 and standard deviation 1 will
19423 lie in the range from 2.5 to 2.8 is @samp{utpn(2.5,2,1) - utpn(2.8,2,1)}
19424 (``the probability that it is greater than 2.5, but not greater than 2.8''),
19425 or equivalently @samp{ltpn(2.8,2,1) - ltpn(2.5,2,1)}.
19432 The @kbd{k B} (@code{calc-utpb}) [@code{utpb}] function uses the
19433 binomial distribution. Push the parameters @var{n}, @var{p}, and
19434 then @var{x} onto the stack; the result (@samp{utpb(x,n,p)}) is the
19435 probability that an event will occur @var{x} or more times out
19436 of @var{n} trials, if its probability of occurring in any given
19437 trial is @var{p}. The @kbd{I k B} [@code{ltpb}] function is
19438 the probability that the event will occur fewer than @var{x} times.
19440 The other probability distribution functions similarly take the
19441 form @kbd{k @var{X}} (@code{calc-utp@var{x}}) [@code{utp@var{x}}]
19442 and @kbd{I k @var{X}} [@code{ltp@var{x}}], for various letters
19443 @var{x}. The arguments to the algebraic functions are the value of
19444 the random variable first, then whatever other parameters define the
19445 distribution. Note these are among the few Calc functions where the
19446 order of the arguments in algebraic form differs from the order of
19447 arguments as found on the stack. (The random variable comes last on
19448 the stack, so that you can type, e.g., @kbd{2 @key{RET} 1 @key{RET} 2.5
19449 k N M-@key{RET} @key{DEL} 2.8 k N -}, using @kbd{M-@key{RET} @key{DEL}} to
19450 recover the original arguments but substitute a new value for @expr{x}.)
19463 The @samp{utpc(x,v)} function uses the chi-square distribution with
19464 @texline @math{\nu}
19466 degrees of freedom. It is the probability that a model is
19467 correct if its chi-square statistic is @expr{x}.
19480 The @samp{utpf(F,v1,v2)} function uses the F distribution, used in
19481 various statistical tests. The parameters
19482 @texline @math{\nu_1}
19483 @infoline @expr{v1}
19485 @texline @math{\nu_2}
19486 @infoline @expr{v2}
19487 are the degrees of freedom in the numerator and denominator,
19488 respectively, used in computing the statistic @expr{F}.
19501 The @samp{utpn(x,m,s)} function uses a normal (Gaussian) distribution
19502 with mean @expr{m} and standard deviation
19503 @texline @math{\sigma}.
19504 @infoline @expr{s}.
19505 It is the probability that such a normal-distributed random variable
19506 would exceed @expr{x}.
19519 The @samp{utpp(n,x)} function uses a Poisson distribution with
19520 mean @expr{x}. It is the probability that @expr{n} or more such
19521 Poisson random events will occur.
19534 The @samp{utpt(t,v)} function uses the Student's ``t'' distribution
19536 @texline @math{\nu}
19538 degrees of freedom. It is the probability that a
19539 t-distributed random variable will be greater than @expr{t}.
19540 (Note: This computes the distribution function
19541 @texline @math{A(t|\nu)}
19542 @infoline @expr{A(t|v)}
19544 @texline @math{A(0|\nu) = 1}
19545 @infoline @expr{A(0|v) = 1}
19547 @texline @math{A(\infty|\nu) \to 0}.
19548 @infoline @expr{A(inf|v) -> 0}.
19549 The @code{UTPT} operation on the HP-48 uses a different definition which
19550 returns half of Calc's value: @samp{UTPT(t,v) = .5*utpt(t,v)}.)
19552 While Calc does not provide inverses of the probability distribution
19553 functions, the @kbd{a R} command can be used to solve for the inverse.
19554 Since the distribution functions are monotonic, @kbd{a R} is guaranteed
19555 to be able to find a solution given any initial guess.
19556 @xref{Numerical Solutions}.
19558 @node Matrix Functions, Algebra, Scientific Functions, Top
19559 @chapter Vector/Matrix Functions
19562 Many of the commands described here begin with the @kbd{v} prefix.
19563 (For convenience, the shift-@kbd{V} prefix is equivalent to @kbd{v}.)
19564 The commands usually apply to both plain vectors and matrices; some
19565 apply only to matrices or only to square matrices. If the argument
19566 has the wrong dimensions the operation is left in symbolic form.
19568 Vectors are entered and displayed using @samp{[a,b,c]} notation.
19569 Matrices are vectors of which all elements are vectors of equal length.
19570 (Though none of the standard Calc commands use this concept, a
19571 three-dimensional matrix or rank-3 tensor could be defined as a
19572 vector of matrices, and so on.)
19575 * Packing and Unpacking::
19576 * Building Vectors::
19577 * Extracting Elements::
19578 * Manipulating Vectors::
19579 * Vector and Matrix Arithmetic::
19581 * Statistical Operations::
19582 * Reducing and Mapping::
19583 * Vector and Matrix Formats::
19586 @node Packing and Unpacking, Building Vectors, Matrix Functions, Matrix Functions
19587 @section Packing and Unpacking
19590 Calc's ``pack'' and ``unpack'' commands collect stack entries to build
19591 composite objects such as vectors and complex numbers. They are
19592 described in this chapter because they are most often used to build
19597 The @kbd{v p} (@code{calc-pack}) [@code{pack}] command collects several
19598 elements from the stack into a matrix, complex number, HMS form, error
19599 form, etc. It uses a numeric prefix argument to specify the kind of
19600 object to be built; this argument is referred to as the ``packing mode.''
19601 If the packing mode is a nonnegative integer, a vector of that
19602 length is created. For example, @kbd{C-u 5 v p} will pop the top
19603 five stack elements and push back a single vector of those five
19604 elements. (@kbd{C-u 0 v p} simply creates an empty vector.)
19606 The same effect can be had by pressing @kbd{[} to push an incomplete
19607 vector on the stack, using @key{TAB} (@code{calc-roll-down}) to sneak
19608 the incomplete object up past a certain number of elements, and
19609 then pressing @kbd{]} to complete the vector.
19611 Negative packing modes create other kinds of composite objects:
19615 Two values are collected to build a complex number. For example,
19616 @kbd{5 @key{RET} 7 C-u -1 v p} creates the complex number
19617 @expr{(5, 7)}. The result is always a rectangular complex
19618 number. The two input values must both be real numbers,
19619 i.e., integers, fractions, or floats. If they are not, Calc
19620 will instead build a formula like @samp{a + (0, 1) b}. (The
19621 other packing modes also create a symbolic answer if the
19622 components are not suitable.)
19625 Two values are collected to build a polar complex number.
19626 The first is the magnitude; the second is the phase expressed
19627 in either degrees or radians according to the current angular
19631 Three values are collected into an HMS form. The first
19632 two values (hours and minutes) must be integers or
19633 integer-valued floats. The third value may be any real
19637 Two values are collected into an error form. The inputs
19638 may be real numbers or formulas.
19641 Two values are collected into a modulo form. The inputs
19642 must be real numbers.
19645 Two values are collected into the interval @samp{[a .. b]}.
19646 The inputs may be real numbers, HMS or date forms, or formulas.
19649 Two values are collected into the interval @samp{[a .. b)}.
19652 Two values are collected into the interval @samp{(a .. b]}.
19655 Two values are collected into the interval @samp{(a .. b)}.
19658 Two integer values are collected into a fraction.
19661 Two values are collected into a floating-point number.
19662 The first is the mantissa; the second, which must be an
19663 integer, is the exponent. The result is the mantissa
19664 times ten to the power of the exponent.
19667 This is treated the same as @mathit{-11} by the @kbd{v p} command.
19668 When unpacking, @mathit{-12} specifies that a floating-point mantissa
19672 A real number is converted into a date form.
19675 Three numbers (year, month, day) are packed into a pure date form.
19678 Six numbers are packed into a date/time form.
19681 With any of the two-input negative packing modes, either or both
19682 of the inputs may be vectors. If both are vectors of the same
19683 length, the result is another vector made by packing corresponding
19684 elements of the input vectors. If one input is a vector and the
19685 other is a plain number, the number is packed along with each vector
19686 element to produce a new vector. For example, @kbd{C-u -4 v p}
19687 could be used to convert a vector of numbers and a vector of errors
19688 into a single vector of error forms; @kbd{C-u -5 v p} could convert
19689 a vector of numbers and a single number @var{M} into a vector of
19690 numbers modulo @var{M}.
19692 If you don't give a prefix argument to @kbd{v p}, it takes
19693 the packing mode from the top of the stack. The elements to
19694 be packed then begin at stack level 2. Thus
19695 @kbd{1 @key{RET} 2 @key{RET} 4 n v p} is another way to
19696 enter the error form @samp{1 +/- 2}.
19698 If the packing mode taken from the stack is a vector, the result is a
19699 matrix with the dimensions specified by the elements of the vector,
19700 which must each be integers. For example, if the packing mode is
19701 @samp{[2, 3]}, then six numbers will be taken from the stack and
19702 returned in the form @samp{[@w{[a, b, c]}, [d, e, f]]}.
19704 If any elements of the vector are negative, other kinds of
19705 packing are done at that level as described above. For
19706 example, @samp{[2, 3, -4]} takes 12 objects and creates a
19707 @texline @math{2\times3}
19709 matrix of error forms: @samp{[[a +/- b, c +/- d ... ]]}.
19710 Also, @samp{[-4, -10]} will convert four integers into an
19711 error form consisting of two fractions: @samp{a:b +/- c:d}.
19717 There is an equivalent algebraic function,
19718 @samp{pack(@var{mode}, @var{items})} where @var{mode} is a
19719 packing mode (an integer or a vector of integers) and @var{items}
19720 is a vector of objects to be packed (re-packed, really) according
19721 to that mode. For example, @samp{pack([3, -4], [a,b,c,d,e,f])}
19722 yields @samp{[a +/- b, @w{c +/- d}, e +/- f]}. The function is
19723 left in symbolic form if the packing mode is invalid, or if the
19724 number of data items does not match the number of items required
19728 @pindex calc-unpack
19729 The @kbd{v u} (@code{calc-unpack}) command takes the vector, complex
19730 number, HMS form, or other composite object on the top of the stack and
19731 ``unpacks'' it, pushing each of its elements onto the stack as separate
19732 objects. Thus, it is the ``inverse'' of @kbd{v p}. If the value
19733 at the top of the stack is a formula, @kbd{v u} unpacks it by pushing
19734 each of the arguments of the top-level operator onto the stack.
19736 You can optionally give a numeric prefix argument to @kbd{v u}
19737 to specify an explicit (un)packing mode. If the packing mode is
19738 negative and the input is actually a vector or matrix, the result
19739 will be two or more similar vectors or matrices of the elements.
19740 For example, given the vector @samp{[@w{a +/- b}, c^2, d +/- 7]},
19741 the result of @kbd{C-u -4 v u} will be the two vectors
19742 @samp{[a, c^2, d]} and @w{@samp{[b, 0, 7]}}.
19744 Note that the prefix argument can have an effect even when the input is
19745 not a vector. For example, if the input is the number @mathit{-5}, then
19746 @kbd{c-u -1 v u} yields @mathit{-5} and 0 (the components of @mathit{-5}
19747 when viewed as a rectangular complex number); @kbd{C-u -2 v u} yields 5
19748 and 180 (assuming Degrees mode); and @kbd{C-u -10 v u} yields @mathit{-5}
19749 and 1 (the numerator and denominator of @mathit{-5}, viewed as a rational
19750 number). Plain @kbd{v u} with this input would complain that the input
19751 is not a composite object.
19753 Unpacking mode @mathit{-11} converts a float into an integer mantissa and
19754 an integer exponent, where the mantissa is not divisible by 10
19755 (except that 0.0 is represented by a mantissa and exponent of 0).
19756 Unpacking mode @mathit{-12} converts a float into a floating-point mantissa
19757 and integer exponent, where the mantissa (for non-zero numbers)
19758 is guaranteed to lie in the range [1 .. 10). In both cases,
19759 the mantissa is shifted left or right (and the exponent adjusted
19760 to compensate) in order to satisfy these constraints.
19762 Positive unpacking modes are treated differently than for @kbd{v p}.
19763 A mode of 1 is much like plain @kbd{v u} with no prefix argument,
19764 except that in addition to the components of the input object,
19765 a suitable packing mode to re-pack the object is also pushed.
19766 Thus, @kbd{C-u 1 v u} followed by @kbd{v p} will re-build the
19769 A mode of 2 unpacks two levels of the object; the resulting
19770 re-packing mode will be a vector of length 2. This might be used
19771 to unpack a matrix, say, or a vector of error forms. Higher
19772 unpacking modes unpack the input even more deeply.
19778 There are two algebraic functions analogous to @kbd{v u}.
19779 The @samp{unpack(@var{mode}, @var{item})} function unpacks the
19780 @var{item} using the given @var{mode}, returning the result as
19781 a vector of components. Here the @var{mode} must be an
19782 integer, not a vector. For example, @samp{unpack(-4, a +/- b)}
19783 returns @samp{[a, b]}, as does @samp{unpack(1, a +/- b)}.
19789 The @code{unpackt} function is like @code{unpack} but instead
19790 of returning a simple vector of items, it returns a vector of
19791 two things: The mode, and the vector of items. For example,
19792 @samp{unpackt(1, 2:3 +/- 1:4)} returns @samp{[-4, [2:3, 1:4]]},
19793 and @samp{unpackt(2, 2:3 +/- 1:4)} returns @samp{[[-4, -10], [2, 3, 1, 4]]}.
19794 The identity for re-building the original object is
19795 @samp{apply(pack, unpackt(@var{n}, @var{x})) = @var{x}}. (The
19796 @code{apply} function builds a function call given the function
19797 name and a vector of arguments.)
19799 @cindex Numerator of a fraction, extracting
19800 Subscript notation is a useful way to extract a particular part
19801 of an object. For example, to get the numerator of a rational
19802 number, you can use @samp{unpack(-10, @var{x})_1}.
19804 @node Building Vectors, Extracting Elements, Packing and Unpacking, Matrix Functions
19805 @section Building Vectors
19808 Vectors and matrices can be added,
19809 subtracted, multiplied, and divided; @pxref{Basic Arithmetic}.
19812 @pindex calc-concat
19817 The @kbd{|} (@code{calc-concat}) [@code{vconcat}] command ``concatenates'' two vectors
19818 into one. For example, after @kbd{@w{[ 1 , 2 ]} [ 3 , 4 ] |}, the stack
19819 will contain the single vector @samp{[1, 2, 3, 4]}. If the arguments
19820 are matrices, the rows of the first matrix are concatenated with the
19821 rows of the second. (In other words, two matrices are just two vectors
19822 of row-vectors as far as @kbd{|} is concerned.)
19824 If either argument to @kbd{|} is a scalar (a non-vector), it is treated
19825 like a one-element vector for purposes of concatenation: @kbd{1 [ 2 , 3 ] |}
19826 produces the vector @samp{[1, 2, 3]}. Likewise, if one argument is a
19827 matrix and the other is a plain vector, the vector is treated as a
19832 The @kbd{H |} (@code{calc-append}) [@code{append}] command concatenates
19833 two vectors without any special cases. Both inputs must be vectors.
19834 Whether or not they are matrices is not taken into account. If either
19835 argument is a scalar, the @code{append} function is left in symbolic form.
19836 See also @code{cons} and @code{rcons} below.
19840 The @kbd{I |} and @kbd{H I |} commands are similar, but they use their
19841 two stack arguments in the opposite order. Thus @kbd{I |} is equivalent
19842 to @kbd{@key{TAB} |}, but possibly more convenient and also a bit faster.
19847 The @kbd{v d} (@code{calc-diag}) [@code{diag}] function builds a diagonal
19848 square matrix. The optional numeric prefix gives the number of rows
19849 and columns in the matrix. If the value at the top of the stack is a
19850 vector, the elements of the vector are used as the diagonal elements; the
19851 prefix, if specified, must match the size of the vector. If the value on
19852 the stack is a scalar, it is used for each element on the diagonal, and
19853 the prefix argument is required.
19855 To build a constant square matrix, e.g., a
19856 @texline @math{3\times3}
19858 matrix filled with ones, use @kbd{0 M-3 v d 1 +}, i.e., build a zero
19859 matrix first and then add a constant value to that matrix. (Another
19860 alternative would be to use @kbd{v b} and @kbd{v a}; see below.)
19865 The @kbd{v i} (@code{calc-ident}) [@code{idn}] function builds an identity
19866 matrix of the specified size. It is a convenient form of @kbd{v d}
19867 where the diagonal element is always one. If no prefix argument is given,
19868 this command prompts for one.
19870 In algebraic notation, @samp{idn(a,n)} acts much like @samp{diag(a,n)},
19871 except that @expr{a} is required to be a scalar (non-vector) quantity.
19872 If @expr{n} is omitted, @samp{idn(a)} represents @expr{a} times an
19873 identity matrix of unknown size. Calc can operate algebraically on
19874 such generic identity matrices, and if one is combined with a matrix
19875 whose size is known, it is converted automatically to an identity
19876 matrix of a suitable matching size. The @kbd{v i} command with an
19877 argument of zero creates a generic identity matrix, @samp{idn(1)}.
19878 Note that in dimensioned Matrix mode (@pxref{Matrix Mode}), generic
19879 identity matrices are immediately expanded to the current default
19885 The @kbd{v x} (@code{calc-index}) [@code{index}] function builds a vector
19886 of consecutive integers from 1 to @var{n}, where @var{n} is the numeric
19887 prefix argument. If you do not provide a prefix argument, you will be
19888 prompted to enter a suitable number. If @var{n} is negative, the result
19889 is a vector of negative integers from @var{n} to @mathit{-1}.
19891 With a prefix argument of just @kbd{C-u}, the @kbd{v x} command takes
19892 three values from the stack: @var{n}, @var{start}, and @var{incr} (with
19893 @var{incr} at top-of-stack). Counting starts at @var{start} and increases
19894 by @var{incr} for successive vector elements. If @var{start} or @var{n}
19895 is in floating-point format, the resulting vector elements will also be
19896 floats. Note that @var{start} and @var{incr} may in fact be any kind
19897 of numbers or formulas.
19899 When @var{start} and @var{incr} are specified, a negative @var{n} has a
19900 different interpretation: It causes a geometric instead of arithmetic
19901 sequence to be generated. For example, @samp{index(-3, a, b)} produces
19902 @samp{[a, a b, a b^2]}. If you omit @var{incr} in the algebraic form,
19903 @samp{index(@var{n}, @var{start})}, the default value for @var{incr}
19904 is one for positive @var{n} or two for negative @var{n}.
19907 @pindex calc-build-vector
19909 The @kbd{v b} (@code{calc-build-vector}) [@code{cvec}] function builds a
19910 vector of @var{n} copies of the value on the top of the stack, where @var{n}
19911 is the numeric prefix argument. In algebraic formulas, @samp{cvec(x,n,m)}
19912 can also be used to build an @var{n}-by-@var{m} matrix of copies of @var{x}.
19913 (Interactively, just use @kbd{v b} twice: once to build a row, then again
19914 to build a matrix of copies of that row.)
19922 The @kbd{v h} (@code{calc-head}) [@code{head}] function returns the first
19923 element of a vector. The @kbd{I v h} (@code{calc-tail}) [@code{tail}]
19924 function returns the vector with its first element removed. In both
19925 cases, the argument must be a non-empty vector.
19930 The @kbd{v k} (@code{calc-cons}) [@code{cons}] function takes a value @var{h}
19931 and a vector @var{t} from the stack, and produces the vector whose head is
19932 @var{h} and whose tail is @var{t}. This is similar to @kbd{|}, except
19933 if @var{h} is itself a vector, @kbd{|} will concatenate the two vectors
19934 whereas @code{cons} will insert @var{h} at the front of the vector @var{t}.
19954 Each of these three functions also accepts the Hyperbolic flag [@code{rhead},
19955 @code{rtail}, @code{rcons}] in which case @var{t} instead represents
19956 the @emph{last} single element of the vector, with @var{h}
19957 representing the remainder of the vector. Thus the vector
19958 @samp{[a, b, c, d] = cons(a, [b, c, d]) = rcons([a, b, c], d)}.
19959 Also, @samp{head([a, b, c, d]) = a}, @samp{tail([a, b, c, d]) = [b, c, d]},
19960 @samp{rhead([a, b, c, d]) = [a, b, c]}, and @samp{rtail([a, b, c, d]) = d}.
19962 @node Extracting Elements, Manipulating Vectors, Building Vectors, Matrix Functions
19963 @section Extracting Vector Elements
19969 The @kbd{v r} (@code{calc-mrow}) [@code{mrow}] command extracts one row of
19970 the matrix on the top of the stack, or one element of the plain vector on
19971 the top of the stack. The row or element is specified by the numeric
19972 prefix argument; the default is to prompt for the row or element number.
19973 The matrix or vector is replaced by the specified row or element in the
19974 form of a vector or scalar, respectively.
19976 @cindex Permutations, applying
19977 With a prefix argument of @kbd{C-u} only, @kbd{v r} takes the index of
19978 the element or row from the top of the stack, and the vector or matrix
19979 from the second-to-top position. If the index is itself a vector of
19980 integers, the result is a vector of the corresponding elements of the
19981 input vector, or a matrix of the corresponding rows of the input matrix.
19982 This command can be used to obtain any permutation of a vector.
19984 With @kbd{C-u}, if the index is an interval form with integer components,
19985 it is interpreted as a range of indices and the corresponding subvector or
19986 submatrix is returned.
19988 @cindex Subscript notation
19990 @pindex calc-subscript
19993 Subscript notation in algebraic formulas (@samp{a_b}) stands for the
19994 Calc function @code{subscr}, which is synonymous with @code{mrow}.
19995 Thus, @samp{[x, y, z]_k} produces @expr{x}, @expr{y}, or @expr{z} if
19996 @expr{k} is one, two, or three, respectively. A double subscript
19997 (@samp{M_i_j}, equivalent to @samp{subscr(subscr(M, i), j)}) will
19998 access the element at row @expr{i}, column @expr{j} of a matrix.
19999 The @kbd{a _} (@code{calc-subscript}) command creates a subscript
20000 formula @samp{a_b} out of two stack entries. (It is on the @kbd{a}
20001 ``algebra'' prefix because subscripted variables are often used
20002 purely as an algebraic notation.)
20005 Given a negative prefix argument, @kbd{v r} instead deletes one row or
20006 element from the matrix or vector on the top of the stack. Thus
20007 @kbd{C-u 2 v r} replaces a matrix with its second row, but @kbd{C-u -2 v r}
20008 replaces the matrix with the same matrix with its second row removed.
20009 In algebraic form this function is called @code{mrrow}.
20012 Given a prefix argument of zero, @kbd{v r} extracts the diagonal elements
20013 of a square matrix in the form of a vector. In algebraic form this
20014 function is called @code{getdiag}.
20020 The @kbd{v c} (@code{calc-mcol}) [@code{mcol} or @code{mrcol}] command is
20021 the analogous operation on columns of a matrix. Given a plain vector
20022 it extracts (or removes) one element, just like @kbd{v r}. If the
20023 index in @kbd{C-u v c} is an interval or vector and the argument is a
20024 matrix, the result is a submatrix with only the specified columns
20025 retained (and possibly permuted in the case of a vector index).
20027 To extract a matrix element at a given row and column, use @kbd{v r} to
20028 extract the row as a vector, then @kbd{v c} to extract the column element
20029 from that vector. In algebraic formulas, it is often more convenient to
20030 use subscript notation: @samp{m_i_j} gives row @expr{i}, column @expr{j}
20031 of matrix @expr{m}.
20034 @pindex calc-subvector
20036 The @kbd{v s} (@code{calc-subvector}) [@code{subvec}] command extracts
20037 a subvector of a vector. The arguments are the vector, the starting
20038 index, and the ending index, with the ending index in the top-of-stack
20039 position. The starting index indicates the first element of the vector
20040 to take. The ending index indicates the first element @emph{past} the
20041 range to be taken. Thus, @samp{subvec([a, b, c, d, e], 2, 4)} produces
20042 the subvector @samp{[b, c]}. You could get the same result using
20043 @samp{mrow([a, b, c, d, e], @w{[2 .. 4)})}.
20045 If either the start or the end index is zero or negative, it is
20046 interpreted as relative to the end of the vector. Thus
20047 @samp{subvec([a, b, c, d, e], 2, -2)} also produces @samp{[b, c]}. In
20048 the algebraic form, the end index can be omitted in which case it
20049 is taken as zero, i.e., elements from the starting element to the
20050 end of the vector are used. The infinity symbol, @code{inf}, also
20051 has this effect when used as the ending index.
20055 With the Inverse flag, @kbd{I v s} [@code{rsubvec}] removes a subvector
20056 from a vector. The arguments are interpreted the same as for the
20057 normal @kbd{v s} command. Thus, @samp{rsubvec([a, b, c, d, e], 2, 4)}
20058 produces @samp{[a, d, e]}. It is always true that @code{subvec} and
20059 @code{rsubvec} return complementary parts of the input vector.
20061 @xref{Selecting Subformulas}, for an alternative way to operate on
20062 vectors one element at a time.
20064 @node Manipulating Vectors, Vector and Matrix Arithmetic, Extracting Elements, Matrix Functions
20065 @section Manipulating Vectors
20069 @pindex calc-vlength
20071 The @kbd{v l} (@code{calc-vlength}) [@code{vlen}] command computes the
20072 length of a vector. The length of a non-vector is considered to be zero.
20073 Note that matrices are just vectors of vectors for the purposes of this
20078 With the Hyperbolic flag, @kbd{H v l} [@code{mdims}] computes a vector
20079 of the dimensions of a vector, matrix, or higher-order object. For
20080 example, @samp{mdims([[a,b,c],[d,e,f]])} returns @samp{[2, 3]} since
20082 @texline @math{2\times3}
20087 @pindex calc-vector-find
20089 The @kbd{v f} (@code{calc-vector-find}) [@code{find}] command searches
20090 along a vector for the first element equal to a given target. The target
20091 is on the top of the stack; the vector is in the second-to-top position.
20092 If a match is found, the result is the index of the matching element.
20093 Otherwise, the result is zero. The numeric prefix argument, if given,
20094 allows you to select any starting index for the search.
20097 @pindex calc-arrange-vector
20099 @cindex Arranging a matrix
20100 @cindex Reshaping a matrix
20101 @cindex Flattening a matrix
20102 The @kbd{v a} (@code{calc-arrange-vector}) [@code{arrange}] command
20103 rearranges a vector to have a certain number of columns and rows. The
20104 numeric prefix argument specifies the number of columns; if you do not
20105 provide an argument, you will be prompted for the number of columns.
20106 The vector or matrix on the top of the stack is @dfn{flattened} into a
20107 plain vector. If the number of columns is nonzero, this vector is
20108 then formed into a matrix by taking successive groups of @var{n} elements.
20109 If the number of columns does not evenly divide the number of elements
20110 in the vector, the last row will be short and the result will not be
20111 suitable for use as a matrix. For example, with the matrix
20112 @samp{[[1, 2], @w{[3, 4]}]} on the stack, @kbd{v a 4} produces
20113 @samp{[[1, 2, 3, 4]]} (a
20114 @texline @math{1\times4}
20116 matrix), @kbd{v a 1} produces @samp{[[1], [2], [3], [4]]} (a
20117 @texline @math{4\times1}
20119 matrix), @kbd{v a 2} produces @samp{[[1, 2], [3, 4]]} (the original
20120 @texline @math{2\times2}
20122 matrix), @w{@kbd{v a 3}} produces @samp{[[1, 2, 3], [4]]} (not a
20123 matrix), and @kbd{v a 0} produces the flattened list
20124 @samp{[1, 2, @w{3, 4}]}.
20126 @cindex Sorting data
20132 The @kbd{V S} (@code{calc-sort}) [@code{sort}] command sorts the elements of
20133 a vector into increasing order. Real numbers, real infinities, and
20134 constant interval forms come first in this ordering; next come other
20135 kinds of numbers, then variables (in alphabetical order), then finally
20136 come formulas and other kinds of objects; these are sorted according
20137 to a kind of lexicographic ordering with the useful property that
20138 one vector is less or greater than another if the first corresponding
20139 unequal elements are less or greater, respectively. Since quoted strings
20140 are stored by Calc internally as vectors of ASCII character codes
20141 (@pxref{Strings}), this means vectors of strings are also sorted into
20142 alphabetical order by this command.
20144 The @kbd{I V S} [@code{rsort}] command sorts a vector into decreasing order.
20146 @cindex Permutation, inverse of
20147 @cindex Inverse of permutation
20148 @cindex Index tables
20149 @cindex Rank tables
20155 The @kbd{V G} (@code{calc-grade}) [@code{grade}, @code{rgrade}] command
20156 produces an index table or permutation vector which, if applied to the
20157 input vector (as the index of @kbd{C-u v r}, say), would sort the vector.
20158 A permutation vector is just a vector of integers from 1 to @var{n}, where
20159 each integer occurs exactly once. One application of this is to sort a
20160 matrix of data rows using one column as the sort key; extract that column,
20161 grade it with @kbd{V G}, then use the result to reorder the original matrix
20162 with @kbd{C-u v r}. Another interesting property of the @code{V G} command
20163 is that, if the input is itself a permutation vector, the result will
20164 be the inverse of the permutation. The inverse of an index table is
20165 a rank table, whose @var{k}th element says where the @var{k}th original
20166 vector element will rest when the vector is sorted. To get a rank
20167 table, just use @kbd{V G V G}.
20169 With the Inverse flag, @kbd{I V G} produces an index table that would
20170 sort the input into decreasing order. Note that @kbd{V S} and @kbd{V G}
20171 use a ``stable'' sorting algorithm, i.e., any two elements which are equal
20172 will not be moved out of their original order. Generally there is no way
20173 to tell with @kbd{V S}, since two elements which are equal look the same,
20174 but with @kbd{V G} this can be an important issue. In the matrix-of-rows
20175 example, suppose you have names and telephone numbers as two columns and
20176 you wish to sort by phone number primarily, and by name when the numbers
20177 are equal. You can sort the data matrix by names first, and then again
20178 by phone numbers. Because the sort is stable, any two rows with equal
20179 phone numbers will remain sorted by name even after the second sort.
20183 @pindex calc-histogram
20185 @mindex histo@idots
20188 The @kbd{V H} (@code{calc-histogram}) [@code{histogram}] command builds a
20189 histogram of a vector of numbers. Vector elements are assumed to be
20190 integers or real numbers in the range [0..@var{n}) for some ``number of
20191 bins'' @var{n}, which is the numeric prefix argument given to the
20192 command. The result is a vector of @var{n} counts of how many times
20193 each value appeared in the original vector. Non-integers in the input
20194 are rounded down to integers. Any vector elements outside the specified
20195 range are ignored. (You can tell if elements have been ignored by noting
20196 that the counts in the result vector don't add up to the length of the
20200 With the Hyperbolic flag, @kbd{H V H} pulls two vectors from the stack.
20201 The second-to-top vector is the list of numbers as before. The top
20202 vector is an equal-sized list of ``weights'' to attach to the elements
20203 of the data vector. For example, if the first data element is 4.2 and
20204 the first weight is 10, then 10 will be added to bin 4 of the result
20205 vector. Without the hyperbolic flag, every element has a weight of one.
20208 @pindex calc-transpose
20210 The @kbd{v t} (@code{calc-transpose}) [@code{trn}] command computes
20211 the transpose of the matrix at the top of the stack. If the argument
20212 is a plain vector, it is treated as a row vector and transposed into
20213 a one-column matrix.
20216 @pindex calc-reverse-vector
20218 The @kbd{v v} (@code{calc-reverse-vector}) [@code{rev}] command reverses
20219 a vector end-for-end. Given a matrix, it reverses the order of the rows.
20220 (To reverse the columns instead, just use @kbd{v t v v v t}. The same
20221 principle can be used to apply other vector commands to the columns of
20225 @pindex calc-mask-vector
20227 The @kbd{v m} (@code{calc-mask-vector}) [@code{vmask}] command uses
20228 one vector as a mask to extract elements of another vector. The mask
20229 is in the second-to-top position; the target vector is on the top of
20230 the stack. These vectors must have the same length. The result is
20231 the same as the target vector, but with all elements which correspond
20232 to zeros in the mask vector deleted. Thus, for example,
20233 @samp{vmask([1, 0, 1, 0, 1], [a, b, c, d, e])} produces @samp{[a, c, e]}.
20234 @xref{Logical Operations}.
20237 @pindex calc-expand-vector
20239 The @kbd{v e} (@code{calc-expand-vector}) [@code{vexp}] command
20240 expands a vector according to another mask vector. The result is a
20241 vector the same length as the mask, but with nonzero elements replaced
20242 by successive elements from the target vector. The length of the target
20243 vector is normally the number of nonzero elements in the mask. If the
20244 target vector is longer, its last few elements are lost. If the target
20245 vector is shorter, the last few nonzero mask elements are left
20246 unreplaced in the result. Thus @samp{vexp([2, 0, 3, 0, 7], [a, b])}
20247 produces @samp{[a, 0, b, 0, 7]}.
20250 With the Hyperbolic flag, @kbd{H v e} takes a filler value from the
20251 top of the stack; the mask and target vectors come from the third and
20252 second elements of the stack. This filler is used where the mask is
20253 zero: @samp{vexp([2, 0, 3, 0, 7], [a, b], z)} produces
20254 @samp{[a, z, c, z, 7]}. If the filler value is itself a vector,
20255 then successive values are taken from it, so that the effect is to
20256 interleave two vectors according to the mask:
20257 @samp{vexp([2, 0, 3, 7, 0, 0], [a, b], [x, y])} produces
20258 @samp{[a, x, b, 7, y, 0]}.
20260 Another variation on the masking idea is to combine @samp{[a, b, c, d, e]}
20261 with the mask @samp{[1, 0, 1, 0, 1]} to produce @samp{[a, 0, c, 0, e]}.
20262 You can accomplish this with @kbd{V M a &}, mapping the logical ``and''
20263 operation across the two vectors. @xref{Logical Operations}. Note that
20264 the @code{? :} operation also discussed there allows other types of
20265 masking using vectors.
20267 @node Vector and Matrix Arithmetic, Set Operations, Manipulating Vectors, Matrix Functions
20268 @section Vector and Matrix Arithmetic
20271 Basic arithmetic operations like addition and multiplication are defined
20272 for vectors and matrices as well as for numbers. Division of matrices, in
20273 the sense of multiplying by the inverse, is supported. (Division by a
20274 matrix actually uses LU-decomposition for greater accuracy and speed.)
20275 @xref{Basic Arithmetic}.
20277 The following functions are applied element-wise if their arguments are
20278 vectors or matrices: @code{change-sign}, @code{conj}, @code{arg},
20279 @code{re}, @code{im}, @code{polar}, @code{rect}, @code{clean},
20280 @code{float}, @code{frac}. @xref{Function Index}.
20283 @pindex calc-conj-transpose
20285 The @kbd{V J} (@code{calc-conj-transpose}) [@code{ctrn}] command computes
20286 the conjugate transpose of its argument, i.e., @samp{conj(trn(x))}.
20291 @kindex A (vectors)
20292 @pindex calc-abs (vectors)
20296 @tindex abs (vectors)
20297 The @kbd{A} (@code{calc-abs}) [@code{abs}] command computes the
20298 Frobenius norm of a vector or matrix argument. This is the square
20299 root of the sum of the squares of the absolute values of the
20300 elements of the vector or matrix. If the vector is interpreted as
20301 a point in two- or three-dimensional space, this is the distance
20302 from that point to the origin.
20307 The @kbd{v n} (@code{calc-rnorm}) [@code{rnorm}] command computes
20308 the row norm, or infinity-norm, of a vector or matrix. For a plain
20309 vector, this is the maximum of the absolute values of the elements.
20310 For a matrix, this is the maximum of the row-absolute-value-sums,
20311 i.e., of the sums of the absolute values of the elements along the
20317 The @kbd{V N} (@code{calc-cnorm}) [@code{cnorm}] command computes
20318 the column norm, or one-norm, of a vector or matrix. For a plain
20319 vector, this is the sum of the absolute values of the elements.
20320 For a matrix, this is the maximum of the column-absolute-value-sums.
20321 General @expr{k}-norms for @expr{k} other than one or infinity are
20327 The @kbd{V C} (@code{calc-cross}) [@code{cross}] command computes the
20328 right-handed cross product of two vectors, each of which must have
20329 exactly three elements.
20334 @kindex & (matrices)
20335 @pindex calc-inv (matrices)
20339 @tindex inv (matrices)
20340 The @kbd{&} (@code{calc-inv}) [@code{inv}] command computes the
20341 inverse of a square matrix. If the matrix is singular, the inverse
20342 operation is left in symbolic form. Matrix inverses are recorded so
20343 that once an inverse (or determinant) of a particular matrix has been
20344 computed, the inverse and determinant of the matrix can be recomputed
20345 quickly in the future.
20347 If the argument to @kbd{&} is a plain number @expr{x}, this
20348 command simply computes @expr{1/x}. This is okay, because the
20349 @samp{/} operator also does a matrix inversion when dividing one
20355 The @kbd{V D} (@code{calc-mdet}) [@code{det}] command computes the
20356 determinant of a square matrix.
20361 The @kbd{V L} (@code{calc-mlud}) [@code{lud}] command computes the
20362 LU decomposition of a matrix. The result is a list of three matrices
20363 which, when multiplied together left-to-right, form the original matrix.
20364 The first is a permutation matrix that arises from pivoting in the
20365 algorithm, the second is lower-triangular with ones on the diagonal,
20366 and the third is upper-triangular.
20369 @pindex calc-mtrace
20371 The @kbd{V T} (@code{calc-mtrace}) [@code{tr}] command computes the
20372 trace of a square matrix. This is defined as the sum of the diagonal
20373 elements of the matrix.
20375 @node Set Operations, Statistical Operations, Vector and Matrix Arithmetic, Matrix Functions
20376 @section Set Operations using Vectors
20379 @cindex Sets, as vectors
20380 Calc includes several commands which interpret vectors as @dfn{sets} of
20381 objects. A set is a collection of objects; any given object can appear
20382 only once in the set. Calc stores sets as vectors of objects in
20383 sorted order. Objects in a Calc set can be any of the usual things,
20384 such as numbers, variables, or formulas. Two set elements are considered
20385 equal if they are identical, except that numerically equal numbers like
20386 the integer 4 and the float 4.0 are considered equal even though they
20387 are not ``identical.'' Variables are treated like plain symbols without
20388 attached values by the set operations; subtracting the set @samp{[b]}
20389 from @samp{[a, b]} always yields the set @samp{[a]} even though if
20390 the variables @samp{a} and @samp{b} both equaled 17, you might
20391 expect the answer @samp{[]}.
20393 If a set contains interval forms, then it is assumed to be a set of
20394 real numbers. In this case, all set operations require the elements
20395 of the set to be only things that are allowed in intervals: Real
20396 numbers, plus and minus infinity, HMS forms, and date forms. If
20397 there are variables or other non-real objects present in a real set,
20398 all set operations on it will be left in unevaluated form.
20400 If the input to a set operation is a plain number or interval form
20401 @var{a}, it is treated like the one-element vector @samp{[@var{a}]}.
20402 The result is always a vector, except that if the set consists of a
20403 single interval, the interval itself is returned instead.
20405 @xref{Logical Operations}, for the @code{in} function which tests if
20406 a certain value is a member of a given set. To test if the set @expr{A}
20407 is a subset of the set @expr{B}, use @samp{vdiff(A, B) = []}.
20410 @pindex calc-remove-duplicates
20412 The @kbd{V +} (@code{calc-remove-duplicates}) [@code{rdup}] command
20413 converts an arbitrary vector into set notation. It works by sorting
20414 the vector as if by @kbd{V S}, then removing duplicates. (For example,
20415 @kbd{[a, 5, 4, a, 4.0]} is sorted to @samp{[4, 4.0, 5, a, a]} and then
20416 reduced to @samp{[4, 5, a]}). Overlapping intervals are merged as
20417 necessary. You rarely need to use @kbd{V +} explicitly, since all the
20418 other set-based commands apply @kbd{V +} to their inputs before using
20422 @pindex calc-set-union
20424 The @kbd{V V} (@code{calc-set-union}) [@code{vunion}] command computes
20425 the union of two sets. An object is in the union of two sets if and
20426 only if it is in either (or both) of the input sets. (You could
20427 accomplish the same thing by concatenating the sets with @kbd{|},
20428 then using @kbd{V +}.)
20431 @pindex calc-set-intersect
20433 The @kbd{V ^} (@code{calc-set-intersect}) [@code{vint}] command computes
20434 the intersection of two sets. An object is in the intersection if
20435 and only if it is in both of the input sets. Thus if the input
20436 sets are disjoint, i.e., if they share no common elements, the result
20437 will be the empty vector @samp{[]}. Note that the characters @kbd{V}
20438 and @kbd{^} were chosen to be close to the conventional mathematical
20440 @texline union@tie{}(@math{A \cup B})
20443 @texline intersection@tie{}(@math{A \cap B}).
20444 @infoline intersection.
20447 @pindex calc-set-difference
20449 The @kbd{V -} (@code{calc-set-difference}) [@code{vdiff}] command computes
20450 the difference between two sets. An object is in the difference
20451 @expr{A - B} if and only if it is in @expr{A} but not in @expr{B}.
20452 Thus subtracting @samp{[y,z]} from a set will remove the elements
20453 @samp{y} and @samp{z} if they are present. You can also think of this
20454 as a general @dfn{set complement} operator; if @expr{A} is the set of
20455 all possible values, then @expr{A - B} is the ``complement'' of @expr{B}.
20456 Obviously this is only practical if the set of all possible values in
20457 your problem is small enough to list in a Calc vector (or simple
20458 enough to express in a few intervals).
20461 @pindex calc-set-xor
20463 The @kbd{V X} (@code{calc-set-xor}) [@code{vxor}] command computes
20464 the ``exclusive-or,'' or ``symmetric difference'' of two sets.
20465 An object is in the symmetric difference of two sets if and only
20466 if it is in one, but @emph{not} both, of the sets. Objects that
20467 occur in both sets ``cancel out.''
20470 @pindex calc-set-complement
20472 The @kbd{V ~} (@code{calc-set-complement}) [@code{vcompl}] command
20473 computes the complement of a set with respect to the real numbers.
20474 Thus @samp{vcompl(x)} is equivalent to @samp{vdiff([-inf .. inf], x)}.
20475 For example, @samp{vcompl([2, (3 .. 4]])} evaluates to
20476 @samp{[[-inf .. 2), (2 .. 3], (4 .. inf]]}.
20479 @pindex calc-set-floor
20481 The @kbd{V F} (@code{calc-set-floor}) [@code{vfloor}] command
20482 reinterprets a set as a set of integers. Any non-integer values,
20483 and intervals that do not enclose any integers, are removed. Open
20484 intervals are converted to equivalent closed intervals. Successive
20485 integers are converted into intervals of integers. For example, the
20486 complement of the set @samp{[2, 6, 7, 8]} is messy, but if you wanted
20487 the complement with respect to the set of integers you could type
20488 @kbd{V ~ V F} to get @samp{[[-inf .. 1], [3 .. 5], [9 .. inf]]}.
20491 @pindex calc-set-enumerate
20493 The @kbd{V E} (@code{calc-set-enumerate}) [@code{venum}] command
20494 converts a set of integers into an explicit vector. Intervals in
20495 the set are expanded out to lists of all integers encompassed by
20496 the intervals. This only works for finite sets (i.e., sets which
20497 do not involve @samp{-inf} or @samp{inf}).
20500 @pindex calc-set-span
20502 The @kbd{V :} (@code{calc-set-span}) [@code{vspan}] command converts any
20503 set of reals into an interval form that encompasses all its elements.
20504 The lower limit will be the smallest element in the set; the upper
20505 limit will be the largest element. For an empty set, @samp{vspan([])}
20506 returns the empty interval @w{@samp{[0 .. 0)}}.
20509 @pindex calc-set-cardinality
20511 The @kbd{V #} (@code{calc-set-cardinality}) [@code{vcard}] command counts
20512 the number of integers in a set. The result is the length of the vector
20513 that would be produced by @kbd{V E}, although the computation is much
20514 more efficient than actually producing that vector.
20516 @cindex Sets, as binary numbers
20517 Another representation for sets that may be more appropriate in some
20518 cases is binary numbers. If you are dealing with sets of integers
20519 in the range 0 to 49, you can use a 50-bit binary number where a
20520 particular bit is 1 if the corresponding element is in the set.
20521 @xref{Binary Functions}, for a list of commands that operate on
20522 binary numbers. Note that many of the above set operations have
20523 direct equivalents in binary arithmetic: @kbd{b o} (@code{calc-or}),
20524 @kbd{b a} (@code{calc-and}), @kbd{b d} (@code{calc-diff}),
20525 @kbd{b x} (@code{calc-xor}), and @kbd{b n} (@code{calc-not}),
20526 respectively. You can use whatever representation for sets is most
20531 @pindex calc-pack-bits
20532 @pindex calc-unpack-bits
20535 The @kbd{b u} (@code{calc-unpack-bits}) [@code{vunpack}] command
20536 converts an integer that represents a set in binary into a set
20537 in vector/interval notation. For example, @samp{vunpack(67)}
20538 returns @samp{[[0 .. 1], 6]}. If the input is negative, the set
20539 it represents is semi-infinite: @samp{vunpack(-4) = [2 .. inf)}.
20540 Use @kbd{V E} afterwards to expand intervals to individual
20541 values if you wish. Note that this command uses the @kbd{b}
20542 (binary) prefix key.
20544 The @kbd{b p} (@code{calc-pack-bits}) [@code{vpack}] command
20545 converts the other way, from a vector or interval representing
20546 a set of nonnegative integers into a binary integer describing
20547 the same set. The set may include positive infinity, but must
20548 not include any negative numbers. The input is interpreted as a
20549 set of integers in the sense of @kbd{V F} (@code{vfloor}). Beware
20550 that a simple input like @samp{[100]} can result in a huge integer
20552 @texline (@math{2^{100}}, a 31-digit integer, in this case).
20553 @infoline (@expr{2^100}, a 31-digit integer, in this case).
20555 @node Statistical Operations, Reducing and Mapping, Set Operations, Matrix Functions
20556 @section Statistical Operations on Vectors
20559 @cindex Statistical functions
20560 The commands in this section take vectors as arguments and compute
20561 various statistical measures on the data stored in the vectors. The
20562 references used in the definitions of these functions are Bevington's
20563 @emph{Data Reduction and Error Analysis for the Physical Sciences},
20564 and @emph{Numerical Recipes} by Press, Flannery, Teukolsky and
20567 The statistical commands use the @kbd{u} prefix key followed by
20568 a shifted letter or other character.
20570 @xref{Manipulating Vectors}, for a description of @kbd{V H}
20571 (@code{calc-histogram}).
20573 @xref{Curve Fitting}, for the @kbd{a F} command for doing
20574 least-squares fits to statistical data.
20576 @xref{Probability Distribution Functions}, for several common
20577 probability distribution functions.
20580 * Single-Variable Statistics::
20581 * Paired-Sample Statistics::
20584 @node Single-Variable Statistics, Paired-Sample Statistics, Statistical Operations, Statistical Operations
20585 @subsection Single-Variable Statistics
20588 These functions do various statistical computations on single
20589 vectors. Given a numeric prefix argument, they actually pop
20590 @var{n} objects from the stack and combine them into a data
20591 vector. Each object may be either a number or a vector; if a
20592 vector, any sub-vectors inside it are ``flattened'' as if by
20593 @kbd{v a 0}; @pxref{Manipulating Vectors}. By default one object
20594 is popped, which (in order to be useful) is usually a vector.
20596 If an argument is a variable name, and the value stored in that
20597 variable is a vector, then the stored vector is used. This method
20598 has the advantage that if your data vector is large, you can avoid
20599 the slow process of manipulating it directly on the stack.
20601 These functions are left in symbolic form if any of their arguments
20602 are not numbers or vectors, e.g., if an argument is a formula, or
20603 a non-vector variable. However, formulas embedded within vector
20604 arguments are accepted; the result is a symbolic representation
20605 of the computation, based on the assumption that the formula does
20606 not itself represent a vector. All varieties of numbers such as
20607 error forms and interval forms are acceptable.
20609 Some of the functions in this section also accept a single error form
20610 or interval as an argument. They then describe a property of the
20611 normal or uniform (respectively) statistical distribution described
20612 by the argument. The arguments are interpreted in the same way as
20613 the @var{M} argument of the random number function @kbd{k r}. In
20614 particular, an interval with integer limits is considered an integer
20615 distribution, so that @samp{[2 .. 6)} is the same as @samp{[2 .. 5]}.
20616 An interval with at least one floating-point limit is a continuous
20617 distribution: @samp{[2.0 .. 6.0)} is @emph{not} the same as
20618 @samp{[2.0 .. 5.0]}!
20621 @pindex calc-vector-count
20623 The @kbd{u #} (@code{calc-vector-count}) [@code{vcount}] command
20624 computes the number of data values represented by the inputs.
20625 For example, @samp{vcount(1, [2, 3], [[4, 5], [], x, y])} returns 7.
20626 If the argument is a single vector with no sub-vectors, this
20627 simply computes the length of the vector.
20631 @pindex calc-vector-sum
20632 @pindex calc-vector-prod
20635 @cindex Summations (statistical)
20636 The @kbd{u +} (@code{calc-vector-sum}) [@code{vsum}] command
20637 computes the sum of the data values. The @kbd{u *}
20638 (@code{calc-vector-prod}) [@code{vprod}] command computes the
20639 product of the data values. If the input is a single flat vector,
20640 these are the same as @kbd{V R +} and @kbd{V R *}
20641 (@pxref{Reducing and Mapping}).
20645 @pindex calc-vector-max
20646 @pindex calc-vector-min
20649 The @kbd{u X} (@code{calc-vector-max}) [@code{vmax}] command
20650 computes the maximum of the data values, and the @kbd{u N}
20651 (@code{calc-vector-min}) [@code{vmin}] command computes the minimum.
20652 If the argument is an interval, this finds the minimum or maximum
20653 value in the interval. (Note that @samp{vmax([2..6)) = 5} as
20654 described above.) If the argument is an error form, this returns
20655 plus or minus infinity.
20658 @pindex calc-vector-mean
20660 @cindex Mean of data values
20661 The @kbd{u M} (@code{calc-vector-mean}) [@code{vmean}] command
20662 computes the average (arithmetic mean) of the data values.
20663 If the inputs are error forms
20664 @texline @math{x \pm \sigma},
20665 @infoline @samp{x +/- s},
20666 this is the weighted mean of the @expr{x} values with weights
20667 @texline @math{1 /\sigma^2}.
20668 @infoline @expr{1 / s^2}.
20671 $$ \mu = { \displaystyle \sum { x_i \over \sigma_i^2 } \over
20672 \displaystyle \sum { 1 \over \sigma_i^2 } } $$
20674 If the inputs are not error forms, this is simply the sum of the
20675 values divided by the count of the values.
20677 Note that a plain number can be considered an error form with
20679 @texline @math{\sigma = 0}.
20680 @infoline @expr{s = 0}.
20681 If the input to @kbd{u M} is a mixture of
20682 plain numbers and error forms, the result is the mean of the
20683 plain numbers, ignoring all values with non-zero errors. (By the
20684 above definitions it's clear that a plain number effectively
20685 has an infinite weight, next to which an error form with a finite
20686 weight is completely negligible.)
20688 This function also works for distributions (error forms or
20689 intervals). The mean of an error form `@var{a} @tfn{+/-} @var{b}' is simply
20690 @expr{a}. The mean of an interval is the mean of the minimum
20691 and maximum values of the interval.
20694 @pindex calc-vector-mean-error
20696 The @kbd{I u M} (@code{calc-vector-mean-error}) [@code{vmeane}]
20697 command computes the mean of the data points expressed as an
20698 error form. This includes the estimated error associated with
20699 the mean. If the inputs are error forms, the error is the square
20700 root of the reciprocal of the sum of the reciprocals of the squares
20701 of the input errors. (I.e., the variance is the reciprocal of the
20702 sum of the reciprocals of the variances.)
20705 $$ \sigma_\mu^2 = {1 \over \displaystyle \sum {1 \over \sigma_i^2}} $$
20707 If the inputs are plain
20708 numbers, the error is equal to the standard deviation of the values
20709 divided by the square root of the number of values. (This works
20710 out to be equivalent to calculating the standard deviation and
20711 then assuming each value's error is equal to this standard
20715 $$ \sigma_\mu^2 = {\sigma^2 \over N} $$
20719 @pindex calc-vector-median
20721 @cindex Median of data values
20722 The @kbd{H u M} (@code{calc-vector-median}) [@code{vmedian}]
20723 command computes the median of the data values. The values are
20724 first sorted into numerical order; the median is the middle
20725 value after sorting. (If the number of data values is even,
20726 the median is taken to be the average of the two middle values.)
20727 The median function is different from the other functions in
20728 this section in that the arguments must all be real numbers;
20729 variables are not accepted even when nested inside vectors.
20730 (Otherwise it is not possible to sort the data values.) If
20731 any of the input values are error forms, their error parts are
20734 The median function also accepts distributions. For both normal
20735 (error form) and uniform (interval) distributions, the median is
20736 the same as the mean.
20739 @pindex calc-vector-harmonic-mean
20741 @cindex Harmonic mean
20742 The @kbd{H I u M} (@code{calc-vector-harmonic-mean}) [@code{vhmean}]
20743 command computes the harmonic mean of the data values. This is
20744 defined as the reciprocal of the arithmetic mean of the reciprocals
20748 $$ { N \over \displaystyle \sum {1 \over x_i} } $$
20752 @pindex calc-vector-geometric-mean
20754 @cindex Geometric mean
20755 The @kbd{u G} (@code{calc-vector-geometric-mean}) [@code{vgmean}]
20756 command computes the geometric mean of the data values. This
20757 is the @var{n}th root of the product of the values. This is also
20758 equal to the @code{exp} of the arithmetic mean of the logarithms
20759 of the data values.
20762 $$ \exp \left ( \sum { \ln x_i } \right ) =
20763 \left ( \prod { x_i } \right)^{1 / N} $$
20768 The @kbd{H u G} [@code{agmean}] command computes the ``arithmetic-geometric
20769 mean'' of two numbers taken from the stack. This is computed by
20770 replacing the two numbers with their arithmetic mean and geometric
20771 mean, then repeating until the two values converge.
20774 $$ a_{i+1} = { a_i + b_i \over 2 } , \qquad b_{i+1} = \sqrt{a_i b_i} $$
20777 @cindex Root-mean-square
20778 Another commonly used mean, the RMS (root-mean-square), can be computed
20779 for a vector of numbers simply by using the @kbd{A} command.
20782 @pindex calc-vector-sdev
20784 @cindex Standard deviation
20785 @cindex Sample statistics
20786 The @kbd{u S} (@code{calc-vector-sdev}) [@code{vsdev}] command
20787 computes the standard
20788 @texline deviation@tie{}@math{\sigma}
20789 @infoline deviation
20790 of the data values. If the values are error forms, the errors are used
20791 as weights just as for @kbd{u M}. This is the @emph{sample} standard
20792 deviation, whose value is the square root of the sum of the squares of
20793 the differences between the values and the mean of the @expr{N} values,
20794 divided by @expr{N-1}.
20797 $$ \sigma^2 = {1 \over N - 1} \sum (x_i - \mu)^2 $$
20800 This function also applies to distributions. The standard deviation
20801 of a single error form is simply the error part. The standard deviation
20802 of a continuous interval happens to equal the difference between the
20804 @texline @math{\sqrt{12}}.
20805 @infoline @expr{sqrt(12)}.
20806 The standard deviation of an integer interval is the same as the
20807 standard deviation of a vector of those integers.
20810 @pindex calc-vector-pop-sdev
20812 @cindex Population statistics
20813 The @kbd{I u S} (@code{calc-vector-pop-sdev}) [@code{vpsdev}]
20814 command computes the @emph{population} standard deviation.
20815 It is defined by the same formula as above but dividing
20816 by @expr{N} instead of by @expr{N-1}. The population standard
20817 deviation is used when the input represents the entire set of
20818 data values in the distribution; the sample standard deviation
20819 is used when the input represents a sample of the set of all
20820 data values, so that the mean computed from the input is itself
20821 only an estimate of the true mean.
20824 $$ \sigma^2 = {1 \over N} \sum (x_i - \mu)^2 $$
20827 For error forms and continuous intervals, @code{vpsdev} works
20828 exactly like @code{vsdev}. For integer intervals, it computes the
20829 population standard deviation of the equivalent vector of integers.
20833 @pindex calc-vector-variance
20834 @pindex calc-vector-pop-variance
20837 @cindex Variance of data values
20838 The @kbd{H u S} (@code{calc-vector-variance}) [@code{vvar}] and
20839 @kbd{H I u S} (@code{calc-vector-pop-variance}) [@code{vpvar}]
20840 commands compute the variance of the data values. The variance
20842 @texline square@tie{}@math{\sigma^2}
20844 of the standard deviation, i.e., the sum of the
20845 squares of the deviations of the data values from the mean.
20846 (This definition also applies when the argument is a distribution.)
20852 The @code{vflat} algebraic function returns a vector of its
20853 arguments, interpreted in the same way as the other functions
20854 in this section. For example, @samp{vflat(1, [2, [3, 4]], 5)}
20855 returns @samp{[1, 2, 3, 4, 5]}.
20857 @node Paired-Sample Statistics, , Single-Variable Statistics, Statistical Operations
20858 @subsection Paired-Sample Statistics
20861 The functions in this section take two arguments, which must be
20862 vectors of equal size. The vectors are each flattened in the same
20863 way as by the single-variable statistical functions. Given a numeric
20864 prefix argument of 1, these functions instead take one object from
20865 the stack, which must be an
20866 @texline @math{N\times2}
20868 matrix of data values. Once again, variable names can be used in place
20869 of actual vectors and matrices.
20872 @pindex calc-vector-covariance
20875 The @kbd{u C} (@code{calc-vector-covariance}) [@code{vcov}] command
20876 computes the sample covariance of two vectors. The covariance
20877 of vectors @var{x} and @var{y} is the sum of the products of the
20878 differences between the elements of @var{x} and the mean of @var{x}
20879 times the differences between the corresponding elements of @var{y}
20880 and the mean of @var{y}, all divided by @expr{N-1}. Note that
20881 the variance of a vector is just the covariance of the vector
20882 with itself. Once again, if the inputs are error forms the
20883 errors are used as weight factors. If both @var{x} and @var{y}
20884 are composed of error forms, the error for a given data point
20885 is taken as the square root of the sum of the squares of the two
20889 $$ \sigma_{x\!y}^2 = {1 \over N-1} \sum (x_i - \mu_x) (y_i - \mu_y) $$
20890 $$ \sigma_{x\!y}^2 =
20891 {\displaystyle {1 \over N-1}
20892 \sum {(x_i - \mu_x) (y_i - \mu_y) \over \sigma_i^2}
20893 \over \displaystyle {1 \over N} \sum {1 \over \sigma_i^2}}
20898 @pindex calc-vector-pop-covariance
20900 The @kbd{I u C} (@code{calc-vector-pop-covariance}) [@code{vpcov}]
20901 command computes the population covariance, which is the same as the
20902 sample covariance computed by @kbd{u C} except dividing by @expr{N}
20903 instead of @expr{N-1}.
20906 @pindex calc-vector-correlation
20908 @cindex Correlation coefficient
20909 @cindex Linear correlation
20910 The @kbd{H u C} (@code{calc-vector-correlation}) [@code{vcorr}]
20911 command computes the linear correlation coefficient of two vectors.
20912 This is defined by the covariance of the vectors divided by the
20913 product of their standard deviations. (There is no difference
20914 between sample or population statistics here.)
20917 $$ r_{x\!y} = { \sigma_{x\!y}^2 \over \sigma_x^2 \sigma_y^2 } $$
20920 @node Reducing and Mapping, Vector and Matrix Formats, Statistical Operations, Matrix Functions
20921 @section Reducing and Mapping Vectors
20924 The commands in this section allow for more general operations on the
20925 elements of vectors.
20930 The simplest of these operations is @kbd{V A} (@code{calc-apply})
20931 [@code{apply}], which applies a given operator to the elements of a vector.
20932 For example, applying the hypothetical function @code{f} to the vector
20933 @w{@samp{[1, 2, 3]}} would produce the function call @samp{f(1, 2, 3)}.
20934 Applying the @code{+} function to the vector @samp{[a, b]} gives
20935 @samp{a + b}. Applying @code{+} to the vector @samp{[a, b, c]} is an
20936 error, since the @code{+} function expects exactly two arguments.
20938 While @kbd{V A} is useful in some cases, you will usually find that either
20939 @kbd{V R} or @kbd{V M}, described below, is closer to what you want.
20942 * Specifying Operators::
20945 * Nesting and Fixed Points::
20946 * Generalized Products::
20949 @node Specifying Operators, Mapping, Reducing and Mapping, Reducing and Mapping
20950 @subsection Specifying Operators
20953 Commands in this section (like @kbd{V A}) prompt you to press the key
20954 corresponding to the desired operator. Press @kbd{?} for a partial
20955 list of the available operators. Generally, an operator is any key or
20956 sequence of keys that would normally take one or more arguments from
20957 the stack and replace them with a result. For example, @kbd{V A H C}
20958 uses the hyperbolic cosine operator, @code{cosh}. (Since @code{cosh}
20959 expects one argument, @kbd{V A H C} requires a vector with a single
20960 element as its argument.)
20962 You can press @kbd{x} at the operator prompt to select any algebraic
20963 function by name to use as the operator. This includes functions you
20964 have defined yourself using the @kbd{Z F} command. (@xref{Algebraic
20965 Definitions}.) If you give a name for which no function has been
20966 defined, the result is left in symbolic form, as in @samp{f(1, 2, 3)}.
20967 Calc will prompt for the number of arguments the function takes if it
20968 can't figure it out on its own (say, because you named a function that
20969 is currently undefined). It is also possible to type a digit key before
20970 the function name to specify the number of arguments, e.g.,
20971 @kbd{V M 3 x f @key{RET}} calls @code{f} with three arguments even if it
20972 looks like it ought to have only two. This technique may be necessary
20973 if the function allows a variable number of arguments. For example,
20974 the @kbd{v e} [@code{vexp}] function accepts two or three arguments;
20975 if you want to map with the three-argument version, you will have to
20976 type @kbd{V M 3 v e}.
20978 It is also possible to apply any formula to a vector by treating that
20979 formula as a function. When prompted for the operator to use, press
20980 @kbd{'} (the apostrophe) and type your formula as an algebraic entry.
20981 You will then be prompted for the argument list, which defaults to a
20982 list of all variables that appear in the formula, sorted into alphabetic
20983 order. For example, suppose you enter the formula @w{@samp{x + 2y^x}}.
20984 The default argument list would be @samp{(x y)}, which means that if
20985 this function is applied to the arguments @samp{[3, 10]} the result will
20986 be @samp{3 + 2*10^3}. (If you plan to use a certain formula in this
20987 way often, you might consider defining it as a function with @kbd{Z F}.)
20989 Another way to specify the arguments to the formula you enter is with
20990 @kbd{$}, @kbd{$$}, and so on. For example, @kbd{V A ' $$ + 2$^$$}
20991 has the same effect as the previous example. The argument list is
20992 automatically taken to be @samp{($$ $)}. (The order of the arguments
20993 may seem backwards, but it is analogous to the way normal algebraic
20994 entry interacts with the stack.)
20996 If you press @kbd{$} at the operator prompt, the effect is similar to
20997 the apostrophe except that the relevant formula is taken from top-of-stack
20998 instead. The actual vector arguments of the @kbd{V A $} or related command
20999 then start at the second-to-top stack position. You will still be
21000 prompted for an argument list.
21002 @cindex Nameless functions
21003 @cindex Generic functions
21004 A function can be written without a name using the notation @samp{<#1 - #2>},
21005 which means ``a function of two arguments that computes the first
21006 argument minus the second argument.'' The symbols @samp{#1} and @samp{#2}
21007 are placeholders for the arguments. You can use any names for these
21008 placeholders if you wish, by including an argument list followed by a
21009 colon: @samp{<x, y : x - y>}. When you type @kbd{V A ' $$ + 2$^$$ @key{RET}},
21010 Calc builds the nameless function @samp{<#1 + 2 #2^#1>} as the function
21011 to map across the vectors. When you type @kbd{V A ' x + 2y^x @key{RET} @key{RET}},
21012 Calc builds the nameless function @w{@samp{<x, y : x + 2 y^x>}}. In both
21013 cases, Calc also writes the nameless function to the Trail so that you
21014 can get it back later if you wish.
21016 If there is only one argument, you can write @samp{#} in place of @samp{#1}.
21017 (Note that @samp{< >} notation is also used for date forms. Calc tells
21018 that @samp{<@var{stuff}>} is a nameless function by the presence of
21019 @samp{#} signs inside @var{stuff}, or by the fact that @var{stuff}
21020 begins with a list of variables followed by a colon.)
21022 You can type a nameless function directly to @kbd{V A '}, or put one on
21023 the stack and use it with @w{@kbd{V A $}}. Calc will not prompt for an
21024 argument list in this case, since the nameless function specifies the
21025 argument list as well as the function itself. In @kbd{V A '}, you can
21026 omit the @samp{< >} marks if you use @samp{#} notation for the arguments,
21027 so that @kbd{V A ' #1+#2 @key{RET}} is the same as @kbd{V A ' <#1+#2> @key{RET}},
21028 which in turn is the same as @kbd{V A ' $$+$ @key{RET}}.
21030 @cindex Lambda expressions
21035 The internal format for @samp{<x, y : x + y>} is @samp{lambda(x, y, x + y)}.
21036 (The word @code{lambda} derives from Lisp notation and the theory of
21037 functions.) The internal format for @samp{<#1 + #2>} is @samp{lambda(ArgA,
21038 ArgB, ArgA + ArgB)}. Note that there is no actual Calc function called
21039 @code{lambda}; the whole point is that the @code{lambda} expression is
21040 used in its symbolic form, not evaluated for an answer until it is applied
21041 to specific arguments by a command like @kbd{V A} or @kbd{V M}.
21043 (Actually, @code{lambda} does have one special property: Its arguments
21044 are never evaluated; for example, putting @samp{<(2/3) #>} on the stack
21045 will not simplify the @samp{2/3} until the nameless function is actually
21074 As usual, commands like @kbd{V A} have algebraic function name equivalents.
21075 For example, @kbd{V A k g} with an argument of @samp{v} is equivalent to
21076 @samp{apply(gcd, v)}. The first argument specifies the operator name,
21077 and is either a variable whose name is the same as the function name,
21078 or a nameless function like @samp{<#^3+1>}. Operators that are normally
21079 written as algebraic symbols have the names @code{add}, @code{sub},
21080 @code{mul}, @code{div}, @code{pow}, @code{neg}, @code{mod}, and
21087 The @code{call} function builds a function call out of several arguments:
21088 @samp{call(gcd, x, y)} is the same as @samp{apply(gcd, [x, y])}, which
21089 in turn is the same as @samp{gcd(x, y)}. The first argument of @code{call},
21090 like the other functions described here, may be either a variable naming a
21091 function, or a nameless function (@samp{call(<#1+2#2>, x, y)} is the same
21094 (Experts will notice that it's not quite proper to use a variable to name
21095 a function, since the name @code{gcd} corresponds to the Lisp variable
21096 @code{var-gcd} but to the Lisp function @code{calcFunc-gcd}. Calc
21097 automatically makes this translation, so you don't have to worry
21100 @node Mapping, Reducing, Specifying Operators, Reducing and Mapping
21101 @subsection Mapping
21107 The @kbd{V M} (@code{calc-map}) [@code{map}] command applies a given
21108 operator elementwise to one or more vectors. For example, mapping
21109 @code{A} [@code{abs}] produces a vector of the absolute values of the
21110 elements in the input vector. Mapping @code{+} pops two vectors from
21111 the stack, which must be of equal length, and produces a vector of the
21112 pairwise sums of the elements. If either argument is a non-vector, it
21113 is duplicated for each element of the other vector. For example,
21114 @kbd{[1,2,3] 2 V M ^} squares the elements of the specified vector.
21115 With the 2 listed first, it would have computed a vector of powers of
21116 two. Mapping a user-defined function pops as many arguments from the
21117 stack as the function requires. If you give an undefined name, you will
21118 be prompted for the number of arguments to use.
21120 If any argument to @kbd{V M} is a matrix, the operator is normally mapped
21121 across all elements of the matrix. For example, given the matrix
21122 @expr{[[1, -2, 3], [-4, 5, -6]]}, @kbd{V M A} takes six absolute values to
21124 @texline @math{3\times2}
21126 matrix, @expr{[[1, 2, 3], [4, 5, 6]]}.
21129 The command @kbd{V M _} [@code{mapr}] (i.e., type an underscore at the
21130 operator prompt) maps by rows instead. For example, @kbd{V M _ A} views
21131 the above matrix as a vector of two 3-element row vectors. It produces
21132 a new vector which contains the absolute values of those row vectors,
21133 namely @expr{[3.74, 8.77]}. (Recall, the absolute value of a vector is
21134 defined as the square root of the sum of the squares of the elements.)
21135 Some operators accept vectors and return new vectors; for example,
21136 @kbd{v v} reverses a vector, so @kbd{V M _ v v} would reverse each row
21137 of the matrix to get a new matrix, @expr{[[3, -2, 1], [-6, 5, -4]]}.
21139 Sometimes a vector of vectors (representing, say, strings, sets, or lists)
21140 happens to look like a matrix. If so, remember to use @kbd{V M _} if you
21141 want to map a function across the whole strings or sets rather than across
21142 their individual elements.
21145 The command @kbd{V M :} [@code{mapc}] maps by columns. Basically, it
21146 transposes the input matrix, maps by rows, and then, if the result is a
21147 matrix, transposes again. For example, @kbd{V M : A} takes the absolute
21148 values of the three columns of the matrix, treating each as a 2-vector,
21149 and @kbd{V M : v v} reverses the columns to get the matrix
21150 @expr{[[-4, 5, -6], [1, -2, 3]]}.
21152 (The symbols @kbd{_} and @kbd{:} were chosen because they had row-like
21153 and column-like appearances, and were not already taken by useful
21154 operators. Also, they appear shifted on most keyboards so they are easy
21155 to type after @kbd{V M}.)
21157 The @kbd{_} and @kbd{:} modifiers have no effect on arguments that are
21158 not matrices (so if none of the arguments are matrices, they have no
21159 effect at all). If some of the arguments are matrices and others are
21160 plain numbers, the plain numbers are held constant for all rows of the
21161 matrix (so that @kbd{2 V M _ ^} squares every row of a matrix; squaring
21162 a vector takes a dot product of the vector with itself).
21164 If some of the arguments are vectors with the same lengths as the
21165 rows (for @kbd{V M _}) or columns (for @kbd{V M :}) of the matrix
21166 arguments, those vectors are also held constant for every row or
21169 Sometimes it is useful to specify another mapping command as the operator
21170 to use with @kbd{V M}. For example, @kbd{V M _ V A +} applies @kbd{V A +}
21171 to each row of the input matrix, which in turn adds the two values on that
21172 row. If you give another vector-operator command as the operator for
21173 @kbd{V M}, it automatically uses map-by-rows mode if you don't specify
21174 otherwise; thus @kbd{V M V A +} is equivalent to @kbd{V M _ V A +}. (If
21175 you really want to map-by-elements another mapping command, you can use
21176 a triple-nested mapping command: @kbd{V M V M V A +} means to map
21177 @kbd{V M V A +} over the rows of the matrix; in turn, @kbd{V A +} is
21178 mapped over the elements of each row.)
21182 Previous versions of Calc had ``map across'' and ``map down'' modes
21183 that are now considered obsolete; the old ``map across'' is now simply
21184 @kbd{V M V A}, and ``map down'' is now @kbd{V M : V A}. The algebraic
21185 functions @code{mapa} and @code{mapd} are still supported, though.
21186 Note also that, while the old mapping modes were persistent (once you
21187 set the mode, it would apply to later mapping commands until you reset
21188 it), the new @kbd{:} and @kbd{_} modifiers apply only to the current
21189 mapping command. The default @kbd{V M} always means map-by-elements.
21191 @xref{Algebraic Manipulation}, for the @kbd{a M} command, which is like
21192 @kbd{V M} but for equations and inequalities instead of vectors.
21193 @xref{Storing Variables}, for the @kbd{s m} command which modifies a
21194 variable's stored value using a @kbd{V M}-like operator.
21196 @node Reducing, Nesting and Fixed Points, Mapping, Reducing and Mapping
21197 @subsection Reducing
21201 @pindex calc-reduce
21203 The @kbd{V R} (@code{calc-reduce}) [@code{reduce}] command applies a given
21204 binary operator across all the elements of a vector. A binary operator is
21205 a function such as @code{+} or @code{max} which takes two arguments. For
21206 example, reducing @code{+} over a vector computes the sum of the elements
21207 of the vector. Reducing @code{-} computes the first element minus each of
21208 the remaining elements. Reducing @code{max} computes the maximum element
21209 and so on. In general, reducing @code{f} over the vector @samp{[a, b, c, d]}
21210 produces @samp{f(f(f(a, b), c), d)}.
21214 The @kbd{I V R} [@code{rreduce}] command is similar to @kbd{V R} except
21215 that works from right to left through the vector. For example, plain
21216 @kbd{V R -} on the vector @samp{[a, b, c, d]} produces @samp{a - b - c - d}
21217 but @kbd{I V R -} on the same vector produces @samp{a - (b - (c - d))},
21218 or @samp{a - b + c - d}. This ``alternating sum'' occurs frequently
21219 in power series expansions.
21223 The @kbd{V U} (@code{calc-accumulate}) [@code{accum}] command does an
21224 accumulation operation. Here Calc does the corresponding reduction
21225 operation, but instead of producing only the final result, it produces
21226 a vector of all the intermediate results. Accumulating @code{+} over
21227 the vector @samp{[a, b, c, d]} produces the vector
21228 @samp{[a, a + b, a + b + c, a + b + c + d]}.
21232 The @kbd{I V U} [@code{raccum}] command does a right-to-left accumulation.
21233 For example, @kbd{I V U -} on the vector @samp{[a, b, c, d]} produces the
21234 vector @samp{[a - b + c - d, b - c + d, c - d, d]}.
21240 As for @kbd{V M}, @kbd{V R} normally reduces a matrix elementwise. For
21241 example, given the matrix @expr{[[a, b, c], [d, e, f]]}, @kbd{V R +} will
21242 compute @expr{a + b + c + d + e + f}. You can type @kbd{V R _} or
21243 @kbd{V R :} to modify this behavior. The @kbd{V R _} [@code{reducea}]
21244 command reduces ``across'' the matrix; it reduces each row of the matrix
21245 as a vector, then collects the results. Thus @kbd{V R _ +} of this
21246 matrix would produce @expr{[a + b + c, d + e + f]}. Similarly, @kbd{V R :}
21247 [@code{reduced}] reduces down; @kbd{V R : +} would produce @expr{[a + d,
21252 There is a third ``by rows'' mode for reduction that is occasionally
21253 useful; @kbd{V R =} [@code{reducer}] simply reduces the operator over
21254 the rows of the matrix themselves. Thus @kbd{V R = +} on the above
21255 matrix would get the same result as @kbd{V R : +}, since adding two
21256 row vectors is equivalent to adding their elements. But @kbd{V R = *}
21257 would multiply the two rows (to get a single number, their dot product),
21258 while @kbd{V R : *} would produce a vector of the products of the columns.
21260 These three matrix reduction modes work with @kbd{V R} and @kbd{I V R},
21261 but they are not currently supported with @kbd{V U} or @kbd{I V U}.
21265 The obsolete reduce-by-columns function, @code{reducec}, is still
21266 supported but there is no way to get it through the @kbd{V R} command.
21268 The commands @kbd{M-# :} and @kbd{M-# _} are equivalent to typing
21269 @kbd{M-# r} to grab a rectangle of data into Calc, and then typing
21270 @kbd{V R : +} or @kbd{V R _ +}, respectively, to sum the columns or
21271 rows of the matrix. @xref{Grabbing From Buffers}.
21273 @node Nesting and Fixed Points, Generalized Products, Reducing, Reducing and Mapping
21274 @subsection Nesting and Fixed Points
21279 The @kbd{H V R} [@code{nest}] command applies a function to a given
21280 argument repeatedly. It takes two values, @samp{a} and @samp{n}, from
21281 the stack, where @samp{n} must be an integer. It then applies the
21282 function nested @samp{n} times; if the function is @samp{f} and @samp{n}
21283 is 3, the result is @samp{f(f(f(a)))}. The number @samp{n} may be
21284 negative if Calc knows an inverse for the function @samp{f}; for
21285 example, @samp{nest(sin, a, -2)} returns @samp{arcsin(arcsin(a))}.
21289 The @kbd{H V U} [@code{anest}] command is an accumulating version of
21290 @code{nest}: It returns a vector of @samp{n+1} values, e.g.,
21291 @samp{[a, f(a), f(f(a)), f(f(f(a)))]}. If @samp{n} is negative and
21292 @samp{F} is the inverse of @samp{f}, then the result is of the
21293 form @samp{[a, F(a), F(F(a)), F(F(F(a)))]}.
21297 @cindex Fixed points
21298 The @kbd{H I V R} [@code{fixp}] command is like @kbd{H V R}, except
21299 that it takes only an @samp{a} value from the stack; the function is
21300 applied until it reaches a ``fixed point,'' i.e., until the result
21305 The @kbd{H I V U} [@code{afixp}] command is an accumulating @code{fixp}.
21306 The first element of the return vector will be the initial value @samp{a};
21307 the last element will be the final result that would have been returned
21310 For example, 0.739085 is a fixed point of the cosine function (in radians):
21311 @samp{cos(0.739085) = 0.739085}. You can find this value by putting, say,
21312 1.0 on the stack and typing @kbd{H I V U C}. (We use the accumulating
21313 version so we can see the intermediate results: @samp{[1, 0.540302, 0.857553,
21314 0.65329, ...]}. With a precision of six, this command will take 36 steps
21315 to converge to 0.739085.)
21317 Newton's method for finding roots is a classic example of iteration
21318 to a fixed point. To find the square root of five starting with an
21319 initial guess, Newton's method would look for a fixed point of the
21320 function @samp{(x + 5/x) / 2}. Putting a guess of 1 on the stack
21321 and typing @kbd{H I V R ' ($ + 5/$)/2 @key{RET}} quickly yields the result
21322 2.23607. This is equivalent to using the @kbd{a R} (@code{calc-find-root})
21323 command to find a root of the equation @samp{x^2 = 5}.
21325 These examples used numbers for @samp{a} values. Calc keeps applying
21326 the function until two successive results are equal to within the
21327 current precision. For complex numbers, both the real parts and the
21328 imaginary parts must be equal to within the current precision. If
21329 @samp{a} is a formula (say, a variable name), then the function is
21330 applied until two successive results are exactly the same formula.
21331 It is up to you to ensure that the function will eventually converge;
21332 if it doesn't, you may have to press @kbd{C-g} to stop the Calculator.
21334 The algebraic @code{fixp} function takes two optional arguments, @samp{n}
21335 and @samp{tol}. The first is the maximum number of steps to be allowed,
21336 and must be either an integer or the symbol @samp{inf} (infinity, the
21337 default). The second is a convergence tolerance. If a tolerance is
21338 specified, all results during the calculation must be numbers, not
21339 formulas, and the iteration stops when the magnitude of the difference
21340 between two successive results is less than or equal to the tolerance.
21341 (This implies that a tolerance of zero iterates until the results are
21344 Putting it all together, @samp{fixp(<(# + A/#)/2>, B, 20, 1e-10)}
21345 computes the square root of @samp{A} given the initial guess @samp{B},
21346 stopping when the result is correct within the specified tolerance, or
21347 when 20 steps have been taken, whichever is sooner.
21349 @node Generalized Products, , Nesting and Fixed Points, Reducing and Mapping
21350 @subsection Generalized Products
21353 @pindex calc-outer-product
21355 The @kbd{V O} (@code{calc-outer-product}) [@code{outer}] command applies
21356 a given binary operator to all possible pairs of elements from two
21357 vectors, to produce a matrix. For example, @kbd{V O *} with @samp{[a, b]}
21358 and @samp{[x, y, z]} on the stack produces a multiplication table:
21359 @samp{[[a x, a y, a z], [b x, b y, b z]]}. Element @var{r},@var{c} of
21360 the result matrix is obtained by applying the operator to element @var{r}
21361 of the lefthand vector and element @var{c} of the righthand vector.
21364 @pindex calc-inner-product
21366 The @kbd{V I} (@code{calc-inner-product}) [@code{inner}] command computes
21367 the generalized inner product of two vectors or matrices, given a
21368 ``multiplicative'' operator and an ``additive'' operator. These can each
21369 actually be any binary operators; if they are @samp{*} and @samp{+},
21370 respectively, the result is a standard matrix multiplication. Element
21371 @var{r},@var{c} of the result matrix is obtained by mapping the
21372 multiplicative operator across row @var{r} of the lefthand matrix and
21373 column @var{c} of the righthand matrix, and then reducing with the additive
21374 operator. Just as for the standard @kbd{*} command, this can also do a
21375 vector-matrix or matrix-vector inner product, or a vector-vector
21376 generalized dot product.
21378 Since @kbd{V I} requires two operators, it prompts twice. In each case,
21379 you can use any of the usual methods for entering the operator. If you
21380 use @kbd{$} twice to take both operator formulas from the stack, the
21381 first (multiplicative) operator is taken from the top of the stack
21382 and the second (additive) operator is taken from second-to-top.
21384 @node Vector and Matrix Formats, , Reducing and Mapping, Matrix Functions
21385 @section Vector and Matrix Display Formats
21388 Commands for controlling vector and matrix display use the @kbd{v} prefix
21389 instead of the usual @kbd{d} prefix. But they are display modes; in
21390 particular, they are influenced by the @kbd{I} and @kbd{H} prefix keys
21391 in the same way (@pxref{Display Modes}). Matrix display is also
21392 influenced by the @kbd{d O} (@code{calc-flat-language}) mode;
21393 @pxref{Normal Language Modes}.
21396 @pindex calc-matrix-left-justify
21398 @pindex calc-matrix-center-justify
21400 @pindex calc-matrix-right-justify
21401 The commands @kbd{v <} (@code{calc-matrix-left-justify}), @kbd{v >}
21402 (@code{calc-matrix-right-justify}), and @w{@kbd{v =}}
21403 (@code{calc-matrix-center-justify}) control whether matrix elements
21404 are justified to the left, right, or center of their columns.
21407 @pindex calc-vector-brackets
21409 @pindex calc-vector-braces
21411 @pindex calc-vector-parens
21412 The @kbd{v [} (@code{calc-vector-brackets}) command turns the square
21413 brackets that surround vectors and matrices displayed in the stack on
21414 and off. The @kbd{v @{} (@code{calc-vector-braces}) and @kbd{v (}
21415 (@code{calc-vector-parens}) commands use curly braces or parentheses,
21416 respectively, instead of square brackets. For example, @kbd{v @{} might
21417 be used in preparation for yanking a matrix into a buffer running
21418 Mathematica. (In fact, the Mathematica language mode uses this mode;
21419 @pxref{Mathematica Language Mode}.) Note that, regardless of the
21420 display mode, either brackets or braces may be used to enter vectors,
21421 and parentheses may never be used for this purpose.
21424 @pindex calc-matrix-brackets
21425 The @kbd{v ]} (@code{calc-matrix-brackets}) command controls the
21426 ``big'' style display of matrices. It prompts for a string of code
21427 letters; currently implemented letters are @code{R}, which enables
21428 brackets on each row of the matrix; @code{O}, which enables outer
21429 brackets in opposite corners of the matrix; and @code{C}, which
21430 enables commas or semicolons at the ends of all rows but the last.
21431 The default format is @samp{RO}. (Before Calc 2.00, the format
21432 was fixed at @samp{ROC}.) Here are some example matrices:
21436 [ [ 123, 0, 0 ] [ [ 123, 0, 0 ],
21437 [ 0, 123, 0 ] [ 0, 123, 0 ],
21438 [ 0, 0, 123 ] ] [ 0, 0, 123 ] ]
21447 [ 123, 0, 0 [ 123, 0, 0 ;
21448 0, 123, 0 0, 123, 0 ;
21449 0, 0, 123 ] 0, 0, 123 ]
21458 [ 123, 0, 0 ] 123, 0, 0
21459 [ 0, 123, 0 ] 0, 123, 0
21460 [ 0, 0, 123 ] 0, 0, 123
21467 Note that of the formats shown here, @samp{RO}, @samp{ROC}, and
21468 @samp{OC} are all recognized as matrices during reading, while
21469 the others are useful for display only.
21472 @pindex calc-vector-commas
21473 The @kbd{v ,} (@code{calc-vector-commas}) command turns commas on and
21474 off in vector and matrix display.
21476 In vectors of length one, and in all vectors when commas have been
21477 turned off, Calc adds extra parentheses around formulas that might
21478 otherwise be ambiguous. For example, @samp{[a b]} could be a vector
21479 of the one formula @samp{a b}, or it could be a vector of two
21480 variables with commas turned off. Calc will display the former
21481 case as @samp{[(a b)]}. You can disable these extra parentheses
21482 (to make the output less cluttered at the expense of allowing some
21483 ambiguity) by adding the letter @code{P} to the control string you
21484 give to @kbd{v ]} (as described above).
21487 @pindex calc-full-vectors
21488 The @kbd{v .} (@code{calc-full-vectors}) command turns abbreviated
21489 display of long vectors on and off. In this mode, vectors of six
21490 or more elements, or matrices of six or more rows or columns, will
21491 be displayed in an abbreviated form that displays only the first
21492 three elements and the last element: @samp{[a, b, c, ..., z]}.
21493 When very large vectors are involved this will substantially
21494 improve Calc's display speed.
21497 @pindex calc-full-trail-vectors
21498 The @kbd{t .} (@code{calc-full-trail-vectors}) command controls a
21499 similar mode for recording vectors in the Trail. If you turn on
21500 this mode, vectors of six or more elements and matrices of six or
21501 more rows or columns will be abbreviated when they are put in the
21502 Trail. The @kbd{t y} (@code{calc-trail-yank}) command will be
21503 unable to recover those vectors. If you are working with very
21504 large vectors, this mode will improve the speed of all operations
21505 that involve the trail.
21508 @pindex calc-break-vectors
21509 The @kbd{v /} (@code{calc-break-vectors}) command turns multi-line
21510 vector display on and off. Normally, matrices are displayed with one
21511 row per line but all other types of vectors are displayed in a single
21512 line. This mode causes all vectors, whether matrices or not, to be
21513 displayed with a single element per line. Sub-vectors within the
21514 vectors will still use the normal linear form.
21516 @node Algebra, Units, Matrix Functions, Top
21520 This section covers the Calc features that help you work with
21521 algebraic formulas. First, the general sub-formula selection
21522 mechanism is described; this works in conjunction with any Calc
21523 commands. Then, commands for specific algebraic operations are
21524 described. Finally, the flexible @dfn{rewrite rule} mechanism
21527 The algebraic commands use the @kbd{a} key prefix; selection
21528 commands use the @kbd{j} (for ``just a letter that wasn't used
21529 for anything else'') prefix.
21531 @xref{Editing Stack Entries}, to see how to manipulate formulas
21532 using regular Emacs editing commands.
21534 When doing algebraic work, you may find several of the Calculator's
21535 modes to be helpful, including Algebraic Simplification mode (@kbd{m A})
21536 or No-Simplification mode (@kbd{m O}),
21537 Algebraic entry mode (@kbd{m a}), Fraction mode (@kbd{m f}), and
21538 Symbolic mode (@kbd{m s}). @xref{Mode Settings}, for discussions
21539 of these modes. You may also wish to select Big display mode (@kbd{d B}).
21540 @xref{Normal Language Modes}.
21543 * Selecting Subformulas::
21544 * Algebraic Manipulation::
21545 * Simplifying Formulas::
21548 * Solving Equations::
21549 * Numerical Solutions::
21552 * Logical Operations::
21556 @node Selecting Subformulas, Algebraic Manipulation, Algebra, Algebra
21557 @section Selecting Sub-Formulas
21561 @cindex Sub-formulas
21562 @cindex Parts of formulas
21563 When working with an algebraic formula it is often necessary to
21564 manipulate a portion of the formula rather than the formula as a
21565 whole. Calc allows you to ``select'' a portion of any formula on
21566 the stack. Commands which would normally operate on that stack
21567 entry will now operate only on the sub-formula, leaving the
21568 surrounding part of the stack entry alone.
21570 One common non-algebraic use for selection involves vectors. To work
21571 on one element of a vector in-place, simply select that element as a
21572 ``sub-formula'' of the vector.
21575 * Making Selections::
21576 * Changing Selections::
21577 * Displaying Selections::
21578 * Operating on Selections::
21579 * Rearranging with Selections::
21582 @node Making Selections, Changing Selections, Selecting Subformulas, Selecting Subformulas
21583 @subsection Making Selections
21587 @pindex calc-select-here
21588 To select a sub-formula, move the Emacs cursor to any character in that
21589 sub-formula, and press @w{@kbd{j s}} (@code{calc-select-here}). Calc will
21590 highlight the smallest portion of the formula that contains that
21591 character. By default the sub-formula is highlighted by blanking out
21592 all of the rest of the formula with dots. Selection works in any
21593 display mode but is perhaps easiest in Big mode (@kbd{d B}).
21594 Suppose you enter the following formula:
21606 (by typing @kbd{' ((a+b)^3 + sqrt(c)) / (2x+1)}). If you move the
21607 cursor to the letter @samp{b} and press @w{@kbd{j s}}, the display changes
21620 Every character not part of the sub-formula @samp{b} has been changed
21621 to a dot. The @samp{*} next to the line number is to remind you that
21622 the formula has a portion of it selected. (In this case, it's very
21623 obvious, but it might not always be. If Embedded mode is enabled,
21624 the word @samp{Sel} also appears in the mode line because the stack
21625 may not be visible. @pxref{Embedded Mode}.)
21627 If you had instead placed the cursor on the parenthesis immediately to
21628 the right of the @samp{b}, the selection would have been:
21640 The portion selected is always large enough to be considered a complete
21641 formula all by itself, so selecting the parenthesis selects the whole
21642 formula that it encloses. Putting the cursor on the @samp{+} sign
21643 would have had the same effect.
21645 (Strictly speaking, the Emacs cursor is really the manifestation of
21646 the Emacs ``point,'' which is a position @emph{between} two characters
21647 in the buffer. So purists would say that Calc selects the smallest
21648 sub-formula which contains the character to the right of ``point.'')
21650 If you supply a numeric prefix argument @var{n}, the selection is
21651 expanded to the @var{n}th enclosing sub-formula. Thus, positioning
21652 the cursor on the @samp{b} and typing @kbd{C-u 1 j s} will select
21653 @samp{a + b}; typing @kbd{C-u 2 j s} will select @samp{(a + b)^3},
21656 If the cursor is not on any part of the formula, or if you give a
21657 numeric prefix that is too large, the entire formula is selected.
21659 If the cursor is on the @samp{.} line that marks the top of the stack
21660 (i.e., its normal ``rest position''), this command selects the entire
21661 formula at stack level 1. Most selection commands similarly operate
21662 on the formula at the top of the stack if you haven't positioned the
21663 cursor on any stack entry.
21666 @pindex calc-select-additional
21667 The @kbd{j a} (@code{calc-select-additional}) command enlarges the
21668 current selection to encompass the cursor. To select the smallest
21669 sub-formula defined by two different points, move to the first and
21670 press @kbd{j s}, then move to the other and press @kbd{j a}. This
21671 is roughly analogous to using @kbd{C-@@} (@code{set-mark-command}) to
21672 select the two ends of a region of text during normal Emacs editing.
21675 @pindex calc-select-once
21676 The @kbd{j o} (@code{calc-select-once}) command selects a formula in
21677 exactly the same way as @kbd{j s}, except that the selection will
21678 last only as long as the next command that uses it. For example,
21679 @kbd{j o 1 +} is a handy way to add one to the sub-formula indicated
21682 (A somewhat more precise definition: The @kbd{j o} command sets a flag
21683 such that the next command involving selected stack entries will clear
21684 the selections on those stack entries afterwards. All other selection
21685 commands except @kbd{j a} and @kbd{j O} clear this flag.)
21689 @pindex calc-select-here-maybe
21690 @pindex calc-select-once-maybe
21691 The @kbd{j S} (@code{calc-select-here-maybe}) and @kbd{j O}
21692 (@code{calc-select-once-maybe}) commands are equivalent to @kbd{j s}
21693 and @kbd{j o}, respectively, except that if the formula already
21694 has a selection they have no effect. This is analogous to the
21695 behavior of some commands such as @kbd{j r} (@code{calc-rewrite-selection};
21696 @pxref{Selections with Rewrite Rules}) and is mainly intended to be
21697 used in keyboard macros that implement your own selection-oriented
21700 Selection of sub-formulas normally treats associative terms like
21701 @samp{a + b - c + d} and @samp{x * y * z} as single levels of the formula.
21702 If you place the cursor anywhere inside @samp{a + b - c + d} except
21703 on one of the variable names and use @kbd{j s}, you will select the
21704 entire four-term sum.
21707 @pindex calc-break-selections
21708 The @kbd{j b} (@code{calc-break-selections}) command controls a mode
21709 in which the ``deep structure'' of these associative formulas shows
21710 through. Calc actually stores the above formulas as @samp{((a + b) - c) + d}
21711 and @samp{x * (y * z)}. (Note that for certain obscure reasons, Calc
21712 treats multiplication as right-associative.) Once you have enabled
21713 @kbd{j b} mode, selecting with the cursor on the @samp{-} sign would
21714 only select the @samp{a + b - c} portion, which makes sense when the
21715 deep structure of the sum is considered. There is no way to select
21716 the @samp{b - c + d} portion; although this might initially look
21717 like just as legitimate a sub-formula as @samp{a + b - c}, the deep
21718 structure shows that it isn't. The @kbd{d U} command can be used
21719 to view the deep structure of any formula (@pxref{Normal Language Modes}).
21721 When @kbd{j b} mode has not been enabled, the deep structure is
21722 generally hidden by the selection commands---what you see is what
21726 @pindex calc-unselect
21727 The @kbd{j u} (@code{calc-unselect}) command unselects the formula
21728 that the cursor is on. If there was no selection in the formula,
21729 this command has no effect. With a numeric prefix argument, it
21730 unselects the @var{n}th stack element rather than using the cursor
21734 @pindex calc-clear-selections
21735 The @kbd{j c} (@code{calc-clear-selections}) command unselects all
21738 @node Changing Selections, Displaying Selections, Making Selections, Selecting Subformulas
21739 @subsection Changing Selections
21743 @pindex calc-select-more
21744 Once you have selected a sub-formula, you can expand it using the
21745 @w{@kbd{j m}} (@code{calc-select-more}) command. If @samp{a + b} is
21746 selected, pressing @w{@kbd{j m}} repeatedly works as follows:
21751 (a + b) . . . (a + b) + V c (a + b) + V c
21752 1* ............... 1* ............... 1* ---------------
21753 . . . . . . . . 2 x + 1
21758 In the last example, the entire formula is selected. This is roughly
21759 the same as having no selection at all, but because there are subtle
21760 differences the @samp{*} character is still there on the line number.
21762 With a numeric prefix argument @var{n}, @kbd{j m} expands @var{n}
21763 times (or until the entire formula is selected). Note that @kbd{j s}
21764 with argument @var{n} is equivalent to plain @kbd{j s} followed by
21765 @kbd{j m} with argument @var{n}. If @w{@kbd{j m}} is used when there
21766 is no current selection, it is equivalent to @w{@kbd{j s}}.
21768 Even though @kbd{j m} does not explicitly use the location of the
21769 cursor within the formula, it nevertheless uses the cursor to determine
21770 which stack element to operate on. As usual, @kbd{j m} when the cursor
21771 is not on any stack element operates on the top stack element.
21774 @pindex calc-select-less
21775 The @kbd{j l} (@code{calc-select-less}) command reduces the current
21776 selection around the cursor position. That is, it selects the
21777 immediate sub-formula of the current selection which contains the
21778 cursor, the opposite of @kbd{j m}. If the cursor is not inside the
21779 current selection, the command de-selects the formula.
21782 @pindex calc-select-part
21783 The @kbd{j 1} through @kbd{j 9} (@code{calc-select-part}) commands
21784 select the @var{n}th sub-formula of the current selection. They are
21785 like @kbd{j l} (@code{calc-select-less}) except they use counting
21786 rather than the cursor position to decide which sub-formula to select.
21787 For example, if the current selection is @kbd{a + b + c} or
21788 @kbd{f(a, b, c)} or @kbd{[a, b, c]}, then @kbd{j 1} selects @samp{a},
21789 @kbd{j 2} selects @samp{b}, and @kbd{j 3} selects @samp{c}; in each of
21790 these cases, @kbd{j 4} through @kbd{j 9} would be errors.
21792 If there is no current selection, @kbd{j 1} through @kbd{j 9} select
21793 the @var{n}th top-level sub-formula. (In other words, they act as if
21794 the entire stack entry were selected first.) To select the @var{n}th
21795 sub-formula where @var{n} is greater than nine, you must instead invoke
21796 @w{@kbd{j 1}} with @var{n} as a numeric prefix argument.
21800 @pindex calc-select-next
21801 @pindex calc-select-previous
21802 The @kbd{j n} (@code{calc-select-next}) and @kbd{j p}
21803 (@code{calc-select-previous}) commands change the current selection
21804 to the next or previous sub-formula at the same level. For example,
21805 if @samp{b} is selected in @w{@samp{2 + a*b*c + x}}, then @kbd{j n}
21806 selects @samp{c}. Further @kbd{j n} commands would be in error because,
21807 even though there is something to the right of @samp{c} (namely, @samp{x}),
21808 it is not at the same level; in this case, it is not a term of the
21809 same product as @samp{b} and @samp{c}. However, @kbd{j m} (to select
21810 the whole product @samp{a*b*c} as a term of the sum) followed by
21811 @w{@kbd{j n}} would successfully select the @samp{x}.
21813 Similarly, @kbd{j p} moves the selection from the @samp{b} in this
21814 sample formula to the @samp{a}. Both commands accept numeric prefix
21815 arguments to move several steps at a time.
21817 It is interesting to compare Calc's selection commands with the
21818 Emacs Info system's commands for navigating through hierarchically
21819 organized documentation. Calc's @kbd{j n} command is completely
21820 analogous to Info's @kbd{n} command. Likewise, @kbd{j p} maps to
21821 @kbd{p}, @kbd{j 2} maps to @kbd{2}, and Info's @kbd{u} is like @kbd{j m}.
21822 (Note that @kbd{j u} stands for @code{calc-unselect}, not ``up''.)
21823 The Info @kbd{m} command is somewhat similar to Calc's @kbd{j s} and
21824 @kbd{j l}; in each case, you can jump directly to a sub-component
21825 of the hierarchy simply by pointing to it with the cursor.
21827 @node Displaying Selections, Operating on Selections, Changing Selections, Selecting Subformulas
21828 @subsection Displaying Selections
21832 @pindex calc-show-selections
21833 The @kbd{j d} (@code{calc-show-selections}) command controls how
21834 selected sub-formulas are displayed. One of the alternatives is
21835 illustrated in the above examples; if we press @kbd{j d} we switch
21836 to the other style in which the selected portion itself is obscured
21842 (a + b) . . . ## # ## + V c
21843 1* ............... 1* ---------------
21848 @node Operating on Selections, Rearranging with Selections, Displaying Selections, Selecting Subformulas
21849 @subsection Operating on Selections
21852 Once a selection is made, all Calc commands that manipulate items
21853 on the stack will operate on the selected portions of the items
21854 instead. (Note that several stack elements may have selections
21855 at once, though there can be only one selection at a time in any
21856 given stack element.)
21859 @pindex calc-enable-selections
21860 The @kbd{j e} (@code{calc-enable-selections}) command disables the
21861 effect that selections have on Calc commands. The current selections
21862 still exist, but Calc commands operate on whole stack elements anyway.
21863 This mode can be identified by the fact that the @samp{*} markers on
21864 the line numbers are gone, even though selections are visible. To
21865 reactivate the selections, press @kbd{j e} again.
21867 To extract a sub-formula as a new formula, simply select the
21868 sub-formula and press @key{RET}. This normally duplicates the top
21869 stack element; here it duplicates only the selected portion of that
21872 To replace a sub-formula with something different, you can enter the
21873 new value onto the stack and press @key{TAB}. This normally exchanges
21874 the top two stack elements; here it swaps the value you entered into
21875 the selected portion of the formula, returning the old selected
21876 portion to the top of the stack.
21881 (a + b) . . . 17 x y . . . 17 x y + V c
21882 2* ............... 2* ............. 2: -------------
21883 . . . . . . . . 2 x + 1
21886 1: 17 x y 1: (a + b) 1: (a + b)
21890 In this example we select a sub-formula of our original example,
21891 enter a new formula, @key{TAB} it into place, then deselect to see
21892 the complete, edited formula.
21894 If you want to swap whole formulas around even though they contain
21895 selections, just use @kbd{j e} before and after.
21898 @pindex calc-enter-selection
21899 The @kbd{j '} (@code{calc-enter-selection}) command is another way
21900 to replace a selected sub-formula. This command does an algebraic
21901 entry just like the regular @kbd{'} key. When you press @key{RET},
21902 the formula you type replaces the original selection. You can use
21903 the @samp{$} symbol in the formula to refer to the original
21904 selection. If there is no selection in the formula under the cursor,
21905 the cursor is used to make a temporary selection for the purposes of
21906 the command. Thus, to change a term of a formula, all you have to
21907 do is move the Emacs cursor to that term and press @kbd{j '}.
21910 @pindex calc-edit-selection
21911 The @kbd{j `} (@code{calc-edit-selection}) command is a similar
21912 analogue of the @kbd{`} (@code{calc-edit}) command. It edits the
21913 selected sub-formula in a separate buffer. If there is no
21914 selection, it edits the sub-formula indicated by the cursor.
21916 To delete a sub-formula, press @key{DEL}. This generally replaces
21917 the sub-formula with the constant zero, but in a few suitable contexts
21918 it uses the constant one instead. The @key{DEL} key automatically
21919 deselects and re-simplifies the entire formula afterwards. Thus:
21924 17 x y + # # 17 x y 17 # y 17 y
21925 1* ------------- 1: ------- 1* ------- 1: -------
21926 2 x + 1 2 x + 1 2 x + 1 2 x + 1
21930 In this example, we first delete the @samp{sqrt(c)} term; Calc
21931 accomplishes this by replacing @samp{sqrt(c)} with zero and
21932 resimplifying. We then delete the @kbd{x} in the numerator;
21933 since this is part of a product, Calc replaces it with @samp{1}
21936 If you select an element of a vector and press @key{DEL}, that
21937 element is deleted from the vector. If you delete one side of
21938 an equation or inequality, only the opposite side remains.
21940 @kindex j @key{DEL}
21941 @pindex calc-del-selection
21942 The @kbd{j @key{DEL}} (@code{calc-del-selection}) command is like
21943 @key{DEL} but with the auto-selecting behavior of @kbd{j '} and
21944 @kbd{j `}. It deletes the selected portion of the formula
21945 indicated by the cursor, or, in the absence of a selection, it
21946 deletes the sub-formula indicated by the cursor position.
21948 @kindex j @key{RET}
21949 @pindex calc-grab-selection
21950 (There is also an auto-selecting @kbd{j @key{RET}} (@code{calc-copy-selection})
21953 Normal arithmetic operations also apply to sub-formulas. Here we
21954 select the denominator, press @kbd{5 -} to subtract five from the
21955 denominator, press @kbd{n} to negate the denominator, then
21956 press @kbd{Q} to take the square root.
21960 .. . .. . .. . .. .
21961 1* ....... 1* ....... 1* ....... 1* ..........
21962 2 x + 1 2 x - 4 4 - 2 x _________
21967 Certain types of operations on selections are not allowed. For
21968 example, for an arithmetic function like @kbd{-} no more than one of
21969 the arguments may be a selected sub-formula. (As the above example
21970 shows, the result of the subtraction is spliced back into the argument
21971 which had the selection; if there were more than one selection involved,
21972 this would not be well-defined.) If you try to subtract two selections,
21973 the command will abort with an error message.
21975 Operations on sub-formulas sometimes leave the formula as a whole
21976 in an ``un-natural'' state. Consider negating the @samp{2 x} term
21977 of our sample formula by selecting it and pressing @kbd{n}
21978 (@code{calc-change-sign}).
21983 1* .......... 1* ...........
21984 ......... ..........
21985 . . . 2 x . . . -2 x
21989 Unselecting the sub-formula reveals that the minus sign, which would
21990 normally have cancelled out with the subtraction automatically, has
21991 not been able to do so because the subtraction was not part of the
21992 selected portion. Pressing @kbd{=} (@code{calc-evaluate}) or doing
21993 any other mathematical operation on the whole formula will cause it
21999 1: ----------- 1: ----------
22000 __________ _________
22001 V 4 - -2 x V 4 + 2 x
22005 @node Rearranging with Selections, , Operating on Selections, Selecting Subformulas
22006 @subsection Rearranging Formulas using Selections
22010 @pindex calc-commute-right
22011 The @kbd{j R} (@code{calc-commute-right}) command moves the selected
22012 sub-formula to the right in its surrounding formula. Generally the
22013 selection is one term of a sum or product; the sum or product is
22014 rearranged according to the commutative laws of algebra.
22016 As with @kbd{j '} and @kbd{j @key{DEL}}, the term under the cursor is used
22017 if there is no selection in the current formula. All commands described
22018 in this section share this property. In this example, we place the
22019 cursor on the @samp{a} and type @kbd{j R}, then repeat.
22022 1: a + b - c 1: b + a - c 1: b - c + a
22026 Note that in the final step above, the @samp{a} is switched with
22027 the @samp{c} but the signs are adjusted accordingly. When moving
22028 terms of sums and products, @kbd{j R} will never change the
22029 mathematical meaning of the formula.
22031 The selected term may also be an element of a vector or an argument
22032 of a function. The term is exchanged with the one to its right.
22033 In this case, the ``meaning'' of the vector or function may of
22034 course be drastically changed.
22037 1: [a, b, c] 1: [b, a, c] 1: [b, c, a]
22039 1: f(a, b, c) 1: f(b, a, c) 1: f(b, c, a)
22043 @pindex calc-commute-left
22044 The @kbd{j L} (@code{calc-commute-left}) command is like @kbd{j R}
22045 except that it swaps the selected term with the one to its left.
22047 With numeric prefix arguments, these commands move the selected
22048 term several steps at a time. It is an error to try to move a
22049 term left or right past the end of its enclosing formula.
22050 With numeric prefix arguments of zero, these commands move the
22051 selected term as far as possible in the given direction.
22054 @pindex calc-sel-distribute
22055 The @kbd{j D} (@code{calc-sel-distribute}) command mixes the selected
22056 sum or product into the surrounding formula using the distributive
22057 law. For example, in @samp{a * (b - c)} with the @samp{b - c}
22058 selected, the result is @samp{a b - a c}. This also distributes
22059 products or quotients into surrounding powers, and can also do
22060 transformations like @samp{exp(a + b)} to @samp{exp(a) exp(b)},
22061 where @samp{a + b} is the selected term, and @samp{ln(a ^ b)}
22062 to @samp{ln(a) b}, where @samp{a ^ b} is the selected term.
22064 For multiple-term sums or products, @kbd{j D} takes off one term
22065 at a time: @samp{a * (b + c - d)} goes to @samp{a * (c - d) + a b}
22066 with the @samp{c - d} selected so that you can type @kbd{j D}
22067 repeatedly to expand completely. The @kbd{j D} command allows a
22068 numeric prefix argument which specifies the maximum number of
22069 times to expand at once; the default is one time only.
22071 @vindex DistribRules
22072 The @kbd{j D} command is implemented using rewrite rules.
22073 @xref{Selections with Rewrite Rules}. The rules are stored in
22074 the Calc variable @code{DistribRules}. A convenient way to view
22075 these rules is to use @kbd{s e} (@code{calc-edit-variable}) which
22076 displays and edits the stored value of a variable. Press @kbd{C-c C-c}
22077 to return from editing mode; be careful not to make any actual changes
22078 or else you will affect the behavior of future @kbd{j D} commands!
22080 To extend @kbd{j D} to handle new cases, just edit @code{DistribRules}
22081 as described above. You can then use the @kbd{s p} command to save
22082 this variable's value permanently for future Calc sessions.
22083 @xref{Operations on Variables}.
22086 @pindex calc-sel-merge
22088 The @kbd{j M} (@code{calc-sel-merge}) command is the complement
22089 of @kbd{j D}; given @samp{a b - a c} with either @samp{a b} or
22090 @samp{a c} selected, the result is @samp{a * (b - c)}. Once
22091 again, @kbd{j M} can also merge calls to functions like @code{exp}
22092 and @code{ln}; examine the variable @code{MergeRules} to see all
22093 the relevant rules.
22096 @pindex calc-sel-commute
22097 @vindex CommuteRules
22098 The @kbd{j C} (@code{calc-sel-commute}) command swaps the arguments
22099 of the selected sum, product, or equation. It always behaves as
22100 if @kbd{j b} mode were in effect, i.e., the sum @samp{a + b + c} is
22101 treated as the nested sums @samp{(a + b) + c} by this command.
22102 If you put the cursor on the first @samp{+}, the result is
22103 @samp{(b + a) + c}; if you put the cursor on the second @samp{+}, the
22104 result is @samp{c + (a + b)} (which the default simplifications
22105 will rearrange to @samp{(c + a) + b}). The relevant rules are stored
22106 in the variable @code{CommuteRules}.
22108 You may need to turn default simplifications off (with the @kbd{m O}
22109 command) in order to get the full benefit of @kbd{j C}. For example,
22110 commuting @samp{a - b} produces @samp{-b + a}, but the default
22111 simplifications will ``simplify'' this right back to @samp{a - b} if
22112 you don't turn them off. The same is true of some of the other
22113 manipulations described in this section.
22116 @pindex calc-sel-negate
22117 @vindex NegateRules
22118 The @kbd{j N} (@code{calc-sel-negate}) command replaces the selected
22119 term with the negative of that term, then adjusts the surrounding
22120 formula in order to preserve the meaning. For example, given
22121 @samp{exp(a - b)} where @samp{a - b} is selected, the result is
22122 @samp{1 / exp(b - a)}. By contrast, selecting a term and using the
22123 regular @kbd{n} (@code{calc-change-sign}) command negates the
22124 term without adjusting the surroundings, thus changing the meaning
22125 of the formula as a whole. The rules variable is @code{NegateRules}.
22128 @pindex calc-sel-invert
22129 @vindex InvertRules
22130 The @kbd{j &} (@code{calc-sel-invert}) command is similar to @kbd{j N}
22131 except it takes the reciprocal of the selected term. For example,
22132 given @samp{a - ln(b)} with @samp{b} selected, the result is
22133 @samp{a + ln(1/b)}. The rules variable is @code{InvertRules}.
22136 @pindex calc-sel-jump-equals
22138 The @kbd{j E} (@code{calc-sel-jump-equals}) command moves the
22139 selected term from one side of an equation to the other. Given
22140 @samp{a + b = c + d} with @samp{c} selected, the result is
22141 @samp{a + b - c = d}. This command also works if the selected
22142 term is part of a @samp{*}, @samp{/}, or @samp{^} formula. The
22143 relevant rules variable is @code{JumpRules}.
22147 @pindex calc-sel-isolate
22148 The @kbd{j I} (@code{calc-sel-isolate}) command isolates the
22149 selected term on its side of an equation. It uses the @kbd{a S}
22150 (@code{calc-solve-for}) command to solve the equation, and the
22151 Hyperbolic flag affects it in the same way. @xref{Solving Equations}.
22152 When it applies, @kbd{j I} is often easier to use than @kbd{j E}.
22153 It understands more rules of algebra, and works for inequalities
22154 as well as equations.
22158 @pindex calc-sel-mult-both-sides
22159 @pindex calc-sel-div-both-sides
22160 The @kbd{j *} (@code{calc-sel-mult-both-sides}) command prompts for a
22161 formula using algebraic entry, then multiplies both sides of the
22162 selected quotient or equation by that formula. It simplifies each
22163 side with @kbd{a s} (@code{calc-simplify}) before re-forming the
22164 quotient or equation. You can suppress this simplification by
22165 providing any numeric prefix argument. There is also a @kbd{j /}
22166 (@code{calc-sel-div-both-sides}) which is similar to @kbd{j *} but
22167 dividing instead of multiplying by the factor you enter.
22169 As a special feature, if the numerator of the quotient is 1, then
22170 the denominator is expanded at the top level using the distributive
22171 law (i.e., using the @kbd{C-u -1 a x} command). Suppose the
22172 formula on the stack is @samp{1 / (sqrt(a) + 1)}, and you wish
22173 to eliminate the square root in the denominator by multiplying both
22174 sides by @samp{sqrt(a) - 1}. Calc's default simplifications would
22175 change the result @samp{(sqrt(a) - 1) / (sqrt(a) - 1) (sqrt(a) + 1)}
22176 right back to the original form by cancellation; Calc expands the
22177 denominator to @samp{sqrt(a) (sqrt(a) - 1) + sqrt(a) - 1} to prevent
22178 this. (You would now want to use an @kbd{a x} command to expand
22179 the rest of the way, whereupon the denominator would cancel out to
22180 the desired form, @samp{a - 1}.) When the numerator is not 1, this
22181 initial expansion is not necessary because Calc's default
22182 simplifications will not notice the potential cancellation.
22184 If the selection is an inequality, @kbd{j *} and @kbd{j /} will
22185 accept any factor, but will warn unless they can prove the factor
22186 is either positive or negative. (In the latter case the direction
22187 of the inequality will be switched appropriately.) @xref{Declarations},
22188 for ways to inform Calc that a given variable is positive or
22189 negative. If Calc can't tell for sure what the sign of the factor
22190 will be, it will assume it is positive and display a warning
22193 For selections that are not quotients, equations, or inequalities,
22194 these commands pull out a multiplicative factor: They divide (or
22195 multiply) by the entered formula, simplify, then multiply (or divide)
22196 back by the formula.
22200 @pindex calc-sel-add-both-sides
22201 @pindex calc-sel-sub-both-sides
22202 The @kbd{j +} (@code{calc-sel-add-both-sides}) and @kbd{j -}
22203 (@code{calc-sel-sub-both-sides}) commands analogously add to or
22204 subtract from both sides of an equation or inequality. For other
22205 types of selections, they extract an additive factor. A numeric
22206 prefix argument suppresses simplification of the intermediate
22210 @pindex calc-sel-unpack
22211 The @kbd{j U} (@code{calc-sel-unpack}) command replaces the
22212 selected function call with its argument. For example, given
22213 @samp{a + sin(x^2)} with @samp{sin(x^2)} selected, the result
22214 is @samp{a + x^2}. (The @samp{x^2} will remain selected; if you
22215 wanted to change the @code{sin} to @code{cos}, just press @kbd{C}
22216 now to take the cosine of the selected part.)
22219 @pindex calc-sel-evaluate
22220 The @kbd{j v} (@code{calc-sel-evaluate}) command performs the
22221 normal default simplifications on the selected sub-formula.
22222 These are the simplifications that are normally done automatically
22223 on all results, but which may have been partially inhibited by
22224 previous selection-related operations, or turned off altogether
22225 by the @kbd{m O} command. This command is just an auto-selecting
22226 version of the @w{@kbd{a v}} command (@pxref{Algebraic Manipulation}).
22228 With a numeric prefix argument of 2, @kbd{C-u 2 j v} applies
22229 the @kbd{a s} (@code{calc-simplify}) command to the selected
22230 sub-formula. With a prefix argument of 3 or more, e.g., @kbd{C-u j v}
22231 applies the @kbd{a e} (@code{calc-simplify-extended}) command.
22232 @xref{Simplifying Formulas}. With a negative prefix argument
22233 it simplifies at the top level only, just as with @kbd{a v}.
22234 Here the ``top'' level refers to the top level of the selected
22238 @pindex calc-sel-expand-formula
22239 The @kbd{j "} (@code{calc-sel-expand-formula}) command is to @kbd{a "}
22240 (@pxref{Algebraic Manipulation}) what @kbd{j v} is to @kbd{a v}.
22242 You can use the @kbd{j r} (@code{calc-rewrite-selection}) command
22243 to define other algebraic operations on sub-formulas. @xref{Rewrite Rules}.
22245 @node Algebraic Manipulation, Simplifying Formulas, Selecting Subformulas, Algebra
22246 @section Algebraic Manipulation
22249 The commands in this section perform general-purpose algebraic
22250 manipulations. They work on the whole formula at the top of the
22251 stack (unless, of course, you have made a selection in that
22254 Many algebra commands prompt for a variable name or formula. If you
22255 answer the prompt with a blank line, the variable or formula is taken
22256 from top-of-stack, and the normal argument for the command is taken
22257 from the second-to-top stack level.
22260 @pindex calc-alg-evaluate
22261 The @kbd{a v} (@code{calc-alg-evaluate}) command performs the normal
22262 default simplifications on a formula; for example, @samp{a - -b} is
22263 changed to @samp{a + b}. These simplifications are normally done
22264 automatically on all Calc results, so this command is useful only if
22265 you have turned default simplifications off with an @kbd{m O}
22266 command. @xref{Simplification Modes}.
22268 It is often more convenient to type @kbd{=}, which is like @kbd{a v}
22269 but which also substitutes stored values for variables in the formula.
22270 Use @kbd{a v} if you want the variables to ignore their stored values.
22272 If you give a numeric prefix argument of 2 to @kbd{a v}, it simplifies
22273 as if in Algebraic Simplification mode. This is equivalent to typing
22274 @kbd{a s}; @pxref{Simplifying Formulas}. If you give a numeric prefix
22275 of 3 or more, it uses Extended Simplification mode (@kbd{a e}).
22277 If you give a negative prefix argument @mathit{-1}, @mathit{-2}, or @mathit{-3},
22278 it simplifies in the corresponding mode but only works on the top-level
22279 function call of the formula. For example, @samp{(2 + 3) * (2 + 3)} will
22280 simplify to @samp{(2 + 3)^2}, without simplifying the sub-formulas
22281 @samp{2 + 3}. As another example, typing @kbd{V R +} to sum the vector
22282 @samp{[1, 2, 3, 4]} produces the formula @samp{reduce(add, [1, 2, 3, 4])}
22283 in No-Simplify mode. Using @kbd{a v} will evaluate this all the way to
22284 10; using @kbd{C-u - a v} will evaluate it only to @samp{1 + 2 + 3 + 4}.
22285 (@xref{Reducing and Mapping}.)
22289 The @kbd{=} command corresponds to the @code{evalv} function, and
22290 the related @kbd{N} command, which is like @kbd{=} but temporarily
22291 disables Symbolic mode (@kbd{m s}) during the evaluation, corresponds
22292 to the @code{evalvn} function. (These commands interpret their prefix
22293 arguments differently than @kbd{a v}; @kbd{=} treats the prefix as
22294 the number of stack elements to evaluate at once, and @kbd{N} treats
22295 it as a temporary different working precision.)
22297 The @code{evalvn} function can take an alternate working precision
22298 as an optional second argument. This argument can be either an
22299 integer, to set the precision absolutely, or a vector containing
22300 a single integer, to adjust the precision relative to the current
22301 precision. Note that @code{evalvn} with a larger than current
22302 precision will do the calculation at this higher precision, but the
22303 result will as usual be rounded back down to the current precision
22304 afterward. For example, @samp{evalvn(pi - 3.1415)} at a precision
22305 of 12 will return @samp{9.265359e-5}; @samp{evalvn(pi - 3.1415, 30)}
22306 will return @samp{9.26535897932e-5} (computing a 25-digit result which
22307 is then rounded down to 12); and @samp{evalvn(pi - 3.1415, [-2])}
22308 will return @samp{9.2654e-5}.
22311 @pindex calc-expand-formula
22312 The @kbd{a "} (@code{calc-expand-formula}) command expands functions
22313 into their defining formulas wherever possible. For example,
22314 @samp{deg(x^2)} is changed to @samp{180 x^2 / pi}. Most functions,
22315 like @code{sin} and @code{gcd}, are not defined by simple formulas
22316 and so are unaffected by this command. One important class of
22317 functions which @emph{can} be expanded is the user-defined functions
22318 created by the @kbd{Z F} command. @xref{Algebraic Definitions}.
22319 Other functions which @kbd{a "} can expand include the probability
22320 distribution functions, most of the financial functions, and the
22321 hyperbolic and inverse hyperbolic functions. A numeric prefix argument
22322 affects @kbd{a "} in the same way as it does @kbd{a v}: A positive
22323 argument expands all functions in the formula and then simplifies in
22324 various ways; a negative argument expands and simplifies only the
22325 top-level function call.
22328 @pindex calc-map-equation
22330 The @kbd{a M} (@code{calc-map-equation}) [@code{mapeq}] command applies
22331 a given function or operator to one or more equations. It is analogous
22332 to @kbd{V M}, which operates on vectors instead of equations.
22333 @pxref{Reducing and Mapping}. For example, @kbd{a M S} changes
22334 @samp{x = y+1} to @samp{sin(x) = sin(y+1)}, and @kbd{a M +} with
22335 @samp{x = y+1} and @expr{6} on the stack produces @samp{x+6 = y+7}.
22336 With two equations on the stack, @kbd{a M +} would add the lefthand
22337 sides together and the righthand sides together to get the two
22338 respective sides of a new equation.
22340 Mapping also works on inequalities. Mapping two similar inequalities
22341 produces another inequality of the same type. Mapping an inequality
22342 with an equation produces an inequality of the same type. Mapping a
22343 @samp{<=} with a @samp{<} or @samp{!=} (not-equal) produces a @samp{<}.
22344 If inequalities with opposite direction (e.g., @samp{<} and @samp{>})
22345 are mapped, the direction of the second inequality is reversed to
22346 match the first: Using @kbd{a M +} on @samp{a < b} and @samp{a > 2}
22347 reverses the latter to get @samp{2 < a}, which then allows the
22348 combination @samp{a + 2 < b + a}, which the @kbd{a s} command can
22349 then simplify to get @samp{2 < b}.
22351 Using @kbd{a M *}, @kbd{a M /}, @kbd{a M n}, or @kbd{a M &} to negate
22352 or invert an inequality will reverse the direction of the inequality.
22353 Other adjustments to inequalities are @emph{not} done automatically;
22354 @kbd{a M S} will change @w{@samp{x < y}} to @samp{sin(x) < sin(y)} even
22355 though this is not true for all values of the variables.
22359 With the Hyperbolic flag, @kbd{H a M} [@code{mapeqp}] does a plain
22360 mapping operation without reversing the direction of any inequalities.
22361 Thus, @kbd{H a M &} would change @kbd{x > 2} to @kbd{1/x > 0.5}.
22362 (This change is mathematically incorrect, but perhaps you were
22363 fixing an inequality which was already incorrect.)
22367 With the Inverse flag, @kbd{I a M} [@code{mapeqr}] always reverses
22368 the direction of the inequality. You might use @kbd{I a M C} to
22369 change @samp{x < y} to @samp{cos(x) > cos(y)} if you know you are
22370 working with small positive angles.
22373 @pindex calc-substitute
22375 The @kbd{a b} (@code{calc-substitute}) [@code{subst}] command substitutes
22377 of some variable or sub-expression of an expression with a new
22378 sub-expression. For example, substituting @samp{sin(x)} with @samp{cos(y)}
22379 in @samp{2 sin(x)^2 + x sin(x) + sin(2 x)} produces
22380 @samp{2 cos(y)^2 + x cos(y) + @w{sin(2 x)}}.
22381 Note that this is a purely structural substitution; the lone @samp{x} and
22382 the @samp{sin(2 x)} stayed the same because they did not look like
22383 @samp{sin(x)}. @xref{Rewrite Rules}, for a more general method for
22384 doing substitutions.
22386 The @kbd{a b} command normally prompts for two formulas, the old
22387 one and the new one. If you enter a blank line for the first
22388 prompt, all three arguments are taken from the stack (new, then old,
22389 then target expression). If you type an old formula but then enter a
22390 blank line for the new one, the new formula is taken from top-of-stack
22391 and the target from second-to-top. If you answer both prompts, the
22392 target is taken from top-of-stack as usual.
22394 Note that @kbd{a b} has no understanding of commutativity or
22395 associativity. The pattern @samp{x+y} will not match the formula
22396 @samp{y+x}. Also, @samp{y+z} will not match inside the formula @samp{x+y+z}
22397 because the @samp{+} operator is left-associative, so the ``deep
22398 structure'' of that formula is @samp{(x+y) + z}. Use @kbd{d U}
22399 (@code{calc-unformatted-language}) mode to see the true structure of
22400 a formula. The rewrite rule mechanism, discussed later, does not have
22403 As an algebraic function, @code{subst} takes three arguments:
22404 Target expression, old, new. Note that @code{subst} is always
22405 evaluated immediately, even if its arguments are variables, so if
22406 you wish to put a call to @code{subst} onto the stack you must
22407 turn the default simplifications off first (with @kbd{m O}).
22409 @node Simplifying Formulas, Polynomials, Algebraic Manipulation, Algebra
22410 @section Simplifying Formulas
22414 @pindex calc-simplify
22416 The @kbd{a s} (@code{calc-simplify}) [@code{simplify}] command applies
22417 various algebraic rules to simplify a formula. This includes rules which
22418 are not part of the default simplifications because they may be too slow
22419 to apply all the time, or may not be desirable all of the time. For
22420 example, non-adjacent terms of sums are combined, as in @samp{a + b + 2 a}
22421 to @samp{b + 3 a}, and some formulas like @samp{sin(arcsin(x))} are
22422 simplified to @samp{x}.
22424 The sections below describe all the various kinds of algebraic
22425 simplifications Calc provides in full detail. None of Calc's
22426 simplification commands are designed to pull rabbits out of hats;
22427 they simply apply certain specific rules to put formulas into
22428 less redundant or more pleasing forms. Serious algebra in Calc
22429 must be done manually, usually with a combination of selections
22430 and rewrite rules. @xref{Rearranging with Selections}.
22431 @xref{Rewrite Rules}.
22433 @xref{Simplification Modes}, for commands to control what level of
22434 simplification occurs automatically. Normally only the ``default
22435 simplifications'' occur.
22438 * Default Simplifications::
22439 * Algebraic Simplifications::
22440 * Unsafe Simplifications::
22441 * Simplification of Units::
22444 @node Default Simplifications, Algebraic Simplifications, Simplifying Formulas, Simplifying Formulas
22445 @subsection Default Simplifications
22448 @cindex Default simplifications
22449 This section describes the ``default simplifications,'' those which are
22450 normally applied to all results. For example, if you enter the variable
22451 @expr{x} on the stack twice and push @kbd{+}, Calc's default
22452 simplifications automatically change @expr{x + x} to @expr{2 x}.
22454 The @kbd{m O} command turns off the default simplifications, so that
22455 @expr{x + x} will remain in this form unless you give an explicit
22456 ``simplify'' command like @kbd{=} or @kbd{a v}. @xref{Algebraic
22457 Manipulation}. The @kbd{m D} command turns the default simplifications
22460 The most basic default simplification is the evaluation of functions.
22461 For example, @expr{2 + 3} is evaluated to @expr{5}, and @expr{@tfn{sqrt}(9)}
22462 is evaluated to @expr{3}. Evaluation does not occur if the arguments
22463 to a function are somehow of the wrong type @expr{@tfn{tan}([2,3,4])}),
22464 range (@expr{@tfn{tan}(90)}), or number (@expr{@tfn{tan}(3,5)}),
22465 or if the function name is not recognized (@expr{@tfn{f}(5)}), or if
22466 Symbolic mode (@pxref{Symbolic Mode}) prevents evaluation
22467 (@expr{@tfn{sqrt}(2)}).
22469 Calc simplifies (evaluates) the arguments to a function before it
22470 simplifies the function itself. Thus @expr{@tfn{sqrt}(5+4)} is
22471 simplified to @expr{@tfn{sqrt}(9)} before the @code{sqrt} function
22472 itself is applied. There are very few exceptions to this rule:
22473 @code{quote}, @code{lambda}, and @code{condition} (the @code{::}
22474 operator) do not evaluate their arguments, @code{if} (the @code{? :}
22475 operator) does not evaluate all of its arguments, and @code{evalto}
22476 does not evaluate its lefthand argument.
22478 Most commands apply the default simplifications to all arguments they
22479 take from the stack, perform a particular operation, then simplify
22480 the result before pushing it back on the stack. In the common special
22481 case of regular arithmetic commands like @kbd{+} and @kbd{Q} [@code{sqrt}],
22482 the arguments are simply popped from the stack and collected into a
22483 suitable function call, which is then simplified (the arguments being
22484 simplified first as part of the process, as described above).
22486 The default simplifications are too numerous to describe completely
22487 here, but this section will describe the ones that apply to the
22488 major arithmetic operators. This list will be rather technical in
22489 nature, and will probably be interesting to you only if you are
22490 a serious user of Calc's algebra facilities.
22496 As well as the simplifications described here, if you have stored
22497 any rewrite rules in the variable @code{EvalRules} then these rules
22498 will also be applied before any built-in default simplifications.
22499 @xref{Automatic Rewrites}, for details.
22505 And now, on with the default simplifications:
22507 Arithmetic operators like @kbd{+} and @kbd{*} always take two
22508 arguments in Calc's internal form. Sums and products of three or
22509 more terms are arranged by the associative law of algebra into
22510 a left-associative form for sums, @expr{((a + b) + c) + d}, and
22511 a right-associative form for products, @expr{a * (b * (c * d))}.
22512 Formulas like @expr{(a + b) + (c + d)} are rearranged to
22513 left-associative form, though this rarely matters since Calc's
22514 algebra commands are designed to hide the inner structure of
22515 sums and products as much as possible. Sums and products in
22516 their proper associative form will be written without parentheses
22517 in the examples below.
22519 Sums and products are @emph{not} rearranged according to the
22520 commutative law (@expr{a + b} to @expr{b + a}) except in a few
22521 special cases described below. Some algebra programs always
22522 rearrange terms into a canonical order, which enables them to
22523 see that @expr{a b + b a} can be simplified to @expr{2 a b}.
22524 Calc assumes you have put the terms into the order you want
22525 and generally leaves that order alone, with the consequence
22526 that formulas like the above will only be simplified if you
22527 explicitly give the @kbd{a s} command. @xref{Algebraic
22530 Differences @expr{a - b} are treated like sums @expr{a + (-b)}
22531 for purposes of simplification; one of the default simplifications
22532 is to rewrite @expr{a + (-b)} or @expr{(-b) + a}, where @expr{-b}
22533 represents a ``negative-looking'' term, into @expr{a - b} form.
22534 ``Negative-looking'' means negative numbers, negated formulas like
22535 @expr{-x}, and products or quotients in which either term is
22538 Other simplifications involving negation are @expr{-(-x)} to @expr{x};
22539 @expr{-(a b)} or @expr{-(a/b)} where either @expr{a} or @expr{b} is
22540 negative-looking, simplified by negating that term, or else where
22541 @expr{a} or @expr{b} is any number, by negating that number;
22542 @expr{-(a + b)} to @expr{-a - b}, and @expr{-(b - a)} to @expr{a - b}.
22543 (This, and rewriting @expr{(-b) + a} to @expr{a - b}, are the only
22544 cases where the order of terms in a sum is changed by the default
22547 The distributive law is used to simplify sums in some cases:
22548 @expr{a x + b x} to @expr{(a + b) x}, where @expr{a} represents
22549 a number or an implicit 1 or @mathit{-1} (as in @expr{x} or @expr{-x})
22550 and similarly for @expr{b}. Use the @kbd{a c}, @w{@kbd{a f}}, or
22551 @kbd{j M} commands to merge sums with non-numeric coefficients
22552 using the distributive law.
22554 The distributive law is only used for sums of two terms, or
22555 for adjacent terms in a larger sum. Thus @expr{a + b + b + c}
22556 is simplified to @expr{a + 2 b + c}, but @expr{a + b + c + b}
22557 is not simplified. The reason is that comparing all terms of a
22558 sum with one another would require time proportional to the
22559 square of the number of terms; Calc relegates potentially slow
22560 operations like this to commands that have to be invoked
22561 explicitly, like @kbd{a s}.
22563 Finally, @expr{a + 0} and @expr{0 + a} are simplified to @expr{a}.
22564 A consequence of the above rules is that @expr{0 - a} is simplified
22571 The products @expr{1 a} and @expr{a 1} are simplified to @expr{a};
22572 @expr{(-1) a} and @expr{a (-1)} are simplified to @expr{-a};
22573 @expr{0 a} and @expr{a 0} are simplified to @expr{0}, except that
22574 in Matrix mode where @expr{a} is not provably scalar the result
22575 is the generic zero matrix @samp{idn(0)}, and that if @expr{a} is
22576 infinite the result is @samp{nan}.
22578 Also, @expr{(-a) b} and @expr{a (-b)} are simplified to @expr{-(a b)},
22579 where this occurs for negated formulas but not for regular negative
22582 Products are commuted only to move numbers to the front:
22583 @expr{a b 2} is commuted to @expr{2 a b}.
22585 The product @expr{a (b + c)} is distributed over the sum only if
22586 @expr{a} and at least one of @expr{b} and @expr{c} are numbers:
22587 @expr{2 (x + 3)} goes to @expr{2 x + 6}. The formula
22588 @expr{(-a) (b - c)}, where @expr{-a} is a negative number, is
22589 rewritten to @expr{a (c - b)}.
22591 The distributive law of products and powers is used for adjacent
22592 terms of the product: @expr{x^a x^b} goes to
22593 @texline @math{x^{a+b}}
22594 @infoline @expr{x^(a+b)}
22595 where @expr{a} is a number, or an implicit 1 (as in @expr{x}),
22596 or the implicit one-half of @expr{@tfn{sqrt}(x)}, and similarly for
22597 @expr{b}. The result is written using @samp{sqrt} or @samp{1/sqrt}
22598 if the sum of the powers is @expr{1/2} or @expr{-1/2}, respectively.
22599 If the sum of the powers is zero, the product is simplified to
22600 @expr{1} or to @samp{idn(1)} if Matrix mode is enabled.
22602 The product of a negative power times anything but another negative
22603 power is changed to use division:
22604 @texline @math{x^{-2} y}
22605 @infoline @expr{x^(-2) y}
22606 goes to @expr{y / x^2} unless Matrix mode is
22607 in effect and neither @expr{x} nor @expr{y} are scalar (in which
22608 case it is considered unsafe to rearrange the order of the terms).
22610 Finally, @expr{a (b/c)} is rewritten to @expr{(a b)/c}, and also
22611 @expr{(a/b) c} is changed to @expr{(a c)/b} unless in Matrix mode.
22617 Simplifications for quotients are analogous to those for products.
22618 The quotient @expr{0 / x} is simplified to @expr{0}, with the same
22619 exceptions that were noted for @expr{0 x}. Likewise, @expr{x / 1}
22620 and @expr{x / (-1)} are simplified to @expr{x} and @expr{-x},
22623 The quotient @expr{x / 0} is left unsimplified or changed to an
22624 infinite quantity, as directed by the current infinite mode.
22625 @xref{Infinite Mode}.
22628 @texline @math{a / b^{-c}}
22629 @infoline @expr{a / b^(-c)}
22630 is changed to @expr{a b^c}, where @expr{-c} is any negative-looking
22631 power. Also, @expr{1 / b^c} is changed to
22632 @texline @math{b^{-c}}
22633 @infoline @expr{b^(-c)}
22634 for any power @expr{c}.
22636 Also, @expr{(-a) / b} and @expr{a / (-b)} go to @expr{-(a/b)};
22637 @expr{(a/b) / c} goes to @expr{a / (b c)}; and @expr{a / (b/c)}
22638 goes to @expr{(a c) / b} unless Matrix mode prevents this
22639 rearrangement. Similarly, @expr{a / (b:c)} is simplified to
22640 @expr{(c:b) a} for any fraction @expr{b:c}.
22642 The distributive law is applied to @expr{(a + b) / c} only if
22643 @expr{c} and at least one of @expr{a} and @expr{b} are numbers.
22644 Quotients of powers and square roots are distributed just as
22645 described for multiplication.
22647 Quotients of products cancel only in the leading terms of the
22648 numerator and denominator. In other words, @expr{a x b / a y b}
22649 is cancelled to @expr{x b / y b} but not to @expr{x / y}. Once
22650 again this is because full cancellation can be slow; use @kbd{a s}
22651 to cancel all terms of the quotient.
22653 Quotients of negative-looking values are simplified according
22654 to @expr{(-a) / (-b)} to @expr{a / b}, @expr{(-a) / (b - c)}
22655 to @expr{a / (c - b)}, and @expr{(a - b) / (-c)} to @expr{(b - a) / c}.
22661 The formula @expr{x^0} is simplified to @expr{1}, or to @samp{idn(1)}
22662 in Matrix mode. The formula @expr{0^x} is simplified to @expr{0}
22663 unless @expr{x} is a negative number or complex number, in which
22664 case the result is an infinity or an unsimplified formula according
22665 to the current infinite mode. Note that @expr{0^0} is an
22666 indeterminate form, as evidenced by the fact that the simplifications
22667 for @expr{x^0} and @expr{0^x} conflict when @expr{x=0}.
22669 Powers of products or quotients @expr{(a b)^c}, @expr{(a/b)^c}
22670 are distributed to @expr{a^c b^c}, @expr{a^c / b^c} only if @expr{c}
22671 is an integer, or if either @expr{a} or @expr{b} are nonnegative
22672 real numbers. Powers of powers @expr{(a^b)^c} are simplified to
22673 @texline @math{a^{b c}}
22674 @infoline @expr{a^(b c)}
22675 only when @expr{c} is an integer and @expr{b c} also
22676 evaluates to an integer. Without these restrictions these simplifications
22677 would not be safe because of problems with principal values.
22679 @texline @math{((-3)^{1/2})^2}
22680 @infoline @expr{((-3)^1:2)^2}
22681 is safe to simplify, but
22682 @texline @math{((-3)^2)^{1/2}}
22683 @infoline @expr{((-3)^2)^1:2}
22684 is not.) @xref{Declarations}, for ways to inform Calc that your
22685 variables satisfy these requirements.
22687 As a special case of this rule, @expr{@tfn{sqrt}(x)^n} is simplified to
22688 @texline @math{x^{n/2}}
22689 @infoline @expr{x^(n/2)}
22690 only for even integers @expr{n}.
22692 If @expr{a} is known to be real, @expr{b} is an even integer, and
22693 @expr{c} is a half- or quarter-integer, then @expr{(a^b)^c} is
22694 simplified to @expr{@tfn{abs}(a^(b c))}.
22696 Also, @expr{(-a)^b} is simplified to @expr{a^b} if @expr{b} is an
22697 even integer, or to @expr{-(a^b)} if @expr{b} is an odd integer,
22698 for any negative-looking expression @expr{-a}.
22700 Square roots @expr{@tfn{sqrt}(x)} generally act like one-half powers
22701 @texline @math{x^{1:2}}
22702 @infoline @expr{x^1:2}
22703 for the purposes of the above-listed simplifications.
22706 @texline @math{1 / x^{1:2}}
22707 @infoline @expr{1 / x^1:2}
22709 @texline @math{x^{-1:2}},
22710 @infoline @expr{x^(-1:2)},
22711 but @expr{1 / @tfn{sqrt}(x)} is left alone.
22717 Generic identity matrices (@pxref{Matrix Mode}) are simplified by the
22718 following rules: @expr{@tfn{idn}(a) + b} to @expr{a + b} if @expr{b}
22719 is provably scalar, or expanded out if @expr{b} is a matrix;
22720 @expr{@tfn{idn}(a) + @tfn{idn}(b)} to @expr{@tfn{idn}(a + b)};
22721 @expr{-@tfn{idn}(a)} to @expr{@tfn{idn}(-a)}; @expr{a @tfn{idn}(b)} to
22722 @expr{@tfn{idn}(a b)} if @expr{a} is provably scalar, or to @expr{a b}
22723 if @expr{a} is provably non-scalar; @expr{@tfn{idn}(a) @tfn{idn}(b)} to
22724 @expr{@tfn{idn}(a b)}; analogous simplifications for quotients involving
22725 @code{idn}; and @expr{@tfn{idn}(a)^n} to @expr{@tfn{idn}(a^n)} where
22726 @expr{n} is an integer.
22732 The @code{floor} function and other integer truncation functions
22733 vanish if the argument is provably integer-valued, so that
22734 @expr{@tfn{floor}(@tfn{round}(x))} simplifies to @expr{@tfn{round}(x)}.
22735 Also, combinations of @code{float}, @code{floor} and its friends,
22736 and @code{ffloor} and its friends, are simplified in appropriate
22737 ways. @xref{Integer Truncation}.
22739 The expression @expr{@tfn{abs}(-x)} changes to @expr{@tfn{abs}(x)}.
22740 The expression @expr{@tfn{abs}(@tfn{abs}(x))} changes to
22741 @expr{@tfn{abs}(x)}; in fact, @expr{@tfn{abs}(x)} changes to @expr{x} or
22742 @expr{-x} if @expr{x} is provably nonnegative or nonpositive
22743 (@pxref{Declarations}).
22745 While most functions do not recognize the variable @code{i} as an
22746 imaginary number, the @code{arg} function does handle the two cases
22747 @expr{@tfn{arg}(@tfn{i})} and @expr{@tfn{arg}(-@tfn{i})} just for convenience.
22749 The expression @expr{@tfn{conj}(@tfn{conj}(x))} simplifies to @expr{x}.
22750 Various other expressions involving @code{conj}, @code{re}, and
22751 @code{im} are simplified, especially if some of the arguments are
22752 provably real or involve the constant @code{i}. For example,
22753 @expr{@tfn{conj}(a + b i)} is changed to
22754 @expr{@tfn{conj}(a) - @tfn{conj}(b) i}, or to @expr{a - b i} if @expr{a}
22755 and @expr{b} are known to be real.
22757 Functions like @code{sin} and @code{arctan} generally don't have
22758 any default simplifications beyond simply evaluating the functions
22759 for suitable numeric arguments and infinity. The @kbd{a s} command
22760 described in the next section does provide some simplifications for
22761 these functions, though.
22763 One important simplification that does occur is that
22764 @expr{@tfn{ln}(@tfn{e})} is simplified to 1, and @expr{@tfn{ln}(@tfn{e}^x)} is
22765 simplified to @expr{x} for any @expr{x}. This occurs even if you have
22766 stored a different value in the Calc variable @samp{e}; but this would
22767 be a bad idea in any case if you were also using natural logarithms!
22769 Among the logical functions, @tfn{(@var{a} <= @var{b})} changes to
22770 @tfn{@var{a} > @var{b}} and so on. Equations and inequalities where both sides
22771 are either negative-looking or zero are simplified by negating both sides
22772 and reversing the inequality. While it might seem reasonable to simplify
22773 @expr{!!x} to @expr{x}, this would not be valid in general because
22774 @expr{!!2} is 1, not 2.
22776 Most other Calc functions have few if any default simplifications
22777 defined, aside of course from evaluation when the arguments are
22780 @node Algebraic Simplifications, Unsafe Simplifications, Default Simplifications, Simplifying Formulas
22781 @subsection Algebraic Simplifications
22784 @cindex Algebraic simplifications
22785 The @kbd{a s} command makes simplifications that may be too slow to
22786 do all the time, or that may not be desirable all of the time.
22787 If you find these simplifications are worthwhile, you can type
22788 @kbd{m A} to have Calc apply them automatically.
22790 This section describes all simplifications that are performed by
22791 the @kbd{a s} command. Note that these occur in addition to the
22792 default simplifications; even if the default simplifications have
22793 been turned off by an @kbd{m O} command, @kbd{a s} will turn them
22794 back on temporarily while it simplifies the formula.
22796 There is a variable, @code{AlgSimpRules}, in which you can put rewrites
22797 to be applied by @kbd{a s}. Its use is analogous to @code{EvalRules},
22798 but without the special restrictions. Basically, the simplifier does
22799 @samp{@w{a r} AlgSimpRules} with an infinite repeat count on the whole
22800 expression being simplified, then it traverses the expression applying
22801 the built-in rules described below. If the result is different from
22802 the original expression, the process repeats with the default
22803 simplifications (including @code{EvalRules}), then @code{AlgSimpRules},
22804 then the built-in simplifications, and so on.
22810 Sums are simplified in two ways. Constant terms are commuted to the
22811 end of the sum, so that @expr{a + 2 + b} changes to @expr{a + b + 2}.
22812 The only exception is that a constant will not be commuted away
22813 from the first position of a difference, i.e., @expr{2 - x} is not
22814 commuted to @expr{-x + 2}.
22816 Also, terms of sums are combined by the distributive law, as in
22817 @expr{x + y + 2 x} to @expr{y + 3 x}. This always occurs for
22818 adjacent terms, but @kbd{a s} compares all pairs of terms including
22825 Products are sorted into a canonical order using the commutative
22826 law. For example, @expr{b c a} is commuted to @expr{a b c}.
22827 This allows easier comparison of products; for example, the default
22828 simplifications will not change @expr{x y + y x} to @expr{2 x y},
22829 but @kbd{a s} will; it first rewrites the sum to @expr{x y + x y},
22830 and then the default simplifications are able to recognize a sum
22831 of identical terms.
22833 The canonical ordering used to sort terms of products has the
22834 property that real-valued numbers, interval forms and infinities
22835 come first, and are sorted into increasing order. The @kbd{V S}
22836 command uses the same ordering when sorting a vector.
22838 Sorting of terms of products is inhibited when Matrix mode is
22839 turned on; in this case, Calc will never exchange the order of
22840 two terms unless it knows at least one of the terms is a scalar.
22842 Products of powers are distributed by comparing all pairs of
22843 terms, using the same method that the default simplifications
22844 use for adjacent terms of products.
22846 Even though sums are not sorted, the commutative law is still
22847 taken into account when terms of a product are being compared.
22848 Thus @expr{(x + y) (y + x)} will be simplified to @expr{(x + y)^2}.
22849 A subtle point is that @expr{(x - y) (y - x)} will @emph{not}
22850 be simplified to @expr{-(x - y)^2}; Calc does not notice that
22851 one term can be written as a constant times the other, even if
22852 that constant is @mathit{-1}.
22854 A fraction times any expression, @expr{(a:b) x}, is changed to
22855 a quotient involving integers: @expr{a x / b}. This is not
22856 done for floating-point numbers like @expr{0.5}, however. This
22857 is one reason why you may find it convenient to turn Fraction mode
22858 on while doing algebra; @pxref{Fraction Mode}.
22864 Quotients are simplified by comparing all terms in the numerator
22865 with all terms in the denominator for possible cancellation using
22866 the distributive law. For example, @expr{a x^2 b / c x^3 d} will
22867 cancel @expr{x^2} from the top and bottom to get @expr{a b / c x d}.
22868 (The terms in the denominator will then be rearranged to @expr{c d x}
22869 as described above.) If there is any common integer or fractional
22870 factor in the numerator and denominator, it is cancelled out;
22871 for example, @expr{(4 x + 6) / 8 x} simplifies to @expr{(2 x + 3) / 4 x}.
22873 Non-constant common factors are not found even by @kbd{a s}. To
22874 cancel the factor @expr{a} in @expr{(a x + a) / a^2} you could first
22875 use @kbd{j M} on the product @expr{a x} to Merge the numerator to
22876 @expr{a (1+x)}, which can then be simplified successfully.
22882 Integer powers of the variable @code{i} are simplified according
22883 to the identity @expr{i^2 = -1}. If you store a new value other
22884 than the complex number @expr{(0,1)} in @code{i}, this simplification
22885 will no longer occur. This is done by @kbd{a s} instead of by default
22886 in case someone (unwisely) uses the name @code{i} for a variable
22887 unrelated to complex numbers; it would be unfortunate if Calc
22888 quietly and automatically changed this formula for reasons the
22889 user might not have been thinking of.
22891 Square roots of integer or rational arguments are simplified in
22892 several ways. (Note that these will be left unevaluated only in
22893 Symbolic mode.) First, square integer or rational factors are
22894 pulled out so that @expr{@tfn{sqrt}(8)} is rewritten as
22895 @texline @math{2\,@tfn{sqrt}(2)}.
22896 @infoline @expr{2 sqrt(2)}.
22897 Conceptually speaking this implies factoring the argument into primes
22898 and moving pairs of primes out of the square root, but for reasons of
22899 efficiency Calc only looks for primes up to 29.
22901 Square roots in the denominator of a quotient are moved to the
22902 numerator: @expr{1 / @tfn{sqrt}(3)} changes to @expr{@tfn{sqrt}(3) / 3}.
22903 The same effect occurs for the square root of a fraction:
22904 @expr{@tfn{sqrt}(2:3)} changes to @expr{@tfn{sqrt}(6) / 3}.
22910 The @code{%} (modulo) operator is simplified in several ways
22911 when the modulus @expr{M} is a positive real number. First, if
22912 the argument is of the form @expr{x + n} for some real number
22913 @expr{n}, then @expr{n} is itself reduced modulo @expr{M}. For
22914 example, @samp{(x - 23) % 10} is simplified to @samp{(x + 7) % 10}.
22916 If the argument is multiplied by a constant, and this constant
22917 has a common integer divisor with the modulus, then this factor is
22918 cancelled out. For example, @samp{12 x % 15} is changed to
22919 @samp{3 (4 x % 5)} by factoring out 3. Also, @samp{(12 x + 1) % 15}
22920 is changed to @samp{3 ((4 x + 1:3) % 5)}. While these forms may
22921 not seem ``simpler,'' they allow Calc to discover useful information
22922 about modulo forms in the presence of declarations.
22924 If the modulus is 1, then Calc can use @code{int} declarations to
22925 evaluate the expression. For example, the idiom @samp{x % 2} is
22926 often used to check whether a number is odd or even. As described
22927 above, @w{@samp{2 n % 2}} and @samp{(2 n + 1) % 2} are simplified to
22928 @samp{2 (n % 1)} and @samp{2 ((n + 1:2) % 1)}, respectively; Calc
22929 can simplify these to 0 and 1 (respectively) if @code{n} has been
22930 declared to be an integer.
22936 Trigonometric functions are simplified in several ways. Whenever a
22937 products of two trigonometric functions can be replaced by a single
22938 function, the replacement is made; for example,
22939 @expr{@tfn{tan}(x) @tfn{cos}(x)} is simplified to @expr{@tfn{sin}(x)}.
22940 Reciprocals of trigonometric functions are replaced by their reciprocal
22941 function; for example, @expr{1/@tfn{sec}(x)} is simplified to
22942 @expr{@tfn{cos}(x)}. The corresponding simplifications for the
22943 hyperbolic functions are also handled.
22945 Trigonometric functions of their inverse functions are
22946 simplified. The expression @expr{@tfn{sin}(@tfn{arcsin}(x))} is
22947 simplified to @expr{x}, and similarly for @code{cos} and @code{tan}.
22948 Trigonometric functions of inverses of different trigonometric
22949 functions can also be simplified, as in @expr{@tfn{sin}(@tfn{arccos}(x))}
22950 to @expr{@tfn{sqrt}(1 - x^2)}.
22952 If the argument to @code{sin} is negative-looking, it is simplified to
22953 @expr{-@tfn{sin}(x)}, and similarly for @code{cos} and @code{tan}.
22954 Finally, certain special values of the argument are recognized;
22955 @pxref{Trigonometric and Hyperbolic Functions}.
22957 Hyperbolic functions of their inverses and of negative-looking
22958 arguments are also handled, as are exponentials of inverse
22959 hyperbolic functions.
22961 No simplifications for inverse trigonometric and hyperbolic
22962 functions are known, except for negative arguments of @code{arcsin},
22963 @code{arctan}, @code{arcsinh}, and @code{arctanh}. Note that
22964 @expr{@tfn{arcsin}(@tfn{sin}(x))} can @emph{not} safely change to
22965 @expr{x}, since this only correct within an integer multiple of
22966 @texline @math{2 \pi}
22967 @infoline @expr{2 pi}
22968 radians or 360 degrees. However, @expr{@tfn{arcsinh}(@tfn{sinh}(x))} is
22969 simplified to @expr{x} if @expr{x} is known to be real.
22971 Several simplifications that apply to logarithms and exponentials
22972 are that @expr{@tfn{exp}(@tfn{ln}(x))},
22973 @texline @tfn{e}@math{^{\ln(x)}},
22974 @infoline @expr{e^@tfn{ln}(x)},
22976 @texline @math{10^{{\rm log10}(x)}}
22977 @infoline @expr{10^@tfn{log10}(x)}
22978 all reduce to @expr{x}. Also, @expr{@tfn{ln}(@tfn{exp}(x))}, etc., can
22979 reduce to @expr{x} if @expr{x} is provably real. The form
22980 @expr{@tfn{exp}(x)^y} is simplified to @expr{@tfn{exp}(x y)}. If @expr{x}
22981 is a suitable multiple of
22982 @texline @math{\pi i}
22983 @infoline @expr{pi i}
22984 (as described above for the trigonometric functions), then
22985 @expr{@tfn{exp}(x)} or @expr{e^x} will be expanded. Finally,
22986 @expr{@tfn{ln}(x)} is simplified to a form involving @code{pi} and
22987 @code{i} where @expr{x} is provably negative, positive imaginary, or
22988 negative imaginary.
22990 The error functions @code{erf} and @code{erfc} are simplified when
22991 their arguments are negative-looking or are calls to the @code{conj}
22998 Equations and inequalities are simplified by cancelling factors
22999 of products, quotients, or sums on both sides. Inequalities
23000 change sign if a negative multiplicative factor is cancelled.
23001 Non-constant multiplicative factors as in @expr{a b = a c} are
23002 cancelled from equations only if they are provably nonzero (generally
23003 because they were declared so; @pxref{Declarations}). Factors
23004 are cancelled from inequalities only if they are nonzero and their
23007 Simplification also replaces an equation or inequality with
23008 1 or 0 (``true'' or ``false'') if it can through the use of
23009 declarations. If @expr{x} is declared to be an integer greater
23010 than 5, then @expr{x < 3}, @expr{x = 3}, and @expr{x = 7.5} are
23011 all simplified to 0, but @expr{x > 3} is simplified to 1.
23012 By a similar analysis, @expr{abs(x) >= 0} is simplified to 1,
23013 as is @expr{x^2 >= 0} if @expr{x} is known to be real.
23015 @node Unsafe Simplifications, Simplification of Units, Algebraic Simplifications, Simplifying Formulas
23016 @subsection ``Unsafe'' Simplifications
23019 @cindex Unsafe simplifications
23020 @cindex Extended simplification
23022 @pindex calc-simplify-extended
23024 @mindex esimpl@idots
23027 The @kbd{a e} (@code{calc-simplify-extended}) [@code{esimplify}] command
23029 except that it applies some additional simplifications which are not
23030 ``safe'' in all cases. Use this only if you know the values in your
23031 formula lie in the restricted ranges for which these simplifications
23032 are valid. The symbolic integrator uses @kbd{a e};
23033 one effect of this is that the integrator's results must be used with
23034 caution. Where an integral table will often attach conditions like
23035 ``for positive @expr{a} only,'' Calc (like most other symbolic
23036 integration programs) will simply produce an unqualified result.
23038 Because @kbd{a e}'s simplifications are unsafe, it is sometimes better
23039 to type @kbd{C-u -3 a v}, which does extended simplification only
23040 on the top level of the formula without affecting the sub-formulas.
23041 In fact, @kbd{C-u -3 j v} allows you to target extended simplification
23042 to any specific part of a formula.
23044 The variable @code{ExtSimpRules} contains rewrites to be applied by
23045 the @kbd{a e} command. These are applied in addition to
23046 @code{EvalRules} and @code{AlgSimpRules}. (The @kbd{a r AlgSimpRules}
23047 step described above is simply followed by an @kbd{a r ExtSimpRules} step.)
23049 Following is a complete list of ``unsafe'' simplifications performed
23056 Inverse trigonometric or hyperbolic functions, called with their
23057 corresponding non-inverse functions as arguments, are simplified
23058 by @kbd{a e}. For example, @expr{@tfn{arcsin}(@tfn{sin}(x))} changes
23059 to @expr{x}. Also, @expr{@tfn{arcsin}(@tfn{cos}(x))} and
23060 @expr{@tfn{arccos}(@tfn{sin}(x))} both change to @expr{@tfn{pi}/2 - x}.
23061 These simplifications are unsafe because they are valid only for
23062 values of @expr{x} in a certain range; outside that range, values
23063 are folded down to the 360-degree range that the inverse trigonometric
23064 functions always produce.
23066 Powers of powers @expr{(x^a)^b} are simplified to
23067 @texline @math{x^{a b}}
23068 @infoline @expr{x^(a b)}
23069 for all @expr{a} and @expr{b}. These results will be valid only
23070 in a restricted range of @expr{x}; for example, in
23071 @texline @math{(x^2)^{1:2}}
23072 @infoline @expr{(x^2)^1:2}
23073 the powers cancel to get @expr{x}, which is valid for positive values
23074 of @expr{x} but not for negative or complex values.
23076 Similarly, @expr{@tfn{sqrt}(x^a)} and @expr{@tfn{sqrt}(x)^a} are both
23077 simplified (possibly unsafely) to
23078 @texline @math{x^{a/2}}.
23079 @infoline @expr{x^(a/2)}.
23081 Forms like @expr{@tfn{sqrt}(1 - sin(x)^2)} are simplified to, e.g.,
23082 @expr{@tfn{cos}(x)}. Calc has identities of this sort for @code{sin},
23083 @code{cos}, @code{tan}, @code{sinh}, and @code{cosh}.
23085 Arguments of square roots are partially factored to look for
23086 squared terms that can be extracted. For example,
23087 @expr{@tfn{sqrt}(a^2 b^3 + a^3 b^2)} simplifies to
23088 @expr{a b @tfn{sqrt}(a+b)}.
23090 The simplifications of @expr{@tfn{ln}(@tfn{exp}(x))},
23091 @expr{@tfn{ln}(@tfn{e}^x)}, and @expr{@tfn{log10}(10^x)} to @expr{x} are also
23092 unsafe because of problems with principal values (although these
23093 simplifications are safe if @expr{x} is known to be real).
23095 Common factors are cancelled from products on both sides of an
23096 equation, even if those factors may be zero: @expr{a x / b x}
23097 to @expr{a / b}. Such factors are never cancelled from
23098 inequalities: Even @kbd{a e} is not bold enough to reduce
23099 @expr{a x < b x} to @expr{a < b} (or @expr{a > b}, depending
23100 on whether you believe @expr{x} is positive or negative).
23101 The @kbd{a M /} command can be used to divide a factor out of
23102 both sides of an inequality.
23104 @node Simplification of Units, , Unsafe Simplifications, Simplifying Formulas
23105 @subsection Simplification of Units
23108 The simplifications described in this section are applied by the
23109 @kbd{u s} (@code{calc-simplify-units}) command. These are in addition
23110 to the regular @kbd{a s} (but not @kbd{a e}) simplifications described
23111 earlier. @xref{Basic Operations on Units}.
23113 The variable @code{UnitSimpRules} contains rewrites to be applied by
23114 the @kbd{u s} command. These are applied in addition to @code{EvalRules}
23115 and @code{AlgSimpRules}.
23117 Scalar mode is automatically put into effect when simplifying units.
23118 @xref{Matrix Mode}.
23120 Sums @expr{a + b} involving units are simplified by extracting the
23121 units of @expr{a} as if by the @kbd{u x} command (call the result
23122 @expr{u_a}), then simplifying the expression @expr{b / u_a}
23123 using @kbd{u b} and @kbd{u s}. If the result has units then the sum
23124 is inconsistent and is left alone. Otherwise, it is rewritten
23125 in terms of the units @expr{u_a}.
23127 If units auto-ranging mode is enabled, products or quotients in
23128 which the first argument is a number which is out of range for the
23129 leading unit are modified accordingly.
23131 When cancelling and combining units in products and quotients,
23132 Calc accounts for unit names that differ only in the prefix letter.
23133 For example, @samp{2 km m} is simplified to @samp{2000 m^2}.
23134 However, compatible but different units like @code{ft} and @code{in}
23135 are not combined in this way.
23137 Quotients @expr{a / b} are simplified in three additional ways. First,
23138 if @expr{b} is a number or a product beginning with a number, Calc
23139 computes the reciprocal of this number and moves it to the numerator.
23141 Second, for each pair of unit names from the numerator and denominator
23142 of a quotient, if the units are compatible (e.g., they are both
23143 units of area) then they are replaced by the ratio between those
23144 units. For example, in @samp{3 s in N / kg cm} the units
23145 @samp{in / cm} will be replaced by @expr{2.54}.
23147 Third, if the units in the quotient exactly cancel out, so that
23148 a @kbd{u b} command on the quotient would produce a dimensionless
23149 number for an answer, then the quotient simplifies to that number.
23151 For powers and square roots, the ``unsafe'' simplifications
23152 @expr{(a b)^c} to @expr{a^c b^c}, @expr{(a/b)^c} to @expr{a^c / b^c},
23153 and @expr{(a^b)^c} to
23154 @texline @math{a^{b c}}
23155 @infoline @expr{a^(b c)}
23156 are done if the powers are real numbers. (These are safe in the context
23157 of units because all numbers involved can reasonably be assumed to be
23160 Also, if a unit name is raised to a fractional power, and the
23161 base units in that unit name all occur to powers which are a
23162 multiple of the denominator of the power, then the unit name
23163 is expanded out into its base units, which can then be simplified
23164 according to the previous paragraph. For example, @samp{acre^1.5}
23165 is simplified by noting that @expr{1.5 = 3:2}, that @samp{acre}
23166 is defined in terms of @samp{m^2}, and that the 2 in the power of
23167 @code{m} is a multiple of 2 in @expr{3:2}. Thus, @code{acre^1.5} is
23168 replaced by approximately
23169 @texline @math{(4046 m^2)^{1.5}}
23170 @infoline @expr{(4046 m^2)^1.5},
23171 which is then changed to
23172 @texline @math{4046^{1.5} \, (m^2)^{1.5}},
23173 @infoline @expr{4046^1.5 (m^2)^1.5},
23174 then to @expr{257440 m^3}.
23176 The functions @code{float}, @code{frac}, @code{clean}, @code{abs},
23177 as well as @code{floor} and the other integer truncation functions,
23178 applied to unit names or products or quotients involving units, are
23179 simplified. For example, @samp{round(1.6 in)} is changed to
23180 @samp{round(1.6) round(in)}; the lefthand term evaluates to 2,
23181 and the righthand term simplifies to @code{in}.
23183 The functions @code{sin}, @code{cos}, and @code{tan} with arguments
23184 that have angular units like @code{rad} or @code{arcmin} are
23185 simplified by converting to base units (radians), then evaluating
23186 with the angular mode temporarily set to radians.
23188 @node Polynomials, Calculus, Simplifying Formulas, Algebra
23189 @section Polynomials
23191 A @dfn{polynomial} is a sum of terms which are coefficients times
23192 various powers of a ``base'' variable. For example, @expr{2 x^2 + 3 x - 4}
23193 is a polynomial in @expr{x}. Some formulas can be considered
23194 polynomials in several different variables: @expr{1 + 2 x + 3 y + 4 x y^2}
23195 is a polynomial in both @expr{x} and @expr{y}. Polynomial coefficients
23196 are often numbers, but they may in general be any formulas not
23197 involving the base variable.
23200 @pindex calc-factor
23202 The @kbd{a f} (@code{calc-factor}) [@code{factor}] command factors a
23203 polynomial into a product of terms. For example, the polynomial
23204 @expr{x^3 + 2 x^2 + x} is factored into @samp{x*(x+1)^2}. As another
23205 example, @expr{a c + b d + b c + a d} is factored into the product
23206 @expr{(a + b) (c + d)}.
23208 Calc currently has three algorithms for factoring. Formulas which are
23209 linear in several variables, such as the second example above, are
23210 merged according to the distributive law. Formulas which are
23211 polynomials in a single variable, with constant integer or fractional
23212 coefficients, are factored into irreducible linear and/or quadratic
23213 terms. The first example above factors into three linear terms
23214 (@expr{x}, @expr{x+1}, and @expr{x+1} again). Finally, formulas
23215 which do not fit the above criteria are handled by the algebraic
23218 Calc's polynomial factorization algorithm works by using the general
23219 root-finding command (@w{@kbd{a P}}) to solve for the roots of the
23220 polynomial. It then looks for roots which are rational numbers
23221 or complex-conjugate pairs, and converts these into linear and
23222 quadratic terms, respectively. Because it uses floating-point
23223 arithmetic, it may be unable to find terms that involve large
23224 integers (whose number of digits approaches the current precision).
23225 Also, irreducible factors of degree higher than quadratic are not
23226 found, and polynomials in more than one variable are not treated.
23227 (A more robust factorization algorithm may be included in a future
23230 @vindex FactorRules
23242 The rewrite-based factorization method uses rules stored in the variable
23243 @code{FactorRules}. @xref{Rewrite Rules}, for a discussion of the
23244 operation of rewrite rules. The default @code{FactorRules} are able
23245 to factor quadratic forms symbolically into two linear terms,
23246 @expr{(a x + b) (c x + d)}. You can edit these rules to include other
23247 cases if you wish. To use the rules, Calc builds the formula
23248 @samp{thecoefs(x, [a, b, c, ...])} where @code{x} is the polynomial
23249 base variable and @code{a}, @code{b}, etc., are polynomial coefficients
23250 (which may be numbers or formulas). The constant term is written first,
23251 i.e., in the @code{a} position. When the rules complete, they should have
23252 changed the formula into the form @samp{thefactors(x, [f1, f2, f3, ...])}
23253 where each @code{fi} should be a factored term, e.g., @samp{x - ai}.
23254 Calc then multiplies these terms together to get the complete
23255 factored form of the polynomial. If the rules do not change the
23256 @code{thecoefs} call to a @code{thefactors} call, @kbd{a f} leaves the
23257 polynomial alone on the assumption that it is unfactorable. (Note that
23258 the function names @code{thecoefs} and @code{thefactors} are used only
23259 as placeholders; there are no actual Calc functions by those names.)
23263 The @kbd{H a f} [@code{factors}] command also factors a polynomial,
23264 but it returns a list of factors instead of an expression which is the
23265 product of the factors. Each factor is represented by a sub-vector
23266 of the factor, and the power with which it appears. For example,
23267 @expr{x^5 + x^4 - 33 x^3 + 63 x^2} factors to @expr{(x + 7) x^2 (x - 3)^2}
23268 in @kbd{a f}, or to @expr{[ [x, 2], [x+7, 1], [x-3, 2] ]} in @kbd{H a f}.
23269 If there is an overall numeric factor, it always comes first in the list.
23270 The functions @code{factor} and @code{factors} allow a second argument
23271 when written in algebraic form; @samp{factor(x,v)} factors @expr{x} with
23272 respect to the specific variable @expr{v}. The default is to factor with
23273 respect to all the variables that appear in @expr{x}.
23276 @pindex calc-collect
23278 The @kbd{a c} (@code{calc-collect}) [@code{collect}] command rearranges a
23280 polynomial in a given variable, ordered in decreasing powers of that
23281 variable. For example, given @expr{1 + 2 x + 3 y + 4 x y^2} on
23282 the stack, @kbd{a c x} would produce @expr{(2 + 4 y^2) x + (1 + 3 y)},
23283 and @kbd{a c y} would produce @expr{(4 x) y^2 + 3 y + (1 + 2 x)}.
23284 The polynomial will be expanded out using the distributive law as
23285 necessary: Collecting @expr{x} in @expr{(x - 1)^3} produces
23286 @expr{x^3 - 3 x^2 + 3 x - 1}. Terms not involving @expr{x} will
23289 The ``variable'' you specify at the prompt can actually be any
23290 expression: @kbd{a c ln(x+1)} will collect together all terms multiplied
23291 by @samp{ln(x+1)} or integer powers thereof. If @samp{x} also appears
23292 in the formula in a context other than @samp{ln(x+1)}, @kbd{a c} will
23293 treat those occurrences as unrelated to @samp{ln(x+1)}, i.e., as constants.
23296 @pindex calc-expand
23298 The @kbd{a x} (@code{calc-expand}) [@code{expand}] command expands an
23299 expression by applying the distributive law everywhere. It applies to
23300 products, quotients, and powers involving sums. By default, it fully
23301 distributes all parts of the expression. With a numeric prefix argument,
23302 the distributive law is applied only the specified number of times, then
23303 the partially expanded expression is left on the stack.
23305 The @kbd{a x} and @kbd{j D} commands are somewhat redundant. Use
23306 @kbd{a x} if you want to expand all products of sums in your formula.
23307 Use @kbd{j D} if you want to expand a particular specified term of
23308 the formula. There is an exactly analogous correspondence between
23309 @kbd{a f} and @kbd{j M}. (The @kbd{j D} and @kbd{j M} commands
23310 also know many other kinds of expansions, such as
23311 @samp{exp(a + b) = exp(a) exp(b)}, which @kbd{a x} and @kbd{a f}
23314 Calc's automatic simplifications will sometimes reverse a partial
23315 expansion. For example, the first step in expanding @expr{(x+1)^3} is
23316 to write @expr{(x+1) (x+1)^2}. If @kbd{a x} stops there and tries
23317 to put this formula onto the stack, though, Calc will automatically
23318 simplify it back to @expr{(x+1)^3} form. The solution is to turn
23319 simplification off first (@pxref{Simplification Modes}), or to run
23320 @kbd{a x} without a numeric prefix argument so that it expands all
23321 the way in one step.
23326 The @kbd{a a} (@code{calc-apart}) [@code{apart}] command expands a
23327 rational function by partial fractions. A rational function is the
23328 quotient of two polynomials; @code{apart} pulls this apart into a
23329 sum of rational functions with simple denominators. In algebraic
23330 notation, the @code{apart} function allows a second argument that
23331 specifies which variable to use as the ``base''; by default, Calc
23332 chooses the base variable automatically.
23335 @pindex calc-normalize-rat
23337 The @kbd{a n} (@code{calc-normalize-rat}) [@code{nrat}] command
23338 attempts to arrange a formula into a quotient of two polynomials.
23339 For example, given @expr{1 + (a + b/c) / d}, the result would be
23340 @expr{(b + a c + c d) / c d}. The quotient is reduced, so that
23341 @kbd{a n} will simplify @expr{(x^2 + 2x + 1) / (x^2 - 1)} by dividing
23342 out the common factor @expr{x + 1}, yielding @expr{(x + 1) / (x - 1)}.
23345 @pindex calc-poly-div
23347 The @kbd{a \} (@code{calc-poly-div}) [@code{pdiv}] command divides
23348 two polynomials @expr{u} and @expr{v}, yielding a new polynomial
23349 @expr{q}. If several variables occur in the inputs, the inputs are
23350 considered multivariate polynomials. (Calc divides by the variable
23351 with the largest power in @expr{u} first, or, in the case of equal
23352 powers, chooses the variables in alphabetical order.) For example,
23353 dividing @expr{x^2 + 3 x + 2} by @expr{x + 2} yields @expr{x + 1}.
23354 The remainder from the division, if any, is reported at the bottom
23355 of the screen and is also placed in the Trail along with the quotient.
23357 Using @code{pdiv} in algebraic notation, you can specify the particular
23358 variable to be used as the base: @code{pdiv(@var{a},@var{b},@var{x})}.
23359 If @code{pdiv} is given only two arguments (as is always the case with
23360 the @kbd{a \} command), then it does a multivariate division as outlined
23364 @pindex calc-poly-rem
23366 The @kbd{a %} (@code{calc-poly-rem}) [@code{prem}] command divides
23367 two polynomials and keeps the remainder @expr{r}. The quotient
23368 @expr{q} is discarded. For any formulas @expr{a} and @expr{b}, the
23369 results of @kbd{a \} and @kbd{a %} satisfy @expr{a = q b + r}.
23370 (This is analogous to plain @kbd{\} and @kbd{%}, which compute the
23371 integer quotient and remainder from dividing two numbers.)
23375 @pindex calc-poly-div-rem
23378 The @kbd{a /} (@code{calc-poly-div-rem}) [@code{pdivrem}] command
23379 divides two polynomials and reports both the quotient and the
23380 remainder as a vector @expr{[q, r]}. The @kbd{H a /} [@code{pdivide}]
23381 command divides two polynomials and constructs the formula
23382 @expr{q + r/b} on the stack. (Naturally if the remainder is zero,
23383 this will immediately simplify to @expr{q}.)
23386 @pindex calc-poly-gcd
23388 The @kbd{a g} (@code{calc-poly-gcd}) [@code{pgcd}] command computes
23389 the greatest common divisor of two polynomials. (The GCD actually
23390 is unique only to within a constant multiplier; Calc attempts to
23391 choose a GCD which will be unsurprising.) For example, the @kbd{a n}
23392 command uses @kbd{a g} to take the GCD of the numerator and denominator
23393 of a quotient, then divides each by the result using @kbd{a \}. (The
23394 definition of GCD ensures that this division can take place without
23395 leaving a remainder.)
23397 While the polynomials used in operations like @kbd{a /} and @kbd{a g}
23398 often have integer coefficients, this is not required. Calc can also
23399 deal with polynomials over the rationals or floating-point reals.
23400 Polynomials with modulo-form coefficients are also useful in many
23401 applications; if you enter @samp{(x^2 + 3 x - 1) mod 5}, Calc
23402 automatically transforms this into a polynomial over the field of
23403 integers mod 5: @samp{(1 mod 5) x^2 + (3 mod 5) x + (4 mod 5)}.
23405 Congratulations and thanks go to Ove Ewerlid
23406 (@code{ewerlid@@mizar.DoCS.UU.SE}), who contributed many of the
23407 polynomial routines used in the above commands.
23409 @xref{Decomposing Polynomials}, for several useful functions for
23410 extracting the individual coefficients of a polynomial.
23412 @node Calculus, Solving Equations, Polynomials, Algebra
23416 The following calculus commands do not automatically simplify their
23417 inputs or outputs using @code{calc-simplify}. You may find it helps
23418 to do this by hand by typing @kbd{a s} or @kbd{a e}. It may also help
23419 to use @kbd{a x} and/or @kbd{a c} to arrange a result in the most
23423 * Differentiation::
23425 * Customizing the Integrator::
23426 * Numerical Integration::
23430 @node Differentiation, Integration, Calculus, Calculus
23431 @subsection Differentiation
23436 @pindex calc-derivative
23439 The @kbd{a d} (@code{calc-derivative}) [@code{deriv}] command computes
23440 the derivative of the expression on the top of the stack with respect to
23441 some variable, which it will prompt you to enter. Normally, variables
23442 in the formula other than the specified differentiation variable are
23443 considered constant, i.e., @samp{deriv(y,x)} is reduced to zero. With
23444 the Hyperbolic flag, the @code{tderiv} (total derivative) operation is used
23445 instead, in which derivatives of variables are not reduced to zero
23446 unless those variables are known to be ``constant,'' i.e., independent
23447 of any other variables. (The built-in special variables like @code{pi}
23448 are considered constant, as are variables that have been declared
23449 @code{const}; @pxref{Declarations}.)
23451 With a numeric prefix argument @var{n}, this command computes the
23452 @var{n}th derivative.
23454 When working with trigonometric functions, it is best to switch to
23455 Radians mode first (with @w{@kbd{m r}}). The derivative of @samp{sin(x)}
23456 in degrees is @samp{(pi/180) cos(x)}, probably not the expected
23459 If you use the @code{deriv} function directly in an algebraic formula,
23460 you can write @samp{deriv(f,x,x0)} which represents the derivative
23461 of @expr{f} with respect to @expr{x}, evaluated at the point
23462 @texline @math{x=x_0}.
23463 @infoline @expr{x=x0}.
23465 If the formula being differentiated contains functions which Calc does
23466 not know, the derivatives of those functions are produced by adding
23467 primes (apostrophe characters). For example, @samp{deriv(f(2x), x)}
23468 produces @samp{2 f'(2 x)}, where the function @code{f'} represents the
23469 derivative of @code{f}.
23471 For functions you have defined with the @kbd{Z F} command, Calc expands
23472 the functions according to their defining formulas unless you have
23473 also defined @code{f'} suitably. For example, suppose we define
23474 @samp{sinc(x) = sin(x)/x} using @kbd{Z F}. If we then differentiate
23475 the formula @samp{sinc(2 x)}, the formula will be expanded to
23476 @samp{sin(2 x) / (2 x)} and differentiated. However, if we also
23477 define @samp{sinc'(x) = dsinc(x)}, say, then Calc will write the
23478 result as @samp{2 dsinc(2 x)}. @xref{Algebraic Definitions}.
23480 For multi-argument functions @samp{f(x,y,z)}, the derivative with respect
23481 to the first argument is written @samp{f'(x,y,z)}; derivatives with
23482 respect to the other arguments are @samp{f'2(x,y,z)} and @samp{f'3(x,y,z)}.
23483 Various higher-order derivatives can be formed in the obvious way, e.g.,
23484 @samp{f'@var{}'(x)} (the second derivative of @code{f}) or
23485 @samp{f'@var{}'2'3(x,y,z)} (@code{f} differentiated with respect to each
23488 @node Integration, Customizing the Integrator, Differentiation, Calculus
23489 @subsection Integration
23493 @pindex calc-integral
23495 The @kbd{a i} (@code{calc-integral}) [@code{integ}] command computes the
23496 indefinite integral of the expression on the top of the stack with
23497 respect to a variable. The integrator is not guaranteed to work for
23498 all integrable functions, but it is able to integrate several large
23499 classes of formulas. In particular, any polynomial or rational function
23500 (a polynomial divided by a polynomial) is acceptable. (Rational functions
23501 don't have to be in explicit quotient form, however;
23502 @texline @math{x/(1+x^{-2})}
23503 @infoline @expr{x/(1+x^-2)}
23504 is not strictly a quotient of polynomials, but it is equivalent to
23505 @expr{x^3/(x^2+1)}, which is.) Also, square roots of terms involving
23506 @expr{x} and @expr{x^2} may appear in rational functions being
23507 integrated. Finally, rational functions involving trigonometric or
23508 hyperbolic functions can be integrated.
23511 If you use the @code{integ} function directly in an algebraic formula,
23512 you can also write @samp{integ(f,x,v)} which expresses the resulting
23513 indefinite integral in terms of variable @code{v} instead of @code{x}.
23514 With four arguments, @samp{integ(f(x),x,a,b)} represents a definite
23515 integral from @code{a} to @code{b}.
23518 If you use the @code{integ} function directly in an algebraic formula,
23519 you can also write @samp{integ(f,x,v)} which expresses the resulting
23520 indefinite integral in terms of variable @code{v} instead of @code{x}.
23521 With four arguments, @samp{integ(f(x),x,a,b)} represents a definite
23522 integral $\int_a^b f(x) \, dx$.
23525 Please note that the current implementation of Calc's integrator sometimes
23526 produces results that are significantly more complex than they need to
23527 be. For example, the integral Calc finds for
23528 @texline @math{1/(x+\sqrt{x^2+1})}
23529 @infoline @expr{1/(x+sqrt(x^2+1))}
23530 is several times more complicated than the answer Mathematica
23531 returns for the same input, although the two forms are numerically
23532 equivalent. Also, any indefinite integral should be considered to have
23533 an arbitrary constant of integration added to it, although Calc does not
23534 write an explicit constant of integration in its result. For example,
23535 Calc's solution for
23536 @texline @math{1/(1+\tan x)}
23537 @infoline @expr{1/(1+tan(x))}
23538 differs from the solution given in the @emph{CRC Math Tables} by a
23540 @texline @math{\pi i / 2}
23541 @infoline @expr{pi i / 2},
23542 due to a different choice of constant of integration.
23544 The Calculator remembers all the integrals it has done. If conditions
23545 change in a way that would invalidate the old integrals, say, a switch
23546 from Degrees to Radians mode, then they will be thrown out. If you
23547 suspect this is not happening when it should, use the
23548 @code{calc-flush-caches} command; @pxref{Caches}.
23551 Calc normally will pursue integration by substitution or integration by
23552 parts up to 3 nested times before abandoning an approach as fruitless.
23553 If the integrator is taking too long, you can lower this limit by storing
23554 a number (like 2) in the variable @code{IntegLimit}. (The @kbd{s I}
23555 command is a convenient way to edit @code{IntegLimit}.) If this variable
23556 has no stored value or does not contain a nonnegative integer, a limit
23557 of 3 is used. The lower this limit is, the greater the chance that Calc
23558 will be unable to integrate a function it could otherwise handle. Raising
23559 this limit allows the Calculator to solve more integrals, though the time
23560 it takes may grow exponentially. You can monitor the integrator's actions
23561 by creating an Emacs buffer called @code{*Trace*}. If such a buffer
23562 exists, the @kbd{a i} command will write a log of its actions there.
23564 If you want to manipulate integrals in a purely symbolic way, you can
23565 set the integration nesting limit to 0 to prevent all but fast
23566 table-lookup solutions of integrals. You might then wish to define
23567 rewrite rules for integration by parts, various kinds of substitutions,
23568 and so on. @xref{Rewrite Rules}.
23570 @node Customizing the Integrator, Numerical Integration, Integration, Calculus
23571 @subsection Customizing the Integrator
23575 Calc has two built-in rewrite rules called @code{IntegRules} and
23576 @code{IntegAfterRules} which you can edit to define new integration
23577 methods. @xref{Rewrite Rules}. At each step of the integration process,
23578 Calc wraps the current integrand in a call to the fictitious function
23579 @samp{integtry(@var{expr},@var{var})}, where @var{expr} is the
23580 integrand and @var{var} is the integration variable. If your rules
23581 rewrite this to be a plain formula (not a call to @code{integtry}), then
23582 Calc will use this formula as the integral of @var{expr}. For example,
23583 the rule @samp{integtry(mysin(x),x) := -mycos(x)} would define a rule to
23584 integrate a function @code{mysin} that acts like the sine function.
23585 Then, putting @samp{4 mysin(2y+1)} on the stack and typing @kbd{a i y}
23586 will produce the integral @samp{-2 mycos(2y+1)}. Note that Calc has
23587 automatically made various transformations on the integral to allow it
23588 to use your rule; integral tables generally give rules for
23589 @samp{mysin(a x + b)}, but you don't need to use this much generality
23590 in your @code{IntegRules}.
23592 @cindex Exponential integral Ei(x)
23597 As a more serious example, the expression @samp{exp(x)/x} cannot be
23598 integrated in terms of the standard functions, so the ``exponential
23599 integral'' function
23600 @texline @math{{\rm Ei}(x)}
23601 @infoline @expr{Ei(x)}
23602 was invented to describe it.
23603 We can get Calc to do this integral in terms of a made-up @code{Ei}
23604 function by adding the rule @samp{[integtry(exp(x)/x, x) := Ei(x)]}
23605 to @code{IntegRules}. Now entering @samp{exp(2x)/x} on the stack
23606 and typing @kbd{a i x} yields @samp{Ei(2 x)}. This new rule will
23607 work with Calc's various built-in integration methods (such as
23608 integration by substitution) to solve a variety of other problems
23609 involving @code{Ei}: For example, now Calc will also be able to
23610 integrate @samp{exp(exp(x))} and @samp{ln(ln(x))} (to get @samp{Ei(exp(x))}
23611 and @samp{x ln(ln(x)) - Ei(ln(x))}, respectively).
23613 Your rule may do further integration by calling @code{integ}. For
23614 example, @samp{integtry(twice(u),x) := twice(integ(u))} allows Calc
23615 to integrate @samp{twice(sin(x))} to get @samp{twice(-cos(x))}.
23616 Note that @code{integ} was called with only one argument. This notation
23617 is allowed only within @code{IntegRules}; it means ``integrate this
23618 with respect to the same integration variable.'' If Calc is unable
23619 to integrate @code{u}, the integration that invoked @code{IntegRules}
23620 also fails. Thus integrating @samp{twice(f(x))} fails, returning the
23621 unevaluated integral @samp{integ(twice(f(x)), x)}. It is still valid
23622 to call @code{integ} with two or more arguments, however; in this case,
23623 if @code{u} is not integrable, @code{twice} itself will still be
23624 integrated: If the above rule is changed to @samp{... := twice(integ(u,x))},
23625 then integrating @samp{twice(f(x))} will yield @samp{twice(integ(f(x),x))}.
23627 If a rule instead produces the formula @samp{integsubst(@var{sexpr},
23628 @var{svar})}, either replacing the top-level @code{integtry} call or
23629 nested anywhere inside the expression, then Calc will apply the
23630 substitution @samp{@var{u} = @var{sexpr}(@var{svar})} to try to
23631 integrate the original @var{expr}. For example, the rule
23632 @samp{sqrt(a) := integsubst(sqrt(x),x)} says that if Calc ever finds
23633 a square root in the integrand, it should attempt the substitution
23634 @samp{u = sqrt(x)}. (This particular rule is unnecessary because
23635 Calc always tries ``obvious'' substitutions where @var{sexpr} actually
23636 appears in the integrand.) The variable @var{svar} may be the same
23637 as the @var{var} that appeared in the call to @code{integtry}, but
23640 When integrating according to an @code{integsubst}, Calc uses the
23641 equation solver to find the inverse of @var{sexpr} (if the integrand
23642 refers to @var{var} anywhere except in subexpressions that exactly
23643 match @var{sexpr}). It uses the differentiator to find the derivative
23644 of @var{sexpr} and/or its inverse (it has two methods that use one
23645 derivative or the other). You can also specify these items by adding
23646 extra arguments to the @code{integsubst} your rules construct; the
23647 general form is @samp{integsubst(@var{sexpr}, @var{svar}, @var{sinv},
23648 @var{sprime})}, where @var{sinv} is the inverse of @var{sexpr} (still
23649 written as a function of @var{svar}), and @var{sprime} is the
23650 derivative of @var{sexpr} with respect to @var{svar}. If you don't
23651 specify these things, and Calc is not able to work them out on its
23652 own with the information it knows, then your substitution rule will
23653 work only in very specific, simple cases.
23655 Calc applies @code{IntegRules} as if by @kbd{C-u 1 a r IntegRules};
23656 in other words, Calc stops rewriting as soon as any rule in your rule
23657 set succeeds. (If it weren't for this, the @samp{integsubst(sqrt(x),x)}
23658 example above would keep on adding layers of @code{integsubst} calls
23661 @vindex IntegSimpRules
23662 Another set of rules, stored in @code{IntegSimpRules}, are applied
23663 every time the integrator uses @kbd{a s} to simplify an intermediate
23664 result. For example, putting the rule @samp{twice(x) := 2 x} into
23665 @code{IntegSimpRules} would tell Calc to convert the @code{twice}
23666 function into a form it knows whenever integration is attempted.
23668 One more way to influence the integrator is to define a function with
23669 the @kbd{Z F} command (@pxref{Algebraic Definitions}). Calc's
23670 integrator automatically expands such functions according to their
23671 defining formulas, even if you originally asked for the function to
23672 be left unevaluated for symbolic arguments. (Certain other Calc
23673 systems, such as the differentiator and the equation solver, also
23676 @vindex IntegAfterRules
23677 Sometimes Calc is able to find a solution to your integral, but it
23678 expresses the result in a way that is unnecessarily complicated. If
23679 this happens, you can either use @code{integsubst} as described
23680 above to try to hint at a more direct path to the desired result, or
23681 you can use @code{IntegAfterRules}. This is an extra rule set that
23682 runs after the main integrator returns its result; basically, Calc does
23683 an @kbd{a r IntegAfterRules} on the result before showing it to you.
23684 (It also does an @kbd{a s}, without @code{IntegSimpRules}, after that
23685 to further simplify the result.) For example, Calc's integrator
23686 sometimes produces expressions of the form @samp{ln(1+x) - ln(1-x)};
23687 the default @code{IntegAfterRules} rewrite this into the more readable
23688 form @samp{2 arctanh(x)}. Note that, unlike @code{IntegRules},
23689 @code{IntegSimpRules} and @code{IntegAfterRules} are applied any number
23690 of times until no further changes are possible. Rewriting by
23691 @code{IntegAfterRules} occurs only after the main integrator has
23692 finished, not at every step as for @code{IntegRules} and
23693 @code{IntegSimpRules}.
23695 @node Numerical Integration, Taylor Series, Customizing the Integrator, Calculus
23696 @subsection Numerical Integration
23700 @pindex calc-num-integral
23702 If you want a purely numerical answer to an integration problem, you can
23703 use the @kbd{a I} (@code{calc-num-integral}) [@code{ninteg}] command. This
23704 command prompts for an integration variable, a lower limit, and an
23705 upper limit. Except for the integration variable, all other variables
23706 that appear in the integrand formula must have stored values. (A stored
23707 value, if any, for the integration variable itself is ignored.)
23709 Numerical integration works by evaluating your formula at many points in
23710 the specified interval. Calc uses an ``open Romberg'' method; this means
23711 that it does not evaluate the formula actually at the endpoints (so that
23712 it is safe to integrate @samp{sin(x)/x} from zero, for example). Also,
23713 the Romberg method works especially well when the function being
23714 integrated is fairly smooth. If the function is not smooth, Calc will
23715 have to evaluate it at quite a few points before it can accurately
23716 determine the value of the integral.
23718 Integration is much faster when the current precision is small. It is
23719 best to set the precision to the smallest acceptable number of digits
23720 before you use @kbd{a I}. If Calc appears to be taking too long, press
23721 @kbd{C-g} to halt it and try a lower precision. If Calc still appears
23722 to need hundreds of evaluations, check to make sure your function is
23723 well-behaved in the specified interval.
23725 It is possible for the lower integration limit to be @samp{-inf} (minus
23726 infinity). Likewise, the upper limit may be plus infinity. Calc
23727 internally transforms the integral into an equivalent one with finite
23728 limits. However, integration to or across singularities is not supported:
23729 The integral of @samp{1/sqrt(x)} from 0 to 1 exists (it can be found
23730 by Calc's symbolic integrator, for example), but @kbd{a I} will fail
23731 because the integrand goes to infinity at one of the endpoints.
23733 @node Taylor Series, , Numerical Integration, Calculus
23734 @subsection Taylor Series
23738 @pindex calc-taylor
23740 The @kbd{a t} (@code{calc-taylor}) [@code{taylor}] command computes a
23741 power series expansion or Taylor series of a function. You specify the
23742 variable and the desired number of terms. You may give an expression of
23743 the form @samp{@var{var} = @var{a}} or @samp{@var{var} - @var{a}} instead
23744 of just a variable to produce a Taylor expansion about the point @var{a}.
23745 You may specify the number of terms with a numeric prefix argument;
23746 otherwise the command will prompt you for the number of terms. Note that
23747 many series expansions have coefficients of zero for some terms, so you
23748 may appear to get fewer terms than you asked for.
23750 If the @kbd{a i} command is unable to find a symbolic integral for a
23751 function, you can get an approximation by integrating the function's
23754 @node Solving Equations, Numerical Solutions, Calculus, Algebra
23755 @section Solving Equations
23759 @pindex calc-solve-for
23761 @cindex Equations, solving
23762 @cindex Solving equations
23763 The @kbd{a S} (@code{calc-solve-for}) [@code{solve}] command rearranges
23764 an equation to solve for a specific variable. An equation is an
23765 expression of the form @expr{L = R}. For example, the command @kbd{a S x}
23766 will rearrange @expr{y = 3x + 6} to the form, @expr{x = y/3 - 2}. If the
23767 input is not an equation, it is treated like an equation of the
23770 This command also works for inequalities, as in @expr{y < 3x + 6}.
23771 Some inequalities cannot be solved where the analogous equation could
23772 be; for example, solving
23773 @texline @math{a < b \, c}
23774 @infoline @expr{a < b c}
23775 for @expr{b} is impossible
23776 without knowing the sign of @expr{c}. In this case, @kbd{a S} will
23778 @texline @math{b \mathbin{\hbox{\code{!=}}} a/c}
23779 @infoline @expr{b != a/c}
23780 (using the not-equal-to operator) to signify that the direction of the
23781 inequality is now unknown. The inequality
23782 @texline @math{a \le b \, c}
23783 @infoline @expr{a <= b c}
23784 is not even partially solved. @xref{Declarations}, for a way to tell
23785 Calc that the signs of the variables in a formula are in fact known.
23787 Two useful commands for working with the result of @kbd{a S} are
23788 @kbd{a .} (@pxref{Logical Operations}), which converts @expr{x = y/3 - 2}
23789 to @expr{y/3 - 2}, and @kbd{s l} (@pxref{Let Command}) which evaluates
23790 another formula with @expr{x} set equal to @expr{y/3 - 2}.
23793 * Multiple Solutions::
23794 * Solving Systems of Equations::
23795 * Decomposing Polynomials::
23798 @node Multiple Solutions, Solving Systems of Equations, Solving Equations, Solving Equations
23799 @subsection Multiple Solutions
23804 Some equations have more than one solution. The Hyperbolic flag
23805 (@code{H a S}) [@code{fsolve}] tells the solver to report the fully
23806 general family of solutions. It will invent variables @code{n1},
23807 @code{n2}, @dots{}, which represent independent arbitrary integers, and
23808 @code{s1}, @code{s2}, @dots{}, which represent independent arbitrary
23809 signs (either @mathit{+1} or @mathit{-1}). If you don't use the Hyperbolic
23810 flag, Calc will use zero in place of all arbitrary integers, and plus
23811 one in place of all arbitrary signs. Note that variables like @code{n1}
23812 and @code{s1} are not given any special interpretation in Calc except by
23813 the equation solver itself. As usual, you can use the @w{@kbd{s l}}
23814 (@code{calc-let}) command to obtain solutions for various actual values
23815 of these variables.
23817 For example, @kbd{' x^2 = y @key{RET} H a S x @key{RET}} solves to
23818 get @samp{x = s1 sqrt(y)}, indicating that the two solutions to the
23819 equation are @samp{sqrt(y)} and @samp{-sqrt(y)}. Another way to
23820 think about it is that the square-root operation is really a
23821 two-valued function; since every Calc function must return a
23822 single result, @code{sqrt} chooses to return the positive result.
23823 Then @kbd{H a S} doctors this result using @code{s1} to indicate
23824 the full set of possible values of the mathematical square-root.
23826 There is a similar phenomenon going the other direction: Suppose
23827 we solve @samp{sqrt(y) = x} for @code{y}. Calc squares both sides
23828 to get @samp{y = x^2}. This is correct, except that it introduces
23829 some dubious solutions. Consider solving @samp{sqrt(y) = -3}:
23830 Calc will report @expr{y = 9} as a valid solution, which is true
23831 in the mathematical sense of square-root, but false (there is no
23832 solution) for the actual Calc positive-valued @code{sqrt}. This
23833 happens for both @kbd{a S} and @kbd{H a S}.
23835 @cindex @code{GenCount} variable
23845 If you store a positive integer in the Calc variable @code{GenCount},
23846 then Calc will generate formulas of the form @samp{as(@var{n})} for
23847 arbitrary signs, and @samp{an(@var{n})} for arbitrary integers,
23848 where @var{n} represents successive values taken by incrementing
23849 @code{GenCount} by one. While the normal arbitrary sign and
23850 integer symbols start over at @code{s1} and @code{n1} with each
23851 new Calc command, the @code{GenCount} approach will give each
23852 arbitrary value a name that is unique throughout the entire Calc
23853 session. Also, the arbitrary values are function calls instead
23854 of variables, which is advantageous in some cases. For example,
23855 you can make a rewrite rule that recognizes all arbitrary signs
23856 using a pattern like @samp{as(n)}. The @kbd{s l} command only works
23857 on variables, but you can use the @kbd{a b} (@code{calc-substitute})
23858 command to substitute actual values for function calls like @samp{as(3)}.
23860 The @kbd{s G} (@code{calc-edit-GenCount}) command is a convenient
23861 way to create or edit this variable. Press @kbd{C-c C-c} to finish.
23863 If you have not stored a value in @code{GenCount}, or if the value
23864 in that variable is not a positive integer, the regular
23865 @code{s1}/@code{n1} notation is used.
23871 With the Inverse flag, @kbd{I a S} [@code{finv}] treats the expression
23872 on top of the stack as a function of the specified variable and solves
23873 to find the inverse function, written in terms of the same variable.
23874 For example, @kbd{I a S x} inverts @expr{2x + 6} to @expr{x/2 - 3}.
23875 You can use both Inverse and Hyperbolic [@code{ffinv}] to obtain a
23876 fully general inverse, as described above.
23879 @pindex calc-poly-roots
23881 Some equations, specifically polynomials, have a known, finite number
23882 of solutions. The @kbd{a P} (@code{calc-poly-roots}) [@code{roots}]
23883 command uses @kbd{H a S} to solve an equation in general form, then, for
23884 all arbitrary-sign variables like @code{s1}, and all arbitrary-integer
23885 variables like @code{n1} for which @code{n1} only usefully varies over
23886 a finite range, it expands these variables out to all their possible
23887 values. The results are collected into a vector, which is returned.
23888 For example, @samp{roots(x^4 = 1, x)} returns the four solutions
23889 @samp{[1, -1, (0, 1), (0, -1)]}. Generally an @var{n}th degree
23890 polynomial will always have @var{n} roots on the complex plane.
23891 (If you have given a @code{real} declaration for the solution
23892 variable, then only the real-valued solutions, if any, will be
23893 reported; @pxref{Declarations}.)
23895 Note that because @kbd{a P} uses @kbd{H a S}, it is able to deliver
23896 symbolic solutions if the polynomial has symbolic coefficients. Also
23897 note that Calc's solver is not able to get exact symbolic solutions
23898 to all polynomials. Polynomials containing powers up to @expr{x^4}
23899 can always be solved exactly; polynomials of higher degree sometimes
23900 can be: @expr{x^6 + x^3 + 1} is converted to @expr{(x^3)^2 + (x^3) + 1},
23901 which can be solved for @expr{x^3} using the quadratic equation, and then
23902 for @expr{x} by taking cube roots. But in many cases, like
23903 @expr{x^6 + x + 1}, Calc does not know how to rewrite the polynomial
23904 into a form it can solve. The @kbd{a P} command can still deliver a
23905 list of numerical roots, however, provided that Symbolic mode (@kbd{m s})
23906 is not turned on. (If you work with Symbolic mode on, recall that the
23907 @kbd{N} (@code{calc-eval-num}) key is a handy way to reevaluate the
23908 formula on the stack with Symbolic mode temporarily off.) Naturally,
23909 @kbd{a P} can only provide numerical roots if the polynomial coefficients
23910 are all numbers (real or complex).
23912 @node Solving Systems of Equations, Decomposing Polynomials, Multiple Solutions, Solving Equations
23913 @subsection Solving Systems of Equations
23916 @cindex Systems of equations, symbolic
23917 You can also use the commands described above to solve systems of
23918 simultaneous equations. Just create a vector of equations, then
23919 specify a vector of variables for which to solve. (You can omit
23920 the surrounding brackets when entering the vector of variables
23923 For example, putting @samp{[x + y = a, x - y = b]} on the stack
23924 and typing @kbd{a S x,y @key{RET}} produces the vector of solutions
23925 @samp{[x = a - (a-b)/2, y = (a-b)/2]}. The result vector will
23926 have the same length as the variables vector, and the variables
23927 will be listed in the same order there. Note that the solutions
23928 are not always simplified as far as possible; the solution for
23929 @expr{x} here could be improved by an application of the @kbd{a n}
23932 Calc's algorithm works by trying to eliminate one variable at a
23933 time by solving one of the equations for that variable and then
23934 substituting into the other equations. Calc will try all the
23935 possibilities, but you can speed things up by noting that Calc
23936 first tries to eliminate the first variable with the first
23937 equation, then the second variable with the second equation,
23938 and so on. It also helps to put the simpler (e.g., more linear)
23939 equations toward the front of the list. Calc's algorithm will
23940 solve any system of linear equations, and also many kinds of
23947 Normally there will be as many variables as equations. If you
23948 give fewer variables than equations (an ``over-determined'' system
23949 of equations), Calc will find a partial solution. For example,
23950 typing @kbd{a S y @key{RET}} with the above system of equations
23951 would produce @samp{[y = a - x]}. There are now several ways to
23952 express this solution in terms of the original variables; Calc uses
23953 the first one that it finds. You can control the choice by adding
23954 variable specifiers of the form @samp{elim(@var{v})} to the
23955 variables list. This says that @var{v} should be eliminated from
23956 the equations; the variable will not appear at all in the solution.
23957 For example, typing @kbd{a S y,elim(x)} would yield
23958 @samp{[y = a - (b+a)/2]}.
23960 If the variables list contains only @code{elim} specifiers,
23961 Calc simply eliminates those variables from the equations
23962 and then returns the resulting set of equations. For example,
23963 @kbd{a S elim(x)} produces @samp{[a - 2 y = b]}. Every variable
23964 eliminated will reduce the number of equations in the system
23967 Again, @kbd{a S} gives you one solution to the system of
23968 equations. If there are several solutions, you can use @kbd{H a S}
23969 to get a general family of solutions, or, if there is a finite
23970 number of solutions, you can use @kbd{a P} to get a list. (In
23971 the latter case, the result will take the form of a matrix where
23972 the rows are different solutions and the columns correspond to the
23973 variables you requested.)
23975 Another way to deal with certain kinds of overdetermined systems of
23976 equations is the @kbd{a F} command, which does least-squares fitting
23977 to satisfy the equations. @xref{Curve Fitting}.
23979 @node Decomposing Polynomials, , Solving Systems of Equations, Solving Equations
23980 @subsection Decomposing Polynomials
23987 The @code{poly} function takes a polynomial and a variable as
23988 arguments, and returns a vector of polynomial coefficients (constant
23989 coefficient first). For example, @samp{poly(x^3 + 2 x, x)} returns
23990 @expr{[0, 2, 0, 1]}. If the input is not a polynomial in @expr{x},
23991 the call to @code{poly} is left in symbolic form. If the input does
23992 not involve the variable @expr{x}, the input is returned in a list
23993 of length one, representing a polynomial with only a constant
23994 coefficient. The call @samp{poly(x, x)} returns the vector @expr{[0, 1]}.
23995 The last element of the returned vector is guaranteed to be nonzero;
23996 note that @samp{poly(0, x)} returns the empty vector @expr{[]}.
23997 Note also that @expr{x} may actually be any formula; for example,
23998 @samp{poly(sin(x)^2 - sin(x) + 3, sin(x))} returns @expr{[3, -1, 1]}.
24000 @cindex Coefficients of polynomial
24001 @cindex Degree of polynomial
24002 To get the @expr{x^k} coefficient of polynomial @expr{p}, use
24003 @samp{poly(p, x)_(k+1)}. To get the degree of polynomial @expr{p},
24004 use @samp{vlen(poly(p, x)) - 1}. For example, @samp{poly((x+1)^4, x)}
24005 returns @samp{[1, 4, 6, 4, 1]}, so @samp{poly((x+1)^4, x)_(2+1)}
24006 gives the @expr{x^2} coefficient of this polynomial, 6.
24012 One important feature of the solver is its ability to recognize
24013 formulas which are ``essentially'' polynomials. This ability is
24014 made available to the user through the @code{gpoly} function, which
24015 is used just like @code{poly}: @samp{gpoly(@var{expr}, @var{var})}.
24016 If @var{expr} is a polynomial in some term which includes @var{var}, then
24017 this function will return a vector @samp{[@var{x}, @var{c}, @var{a}]}
24018 where @var{x} is the term that depends on @var{var}, @var{c} is a
24019 vector of polynomial coefficients (like the one returned by @code{poly}),
24020 and @var{a} is a multiplier which is usually 1. Basically,
24021 @samp{@var{expr} = @var{a}*(@var{c}_1 + @var{c}_2 @var{x} +
24022 @var{c}_3 @var{x}^2 + ...)}. The last element of @var{c} is
24023 guaranteed to be non-zero, and @var{c} will not equal @samp{[1]}
24024 (i.e., the trivial decomposition @var{expr} = @var{x} is not
24025 considered a polynomial). One side effect is that @samp{gpoly(x, x)}
24026 and @samp{gpoly(6, x)}, both of which might be expected to recognize
24027 their arguments as polynomials, will not because the decomposition
24028 is considered trivial.
24030 For example, @samp{gpoly((x-2)^2, x)} returns @samp{[x, [4, -4, 1], 1]},
24031 since the expanded form of this polynomial is @expr{4 - 4 x + x^2}.
24033 The term @var{x} may itself be a polynomial in @var{var}. This is
24034 done to reduce the size of the @var{c} vector. For example,
24035 @samp{gpoly(x^4 + x^2 - 1, x)} returns @samp{[x^2, [-1, 1, 1], 1]},
24036 since a quadratic polynomial in @expr{x^2} is easier to solve than
24037 a quartic polynomial in @expr{x}.
24039 A few more examples of the kinds of polynomials @code{gpoly} can
24043 sin(x) - 1 [sin(x), [-1, 1], 1]
24044 x + 1/x - 1 [x, [1, -1, 1], 1/x]
24045 x + 1/x [x^2, [1, 1], 1/x]
24046 x^3 + 2 x [x^2, [2, 1], x]
24047 x + x^2:3 + sqrt(x) [x^1:6, [1, 1, 0, 1], x^1:2]
24048 x^(2a) + 2 x^a + 5 [x^a, [5, 2, 1], 1]
24049 (exp(-x) + exp(x)) / 2 [e^(2 x), [0.5, 0.5], e^-x]
24052 The @code{poly} and @code{gpoly} functions accept a third integer argument
24053 which specifies the largest degree of polynomial that is acceptable.
24054 If this is @expr{n}, then only @var{c} vectors of length @expr{n+1}
24055 or less will be returned. Otherwise, the @code{poly} or @code{gpoly}
24056 call will remain in symbolic form. For example, the equation solver
24057 can handle quartics and smaller polynomials, so it calls
24058 @samp{gpoly(@var{expr}, @var{var}, 4)} to discover whether @var{expr}
24059 can be treated by its linear, quadratic, cubic, or quartic formulas.
24065 The @code{pdeg} function computes the degree of a polynomial;
24066 @samp{pdeg(p,x)} is the highest power of @code{x} that appears in
24067 @code{p}. This is the same as @samp{vlen(poly(p,x))-1}, but is
24068 much more efficient. If @code{p} is constant with respect to @code{x},
24069 then @samp{pdeg(p,x) = 0}. If @code{p} is not a polynomial in @code{x}
24070 (e.g., @samp{pdeg(2 cos(x), x)}, the function remains unevaluated.
24071 It is possible to omit the second argument @code{x}, in which case
24072 @samp{pdeg(p)} returns the highest total degree of any term of the
24073 polynomial, counting all variables that appear in @code{p}. Note
24074 that @code{pdeg(c) = pdeg(c,x) = 0} for any nonzero constant @code{c};
24075 the degree of the constant zero is considered to be @code{-inf}
24082 The @code{plead} function finds the leading term of a polynomial.
24083 Thus @samp{plead(p,x)} is equivalent to @samp{poly(p,x)_vlen(poly(p,x))},
24084 though again more efficient. In particular, @samp{plead((2x+1)^10, x)}
24085 returns 1024 without expanding out the list of coefficients. The
24086 value of @code{plead(p,x)} will be zero only if @expr{p = 0}.
24092 The @code{pcont} function finds the @dfn{content} of a polynomial. This
24093 is the greatest common divisor of all the coefficients of the polynomial.
24094 With two arguments, @code{pcont(p,x)} effectively uses @samp{poly(p,x)}
24095 to get a list of coefficients, then uses @code{pgcd} (the polynomial
24096 GCD function) to combine these into an answer. For example,
24097 @samp{pcont(4 x y^2 + 6 x^2 y, x)} is @samp{2 y}. The content is
24098 basically the ``biggest'' polynomial that can be divided into @code{p}
24099 exactly. The sign of the content is the same as the sign of the leading
24102 With only one argument, @samp{pcont(p)} computes the numerical
24103 content of the polynomial, i.e., the @code{gcd} of the numerical
24104 coefficients of all the terms in the formula. Note that @code{gcd}
24105 is defined on rational numbers as well as integers; it computes
24106 the @code{gcd} of the numerators and the @code{lcm} of the
24107 denominators. Thus @samp{pcont(4:3 x y^2 + 6 x^2 y)} returns 2:3.
24108 Dividing the polynomial by this number will clear all the
24109 denominators, as well as dividing by any common content in the
24110 numerators. The numerical content of a polynomial is negative only
24111 if all the coefficients in the polynomial are negative.
24117 The @code{pprim} function finds the @dfn{primitive part} of a
24118 polynomial, which is simply the polynomial divided (using @code{pdiv}
24119 if necessary) by its content. If the input polynomial has rational
24120 coefficients, the result will have integer coefficients in simplest
24123 @node Numerical Solutions, Curve Fitting, Solving Equations, Algebra
24124 @section Numerical Solutions
24127 Not all equations can be solved symbolically. The commands in this
24128 section use numerical algorithms that can find a solution to a specific
24129 instance of an equation to any desired accuracy. Note that the
24130 numerical commands are slower than their algebraic cousins; it is a
24131 good idea to try @kbd{a S} before resorting to these commands.
24133 (@xref{Curve Fitting}, for some other, more specialized, operations
24134 on numerical data.)
24139 * Numerical Systems of Equations::
24142 @node Root Finding, Minimization, Numerical Solutions, Numerical Solutions
24143 @subsection Root Finding
24147 @pindex calc-find-root
24149 @cindex Newton's method
24150 @cindex Roots of equations
24151 @cindex Numerical root-finding
24152 The @kbd{a R} (@code{calc-find-root}) [@code{root}] command finds a
24153 numerical solution (or @dfn{root}) of an equation. (This command treats
24154 inequalities the same as equations. If the input is any other kind
24155 of formula, it is interpreted as an equation of the form @expr{X = 0}.)
24157 The @kbd{a R} command requires an initial guess on the top of the
24158 stack, and a formula in the second-to-top position. It prompts for a
24159 solution variable, which must appear in the formula. All other variables
24160 that appear in the formula must have assigned values, i.e., when
24161 a value is assigned to the solution variable and the formula is
24162 evaluated with @kbd{=}, it should evaluate to a number. Any assigned
24163 value for the solution variable itself is ignored and unaffected by
24166 When the command completes, the initial guess is replaced on the stack
24167 by a vector of two numbers: The value of the solution variable that
24168 solves the equation, and the difference between the lefthand and
24169 righthand sides of the equation at that value. Ordinarily, the second
24170 number will be zero or very nearly zero. (Note that Calc uses a
24171 slightly higher precision while finding the root, and thus the second
24172 number may be slightly different from the value you would compute from
24173 the equation yourself.)
24175 The @kbd{v h} (@code{calc-head}) command is a handy way to extract
24176 the first element of the result vector, discarding the error term.
24178 The initial guess can be a real number, in which case Calc searches
24179 for a real solution near that number, or a complex number, in which
24180 case Calc searches the whole complex plane near that number for a
24181 solution, or it can be an interval form which restricts the search
24182 to real numbers inside that interval.
24184 Calc tries to use @kbd{a d} to take the derivative of the equation.
24185 If this succeeds, it uses Newton's method. If the equation is not
24186 differentiable Calc uses a bisection method. (If Newton's method
24187 appears to be going astray, Calc switches over to bisection if it
24188 can, or otherwise gives up. In this case it may help to try again
24189 with a slightly different initial guess.) If the initial guess is a
24190 complex number, the function must be differentiable.
24192 If the formula (or the difference between the sides of an equation)
24193 is negative at one end of the interval you specify and positive at
24194 the other end, the root finder is guaranteed to find a root.
24195 Otherwise, Calc subdivides the interval into small parts looking for
24196 positive and negative values to bracket the root. When your guess is
24197 an interval, Calc will not look outside that interval for a root.
24201 The @kbd{H a R} [@code{wroot}] command is similar to @kbd{a R}, except
24202 that if the initial guess is an interval for which the function has
24203 the same sign at both ends, then rather than subdividing the interval
24204 Calc attempts to widen it to enclose a root. Use this mode if
24205 you are not sure if the function has a root in your interval.
24207 If the function is not differentiable, and you give a simple number
24208 instead of an interval as your initial guess, Calc uses this widening
24209 process even if you did not type the Hyperbolic flag. (If the function
24210 @emph{is} differentiable, Calc uses Newton's method which does not
24211 require a bounding interval in order to work.)
24213 If Calc leaves the @code{root} or @code{wroot} function in symbolic
24214 form on the stack, it will normally display an explanation for why
24215 no root was found. If you miss this explanation, press @kbd{w}
24216 (@code{calc-why}) to get it back.
24218 @node Minimization, Numerical Systems of Equations, Root Finding, Numerical Solutions
24219 @subsection Minimization
24226 @pindex calc-find-minimum
24227 @pindex calc-find-maximum
24230 @cindex Minimization, numerical
24231 The @kbd{a N} (@code{calc-find-minimum}) [@code{minimize}] command
24232 finds a minimum value for a formula. It is very similar in operation
24233 to @kbd{a R} (@code{calc-find-root}): You give the formula and an initial
24234 guess on the stack, and are prompted for the name of a variable. The guess
24235 may be either a number near the desired minimum, or an interval enclosing
24236 the desired minimum. The function returns a vector containing the
24237 value of the variable which minimizes the formula's value, along
24238 with the minimum value itself.
24240 Note that this command looks for a @emph{local} minimum. Many functions
24241 have more than one minimum; some, like
24242 @texline @math{x \sin x},
24243 @infoline @expr{x sin(x)},
24244 have infinitely many. In fact, there is no easy way to define the
24245 ``global'' minimum of
24246 @texline @math{x \sin x}
24247 @infoline @expr{x sin(x)}
24248 but Calc can still locate any particular local minimum
24249 for you. Calc basically goes downhill from the initial guess until it
24250 finds a point at which the function's value is greater both to the left
24251 and to the right. Calc does not use derivatives when minimizing a function.
24253 If your initial guess is an interval and it looks like the minimum
24254 occurs at one or the other endpoint of the interval, Calc will return
24255 that endpoint only if that endpoint is closed; thus, minimizing @expr{17 x}
24256 over @expr{[2..3]} will return @expr{[2, 38]}, but minimizing over
24257 @expr{(2..3]} would report no minimum found. In general, you should
24258 use closed intervals to find literally the minimum value in that
24259 range of @expr{x}, or open intervals to find the local minimum, if
24260 any, that happens to lie in that range.
24262 Most functions are smooth and flat near their minimum values. Because
24263 of this flatness, if the current precision is, say, 12 digits, the
24264 variable can only be determined meaningfully to about six digits. Thus
24265 you should set the precision to twice as many digits as you need in your
24276 The @kbd{H a N} [@code{wminimize}] command, analogously to @kbd{H a R},
24277 expands the guess interval to enclose a minimum rather than requiring
24278 that the minimum lie inside the interval you supply.
24280 The @kbd{a X} (@code{calc-find-maximum}) [@code{maximize}] and
24281 @kbd{H a X} [@code{wmaximize}] commands effectively minimize the
24282 negative of the formula you supply.
24284 The formula must evaluate to a real number at all points inside the
24285 interval (or near the initial guess if the guess is a number). If
24286 the initial guess is a complex number the variable will be minimized
24287 over the complex numbers; if it is real or an interval it will
24288 be minimized over the reals.
24290 @node Numerical Systems of Equations, , Minimization, Numerical Solutions
24291 @subsection Systems of Equations
24294 @cindex Systems of equations, numerical
24295 The @kbd{a R} command can also solve systems of equations. In this
24296 case, the equation should instead be a vector of equations, the
24297 guess should instead be a vector of numbers (intervals are not
24298 supported), and the variable should be a vector of variables. You
24299 can omit the brackets while entering the list of variables. Each
24300 equation must be differentiable by each variable for this mode to
24301 work. The result will be a vector of two vectors: The variable
24302 values that solved the system of equations, and the differences
24303 between the sides of the equations with those variable values.
24304 There must be the same number of equations as variables. Since
24305 only plain numbers are allowed as guesses, the Hyperbolic flag has
24306 no effect when solving a system of equations.
24308 It is also possible to minimize over many variables with @kbd{a N}
24309 (or maximize with @kbd{a X}). Once again the variable name should
24310 be replaced by a vector of variables, and the initial guess should
24311 be an equal-sized vector of initial guesses. But, unlike the case of
24312 multidimensional @kbd{a R}, the formula being minimized should
24313 still be a single formula, @emph{not} a vector. Beware that
24314 multidimensional minimization is currently @emph{very} slow.
24316 @node Curve Fitting, Summations, Numerical Solutions, Algebra
24317 @section Curve Fitting
24320 The @kbd{a F} command fits a set of data to a @dfn{model formula},
24321 such as @expr{y = m x + b} where @expr{m} and @expr{b} are parameters
24322 to be determined. For a typical set of measured data there will be
24323 no single @expr{m} and @expr{b} that exactly fit the data; in this
24324 case, Calc chooses values of the parameters that provide the closest
24329 * Polynomial and Multilinear Fits::
24330 * Error Estimates for Fits::
24331 * Standard Nonlinear Models::
24332 * Curve Fitting Details::
24336 @node Linear Fits, Polynomial and Multilinear Fits, Curve Fitting, Curve Fitting
24337 @subsection Linear Fits
24341 @pindex calc-curve-fit
24343 @cindex Linear regression
24344 @cindex Least-squares fits
24345 The @kbd{a F} (@code{calc-curve-fit}) [@code{fit}] command attempts
24346 to fit a set of data (@expr{x} and @expr{y} vectors of numbers) to a
24347 straight line, polynomial, or other function of @expr{x}. For the
24348 moment we will consider only the case of fitting to a line, and we
24349 will ignore the issue of whether or not the model was in fact a good
24352 In a standard linear least-squares fit, we have a set of @expr{(x,y)}
24353 data points that we wish to fit to the model @expr{y = m x + b}
24354 by adjusting the parameters @expr{m} and @expr{b} to make the @expr{y}
24355 values calculated from the formula be as close as possible to the actual
24356 @expr{y} values in the data set. (In a polynomial fit, the model is
24357 instead, say, @expr{y = a x^3 + b x^2 + c x + d}. In a multilinear fit,
24358 we have data points of the form @expr{(x_1,x_2,x_3,y)} and our model is
24359 @expr{y = a x_1 + b x_2 + c x_3 + d}. These will be discussed later.)
24361 In the model formula, variables like @expr{x} and @expr{x_2} are called
24362 the @dfn{independent variables}, and @expr{y} is the @dfn{dependent
24363 variable}. Variables like @expr{m}, @expr{a}, and @expr{b} are called
24364 the @dfn{parameters} of the model.
24366 The @kbd{a F} command takes the data set to be fitted from the stack.
24367 By default, it expects the data in the form of a matrix. For example,
24368 for a linear or polynomial fit, this would be a
24369 @texline @math{2\times N}
24371 matrix where the first row is a list of @expr{x} values and the second
24372 row has the corresponding @expr{y} values. For the multilinear fit
24373 shown above, the matrix would have four rows (@expr{x_1}, @expr{x_2},
24374 @expr{x_3}, and @expr{y}, respectively).
24376 If you happen to have an
24377 @texline @math{N\times2}
24379 matrix instead of a
24380 @texline @math{2\times N}
24382 matrix, just press @kbd{v t} first to transpose the matrix.
24384 After you type @kbd{a F}, Calc prompts you to select a model. For a
24385 linear fit, press the digit @kbd{1}.
24387 Calc then prompts for you to name the variables. By default it chooses
24388 high letters like @expr{x} and @expr{y} for independent variables and
24389 low letters like @expr{a} and @expr{b} for parameters. (The dependent
24390 variable doesn't need a name.) The two kinds of variables are separated
24391 by a semicolon. Since you generally care more about the names of the
24392 independent variables than of the parameters, Calc also allows you to
24393 name only those and let the parameters use default names.
24395 For example, suppose the data matrix
24400 [ [ 1, 2, 3, 4, 5 ]
24401 [ 5, 7, 9, 11, 13 ] ]
24409 $$ \pmatrix{ 1 & 2 & 3 & 4 & 5 \cr
24410 5 & 7 & 9 & 11 & 13 }
24416 is on the stack and we wish to do a simple linear fit. Type
24417 @kbd{a F}, then @kbd{1} for the model, then @key{RET} to use
24418 the default names. The result will be the formula @expr{3 + 2 x}
24419 on the stack. Calc has created the model expression @kbd{a + b x},
24420 then found the optimal values of @expr{a} and @expr{b} to fit the
24421 data. (In this case, it was able to find an exact fit.) Calc then
24422 substituted those values for @expr{a} and @expr{b} in the model
24425 The @kbd{a F} command puts two entries in the trail. One is, as
24426 always, a copy of the result that went to the stack; the other is
24427 a vector of the actual parameter values, written as equations:
24428 @expr{[a = 3, b = 2]}, in case you'd rather read them in a list
24429 than pick them out of the formula. (You can type @kbd{t y}
24430 to move this vector to the stack; see @ref{Trail Commands}.
24432 Specifying a different independent variable name will affect the
24433 resulting formula: @kbd{a F 1 k @key{RET}} produces @kbd{3 + 2 k}.
24434 Changing the parameter names (say, @kbd{a F 1 k;b,m @key{RET}}) will affect
24435 the equations that go into the trail.
24441 To see what happens when the fit is not exact, we could change
24442 the number 13 in the data matrix to 14 and try the fit again.
24449 Evaluating this formula, say with @kbd{v x 5 @key{RET} @key{TAB} V M $ @key{RET}}, shows
24450 a reasonably close match to the y-values in the data.
24453 [4.8, 7., 9.2, 11.4, 13.6]
24456 Since there is no line which passes through all the @var{n} data points,
24457 Calc has chosen a line that best approximates the data points using
24458 the method of least squares. The idea is to define the @dfn{chi-square}
24463 chi^2 = sum((y_i - (a + b x_i))^2, i, 1, N)
24469 $$ \chi^2 = \sum_{i=1}^N (y_i - (a + b x_i))^2 $$
24474 which is clearly zero if @expr{a + b x} exactly fits all data points,
24475 and increases as various @expr{a + b x_i} values fail to match the
24476 corresponding @expr{y_i} values. There are several reasons why the
24477 summand is squared, one of them being to ensure that
24478 @texline @math{\chi^2 \ge 0}.
24479 @infoline @expr{chi^2 >= 0}.
24480 Least-squares fitting simply chooses the values of @expr{a} and @expr{b}
24481 for which the error
24482 @texline @math{\chi^2}
24483 @infoline @expr{chi^2}
24484 is as small as possible.
24486 Other kinds of models do the same thing but with a different model
24487 formula in place of @expr{a + b x_i}.
24493 A numeric prefix argument causes the @kbd{a F} command to take the
24494 data in some other form than one big matrix. A positive argument @var{n}
24495 will take @var{N} items from the stack, corresponding to the @var{n} rows
24496 of a data matrix. In the linear case, @var{n} must be 2 since there
24497 is always one independent variable and one dependent variable.
24499 A prefix of zero or plain @kbd{C-u} is a compromise; Calc takes two
24500 items from the stack, an @var{n}-row matrix of @expr{x} values, and a
24501 vector of @expr{y} values. If there is only one independent variable,
24502 the @expr{x} values can be either a one-row matrix or a plain vector,
24503 in which case the @kbd{C-u} prefix is the same as a @w{@kbd{C-u 2}} prefix.
24505 @node Polynomial and Multilinear Fits, Error Estimates for Fits, Linear Fits, Curve Fitting
24506 @subsection Polynomial and Multilinear Fits
24509 To fit the data to higher-order polynomials, just type one of the
24510 digits @kbd{2} through @kbd{9} when prompted for a model. For example,
24511 we could fit the original data matrix from the previous section
24512 (with 13, not 14) to a parabola instead of a line by typing
24513 @kbd{a F 2 @key{RET}}.
24516 2.00000000001 x - 1.5e-12 x^2 + 2.99999999999
24519 Note that since the constant and linear terms are enough to fit the
24520 data exactly, it's no surprise that Calc chose a tiny contribution
24521 for @expr{x^2}. (The fact that it's not exactly zero is due only
24522 to roundoff error. Since our data are exact integers, we could get
24523 an exact answer by typing @kbd{m f} first to get Fraction mode.
24524 Then the @expr{x^2} term would vanish altogether. Usually, though,
24525 the data being fitted will be approximate floats so Fraction mode
24528 Doing the @kbd{a F 2} fit on the data set with 14 instead of 13
24529 gives a much larger @expr{x^2} contribution, as Calc bends the
24530 line slightly to improve the fit.
24533 0.142857142855 x^2 + 1.34285714287 x + 3.59999999998
24536 An important result from the theory of polynomial fitting is that it
24537 is always possible to fit @var{n} data points exactly using a polynomial
24538 of degree @mathit{@var{n}-1}, sometimes called an @dfn{interpolating polynomial}.
24539 Using the modified (14) data matrix, a model number of 4 gives
24540 a polynomial that exactly matches all five data points:
24543 0.04167 x^4 - 0.4167 x^3 + 1.458 x^2 - 0.08333 x + 4.
24546 The actual coefficients we get with a precision of 12, like
24547 @expr{0.0416666663588}, clearly suffer from loss of precision.
24548 It is a good idea to increase the working precision to several
24549 digits beyond what you need when you do a fitting operation.
24550 Or, if your data are exact, use Fraction mode to get exact
24553 You can type @kbd{i} instead of a digit at the model prompt to fit
24554 the data exactly to a polynomial. This just counts the number of
24555 columns of the data matrix to choose the degree of the polynomial
24558 Fitting data ``exactly'' to high-degree polynomials is not always
24559 a good idea, though. High-degree polynomials have a tendency to
24560 wiggle uncontrollably in between the fitting data points. Also,
24561 if the exact-fit polynomial is going to be used to interpolate or
24562 extrapolate the data, it is numerically better to use the @kbd{a p}
24563 command described below. @xref{Interpolation}.
24569 Another generalization of the linear model is to assume the
24570 @expr{y} values are a sum of linear contributions from several
24571 @expr{x} values. This is a @dfn{multilinear} fit, and it is also
24572 selected by the @kbd{1} digit key. (Calc decides whether the fit
24573 is linear or multilinear by counting the rows in the data matrix.)
24575 Given the data matrix,
24579 [ [ 1, 2, 3, 4, 5 ]
24581 [ 14.5, 15, 18.5, 22.5, 24 ] ]
24586 the command @kbd{a F 1 @key{RET}} will call the first row @expr{x} and the
24587 second row @expr{y}, and will fit the values in the third row to the
24588 model @expr{a + b x + c y}.
24594 Calc can do multilinear fits with any number of independent variables
24595 (i.e., with any number of data rows).
24601 Yet another variation is @dfn{homogeneous} linear models, in which
24602 the constant term is known to be zero. In the linear case, this
24603 means the model formula is simply @expr{a x}; in the multilinear
24604 case, the model might be @expr{a x + b y + c z}; and in the polynomial
24605 case, the model could be @expr{a x + b x^2 + c x^3}. You can get
24606 a homogeneous linear or multilinear model by pressing the letter
24607 @kbd{h} followed by a regular model key, like @kbd{1} or @kbd{2}.
24609 It is certainly possible to have other constrained linear models,
24610 like @expr{2.3 + a x} or @expr{a - 4 x}. While there is no single
24611 key to select models like these, a later section shows how to enter
24612 any desired model by hand. In the first case, for example, you
24613 would enter @kbd{a F ' 2.3 + a x}.
24615 Another class of models that will work but must be entered by hand
24616 are multinomial fits, e.g., @expr{a + b x + c y + d x^2 + e y^2 + f x y}.
24618 @node Error Estimates for Fits, Standard Nonlinear Models, Polynomial and Multilinear Fits, Curve Fitting
24619 @subsection Error Estimates for Fits
24624 With the Hyperbolic flag, @kbd{H a F} [@code{efit}] performs the same
24625 fitting operation as @kbd{a F}, but reports the coefficients as error
24626 forms instead of plain numbers. Fitting our two data matrices (first
24627 with 13, then with 14) to a line with @kbd{H a F} gives the results,
24631 2.6 +/- 0.382970843103 + 2.2 +/- 0.115470053838 x
24634 In the first case the estimated errors are zero because the linear
24635 fit is perfect. In the second case, the errors are nonzero but
24636 moderately small, because the data are still very close to linear.
24638 It is also possible for the @emph{input} to a fitting operation to
24639 contain error forms. The data values must either all include errors
24640 or all be plain numbers. Error forms can go anywhere but generally
24641 go on the numbers in the last row of the data matrix. If the last
24642 row contains error forms
24643 @texline `@var{y_i}@w{ @tfn{+/-} }@math{\sigma_i}',
24644 @infoline `@var{y_i}@w{ @tfn{+/-} }@var{sigma_i}',
24646 @texline @math{\chi^2}
24647 @infoline @expr{chi^2}
24652 chi^2 = sum(((y_i - (a + b x_i)) / sigma_i)^2, i, 1, N)
24658 $$ \chi^2 = \sum_{i=1}^N \left(y_i - (a + b x_i) \over \sigma_i\right)^2 $$
24663 so that data points with larger error estimates contribute less to
24664 the fitting operation.
24666 If there are error forms on other rows of the data matrix, all the
24667 errors for a given data point are combined; the square root of the
24668 sum of the squares of the errors forms the
24669 @texline @math{\sigma_i}
24670 @infoline @expr{sigma_i}
24671 used for the data point.
24673 Both @kbd{a F} and @kbd{H a F} can accept error forms in the input
24674 matrix, although if you are concerned about error analysis you will
24675 probably use @kbd{H a F} so that the output also contains error
24678 If the input contains error forms but all the
24679 @texline @math{\sigma_i}
24680 @infoline @expr{sigma_i}
24681 values are the same, it is easy to see that the resulting fitted model
24682 will be the same as if the input did not have error forms at all
24683 @texline (@math{\chi^2}
24684 @infoline (@expr{chi^2}
24685 is simply scaled uniformly by
24686 @texline @math{1 / \sigma^2},
24687 @infoline @expr{1 / sigma^2},
24688 which doesn't affect where it has a minimum). But there @emph{will} be
24689 a difference in the estimated errors of the coefficients reported by
24692 Consult any text on statistical modeling of data for a discussion
24693 of where these error estimates come from and how they should be
24702 With the Inverse flag, @kbd{I a F} [@code{xfit}] produces even more
24703 information. The result is a vector of six items:
24707 The model formula with error forms for its coefficients or
24708 parameters. This is the result that @kbd{H a F} would have
24712 A vector of ``raw'' parameter values for the model. These are the
24713 polynomial coefficients or other parameters as plain numbers, in the
24714 same order as the parameters appeared in the final prompt of the
24715 @kbd{I a F} command. For polynomials of degree @expr{d}, this vector
24716 will have length @expr{M = d+1} with the constant term first.
24719 The covariance matrix @expr{C} computed from the fit. This is
24720 an @var{m}x@var{m} symmetric matrix; the diagonal elements
24721 @texline @math{C_{jj}}
24722 @infoline @expr{C_j_j}
24724 @texline @math{\sigma_j^2}
24725 @infoline @expr{sigma_j^2}
24726 of the parameters. The other elements are covariances
24727 @texline @math{\sigma_{ij}^2}
24728 @infoline @expr{sigma_i_j^2}
24729 that describe the correlation between pairs of parameters. (A related
24730 set of numbers, the @dfn{linear correlation coefficients}
24731 @texline @math{r_{ij}},
24732 @infoline @expr{r_i_j},
24734 @texline @math{\sigma_{ij}^2 / \sigma_i \, \sigma_j}.)
24735 @infoline @expr{sigma_i_j^2 / sigma_i sigma_j}.)
24738 A vector of @expr{M} ``parameter filter'' functions whose
24739 meanings are described below. If no filters are necessary this
24740 will instead be an empty vector; this is always the case for the
24741 polynomial and multilinear fits described so far.
24745 @texline @math{\chi^2}
24746 @infoline @expr{chi^2}
24747 for the fit, calculated by the formulas shown above. This gives a
24748 measure of the quality of the fit; statisticians consider
24749 @texline @math{\chi^2 \approx N - M}
24750 @infoline @expr{chi^2 = N - M}
24751 to indicate a moderately good fit (where again @expr{N} is the number of
24752 data points and @expr{M} is the number of parameters).
24755 A measure of goodness of fit expressed as a probability @expr{Q}.
24756 This is computed from the @code{utpc} probability distribution
24758 @texline @math{\chi^2}
24759 @infoline @expr{chi^2}
24760 with @expr{N - M} degrees of freedom. A
24761 value of 0.5 implies a good fit; some texts recommend that often
24762 @expr{Q = 0.1} or even 0.001 can signify an acceptable fit. In
24764 @texline @math{\chi^2}
24765 @infoline @expr{chi^2}
24766 statistics assume the errors in your inputs
24767 follow a normal (Gaussian) distribution; if they don't, you may
24768 have to accept smaller values of @expr{Q}.
24770 The @expr{Q} value is computed only if the input included error
24771 estimates. Otherwise, Calc will report the symbol @code{nan}
24772 for @expr{Q}. The reason is that in this case the
24773 @texline @math{\chi^2}
24774 @infoline @expr{chi^2}
24775 value has effectively been used to estimate the original errors
24776 in the input, and thus there is no redundant information left
24777 over to use for a confidence test.
24780 @node Standard Nonlinear Models, Curve Fitting Details, Error Estimates for Fits, Curve Fitting
24781 @subsection Standard Nonlinear Models
24784 The @kbd{a F} command also accepts other kinds of models besides
24785 lines and polynomials. Some common models have quick single-key
24786 abbreviations; others must be entered by hand as algebraic formulas.
24788 Here is a complete list of the standard models recognized by @kbd{a F}:
24792 Linear or multilinear. @mathit{a + b x + c y + d z}.
24794 Polynomials. @mathit{a + b x + c x^2 + d x^3}.
24796 Exponential. @mathit{a} @tfn{exp}@mathit{(b x)} @tfn{exp}@mathit{(c y)}.
24798 Base-10 exponential. @mathit{a} @tfn{10^}@mathit{(b x)} @tfn{10^}@mathit{(c y)}.
24800 Exponential (alternate notation). @tfn{exp}@mathit{(a + b x + c y)}.
24802 Base-10 exponential (alternate). @tfn{10^}@mathit{(a + b x + c y)}.
24804 Logarithmic. @mathit{a + b} @tfn{ln}@mathit{(x) + c} @tfn{ln}@mathit{(y)}.
24806 Base-10 logarithmic. @mathit{a + b} @tfn{log10}@mathit{(x) + c} @tfn{log10}@mathit{(y)}.
24808 General exponential. @mathit{a b^x c^y}.
24810 Power law. @mathit{a x^b y^c}.
24812 Quadratic. @mathit{a + b (x-c)^2 + d (x-e)^2}.
24815 @texline @math{{a \over b \sqrt{2 \pi}} \exp\left( -{1 \over 2} \left( x - c \over b \right)^2 \right)}.
24816 @infoline @mathit{(a / b sqrt(2 pi)) exp(-0.5*((x-c)/b)^2)}.
24819 All of these models are used in the usual way; just press the appropriate
24820 letter at the model prompt, and choose variable names if you wish. The
24821 result will be a formula as shown in the above table, with the best-fit
24822 values of the parameters substituted. (You may find it easier to read
24823 the parameter values from the vector that is placed in the trail.)
24825 All models except Gaussian and polynomials can generalize as shown to any
24826 number of independent variables. Also, all the built-in models have an
24827 additive or multiplicative parameter shown as @expr{a} in the above table
24828 which can be replaced by zero or one, as appropriate, by typing @kbd{h}
24829 before the model key.
24831 Note that many of these models are essentially equivalent, but express
24832 the parameters slightly differently. For example, @expr{a b^x} and
24833 the other two exponential models are all algebraic rearrangements of
24834 each other. Also, the ``quadratic'' model is just a degree-2 polynomial
24835 with the parameters expressed differently. Use whichever form best
24836 matches the problem.
24838 The HP-28/48 calculators support four different models for curve
24839 fitting, called @code{LIN}, @code{LOG}, @code{EXP}, and @code{PWR}.
24840 These correspond to Calc models @samp{a + b x}, @samp{a + b ln(x)},
24841 @samp{a exp(b x)}, and @samp{a x^b}, respectively. In each case,
24842 @expr{a} is what the HP-48 identifies as the ``intercept,'' and
24843 @expr{b} is what it calls the ``slope.''
24849 If the model you want doesn't appear on this list, press @kbd{'}
24850 (the apostrophe key) at the model prompt to enter any algebraic
24851 formula, such as @kbd{m x - b}, as the model. (Not all models
24852 will work, though---see the next section for details.)
24854 The model can also be an equation like @expr{y = m x + b}.
24855 In this case, Calc thinks of all the rows of the data matrix on
24856 equal terms; this model effectively has two parameters
24857 (@expr{m} and @expr{b}) and two independent variables (@expr{x}
24858 and @expr{y}), with no ``dependent'' variables. Model equations
24859 do not need to take this @expr{y =} form. For example, the
24860 implicit line equation @expr{a x + b y = 1} works fine as a
24863 When you enter a model, Calc makes an alphabetical list of all
24864 the variables that appear in the model. These are used for the
24865 default parameters, independent variables, and dependent variable
24866 (in that order). If you enter a plain formula (not an equation),
24867 Calc assumes the dependent variable does not appear in the formula
24868 and thus does not need a name.
24870 For example, if the model formula has the variables @expr{a,mu,sigma,t,x},
24871 and the data matrix has three rows (meaning two independent variables),
24872 Calc will use @expr{a,mu,sigma} as the default parameters, and the
24873 data rows will be named @expr{t} and @expr{x}, respectively. If you
24874 enter an equation instead of a plain formula, Calc will use @expr{a,mu}
24875 as the parameters, and @expr{sigma,t,x} as the three independent
24878 You can, of course, override these choices by entering something
24879 different at the prompt. If you leave some variables out of the list,
24880 those variables must have stored values and those stored values will
24881 be used as constants in the model. (Stored values for the parameters
24882 and independent variables are ignored by the @kbd{a F} command.)
24883 If you list only independent variables, all the remaining variables
24884 in the model formula will become parameters.
24886 If there are @kbd{$} signs in the model you type, they will stand
24887 for parameters and all other variables (in alphabetical order)
24888 will be independent. Use @kbd{$} for one parameter, @kbd{$$} for
24889 another, and so on. Thus @kbd{$ x + $$} is another way to describe
24892 If you type a @kbd{$} instead of @kbd{'} at the model prompt itself,
24893 Calc will take the model formula from the stack. (The data must then
24894 appear at the second stack level.) The same conventions are used to
24895 choose which variables in the formula are independent by default and
24896 which are parameters.
24898 Models taken from the stack can also be expressed as vectors of
24899 two or three elements, @expr{[@var{model}, @var{vars}]} or
24900 @expr{[@var{model}, @var{vars}, @var{params}]}. Each of @var{vars}
24901 and @var{params} may be either a variable or a vector of variables.
24902 (If @var{params} is omitted, all variables in @var{model} except
24903 those listed as @var{vars} are parameters.)
24905 When you enter a model manually with @kbd{'}, Calc puts a 3-vector
24906 describing the model in the trail so you can get it back if you wish.
24914 Finally, you can store a model in one of the Calc variables
24915 @code{Model1} or @code{Model2}, then use this model by typing
24916 @kbd{a F u} or @kbd{a F U} (respectively). The value stored in
24917 the variable can be any of the formats that @kbd{a F $} would
24918 accept for a model on the stack.
24924 Calc uses the principal values of inverse functions like @code{ln}
24925 and @code{arcsin} when doing fits. For example, when you enter
24926 the model @samp{y = sin(a t + b)} Calc actually uses the easier
24927 form @samp{arcsin(y) = a t + b}. The @code{arcsin} function always
24928 returns results in the range from @mathit{-90} to 90 degrees (or the
24929 equivalent range in radians). Suppose you had data that you
24930 believed to represent roughly three oscillations of a sine wave,
24931 so that the argument of the sine might go from zero to
24932 @texline @math{3\times360}
24933 @infoline @mathit{3*360}
24935 The above model would appear to be a good way to determine the
24936 true frequency and phase of the sine wave, but in practice it
24937 would fail utterly. The righthand side of the actual model
24938 @samp{arcsin(y) = a t + b} will grow smoothly with @expr{t}, but
24939 the lefthand side will bounce back and forth between @mathit{-90} and 90.
24940 No values of @expr{a} and @expr{b} can make the two sides match,
24941 even approximately.
24943 There is no good solution to this problem at present. You could
24944 restrict your data to small enough ranges so that the above problem
24945 doesn't occur (i.e., not straddling any peaks in the sine wave).
24946 Or, in this case, you could use a totally different method such as
24947 Fourier analysis, which is beyond the scope of the @kbd{a F} command.
24948 (Unfortunately, Calc does not currently have any facilities for
24949 taking Fourier and related transforms.)
24951 @node Curve Fitting Details, Interpolation, Standard Nonlinear Models, Curve Fitting
24952 @subsection Curve Fitting Details
24955 Calc's internal least-squares fitter can only handle multilinear
24956 models. More precisely, it can handle any model of the form
24957 @expr{a f(x,y,z) + b g(x,y,z) + c h(x,y,z)}, where @expr{a,b,c}
24958 are the parameters and @expr{x,y,z} are the independent variables
24959 (of course there can be any number of each, not just three).
24961 In a simple multilinear or polynomial fit, it is easy to see how
24962 to convert the model into this form. For example, if the model
24963 is @expr{a + b x + c x^2}, then @expr{f(x) = 1}, @expr{g(x) = x},
24964 and @expr{h(x) = x^2} are suitable functions.
24966 For other models, Calc uses a variety of algebraic manipulations
24967 to try to put the problem into the form
24970 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)
24974 where @expr{Y,A,B,C,F,G,H} are arbitrary functions. It computes
24975 @expr{Y}, @expr{F}, @expr{G}, and @expr{H} for all the data points,
24976 does a standard linear fit to find the values of @expr{A}, @expr{B},
24977 and @expr{C}, then uses the equation solver to solve for @expr{a,b,c}
24978 in terms of @expr{A,B,C}.
24980 A remarkable number of models can be cast into this general form.
24981 We'll look at two examples here to see how it works. The power-law
24982 model @expr{y = a x^b} with two independent variables and two parameters
24983 can be rewritten as follows:
24988 y = exp(ln(a) + b ln(x))
24989 ln(y) = ln(a) + b ln(x)
24993 which matches the desired form with
24994 @texline @math{Y = \ln(y)},
24995 @infoline @expr{Y = ln(y)},
24996 @texline @math{A = \ln(a)},
24997 @infoline @expr{A = ln(a)},
24998 @expr{F = 1}, @expr{B = b}, and
24999 @texline @math{G = \ln(x)}.
25000 @infoline @expr{G = ln(x)}.
25001 Calc thus computes the logarithms of your @expr{y} and @expr{x} values,
25002 does a linear fit for @expr{A} and @expr{B}, then solves to get
25003 @texline @math{a = \exp(A)}
25004 @infoline @expr{a = exp(A)}
25007 Another interesting example is the ``quadratic'' model, which can
25008 be handled by expanding according to the distributive law.
25011 y = a + b*(x - c)^2
25012 y = a + b c^2 - 2 b c x + b x^2
25016 which matches with @expr{Y = y}, @expr{A = a + b c^2}, @expr{F = 1},
25017 @expr{B = -2 b c}, @expr{G = x} (the @mathit{-2} factor could just as easily
25018 have been put into @expr{G} instead of @expr{B}), @expr{C = b}, and
25021 The Gaussian model looks quite complicated, but a closer examination
25022 shows that it's actually similar to the quadratic model but with an
25023 exponential that can be brought to the top and moved into @expr{Y}.
25025 An example of a model that cannot be put into general linear
25026 form is a Gaussian with a constant background added on, i.e.,
25027 @expr{d} + the regular Gaussian formula. If you have a model like
25028 this, your best bet is to replace enough of your parameters with
25029 constants to make the model linearizable, then adjust the constants
25030 manually by doing a series of fits. You can compare the fits by
25031 graphing them, by examining the goodness-of-fit measures returned by
25032 @kbd{I a F}, or by some other method suitable to your application.
25033 Note that some models can be linearized in several ways. The
25034 Gaussian-plus-@var{d} model can be linearized by setting @expr{d}
25035 (the background) to a constant, or by setting @expr{b} (the standard
25036 deviation) and @expr{c} (the mean) to constants.
25038 To fit a model with constants substituted for some parameters, just
25039 store suitable values in those parameter variables, then omit them
25040 from the list of parameters when you answer the variables prompt.
25046 A last desperate step would be to use the general-purpose
25047 @code{minimize} function rather than @code{fit}. After all, both
25048 functions solve the problem of minimizing an expression (the
25049 @texline @math{\chi^2}
25050 @infoline @expr{chi^2}
25051 sum) by adjusting certain parameters in the expression. The @kbd{a F}
25052 command is able to use a vastly more efficient algorithm due to its
25053 special knowledge about linear chi-square sums, but the @kbd{a N}
25054 command can do the same thing by brute force.
25056 A compromise would be to pick out a few parameters without which the
25057 fit is linearizable, and use @code{minimize} on a call to @code{fit}
25058 which efficiently takes care of the rest of the parameters. The thing
25059 to be minimized would be the value of
25060 @texline @math{\chi^2}
25061 @infoline @expr{chi^2}
25062 returned as the fifth result of the @code{xfit} function:
25065 minimize(xfit(gaus(a,b,c,d,x), x, [a,b,c], data)_5, d, guess)
25069 where @code{gaus} represents the Gaussian model with background,
25070 @code{data} represents the data matrix, and @code{guess} represents
25071 the initial guess for @expr{d} that @code{minimize} requires.
25072 This operation will only be, shall we say, extraordinarily slow
25073 rather than astronomically slow (as would be the case if @code{minimize}
25074 were used by itself to solve the problem).
25080 The @kbd{I a F} [@code{xfit}] command is somewhat trickier when
25081 nonlinear models are used. The second item in the result is the
25082 vector of ``raw'' parameters @expr{A}, @expr{B}, @expr{C}. The
25083 covariance matrix is written in terms of those raw parameters.
25084 The fifth item is a vector of @dfn{filter} expressions. This
25085 is the empty vector @samp{[]} if the raw parameters were the same
25086 as the requested parameters, i.e., if @expr{A = a}, @expr{B = b},
25087 and so on (which is always true if the model is already linear
25088 in the parameters as written, e.g., for polynomial fits). If the
25089 parameters had to be rearranged, the fifth item is instead a vector
25090 of one formula per parameter in the original model. The raw
25091 parameters are expressed in these ``filter'' formulas as
25092 @samp{fitdummy(1)} for @expr{A}, @samp{fitdummy(2)} for @expr{B},
25095 When Calc needs to modify the model to return the result, it replaces
25096 @samp{fitdummy(1)} in all the filters with the first item in the raw
25097 parameters list, and so on for the other raw parameters, then
25098 evaluates the resulting filter formulas to get the actual parameter
25099 values to be substituted into the original model. In the case of
25100 @kbd{H a F} and @kbd{I a F} where the parameters must be error forms,
25101 Calc uses the square roots of the diagonal entries of the covariance
25102 matrix as error values for the raw parameters, then lets Calc's
25103 standard error-form arithmetic take it from there.
25105 If you use @kbd{I a F} with a nonlinear model, be sure to remember
25106 that the covariance matrix is in terms of the raw parameters,
25107 @emph{not} the actual requested parameters. It's up to you to
25108 figure out how to interpret the covariances in the presence of
25109 nontrivial filter functions.
25111 Things are also complicated when the input contains error forms.
25112 Suppose there are three independent and dependent variables, @expr{x},
25113 @expr{y}, and @expr{z}, one or more of which are error forms in the
25114 data. Calc combines all the error values by taking the square root
25115 of the sum of the squares of the errors. It then changes @expr{x}
25116 and @expr{y} to be plain numbers, and makes @expr{z} into an error
25117 form with this combined error. The @expr{Y(x,y,z)} part of the
25118 linearized model is evaluated, and the result should be an error
25119 form. The error part of that result is used for
25120 @texline @math{\sigma_i}
25121 @infoline @expr{sigma_i}
25122 for the data point. If for some reason @expr{Y(x,y,z)} does not return
25123 an error form, the combined error from @expr{z} is used directly for
25124 @texline @math{\sigma_i}.
25125 @infoline @expr{sigma_i}.
25126 Finally, @expr{z} is also stripped of its error
25127 for use in computing @expr{F(x,y,z)}, @expr{G(x,y,z)} and so on;
25128 the righthand side of the linearized model is computed in regular
25129 arithmetic with no error forms.
25131 (While these rules may seem complicated, they are designed to do
25132 the most reasonable thing in the typical case that @expr{Y(x,y,z)}
25133 depends only on the dependent variable @expr{z}, and in fact is
25134 often simply equal to @expr{z}. For common cases like polynomials
25135 and multilinear models, the combined error is simply used as the
25136 @texline @math{\sigma}
25137 @infoline @expr{sigma}
25138 for the data point with no further ado.)
25145 It may be the case that the model you wish to use is linearizable,
25146 but Calc's built-in rules are unable to figure it out. Calc uses
25147 its algebraic rewrite mechanism to linearize a model. The rewrite
25148 rules are kept in the variable @code{FitRules}. You can edit this
25149 variable using the @kbd{s e FitRules} command; in fact, there is
25150 a special @kbd{s F} command just for editing @code{FitRules}.
25151 @xref{Operations on Variables}.
25153 @xref{Rewrite Rules}, for a discussion of rewrite rules.
25187 Calc uses @code{FitRules} as follows. First, it converts the model
25188 to an equation if necessary and encloses the model equation in a
25189 call to the function @code{fitmodel} (which is not actually a defined
25190 function in Calc; it is only used as a placeholder by the rewrite rules).
25191 Parameter variables are renamed to function calls @samp{fitparam(1)},
25192 @samp{fitparam(2)}, and so on, and independent variables are renamed
25193 to @samp{fitvar(1)}, @samp{fitvar(2)}, etc. The dependent variable
25194 is the highest-numbered @code{fitvar}. For example, the power law
25195 model @expr{a x^b} is converted to @expr{y = a x^b}, then to
25199 fitmodel(fitvar(2) = fitparam(1) fitvar(1)^fitparam(2))
25203 Calc then applies the rewrites as if by @samp{C-u 0 a r FitRules}.
25204 (The zero prefix means that rewriting should continue until no further
25205 changes are possible.)
25207 When rewriting is complete, the @code{fitmodel} call should have
25208 been replaced by a @code{fitsystem} call that looks like this:
25211 fitsystem(@var{Y}, @var{FGH}, @var{abc})
25215 where @var{Y} is a formula that describes the function @expr{Y(x,y,z)},
25216 @var{FGH} is the vector of formulas @expr{[F(x,y,z), G(x,y,z), H(x,y,z)]},
25217 and @var{abc} is the vector of parameter filters which refer to the
25218 raw parameters as @samp{fitdummy(1)} for @expr{A}, @samp{fitdummy(2)}
25219 for @expr{B}, etc. While the number of raw parameters (the length of
25220 the @var{FGH} vector) is usually the same as the number of original
25221 parameters (the length of the @var{abc} vector), this is not required.
25223 The power law model eventually boils down to
25227 fitsystem(ln(fitvar(2)),
25228 [1, ln(fitvar(1))],
25229 [exp(fitdummy(1)), fitdummy(2)])
25233 The actual implementation of @code{FitRules} is complicated; it
25234 proceeds in four phases. First, common rearrangements are done
25235 to try to bring linear terms together and to isolate functions like
25236 @code{exp} and @code{ln} either all the way ``out'' (so that they
25237 can be put into @var{Y}) or all the way ``in'' (so that they can
25238 be put into @var{abc} or @var{FGH}). In particular, all
25239 non-constant powers are converted to logs-and-exponentials form,
25240 and the distributive law is used to expand products of sums.
25241 Quotients are rewritten to use the @samp{fitinv} function, where
25242 @samp{fitinv(x)} represents @expr{1/x} while the @code{FitRules}
25243 are operating. (The use of @code{fitinv} makes recognition of
25244 linear-looking forms easier.) If you modify @code{FitRules}, you
25245 will probably only need to modify the rules for this phase.
25247 Phase two, whose rules can actually also apply during phases one
25248 and three, first rewrites @code{fitmodel} to a two-argument
25249 form @samp{fitmodel(@var{Y}, @var{model})}, where @var{Y} is
25250 initially zero and @var{model} has been changed from @expr{a=b}
25251 to @expr{a-b} form. It then tries to peel off invertible functions
25252 from the outside of @var{model} and put them into @var{Y} instead,
25253 calling the equation solver to invert the functions. Finally, when
25254 this is no longer possible, the @code{fitmodel} is changed to a
25255 four-argument @code{fitsystem}, where the fourth argument is
25256 @var{model} and the @var{FGH} and @var{abc} vectors are initially
25257 empty. (The last vector is really @var{ABC}, corresponding to
25258 raw parameters, for now.)
25260 Phase three converts a sum of items in the @var{model} to a sum
25261 of @samp{fitpart(@var{a}, @var{b}, @var{c})} terms which represent
25262 terms @samp{@var{a}*@var{b}*@var{c}} of the sum, where @var{a}
25263 is all factors that do not involve any variables, @var{b} is all
25264 factors that involve only parameters, and @var{c} is the factors
25265 that involve only independent variables. (If this decomposition
25266 is not possible, the rule set will not complete and Calc will
25267 complain that the model is too complex.) Then @code{fitpart}s
25268 with equal @var{b} or @var{c} components are merged back together
25269 using the distributive law in order to minimize the number of
25270 raw parameters needed.
25272 Phase four moves the @code{fitpart} terms into the @var{FGH} and
25273 @var{ABC} vectors. Also, some of the algebraic expansions that
25274 were done in phase 1 are undone now to make the formulas more
25275 computationally efficient. Finally, it calls the solver one more
25276 time to convert the @var{ABC} vector to an @var{abc} vector, and
25277 removes the fourth @var{model} argument (which by now will be zero)
25278 to obtain the three-argument @code{fitsystem} that the linear
25279 least-squares solver wants to see.
25285 @mindex hasfit@idots
25287 @tindex hasfitparams
25295 Two functions which are useful in connection with @code{FitRules}
25296 are @samp{hasfitparams(x)} and @samp{hasfitvars(x)}, which check
25297 whether @expr{x} refers to any parameters or independent variables,
25298 respectively. Specifically, these functions return ``true'' if the
25299 argument contains any @code{fitparam} (or @code{fitvar}) function
25300 calls, and ``false'' otherwise. (Recall that ``true'' means a
25301 nonzero number, and ``false'' means zero. The actual nonzero number
25302 returned is the largest @var{n} from all the @samp{fitparam(@var{n})}s
25303 or @samp{fitvar(@var{n})}s, respectively, that appear in the formula.)
25309 The @code{fit} function in algebraic notation normally takes four
25310 arguments, @samp{fit(@var{model}, @var{vars}, @var{params}, @var{data})},
25311 where @var{model} is the model formula as it would be typed after
25312 @kbd{a F '}, @var{vars} is the independent variable or a vector of
25313 independent variables, @var{params} likewise gives the parameter(s),
25314 and @var{data} is the data matrix. Note that the length of @var{vars}
25315 must be equal to the number of rows in @var{data} if @var{model} is
25316 an equation, or one less than the number of rows if @var{model} is
25317 a plain formula. (Actually, a name for the dependent variable is
25318 allowed but will be ignored in the plain-formula case.)
25320 If @var{params} is omitted, the parameters are all variables in
25321 @var{model} except those that appear in @var{vars}. If @var{vars}
25322 is also omitted, Calc sorts all the variables that appear in
25323 @var{model} alphabetically and uses the higher ones for @var{vars}
25324 and the lower ones for @var{params}.
25326 Alternatively, @samp{fit(@var{modelvec}, @var{data})} is allowed
25327 where @var{modelvec} is a 2- or 3-vector describing the model
25328 and variables, as discussed previously.
25330 If Calc is unable to do the fit, the @code{fit} function is left
25331 in symbolic form, ordinarily with an explanatory message. The
25332 message will be ``Model expression is too complex'' if the
25333 linearizer was unable to put the model into the required form.
25335 The @code{efit} (corresponding to @kbd{H a F}) and @code{xfit}
25336 (for @kbd{I a F}) functions are completely analogous.
25338 @node Interpolation, , Curve Fitting Details, Curve Fitting
25339 @subsection Polynomial Interpolation
25342 @pindex calc-poly-interp
25344 The @kbd{a p} (@code{calc-poly-interp}) [@code{polint}] command does
25345 a polynomial interpolation at a particular @expr{x} value. It takes
25346 two arguments from the stack: A data matrix of the sort used by
25347 @kbd{a F}, and a single number which represents the desired @expr{x}
25348 value. Calc effectively does an exact polynomial fit as if by @kbd{a F i},
25349 then substitutes the @expr{x} value into the result in order to get an
25350 approximate @expr{y} value based on the fit. (Calc does not actually
25351 use @kbd{a F i}, however; it uses a direct method which is both more
25352 efficient and more numerically stable.)
25354 The result of @kbd{a p} is actually a vector of two values: The @expr{y}
25355 value approximation, and an error measure @expr{dy} that reflects Calc's
25356 estimation of the probable error of the approximation at that value of
25357 @expr{x}. If the input @expr{x} is equal to any of the @expr{x} values
25358 in the data matrix, the output @expr{y} will be the corresponding @expr{y}
25359 value from the matrix, and the output @expr{dy} will be exactly zero.
25361 A prefix argument of 2 causes @kbd{a p} to take separate x- and
25362 y-vectors from the stack instead of one data matrix.
25364 If @expr{x} is a vector of numbers, @kbd{a p} will return a matrix of
25365 interpolated results for each of those @expr{x} values. (The matrix will
25366 have two columns, the @expr{y} values and the @expr{dy} values.)
25367 If @expr{x} is a formula instead of a number, the @code{polint} function
25368 remains in symbolic form; use the @kbd{a "} command to expand it out to
25369 a formula that describes the fit in symbolic terms.
25371 In all cases, the @kbd{a p} command leaves the data vectors or matrix
25372 on the stack. Only the @expr{x} value is replaced by the result.
25376 The @kbd{H a p} [@code{ratint}] command does a rational function
25377 interpolation. It is used exactly like @kbd{a p}, except that it
25378 uses as its model the quotient of two polynomials. If there are
25379 @expr{N} data points, the numerator and denominator polynomials will
25380 each have degree @expr{N/2} (if @expr{N} is odd, the denominator will
25381 have degree one higher than the numerator).
25383 Rational approximations have the advantage that they can accurately
25384 describe functions that have poles (points at which the function's value
25385 goes to infinity, so that the denominator polynomial of the approximation
25386 goes to zero). If @expr{x} corresponds to a pole of the fitted rational
25387 function, then the result will be a division by zero. If Infinite mode
25388 is enabled, the result will be @samp{[uinf, uinf]}.
25390 There is no way to get the actual coefficients of the rational function
25391 used by @kbd{H a p}. (The algorithm never generates these coefficients
25392 explicitly, and quotients of polynomials are beyond @w{@kbd{a F}}'s
25393 capabilities to fit.)
25395 @node Summations, Logical Operations, Curve Fitting, Algebra
25396 @section Summations
25399 @cindex Summation of a series
25401 @pindex calc-summation
25403 The @kbd{a +} (@code{calc-summation}) [@code{sum}] command computes
25404 the sum of a formula over a certain range of index values. The formula
25405 is taken from the top of the stack; the command prompts for the
25406 name of the summation index variable, the lower limit of the
25407 sum (any formula), and the upper limit of the sum. If you
25408 enter a blank line at any of these prompts, that prompt and
25409 any later ones are answered by reading additional elements from
25410 the stack. Thus, @kbd{' k^2 @key{RET} ' k @key{RET} 1 @key{RET} 5 @key{RET} a + @key{RET}}
25411 produces the result 55.
25414 $$ \sum_{k=1}^5 k^2 = 55 $$
25417 The choice of index variable is arbitrary, but it's best not to
25418 use a variable with a stored value. In particular, while
25419 @code{i} is often a favorite index variable, it should be avoided
25420 in Calc because @code{i} has the imaginary constant @expr{(0, 1)}
25421 as a value. If you pressed @kbd{=} on a sum over @code{i}, it would
25422 be changed to a nonsensical sum over the ``variable'' @expr{(0, 1)}!
25423 If you really want to use @code{i} as an index variable, use
25424 @w{@kbd{s u i @key{RET}}} first to ``unstore'' this variable.
25425 (@xref{Storing Variables}.)
25427 A numeric prefix argument steps the index by that amount rather
25428 than by one. Thus @kbd{' a_k @key{RET} C-u -2 a + k @key{RET} 10 @key{RET} 0 @key{RET}}
25429 yields @samp{a_10 + a_8 + a_6 + a_4 + a_2 + a_0}. A prefix
25430 argument of plain @kbd{C-u} causes @kbd{a +} to prompt for the
25431 step value, in which case you can enter any formula or enter
25432 a blank line to take the step value from the stack. With the
25433 @kbd{C-u} prefix, @kbd{a +} can take up to five arguments from
25434 the stack: The formula, the variable, the lower limit, the
25435 upper limit, and (at the top of the stack), the step value.
25437 Calc knows how to do certain sums in closed form. For example,
25438 @samp{sum(6 k^2, k, 1, n) = @w{2 n^3} + 3 n^2 + n}. In particular,
25439 this is possible if the formula being summed is polynomial or
25440 exponential in the index variable. Sums of logarithms are
25441 transformed into logarithms of products. Sums of trigonometric
25442 and hyperbolic functions are transformed to sums of exponentials
25443 and then done in closed form. Also, of course, sums in which the
25444 lower and upper limits are both numbers can always be evaluated
25445 just by grinding them out, although Calc will use closed forms
25446 whenever it can for the sake of efficiency.
25448 The notation for sums in algebraic formulas is
25449 @samp{sum(@var{expr}, @var{var}, @var{low}, @var{high}, @var{step})}.
25450 If @var{step} is omitted, it defaults to one. If @var{high} is
25451 omitted, @var{low} is actually the upper limit and the lower limit
25452 is one. If @var{low} is also omitted, the limits are @samp{-inf}
25453 and @samp{inf}, respectively.
25455 Infinite sums can sometimes be evaluated: @samp{sum(.5^k, k, 1, inf)}
25456 returns @expr{1}. This is done by evaluating the sum in closed
25457 form (to @samp{1. - 0.5^n} in this case), then evaluating this
25458 formula with @code{n} set to @code{inf}. Calc's usual rules
25459 for ``infinite'' arithmetic can find the answer from there. If
25460 infinite arithmetic yields a @samp{nan}, or if the sum cannot be
25461 solved in closed form, Calc leaves the @code{sum} function in
25462 symbolic form. @xref{Infinities}.
25464 As a special feature, if the limits are infinite (or omitted, as
25465 described above) but the formula includes vectors subscripted by
25466 expressions that involve the iteration variable, Calc narrows
25467 the limits to include only the range of integers which result in
25468 valid subscripts for the vector. For example, the sum
25469 @samp{sum(k [a,b,c,d,e,f,g]_(2k),k)} evaluates to @samp{b + 2 d + 3 f}.
25471 The limits of a sum do not need to be integers. For example,
25472 @samp{sum(a_k, k, 0, 2 n, n)} produces @samp{a_0 + a_n + a_(2 n)}.
25473 Calc computes the number of iterations using the formula
25474 @samp{1 + (@var{high} - @var{low}) / @var{step}}, which must,
25475 after simplification as if by @kbd{a s}, evaluate to an integer.
25477 If the number of iterations according to the above formula does
25478 not come out to an integer, the sum is invalid and will be left
25479 in symbolic form. However, closed forms are still supplied, and
25480 you are on your honor not to misuse the resulting formulas by
25481 substituting mismatched bounds into them. For example,
25482 @samp{sum(k, k, 1, 10, 2)} is invalid, but Calc will go ahead and
25483 evaluate the closed form solution for the limits 1 and 10 to get
25484 the rather dubious answer, 29.25.
25486 If the lower limit is greater than the upper limit (assuming a
25487 positive step size), the result is generally zero. However,
25488 Calc only guarantees a zero result when the upper limit is
25489 exactly one step less than the lower limit, i.e., if the number
25490 of iterations is @mathit{-1}. Thus @samp{sum(f(k), k, n, n-1)} is zero
25491 but the sum from @samp{n} to @samp{n-2} may report a nonzero value
25492 if Calc used a closed form solution.
25494 Calc's logical predicates like @expr{a < b} return 1 for ``true''
25495 and 0 for ``false.'' @xref{Logical Operations}. This can be
25496 used to advantage for building conditional sums. For example,
25497 @samp{sum(prime(k)*k^2, k, 1, 20)} is the sum of the squares of all
25498 prime numbers from 1 to 20; the @code{prime} predicate returns 1 if
25499 its argument is prime and 0 otherwise. You can read this expression
25500 as ``the sum of @expr{k^2}, where @expr{k} is prime.'' Indeed,
25501 @samp{sum(prime(k)*k^2, k)} would represent the sum of @emph{all} primes
25502 squared, since the limits default to plus and minus infinity, but
25503 there are no such sums that Calc's built-in rules can do in
25506 As another example, @samp{sum((k != k_0) * f(k), k, 1, n)} is the
25507 sum of @expr{f(k)} for all @expr{k} from 1 to @expr{n}, excluding
25508 one value @expr{k_0}. Slightly more tricky is the summand
25509 @samp{(k != k_0) / (k - k_0)}, which is an attempt to describe
25510 the sum of all @expr{1/(k-k_0)} except at @expr{k = k_0}, where
25511 this would be a division by zero. But at @expr{k = k_0}, this
25512 formula works out to the indeterminate form @expr{0 / 0}, which
25513 Calc will not assume is zero. Better would be to use
25514 @samp{(k != k_0) ? 1/(k-k_0) : 0}; the @samp{? :} operator does
25515 an ``if-then-else'' test: This expression says, ``if
25516 @texline @math{k \ne k_0},
25517 @infoline @expr{k != k_0},
25518 then @expr{1/(k-k_0)}, else zero.'' Now the formula @expr{1/(k-k_0)}
25519 will not even be evaluated by Calc when @expr{k = k_0}.
25521 @cindex Alternating sums
25523 @pindex calc-alt-summation
25525 The @kbd{a -} (@code{calc-alt-summation}) [@code{asum}] command
25526 computes an alternating sum. Successive terms of the sequence
25527 are given alternating signs, with the first term (corresponding
25528 to the lower index value) being positive. Alternating sums
25529 are converted to normal sums with an extra term of the form
25530 @samp{(-1)^(k-@var{low})}. This formula is adjusted appropriately
25531 if the step value is other than one. For example, the Taylor
25532 series for the sine function is @samp{asum(x^k / k!, k, 1, inf, 2)}.
25533 (Calc cannot evaluate this infinite series, but it can approximate
25534 it if you replace @code{inf} with any particular odd number.)
25535 Calc converts this series to a regular sum with a step of one,
25536 namely @samp{sum((-1)^k x^(2k+1) / (2k+1)!, k, 0, inf)}.
25538 @cindex Product of a sequence
25540 @pindex calc-product
25542 The @kbd{a *} (@code{calc-product}) [@code{prod}] command is
25543 the analogous way to take a product of many terms. Calc also knows
25544 some closed forms for products, such as @samp{prod(k, k, 1, n) = n!}.
25545 Conditional products can be written @samp{prod(k^prime(k), k, 1, n)}
25546 or @samp{prod(prime(k) ? k : 1, k, 1, n)}.
25549 @pindex calc-tabulate
25551 The @kbd{a T} (@code{calc-tabulate}) [@code{table}] command
25552 evaluates a formula at a series of iterated index values, just
25553 like @code{sum} and @code{prod}, but its result is simply a
25554 vector of the results. For example, @samp{table(a_i, i, 1, 7, 2)}
25555 produces @samp{[a_1, a_3, a_5, a_7]}.
25557 @node Logical Operations, Rewrite Rules, Summations, Algebra
25558 @section Logical Operations
25561 The following commands and algebraic functions return true/false values,
25562 where 1 represents ``true'' and 0 represents ``false.'' In cases where
25563 a truth value is required (such as for the condition part of a rewrite
25564 rule, or as the condition for a @w{@kbd{Z [ Z ]}} control structure), any
25565 nonzero value is accepted to mean ``true.'' (Specifically, anything
25566 for which @code{dnonzero} returns 1 is ``true,'' and anything for
25567 which @code{dnonzero} returns 0 or cannot decide is assumed ``false.''
25568 Note that this means that @w{@kbd{Z [ Z ]}} will execute the ``then''
25569 portion if its condition is provably true, but it will execute the
25570 ``else'' portion for any condition like @expr{a = b} that is not
25571 provably true, even if it might be true. Algebraic functions that
25572 have conditions as arguments, like @code{? :} and @code{&&}, remain
25573 unevaluated if the condition is neither provably true nor provably
25574 false. @xref{Declarations}.)
25577 @pindex calc-equal-to
25581 The @kbd{a =} (@code{calc-equal-to}) command, or @samp{eq(a,b)} function
25582 (which can also be written @samp{a = b} or @samp{a == b} in an algebraic
25583 formula) is true if @expr{a} and @expr{b} are equal, either because they
25584 are identical expressions, or because they are numbers which are
25585 numerically equal. (Thus the integer 1 is considered equal to the float
25586 1.0.) If the equality of @expr{a} and @expr{b} cannot be determined,
25587 the comparison is left in symbolic form. Note that as a command, this
25588 operation pops two values from the stack and pushes back either a 1 or
25589 a 0, or a formula @samp{a = b} if the values' equality cannot be determined.
25591 Many Calc commands use @samp{=} formulas to represent @dfn{equations}.
25592 For example, the @kbd{a S} (@code{calc-solve-for}) command rearranges
25593 an equation to solve for a given variable. The @kbd{a M}
25594 (@code{calc-map-equation}) command can be used to apply any
25595 function to both sides of an equation; for example, @kbd{2 a M *}
25596 multiplies both sides of the equation by two. Note that just
25597 @kbd{2 *} would not do the same thing; it would produce the formula
25598 @samp{2 (a = b)} which represents 2 if the equality is true or
25601 The @code{eq} function with more than two arguments (e.g., @kbd{C-u 3 a =}
25602 or @samp{a = b = c}) tests if all of its arguments are equal. In
25603 algebraic notation, the @samp{=} operator is unusual in that it is
25604 neither left- nor right-associative: @samp{a = b = c} is not the
25605 same as @samp{(a = b) = c} or @samp{a = (b = c)} (which each compare
25606 one variable with the 1 or 0 that results from comparing two other
25610 @pindex calc-not-equal-to
25613 The @kbd{a #} (@code{calc-not-equal-to}) command, or @samp{neq(a,b)} or
25614 @samp{a != b} function, is true if @expr{a} and @expr{b} are not equal.
25615 This also works with more than two arguments; @samp{a != b != c != d}
25616 tests that all four of @expr{a}, @expr{b}, @expr{c}, and @expr{d} are
25633 @pindex calc-less-than
25634 @pindex calc-greater-than
25635 @pindex calc-less-equal
25636 @pindex calc-greater-equal
25665 The @kbd{a <} (@code{calc-less-than}) [@samp{lt(a,b)} or @samp{a < b}]
25666 operation is true if @expr{a} is less than @expr{b}. Similar functions
25667 are @kbd{a >} (@code{calc-greater-than}) [@samp{gt(a,b)} or @samp{a > b}],
25668 @kbd{a [} (@code{calc-less-equal}) [@samp{leq(a,b)} or @samp{a <= b}], and
25669 @kbd{a ]} (@code{calc-greater-equal}) [@samp{geq(a,b)} or @samp{a >= b}].
25671 While the inequality functions like @code{lt} do not accept more
25672 than two arguments, the syntax @w{@samp{a <= b < c}} is translated to an
25673 equivalent expression involving intervals: @samp{b in [a .. c)}.
25674 (See the description of @code{in} below.) All four combinations
25675 of @samp{<} and @samp{<=} are allowed, or any of the four combinations
25676 of @samp{>} and @samp{>=}. Four-argument constructions like
25677 @samp{a < b < c < d}, and mixtures like @w{@samp{a < b = c}} that
25678 involve both equalities and inequalities, are not allowed.
25681 @pindex calc-remove-equal
25683 The @kbd{a .} (@code{calc-remove-equal}) [@code{rmeq}] command extracts
25684 the righthand side of the equation or inequality on the top of the
25685 stack. It also works elementwise on vectors. For example, if
25686 @samp{[x = 2.34, y = z / 2]} is on the stack, then @kbd{a .} produces
25687 @samp{[2.34, z / 2]}. As a special case, if the righthand side is a
25688 variable and the lefthand side is a number (as in @samp{2.34 = x}), then
25689 Calc keeps the lefthand side instead. Finally, this command works with
25690 assignments @samp{x := 2.34} as well as equations, always taking the
25691 the righthand side, and for @samp{=>} (evaluates-to) operators, always
25692 taking the lefthand side.
25695 @pindex calc-logical-and
25698 The @kbd{a &} (@code{calc-logical-and}) [@samp{land(a,b)} or @samp{a && b}]
25699 function is true if both of its arguments are true, i.e., are
25700 non-zero numbers. In this case, the result will be either @expr{a} or
25701 @expr{b}, chosen arbitrarily. If either argument is zero, the result is
25702 zero. Otherwise, the formula is left in symbolic form.
25705 @pindex calc-logical-or
25708 The @kbd{a |} (@code{calc-logical-or}) [@samp{lor(a,b)} or @samp{a || b}]
25709 function is true if either or both of its arguments are true (nonzero).
25710 The result is whichever argument was nonzero, choosing arbitrarily if both
25711 are nonzero. If both @expr{a} and @expr{b} are zero, the result is
25715 @pindex calc-logical-not
25718 The @kbd{a !} (@code{calc-logical-not}) [@samp{lnot(a)} or @samp{!@: a}]
25719 function is true if @expr{a} is false (zero), or false if @expr{a} is
25720 true (nonzero). It is left in symbolic form if @expr{a} is not a
25724 @pindex calc-logical-if
25734 @cindex Arguments, not evaluated
25735 The @kbd{a :} (@code{calc-logical-if}) [@samp{if(a,b,c)} or @samp{a ? b :@: c}]
25736 function is equal to either @expr{b} or @expr{c} if @expr{a} is a nonzero
25737 number or zero, respectively. If @expr{a} is not a number, the test is
25738 left in symbolic form and neither @expr{b} nor @expr{c} is evaluated in
25739 any way. In algebraic formulas, this is one of the few Calc functions
25740 whose arguments are not automatically evaluated when the function itself
25741 is evaluated. The others are @code{lambda}, @code{quote}, and
25744 One minor surprise to watch out for is that the formula @samp{a?3:4}
25745 will not work because the @samp{3:4} is parsed as a fraction instead of
25746 as three separate symbols. Type something like @samp{a ? 3 : 4} or
25747 @samp{a?(3):4} instead.
25749 As a special case, if @expr{a} evaluates to a vector, then both @expr{b}
25750 and @expr{c} are evaluated; the result is a vector of the same length
25751 as @expr{a} whose elements are chosen from corresponding elements of
25752 @expr{b} and @expr{c} according to whether each element of @expr{a}
25753 is zero or nonzero. Each of @expr{b} and @expr{c} must be either a
25754 vector of the same length as @expr{a}, or a non-vector which is matched
25755 with all elements of @expr{a}.
25758 @pindex calc-in-set
25760 The @kbd{a @{} (@code{calc-in-set}) [@samp{in(a,b)}] function is true if
25761 the number @expr{a} is in the set of numbers represented by @expr{b}.
25762 If @expr{b} is an interval form, @expr{a} must be one of the values
25763 encompassed by the interval. If @expr{b} is a vector, @expr{a} must be
25764 equal to one of the elements of the vector. (If any vector elements are
25765 intervals, @expr{a} must be in any of the intervals.) If @expr{b} is a
25766 plain number, @expr{a} must be numerically equal to @expr{b}.
25767 @xref{Set Operations}, for a group of commands that manipulate sets
25774 The @samp{typeof(a)} function produces an integer or variable which
25775 characterizes @expr{a}. If @expr{a} is a number, vector, or variable,
25776 the result will be one of the following numbers:
25781 3 Floating-point number
25783 5 Rectangular complex number
25784 6 Polar complex number
25790 12 Infinity (inf, uinf, or nan)
25792 101 Vector (but not a matrix)
25796 Otherwise, @expr{a} is a formula, and the result is a variable which
25797 represents the name of the top-level function call.
25811 The @samp{integer(a)} function returns true if @expr{a} is an integer.
25812 The @samp{real(a)} function
25813 is true if @expr{a} is a real number, either integer, fraction, or
25814 float. The @samp{constant(a)} function returns true if @expr{a} is
25815 any of the objects for which @code{typeof} would produce an integer
25816 code result except for variables, and provided that the components of
25817 an object like a vector or error form are themselves constant.
25818 Note that infinities do not satisfy any of these tests, nor do
25819 special constants like @code{pi} and @code{e}.
25821 @xref{Declarations}, for a set of similar functions that recognize
25822 formulas as well as actual numbers. For example, @samp{dint(floor(x))}
25823 is true because @samp{floor(x)} is provably integer-valued, but
25824 @samp{integer(floor(x))} does not because @samp{floor(x)} is not
25825 literally an integer constant.
25831 The @samp{refers(a,b)} function is true if the variable (or sub-expression)
25832 @expr{b} appears in @expr{a}, or false otherwise. Unlike the other
25833 tests described here, this function returns a definite ``no'' answer
25834 even if its arguments are still in symbolic form. The only case where
25835 @code{refers} will be left unevaluated is if @expr{a} is a plain
25836 variable (different from @expr{b}).
25842 The @samp{negative(a)} function returns true if @expr{a} ``looks'' negative,
25843 because it is a negative number, because it is of the form @expr{-x},
25844 or because it is a product or quotient with a term that looks negative.
25845 This is most useful in rewrite rules. Beware that @samp{negative(a)}
25846 evaluates to 1 or 0 for @emph{any} argument @expr{a}, so it can only
25847 be stored in a formula if the default simplifications are turned off
25848 first with @kbd{m O} (or if it appears in an unevaluated context such
25849 as a rewrite rule condition).
25855 The @samp{variable(a)} function is true if @expr{a} is a variable,
25856 or false if not. If @expr{a} is a function call, this test is left
25857 in symbolic form. Built-in variables like @code{pi} and @code{inf}
25858 are considered variables like any others by this test.
25864 The @samp{nonvar(a)} function is true if @expr{a} is a non-variable.
25865 If its argument is a variable it is left unsimplified; it never
25866 actually returns zero. However, since Calc's condition-testing
25867 commands consider ``false'' anything not provably true, this is
25886 @cindex Linearity testing
25887 The functions @code{lin}, @code{linnt}, @code{islin}, and @code{islinnt}
25888 check if an expression is ``linear,'' i.e., can be written in the form
25889 @expr{a + b x} for some constants @expr{a} and @expr{b}, and some
25890 variable or subformula @expr{x}. The function @samp{islin(f,x)} checks
25891 if formula @expr{f} is linear in @expr{x}, returning 1 if so. For
25892 example, @samp{islin(x,x)}, @samp{islin(-x,x)}, @samp{islin(3,x)}, and
25893 @samp{islin(x y / 3 - 2, x)} all return 1. The @samp{lin(f,x)} function
25894 is similar, except that instead of returning 1 it returns the vector
25895 @expr{[a, b, x]}. For the above examples, this vector would be
25896 @expr{[0, 1, x]}, @expr{[0, -1, x]}, @expr{[3, 0, x]}, and
25897 @expr{[-2, y/3, x]}, respectively. Both @code{lin} and @code{islin}
25898 generally remain unevaluated for expressions which are not linear,
25899 e.g., @samp{lin(2 x^2, x)} and @samp{lin(sin(x), x)}. The second
25900 argument can also be a formula; @samp{islin(2 + 3 sin(x), sin(x))}
25903 The @code{linnt} and @code{islinnt} functions perform a similar check,
25904 but require a ``non-trivial'' linear form, which means that the
25905 @expr{b} coefficient must be non-zero. For example, @samp{lin(2,x)}
25906 returns @expr{[2, 0, x]} and @samp{lin(y,x)} returns @expr{[y, 0, x]},
25907 but @samp{linnt(2,x)} and @samp{linnt(y,x)} are left unevaluated
25908 (in other words, these formulas are considered to be only ``trivially''
25909 linear in @expr{x}).
25911 All four linearity-testing functions allow you to omit the second
25912 argument, in which case the input may be linear in any non-constant
25913 formula. Here, the @expr{a=0}, @expr{b=1} case is also considered
25914 trivial, and only constant values for @expr{a} and @expr{b} are
25915 recognized. Thus, @samp{lin(2 x y)} returns @expr{[0, 2, x y]},
25916 @samp{lin(2 - x y)} returns @expr{[2, -1, x y]}, and @samp{lin(x y)}
25917 returns @expr{[0, 1, x y]}. The @code{linnt} function would allow the
25918 first two cases but not the third. Also, neither @code{lin} nor
25919 @code{linnt} accept plain constants as linear in the one-argument
25920 case: @samp{islin(2,x)} is true, but @samp{islin(2)} is false.
25926 The @samp{istrue(a)} function returns 1 if @expr{a} is a nonzero
25927 number or provably nonzero formula, or 0 if @expr{a} is anything else.
25928 Calls to @code{istrue} can only be manipulated if @kbd{m O} mode is
25929 used to make sure they are not evaluated prematurely. (Note that
25930 declarations are used when deciding whether a formula is true;
25931 @code{istrue} returns 1 when @code{dnonzero} would return 1, and
25932 it returns 0 when @code{dnonzero} would return 0 or leave itself
25935 @node Rewrite Rules, , Logical Operations, Algebra
25936 @section Rewrite Rules
25939 @cindex Rewrite rules
25940 @cindex Transformations
25941 @cindex Pattern matching
25943 @pindex calc-rewrite
25945 The @kbd{a r} (@code{calc-rewrite}) [@code{rewrite}] command makes
25946 substitutions in a formula according to a specified pattern or patterns
25947 known as @dfn{rewrite rules}. Whereas @kbd{a b} (@code{calc-substitute})
25948 matches literally, so that substituting @samp{sin(x)} with @samp{cos(x)}
25949 matches only the @code{sin} function applied to the variable @code{x},
25950 rewrite rules match general kinds of formulas; rewriting using the rule
25951 @samp{sin(x) := cos(x)} matches @code{sin} of any argument and replaces
25952 it with @code{cos} of that same argument. The only significance of the
25953 name @code{x} is that the same name is used on both sides of the rule.
25955 Rewrite rules rearrange formulas already in Calc's memory.
25956 @xref{Syntax Tables}, to read about @dfn{syntax rules}, which are
25957 similar to algebraic rewrite rules but operate when new algebraic
25958 entries are being parsed, converting strings of characters into
25962 * Entering Rewrite Rules::
25963 * Basic Rewrite Rules::
25964 * Conditional Rewrite Rules::
25965 * Algebraic Properties of Rewrite Rules::
25966 * Other Features of Rewrite Rules::
25967 * Composing Patterns in Rewrite Rules::
25968 * Nested Formulas with Rewrite Rules::
25969 * Multi-Phase Rewrite Rules::
25970 * Selections with Rewrite Rules::
25971 * Matching Commands::
25972 * Automatic Rewrites::
25973 * Debugging Rewrites::
25974 * Examples of Rewrite Rules::
25977 @node Entering Rewrite Rules, Basic Rewrite Rules, Rewrite Rules, Rewrite Rules
25978 @subsection Entering Rewrite Rules
25981 Rewrite rules normally use the ``assignment'' operator
25982 @samp{@var{old} := @var{new}}.
25983 This operator is equivalent to the function call @samp{assign(old, new)}.
25984 The @code{assign} function is undefined by itself in Calc, so an
25985 assignment formula such as a rewrite rule will be left alone by ordinary
25986 Calc commands. But certain commands, like the rewrite system, interpret
25987 assignments in special ways.
25989 For example, the rule @samp{sin(x)^2 := 1-cos(x)^2} says to replace
25990 every occurrence of the sine of something, squared, with one minus the
25991 square of the cosine of that same thing. All by itself as a formula
25992 on the stack it does nothing, but when given to the @kbd{a r} command
25993 it turns that command into a sine-squared-to-cosine-squared converter.
25995 To specify a set of rules to be applied all at once, make a vector of
25998 When @kbd{a r} prompts you to enter the rewrite rules, you can answer
26003 With a rule: @kbd{f(x) := g(x) @key{RET}}.
26005 With a vector of rules: @kbd{[f1(x) := g1(x), f2(x) := g2(x)] @key{RET}}.
26006 (You can omit the enclosing square brackets if you wish.)
26008 With the name of a variable that contains the rule or rules vector:
26009 @kbd{myrules @key{RET}}.
26011 With any formula except a rule, a vector, or a variable name; this
26012 will be interpreted as the @var{old} half of a rewrite rule,
26013 and you will be prompted a second time for the @var{new} half:
26014 @kbd{f(x) @key{RET} g(x) @key{RET}}.
26016 With a blank line, in which case the rule, rules vector, or variable
26017 will be taken from the top of the stack (and the formula to be
26018 rewritten will come from the second-to-top position).
26021 If you enter the rules directly (as opposed to using rules stored
26022 in a variable), those rules will be put into the Trail so that you
26023 can retrieve them later. @xref{Trail Commands}.
26025 It is most convenient to store rules you use often in a variable and
26026 invoke them by giving the variable name. The @kbd{s e}
26027 (@code{calc-edit-variable}) command is an easy way to create or edit a
26028 rule set stored in a variable. You may also wish to use @kbd{s p}
26029 (@code{calc-permanent-variable}) to save your rules permanently;
26030 @pxref{Operations on Variables}.
26032 Rewrite rules are compiled into a special internal form for faster
26033 matching. If you enter a rule set directly it must be recompiled
26034 every time. If you store the rules in a variable and refer to them
26035 through that variable, they will be compiled once and saved away
26036 along with the variable for later reference. This is another good
26037 reason to store your rules in a variable.
26039 Calc also accepts an obsolete notation for rules, as vectors
26040 @samp{[@var{old}, @var{new}]}. But because it is easily confused with a
26041 vector of two rules, the use of this notation is no longer recommended.
26043 @node Basic Rewrite Rules, Conditional Rewrite Rules, Entering Rewrite Rules, Rewrite Rules
26044 @subsection Basic Rewrite Rules
26047 To match a particular formula @expr{x} with a particular rewrite rule
26048 @samp{@var{old} := @var{new}}, Calc compares the structure of @expr{x} with
26049 the structure of @var{old}. Variables that appear in @var{old} are
26050 treated as @dfn{meta-variables}; the corresponding positions in @expr{x}
26051 may contain any sub-formulas. For example, the pattern @samp{f(x,y)}
26052 would match the expression @samp{f(12, a+1)} with the meta-variable
26053 @samp{x} corresponding to 12 and with @samp{y} corresponding to
26054 @samp{a+1}. However, this pattern would not match @samp{f(12)} or
26055 @samp{g(12, a+1)}, since there is no assignment of the meta-variables
26056 that will make the pattern match these expressions. Notice that if
26057 the pattern is a single meta-variable, it will match any expression.
26059 If a given meta-variable appears more than once in @var{old}, the
26060 corresponding sub-formulas of @expr{x} must be identical. Thus
26061 the pattern @samp{f(x,x)} would match @samp{f(12, 12)} and
26062 @samp{f(a+1, a+1)} but not @samp{f(12, a+1)} or @samp{f(a+b, b+a)}.
26063 (@xref{Conditional Rewrite Rules}, for a way to match the latter.)
26065 Things other than variables must match exactly between the pattern
26066 and the target formula. To match a particular variable exactly, use
26067 the pseudo-function @samp{quote(v)} in the pattern. For example, the
26068 pattern @samp{x+quote(y)} matches @samp{x+y}, @samp{2+y}, or
26071 The special variable names @samp{e}, @samp{pi}, @samp{i}, @samp{phi},
26072 @samp{gamma}, @samp{inf}, @samp{uinf}, and @samp{nan} always match
26073 literally. Thus the pattern @samp{sin(d + e + f)} acts exactly like
26074 @samp{sin(d + quote(e) + f)}.
26076 If the @var{old} pattern is found to match a given formula, that
26077 formula is replaced by @var{new}, where any occurrences in @var{new}
26078 of meta-variables from the pattern are replaced with the sub-formulas
26079 that they matched. Thus, applying the rule @samp{f(x,y) := g(y+x,x)}
26080 to @samp{f(12, a+1)} would produce @samp{g(a+13, 12)}.
26082 The normal @kbd{a r} command applies rewrite rules over and over
26083 throughout the target formula until no further changes are possible
26084 (up to a limit of 100 times). Use @kbd{C-u 1 a r} to make only one
26087 @node Conditional Rewrite Rules, Algebraic Properties of Rewrite Rules, Basic Rewrite Rules, Rewrite Rules
26088 @subsection Conditional Rewrite Rules
26091 A rewrite rule can also be @dfn{conditional}, written in the form
26092 @samp{@var{old} := @var{new} :: @var{cond}}. (There is also the obsolete
26093 form @samp{[@var{old}, @var{new}, @var{cond}]}.) If a @var{cond} part
26095 rule, this is an additional condition that must be satisfied before
26096 the rule is accepted. Once @var{old} has been successfully matched
26097 to the target expression, @var{cond} is evaluated (with all the
26098 meta-variables substituted for the values they matched) and simplified
26099 with @kbd{a s} (@code{calc-simplify}). If the result is a nonzero
26100 number or any other object known to be nonzero (@pxref{Declarations}),
26101 the rule is accepted. If the result is zero or if it is a symbolic
26102 formula that is not known to be nonzero, the rule is rejected.
26103 @xref{Logical Operations}, for a number of functions that return
26104 1 or 0 according to the results of various tests.
26106 For example, the formula @samp{n > 0} simplifies to 1 or 0 if @expr{n}
26107 is replaced by a positive or nonpositive number, respectively (or if
26108 @expr{n} has been declared to be positive or nonpositive). Thus,
26109 the rule @samp{f(x,y) := g(y+x,x) :: x+y > 0} would apply to
26110 @samp{f(0, 4)} but not to @samp{f(-3, 2)} or @samp{f(12, a+1)}
26111 (assuming no outstanding declarations for @expr{a}). In the case of
26112 @samp{f(-3, 2)}, the condition can be shown not to be satisfied; in
26113 the case of @samp{f(12, a+1)}, the condition merely cannot be shown
26114 to be satisfied, but that is enough to reject the rule.
26116 While Calc will use declarations to reason about variables in the
26117 formula being rewritten, declarations do not apply to meta-variables.
26118 For example, the rule @samp{f(a) := g(a+1)} will match for any values
26119 of @samp{a}, such as complex numbers, vectors, or formulas, even if
26120 @samp{a} has been declared to be real or scalar. If you want the
26121 meta-variable @samp{a} to match only literal real numbers, use
26122 @samp{f(a) := g(a+1) :: real(a)}. If you want @samp{a} to match only
26123 reals and formulas which are provably real, use @samp{dreal(a)} as
26126 The @samp{::} operator is a shorthand for the @code{condition}
26127 function; @samp{@var{old} := @var{new} :: @var{cond}} is equivalent to
26128 the formula @samp{condition(assign(@var{old}, @var{new}), @var{cond})}.
26130 If you have several conditions, you can use @samp{... :: c1 :: c2 :: c3}
26131 or @samp{... :: c1 && c2 && c3}. The two are entirely equivalent.
26133 It is also possible to embed conditions inside the pattern:
26134 @samp{f(x :: x>0, y) := g(y+x, x)}. This is purely a notational
26135 convenience, though; where a condition appears in a rule has no
26136 effect on when it is tested. The rewrite-rule compiler automatically
26137 decides when it is best to test each condition while a rule is being
26140 Certain conditions are handled as special cases by the rewrite rule
26141 system and are tested very efficiently: Where @expr{x} is any
26142 meta-variable, these conditions are @samp{integer(x)}, @samp{real(x)},
26143 @samp{constant(x)}, @samp{negative(x)}, @samp{x >= y} where @expr{y}
26144 is either a constant or another meta-variable and @samp{>=} may be
26145 replaced by any of the six relational operators, and @samp{x % a = b}
26146 where @expr{a} and @expr{b} are constants. Other conditions, like
26147 @samp{x >= y+1} or @samp{dreal(x)}, will be less efficient to check
26148 since Calc must bring the whole evaluator and simplifier into play.
26150 An interesting property of @samp{::} is that neither of its arguments
26151 will be touched by Calc's default simplifications. This is important
26152 because conditions often are expressions that cannot safely be
26153 evaluated early. For example, the @code{typeof} function never
26154 remains in symbolic form; entering @samp{typeof(a)} will put the
26155 number 100 (the type code for variables like @samp{a}) on the stack.
26156 But putting the condition @samp{... :: typeof(a) = 6} on the stack
26157 is safe since @samp{::} prevents the @code{typeof} from being
26158 evaluated until the condition is actually used by the rewrite system.
26160 Since @samp{::} protects its lefthand side, too, you can use a dummy
26161 condition to protect a rule that must itself not evaluate early.
26162 For example, it's not safe to put @samp{a(f,x) := apply(f, [x])} on
26163 the stack because it will immediately evaluate to @samp{a(f,x) := f(x)},
26164 where the meta-variable-ness of @code{f} on the righthand side has been
26165 lost. But @samp{a(f,x) := apply(f, [x]) :: 1} is safe, and of course
26166 the condition @samp{1} is always true (nonzero) so it has no effect on
26167 the functioning of the rule. (The rewrite compiler will ensure that
26168 it doesn't even impact the speed of matching the rule.)
26170 @node Algebraic Properties of Rewrite Rules, Other Features of Rewrite Rules, Conditional Rewrite Rules, Rewrite Rules
26171 @subsection Algebraic Properties of Rewrite Rules
26174 The rewrite mechanism understands the algebraic properties of functions
26175 like @samp{+} and @samp{*}. In particular, pattern matching takes
26176 the associativity and commutativity of the following functions into
26180 + - * = != && || and or xor vint vunion vxor gcd lcm max min beta
26183 For example, the rewrite rule:
26186 a x + b x := (a + b) x
26190 will match formulas of the form,
26193 a x + b x, x a + x b, a x + x b, x a + b x
26196 Rewrites also understand the relationship between the @samp{+} and @samp{-}
26197 operators. The above rewrite rule will also match the formulas,
26200 a x - b x, x a - x b, a x - x b, x a - b x
26204 by matching @samp{b} in the pattern to @samp{-b} from the formula.
26206 Applied to a sum of many terms like @samp{r + a x + s + b x + t}, this
26207 pattern will check all pairs of terms for possible matches. The rewrite
26208 will take whichever suitable pair it discovers first.
26210 In general, a pattern using an associative operator like @samp{a + b}
26211 will try @var{2 n} different ways to match a sum of @var{n} terms
26212 like @samp{x + y + z - w}. First, @samp{a} is matched against each
26213 of @samp{x}, @samp{y}, @samp{z}, and @samp{-w} in turn, with @samp{b}
26214 being matched to the remainders @samp{y + z - w}, @samp{x + z - w}, etc.
26215 If none of these succeed, then @samp{b} is matched against each of the
26216 four terms with @samp{a} matching the remainder. Half-and-half matches,
26217 like @samp{(x + y) + (z - w)}, are not tried.
26219 Note that @samp{*} is not commutative when applied to matrices, but
26220 rewrite rules pretend that it is. If you type @kbd{m v} to enable
26221 Matrix mode (@pxref{Matrix Mode}), rewrite rules will match @samp{*}
26222 literally, ignoring its usual commutativity property. (In the
26223 current implementation, the associativity also vanishes---it is as
26224 if the pattern had been enclosed in a @code{plain} marker; see below.)
26225 If you are applying rewrites to formulas with matrices, it's best to
26226 enable Matrix mode first to prevent algebraically incorrect rewrites
26229 The pattern @samp{-x} will actually match any expression. For example,
26237 will rewrite @samp{f(a)} to @samp{-f(-a)}. To avoid this, either use
26238 a @code{plain} marker as described below, or add a @samp{negative(x)}
26239 condition. The @code{negative} function is true if its argument
26240 ``looks'' negative, for example, because it is a negative number or
26241 because it is a formula like @samp{-x}. The new rule using this
26245 f(x) := -f(-x) :: negative(x) @r{or, equivalently,}
26246 f(-x) := -f(x) :: negative(-x)
26249 In the same way, the pattern @samp{x - y} will match the sum @samp{a + b}
26250 by matching @samp{y} to @samp{-b}.
26252 The pattern @samp{a b} will also match the formula @samp{x/y} if
26253 @samp{y} is a number. Thus the rule @samp{a x + @w{b x} := (a+b) x}
26254 will also convert @samp{a x + x / 2} to @samp{(a + 0.5) x} (or
26255 @samp{(a + 1:2) x}, depending on the current fraction mode).
26257 Calc will @emph{not} take other liberties with @samp{*}, @samp{/}, and
26258 @samp{^}. For example, the pattern @samp{f(a b)} will not match
26259 @samp{f(x^2)}, and @samp{f(a + b)} will not match @samp{f(2 x)}, even
26260 though conceivably these patterns could match with @samp{a = b = x}.
26261 Nor will @samp{f(a b)} match @samp{f(x / y)} if @samp{y} is not a
26262 constant, even though it could be considered to match with @samp{a = x}
26263 and @samp{b = 1/y}. The reasons are partly for efficiency, and partly
26264 because while few mathematical operations are substantively different
26265 for addition and subtraction, often it is preferable to treat the cases
26266 of multiplication, division, and integer powers separately.
26268 Even more subtle is the rule set
26271 [ f(a) + f(b) := f(a + b), -f(a) := f(-a) ]
26275 attempting to match @samp{f(x) - f(y)}. You might think that Calc
26276 will view this subtraction as @samp{f(x) + (-f(y))} and then apply
26277 the above two rules in turn, but actually this will not work because
26278 Calc only does this when considering rules for @samp{+} (like the
26279 first rule in this set). So it will see first that @samp{f(x) + (-f(y))}
26280 does not match @samp{f(a) + f(b)} for any assignments of the
26281 meta-variables, and then it will see that @samp{f(x) - f(y)} does
26282 not match @samp{-f(a)} for any assignment of @samp{a}. Because Calc
26283 tries only one rule at a time, it will not be able to rewrite
26284 @samp{f(x) - f(y)} with this rule set. An explicit @samp{f(a) - f(b)}
26285 rule will have to be added.
26287 Another thing patterns will @emph{not} do is break up complex numbers.
26288 The pattern @samp{myconj(a + @w{b i)} := a - b i} will work for formulas
26289 involving the special constant @samp{i} (such as @samp{3 - 4 i}), but
26290 it will not match actual complex numbers like @samp{(3, -4)}. A version
26291 of the above rule for complex numbers would be
26294 myconj(a) := re(a) - im(a) (0,1) :: im(a) != 0
26298 (Because the @code{re} and @code{im} functions understand the properties
26299 of the special constant @samp{i}, this rule will also work for
26300 @samp{3 - 4 i}. In fact, this particular rule would probably be better
26301 without the @samp{im(a) != 0} condition, since if @samp{im(a) = 0} the
26302 righthand side of the rule will still give the correct answer for the
26303 conjugate of a real number.)
26305 It is also possible to specify optional arguments in patterns. The rule
26308 opt(a) x + opt(b) (x^opt(c) + opt(d)) := f(a, b, c, d)
26312 will match the formula
26319 in a fairly straightforward manner, but it will also match reduced
26323 x + x^2, 2(x + 1) - x, x + x
26327 producing, respectively,
26330 f(1, 1, 2, 0), f(-1, 2, 1, 1), f(1, 1, 1, 0)
26333 (The latter two formulas can be entered only if default simplifications
26334 have been turned off with @kbd{m O}.)
26336 The default value for a term of a sum is zero. The default value
26337 for a part of a product, for a power, or for the denominator of a
26338 quotient, is one. Also, @samp{-x} matches the pattern @samp{opt(a) b}
26339 with @samp{a = -1}.
26341 In particular, the distributive-law rule can be refined to
26344 opt(a) x + opt(b) x := (a + b) x
26348 so that it will convert, e.g., @samp{a x - x}, to @samp{(a - 1) x}.
26350 The pattern @samp{opt(a) + opt(b) x} matches almost any formulas which
26351 are linear in @samp{x}. You can also use the @code{lin} and @code{islin}
26352 functions with rewrite conditions to test for this; @pxref{Logical
26353 Operations}. These functions are not as convenient to use in rewrite
26354 rules, but they recognize more kinds of formulas as linear:
26355 @samp{x/z} is considered linear with @expr{b = 1/z} by @code{lin},
26356 but it will not match the above pattern because that pattern calls
26357 for a multiplication, not a division.
26359 As another example, the obvious rule to replace @samp{sin(x)^2 + cos(x)^2}
26363 sin(x)^2 + cos(x)^2 := 1
26367 misses many cases because the sine and cosine may both be multiplied by
26368 an equal factor. Here's a more successful rule:
26371 opt(a) sin(x)^2 + opt(a) cos(x)^2 := a
26374 Note that this rule will @emph{not} match @samp{sin(x)^2 + 6 cos(x)^2}
26375 because one @expr{a} would have ``matched'' 1 while the other matched 6.
26377 Calc automatically converts a rule like
26387 f(temp, x) := g(x) :: temp = x-1
26391 (where @code{temp} stands for a new, invented meta-variable that
26392 doesn't actually have a name). This modified rule will successfully
26393 match @samp{f(6, 7)}, binding @samp{temp} and @samp{x} to 6 and 7,
26394 respectively, then verifying that they differ by one even though
26395 @samp{6} does not superficially look like @samp{x-1}.
26397 However, Calc does not solve equations to interpret a rule. The
26401 f(x-1, x+1) := g(x)
26405 will not work. That is, it will match @samp{f(a - 1 + b, a + 1 + b)}
26406 but not @samp{f(6, 8)}. Calc always interprets at least one occurrence
26407 of a variable by literal matching. If the variable appears ``isolated''
26408 then Calc is smart enough to use it for literal matching. But in this
26409 last example, Calc is forced to rewrite the rule to @samp{f(x-1, temp)
26410 := g(x) :: temp = x+1} where the @samp{x-1} term must correspond to an
26411 actual ``something-minus-one'' in the target formula.
26413 A successful way to write this would be @samp{f(x, x+2) := g(x+1)}.
26414 You could make this resemble the original form more closely by using
26415 @code{let} notation, which is described in the next section:
26418 f(xm1, x+1) := g(x) :: let(x := xm1+1)
26421 Calc does this rewriting or ``conditionalizing'' for any sub-pattern
26422 which involves only the functions in the following list, operating
26423 only on constants and meta-variables which have already been matched
26424 elsewhere in the pattern. When matching a function call, Calc is
26425 careful to match arguments which are plain variables before arguments
26426 which are calls to any of the functions below, so that a pattern like
26427 @samp{f(x-1, x)} can be conditionalized even though the isolated
26428 @samp{x} comes after the @samp{x-1}.
26431 + - * / \ % ^ abs sign round rounde roundu trunc floor ceil
26432 max min re im conj arg
26435 You can suppress all of the special treatments described in this
26436 section by surrounding a function call with a @code{plain} marker.
26437 This marker causes the function call which is its argument to be
26438 matched literally, without regard to commutativity, associativity,
26439 negation, or conditionalization. When you use @code{plain}, the
26440 ``deep structure'' of the formula being matched can show through.
26444 plain(a - a b) := f(a, b)
26448 will match only literal subtractions. However, the @code{plain}
26449 marker does not affect its arguments' arguments. In this case,
26450 commutativity and associativity is still considered while matching
26451 the @w{@samp{a b}} sub-pattern, so the whole pattern will match
26452 @samp{x - y x} as well as @samp{x - x y}. We could go still
26456 plain(a - plain(a b)) := f(a, b)
26460 which would do a completely strict match for the pattern.
26462 By contrast, the @code{quote} marker means that not only the
26463 function name but also the arguments must be literally the same.
26464 The above pattern will match @samp{x - x y} but
26467 quote(a - a b) := f(a, b)
26471 will match only the single formula @samp{a - a b}. Also,
26474 quote(a - quote(a b)) := f(a, b)
26478 will match only @samp{a - quote(a b)}---probably not the desired
26481 A certain amount of algebra is also done when substituting the
26482 meta-variables on the righthand side of a rule. For example,
26486 a + f(b) := f(a + b)
26490 matching @samp{f(x) - y} would produce @samp{f((-y) + x)} if
26491 taken literally, but the rewrite mechanism will simplify the
26492 righthand side to @samp{f(x - y)} automatically. (Of course,
26493 the default simplifications would do this anyway, so this
26494 special simplification is only noticeable if you have turned the
26495 default simplifications off.) This rewriting is done only when
26496 a meta-variable expands to a ``negative-looking'' expression.
26497 If this simplification is not desirable, you can use a @code{plain}
26498 marker on the righthand side:
26501 a + f(b) := f(plain(a + b))
26505 In this example, we are still allowing the pattern-matcher to
26506 use all the algebra it can muster, but the righthand side will
26507 always simplify to a literal addition like @samp{f((-y) + x)}.
26509 @node Other Features of Rewrite Rules, Composing Patterns in Rewrite Rules, Algebraic Properties of Rewrite Rules, Rewrite Rules
26510 @subsection Other Features of Rewrite Rules
26513 Certain ``function names'' serve as markers in rewrite rules.
26514 Here is a complete list of these markers. First are listed the
26515 markers that work inside a pattern; then come the markers that
26516 work in the righthand side of a rule.
26522 One kind of marker, @samp{import(x)}, takes the place of a whole
26523 rule. Here @expr{x} is the name of a variable containing another
26524 rule set; those rules are ``spliced into'' the rule set that
26525 imports them. For example, if @samp{[f(a+b) := f(a) + f(b),
26526 f(a b) := a f(b) :: real(a)]} is stored in variable @samp{linearF},
26527 then the rule set @samp{[f(0) := 0, import(linearF)]} will apply
26528 all three rules. It is possible to modify the imported rules
26529 slightly: @samp{import(x, v1, x1, v2, x2, @dots{})} imports
26530 the rule set @expr{x} with all occurrences of
26531 @texline @math{v_1},
26532 @infoline @expr{v1},
26533 as either a variable name or a function name, replaced with
26534 @texline @math{x_1}
26535 @infoline @expr{x1}
26537 @texline @math{v_1}
26538 @infoline @expr{v1}
26539 is used as a function name, then
26540 @texline @math{x_1}
26541 @infoline @expr{x1}
26542 must be either a function name itself or a @w{@samp{< >}} nameless
26543 function; @pxref{Specifying Operators}.) For example, @samp{[g(0) := 0,
26544 import(linearF, f, g)]} applies the linearity rules to the function
26545 @samp{g} instead of @samp{f}. Imports can be nested, but the
26546 import-with-renaming feature may fail to rename sub-imports properly.
26548 The special functions allowed in patterns are:
26556 This pattern matches exactly @expr{x}; variable names in @expr{x} are
26557 not interpreted as meta-variables. The only flexibility is that
26558 numbers are compared for numeric equality, so that the pattern
26559 @samp{f(quote(12))} will match both @samp{f(12)} and @samp{f(12.0)}.
26560 (Numbers are always treated this way by the rewrite mechanism:
26561 The rule @samp{f(x,x) := g(x)} will match @samp{f(12, 12.0)}.
26562 The rewrite may produce either @samp{g(12)} or @samp{g(12.0)}
26563 as a result in this case.)
26570 Here @expr{x} must be a function call @samp{f(x1,x2,@dots{})}. This
26571 pattern matches a call to function @expr{f} with the specified
26572 argument patterns. No special knowledge of the properties of the
26573 function @expr{f} is used in this case; @samp{+} is not commutative or
26574 associative. Unlike @code{quote}, the arguments @samp{x1,x2,@dots{}}
26575 are treated as patterns. If you wish them to be treated ``plainly''
26576 as well, you must enclose them with more @code{plain} markers:
26577 @samp{plain(plain(@w{-a}) + plain(b c))}.
26584 Here @expr{x} must be a variable name. This must appear as an
26585 argument to a function or an element of a vector; it specifies that
26586 the argument or element is optional.
26587 As an argument to @samp{+}, @samp{-}, @samp{*}, @samp{&&}, or @samp{||},
26588 or as the second argument to @samp{/} or @samp{^}, the value @var{def}
26589 may be omitted. The pattern @samp{x + opt(y)} matches a sum by
26590 binding one summand to @expr{x} and the other to @expr{y}, and it
26591 matches anything else by binding the whole expression to @expr{x} and
26592 zero to @expr{y}. The other operators above work similarly.
26594 For general miscellaneous functions, the default value @code{def}
26595 must be specified. Optional arguments are dropped starting with
26596 the rightmost one during matching. For example, the pattern
26597 @samp{f(opt(a,0), b, opt(c,b))} will match @samp{f(b)}, @samp{f(a,b)},
26598 or @samp{f(a,b,c)}. Default values of zero and @expr{b} are
26599 supplied in this example for the omitted arguments. Note that
26600 the literal variable @expr{b} will be the default in the latter
26601 case, @emph{not} the value that matched the meta-variable @expr{b}.
26602 In other words, the default @var{def} is effectively quoted.
26604 @item condition(x,c)
26610 This matches the pattern @expr{x}, with the attached condition
26611 @expr{c}. It is the same as @samp{x :: c}.
26619 This matches anything that matches both pattern @expr{x} and
26620 pattern @expr{y}. It is the same as @samp{x &&& y}.
26621 @pxref{Composing Patterns in Rewrite Rules}.
26629 This matches anything that matches either pattern @expr{x} or
26630 pattern @expr{y}. It is the same as @w{@samp{x ||| y}}.
26638 This matches anything that does not match pattern @expr{x}.
26639 It is the same as @samp{!!! x}.
26645 @tindex cons (rewrites)
26646 This matches any vector of one or more elements. The first
26647 element is matched to @expr{h}; a vector of the remaining
26648 elements is matched to @expr{t}. Note that vectors of fixed
26649 length can also be matched as actual vectors: The rule
26650 @samp{cons(a,cons(b,[])) := cons(a+b,[])} is equivalent
26651 to the rule @samp{[a,b] := [a+b]}.
26657 @tindex rcons (rewrites)
26658 This is like @code{cons}, except that the @emph{last} element
26659 is matched to @expr{h}, with the remaining elements matched
26662 @item apply(f,args)
26666 @tindex apply (rewrites)
26667 This matches any function call. The name of the function, in
26668 the form of a variable, is matched to @expr{f}. The arguments
26669 of the function, as a vector of zero or more objects, are
26670 matched to @samp{args}. Constants, variables, and vectors
26671 do @emph{not} match an @code{apply} pattern. For example,
26672 @samp{apply(f,x)} matches any function call, @samp{apply(quote(f),x)}
26673 matches any call to the function @samp{f}, @samp{apply(f,[a,b])}
26674 matches any function call with exactly two arguments, and
26675 @samp{apply(quote(f), cons(a,cons(b,x)))} matches any call
26676 to the function @samp{f} with two or more arguments. Another
26677 way to implement the latter, if the rest of the rule does not
26678 need to refer to the first two arguments of @samp{f} by name,
26679 would be @samp{apply(quote(f), x :: vlen(x) >= 2)}.
26680 Here's a more interesting sample use of @code{apply}:
26683 apply(f,[x+n]) := n + apply(f,[x])
26684 :: in(f, [floor,ceil,round,trunc]) :: integer(n)
26687 Note, however, that this will be slower to match than a rule
26688 set with four separate rules. The reason is that Calc sorts
26689 the rules of a rule set according to top-level function name;
26690 if the top-level function is @code{apply}, Calc must try the
26691 rule for every single formula and sub-formula. If the top-level
26692 function in the pattern is, say, @code{floor}, then Calc invokes
26693 the rule only for sub-formulas which are calls to @code{floor}.
26695 Formulas normally written with operators like @code{+} are still
26696 considered function calls: @code{apply(f,x)} matches @samp{a+b}
26697 with @samp{f = add}, @samp{x = [a,b]}.
26699 You must use @code{apply} for meta-variables with function names
26700 on both sides of a rewrite rule: @samp{apply(f, [x]) := f(x+1)}
26701 is @emph{not} correct, because it rewrites @samp{spam(6)} into
26702 @samp{f(7)}. The righthand side should be @samp{apply(f, [x+1])}.
26703 Also note that you will have to use No-Simplify mode (@kbd{m O})
26704 when entering this rule so that the @code{apply} isn't
26705 evaluated immediately to get the new rule @samp{f(x) := f(x+1)}.
26706 Or, use @kbd{s e} to enter the rule without going through the stack,
26707 or enter the rule as @samp{apply(f, [x]) := apply(f, [x+1]) @w{:: 1}}.
26708 @xref{Conditional Rewrite Rules}.
26715 This is used for applying rules to formulas with selections;
26716 @pxref{Selections with Rewrite Rules}.
26719 Special functions for the righthand sides of rules are:
26723 The notation @samp{quote(x)} is changed to @samp{x} when the
26724 righthand side is used. As far as the rewrite rule is concerned,
26725 @code{quote} is invisible. However, @code{quote} has the special
26726 property in Calc that its argument is not evaluated. Thus,
26727 while it will not work to put the rule @samp{t(a) := typeof(a)}
26728 on the stack because @samp{typeof(a)} is evaluated immediately
26729 to produce @samp{t(a) := 100}, you can use @code{quote} to
26730 protect the righthand side: @samp{t(a) := quote(typeof(a))}.
26731 (@xref{Conditional Rewrite Rules}, for another trick for
26732 protecting rules from evaluation.)
26735 Special properties of and simplifications for the function call
26736 @expr{x} are not used. One interesting case where @code{plain}
26737 is useful is the rule, @samp{q(x) := quote(x)}, trying to expand a
26738 shorthand notation for the @code{quote} function. This rule will
26739 not work as shown; instead of replacing @samp{q(foo)} with
26740 @samp{quote(foo)}, it will replace it with @samp{foo}! The correct
26741 rule would be @samp{q(x) := plain(quote(x))}.
26744 Where @expr{t} is a vector, this is converted into an expanded
26745 vector during rewrite processing. Note that @code{cons} is a regular
26746 Calc function which normally does this anyway; the only way @code{cons}
26747 is treated specially by rewrites is that @code{cons} on the righthand
26748 side of a rule will be evaluated even if default simplifications
26749 have been turned off.
26752 Analogous to @code{cons} except putting @expr{h} at the @emph{end} of
26753 the vector @expr{t}.
26755 @item apply(f,args)
26756 Where @expr{f} is a variable and @var{args} is a vector, this
26757 is converted to a function call. Once again, note that @code{apply}
26758 is also a regular Calc function.
26765 The formula @expr{x} is handled in the usual way, then the
26766 default simplifications are applied to it even if they have
26767 been turned off normally. This allows you to treat any function
26768 similarly to the way @code{cons} and @code{apply} are always
26769 treated. However, there is a slight difference: @samp{cons(2+3, [])}
26770 with default simplifications off will be converted to @samp{[2+3]},
26771 whereas @samp{eval(cons(2+3, []))} will be converted to @samp{[5]}.
26778 The formula @expr{x} has meta-variables substituted in the usual
26779 way, then algebraically simplified as if by the @kbd{a s} command.
26781 @item evalextsimp(x)
26785 @tindex evalextsimp
26786 The formula @expr{x} has meta-variables substituted in the normal
26787 way, then ``extendedly'' simplified as if by the @kbd{a e} command.
26790 @xref{Selections with Rewrite Rules}.
26793 There are also some special functions you can use in conditions.
26801 The expression @expr{x} is evaluated with meta-variables substituted.
26802 The @kbd{a s} command's simplifications are @emph{not} applied by
26803 default, but @expr{x} can include calls to @code{evalsimp} or
26804 @code{evalextsimp} as described above to invoke higher levels
26805 of simplification. The
26806 result of @expr{x} is then bound to the meta-variable @expr{v}. As
26807 usual, if this meta-variable has already been matched to something
26808 else the two values must be equal; if the meta-variable is new then
26809 it is bound to the result of the expression. This variable can then
26810 appear in later conditions, and on the righthand side of the rule.
26811 In fact, @expr{v} may be any pattern in which case the result of
26812 evaluating @expr{x} is matched to that pattern, binding any
26813 meta-variables that appear in that pattern. Note that @code{let}
26814 can only appear by itself as a condition, or as one term of an
26815 @samp{&&} which is a whole condition: It cannot be inside
26816 an @samp{||} term or otherwise buried.
26818 The alternate, equivalent form @samp{let(v, x)} is also recognized.
26819 Note that the use of @samp{:=} by @code{let}, while still being
26820 assignment-like in character, is unrelated to the use of @samp{:=}
26821 in the main part of a rewrite rule.
26823 As an example, @samp{f(a) := g(ia) :: let(ia := 1/a) :: constant(ia)}
26824 replaces @samp{f(a)} with @samp{g} of the inverse of @samp{a}, if
26825 that inverse exists and is constant. For example, if @samp{a} is a
26826 singular matrix the operation @samp{1/a} is left unsimplified and
26827 @samp{constant(ia)} fails, but if @samp{a} is an invertible matrix
26828 then the rule succeeds. Without @code{let} there would be no way
26829 to express this rule that didn't have to invert the matrix twice.
26830 Note that, because the meta-variable @samp{ia} is otherwise unbound
26831 in this rule, the @code{let} condition itself always ``succeeds''
26832 because no matter what @samp{1/a} evaluates to, it can successfully
26833 be bound to @code{ia}.
26835 Here's another example, for integrating cosines of linear
26836 terms: @samp{myint(cos(y),x) := sin(y)/b :: let([a,b,x] := lin(y,x))}.
26837 The @code{lin} function returns a 3-vector if its argument is linear,
26838 or leaves itself unevaluated if not. But an unevaluated @code{lin}
26839 call will not match the 3-vector on the lefthand side of the @code{let},
26840 so this @code{let} both verifies that @code{y} is linear, and binds
26841 the coefficients @code{a} and @code{b} for use elsewhere in the rule.
26842 (It would have been possible to use @samp{sin(a x + b)/b} for the
26843 righthand side instead, but using @samp{sin(y)/b} avoids gratuitous
26844 rearrangement of the argument of the sine.)
26850 Similarly, here is a rule that implements an inverse-@code{erf}
26851 function. It uses @code{root} to search for a solution. If
26852 @code{root} succeeds, it will return a vector of two numbers
26853 where the first number is the desired solution. If no solution
26854 is found, @code{root} remains in symbolic form. So we use
26855 @code{let} to check that the result was indeed a vector.
26858 ierf(x) := y :: let([y,z] := root(erf(a) = x, a, .5))
26862 The meta-variable @var{v}, which must already have been matched
26863 to something elsewhere in the rule, is compared against pattern
26864 @var{p}. Since @code{matches} is a standard Calc function, it
26865 can appear anywhere in a condition. But if it appears alone or
26866 as a term of a top-level @samp{&&}, then you get the special
26867 extra feature that meta-variables which are bound to things
26868 inside @var{p} can be used elsewhere in the surrounding rewrite
26871 The only real difference between @samp{let(p := v)} and
26872 @samp{matches(v, p)} is that the former evaluates @samp{v} using
26873 the default simplifications, while the latter does not.
26877 This is actually a variable, not a function. If @code{remember}
26878 appears as a condition in a rule, then when that rule succeeds
26879 the original expression and rewritten expression are added to the
26880 front of the rule set that contained the rule. If the rule set
26881 was not stored in a variable, @code{remember} is ignored. The
26882 lefthand side is enclosed in @code{quote} in the added rule if it
26883 contains any variables.
26885 For example, the rule @samp{f(n) := n f(n-1) :: remember} applied
26886 to @samp{f(7)} will add the rule @samp{f(7) := 7 f(6)} to the front
26887 of the rule set. The rule set @code{EvalRules} works slightly
26888 differently: There, the evaluation of @samp{f(6)} will complete before
26889 the result is added to the rule set, in this case as @samp{f(7) := 5040}.
26890 Thus @code{remember} is most useful inside @code{EvalRules}.
26892 It is up to you to ensure that the optimization performed by
26893 @code{remember} is safe. For example, the rule @samp{foo(n) := n
26894 :: evalv(eatfoo) > 0 :: remember} is a bad idea (@code{evalv} is
26895 the function equivalent of the @kbd{=} command); if the variable
26896 @code{eatfoo} ever contains 1, rules like @samp{foo(7) := 7} will
26897 be added to the rule set and will continue to operate even if
26898 @code{eatfoo} is later changed to 0.
26905 Remember the match as described above, but only if condition @expr{c}
26906 is true. For example, @samp{remember(n % 4 = 0)} in the above factorial
26907 rule remembers only every fourth result. Note that @samp{remember(1)}
26908 is equivalent to @samp{remember}, and @samp{remember(0)} has no effect.
26911 @node Composing Patterns in Rewrite Rules, Nested Formulas with Rewrite Rules, Other Features of Rewrite Rules, Rewrite Rules
26912 @subsection Composing Patterns in Rewrite Rules
26915 There are three operators, @samp{&&&}, @samp{|||}, and @samp{!!!},
26916 that combine rewrite patterns to make larger patterns. The
26917 combinations are ``and,'' ``or,'' and ``not,'' respectively, and
26918 these operators are the pattern equivalents of @samp{&&}, @samp{||}
26919 and @samp{!} (which operate on zero-or-nonzero logical values).
26921 Note that @samp{&&&}, @samp{|||}, and @samp{!!!} are left in symbolic
26922 form by all regular Calc features; they have special meaning only in
26923 the context of rewrite rule patterns.
26925 The pattern @samp{@var{p1} &&& @var{p2}} matches anything that
26926 matches both @var{p1} and @var{p2}. One especially useful case is
26927 when one of @var{p1} or @var{p2} is a meta-variable. For example,
26928 here is a rule that operates on error forms:
26931 f(x &&& a +/- b, x) := g(x)
26934 This does the same thing, but is arguably simpler than, the rule
26937 f(a +/- b, a +/- b) := g(a +/- b)
26944 Here's another interesting example:
26947 ends(cons(a, x) &&& rcons(y, b)) := [a, b]
26951 which effectively clips out the middle of a vector leaving just
26952 the first and last elements. This rule will change a one-element
26953 vector @samp{[a]} to @samp{[a, a]}. The similar rule
26956 ends(cons(a, rcons(y, b))) := [a, b]
26960 would do the same thing except that it would fail to match a
26961 one-element vector.
26967 The pattern @samp{@var{p1} ||| @var{p2}} matches anything that
26968 matches either @var{p1} or @var{p2}. Calc first tries matching
26969 against @var{p1}; if that fails, it goes on to try @var{p2}.
26975 A simple example of @samp{|||} is
26978 curve(inf ||| -inf) := 0
26982 which converts both @samp{curve(inf)} and @samp{curve(-inf)} to zero.
26984 Here is a larger example:
26987 log(a, b) ||| (ln(a) :: let(b := e)) := mylog(a, b)
26990 This matches both generalized and natural logarithms in a single rule.
26991 Note that the @samp{::} term must be enclosed in parentheses because
26992 that operator has lower precedence than @samp{|||} or @samp{:=}.
26994 (In practice this rule would probably include a third alternative,
26995 omitted here for brevity, to take care of @code{log10}.)
26997 While Calc generally treats interior conditions exactly the same as
26998 conditions on the outside of a rule, it does guarantee that if all the
26999 variables in the condition are special names like @code{e}, or already
27000 bound in the pattern to which the condition is attached (say, if
27001 @samp{a} had appeared in this condition), then Calc will process this
27002 condition right after matching the pattern to the left of the @samp{::}.
27003 Thus, we know that @samp{b} will be bound to @samp{e} only if the
27004 @code{ln} branch of the @samp{|||} was taken.
27006 Note that this rule was careful to bind the same set of meta-variables
27007 on both sides of the @samp{|||}. Calc does not check this, but if
27008 you bind a certain meta-variable only in one branch and then use that
27009 meta-variable elsewhere in the rule, results are unpredictable:
27012 f(a,b) ||| g(b) := h(a,b)
27015 Here if the pattern matches @samp{g(17)}, Calc makes no promises about
27016 the value that will be substituted for @samp{a} on the righthand side.
27022 The pattern @samp{!!! @var{pat}} matches anything that does not
27023 match @var{pat}. Any meta-variables that are bound while matching
27024 @var{pat} remain unbound outside of @var{pat}.
27029 f(x &&& !!! a +/- b, !!![]) := g(x)
27033 converts @code{f} whose first argument is anything @emph{except} an
27034 error form, and whose second argument is not the empty vector, into
27035 a similar call to @code{g} (but without the second argument).
27037 If we know that the second argument will be a vector (empty or not),
27038 then an equivalent rule would be:
27041 f(x, y) := g(x) :: typeof(x) != 7 :: vlen(y) > 0
27045 where of course 7 is the @code{typeof} code for error forms.
27046 Another final condition, that works for any kind of @samp{y},
27047 would be @samp{!istrue(y == [])}. (The @code{istrue} function
27048 returns an explicit 0 if its argument was left in symbolic form;
27049 plain @samp{!(y == [])} or @samp{y != []} would not work to replace
27050 @samp{!!![]} since these would be left unsimplified, and thus cause
27051 the rule to fail, if @samp{y} was something like a variable name.)
27053 It is possible for a @samp{!!!} to refer to meta-variables bound
27054 elsewhere in the pattern. For example,
27061 matches any call to @code{f} with different arguments, changing
27062 this to @code{g} with only the first argument.
27064 If a function call is to be matched and one of the argument patterns
27065 contains a @samp{!!!} somewhere inside it, that argument will be
27073 will be careful to bind @samp{a} to the second argument of @code{f}
27074 before testing the first argument. If Calc had tried to match the
27075 first argument of @code{f} first, the results would have been
27076 disastrous: since @code{a} was unbound so far, the pattern @samp{a}
27077 would have matched anything at all, and the pattern @samp{!!!a}
27078 therefore would @emph{not} have matched anything at all!
27080 @node Nested Formulas with Rewrite Rules, Multi-Phase Rewrite Rules, Composing Patterns in Rewrite Rules, Rewrite Rules
27081 @subsection Nested Formulas with Rewrite Rules
27084 When @kbd{a r} (@code{calc-rewrite}) is used, it takes an expression from
27085 the top of the stack and attempts to match any of the specified rules
27086 to any part of the expression, starting with the whole expression
27087 and then, if that fails, trying deeper and deeper sub-expressions.
27088 For each part of the expression, the rules are tried in the order
27089 they appear in the rules vector. The first rule to match the first
27090 sub-expression wins; it replaces the matched sub-expression according
27091 to the @var{new} part of the rule.
27093 Often, the rule set will match and change the formula several times.
27094 The top-level formula is first matched and substituted repeatedly until
27095 it no longer matches the pattern; then, sub-formulas are tried, and
27096 so on. Once every part of the formula has gotten its chance, the
27097 rewrite mechanism starts over again with the top-level formula
27098 (in case a substitution of one of its arguments has caused it again
27099 to match). This continues until no further matches can be made
27100 anywhere in the formula.
27102 It is possible for a rule set to get into an infinite loop. The
27103 most obvious case, replacing a formula with itself, is not a problem
27104 because a rule is not considered to ``succeed'' unless the righthand
27105 side actually comes out to something different than the original
27106 formula or sub-formula that was matched. But if you accidentally
27107 had both @samp{ln(a b) := ln(a) + ln(b)} and the reverse
27108 @samp{ln(a) + ln(b) := ln(a b)} in your rule set, Calc would
27109 run forever switching a formula back and forth between the two
27112 To avoid disaster, Calc normally stops after 100 changes have been
27113 made to the formula. This will be enough for most multiple rewrites,
27114 but it will keep an endless loop of rewrites from locking up the
27115 computer forever. (On most systems, you can also type @kbd{C-g} to
27116 halt any Emacs command prematurely.)
27118 To change this limit, give a positive numeric prefix argument.
27119 In particular, @kbd{M-1 a r} applies only one rewrite at a time,
27120 useful when you are first testing your rule (or just if repeated
27121 rewriting is not what is called for by your application).
27130 You can also put a ``function call'' @samp{iterations(@var{n})}
27131 in place of a rule anywhere in your rules vector (but usually at
27132 the top). Then, @var{n} will be used instead of 100 as the default
27133 number of iterations for this rule set. You can use
27134 @samp{iterations(inf)} if you want no iteration limit by default.
27135 A prefix argument will override the @code{iterations} limit in the
27143 More precisely, the limit controls the number of ``iterations,''
27144 where each iteration is a successful matching of a rule pattern whose
27145 righthand side, after substituting meta-variables and applying the
27146 default simplifications, is different from the original sub-formula
27149 A prefix argument of zero sets the limit to infinity. Use with caution!
27151 Given a negative numeric prefix argument, @kbd{a r} will match and
27152 substitute the top-level expression up to that many times, but
27153 will not attempt to match the rules to any sub-expressions.
27155 In a formula, @code{rewrite(@var{expr}, @var{rules}, @var{n})}
27156 does a rewriting operation. Here @var{expr} is the expression
27157 being rewritten, @var{rules} is the rule, vector of rules, or
27158 variable containing the rules, and @var{n} is the optional
27159 iteration limit, which may be a positive integer, a negative
27160 integer, or @samp{inf} or @samp{-inf}. If @var{n} is omitted
27161 the @code{iterations} value from the rule set is used; if both
27162 are omitted, 100 is used.
27164 @node Multi-Phase Rewrite Rules, Selections with Rewrite Rules, Nested Formulas with Rewrite Rules, Rewrite Rules
27165 @subsection Multi-Phase Rewrite Rules
27168 It is possible to separate a rewrite rule set into several @dfn{phases}.
27169 During each phase, certain rules will be enabled while certain others
27170 will be disabled. A @dfn{phase schedule} controls the order in which
27171 phases occur during the rewriting process.
27178 If a call to the marker function @code{phase} appears in the rules
27179 vector in place of a rule, all rules following that point will be
27180 members of the phase(s) identified in the arguments to @code{phase}.
27181 Phases are given integer numbers. The markers @samp{phase()} and
27182 @samp{phase(all)} both mean the following rules belong to all phases;
27183 this is the default at the start of the rule set.
27185 If you do not explicitly schedule the phases, Calc sorts all phase
27186 numbers that appear in the rule set and executes the phases in
27187 ascending order. For example, the rule set
27204 has three phases, 1 through 3. Phase 1 consists of the @code{f0},
27205 @code{f1}, and @code{f4} rules (in that order). Phase 2 consists of
27206 @code{f0}, @code{f2}, and @code{f4}. Phase 3 consists of @code{f0}
27209 When Calc rewrites a formula using this rule set, it first rewrites
27210 the formula using only the phase 1 rules until no further changes are
27211 possible. Then it switches to the phase 2 rule set and continues
27212 until no further changes occur, then finally rewrites with phase 3.
27213 When no more phase 3 rules apply, rewriting finishes. (This is
27214 assuming @kbd{a r} with a large enough prefix argument to allow the
27215 rewriting to run to completion; the sequence just described stops
27216 early if the number of iterations specified in the prefix argument,
27217 100 by default, is reached.)
27219 During each phase, Calc descends through the nested levels of the
27220 formula as described previously. (@xref{Nested Formulas with Rewrite
27221 Rules}.) Rewriting starts at the top of the formula, then works its
27222 way down to the parts, then goes back to the top and works down again.
27223 The phase 2 rules do not begin until no phase 1 rules apply anywhere
27230 A @code{schedule} marker appearing in the rule set (anywhere, but
27231 conventionally at the top) changes the default schedule of phases.
27232 In the simplest case, @code{schedule} has a sequence of phase numbers
27233 for arguments; each phase number is invoked in turn until the
27234 arguments to @code{schedule} are exhausted. Thus adding
27235 @samp{schedule(3,2,1)} at the top of the above rule set would
27236 reverse the order of the phases; @samp{schedule(1,2,3)} would have
27237 no effect since this is the default schedule; and @samp{schedule(1,2,1,3)}
27238 would give phase 1 a second chance after phase 2 has completed, before
27239 moving on to phase 3.
27241 Any argument to @code{schedule} can instead be a vector of phase
27242 numbers (or even of sub-vectors). Then the sub-sequence of phases
27243 described by the vector are tried repeatedly until no change occurs
27244 in any phase in the sequence. For example, @samp{schedule([1, 2], 3)}
27245 tries phase 1, then phase 2, then, if either phase made any changes
27246 to the formula, repeats these two phases until they can make no
27247 further progress. Finally, it goes on to phase 3 for finishing
27250 Also, items in @code{schedule} can be variable names as well as
27251 numbers. A variable name is interpreted as the name of a function
27252 to call on the whole formula. For example, @samp{schedule(1, simplify)}
27253 says to apply the phase-1 rules (presumably, all of them), then to
27254 call @code{simplify} which is the function name equivalent of @kbd{a s}.
27255 Likewise, @samp{schedule([1, simplify])} says to alternate between
27256 phase 1 and @kbd{a s} until no further changes occur.
27258 Phases can be used purely to improve efficiency; if it is known that
27259 a certain group of rules will apply only at the beginning of rewriting,
27260 and a certain other group will apply only at the end, then rewriting
27261 will be faster if these groups are identified as separate phases.
27262 Once the phase 1 rules are done, Calc can put them aside and no longer
27263 spend any time on them while it works on phase 2.
27265 There are also some problems that can only be solved with several
27266 rewrite phases. For a real-world example of a multi-phase rule set,
27267 examine the set @code{FitRules}, which is used by the curve-fitting
27268 command to convert a model expression to linear form.
27269 @xref{Curve Fitting Details}. This set is divided into four phases.
27270 The first phase rewrites certain kinds of expressions to be more
27271 easily linearizable, but less computationally efficient. After the
27272 linear components have been picked out, the final phase includes the
27273 opposite rewrites to put each component back into an efficient form.
27274 If both sets of rules were included in one big phase, Calc could get
27275 into an infinite loop going back and forth between the two forms.
27277 Elsewhere in @code{FitRules}, the components are first isolated,
27278 then recombined where possible to reduce the complexity of the linear
27279 fit, then finally packaged one component at a time into vectors.
27280 If the packaging rules were allowed to begin before the recombining
27281 rules were finished, some components might be put away into vectors
27282 before they had a chance to recombine. By putting these rules in
27283 two separate phases, this problem is neatly avoided.
27285 @node Selections with Rewrite Rules, Matching Commands, Multi-Phase Rewrite Rules, Rewrite Rules
27286 @subsection Selections with Rewrite Rules
27289 If a sub-formula of the current formula is selected (as by @kbd{j s};
27290 @pxref{Selecting Subformulas}), the @kbd{a r} (@code{calc-rewrite})
27291 command applies only to that sub-formula. Together with a negative
27292 prefix argument, you can use this fact to apply a rewrite to one
27293 specific part of a formula without affecting any other parts.
27296 @pindex calc-rewrite-selection
27297 The @kbd{j r} (@code{calc-rewrite-selection}) command allows more
27298 sophisticated operations on selections. This command prompts for
27299 the rules in the same way as @kbd{a r}, but it then applies those
27300 rules to the whole formula in question even though a sub-formula
27301 of it has been selected. However, the selected sub-formula will
27302 first have been surrounded by a @samp{select( )} function call.
27303 (Calc's evaluator does not understand the function name @code{select};
27304 this is only a tag used by the @kbd{j r} command.)
27306 For example, suppose the formula on the stack is @samp{2 (a + b)^2}
27307 and the sub-formula @samp{a + b} is selected. This formula will
27308 be rewritten to @samp{2 select(a + b)^2} and then the rewrite
27309 rules will be applied in the usual way. The rewrite rules can
27310 include references to @code{select} to tell where in the pattern
27311 the selected sub-formula should appear.
27313 If there is still exactly one @samp{select( )} function call in
27314 the formula after rewriting is done, it indicates which part of
27315 the formula should be selected afterwards. Otherwise, the
27316 formula will be unselected.
27318 You can make @kbd{j r} act much like @kbd{a r} by enclosing both parts
27319 of the rewrite rule with @samp{select()}. However, @kbd{j r}
27320 allows you to use the current selection in more flexible ways.
27321 Suppose you wished to make a rule which removed the exponent from
27322 the selected term; the rule @samp{select(a)^x := select(a)} would
27323 work. In the above example, it would rewrite @samp{2 select(a + b)^2}
27324 to @samp{2 select(a + b)}. This would then be returned to the
27325 stack as @samp{2 (a + b)} with the @samp{a + b} selected.
27327 The @kbd{j r} command uses one iteration by default, unlike
27328 @kbd{a r} which defaults to 100 iterations. A numeric prefix
27329 argument affects @kbd{j r} in the same way as @kbd{a r}.
27330 @xref{Nested Formulas with Rewrite Rules}.
27332 As with other selection commands, @kbd{j r} operates on the stack
27333 entry that contains the cursor. (If the cursor is on the top-of-stack
27334 @samp{.} marker, it works as if the cursor were on the formula
27337 If you don't specify a set of rules, the rules are taken from the
27338 top of the stack, just as with @kbd{a r}. In this case, the
27339 cursor must indicate stack entry 2 or above as the formula to be
27340 rewritten (otherwise the same formula would be used as both the
27341 target and the rewrite rules).
27343 If the indicated formula has no selection, the cursor position within
27344 the formula temporarily selects a sub-formula for the purposes of this
27345 command. If the cursor is not on any sub-formula (e.g., it is in
27346 the line-number area to the left of the formula), the @samp{select( )}
27347 markers are ignored by the rewrite mechanism and the rules are allowed
27348 to apply anywhere in the formula.
27350 As a special feature, the normal @kbd{a r} command also ignores
27351 @samp{select( )} calls in rewrite rules. For example, if you used the
27352 above rule @samp{select(a)^x := select(a)} with @kbd{a r}, it would apply
27353 the rule as if it were @samp{a^x := a}. Thus, you can write general
27354 purpose rules with @samp{select( )} hints inside them so that they
27355 will ``do the right thing'' in both @kbd{a r} and @kbd{j r},
27356 both with and without selections.
27358 @node Matching Commands, Automatic Rewrites, Selections with Rewrite Rules, Rewrite Rules
27359 @subsection Matching Commands
27365 The @kbd{a m} (@code{calc-match}) [@code{match}] function takes a
27366 vector of formulas and a rewrite-rule-style pattern, and produces
27367 a vector of all formulas which match the pattern. The command
27368 prompts you to enter the pattern; as for @kbd{a r}, you can enter
27369 a single pattern (i.e., a formula with meta-variables), or a
27370 vector of patterns, or a variable which contains patterns, or
27371 you can give a blank response in which case the patterns are taken
27372 from the top of the stack. The pattern set will be compiled once
27373 and saved if it is stored in a variable. If there are several
27374 patterns in the set, vector elements are kept if they match any
27377 For example, @samp{match(a+b, [x, x+y, x-y, 7, x+y+z])}
27378 will return @samp{[x+y, x-y, x+y+z]}.
27380 The @code{import} mechanism is not available for pattern sets.
27382 The @kbd{a m} command can also be used to extract all vector elements
27383 which satisfy any condition: The pattern @samp{x :: x>0} will select
27384 all the positive vector elements.
27388 With the Inverse flag [@code{matchnot}], this command extracts all
27389 vector elements which do @emph{not} match the given pattern.
27395 There is also a function @samp{matches(@var{x}, @var{p})} which
27396 evaluates to 1 if expression @var{x} matches pattern @var{p}, or
27397 to 0 otherwise. This is sometimes useful for including into the
27398 conditional clauses of other rewrite rules.
27404 The function @code{vmatches} is just like @code{matches}, except
27405 that if the match succeeds it returns a vector of assignments to
27406 the meta-variables instead of the number 1. For example,
27407 @samp{vmatches(f(1,2), f(a,b))} returns @samp{[a := 1, b := 2]}.
27408 If the match fails, the function returns the number 0.
27410 @node Automatic Rewrites, Debugging Rewrites, Matching Commands, Rewrite Rules
27411 @subsection Automatic Rewrites
27414 @cindex @code{EvalRules} variable
27416 It is possible to get Calc to apply a set of rewrite rules on all
27417 results, effectively adding to the built-in set of default
27418 simplifications. To do this, simply store your rule set in the
27419 variable @code{EvalRules}. There is a convenient @kbd{s E} command
27420 for editing @code{EvalRules}; @pxref{Operations on Variables}.
27422 For example, suppose you want @samp{sin(a + b)} to be expanded out
27423 to @samp{sin(b) cos(a) + cos(b) sin(a)} wherever it appears, and
27424 similarly for @samp{cos(a + b)}. The corresponding rewrite rule
27429 [ sin(a + b) := cos(a) sin(b) + sin(a) cos(b),
27430 cos(a + b) := cos(a) cos(b) - sin(a) sin(b) ]
27434 To apply these manually, you could put them in a variable called
27435 @code{trigexp} and then use @kbd{a r trigexp} every time you wanted
27436 to expand trig functions. But if instead you store them in the
27437 variable @code{EvalRules}, they will automatically be applied to all
27438 sines and cosines of sums. Then, with @samp{2 x} and @samp{45} on
27439 the stack, typing @kbd{+ S} will (assuming Degrees mode) result in
27440 @samp{0.7071 sin(2 x) + 0.7071 cos(2 x)} automatically.
27442 As each level of a formula is evaluated, the rules from
27443 @code{EvalRules} are applied before the default simplifications.
27444 Rewriting continues until no further @code{EvalRules} apply.
27445 Note that this is different from the usual order of application of
27446 rewrite rules: @code{EvalRules} works from the bottom up, simplifying
27447 the arguments to a function before the function itself, while @kbd{a r}
27448 applies rules from the top down.
27450 Because the @code{EvalRules} are tried first, you can use them to
27451 override the normal behavior of any built-in Calc function.
27453 It is important not to write a rule that will get into an infinite
27454 loop. For example, the rule set @samp{[f(0) := 1, f(n) := n f(n-1)]}
27455 appears to be a good definition of a factorial function, but it is
27456 unsafe. Imagine what happens if @samp{f(2.5)} is simplified. Calc
27457 will continue to subtract 1 from this argument forever without reaching
27458 zero. A safer second rule would be @samp{f(n) := n f(n-1) :: n>0}.
27459 Another dangerous rule is @samp{g(x, y) := g(y, x)}. Rewriting
27460 @samp{g(2, 4)}, this would bounce back and forth between that and
27461 @samp{g(4, 2)} forever. If an infinite loop in @code{EvalRules}
27462 occurs, Emacs will eventually stop with a ``Computation got stuck
27463 or ran too long'' message.
27465 Another subtle difference between @code{EvalRules} and regular rewrites
27466 concerns rules that rewrite a formula into an identical formula. For
27467 example, @samp{f(n) := f(floor(n))} ``fails to match'' when @expr{n} is
27468 already an integer. But in @code{EvalRules} this case is detected only
27469 if the righthand side literally becomes the original formula before any
27470 further simplification. This means that @samp{f(n) := f(floor(n))} will
27471 get into an infinite loop if it occurs in @code{EvalRules}. Calc will
27472 replace @samp{f(6)} with @samp{f(floor(6))}, which is different from
27473 @samp{f(6)}, so it will consider the rule to have matched and will
27474 continue simplifying that formula; first the argument is simplified
27475 to get @samp{f(6)}, then the rule matches again to get @samp{f(floor(6))}
27476 again, ad infinitum. A much safer rule would check its argument first,
27477 say, with @samp{f(n) := f(floor(n)) :: !dint(n)}.
27479 (What really happens is that the rewrite mechanism substitutes the
27480 meta-variables in the righthand side of a rule, compares to see if the
27481 result is the same as the original formula and fails if so, then uses
27482 the default simplifications to simplify the result and compares again
27483 (and again fails if the formula has simplified back to its original
27484 form). The only special wrinkle for the @code{EvalRules} is that the
27485 same rules will come back into play when the default simplifications
27486 are used. What Calc wants to do is build @samp{f(floor(6))}, see that
27487 this is different from the original formula, simplify to @samp{f(6)},
27488 see that this is the same as the original formula, and thus halt the
27489 rewriting. But while simplifying, @samp{f(6)} will again trigger
27490 the same @code{EvalRules} rule and Calc will get into a loop inside
27491 the rewrite mechanism itself.)
27493 The @code{phase}, @code{schedule}, and @code{iterations} markers do
27494 not work in @code{EvalRules}. If the rule set is divided into phases,
27495 only the phase 1 rules are applied, and the schedule is ignored.
27496 The rules are always repeated as many times as possible.
27498 The @code{EvalRules} are applied to all function calls in a formula,
27499 but not to numbers (and other number-like objects like error forms),
27500 nor to vectors or individual variable names. (Though they will apply
27501 to @emph{components} of vectors and error forms when appropriate.) You
27502 might try to make a variable @code{phihat} which automatically expands
27503 to its definition without the need to press @kbd{=} by writing the
27504 rule @samp{quote(phihat) := (1-sqrt(5))/2}, but unfortunately this rule
27505 will not work as part of @code{EvalRules}.
27507 Finally, another limitation is that Calc sometimes calls its built-in
27508 functions directly rather than going through the default simplifications.
27509 When it does this, @code{EvalRules} will not be able to override those
27510 functions. For example, when you take the absolute value of the complex
27511 number @expr{(2, 3)}, Calc computes @samp{sqrt(2*2 + 3*3)} by calling
27512 the multiplication, addition, and square root functions directly rather
27513 than applying the default simplifications to this formula. So an
27514 @code{EvalRules} rule that (perversely) rewrites @samp{sqrt(13) := 6}
27515 would not apply. (However, if you put Calc into Symbolic mode so that
27516 @samp{sqrt(13)} will be left in symbolic form by the built-in square
27517 root function, your rule will be able to apply. But if the complex
27518 number were @expr{(3,4)}, so that @samp{sqrt(25)} must be calculated,
27519 then Symbolic mode will not help because @samp{sqrt(25)} can be
27520 evaluated exactly to 5.)
27522 One subtle restriction that normally only manifests itself with
27523 @code{EvalRules} is that while a given rewrite rule is in the process
27524 of being checked, that same rule cannot be recursively applied. Calc
27525 effectively removes the rule from its rule set while checking the rule,
27526 then puts it back once the match succeeds or fails. (The technical
27527 reason for this is that compiled pattern programs are not reentrant.)
27528 For example, consider the rule @samp{foo(x) := x :: foo(x/2) > 0}
27529 attempting to match @samp{foo(8)}. This rule will be inactive while
27530 the condition @samp{foo(4) > 0} is checked, even though it might be
27531 an integral part of evaluating that condition. Note that this is not
27532 a problem for the more usual recursive type of rule, such as
27533 @samp{foo(x) := foo(x/2)}, because there the rule has succeeded and
27534 been reactivated by the time the righthand side is evaluated.
27536 If @code{EvalRules} has no stored value (its default state), or if
27537 anything but a vector is stored in it, then it is ignored.
27539 Even though Calc's rewrite mechanism is designed to compare rewrite
27540 rules to formulas as quickly as possible, storing rules in
27541 @code{EvalRules} may make Calc run substantially slower. This is
27542 particularly true of rules where the top-level call is a commonly used
27543 function, or is not fixed. The rule @samp{f(n) := n f(n-1) :: n>0} will
27544 only activate the rewrite mechanism for calls to the function @code{f},
27545 but @samp{lg(n) + lg(m) := lg(n m)} will check every @samp{+} operator.
27548 apply(f, [a*b]) := apply(f, [a]) + apply(f, [b]) :: in(f, [ln, log10])
27552 may seem more ``efficient'' than two separate rules for @code{ln} and
27553 @code{log10}, but actually it is vastly less efficient because rules
27554 with @code{apply} as the top-level pattern must be tested against
27555 @emph{every} function call that is simplified.
27557 @cindex @code{AlgSimpRules} variable
27558 @vindex AlgSimpRules
27559 Suppose you want @samp{sin(a + b)} to be expanded out not all the time,
27560 but only when @kbd{a s} is used to simplify the formula. The variable
27561 @code{AlgSimpRules} holds rules for this purpose. The @kbd{a s} command
27562 will apply @code{EvalRules} and @code{AlgSimpRules} to the formula, as
27563 well as all of its built-in simplifications.
27565 Most of the special limitations for @code{EvalRules} don't apply to
27566 @code{AlgSimpRules}. Calc simply does an @kbd{a r AlgSimpRules}
27567 command with an infinite repeat count as the first step of @kbd{a s}.
27568 It then applies its own built-in simplifications throughout the
27569 formula, and then repeats these two steps (along with applying the
27570 default simplifications) until no further changes are possible.
27572 @cindex @code{ExtSimpRules} variable
27573 @cindex @code{UnitSimpRules} variable
27574 @vindex ExtSimpRules
27575 @vindex UnitSimpRules
27576 There are also @code{ExtSimpRules} and @code{UnitSimpRules} variables
27577 that are used by @kbd{a e} and @kbd{u s}, respectively; these commands
27578 also apply @code{EvalRules} and @code{AlgSimpRules}. The variable
27579 @code{IntegSimpRules} contains simplification rules that are used
27580 only during integration by @kbd{a i}.
27582 @node Debugging Rewrites, Examples of Rewrite Rules, Automatic Rewrites, Rewrite Rules
27583 @subsection Debugging Rewrites
27586 If a buffer named @samp{*Trace*} exists, the rewrite mechanism will
27587 record some useful information there as it operates. The original
27588 formula is written there, as is the result of each successful rewrite,
27589 and the final result of the rewriting. All phase changes are also
27592 Calc always appends to @samp{*Trace*}. You must empty this buffer
27593 yourself periodically if it is in danger of growing unwieldy.
27595 Note that the rewriting mechanism is substantially slower when the
27596 @samp{*Trace*} buffer exists, even if the buffer is not visible on
27597 the screen. Once you are done, you will probably want to kill this
27598 buffer (with @kbd{C-x k *Trace* @key{RET}}). If you leave it in
27599 existence and forget about it, all your future rewrite commands will
27600 be needlessly slow.
27602 @node Examples of Rewrite Rules, , Debugging Rewrites, Rewrite Rules
27603 @subsection Examples of Rewrite Rules
27606 Returning to the example of substituting the pattern
27607 @samp{sin(x)^2 + cos(x)^2} with 1, we saw that the rule
27608 @samp{opt(a) sin(x)^2 + opt(a) cos(x)^2 := a} does a good job of
27609 finding suitable cases. Another solution would be to use the rule
27610 @samp{cos(x)^2 := 1 - sin(x)^2}, followed by algebraic simplification
27611 if necessary. This rule will be the most effective way to do the job,
27612 but at the expense of making some changes that you might not desire.
27614 Another algebraic rewrite rule is @samp{exp(x+y) := exp(x) exp(y)}.
27615 To make this work with the @w{@kbd{j r}} command so that it can be
27616 easily targeted to a particular exponential in a large formula,
27617 you might wish to write the rule as @samp{select(exp(x+y)) :=
27618 select(exp(x) exp(y))}. The @samp{select} markers will be
27619 ignored by the regular @kbd{a r} command
27620 (@pxref{Selections with Rewrite Rules}).
27622 A surprisingly useful rewrite rule is @samp{a/(b-c) := a*(b+c)/(b^2-c^2)}.
27623 This will simplify the formula whenever @expr{b} and/or @expr{c} can
27624 be made simpler by squaring. For example, applying this rule to
27625 @samp{2 / (sqrt(2) + 3)} yields @samp{6:7 - 2:7 sqrt(2)} (assuming
27626 Symbolic mode has been enabled to keep the square root from being
27627 evaluated to a floating-point approximation). This rule is also
27628 useful when working with symbolic complex numbers, e.g.,
27629 @samp{(a + b i) / (c + d i)}.
27631 As another example, we could define our own ``triangular numbers'' function
27632 with the rules @samp{[tri(0) := 0, tri(n) := n + tri(n-1) :: n>0]}. Enter
27633 this vector and store it in a variable: @kbd{@w{s t} trirules}. Now, given
27634 a suitable formula like @samp{tri(5)} on the stack, type @samp{a r trirules}
27635 to apply these rules repeatedly. After six applications, @kbd{a r} will
27636 stop with 15 on the stack. Once these rules are debugged, it would probably
27637 be most useful to add them to @code{EvalRules} so that Calc will evaluate
27638 the new @code{tri} function automatically. We could then use @kbd{Z K} on
27639 the keyboard macro @kbd{' tri($) @key{RET}} to make a command that applies
27640 @code{tri} to the value on the top of the stack. @xref{Programming}.
27642 @cindex Quaternions
27643 The following rule set, contributed by
27644 @texline Fran\c cois
27646 Pinard, implements @dfn{quaternions}, a generalization of the concept of
27647 complex numbers. Quaternions have four components, and are here
27648 represented by function calls @samp{quat(@var{w}, [@var{x}, @var{y},
27649 @var{z}])} with ``real part'' @var{w} and the three ``imaginary'' parts
27650 collected into a vector. Various arithmetical operations on quaternions
27651 are supported. To use these rules, either add them to @code{EvalRules},
27652 or create a command based on @kbd{a r} for simplifying quaternion
27653 formulas. A convenient way to enter quaternions would be a command
27654 defined by a keyboard macro containing: @kbd{' quat($$$$, [$$$, $$, $])
27658 [ quat(w, x, y, z) := quat(w, [x, y, z]),
27659 quat(w, [0, 0, 0]) := w,
27660 abs(quat(w, v)) := hypot(w, v),
27661 -quat(w, v) := quat(-w, -v),
27662 r + quat(w, v) := quat(r + w, v) :: real(r),
27663 r - quat(w, v) := quat(r - w, -v) :: real(r),
27664 quat(w1, v1) + quat(w2, v2) := quat(w1 + w2, v1 + v2),
27665 r * quat(w, v) := quat(r * w, r * v) :: real(r),
27666 plain(quat(w1, v1) * quat(w2, v2))
27667 := quat(w1 * w2 - v1 * v2, w1 * v2 + w2 * v1 + cross(v1, v2)),
27668 quat(w1, v1) / r := quat(w1 / r, v1 / r) :: real(r),
27669 z / quat(w, v) := z * quatinv(quat(w, v)),
27670 quatinv(quat(w, v)) := quat(w, -v) / (w^2 + v^2),
27671 quatsqr(quat(w, v)) := quat(w^2 - v^2, 2 * w * v),
27672 quat(w, v)^k := quatsqr(quat(w, v)^(k / 2))
27673 :: integer(k) :: k > 0 :: k % 2 = 0,
27674 quat(w, v)^k := quatsqr(quat(w, v)^((k - 1) / 2)) * quat(w, v)
27675 :: integer(k) :: k > 2,
27676 quat(w, v)^-k := quatinv(quat(w, v)^k) :: integer(k) :: k > 0 ]
27679 Quaternions, like matrices, have non-commutative multiplication.
27680 In other words, @expr{q1 * q2 = q2 * q1} is not necessarily true if
27681 @expr{q1} and @expr{q2} are @code{quat} forms. The @samp{quat*quat}
27682 rule above uses @code{plain} to prevent Calc from rearranging the
27683 product. It may also be wise to add the line @samp{[quat(), matrix]}
27684 to the @code{Decls} matrix, to ensure that Calc's other algebraic
27685 operations will not rearrange a quaternion product. @xref{Declarations}.
27687 These rules also accept a four-argument @code{quat} form, converting
27688 it to the preferred form in the first rule. If you would rather see
27689 results in the four-argument form, just append the two items
27690 @samp{phase(2), quat(w, [x, y, z]) := quat(w, x, y, z)} to the end
27691 of the rule set. (But remember that multi-phase rule sets don't work
27692 in @code{EvalRules}.)
27694 @node Units, Store and Recall, Algebra, Top
27695 @chapter Operating on Units
27698 One special interpretation of algebraic formulas is as numbers with units.
27699 For example, the formula @samp{5 m / s^2} can be read ``five meters
27700 per second squared.'' The commands in this chapter help you
27701 manipulate units expressions in this form. Units-related commands
27702 begin with the @kbd{u} prefix key.
27705 * Basic Operations on Units::
27706 * The Units Table::
27707 * Predefined Units::
27708 * User-Defined Units::
27711 @node Basic Operations on Units, The Units Table, Units, Units
27712 @section Basic Operations on Units
27715 A @dfn{units expression} is a formula which is basically a number
27716 multiplied and/or divided by one or more @dfn{unit names}, which may
27717 optionally be raised to integer powers. Actually, the value part need not
27718 be a number; any product or quotient involving unit names is a units
27719 expression. Many of the units commands will also accept any formula,
27720 where the command applies to all units expressions which appear in the
27723 A unit name is a variable whose name appears in the @dfn{unit table},
27724 or a variable whose name is a prefix character like @samp{k} (for ``kilo'')
27725 or @samp{u} (for ``micro'') followed by a name in the unit table.
27726 A substantial table of built-in units is provided with Calc;
27727 @pxref{Predefined Units}. You can also define your own unit names;
27728 @pxref{User-Defined Units}.
27730 Note that if the value part of a units expression is exactly @samp{1},
27731 it will be removed by the Calculator's automatic algebra routines: The
27732 formula @samp{1 mm} is ``simplified'' to @samp{mm}. This is only a
27733 display anomaly, however; @samp{mm} will work just fine as a
27734 representation of one millimeter.
27736 You may find that Algebraic mode (@pxref{Algebraic Entry}) makes working
27737 with units expressions easier. Otherwise, you will have to remember
27738 to hit the apostrophe key every time you wish to enter units.
27741 @pindex calc-simplify-units
27743 @mindex usimpl@idots
27746 The @kbd{u s} (@code{calc-simplify-units}) [@code{usimplify}] command
27748 expression. It uses @kbd{a s} (@code{calc-simplify}) to simplify the
27749 expression first as a regular algebraic formula; it then looks for
27750 features that can be further simplified by converting one object's units
27751 to be compatible with another's. For example, @samp{5 m + 23 mm} will
27752 simplify to @samp{5.023 m}. When different but compatible units are
27753 added, the righthand term's units are converted to match those of the
27754 lefthand term. @xref{Simplification Modes}, for a way to have this done
27755 automatically at all times.
27757 Units simplification also handles quotients of two units with the same
27758 dimensionality, as in @w{@samp{2 in s/L cm}} to @samp{5.08 s/L}; fractional
27759 powers of unit expressions, as in @samp{sqrt(9 mm^2)} to @samp{3 mm} and
27760 @samp{sqrt(9 acre)} to a quantity in meters; and @code{floor},
27761 @code{ceil}, @code{round}, @code{rounde}, @code{roundu}, @code{trunc},
27762 @code{float}, @code{frac}, @code{abs}, and @code{clean}
27763 applied to units expressions, in which case
27764 the operation in question is applied only to the numeric part of the
27765 expression. Finally, trigonometric functions of quantities with units
27766 of angle are evaluated, regardless of the current angular mode.
27769 @pindex calc-convert-units
27770 The @kbd{u c} (@code{calc-convert-units}) command converts a units
27771 expression to new, compatible units. For example, given the units
27772 expression @samp{55 mph}, typing @kbd{u c m/s @key{RET}} produces
27773 @samp{24.5872 m/s}. If the units you request are inconsistent with
27774 the original units, the number will be converted into your units
27775 times whatever ``remainder'' units are left over. For example,
27776 converting @samp{55 mph} into acres produces @samp{6.08e-3 acre / m s}.
27777 (Recall that multiplication binds more strongly than division in Calc
27778 formulas, so the units here are acres per meter-second.) Remainder
27779 units are expressed in terms of ``fundamental'' units like @samp{m} and
27780 @samp{s}, regardless of the input units.
27782 One special exception is that if you specify a single unit name, and
27783 a compatible unit appears somewhere in the units expression, then
27784 that compatible unit will be converted to the new unit and the
27785 remaining units in the expression will be left alone. For example,
27786 given the input @samp{980 cm/s^2}, the command @kbd{u c ms} will
27787 change the @samp{s} to @samp{ms} to get @samp{9.8e-4 cm/ms^2}.
27788 The ``remainder unit'' @samp{cm} is left alone rather than being
27789 changed to the base unit @samp{m}.
27791 You can use explicit unit conversion instead of the @kbd{u s} command
27792 to gain more control over the units of the result of an expression.
27793 For example, given @samp{5 m + 23 mm}, you can type @kbd{u c m} or
27794 @kbd{u c mm} to express the result in either meters or millimeters.
27795 (For that matter, you could type @kbd{u c fath} to express the result
27796 in fathoms, if you preferred!)
27798 In place of a specific set of units, you can also enter one of the
27799 units system names @code{si}, @code{mks} (equivalent), or @code{cgs}.
27800 For example, @kbd{u c si @key{RET}} converts the expression into
27801 International System of Units (SI) base units. Also, @kbd{u c base}
27802 converts to Calc's base units, which are the same as @code{si} units
27803 except that @code{base} uses @samp{g} as the fundamental unit of mass
27804 whereas @code{si} uses @samp{kg}.
27806 @cindex Composite units
27807 The @kbd{u c} command also accepts @dfn{composite units}, which
27808 are expressed as the sum of several compatible unit names. For
27809 example, converting @samp{30.5 in} to units @samp{mi+ft+in} (miles,
27810 feet, and inches) produces @samp{2 ft + 6.5 in}. Calc first
27811 sorts the unit names into order of decreasing relative size.
27812 It then accounts for as much of the input quantity as it can
27813 using an integer number times the largest unit, then moves on
27814 to the next smaller unit, and so on. Only the smallest unit
27815 may have a non-integer amount attached in the result. A few
27816 standard unit names exist for common combinations, such as
27817 @code{mfi} for @samp{mi+ft+in}, and @code{tpo} for @samp{ton+lb+oz}.
27818 Composite units are expanded as if by @kbd{a x}, so that
27819 @samp{(ft+in)/hr} is first converted to @samp{ft/hr+in/hr}.
27821 If the value on the stack does not contain any units, @kbd{u c} will
27822 prompt first for the old units which this value should be considered
27823 to have, then for the new units. Assuming the old and new units you
27824 give are consistent with each other, the result also will not contain
27825 any units. For example, @kbd{@w{u c} cm @key{RET} in @key{RET}} converts the number
27826 2 on the stack to 5.08.
27829 @pindex calc-base-units
27830 The @kbd{u b} (@code{calc-base-units}) command is shorthand for
27831 @kbd{u c base}; it converts the units expression on the top of the
27832 stack into @code{base} units. If @kbd{u s} does not simplify a
27833 units expression as far as you would like, try @kbd{u b}.
27835 The @kbd{u c} and @kbd{u b} commands treat temperature units (like
27836 @samp{degC} and @samp{K}) as relative temperatures. For example,
27837 @kbd{u c} converts @samp{10 degC} to @samp{18 degF}: A change of 10
27838 degrees Celsius corresponds to a change of 18 degrees Fahrenheit.
27841 @pindex calc-convert-temperature
27842 @cindex Temperature conversion
27843 The @kbd{u t} (@code{calc-convert-temperature}) command converts
27844 absolute temperatures. The value on the stack must be a simple units
27845 expression with units of temperature only. This command would convert
27846 @samp{10 degC} to @samp{50 degF}, the equivalent temperature on the
27850 @pindex calc-remove-units
27852 @pindex calc-extract-units
27853 The @kbd{u r} (@code{calc-remove-units}) command removes units from the
27854 formula at the top of the stack. The @kbd{u x}
27855 (@code{calc-extract-units}) command extracts only the units portion of a
27856 formula. These commands essentially replace every term of the formula
27857 that does or doesn't (respectively) look like a unit name by the
27858 constant 1, then resimplify the formula.
27861 @pindex calc-autorange-units
27862 The @kbd{u a} (@code{calc-autorange-units}) command turns on and off a
27863 mode in which unit prefixes like @code{k} (``kilo'') are automatically
27864 applied to keep the numeric part of a units expression in a reasonable
27865 range. This mode affects @kbd{u s} and all units conversion commands
27866 except @kbd{u b}. For example, with autoranging on, @samp{12345 Hz}
27867 will be simplified to @samp{12.345 kHz}. Autoranging is useful for
27868 some kinds of units (like @code{Hz} and @code{m}), but is probably
27869 undesirable for non-metric units like @code{ft} and @code{tbsp}.
27870 (Composite units are more appropriate for those; see above.)
27872 Autoranging always applies the prefix to the leftmost unit name.
27873 Calc chooses the largest prefix that causes the number to be greater
27874 than or equal to 1.0. Thus an increasing sequence of adjusted times
27875 would be @samp{1 ms, 10 ms, 100 ms, 1 s, 10 s, 100 s, 1 ks}.
27876 Generally the rule of thumb is that the number will be adjusted
27877 to be in the interval @samp{[1 .. 1000)}, although there are several
27878 exceptions to this rule. First, if the unit has a power then this
27879 is not possible; @samp{0.1 s^2} simplifies to @samp{100000 ms^2}.
27880 Second, the ``centi-'' prefix is allowed to form @code{cm} (centimeters),
27881 but will not apply to other units. The ``deci-,'' ``deka-,'' and
27882 ``hecto-'' prefixes are never used. Thus the allowable interval is
27883 @samp{[1 .. 10)} for millimeters and @samp{[1 .. 100)} for centimeters.
27884 Finally, a prefix will not be added to a unit if the resulting name
27885 is also the actual name of another unit; @samp{1e-15 t} would normally
27886 be considered a ``femto-ton,'' but it is written as @samp{1000 at}
27887 (1000 atto-tons) instead because @code{ft} would be confused with feet.
27889 @node The Units Table, Predefined Units, Basic Operations on Units, Units
27890 @section The Units Table
27894 @pindex calc-enter-units-table
27895 The @kbd{u v} (@code{calc-enter-units-table}) command displays the units table
27896 in another buffer called @code{*Units Table*}. Each entry in this table
27897 gives the unit name as it would appear in an expression, the definition
27898 of the unit in terms of simpler units, and a full name or description of
27899 the unit. Fundamental units are defined as themselves; these are the
27900 units produced by the @kbd{u b} command. The fundamental units are
27901 meters, seconds, grams, kelvins, amperes, candelas, moles, radians,
27904 The Units Table buffer also displays the Unit Prefix Table. Note that
27905 two prefixes, ``kilo'' and ``hecto,'' accept either upper- or lower-case
27906 prefix letters. @samp{Meg} is also accepted as a synonym for the @samp{M}
27907 prefix. Whenever a unit name can be interpreted as either a built-in name
27908 or a prefix followed by another built-in name, the former interpretation
27909 wins. For example, @samp{2 pt} means two pints, not two pico-tons.
27911 The Units Table buffer, once created, is not rebuilt unless you define
27912 new units. To force the buffer to be rebuilt, give any numeric prefix
27913 argument to @kbd{u v}.
27916 @pindex calc-view-units-table
27917 The @kbd{u V} (@code{calc-view-units-table}) command is like @kbd{u v} except
27918 that the cursor is not moved into the Units Table buffer. You can
27919 type @kbd{u V} again to remove the Units Table from the display. To
27920 return from the Units Table buffer after a @kbd{u v}, type @kbd{M-# c}
27921 again or use the regular Emacs @w{@kbd{C-x o}} (@code{other-window})
27922 command. You can also kill the buffer with @kbd{C-x k} if you wish;
27923 the actual units table is safely stored inside the Calculator.
27926 @pindex calc-get-unit-definition
27927 The @kbd{u g} (@code{calc-get-unit-definition}) command retrieves a unit's
27928 defining expression and pushes it onto the Calculator stack. For example,
27929 @kbd{u g in} will produce the expression @samp{2.54 cm}. This is the
27930 same definition for the unit that would appear in the Units Table buffer.
27931 Note that this command works only for actual unit names; @kbd{u g km}
27932 will report that no such unit exists, for example, because @code{km} is
27933 really the unit @code{m} with a @code{k} (``kilo'') prefix. To see a
27934 definition of a unit in terms of base units, it is easier to push the
27935 unit name on the stack and then reduce it to base units with @kbd{u b}.
27938 @pindex calc-explain-units
27939 The @kbd{u e} (@code{calc-explain-units}) command displays an English
27940 description of the units of the expression on the stack. For example,
27941 for the expression @samp{62 km^2 g / s^2 mol K}, the description is
27942 ``Square-Kilometer Gram per (Second-squared Mole Degree-Kelvin).'' This
27943 command uses the English descriptions that appear in the righthand
27944 column of the Units Table.
27946 @node Predefined Units, User-Defined Units, The Units Table, Units
27947 @section Predefined Units
27950 Since the exact definitions of many kinds of units have evolved over the
27951 years, and since certain countries sometimes have local differences in
27952 their definitions, it is a good idea to examine Calc's definition of a
27953 unit before depending on its exact value. For example, there are three
27954 different units for gallons, corresponding to the US (@code{gal}),
27955 Canadian (@code{galC}), and British (@code{galUK}) definitions. Also,
27956 note that @code{oz} is a standard ounce of mass, @code{ozt} is a Troy
27957 ounce, and @code{ozfl} is a fluid ounce.
27959 The temperature units corresponding to degrees Kelvin and Centigrade
27960 (Celsius) are the same in this table, since most units commands treat
27961 temperatures as being relative. The @code{calc-convert-temperature}
27962 command has special rules for handling the different absolute magnitudes
27963 of the various temperature scales.
27965 The unit of volume ``liters'' can be referred to by either the lower-case
27966 @code{l} or the upper-case @code{L}.
27968 The unit @code{A} stands for Amperes; the name @code{Ang} is used
27976 The unit @code{pt} stands for pints; the name @code{point} stands for
27977 a typographical point, defined by @samp{72 point = 1 in}. There is
27978 also @code{tpt}, which stands for a printer's point as defined by the
27979 @TeX{} typesetting system: @samp{72.27 tpt = 1 in}.
27981 The unit @code{e} stands for the elementary (electron) unit of charge;
27982 because algebra command could mistake this for the special constant
27983 @expr{e}, Calc provides the alternate unit name @code{ech} which is
27984 preferable to @code{e}.
27986 The name @code{g} stands for one gram of mass; there is also @code{gf},
27987 one gram of force. (Likewise for @kbd{lb}, pounds, and @kbd{lbf}.)
27988 Meanwhile, one ``@expr{g}'' of acceleration is denoted @code{ga}.
27990 The unit @code{ton} is a U.S. ton of @samp{2000 lb}, and @code{t} is
27991 a metric ton of @samp{1000 kg}.
27993 The names @code{s} (or @code{sec}) and @code{min} refer to units of
27994 time; @code{arcsec} and @code{arcmin} are units of angle.
27996 Some ``units'' are really physical constants; for example, @code{c}
27997 represents the speed of light, and @code{h} represents Planck's
27998 constant. You can use these just like other units: converting
27999 @samp{.5 c} to @samp{m/s} expresses one-half the speed of light in
28000 meters per second. You can also use this merely as a handy reference;
28001 the @kbd{u g} command gets the definition of one of these constants
28002 in its normal terms, and @kbd{u b} expresses the definition in base
28005 Two units, @code{pi} and @code{fsc} (the fine structure constant,
28006 approximately @mathit{1/137}) are dimensionless. The units simplification
28007 commands simply treat these names as equivalent to their corresponding
28008 values. However you can, for example, use @kbd{u c} to convert a pure
28009 number into multiples of the fine structure constant, or @kbd{u b} to
28010 convert this back into a pure number. (When @kbd{u c} prompts for the
28011 ``old units,'' just enter a blank line to signify that the value
28012 really is unitless.)
28014 @c Describe angular units, luminosity vs. steradians problem.
28016 @node User-Defined Units, , Predefined Units, Units
28017 @section User-Defined Units
28020 Calc provides ways to get quick access to your selected ``favorite''
28021 units, as well as ways to define your own new units.
28024 @pindex calc-quick-units
28026 @cindex @code{Units} variable
28027 @cindex Quick units
28028 To select your favorite units, store a vector of unit names or
28029 expressions in the Calc variable @code{Units}. The @kbd{u 1}
28030 through @kbd{u 9} commands (@code{calc-quick-units}) provide access
28031 to these units. If the value on the top of the stack is a plain
28032 number (with no units attached), then @kbd{u 1} gives it the
28033 specified units. (Basically, it multiplies the number by the
28034 first item in the @code{Units} vector.) If the number on the
28035 stack @emph{does} have units, then @kbd{u 1} converts that number
28036 to the new units. For example, suppose the vector @samp{[in, ft]}
28037 is stored in @code{Units}. Then @kbd{30 u 1} will create the
28038 expression @samp{30 in}, and @kbd{u 2} will convert that expression
28041 The @kbd{u 0} command accesses the tenth element of @code{Units}.
28042 Only ten quick units may be defined at a time. If the @code{Units}
28043 variable has no stored value (the default), or if its value is not
28044 a vector, then the quick-units commands will not function. The
28045 @kbd{s U} command is a convenient way to edit the @code{Units}
28046 variable; @pxref{Operations on Variables}.
28049 @pindex calc-define-unit
28050 @cindex User-defined units
28051 The @kbd{u d} (@code{calc-define-unit}) command records the units
28052 expression on the top of the stack as the definition for a new,
28053 user-defined unit. For example, putting @samp{16.5 ft} on the stack and
28054 typing @kbd{u d rod} defines the new unit @samp{rod} to be equivalent to
28055 16.5 feet. The unit conversion and simplification commands will now
28056 treat @code{rod} just like any other unit of length. You will also be
28057 prompted for an optional English description of the unit, which will
28058 appear in the Units Table.
28061 @pindex calc-undefine-unit
28062 The @kbd{u u} (@code{calc-undefine-unit}) command removes a user-defined
28063 unit. It is not possible to remove one of the predefined units,
28066 If you define a unit with an existing unit name, your new definition
28067 will replace the original definition of that unit. If the unit was a
28068 predefined unit, the old definition will not be replaced, only
28069 ``shadowed.'' The built-in definition will reappear if you later use
28070 @kbd{u u} to remove the shadowing definition.
28072 To create a new fundamental unit, use either 1 or the unit name itself
28073 as the defining expression. Otherwise the expression can involve any
28074 other units that you like (except for composite units like @samp{mfi}).
28075 You can create a new composite unit with a sum of other units as the
28076 defining expression. The next unit operation like @kbd{u c} or @kbd{u v}
28077 will rebuild the internal unit table incorporating your modifications.
28078 Note that erroneous definitions (such as two units defined in terms of
28079 each other) will not be detected until the unit table is next rebuilt;
28080 @kbd{u v} is a convenient way to force this to happen.
28082 Temperature units are treated specially inside the Calculator; it is not
28083 possible to create user-defined temperature units.
28086 @pindex calc-permanent-units
28087 @cindex Calc init file, user-defined units
28088 The @kbd{u p} (@code{calc-permanent-units}) command stores the user-defined
28089 units in your Calc init file (the file given by the variable
28090 @code{calc-settings-file}, typically @file{~/.calc.el}), so that the
28091 units will still be available in subsequent Emacs sessions. If there
28092 was already a set of user-defined units in your Calc init file, it
28093 is replaced by the new set. (@xref{General Mode Commands}, for a way to
28094 tell Calc to use a different file for the Calc init file.)
28096 @node Store and Recall, Graphics, Units, Top
28097 @chapter Storing and Recalling
28100 Calculator variables are really just Lisp variables that contain numbers
28101 or formulas in a form that Calc can understand. The commands in this
28102 section allow you to manipulate variables conveniently. Commands related
28103 to variables use the @kbd{s} prefix key.
28106 * Storing Variables::
28107 * Recalling Variables::
28108 * Operations on Variables::
28110 * Evaluates-To Operator::
28113 @node Storing Variables, Recalling Variables, Store and Recall, Store and Recall
28114 @section Storing Variables
28119 @cindex Storing variables
28120 @cindex Quick variables
28123 The @kbd{s s} (@code{calc-store}) command stores the value at the top of
28124 the stack into a specified variable. It prompts you to enter the
28125 name of the variable. If you press a single digit, the value is stored
28126 immediately in one of the ``quick'' variables @code{q0} through
28127 @code{q9}. Or you can enter any variable name.
28130 @pindex calc-store-into
28131 The @kbd{s s} command leaves the stored value on the stack. There is
28132 also an @kbd{s t} (@code{calc-store-into}) command, which removes a
28133 value from the stack and stores it in a variable.
28135 If the top of stack value is an equation @samp{a = 7} or assignment
28136 @samp{a := 7} with a variable on the lefthand side, then Calc will
28137 assign that variable with that value by default, i.e., if you type
28138 @kbd{s s @key{RET}} or @kbd{s t @key{RET}}. In this example, the
28139 value 7 would be stored in the variable @samp{a}. (If you do type
28140 a variable name at the prompt, the top-of-stack value is stored in
28141 its entirety, even if it is an equation: @samp{s s b @key{RET}}
28142 with @samp{a := 7} on the stack stores @samp{a := 7} in @code{b}.)
28144 In fact, the top of stack value can be a vector of equations or
28145 assignments with different variables on their lefthand sides; the
28146 default will be to store all the variables with their corresponding
28147 righthand sides simultaneously.
28149 It is also possible to type an equation or assignment directly at
28150 the prompt for the @kbd{s s} or @kbd{s t} command: @kbd{s s foo = 7}.
28151 In this case the expression to the right of the @kbd{=} or @kbd{:=}
28152 symbol is evaluated as if by the @kbd{=} command, and that value is
28153 stored in the variable. No value is taken from the stack; @kbd{s s}
28154 and @kbd{s t} are equivalent when used in this way.
28158 The prefix keys @kbd{s} and @kbd{t} may be followed immediately by a
28159 digit; @kbd{s 9} is equivalent to @kbd{s s 9}, and @kbd{t 9} is
28160 equivalent to @kbd{s t 9}. (The @kbd{t} prefix is otherwise used
28161 for trail and time/date commands.)
28197 @pindex calc-store-plus
28198 @pindex calc-store-minus
28199 @pindex calc-store-times
28200 @pindex calc-store-div
28201 @pindex calc-store-power
28202 @pindex calc-store-concat
28203 @pindex calc-store-neg
28204 @pindex calc-store-inv
28205 @pindex calc-store-decr
28206 @pindex calc-store-incr
28207 There are also several ``arithmetic store'' commands. For example,
28208 @kbd{s +} removes a value from the stack and adds it to the specified
28209 variable. The other arithmetic stores are @kbd{s -}, @kbd{s *}, @kbd{s /},
28210 @kbd{s ^}, and @w{@kbd{s |}} (vector concatenation), plus @kbd{s n} and
28211 @kbd{s &} which negate or invert the value in a variable, and @w{@kbd{s [}}
28212 and @kbd{s ]} which decrease or increase a variable by one.
28214 All the arithmetic stores accept the Inverse prefix to reverse the
28215 order of the operands. If @expr{v} represents the contents of the
28216 variable, and @expr{a} is the value drawn from the stack, then regular
28217 @w{@kbd{s -}} assigns
28218 @texline @math{v \coloneq v - a},
28219 @infoline @expr{v := v - a},
28220 but @kbd{I s -} assigns
28221 @texline @math{v \coloneq a - v}.
28222 @infoline @expr{v := a - v}.
28223 While @kbd{I s *} might seem pointless, it is
28224 useful if matrix multiplication is involved. Actually, all the
28225 arithmetic stores use formulas designed to behave usefully both
28226 forwards and backwards:
28230 s + v := v + a v := a + v
28231 s - v := v - a v := a - v
28232 s * v := v * a v := a * v
28233 s / v := v / a v := a / v
28234 s ^ v := v ^ a v := a ^ v
28235 s | v := v | a v := a | v
28236 s n v := v / (-1) v := (-1) / v
28237 s & v := v ^ (-1) v := (-1) ^ v
28238 s [ v := v - 1 v := 1 - v
28239 s ] v := v - (-1) v := (-1) - v
28243 In the last four cases, a numeric prefix argument will be used in
28244 place of the number one. (For example, @kbd{M-2 s ]} increases
28245 a variable by 2, and @kbd{M-2 I s ]} replaces a variable by
28246 minus-two minus the variable.
28248 The first six arithmetic stores can also be typed @kbd{s t +}, @kbd{s t -},
28249 etc. The commands @kbd{s s +}, @kbd{s s -}, and so on are analogous
28250 arithmetic stores that don't remove the value @expr{a} from the stack.
28252 All arithmetic stores report the new value of the variable in the
28253 Trail for your information. They signal an error if the variable
28254 previously had no stored value. If default simplifications have been
28255 turned off, the arithmetic stores temporarily turn them on for numeric
28256 arguments only (i.e., they temporarily do an @kbd{m N} command).
28257 @xref{Simplification Modes}. Large vectors put in the trail by
28258 these commands always use abbreviated (@kbd{t .}) mode.
28261 @pindex calc-store-map
28262 The @kbd{s m} command is a general way to adjust a variable's value
28263 using any Calc function. It is a ``mapping'' command analogous to
28264 @kbd{V M}, @kbd{V R}, etc. @xref{Reducing and Mapping}, to see
28265 how to specify a function for a mapping command. Basically,
28266 all you do is type the Calc command key that would invoke that
28267 function normally. For example, @kbd{s m n} applies the @kbd{n}
28268 key to negate the contents of the variable, so @kbd{s m n} is
28269 equivalent to @kbd{s n}. Also, @kbd{s m Q} takes the square root
28270 of the value stored in a variable, @kbd{s m v v} uses @kbd{v v} to
28271 reverse the vector stored in the variable, and @kbd{s m H I S}
28272 takes the hyperbolic arcsine of the variable contents.
28274 If the mapping function takes two or more arguments, the additional
28275 arguments are taken from the stack; the old value of the variable
28276 is provided as the first argument. Thus @kbd{s m -} with @expr{a}
28277 on the stack computes @expr{v - a}, just like @kbd{s -}. With the
28278 Inverse prefix, the variable's original value becomes the @emph{last}
28279 argument instead of the first. Thus @kbd{I s m -} is also
28280 equivalent to @kbd{I s -}.
28283 @pindex calc-store-exchange
28284 The @kbd{s x} (@code{calc-store-exchange}) command exchanges the value
28285 of a variable with the value on the top of the stack. Naturally, the
28286 variable must already have a stored value for this to work.
28288 You can type an equation or assignment at the @kbd{s x} prompt. The
28289 command @kbd{s x a=6} takes no values from the stack; instead, it
28290 pushes the old value of @samp{a} on the stack and stores @samp{a = 6}.
28293 @pindex calc-unstore
28294 @cindex Void variables
28295 @cindex Un-storing variables
28296 Until you store something in them, variables are ``void,'' that is, they
28297 contain no value at all. If they appear in an algebraic formula they
28298 will be left alone even if you press @kbd{=} (@code{calc-evaluate}).
28299 The @kbd{s u} (@code{calc-unstore}) command returns a variable to the
28302 The only variables with predefined values are the ``special constants''
28303 @code{pi}, @code{e}, @code{i}, @code{phi}, and @code{gamma}. You are free
28304 to unstore these variables or to store new values into them if you like,
28305 although some of the algebraic-manipulation functions may assume these
28306 variables represent their standard values. Calc displays a warning if
28307 you change the value of one of these variables, or of one of the other
28308 special variables @code{inf}, @code{uinf}, and @code{nan} (which are
28311 Note that @code{pi} doesn't actually have 3.14159265359 stored
28312 in it, but rather a special magic value that evaluates to @cpi{}
28313 at the current precision. Likewise @code{e}, @code{i}, and
28314 @code{phi} evaluate according to the current precision or polar mode.
28315 If you recall a value from @code{pi} and store it back, this magic
28316 property will be lost.
28319 @pindex calc-copy-variable
28320 The @kbd{s c} (@code{calc-copy-variable}) command copies the stored
28321 value of one variable to another. It differs from a simple @kbd{s r}
28322 followed by an @kbd{s t} in two important ways. First, the value never
28323 goes on the stack and thus is never rounded, evaluated, or simplified
28324 in any way; it is not even rounded down to the current precision.
28325 Second, the ``magic'' contents of a variable like @code{e} can
28326 be copied into another variable with this command, perhaps because
28327 you need to unstore @code{e} right now but you wish to put it
28328 back when you're done. The @kbd{s c} command is the only way to
28329 manipulate these magic values intact.
28331 @node Recalling Variables, Operations on Variables, Storing Variables, Store and Recall
28332 @section Recalling Variables
28336 @pindex calc-recall
28337 @cindex Recalling variables
28338 The most straightforward way to extract the stored value from a variable
28339 is to use the @kbd{s r} (@code{calc-recall}) command. This command prompts
28340 for a variable name (similarly to @code{calc-store}), looks up the value
28341 of the specified variable, and pushes that value onto the stack. It is
28342 an error to try to recall a void variable.
28344 It is also possible to recall the value from a variable by evaluating a
28345 formula containing that variable. For example, @kbd{' a @key{RET} =} is
28346 the same as @kbd{s r a @key{RET}} except that if the variable is void, the
28347 former will simply leave the formula @samp{a} on the stack whereas the
28348 latter will produce an error message.
28351 The @kbd{r} prefix may be followed by a digit, so that @kbd{r 9} is
28352 equivalent to @kbd{s r 9}. (The @kbd{r} prefix is otherwise unused
28353 in the current version of Calc.)
28355 @node Operations on Variables, Let Command, Recalling Variables, Store and Recall
28356 @section Other Operations on Variables
28360 @pindex calc-edit-variable
28361 The @kbd{s e} (@code{calc-edit-variable}) command edits the stored
28362 value of a variable without ever putting that value on the stack
28363 or simplifying or evaluating the value. It prompts for the name of
28364 the variable to edit. If the variable has no stored value, the
28365 editing buffer will start out empty. If the editing buffer is
28366 empty when you press @kbd{C-c C-c} to finish, the variable will
28367 be made void. @xref{Editing Stack Entries}, for a general
28368 description of editing.
28370 The @kbd{s e} command is especially useful for creating and editing
28371 rewrite rules which are stored in variables. Sometimes these rules
28372 contain formulas which must not be evaluated until the rules are
28373 actually used. (For example, they may refer to @samp{deriv(x,y)},
28374 where @code{x} will someday become some expression involving @code{y};
28375 if you let Calc evaluate the rule while you are defining it, Calc will
28376 replace @samp{deriv(x,y)} with 0 because the formula @code{x} does
28377 not itself refer to @code{y}.) By contrast, recalling the variable,
28378 editing with @kbd{`}, and storing will evaluate the variable's value
28379 as a side effect of putting the value on the stack.
28427 @pindex calc-store-AlgSimpRules
28428 @pindex calc-store-Decls
28429 @pindex calc-store-EvalRules
28430 @pindex calc-store-FitRules
28431 @pindex calc-store-GenCount
28432 @pindex calc-store-Holidays
28433 @pindex calc-store-IntegLimit
28434 @pindex calc-store-LineStyles
28435 @pindex calc-store-PointStyles
28436 @pindex calc-store-PlotRejects
28437 @pindex calc-store-TimeZone
28438 @pindex calc-store-Units
28439 @pindex calc-store-ExtSimpRules
28440 There are several special-purpose variable-editing commands that
28441 use the @kbd{s} prefix followed by a shifted letter:
28445 Edit @code{AlgSimpRules}. @xref{Algebraic Simplifications}.
28447 Edit @code{Decls}. @xref{Declarations}.
28449 Edit @code{EvalRules}. @xref{Default Simplifications}.
28451 Edit @code{FitRules}. @xref{Curve Fitting}.
28453 Edit @code{GenCount}. @xref{Solving Equations}.
28455 Edit @code{Holidays}. @xref{Business Days}.
28457 Edit @code{IntegLimit}. @xref{Calculus}.
28459 Edit @code{LineStyles}. @xref{Graphics}.
28461 Edit @code{PointStyles}. @xref{Graphics}.
28463 Edit @code{PlotRejects}. @xref{Graphics}.
28465 Edit @code{TimeZone}. @xref{Time Zones}.
28467 Edit @code{Units}. @xref{User-Defined Units}.
28469 Edit @code{ExtSimpRules}. @xref{Unsafe Simplifications}.
28472 These commands are just versions of @kbd{s e} that use fixed variable
28473 names rather than prompting for the variable name.
28476 @pindex calc-permanent-variable
28477 @cindex Storing variables
28478 @cindex Permanent variables
28479 @cindex Calc init file, variables
28480 The @kbd{s p} (@code{calc-permanent-variable}) command saves a
28481 variable's value permanently in your Calc init file (the file given by
28482 the variable @code{calc-settings-file}, typically @file{~/.calc.el}), so
28483 that its value will still be available in future Emacs sessions. You
28484 can re-execute @w{@kbd{s p}} later on to update the saved value, but the
28485 only way to remove a saved variable is to edit your calc init file
28486 by hand. (@xref{General Mode Commands}, for a way to tell Calc to
28487 use a different file for the Calc init file.)
28489 If you do not specify the name of a variable to save (i.e.,
28490 @kbd{s p @key{RET}}), all Calc variables with defined values
28491 are saved except for the special constants @code{pi}, @code{e},
28492 @code{i}, @code{phi}, and @code{gamma}; the variables @code{TimeZone}
28493 and @code{PlotRejects};
28494 @code{FitRules}, @code{DistribRules}, and other built-in rewrite
28495 rules; and @code{PlotData@var{n}} variables generated
28496 by the graphics commands. (You can still save these variables by
28497 explicitly naming them in an @kbd{s p} command.)
28500 @pindex calc-insert-variables
28501 The @kbd{s i} (@code{calc-insert-variables}) command writes
28502 the values of all Calc variables into a specified buffer.
28503 The variables are written with the prefix @code{var-} in the form of
28504 Lisp @code{setq} commands
28505 which store the values in string form. You can place these commands
28506 in your Calc init file (or @file{.emacs}) if you wish, though in this case it
28507 would be easier to use @kbd{s p @key{RET}}. (Note that @kbd{s i}
28508 omits the same set of variables as @w{@kbd{s p @key{RET}}}; the difference
28509 is that @kbd{s i} will store the variables in any buffer, and it also
28510 stores in a more human-readable format.)
28512 @node Let Command, Evaluates-To Operator, Operations on Variables, Store and Recall
28513 @section The Let Command
28518 @cindex Variables, temporary assignment
28519 @cindex Temporary assignment to variables
28520 If you have an expression like @samp{a+b^2} on the stack and you wish to
28521 compute its value where @expr{b=3}, you can simply store 3 in @expr{b} and
28522 then press @kbd{=} to reevaluate the formula. This has the side-effect
28523 of leaving the stored value of 3 in @expr{b} for future operations.
28525 The @kbd{s l} (@code{calc-let}) command evaluates a formula under a
28526 @emph{temporary} assignment of a variable. It stores the value on the
28527 top of the stack into the specified variable, then evaluates the
28528 second-to-top stack entry, then restores the original value (or lack of one)
28529 in the variable. Thus after @kbd{'@w{ }a+b^2 @key{RET} 3 s l b @key{RET}},
28530 the stack will contain the formula @samp{a + 9}. The subsequent command
28531 @kbd{@w{5 s l a} @key{RET}} will replace this formula with the number 14.
28532 The variables @samp{a} and @samp{b} are not permanently affected in any way
28535 The value on the top of the stack may be an equation or assignment, or
28536 a vector of equations or assignments, in which case the default will be
28537 analogous to the case of @kbd{s t @key{RET}}. @xref{Storing Variables}.
28539 Also, you can answer the variable-name prompt with an equation or
28540 assignment: @kbd{s l b=3 @key{RET}} is the same as storing 3 on the stack
28541 and typing @kbd{s l b @key{RET}}.
28543 The @kbd{a b} (@code{calc-substitute}) command is another way to substitute
28544 a variable with a value in a formula. It does an actual substitution
28545 rather than temporarily assigning the variable and evaluating. For
28546 example, letting @expr{n=2} in @samp{f(n pi)} with @kbd{a b} will
28547 produce @samp{f(2 pi)}, whereas @kbd{s l} would give @samp{f(6.28)}
28548 since the evaluation step will also evaluate @code{pi}.
28550 @node Evaluates-To Operator, , Let Command, Store and Recall
28551 @section The Evaluates-To Operator
28556 @cindex Evaluates-to operator
28557 @cindex @samp{=>} operator
28558 The special algebraic symbol @samp{=>} is known as the @dfn{evaluates-to
28559 operator}. (It will show up as an @code{evalto} function call in
28560 other language modes like Pascal and La@TeX{}.) This is a binary
28561 operator, that is, it has a lefthand and a righthand argument,
28562 although it can be entered with the righthand argument omitted.
28564 A formula like @samp{@var{a} => @var{b}} is evaluated by Calc as
28565 follows: First, @var{a} is not simplified or modified in any
28566 way. The previous value of argument @var{b} is thrown away; the
28567 formula @var{a} is then copied and evaluated as if by the @kbd{=}
28568 command according to all current modes and stored variable values,
28569 and the result is installed as the new value of @var{b}.
28571 For example, suppose you enter the algebraic formula @samp{2 + 3 => 17}.
28572 The number 17 is ignored, and the lefthand argument is left in its
28573 unevaluated form; the result is the formula @samp{2 + 3 => 5}.
28576 @pindex calc-evalto
28577 You can enter an @samp{=>} formula either directly using algebraic
28578 entry (in which case the righthand side may be omitted since it is
28579 going to be replaced right away anyhow), or by using the @kbd{s =}
28580 (@code{calc-evalto}) command, which takes @var{a} from the stack
28581 and replaces it with @samp{@var{a} => @var{b}}.
28583 Calc keeps track of all @samp{=>} operators on the stack, and
28584 recomputes them whenever anything changes that might affect their
28585 values, i.e., a mode setting or variable value. This occurs only
28586 if the @samp{=>} operator is at the top level of the formula, or
28587 if it is part of a top-level vector. In other words, pushing
28588 @samp{2 + (a => 17)} will change the 17 to the actual value of
28589 @samp{a} when you enter the formula, but the result will not be
28590 dynamically updated when @samp{a} is changed later because the
28591 @samp{=>} operator is buried inside a sum. However, a vector
28592 of @samp{=>} operators will be recomputed, since it is convenient
28593 to push a vector like @samp{[a =>, b =>, c =>]} on the stack to
28594 make a concise display of all the variables in your problem.
28595 (Another way to do this would be to use @samp{[a, b, c] =>},
28596 which provides a slightly different format of display. You
28597 can use whichever you find easiest to read.)
28600 @pindex calc-auto-recompute
28601 The @kbd{m C} (@code{calc-auto-recompute}) command allows you to
28602 turn this automatic recomputation on or off. If you turn
28603 recomputation off, you must explicitly recompute an @samp{=>}
28604 operator on the stack in one of the usual ways, such as by
28605 pressing @kbd{=}. Turning recomputation off temporarily can save
28606 a lot of time if you will be changing several modes or variables
28607 before you look at the @samp{=>} entries again.
28609 Most commands are not especially useful with @samp{=>} operators
28610 as arguments. For example, given @samp{x + 2 => 17}, it won't
28611 work to type @kbd{1 +} to get @samp{x + 3 => 18}. If you want
28612 to operate on the lefthand side of the @samp{=>} operator on
28613 the top of the stack, type @kbd{j 1} (that's the digit ``one'')
28614 to select the lefthand side, execute your commands, then type
28615 @kbd{j u} to unselect.
28617 All current modes apply when an @samp{=>} operator is computed,
28618 including the current simplification mode. Recall that the
28619 formula @samp{x + y + x} is not handled by Calc's default
28620 simplifications, but the @kbd{a s} command will reduce it to
28621 the simpler form @samp{y + 2 x}. You can also type @kbd{m A}
28622 to enable an Algebraic Simplification mode in which the
28623 equivalent of @kbd{a s} is used on all of Calc's results.
28624 If you enter @samp{x + y + x =>} normally, the result will
28625 be @samp{x + y + x => x + y + x}. If you change to
28626 Algebraic Simplification mode, the result will be
28627 @samp{x + y + x => y + 2 x}. However, just pressing @kbd{a s}
28628 once will have no effect on @samp{x + y + x => x + y + x},
28629 because the righthand side depends only on the lefthand side
28630 and the current mode settings, and the lefthand side is not
28631 affected by commands like @kbd{a s}.
28633 The ``let'' command (@kbd{s l}) has an interesting interaction
28634 with the @samp{=>} operator. The @kbd{s l} command evaluates the
28635 second-to-top stack entry with the top stack entry supplying
28636 a temporary value for a given variable. As you might expect,
28637 if that stack entry is an @samp{=>} operator its righthand
28638 side will temporarily show this value for the variable. In
28639 fact, all @samp{=>}s on the stack will be updated if they refer
28640 to that variable. But this change is temporary in the sense
28641 that the next command that causes Calc to look at those stack
28642 entries will make them revert to the old variable value.
28646 2: a => a 2: a => 17 2: a => a
28647 1: a + 1 => a + 1 1: a + 1 => 18 1: a + 1 => a + 1
28650 17 s l a @key{RET} p 8 @key{RET}
28654 Here the @kbd{p 8} command changes the current precision,
28655 thus causing the @samp{=>} forms to be recomputed after the
28656 influence of the ``let'' is gone. The @kbd{d @key{SPC}} command
28657 (@code{calc-refresh}) is a handy way to force the @samp{=>}
28658 operators on the stack to be recomputed without any other
28662 @pindex calc-assign
28665 Embedded mode also uses @samp{=>} operators. In Embedded mode,
28666 the lefthand side of an @samp{=>} operator can refer to variables
28667 assigned elsewhere in the file by @samp{:=} operators. The
28668 assignment operator @samp{a := 17} does not actually do anything
28669 by itself. But Embedded mode recognizes it and marks it as a sort
28670 of file-local definition of the variable. You can enter @samp{:=}
28671 operators in Algebraic mode, or by using the @kbd{s :}
28672 (@code{calc-assign}) [@code{assign}] command which takes a variable
28673 and value from the stack and replaces them with an assignment.
28675 @xref{TeX and LaTeX Language Modes}, for the way @samp{=>} appears in
28676 @TeX{} language output. The @dfn{eqn} mode gives similar
28677 treatment to @samp{=>}.
28679 @node Graphics, Kill and Yank, Store and Recall, Top
28683 The commands for graphing data begin with the @kbd{g} prefix key. Calc
28684 uses GNUPLOT 2.0 or 3.0 to do graphics. These commands will only work
28685 if GNUPLOT is available on your system. (While GNUPLOT sounds like
28686 a relative of GNU Emacs, it is actually completely unrelated.
28687 However, it is free software and can be obtained from the Free
28688 Software Foundation's machine @samp{prep.ai.mit.edu}.)
28690 @vindex calc-gnuplot-name
28691 If you have GNUPLOT installed on your system but Calc is unable to
28692 find it, you may need to set the @code{calc-gnuplot-name} variable
28693 in your Calc init file or @file{.emacs}. You may also need to set some Lisp
28694 variables to show Calc how to run GNUPLOT on your system; these
28695 are described under @kbd{g D} and @kbd{g O} below. If you are
28696 using the X window system, Calc will configure GNUPLOT for you
28697 automatically. If you have GNUPLOT 3.0 and you are not using X,
28698 Calc will configure GNUPLOT to display graphs using simple character
28699 graphics that will work on any terminal.
28703 * Three Dimensional Graphics::
28704 * Managing Curves::
28705 * Graphics Options::
28709 @node Basic Graphics, Three Dimensional Graphics, Graphics, Graphics
28710 @section Basic Graphics
28714 @pindex calc-graph-fast
28715 The easiest graphics command is @kbd{g f} (@code{calc-graph-fast}).
28716 This command takes two vectors of equal length from the stack.
28717 The vector at the top of the stack represents the ``y'' values of
28718 the various data points. The vector in the second-to-top position
28719 represents the corresponding ``x'' values. This command runs
28720 GNUPLOT (if it has not already been started by previous graphing
28721 commands) and displays the set of data points. The points will
28722 be connected by lines, and there will also be some kind of symbol
28723 to indicate the points themselves.
28725 The ``x'' entry may instead be an interval form, in which case suitable
28726 ``x'' values are interpolated between the minimum and maximum values of
28727 the interval (whether the interval is open or closed is ignored).
28729 The ``x'' entry may also be a number, in which case Calc uses the
28730 sequence of ``x'' values @expr{x}, @expr{x+1}, @expr{x+2}, etc.
28731 (Generally the number 0 or 1 would be used for @expr{x} in this case.)
28733 The ``y'' entry may be any formula instead of a vector. Calc effectively
28734 uses @kbd{N} (@code{calc-eval-num}) to evaluate variables in the formula;
28735 the result of this must be a formula in a single (unassigned) variable.
28736 The formula is plotted with this variable taking on the various ``x''
28737 values. Graphs of formulas by default use lines without symbols at the
28738 computed data points. Note that if neither ``x'' nor ``y'' is a vector,
28739 Calc guesses at a reasonable number of data points to use. See the
28740 @kbd{g N} command below. (The ``x'' values must be either a vector
28741 or an interval if ``y'' is a formula.)
28747 If ``y'' is (or evaluates to) a formula of the form
28748 @samp{xy(@var{x}, @var{y})} then the result is a
28749 parametric plot. The two arguments of the fictitious @code{xy} function
28750 are used as the ``x'' and ``y'' coordinates of the curve, respectively.
28751 In this case the ``x'' vector or interval you specified is not directly
28752 visible in the graph. For example, if ``x'' is the interval @samp{[0..360]}
28753 and ``y'' is the formula @samp{xy(sin(t), cos(t))}, the resulting graph
28756 Also, ``x'' and ``y'' may each be variable names, in which case Calc
28757 looks for suitable vectors, intervals, or formulas stored in those
28760 The ``x'' and ``y'' values for the data points (as pulled from the vectors,
28761 calculated from the formulas, or interpolated from the intervals) should
28762 be real numbers (integers, fractions, or floats). If either the ``x''
28763 value or the ``y'' value of a given data point is not a real number, that
28764 data point will be omitted from the graph. The points on either side
28765 of the invalid point will @emph{not} be connected by a line.
28767 See the documentation for @kbd{g a} below for a description of the way
28768 numeric prefix arguments affect @kbd{g f}.
28770 @cindex @code{PlotRejects} variable
28771 @vindex PlotRejects
28772 If you store an empty vector in the variable @code{PlotRejects}
28773 (i.e., @kbd{[ ] s t PlotRejects}), Calc will append information to
28774 this vector for every data point which was rejected because its
28775 ``x'' or ``y'' values were not real numbers. The result will be
28776 a matrix where each row holds the curve number, data point number,
28777 ``x'' value, and ``y'' value for a rejected data point.
28778 @xref{Evaluates-To Operator}, for a handy way to keep tabs on the
28779 current value of @code{PlotRejects}. @xref{Operations on Variables},
28780 for the @kbd{s R} command which is another easy way to examine
28781 @code{PlotRejects}.
28784 @pindex calc-graph-clear
28785 To clear the graphics display, type @kbd{g c} (@code{calc-graph-clear}).
28786 If the GNUPLOT output device is an X window, the window will go away.
28787 Effects on other kinds of output devices will vary. You don't need
28788 to use @kbd{g c} if you don't want to---if you give another @kbd{g f}
28789 or @kbd{g p} command later on, it will reuse the existing graphics
28790 window if there is one.
28792 @node Three Dimensional Graphics, Managing Curves, Basic Graphics, Graphics
28793 @section Three-Dimensional Graphics
28796 @pindex calc-graph-fast-3d
28797 The @kbd{g F} (@code{calc-graph-fast-3d}) command makes a three-dimensional
28798 graph. It works only if you have GNUPLOT 3.0 or later; with GNUPLOT 2.0,
28799 you will see a GNUPLOT error message if you try this command.
28801 The @kbd{g F} command takes three values from the stack, called ``x'',
28802 ``y'', and ``z'', respectively. As was the case for 2D graphs, there
28803 are several options for these values.
28805 In the first case, ``x'' and ``y'' are each vectors (not necessarily of
28806 the same length); either or both may instead be interval forms. The
28807 ``z'' value must be a matrix with the same number of rows as elements
28808 in ``x'', and the same number of columns as elements in ``y''. The
28809 result is a surface plot where
28810 @texline @math{z_{ij}}
28811 @infoline @expr{z_ij}
28812 is the height of the point
28813 at coordinate @expr{(x_i, y_j)} on the surface. The 3D graph will
28814 be displayed from a certain default viewpoint; you can change this
28815 viewpoint by adding a @samp{set view} to the @samp{*Gnuplot Commands*}
28816 buffer as described later. See the GNUPLOT 3.0 documentation for a
28817 description of the @samp{set view} command.
28819 Each point in the matrix will be displayed as a dot in the graph,
28820 and these points will be connected by a grid of lines (@dfn{isolines}).
28822 In the second case, ``x'', ``y'', and ``z'' are all vectors of equal
28823 length. The resulting graph displays a 3D line instead of a surface,
28824 where the coordinates of points along the line are successive triplets
28825 of values from the input vectors.
28827 In the third case, ``x'' and ``y'' are vectors or interval forms, and
28828 ``z'' is any formula involving two variables (not counting variables
28829 with assigned values). These variables are sorted into alphabetical
28830 order; the first takes on values from ``x'' and the second takes on
28831 values from ``y'' to form a matrix of results that are graphed as a
28838 If the ``z'' formula evaluates to a call to the fictitious function
28839 @samp{xyz(@var{x}, @var{y}, @var{z})}, then the result is a
28840 ``parametric surface.'' In this case, the axes of the graph are
28841 taken from the @var{x} and @var{y} values in these calls, and the
28842 ``x'' and ``y'' values from the input vectors or intervals are used only
28843 to specify the range of inputs to the formula. For example, plotting
28844 @samp{[0..360], [0..180], xyz(sin(x)*sin(y), cos(x)*sin(y), cos(y))}
28845 will draw a sphere. (Since the default resolution for 3D plots is
28846 5 steps in each of ``x'' and ``y'', this will draw a very crude
28847 sphere. You could use the @kbd{g N} command, described below, to
28848 increase this resolution, or specify the ``x'' and ``y'' values as
28849 vectors with more than 5 elements.
28851 It is also possible to have a function in a regular @kbd{g f} plot
28852 evaluate to an @code{xyz} call. Since @kbd{g f} plots a line, not
28853 a surface, the result will be a 3D parametric line. For example,
28854 @samp{[[0..720], xyz(sin(x), cos(x), x)]} will plot two turns of a
28855 helix (a three-dimensional spiral).
28857 As for @kbd{g f}, each of ``x'', ``y'', and ``z'' may instead be
28858 variables containing the relevant data.
28860 @node Managing Curves, Graphics Options, Three Dimensional Graphics, Graphics
28861 @section Managing Curves
28864 The @kbd{g f} command is really shorthand for the following commands:
28865 @kbd{C-u g d g a g p}. Likewise, @w{@kbd{g F}} is shorthand for
28866 @kbd{C-u g d g A g p}. You can gain more control over your graph
28867 by using these commands directly.
28870 @pindex calc-graph-add
28871 The @kbd{g a} (@code{calc-graph-add}) command adds the ``curve''
28872 represented by the two values on the top of the stack to the current
28873 graph. You can have any number of curves in the same graph. When
28874 you give the @kbd{g p} command, all the curves will be drawn superimposed
28877 The @kbd{g a} command (and many others that affect the current graph)
28878 will cause a special buffer, @samp{*Gnuplot Commands*}, to be displayed
28879 in another window. This buffer is a template of the commands that will
28880 be sent to GNUPLOT when it is time to draw the graph. The first
28881 @kbd{g a} command adds a @code{plot} command to this buffer. Succeeding
28882 @kbd{g a} commands add extra curves onto that @code{plot} command.
28883 Other graph-related commands put other GNUPLOT commands into this
28884 buffer. In normal usage you never need to work with this buffer
28885 directly, but you can if you wish. The only constraint is that there
28886 must be only one @code{plot} command, and it must be the last command
28887 in the buffer. If you want to save and later restore a complete graph
28888 configuration, you can use regular Emacs commands to save and restore
28889 the contents of the @samp{*Gnuplot Commands*} buffer.
28893 If the values on the stack are not variable names, @kbd{g a} will invent
28894 variable names for them (of the form @samp{PlotData@var{n}}) and store
28895 the values in those variables. The ``x'' and ``y'' variables are what
28896 go into the @code{plot} command in the template. If you add a curve
28897 that uses a certain variable and then later change that variable, you
28898 can replot the graph without having to delete and re-add the curve.
28899 That's because the variable name, not the vector, interval or formula
28900 itself, is what was added by @kbd{g a}.
28902 A numeric prefix argument on @kbd{g a} or @kbd{g f} changes the way
28903 stack entries are interpreted as curves. With a positive prefix
28904 argument @expr{n}, the top @expr{n} stack entries are ``y'' values
28905 for @expr{n} different curves which share a common ``x'' value in
28906 the @expr{n+1}st stack entry. (Thus @kbd{g a} with no prefix
28907 argument is equivalent to @kbd{C-u 1 g a}.)
28909 A prefix of zero or plain @kbd{C-u} means to take two stack entries,
28910 ``x'' and ``y'' as usual, but to interpret ``y'' as a vector of
28911 ``y'' values for several curves that share a common ``x''.
28913 A negative prefix argument tells Calc to read @expr{n} vectors from
28914 the stack; each vector @expr{[x, y]} describes an independent curve.
28915 This is the only form of @kbd{g a} that creates several curves at once
28916 that don't have common ``x'' values. (Of course, the range of ``x''
28917 values covered by all the curves ought to be roughly the same if
28918 they are to look nice on the same graph.)
28920 For example, to plot
28921 @texline @math{\sin n x}
28922 @infoline @expr{sin(n x)}
28923 for integers @expr{n}
28924 from 1 to 5, you could use @kbd{v x} to create a vector of integers
28925 (@expr{n}), then @kbd{V M '} or @kbd{V M $} to map @samp{sin(n x)}
28926 across this vector. The resulting vector of formulas is suitable
28927 for use as the ``y'' argument to a @kbd{C-u g a} or @kbd{C-u g f}
28931 @pindex calc-graph-add-3d
28932 The @kbd{g A} (@code{calc-graph-add-3d}) command adds a 3D curve
28933 to the graph. It is not valid to intermix 2D and 3D curves in a
28934 single graph. This command takes three arguments, ``x'', ``y'',
28935 and ``z'', from the stack. With a positive prefix @expr{n}, it
28936 takes @expr{n+2} arguments (common ``x'' and ``y'', plus @expr{n}
28937 separate ``z''s). With a zero prefix, it takes three stack entries
28938 but the ``z'' entry is a vector of curve values. With a negative
28939 prefix @expr{-n}, it takes @expr{n} vectors of the form @expr{[x, y, z]}.
28940 The @kbd{g A} command works by adding a @code{splot} (surface-plot)
28941 command to the @samp{*Gnuplot Commands*} buffer.
28943 (Although @kbd{g a} adds a 2D @code{plot} command to the
28944 @samp{*Gnuplot Commands*} buffer, Calc changes this to @code{splot}
28945 before sending it to GNUPLOT if it notices that the data points are
28946 evaluating to @code{xyz} calls. It will not work to mix 2D and 3D
28947 @kbd{g a} curves in a single graph, although Calc does not currently
28951 @pindex calc-graph-delete
28952 The @kbd{g d} (@code{calc-graph-delete}) command deletes the most
28953 recently added curve from the graph. It has no effect if there are
28954 no curves in the graph. With a numeric prefix argument of any kind,
28955 it deletes all of the curves from the graph.
28958 @pindex calc-graph-hide
28959 The @kbd{g H} (@code{calc-graph-hide}) command ``hides'' or ``unhides''
28960 the most recently added curve. A hidden curve will not appear in
28961 the actual plot, but information about it such as its name and line and
28962 point styles will be retained.
28965 @pindex calc-graph-juggle
28966 The @kbd{g j} (@code{calc-graph-juggle}) command moves the curve
28967 at the end of the list (the ``most recently added curve'') to the
28968 front of the list. The next-most-recent curve is thus exposed for
28969 @w{@kbd{g d}} or similar commands to use. With @kbd{g j} you can work
28970 with any curve in the graph even though curve-related commands only
28971 affect the last curve in the list.
28974 @pindex calc-graph-plot
28975 The @kbd{g p} (@code{calc-graph-plot}) command uses GNUPLOT to draw
28976 the graph described in the @samp{*Gnuplot Commands*} buffer. Any
28977 GNUPLOT parameters which are not defined by commands in this buffer
28978 are reset to their default values. The variables named in the @code{plot}
28979 command are written to a temporary data file and the variable names
28980 are then replaced by the file name in the template. The resulting
28981 plotting commands are fed to the GNUPLOT program. See the documentation
28982 for the GNUPLOT program for more specific information. All temporary
28983 files are removed when Emacs or GNUPLOT exits.
28985 If you give a formula for ``y'', Calc will remember all the values that
28986 it calculates for the formula so that later plots can reuse these values.
28987 Calc throws out these saved values when you change any circumstances
28988 that may affect the data, such as switching from Degrees to Radians
28989 mode, or changing the value of a parameter in the formula. You can
28990 force Calc to recompute the data from scratch by giving a negative
28991 numeric prefix argument to @kbd{g p}.
28993 Calc uses a fairly rough step size when graphing formulas over intervals.
28994 This is to ensure quick response. You can ``refine'' a plot by giving
28995 a positive numeric prefix argument to @kbd{g p}. Calc goes through
28996 the data points it has computed and saved from previous plots of the
28997 function, and computes and inserts a new data point midway between
28998 each of the existing points. You can refine a plot any number of times,
28999 but beware that the amount of calculation involved doubles each time.
29001 Calc does not remember computed values for 3D graphs. This means the
29002 numerix prefix argument, if any, to @kbd{g p} is effectively ignored if
29003 the current graph is three-dimensional.
29006 @pindex calc-graph-print
29007 The @kbd{g P} (@code{calc-graph-print}) command is like @kbd{g p},
29008 except that it sends the output to a printer instead of to the
29009 screen. More precisely, @kbd{g p} looks for @samp{set terminal}
29010 or @samp{set output} commands in the @samp{*Gnuplot Commands*} buffer;
29011 lacking these it uses the default settings. However, @kbd{g P}
29012 ignores @samp{set terminal} and @samp{set output} commands and
29013 uses a different set of default values. All of these values are
29014 controlled by the @kbd{g D} and @kbd{g O} commands discussed below.
29015 Provided everything is set up properly, @kbd{g p} will plot to
29016 the screen unless you have specified otherwise and @kbd{g P} will
29017 always plot to the printer.
29019 @node Graphics Options, Devices, Managing Curves, Graphics
29020 @section Graphics Options
29024 @pindex calc-graph-grid
29025 The @kbd{g g} (@code{calc-graph-grid}) command turns the ``grid''
29026 on and off. It is off by default; tick marks appear only at the
29027 edges of the graph. With the grid turned on, dotted lines appear
29028 across the graph at each tick mark. Note that this command only
29029 changes the setting in @samp{*Gnuplot Commands*}; to see the effects
29030 of the change you must give another @kbd{g p} command.
29033 @pindex calc-graph-border
29034 The @kbd{g b} (@code{calc-graph-border}) command turns the border
29035 (the box that surrounds the graph) on and off. It is on by default.
29036 This command will only work with GNUPLOT 3.0 and later versions.
29039 @pindex calc-graph-key
29040 The @kbd{g k} (@code{calc-graph-key}) command turns the ``key''
29041 on and off. The key is a chart in the corner of the graph that
29042 shows the correspondence between curves and line styles. It is
29043 off by default, and is only really useful if you have several
29044 curves on the same graph.
29047 @pindex calc-graph-num-points
29048 The @kbd{g N} (@code{calc-graph-num-points}) command allows you
29049 to select the number of data points in the graph. This only affects
29050 curves where neither ``x'' nor ``y'' is specified as a vector.
29051 Enter a blank line to revert to the default value (initially 15).
29052 With no prefix argument, this command affects only the current graph.
29053 With a positive prefix argument this command changes or, if you enter
29054 a blank line, displays the default number of points used for all
29055 graphs created by @kbd{g a} that don't specify the resolution explicitly.
29056 With a negative prefix argument, this command changes or displays
29057 the default value (initially 5) used for 3D graphs created by @kbd{g A}.
29058 Note that a 3D setting of 5 means that a total of @expr{5^2 = 25} points
29059 will be computed for the surface.
29061 Data values in the graph of a function are normally computed to a
29062 precision of five digits, regardless of the current precision at the
29063 time. This is usually more than adequate, but there are cases where
29064 it will not be. For example, plotting @expr{1 + x} with @expr{x} in the
29065 interval @samp{[0 ..@: 1e-6]} will round all the data points down
29066 to 1.0! Putting the command @samp{set precision @var{n}} in the
29067 @samp{*Gnuplot Commands*} buffer will cause the data to be computed
29068 at precision @var{n} instead of 5. Since this is such a rare case,
29069 there is no keystroke-based command to set the precision.
29072 @pindex calc-graph-header
29073 The @kbd{g h} (@code{calc-graph-header}) command sets the title
29074 for the graph. This will show up centered above the graph.
29075 The default title is blank (no title).
29078 @pindex calc-graph-name
29079 The @kbd{g n} (@code{calc-graph-name}) command sets the title of an
29080 individual curve. Like the other curve-manipulating commands, it
29081 affects the most recently added curve, i.e., the last curve on the
29082 list in the @samp{*Gnuplot Commands*} buffer. To set the title of
29083 the other curves you must first juggle them to the end of the list
29084 with @kbd{g j}, or edit the @samp{*Gnuplot Commands*} buffer by hand.
29085 Curve titles appear in the key; if the key is turned off they are
29090 @pindex calc-graph-title-x
29091 @pindex calc-graph-title-y
29092 The @kbd{g t} (@code{calc-graph-title-x}) and @kbd{g T}
29093 (@code{calc-graph-title-y}) commands set the titles on the ``x''
29094 and ``y'' axes, respectively. These titles appear next to the
29095 tick marks on the left and bottom edges of the graph, respectively.
29096 Calc does not have commands to control the tick marks themselves,
29097 but you can edit them into the @samp{*Gnuplot Commands*} buffer if
29098 you wish. See the GNUPLOT documentation for details.
29102 @pindex calc-graph-range-x
29103 @pindex calc-graph-range-y
29104 The @kbd{g r} (@code{calc-graph-range-x}) and @kbd{g R}
29105 (@code{calc-graph-range-y}) commands set the range of values on the
29106 ``x'' and ``y'' axes, respectively. You are prompted to enter a
29107 suitable range. This should be either a pair of numbers of the
29108 form, @samp{@var{min}:@var{max}}, or a blank line to revert to the
29109 default behavior of setting the range based on the range of values
29110 in the data, or @samp{$} to take the range from the top of the stack.
29111 Ranges on the stack can be represented as either interval forms or
29112 vectors: @samp{[@var{min} ..@: @var{max}]} or @samp{[@var{min}, @var{max}]}.
29116 @pindex calc-graph-log-x
29117 @pindex calc-graph-log-y
29118 The @kbd{g l} (@code{calc-graph-log-x}) and @kbd{g L} (@code{calc-graph-log-y})
29119 commands allow you to set either or both of the axes of the graph to
29120 be logarithmic instead of linear.
29125 @pindex calc-graph-log-z
29126 @pindex calc-graph-range-z
29127 @pindex calc-graph-title-z
29128 For 3D plots, @kbd{g C-t}, @kbd{g C-r}, and @kbd{g C-l} (those are
29129 letters with the Control key held down) are the corresponding commands
29130 for the ``z'' axis.
29134 @pindex calc-graph-zero-x
29135 @pindex calc-graph-zero-y
29136 The @kbd{g z} (@code{calc-graph-zero-x}) and @kbd{g Z}
29137 (@code{calc-graph-zero-y}) commands control whether a dotted line is
29138 drawn to indicate the ``x'' and/or ``y'' zero axes. (These are the same
29139 dotted lines that would be drawn there anyway if you used @kbd{g g} to
29140 turn the ``grid'' feature on.) Zero-axis lines are on by default, and
29141 may be turned off only in GNUPLOT 3.0 and later versions. They are
29142 not available for 3D plots.
29145 @pindex calc-graph-line-style
29146 The @kbd{g s} (@code{calc-graph-line-style}) command turns the connecting
29147 lines on or off for the most recently added curve, and optionally selects
29148 the style of lines to be used for that curve. Plain @kbd{g s} simply
29149 toggles the lines on and off. With a numeric prefix argument, @kbd{g s}
29150 turns lines on and sets a particular line style. Line style numbers
29151 start at one and their meanings vary depending on the output device.
29152 GNUPLOT guarantees that there will be at least six different line styles
29153 available for any device.
29156 @pindex calc-graph-point-style
29157 The @kbd{g S} (@code{calc-graph-point-style}) command similarly turns
29158 the symbols at the data points on or off, or sets the point style.
29159 If you turn both lines and points off, the data points will show as
29162 @cindex @code{LineStyles} variable
29163 @cindex @code{PointStyles} variable
29165 @vindex PointStyles
29166 Another way to specify curve styles is with the @code{LineStyles} and
29167 @code{PointStyles} variables. These variables initially have no stored
29168 values, but if you store a vector of integers in one of these variables,
29169 the @kbd{g a} and @kbd{g f} commands will use those style numbers
29170 instead of the defaults for new curves that are added to the graph.
29171 An entry should be a positive integer for a specific style, or 0 to let
29172 the style be chosen automatically, or @mathit{-1} to turn off lines or points
29173 altogether. If there are more curves than elements in the vector, the
29174 last few curves will continue to have the default styles. Of course,
29175 you can later use @kbd{g s} and @kbd{g S} to change any of these styles.
29177 For example, @kbd{'[2 -1 3] @key{RET} s t LineStyles} causes the first curve
29178 to have lines in style number 2, the second curve to have no connecting
29179 lines, and the third curve to have lines in style 3. Point styles will
29180 still be assigned automatically, but you could store another vector in
29181 @code{PointStyles} to define them, too.
29183 @node Devices, , Graphics Options, Graphics
29184 @section Graphical Devices
29188 @pindex calc-graph-device
29189 The @kbd{g D} (@code{calc-graph-device}) command sets the device name
29190 (or ``terminal name'' in GNUPLOT lingo) to be used by @kbd{g p} commands
29191 on this graph. It does not affect the permanent default device name.
29192 If you enter a blank name, the device name reverts to the default.
29193 Enter @samp{?} to see a list of supported devices.
29195 With a positive numeric prefix argument, @kbd{g D} instead sets
29196 the default device name, used by all plots in the future which do
29197 not override it with a plain @kbd{g D} command. If you enter a
29198 blank line this command shows you the current default. The special
29199 name @code{default} signifies that Calc should choose @code{x11} if
29200 the X window system is in use (as indicated by the presence of a
29201 @code{DISPLAY} environment variable), or otherwise @code{dumb} under
29202 GNUPLOT 3.0 and later, or @code{postscript} under GNUPLOT 2.0.
29203 This is the initial default value.
29205 The @code{dumb} device is an interface to ``dumb terminals,'' i.e.,
29206 terminals with no special graphics facilities. It writes a crude
29207 picture of the graph composed of characters like @code{-} and @code{|}
29208 to a buffer called @samp{*Gnuplot Trail*}, which Calc then displays.
29209 The graph is made the same size as the Emacs screen, which on most
29210 dumb terminals will be
29211 @texline @math{80\times24}
29213 characters. The graph is displayed in
29214 an Emacs ``recursive edit''; type @kbd{q} or @kbd{C-c C-c} to exit
29215 the recursive edit and return to Calc. Note that the @code{dumb}
29216 device is present only in GNUPLOT 3.0 and later versions.
29218 The word @code{dumb} may be followed by two numbers separated by
29219 spaces. These are the desired width and height of the graph in
29220 characters. Also, the device name @code{big} is like @code{dumb}
29221 but creates a graph four times the width and height of the Emacs
29222 screen. You will then have to scroll around to view the entire
29223 graph. In the @samp{*Gnuplot Trail*} buffer, @key{SPC}, @key{DEL},
29224 @kbd{<}, and @kbd{>} are defined to scroll by one screenful in each
29225 of the four directions.
29227 With a negative numeric prefix argument, @kbd{g D} sets or displays
29228 the device name used by @kbd{g P} (@code{calc-graph-print}). This
29229 is initially @code{postscript}. If you don't have a PostScript
29230 printer, you may decide once again to use @code{dumb} to create a
29231 plot on any text-only printer.
29234 @pindex calc-graph-output
29235 The @kbd{g O} (@code{calc-graph-output}) command sets the name of
29236 the output file used by GNUPLOT. For some devices, notably @code{x11},
29237 there is no output file and this information is not used. Many other
29238 ``devices'' are really file formats like @code{postscript}; in these
29239 cases the output in the desired format goes into the file you name
29240 with @kbd{g O}. Type @kbd{g O stdout @key{RET}} to set GNUPLOT to write
29241 to its standard output stream, i.e., to @samp{*Gnuplot Trail*}.
29242 This is the default setting.
29244 Another special output name is @code{tty}, which means that GNUPLOT
29245 is going to write graphics commands directly to its standard output,
29246 which you wish Emacs to pass through to your terminal. Tektronix
29247 graphics terminals, among other devices, operate this way. Calc does
29248 this by telling GNUPLOT to write to a temporary file, then running a
29249 sub-shell executing the command @samp{cat tempfile >/dev/tty}. On
29250 typical Unix systems, this will copy the temporary file directly to
29251 the terminal, bypassing Emacs entirely. You will have to type @kbd{C-l}
29252 to Emacs afterwards to refresh the screen.
29254 Once again, @kbd{g O} with a positive or negative prefix argument
29255 sets the default or printer output file names, respectively. In each
29256 case you can specify @code{auto}, which causes Calc to invent a temporary
29257 file name for each @kbd{g p} (or @kbd{g P}) command. This temporary file
29258 will be deleted once it has been displayed or printed. If the output file
29259 name is not @code{auto}, the file is not automatically deleted.
29261 The default and printer devices and output files can be saved
29262 permanently by the @kbd{m m} (@code{calc-save-modes}) command. The
29263 default number of data points (see @kbd{g N}) and the X geometry
29264 (see @kbd{g X}) are also saved. Other graph information is @emph{not}
29265 saved; you can save a graph's configuration simply by saving the contents
29266 of the @samp{*Gnuplot Commands*} buffer.
29268 @vindex calc-gnuplot-plot-command
29269 @vindex calc-gnuplot-default-device
29270 @vindex calc-gnuplot-default-output
29271 @vindex calc-gnuplot-print-command
29272 @vindex calc-gnuplot-print-device
29273 @vindex calc-gnuplot-print-output
29274 You may wish to configure the default and
29275 printer devices and output files for the whole system. The relevant
29276 Lisp variables are @code{calc-gnuplot-default-device} and @code{-output},
29277 and @code{calc-gnuplot-print-device} and @code{-output}. The output
29278 file names must be either strings as described above, or Lisp
29279 expressions which are evaluated on the fly to get the output file names.
29281 Other important Lisp variables are @code{calc-gnuplot-plot-command} and
29282 @code{calc-gnuplot-print-command}, which give the system commands to
29283 display or print the output of GNUPLOT, respectively. These may be
29284 @code{nil} if no command is necessary, or strings which can include
29285 @samp{%s} to signify the name of the file to be displayed or printed.
29286 Or, these variables may contain Lisp expressions which are evaluated
29287 to display or print the output. These variables are customizable
29288 (@pxref{Customizable Variables}).
29291 @pindex calc-graph-display
29292 The @kbd{g x} (@code{calc-graph-display}) command lets you specify
29293 on which X window system display your graphs should be drawn. Enter
29294 a blank line to see the current display name. This command has no
29295 effect unless the current device is @code{x11}.
29298 @pindex calc-graph-geometry
29299 The @kbd{g X} (@code{calc-graph-geometry}) command is a similar
29300 command for specifying the position and size of the X window.
29301 The normal value is @code{default}, which generally means your
29302 window manager will let you place the window interactively.
29303 Entering @samp{800x500+0+0} would create an 800-by-500 pixel
29304 window in the upper-left corner of the screen.
29306 The buffer called @samp{*Gnuplot Trail*} holds a transcript of the
29307 session with GNUPLOT. This shows the commands Calc has ``typed'' to
29308 GNUPLOT and the responses it has received. Calc tries to notice when an
29309 error message has appeared here and display the buffer for you when
29310 this happens. You can check this buffer yourself if you suspect
29311 something has gone wrong.
29314 @pindex calc-graph-command
29315 The @kbd{g C} (@code{calc-graph-command}) command prompts you to
29316 enter any line of text, then simply sends that line to the current
29317 GNUPLOT process. The @samp{*Gnuplot Trail*} buffer looks deceptively
29318 like a Shell buffer but you can't type commands in it yourself.
29319 Instead, you must use @kbd{g C} for this purpose.
29323 @pindex calc-graph-view-commands
29324 @pindex calc-graph-view-trail
29325 The @kbd{g v} (@code{calc-graph-view-commands}) and @kbd{g V}
29326 (@code{calc-graph-view-trail}) commands display the @samp{*Gnuplot Commands*}
29327 and @samp{*Gnuplot Trail*} buffers, respectively, in another window.
29328 This happens automatically when Calc thinks there is something you
29329 will want to see in either of these buffers. If you type @kbd{g v}
29330 or @kbd{g V} when the relevant buffer is already displayed, the
29331 buffer is hidden again.
29333 One reason to use @kbd{g v} is to add your own commands to the
29334 @samp{*Gnuplot Commands*} buffer. Press @kbd{g v}, then use
29335 @kbd{C-x o} to switch into that window. For example, GNUPLOT has
29336 @samp{set label} and @samp{set arrow} commands that allow you to
29337 annotate your plots. Since Calc doesn't understand these commands,
29338 you have to add them to the @samp{*Gnuplot Commands*} buffer
29339 yourself, then use @w{@kbd{g p}} to replot using these new commands. Note
29340 that your commands must appear @emph{before} the @code{plot} command.
29341 To get help on any GNUPLOT feature, type, e.g., @kbd{g C help set label}.
29342 You may have to type @kbd{g C @key{RET}} a few times to clear the
29343 ``press return for more'' or ``subtopic of @dots{}'' requests.
29344 Note that Calc always sends commands (like @samp{set nolabel}) to
29345 reset all plotting parameters to the defaults before each plot, so
29346 to delete a label all you need to do is delete the @samp{set label}
29347 line you added (or comment it out with @samp{#}) and then replot
29351 @pindex calc-graph-quit
29352 You can use @kbd{g q} (@code{calc-graph-quit}) to kill the GNUPLOT
29353 process that is running. The next graphing command you give will
29354 start a fresh GNUPLOT process. The word @samp{Graph} appears in
29355 the Calc window's mode line whenever a GNUPLOT process is currently
29356 running. The GNUPLOT process is automatically killed when you
29357 exit Emacs if you haven't killed it manually by then.
29360 @pindex calc-graph-kill
29361 The @kbd{g K} (@code{calc-graph-kill}) command is like @kbd{g q}
29362 except that it also views the @samp{*Gnuplot Trail*} buffer so that
29363 you can see the process being killed. This is better if you are
29364 killing GNUPLOT because you think it has gotten stuck.
29366 @node Kill and Yank, Keypad Mode, Graphics, Top
29367 @chapter Kill and Yank Functions
29370 The commands in this chapter move information between the Calculator and
29371 other Emacs editing buffers.
29373 In many cases Embedded mode is an easier and more natural way to
29374 work with Calc from a regular editing buffer. @xref{Embedded Mode}.
29377 * Killing From Stack::
29378 * Yanking Into Stack::
29379 * Grabbing From Buffers::
29380 * Yanking Into Buffers::
29381 * X Cut and Paste::
29384 @node Killing From Stack, Yanking Into Stack, Kill and Yank, Kill and Yank
29385 @section Killing from the Stack
29391 @pindex calc-copy-as-kill
29393 @pindex calc-kill-region
29395 @pindex calc-copy-region-as-kill
29397 @dfn{Kill} commands are Emacs commands that insert text into the
29398 ``kill ring,'' from which it can later be ``yanked'' by a @kbd{C-y}
29399 command. Three common kill commands in normal Emacs are @kbd{C-k}, which
29400 kills one line, @kbd{C-w}, which kills the region between mark and point,
29401 and @kbd{M-w}, which puts the region into the kill ring without actually
29402 deleting it. All of these commands work in the Calculator, too. Also,
29403 @kbd{M-k} has been provided to complete the set; it puts the current line
29404 into the kill ring without deleting anything.
29406 The kill commands are unusual in that they pay attention to the location
29407 of the cursor in the Calculator buffer. If the cursor is on or below the
29408 bottom line, the kill commands operate on the top of the stack. Otherwise,
29409 they operate on whatever stack element the cursor is on. Calc's kill
29410 commands always operate on whole stack entries. (They act the same as their
29411 standard Emacs cousins except they ``round up'' the specified region to
29412 encompass full lines.) The text is copied into the kill ring exactly as
29413 it appears on the screen, including line numbers if they are enabled.
29415 A numeric prefix argument to @kbd{C-k} or @kbd{M-k} affects the number
29416 of lines killed. A positive argument kills the current line and @expr{n-1}
29417 lines below it. A negative argument kills the @expr{-n} lines above the
29418 current line. Again this mirrors the behavior of the standard Emacs
29419 @kbd{C-k} command. Although a whole line is always deleted, @kbd{C-k}
29420 with no argument copies only the number itself into the kill ring, whereas
29421 @kbd{C-k} with a prefix argument of 1 copies the number with its trailing
29424 @node Yanking Into Stack, Grabbing From Buffers, Killing From Stack, Kill and Yank
29425 @section Yanking into the Stack
29430 The @kbd{C-y} command yanks the most recently killed text back into the
29431 Calculator. It pushes this value onto the top of the stack regardless of
29432 the cursor position. In general it re-parses the killed text as a number
29433 or formula (or a list of these separated by commas or newlines). However if
29434 the thing being yanked is something that was just killed from the Calculator
29435 itself, its full internal structure is yanked. For example, if you have
29436 set the floating-point display mode to show only four significant digits,
29437 then killing and re-yanking 3.14159 (which displays as 3.142) will yank the
29438 full 3.14159, even though yanking it into any other buffer would yank the
29439 number in its displayed form, 3.142. (Since the default display modes
29440 show all objects to their full precision, this feature normally makes no
29443 @node Grabbing From Buffers, Yanking Into Buffers, Yanking Into Stack, Kill and Yank
29444 @section Grabbing from Other Buffers
29448 @pindex calc-grab-region
29449 The @kbd{M-# g} (@code{calc-grab-region}) command takes the text between
29450 point and mark in the current buffer and attempts to parse it as a
29451 vector of values. Basically, it wraps the text in vector brackets
29452 @samp{[ ]} unless the text already is enclosed in vector brackets,
29453 then reads the text as if it were an algebraic entry. The contents
29454 of the vector may be numbers, formulas, or any other Calc objects.
29455 If the @kbd{M-# g} command works successfully, it does an automatic
29456 @kbd{M-# c} to enter the Calculator buffer.
29458 A numeric prefix argument grabs the specified number of lines around
29459 point, ignoring the mark. A positive prefix grabs from point to the
29460 @expr{n}th following newline (so that @kbd{M-1 M-# g} grabs from point
29461 to the end of the current line); a negative prefix grabs from point
29462 back to the @expr{n+1}st preceding newline. In these cases the text
29463 that is grabbed is exactly the same as the text that @kbd{C-k} would
29464 delete given that prefix argument.
29466 A prefix of zero grabs the current line; point may be anywhere on the
29469 A plain @kbd{C-u} prefix interprets the region between point and mark
29470 as a single number or formula rather than a vector. For example,
29471 @kbd{M-# g} on the text @samp{2 a b} produces the vector of three
29472 values @samp{[2, a, b]}, but @kbd{C-u M-# g} on the same region
29473 reads a formula which is a product of three things: @samp{2 a b}.
29474 (The text @samp{a + b}, on the other hand, will be grabbed as a
29475 vector of one element by plain @kbd{M-# g} because the interpretation
29476 @samp{[a, +, b]} would be a syntax error.)
29478 If a different language has been specified (@pxref{Language Modes}),
29479 the grabbed text will be interpreted according to that language.
29482 @pindex calc-grab-rectangle
29483 The @kbd{M-# r} (@code{calc-grab-rectangle}) command takes the text between
29484 point and mark and attempts to parse it as a matrix. If point and mark
29485 are both in the leftmost column, the lines in between are parsed in their
29486 entirety. Otherwise, point and mark define the corners of a rectangle
29487 whose contents are parsed.
29489 Each line of the grabbed area becomes a row of the matrix. The result
29490 will actually be a vector of vectors, which Calc will treat as a matrix
29491 only if every row contains the same number of values.
29493 If a line contains a portion surrounded by square brackets (or curly
29494 braces), that portion is interpreted as a vector which becomes a row
29495 of the matrix. Any text surrounding the bracketed portion on the line
29498 Otherwise, the entire line is interpreted as a row vector as if it
29499 were surrounded by square brackets. Leading line numbers (in the
29500 format used in the Calc stack buffer) are ignored. If you wish to
29501 force this interpretation (even if the line contains bracketed
29502 portions), give a negative numeric prefix argument to the
29503 @kbd{M-# r} command.
29505 If you give a numeric prefix argument of zero or plain @kbd{C-u}, each
29506 line is instead interpreted as a single formula which is converted into
29507 a one-element vector. Thus the result of @kbd{C-u M-# r} will be a
29508 one-column matrix. For example, suppose one line of the data is the
29509 expression @samp{2 a}. A plain @w{@kbd{M-# r}} will interpret this as
29510 @samp{[2 a]}, which in turn is read as a two-element vector that forms
29511 one row of the matrix. But a @kbd{C-u M-# r} will interpret this row
29514 If you give a positive numeric prefix argument @var{n}, then each line
29515 will be split up into columns of width @var{n}; each column is parsed
29516 separately as a matrix element. If a line contained
29517 @w{@samp{2 +/- 3 4 +/- 5}}, then grabbing with a prefix argument of 8
29518 would correctly split the line into two error forms.
29520 @xref{Matrix Functions}, to see how to pull the matrix apart into its
29521 constituent rows and columns. (If it is a
29522 @texline @math{1\times1}
29524 matrix, just hit @kbd{v u} (@code{calc-unpack}) twice.)
29528 @pindex calc-grab-sum-across
29529 @pindex calc-grab-sum-down
29530 @cindex Summing rows and columns of data
29531 The @kbd{M-# :} (@code{calc-grab-sum-down}) command is a handy way to
29532 grab a rectangle of data and sum its columns. It is equivalent to
29533 typing @kbd{M-# r}, followed by @kbd{V R : +} (the vector reduction
29534 command that sums the columns of a matrix; @pxref{Reducing}). The
29535 result of the command will be a vector of numbers, one for each column
29536 in the input data. The @kbd{M-# _} (@code{calc-grab-sum-across}) command
29537 similarly grabs a rectangle and sums its rows by executing @w{@kbd{V R _ +}}.
29539 As well as being more convenient, @kbd{M-# :} and @kbd{M-# _} are also
29540 much faster because they don't actually place the grabbed vector on
29541 the stack. In a @kbd{M-# r V R : +} sequence, formatting the vector
29542 for display on the stack takes a large fraction of the total time
29543 (unless you have planned ahead and used @kbd{v .} and @kbd{t .} modes).
29545 For example, suppose we have a column of numbers in a file which we
29546 wish to sum. Go to one corner of the column and press @kbd{C-@@} to
29547 set the mark; go to the other corner and type @kbd{M-# :}. Since there
29548 is only one column, the result will be a vector of one number, the sum.
29549 (You can type @kbd{v u} to unpack this vector into a plain number if
29550 you want to do further arithmetic with it.)
29552 To compute the product of the column of numbers, we would have to do
29553 it ``by hand'' since there's no special grab-and-multiply command.
29554 Use @kbd{M-# r} to grab the column of numbers into the calculator in
29555 the form of a column matrix. The statistics command @kbd{u *} is a
29556 handy way to find the product of a vector or matrix of numbers.
29557 @xref{Statistical Operations}. Another approach would be to use
29558 an explicit column reduction command, @kbd{V R : *}.
29560 @node Yanking Into Buffers, X Cut and Paste, Grabbing From Buffers, Kill and Yank
29561 @section Yanking into Other Buffers
29565 @pindex calc-copy-to-buffer
29566 The plain @kbd{y} (@code{calc-copy-to-buffer}) command inserts the number
29567 at the top of the stack into the most recently used normal editing buffer.
29568 (More specifically, this is the most recently used buffer which is displayed
29569 in a window and whose name does not begin with @samp{*}. If there is no
29570 such buffer, this is the most recently used buffer except for Calculator
29571 and Calc Trail buffers.) The number is inserted exactly as it appears and
29572 without a newline. (If line-numbering is enabled, the line number is
29573 normally not included.) The number is @emph{not} removed from the stack.
29575 With a prefix argument, @kbd{y} inserts several numbers, one per line.
29576 A positive argument inserts the specified number of values from the top
29577 of the stack. A negative argument inserts the @expr{n}th value from the
29578 top of the stack. An argument of zero inserts the entire stack. Note
29579 that @kbd{y} with an argument of 1 is slightly different from @kbd{y}
29580 with no argument; the former always copies full lines, whereas the
29581 latter strips off the trailing newline.
29583 With a lone @kbd{C-u} as a prefix argument, @kbd{y} @emph{replaces} the
29584 region in the other buffer with the yanked text, then quits the
29585 Calculator, leaving you in that buffer. A typical use would be to use
29586 @kbd{M-# g} to read a region of data into the Calculator, operate on the
29587 data to produce a new matrix, then type @kbd{C-u y} to replace the
29588 original data with the new data. One might wish to alter the matrix
29589 display style (@pxref{Vector and Matrix Formats}) or change the current
29590 display language (@pxref{Language Modes}) before doing this. Also, note
29591 that this command replaces a linear region of text (as grabbed by
29592 @kbd{M-# g}), not a rectangle (as grabbed by @kbd{M-# r}).
29594 If the editing buffer is in overwrite (as opposed to insert) mode,
29595 and the @kbd{C-u} prefix was not used, then the yanked number will
29596 overwrite the characters following point rather than being inserted
29597 before those characters. The usual conventions of overwrite mode
29598 are observed; for example, characters will be inserted at the end of
29599 a line rather than overflowing onto the next line. Yanking a multi-line
29600 object such as a matrix in overwrite mode overwrites the next @var{n}
29601 lines in the buffer, lengthening or shortening each line as necessary.
29602 Finally, if the thing being yanked is a simple integer or floating-point
29603 number (like @samp{-1.2345e-3}) and the characters following point also
29604 make up such a number, then Calc will replace that number with the new
29605 number, lengthening or shortening as necessary. The concept of
29606 ``overwrite mode'' has thus been generalized from overwriting characters
29607 to overwriting one complete number with another.
29610 The @kbd{M-# y} key sequence is equivalent to @kbd{y} except that
29611 it can be typed anywhere, not just in Calc. This provides an easy
29612 way to guarantee that Calc knows which editing buffer you want to use!
29614 @node X Cut and Paste, , Yanking Into Buffers, Kill and Yank
29615 @section X Cut and Paste
29618 If you are using Emacs with the X window system, there is an easier
29619 way to move small amounts of data into and out of the calculator:
29620 Use the mouse-oriented cut and paste facilities of X.
29622 The default bindings for a three-button mouse cause the left button
29623 to move the Emacs cursor to the given place, the right button to
29624 select the text between the cursor and the clicked location, and
29625 the middle button to yank the selection into the buffer at the
29626 clicked location. So, if you have a Calc window and an editing
29627 window on your Emacs screen, you can use left-click/right-click
29628 to select a number, vector, or formula from one window, then
29629 middle-click to paste that value into the other window. When you
29630 paste text into the Calc window, Calc interprets it as an algebraic
29631 entry. It doesn't matter where you click in the Calc window; the
29632 new value is always pushed onto the top of the stack.
29634 The @code{xterm} program that is typically used for general-purpose
29635 shell windows in X interprets the mouse buttons in the same way.
29636 So you can use the mouse to move data between Calc and any other
29637 Unix program. One nice feature of @code{xterm} is that a double
29638 left-click selects one word, and a triple left-click selects a
29639 whole line. So you can usually transfer a single number into Calc
29640 just by double-clicking on it in the shell, then middle-clicking
29641 in the Calc window.
29643 @node Keypad Mode, Embedded Mode, Kill and Yank, Introduction
29644 @chapter Keypad Mode
29648 @pindex calc-keypad
29649 The @kbd{M-# k} (@code{calc-keypad}) command starts the Calculator
29650 and displays a picture of a calculator-style keypad. If you are using
29651 the X window system, you can click on any of the ``keys'' in the
29652 keypad using the left mouse button to operate the calculator.
29653 The original window remains the selected window; in Keypad mode
29654 you can type in your file while simultaneously performing
29655 calculations with the mouse.
29657 @pindex full-calc-keypad
29658 If you have used @kbd{M-# b} first, @kbd{M-# k} instead invokes
29659 the @code{full-calc-keypad} command, which takes over the whole
29660 Emacs screen and displays the keypad, the Calc stack, and the Calc
29661 trail all at once. This mode would normally be used when running
29662 Calc standalone (@pxref{Standalone Operation}).
29664 If you aren't using the X window system, you must switch into
29665 the @samp{*Calc Keypad*} window, place the cursor on the desired
29666 ``key,'' and type @key{SPC} or @key{RET}. If you think this
29667 is easier than using Calc normally, go right ahead.
29669 Calc commands are more or less the same in Keypad mode. Certain
29670 keypad keys differ slightly from the corresponding normal Calc
29671 keystrokes; all such deviations are described below.
29673 Keypad mode includes many more commands than will fit on the keypad
29674 at once. Click the right mouse button [@code{calc-keypad-menu}]
29675 to switch to the next menu. The bottom five rows of the keypad
29676 stay the same; the top three rows change to a new set of commands.
29677 To return to earlier menus, click the middle mouse button
29678 [@code{calc-keypad-menu-back}] or simply advance through the menus
29679 until you wrap around. Typing @key{TAB} inside the keypad window
29680 is equivalent to clicking the right mouse button there.
29682 You can always click the @key{EXEC} button and type any normal
29683 Calc key sequence. This is equivalent to switching into the
29684 Calc buffer, typing the keys, then switching back to your
29688 * Keypad Main Menu::
29689 * Keypad Functions Menu::
29690 * Keypad Binary Menu::
29691 * Keypad Vectors Menu::
29692 * Keypad Modes Menu::
29695 @node Keypad Main Menu, Keypad Functions Menu, Keypad Mode, Keypad Mode
29700 |----+-----Calc 2.00-----+----1
29701 |FLR |CEIL|RND |TRNC|CLN2|FLT |
29702 |----+----+----+----+----+----|
29703 | LN |EXP | |ABS |IDIV|MOD |
29704 |----+----+----+----+----+----|
29705 |SIN |COS |TAN |SQRT|y^x |1/x |
29706 |----+----+----+----+----+----|
29707 | ENTER |+/- |EEX |UNDO| <- |
29708 |-----+---+-+--+--+-+---++----|
29709 | INV | 7 | 8 | 9 | / |
29710 |-----+-----+-----+-----+-----|
29711 | HYP | 4 | 5 | 6 | * |
29712 |-----+-----+-----+-----+-----|
29713 |EXEC | 1 | 2 | 3 | - |
29714 |-----+-----+-----+-----+-----|
29715 | OFF | 0 | . | PI | + |
29716 |-----+-----+-----+-----+-----+
29721 This is the menu that appears the first time you start Keypad mode.
29722 It will show up in a vertical window on the right side of your screen.
29723 Above this menu is the traditional Calc stack display. On a 24-line
29724 screen you will be able to see the top three stack entries.
29726 The ten digit keys, decimal point, and @key{EEX} key are used for
29727 entering numbers in the obvious way. @key{EEX} begins entry of an
29728 exponent in scientific notation. Just as with regular Calc, the
29729 number is pushed onto the stack as soon as you press @key{ENTER}
29730 or any other function key.
29732 The @key{+/-} key corresponds to normal Calc's @kbd{n} key. During
29733 numeric entry it changes the sign of the number or of the exponent.
29734 At other times it changes the sign of the number on the top of the
29737 The @key{INV} and @key{HYP} keys modify other keys. As well as
29738 having the effects described elsewhere in this manual, Keypad mode
29739 defines several other ``inverse'' operations. These are described
29740 below and in the following sections.
29742 The @key{ENTER} key finishes the current numeric entry, or otherwise
29743 duplicates the top entry on the stack.
29745 The @key{UNDO} key undoes the most recent Calc operation.
29746 @kbd{INV UNDO} is the ``redo'' command, and @kbd{HYP UNDO} is
29747 ``last arguments'' (@kbd{M-@key{RET}}).
29749 The @key{<-} key acts as a ``backspace'' during numeric entry.
29750 At other times it removes the top stack entry. @kbd{INV <-}
29751 clears the entire stack. @kbd{HYP <-} takes an integer from
29752 the stack, then removes that many additional stack elements.
29754 The @key{EXEC} key prompts you to enter any keystroke sequence
29755 that would normally work in Calc mode. This can include a
29756 numeric prefix if you wish. It is also possible simply to
29757 switch into the Calc window and type commands in it; there is
29758 nothing ``magic'' about this window when Keypad mode is active.
29760 The other keys in this display perform their obvious calculator
29761 functions. @key{CLN2} rounds the top-of-stack by temporarily
29762 reducing the precision by 2 digits. @key{FLT} converts an
29763 integer or fraction on the top of the stack to floating-point.
29765 The @key{INV} and @key{HYP} keys combined with several of these keys
29766 give you access to some common functions even if the appropriate menu
29767 is not displayed. Obviously you don't need to learn these keys
29768 unless you find yourself wasting time switching among the menus.
29772 is the same as @key{1/x}.
29774 is the same as @key{SQRT}.
29776 is the same as @key{CONJ}.
29778 is the same as @key{y^x}.
29780 is the same as @key{INV y^x} (the @expr{x}th root of @expr{y}).
29782 are the same as @key{SIN} / @kbd{INV SIN}.
29784 are the same as @key{COS} / @kbd{INV COS}.
29786 are the same as @key{TAN} / @kbd{INV TAN}.
29788 are the same as @key{LN} / @kbd{HYP LN}.
29790 are the same as @key{EXP} / @kbd{HYP EXP}.
29792 is the same as @key{ABS}.
29794 is the same as @key{RND} (@code{calc-round}).
29796 is the same as @key{CLN2}.
29798 is the same as @key{FLT} (@code{calc-float}).
29800 is the same as @key{IMAG}.
29802 is the same as @key{PREC}.
29804 is the same as @key{SWAP}.
29806 is the same as @key{RLL3}.
29807 @item INV HYP ENTER
29808 is the same as @key{OVER}.
29810 packs the top two stack entries as an error form.
29812 packs the top two stack entries as a modulo form.
29814 creates an interval form; this removes an integer which is one
29815 of 0 @samp{[]}, 1 @samp{[)}, 2 @samp{(]} or 3 @samp{()}, followed
29816 by the two limits of the interval.
29819 The @kbd{OFF} key turns Calc off; typing @kbd{M-# k} or @kbd{M-# M-#}
29820 again has the same effect. This is analogous to typing @kbd{q} or
29821 hitting @kbd{M-# c} again in the normal calculator. If Calc is
29822 running standalone (the @code{full-calc-keypad} command appeared in the
29823 command line that started Emacs), then @kbd{OFF} is replaced with
29824 @kbd{EXIT}; clicking on this actually exits Emacs itself.
29826 @node Keypad Functions Menu, Keypad Binary Menu, Keypad Main Menu, Keypad Mode
29827 @section Functions Menu
29831 |----+----+----+----+----+----2
29832 |IGAM|BETA|IBET|ERF |BESJ|BESY|
29833 |----+----+----+----+----+----|
29834 |IMAG|CONJ| RE |ATN2|RAND|RAGN|
29835 |----+----+----+----+----+----|
29836 |GCD |FACT|DFCT|BNOM|PERM|NXTP|
29837 |----+----+----+----+----+----|
29842 This menu provides various operations from the @kbd{f} and @kbd{k}
29845 @key{IMAG} multiplies the number on the stack by the imaginary
29846 number @expr{i = (0, 1)}.
29848 @key{RE} extracts the real part a complex number. @kbd{INV RE}
29849 extracts the imaginary part.
29851 @key{RAND} takes a number from the top of the stack and computes
29852 a random number greater than or equal to zero but less than that
29853 number. (@xref{Random Numbers}.) @key{RAGN} is the ``random
29854 again'' command; it computes another random number using the
29855 same limit as last time.
29857 @key{INV GCD} computes the LCM (least common multiple) function.
29859 @key{INV FACT} is the gamma function.
29860 @texline @math{\Gamma(x) = (x-1)!}.
29861 @infoline @expr{gamma(x) = (x-1)!}.
29863 @key{PERM} is the number-of-permutations function, which is on the
29864 @kbd{H k c} key in normal Calc.
29866 @key{NXTP} finds the next prime after a number. @kbd{INV NXTP}
29867 finds the previous prime.
29869 @node Keypad Binary Menu, Keypad Vectors Menu, Keypad Functions Menu, Keypad Mode
29870 @section Binary Menu
29874 |----+----+----+----+----+----3
29875 |AND | OR |XOR |NOT |LSH |RSH |
29876 |----+----+----+----+----+----|
29877 |DEC |HEX |OCT |BIN |WSIZ|ARSH|
29878 |----+----+----+----+----+----|
29879 | A | B | C | D | E | F |
29880 |----+----+----+----+----+----|
29885 The keys in this menu perform operations on binary integers.
29886 Note that both logical and arithmetic right-shifts are provided.
29887 @key{INV LSH} rotates one bit to the left.
29889 The ``difference'' function (normally on @kbd{b d}) is on @key{INV AND}.
29890 The ``clip'' function (normally on @w{@kbd{b c}}) is on @key{INV NOT}.
29892 The @key{DEC}, @key{HEX}, @key{OCT}, and @key{BIN} keys select the
29893 current radix for display and entry of numbers: Decimal, hexadecimal,
29894 octal, or binary. The six letter keys @key{A} through @key{F} are used
29895 for entering hexadecimal numbers.
29897 The @key{WSIZ} key displays the current word size for binary operations
29898 and allows you to enter a new word size. You can respond to the prompt
29899 using either the keyboard or the digits and @key{ENTER} from the keypad.
29900 The initial word size is 32 bits.
29902 @node Keypad Vectors Menu, Keypad Modes Menu, Keypad Binary Menu, Keypad Mode
29903 @section Vectors Menu
29907 |----+----+----+----+----+----4
29908 |SUM |PROD|MAX |MAP*|MAP^|MAP$|
29909 |----+----+----+----+----+----|
29910 |MINV|MDET|MTRN|IDNT|CRSS|"x" |
29911 |----+----+----+----+----+----|
29912 |PACK|UNPK|INDX|BLD |LEN |... |
29913 |----+----+----+----+----+----|
29918 The keys in this menu operate on vectors and matrices.
29920 @key{PACK} removes an integer @var{n} from the top of the stack;
29921 the next @var{n} stack elements are removed and packed into a vector,
29922 which is replaced onto the stack. Thus the sequence
29923 @kbd{1 ENTER 3 ENTER 5 ENTER 3 PACK} enters the vector
29924 @samp{[1, 3, 5]} onto the stack. To enter a matrix, build each row
29925 on the stack as a vector, then use a final @key{PACK} to collect the
29926 rows into a matrix.
29928 @key{UNPK} unpacks the vector on the stack, pushing each of its
29929 components separately.
29931 @key{INDX} removes an integer @var{n}, then builds a vector of
29932 integers from 1 to @var{n}. @kbd{INV INDX} takes three numbers
29933 from the stack: The vector size @var{n}, the starting number,
29934 and the increment. @kbd{BLD} takes an integer @var{n} and any
29935 value @var{x} and builds a vector of @var{n} copies of @var{x}.
29937 @key{IDNT} removes an integer @var{n}, then builds an @var{n}-by-@var{n}
29940 @key{LEN} replaces a vector by its length, an integer.
29942 @key{...} turns on or off ``abbreviated'' display mode for large vectors.
29944 @key{MINV}, @key{MDET}, @key{MTRN}, and @key{CROSS} are the matrix
29945 inverse, determinant, and transpose, and vector cross product.
29947 @key{SUM} replaces a vector by the sum of its elements. It is
29948 equivalent to @kbd{u +} in normal Calc (@pxref{Statistical Operations}).
29949 @key{PROD} computes the product of the elements of a vector, and
29950 @key{MAX} computes the maximum of all the elements of a vector.
29952 @key{INV SUM} computes the alternating sum of the first element
29953 minus the second, plus the third, minus the fourth, and so on.
29954 @key{INV MAX} computes the minimum of the vector elements.
29956 @key{HYP SUM} computes the mean of the vector elements.
29957 @key{HYP PROD} computes the sample standard deviation.
29958 @key{HYP MAX} computes the median.
29960 @key{MAP*} multiplies two vectors elementwise. It is equivalent
29961 to the @kbd{V M *} command. @key{MAP^} computes powers elementwise.
29962 The arguments must be vectors of equal length, or one must be a vector
29963 and the other must be a plain number. For example, @kbd{2 MAP^} squares
29964 all the elements of a vector.
29966 @key{MAP$} maps the formula on the top of the stack across the
29967 vector in the second-to-top position. If the formula contains
29968 several variables, Calc takes that many vectors starting at the
29969 second-to-top position and matches them to the variables in
29970 alphabetical order. The result is a vector of the same size as
29971 the input vectors, whose elements are the formula evaluated with
29972 the variables set to the various sets of numbers in those vectors.
29973 For example, you could simulate @key{MAP^} using @key{MAP$} with
29974 the formula @samp{x^y}.
29976 The @kbd{"x"} key pushes the variable name @expr{x} onto the
29977 stack. To build the formula @expr{x^2 + 6}, you would use the
29978 key sequence @kbd{"x" 2 y^x 6 +}. This formula would then be
29979 suitable for use with the @key{MAP$} key described above.
29980 With @key{INV}, @key{HYP}, or @key{INV} and @key{HYP}, the
29981 @kbd{"x"} key pushes the variable names @expr{y}, @expr{z}, and
29982 @expr{t}, respectively.
29984 @node Keypad Modes Menu, , Keypad Vectors Menu, Keypad Mode
29985 @section Modes Menu
29989 |----+----+----+----+----+----5
29990 |FLT |FIX |SCI |ENG |GRP | |
29991 |----+----+----+----+----+----|
29992 |RAD |DEG |FRAC|POLR|SYMB|PREC|
29993 |----+----+----+----+----+----|
29994 |SWAP|RLL3|RLL4|OVER|STO |RCL |
29995 |----+----+----+----+----+----|
30000 The keys in this menu manipulate modes, variables, and the stack.
30002 The @key{FLT}, @key{FIX}, @key{SCI}, and @key{ENG} keys select
30003 floating-point, fixed-point, scientific, or engineering notation.
30004 @key{FIX} displays two digits after the decimal by default; the
30005 others display full precision. With the @key{INV} prefix, these
30006 keys pop a number-of-digits argument from the stack.
30008 The @key{GRP} key turns grouping of digits with commas on or off.
30009 @kbd{INV GRP} enables grouping to the right of the decimal point as
30010 well as to the left.
30012 The @key{RAD} and @key{DEG} keys switch between radians and degrees
30013 for trigonometric functions.
30015 The @key{FRAC} key turns Fraction mode on or off. This affects
30016 whether commands like @kbd{/} with integer arguments produce
30017 fractional or floating-point results.
30019 The @key{POLR} key turns Polar mode on or off, determining whether
30020 polar or rectangular complex numbers are used by default.
30022 The @key{SYMB} key turns Symbolic mode on or off, in which
30023 operations that would produce inexact floating-point results
30024 are left unevaluated as algebraic formulas.
30026 The @key{PREC} key selects the current precision. Answer with
30027 the keyboard or with the keypad digit and @key{ENTER} keys.
30029 The @key{SWAP} key exchanges the top two stack elements.
30030 The @key{RLL3} key rotates the top three stack elements upwards.
30031 The @key{RLL4} key rotates the top four stack elements upwards.
30032 The @key{OVER} key duplicates the second-to-top stack element.
30034 The @key{STO} and @key{RCL} keys are analogous to @kbd{s t} and
30035 @kbd{s r} in regular Calc. @xref{Store and Recall}. Click the
30036 @key{STO} or @key{RCL} key, then one of the ten digits. (Named
30037 variables are not available in Keypad mode.) You can also use,
30038 for example, @kbd{STO + 3} to add to register 3.
30040 @node Embedded Mode, Programming, Keypad Mode, Top
30041 @chapter Embedded Mode
30044 Embedded mode in Calc provides an alternative to copying numbers
30045 and formulas back and forth between editing buffers and the Calc
30046 stack. In Embedded mode, your editing buffer becomes temporarily
30047 linked to the stack and this copying is taken care of automatically.
30050 * Basic Embedded Mode::
30051 * More About Embedded Mode::
30052 * Assignments in Embedded Mode::
30053 * Mode Settings in Embedded Mode::
30054 * Customizing Embedded Mode::
30057 @node Basic Embedded Mode, More About Embedded Mode, Embedded Mode, Embedded Mode
30058 @section Basic Embedded Mode
30062 @pindex calc-embedded
30063 To enter Embedded mode, position the Emacs point (cursor) on a
30064 formula in any buffer and press @kbd{M-# e} (@code{calc-embedded}).
30065 Note that @kbd{M-# e} is not to be used in the Calc stack buffer
30066 like most Calc commands, but rather in regular editing buffers that
30067 are visiting your own files.
30069 Calc will try to guess an appropriate language based on the major mode
30070 of the editing buffer. (@xref{Language Modes}.) If the current buffer is
30071 in @code{latex-mode}, for example, Calc will set its language to La@TeX{}.
30072 Similarly, Calc will use @TeX{} language for @code{tex-mode},
30073 @code{plain-tex-mode} and @code{context-mode}, C language for
30074 @code{c-mode} and @code{c++-mode}, FORTRAN language for
30075 @code{fortran-mode} and @code{f90-mode}, Pascal for @code{pascal-mode},
30076 and eqn for @code{nroff-mode} (@pxref{Customizable Variables}).
30077 These can be overridden with Calc's mode
30078 changing commands (@pxref{Mode Settings in Embedded Mode}). If no
30079 suitable language is available, Calc will continue with its current language.
30081 Calc normally scans backward and forward in the buffer for the
30082 nearest opening and closing @dfn{formula delimiters}. The simplest
30083 delimiters are blank lines. Other delimiters that Embedded mode
30088 The @TeX{} and La@TeX{} math delimiters @samp{$ $}, @samp{$$ $$},
30089 @samp{\[ \]}, and @samp{\( \)};
30091 Lines beginning with @samp{\begin} and @samp{\end} (except matrix delimiters);
30093 Lines beginning with @samp{@@} (Texinfo delimiters).
30095 Lines beginning with @samp{.EQ} and @samp{.EN} (@dfn{eqn} delimiters);
30097 Lines containing a single @samp{%} or @samp{.\"} symbol and nothing else.
30100 @xref{Customizing Embedded Mode}, to see how to make Calc recognize
30101 your own favorite delimiters. Delimiters like @samp{$ $} can appear
30102 on their own separate lines or in-line with the formula.
30104 If you give a positive or negative numeric prefix argument, Calc
30105 instead uses the current point as one end of the formula, and moves
30106 forward or backward (respectively) by that many lines to find the
30107 other end. Explicit delimiters are not necessary in this case.
30109 With a prefix argument of zero, Calc uses the current region
30110 (delimited by point and mark) instead of formula delimiters.
30113 @pindex calc-embedded-word
30114 With a prefix argument of @kbd{C-u} only, Calc scans for the first
30115 non-numeric character (i.e., the first character that is not a
30116 digit, sign, decimal point, or upper- or lower-case @samp{e})
30117 forward and backward to delimit the formula. @kbd{M-# w}
30118 (@code{calc-embedded-word}) is equivalent to @kbd{C-u M-# e}.
30120 When you enable Embedded mode for a formula, Calc reads the text
30121 between the delimiters and tries to interpret it as a Calc formula.
30122 Calc can generally identify @TeX{} formulas and
30123 Big-style formulas even if the language mode is wrong. If Calc
30124 can't make sense of the formula, it beeps and refuses to enter
30125 Embedded mode. But if the current language is wrong, Calc can
30126 sometimes parse the formula successfully (but incorrectly);
30127 for example, the C expression @samp{atan(a[1])} can be parsed
30128 in Normal language mode, but the @code{atan} won't correspond to
30129 the built-in @code{arctan} function, and the @samp{a[1]} will be
30130 interpreted as @samp{a} times the vector @samp{[1]}!
30132 If you press @kbd{M-# e} or @kbd{M-# w} to activate an embedded
30133 formula which is blank, say with the cursor on the space between
30134 the two delimiters @samp{$ $}, Calc will immediately prompt for
30135 an algebraic entry.
30137 Only one formula in one buffer can be enabled at a time. If you
30138 move to another area of the current buffer and give Calc commands,
30139 Calc turns Embedded mode off for the old formula and then tries
30140 to restart Embedded mode at the new position. Other buffers are
30141 not affected by Embedded mode.
30143 When Embedded mode begins, Calc pushes the current formula onto
30144 the stack. No Calc stack window is created; however, Calc copies
30145 the top-of-stack position into the original buffer at all times.
30146 You can create a Calc window by hand with @kbd{M-# o} if you
30147 find you need to see the entire stack.
30149 For example, typing @kbd{M-# e} while somewhere in the formula
30150 @samp{n>2} in the following line enables Embedded mode on that
30154 We define $F_n = F_(n-1)+F_(n-2)$ for all $n>2$.
30158 The formula @expr{n>2} will be pushed onto the Calc stack, and
30159 the top of stack will be copied back into the editing buffer.
30160 This means that spaces will appear around the @samp{>} symbol
30161 to match Calc's usual display style:
30164 We define $F_n = F_(n-1)+F_(n-2)$ for all $n > 2$.
30168 No spaces have appeared around the @samp{+} sign because it's
30169 in a different formula, one which we have not yet touched with
30172 Now that Embedded mode is enabled, keys you type in this buffer
30173 are interpreted as Calc commands. At this point we might use
30174 the ``commute'' command @kbd{j C} to reverse the inequality.
30175 This is a selection-based command for which we first need to
30176 move the cursor onto the operator (@samp{>} in this case) that
30177 needs to be commuted.
30180 We define $F_n = F_(n-1)+F_(n-2)$ for all $2 < n$.
30183 The @kbd{M-# o} command is a useful way to open a Calc window
30184 without actually selecting that window. Giving this command
30185 verifies that @samp{2 < n} is also on the Calc stack. Typing
30186 @kbd{17 @key{RET}} would produce:
30189 We define $F_n = F_(n-1)+F_(n-2)$ for all $17$.
30193 with @samp{2 < n} and @samp{17} on the stack; typing @key{TAB}
30194 at this point will exchange the two stack values and restore
30195 @samp{2 < n} to the embedded formula. Even though you can't
30196 normally see the stack in Embedded mode, it is still there and
30197 it still operates in the same way. But, as with old-fashioned
30198 RPN calculators, you can only see the value at the top of the
30199 stack at any given time (unless you use @kbd{M-# o}).
30201 Typing @kbd{M-# e} again turns Embedded mode off. The Calc
30202 window reveals that the formula @w{@samp{2 < n}} is automatically
30203 removed from the stack, but the @samp{17} is not. Entering
30204 Embedded mode always pushes one thing onto the stack, and
30205 leaving Embedded mode always removes one thing. Anything else
30206 that happens on the stack is entirely your business as far as
30207 Embedded mode is concerned.
30209 If you press @kbd{M-# e} in the wrong place by accident, it is
30210 possible that Calc will be able to parse the nearby text as a
30211 formula and will mangle that text in an attempt to redisplay it
30212 ``properly'' in the current language mode. If this happens,
30213 press @kbd{M-# e} again to exit Embedded mode, then give the
30214 regular Emacs ``undo'' command (@kbd{C-_} or @kbd{C-x u}) to put
30215 the text back the way it was before Calc edited it. Note that Calc's
30216 own Undo command (typed before you turn Embedded mode back off)
30217 will not do you any good, because as far as Calc is concerned
30218 you haven't done anything with this formula yet.
30220 @node More About Embedded Mode, Assignments in Embedded Mode, Basic Embedded Mode, Embedded Mode
30221 @section More About Embedded Mode
30224 When Embedded mode ``activates'' a formula, i.e., when it examines
30225 the formula for the first time since the buffer was created or
30226 loaded, Calc tries to sense the language in which the formula was
30227 written. If the formula contains any La@TeX{}-like @samp{\} sequences,
30228 it is parsed (i.e., read) in La@TeX{} mode. If the formula appears to
30229 be written in multi-line Big mode, it is parsed in Big mode. Otherwise,
30230 it is parsed according to the current language mode.
30232 Note that Calc does not change the current language mode according
30233 the formula it reads in. Even though it can read a La@TeX{} formula when
30234 not in La@TeX{} mode, it will immediately rewrite this formula using
30235 whatever language mode is in effect.
30242 @pindex calc-show-plain
30243 Calc's parser is unable to read certain kinds of formulas. For
30244 example, with @kbd{v ]} (@code{calc-matrix-brackets}) you can
30245 specify matrix display styles which the parser is unable to
30246 recognize as matrices. The @kbd{d p} (@code{calc-show-plain})
30247 command turns on a mode in which a ``plain'' version of a
30248 formula is placed in front of the fully-formatted version.
30249 When Calc reads a formula that has such a plain version in
30250 front, it reads the plain version and ignores the formatted
30253 Plain formulas are preceded and followed by @samp{%%%} signs
30254 by default. This notation has the advantage that the @samp{%}
30255 character begins a comment in @TeX{} and La@TeX{}, so if your formula is
30256 embedded in a @TeX{} or La@TeX{} document its plain version will be
30257 invisible in the final printed copy. @xref{Customizing
30258 Embedded Mode}, to see how to change the ``plain'' formula
30259 delimiters, say to something that @dfn{eqn} or some other
30260 formatter will treat as a comment.
30262 There are several notations which Calc's parser for ``big''
30263 formatted formulas can't yet recognize. In particular, it can't
30264 read the large symbols for @code{sum}, @code{prod}, and @code{integ},
30265 and it can't handle @samp{=>} with the righthand argument omitted.
30266 Also, Calc won't recognize special formats you have defined with
30267 the @kbd{Z C} command (@pxref{User-Defined Compositions}). In
30268 these cases it is important to use ``plain'' mode to make sure
30269 Calc will be able to read your formula later.
30271 Another example where ``plain'' mode is important is if you have
30272 specified a float mode with few digits of precision. Normally
30273 any digits that are computed but not displayed will simply be
30274 lost when you save and re-load your embedded buffer, but ``plain''
30275 mode allows you to make sure that the complete number is present
30276 in the file as well as the rounded-down number.
30282 Embedded buffers remember active formulas for as long as they
30283 exist in Emacs memory. Suppose you have an embedded formula
30284 which is @cpi{} to the normal 12 decimal places, and then
30285 type @w{@kbd{C-u 5 d n}} to display only five decimal places.
30286 If you then type @kbd{d n}, all 12 places reappear because the
30287 full number is still there on the Calc stack. More surprisingly,
30288 even if you exit Embedded mode and later re-enter it for that
30289 formula, typing @kbd{d n} will restore all 12 places because
30290 each buffer remembers all its active formulas. However, if you
30291 save the buffer in a file and reload it in a new Emacs session,
30292 all non-displayed digits will have been lost unless you used
30299 In some applications of Embedded mode, you will want to have a
30300 sequence of copies of a formula that show its evolution as you
30301 work on it. For example, you might want to have a sequence
30302 like this in your file (elaborating here on the example from
30303 the ``Getting Started'' chapter):
30312 @r{(the derivative of }ln(ln(x))@r{)}
30314 whose value at x = 2 is
30324 @pindex calc-embedded-duplicate
30325 The @kbd{M-# d} (@code{calc-embedded-duplicate}) command is a
30326 handy way to make sequences like this. If you type @kbd{M-# d},
30327 the formula under the cursor (which may or may not have Embedded
30328 mode enabled for it at the time) is copied immediately below and
30329 Embedded mode is then enabled for that copy.
30331 For this example, you would start with just
30340 and press @kbd{M-# d} with the cursor on this formula. The result
30353 with the second copy of the formula enabled in Embedded mode.
30354 You can now press @kbd{a d x @key{RET}} to take the derivative, and
30355 @kbd{M-# d M-# d} to make two more copies of the derivative.
30356 To complete the computations, type @kbd{3 s l x @key{RET}} to evaluate
30357 the last formula, then move up to the second-to-last formula
30358 and type @kbd{2 s l x @key{RET}}.
30360 Finally, you would want to press @kbd{M-# e} to exit Embedded
30361 mode, then go up and insert the necessary text in between the
30362 various formulas and numbers.
30370 @pindex calc-embedded-new-formula
30371 The @kbd{M-# f} (@code{calc-embedded-new-formula}) command
30372 creates a new embedded formula at the current point. It inserts
30373 some default delimiters, which are usually just blank lines,
30374 and then does an algebraic entry to get the formula (which is
30375 then enabled for Embedded mode). This is just shorthand for
30376 typing the delimiters yourself, positioning the cursor between
30377 the new delimiters, and pressing @kbd{M-# e}. The key sequence
30378 @kbd{M-# '} is equivalent to @kbd{M-# f}.
30382 @pindex calc-embedded-next
30383 @pindex calc-embedded-previous
30384 The @kbd{M-# n} (@code{calc-embedded-next}) and @kbd{M-# p}
30385 (@code{calc-embedded-previous}) commands move the cursor to the
30386 next or previous active embedded formula in the buffer. They
30387 can take positive or negative prefix arguments to move by several
30388 formulas. Note that these commands do not actually examine the
30389 text of the buffer looking for formulas; they only see formulas
30390 which have previously been activated in Embedded mode. In fact,
30391 @kbd{M-# n} and @kbd{M-# p} are a useful way to tell which
30392 embedded formulas are currently active. Also, note that these
30393 commands do not enable Embedded mode on the next or previous
30394 formula, they just move the cursor. (By the way, @kbd{M-# n} is
30395 not as awkward to type as it may seem, because @kbd{M-#} ignores
30396 Shift and Meta on the second keystroke: @kbd{M-# M-N} can be typed
30397 by holding down Shift and Meta and alternately typing two keys.)
30400 @pindex calc-embedded-edit
30401 The @kbd{M-# `} (@code{calc-embedded-edit}) command edits the
30402 embedded formula at the current point as if by @kbd{`} (@code{calc-edit}).
30403 Embedded mode does not have to be enabled for this to work. Press
30404 @kbd{C-c C-c} to finish the edit, or @kbd{C-x k} to cancel.
30406 @node Assignments in Embedded Mode, Mode Settings in Embedded Mode, More About Embedded Mode, Embedded Mode
30407 @section Assignments in Embedded Mode
30410 The @samp{:=} (assignment) and @samp{=>} (``evaluates-to'') operators
30411 are especially useful in Embedded mode. They allow you to make
30412 a definition in one formula, then refer to that definition in
30413 other formulas embedded in the same buffer.
30415 An embedded formula which is an assignment to a variable, as in
30422 records @expr{5} as the stored value of @code{foo} for the
30423 purposes of Embedded mode operations in the current buffer. It
30424 does @emph{not} actually store @expr{5} as the ``global'' value
30425 of @code{foo}, however. Regular Calc operations, and Embedded
30426 formulas in other buffers, will not see this assignment.
30428 One way to use this assigned value is simply to create an
30429 Embedded formula elsewhere that refers to @code{foo}, and to press
30430 @kbd{=} in that formula. However, this permanently replaces the
30431 @code{foo} in the formula with its current value. More interesting
30432 is to use @samp{=>} elsewhere:
30438 @xref{Evaluates-To Operator}, for a general discussion of @samp{=>}.
30440 If you move back and change the assignment to @code{foo}, any
30441 @samp{=>} formulas which refer to it are automatically updated.
30449 The obvious question then is, @emph{how} can one easily change the
30450 assignment to @code{foo}? If you simply select the formula in
30451 Embedded mode and type 17, the assignment itself will be replaced
30452 by the 17. The effect on the other formula will be that the
30453 variable @code{foo} becomes unassigned:
30461 The right thing to do is first to use a selection command (@kbd{j 2}
30462 will do the trick) to select the righthand side of the assignment.
30463 Then, @kbd{17 @key{TAB} @key{DEL}} will swap the 17 into place (@pxref{Selecting
30464 Subformulas}, to see how this works).
30467 @pindex calc-embedded-select
30468 The @kbd{M-# j} (@code{calc-embedded-select}) command provides an
30469 easy way to operate on assignments. It is just like @kbd{M-# e},
30470 except that if the enabled formula is an assignment, it uses
30471 @kbd{j 2} to select the righthand side. If the enabled formula
30472 is an evaluates-to, it uses @kbd{j 1} to select the lefthand side.
30473 A formula can also be a combination of both:
30476 bar := foo + 3 => 20
30480 in which case @kbd{M-# j} will select the middle part (@samp{foo + 3}).
30482 The formula is automatically deselected when you leave Embedded
30487 @pindex calc-embedded-update
30488 Another way to change the assignment to @code{foo} would simply be
30489 to edit the number using regular Emacs editing rather than Embedded
30490 mode. Then, we have to find a way to get Embedded mode to notice
30491 the change. The @kbd{M-# u} or @kbd{M-# =}
30492 (@code{calc-embedded-update-formula}) command is a convenient way
30501 Pressing @kbd{M-# u} is much like pressing @kbd{M-# e = M-# e}, that
30502 is, temporarily enabling Embedded mode for the formula under the
30503 cursor and then evaluating it with @kbd{=}. But @kbd{M-# u} does
30504 not actually use @kbd{M-# e}, and in fact another formula somewhere
30505 else can be enabled in Embedded mode while you use @kbd{M-# u} and
30506 that formula will not be disturbed.
30508 With a numeric prefix argument, @kbd{M-# u} updates all active
30509 @samp{=>} formulas in the buffer. Formulas which have not yet
30510 been activated in Embedded mode, and formulas which do not have
30511 @samp{=>} as their top-level operator, are not affected by this.
30512 (This is useful only if you have used @kbd{m C}; see below.)
30514 With a plain @kbd{C-u} prefix, @kbd{C-u M-# u} updates only in the
30515 region between mark and point rather than in the whole buffer.
30517 @kbd{M-# u} is also a handy way to activate a formula, such as an
30518 @samp{=>} formula that has freshly been typed in or loaded from a
30522 @pindex calc-embedded-activate
30523 The @kbd{M-# a} (@code{calc-embedded-activate}) command scans
30524 through the current buffer and activates all embedded formulas
30525 that contain @samp{:=} or @samp{=>} symbols. This does not mean
30526 that Embedded mode is actually turned on, but only that the
30527 formulas' positions are registered with Embedded mode so that
30528 the @samp{=>} values can be properly updated as assignments are
30531 It is a good idea to type @kbd{M-# a} right after loading a file
30532 that uses embedded @samp{=>} operators. Emacs includes a nifty
30533 ``buffer-local variables'' feature that you can use to do this
30534 automatically. The idea is to place near the end of your file
30535 a few lines that look like this:
30538 --- Local Variables: ---
30539 --- eval:(calc-embedded-activate) ---
30544 where the leading and trailing @samp{---} can be replaced by
30545 any suitable strings (which must be the same on all three lines)
30546 or omitted altogether; in a @TeX{} or La@TeX{} file, @samp{%} would be a good
30547 leading string and no trailing string would be necessary. In a
30548 C program, @samp{/*} and @samp{*/} would be good leading and
30551 When Emacs loads a file into memory, it checks for a Local Variables
30552 section like this one at the end of the file. If it finds this
30553 section, it does the specified things (in this case, running
30554 @kbd{M-# a} automatically) before editing of the file begins.
30555 The Local Variables section must be within 3000 characters of the
30556 end of the file for Emacs to find it, and it must be in the last
30557 page of the file if the file has any page separators.
30558 @xref{File Variables, , Local Variables in Files, emacs, the
30561 Note that @kbd{M-# a} does not update the formulas it finds.
30562 To do this, type, say, @kbd{M-1 M-# u} after @w{@kbd{M-# a}}.
30563 Generally this should not be a problem, though, because the
30564 formulas will have been up-to-date already when the file was
30567 Normally, @kbd{M-# a} activates all the formulas it finds, but
30568 any previous active formulas remain active as well. With a
30569 positive numeric prefix argument, @kbd{M-# a} first deactivates
30570 all current active formulas, then actives the ones it finds in
30571 its scan of the buffer. With a negative prefix argument,
30572 @kbd{M-# a} simply deactivates all formulas.
30574 Embedded mode has two symbols, @samp{Active} and @samp{~Active},
30575 which it puts next to the major mode name in a buffer's mode line.
30576 It puts @samp{Active} if it has reason to believe that all
30577 formulas in the buffer are active, because you have typed @kbd{M-# a}
30578 and Calc has not since had to deactivate any formulas (which can
30579 happen if Calc goes to update an @samp{=>} formula somewhere because
30580 a variable changed, and finds that the formula is no longer there
30581 due to some kind of editing outside of Embedded mode). Calc puts
30582 @samp{~Active} in the mode line if some, but probably not all,
30583 formulas in the buffer are active. This happens if you activate
30584 a few formulas one at a time but never use @kbd{M-# a}, or if you
30585 used @kbd{M-# a} but then Calc had to deactivate a formula
30586 because it lost track of it. If neither of these symbols appears
30587 in the mode line, no embedded formulas are active in the buffer
30588 (e.g., before Embedded mode has been used, or after a @kbd{M-- M-# a}).
30590 Embedded formulas can refer to assignments both before and after them
30591 in the buffer. If there are several assignments to a variable, the
30592 nearest preceding assignment is used if there is one, otherwise the
30593 following assignment is used.
30607 As well as simple variables, you can also assign to subscript
30608 expressions of the form @samp{@var{var}_@var{number}} (as in
30609 @code{x_0}), or @samp{@var{var}_@var{var}} (as in @code{x_max}).
30610 Assignments to other kinds of objects can be represented by Calc,
30611 but the automatic linkage between assignments and references works
30612 only for plain variables and these two kinds of subscript expressions.
30614 If there are no assignments to a given variable, the global
30615 stored value for the variable is used (@pxref{Storing Variables}),
30616 or, if no value is stored, the variable is left in symbolic form.
30617 Note that global stored values will be lost when the file is saved
30618 and loaded in a later Emacs session, unless you have used the
30619 @kbd{s p} (@code{calc-permanent-variable}) command to save them;
30620 @pxref{Operations on Variables}.
30622 The @kbd{m C} (@code{calc-auto-recompute}) command turns automatic
30623 recomputation of @samp{=>} forms on and off. If you turn automatic
30624 recomputation off, you will have to use @kbd{M-# u} to update these
30625 formulas manually after an assignment has been changed. If you
30626 plan to change several assignments at once, it may be more efficient
30627 to type @kbd{m C}, change all the assignments, then use @kbd{M-1 M-# u}
30628 to update the entire buffer afterwards. The @kbd{m C} command also
30629 controls @samp{=>} formulas on the stack; @pxref{Evaluates-To
30630 Operator}. When you turn automatic recomputation back on, the
30631 stack will be updated but the Embedded buffer will not; you must
30632 use @kbd{M-# u} to update the buffer by hand.
30634 @node Mode Settings in Embedded Mode, Customizing Embedded Mode, Assignments in Embedded Mode, Embedded Mode
30635 @section Mode Settings in Embedded Mode
30638 @pindex calc-embedded-preserve-modes
30640 The mode settings can be changed while Calc is in embedded mode, but
30641 by default they will revert to their original values when embedded mode
30642 is ended. However, the modes saved when the mode-recording mode is
30643 @code{Save} (see below) and the modes in effect when the @kbd{m e}
30644 (@code{calc-embedded-preserve-modes}) command is given
30645 will be preserved when embedded mode is ended.
30647 Embedded mode has a rather complicated mechanism for handling mode
30648 settings in Embedded formulas. It is possible to put annotations
30649 in the file that specify mode settings either global to the entire
30650 file or local to a particular formula or formulas. In the latter
30651 case, different modes can be specified for use when a formula
30652 is the enabled Embedded mode formula.
30654 When you give any mode-setting command, like @kbd{m f} (for Fraction
30655 mode) or @kbd{d s} (for scientific notation), Embedded mode adds
30656 a line like the following one to the file just before the opening
30657 delimiter of the formula.
30660 % [calc-mode: fractions: t]
30661 % [calc-mode: float-format: (sci 0)]
30664 When Calc interprets an embedded formula, it scans the text before
30665 the formula for mode-setting annotations like these and sets the
30666 Calc buffer to match these modes. Modes not explicitly described
30667 in the file are not changed. Calc scans all the way to the top of
30668 the file, or up to a line of the form
30675 which you can insert at strategic places in the file if this backward
30676 scan is getting too slow, or just to provide a barrier between one
30677 ``zone'' of mode settings and another.
30679 If the file contains several annotations for the same mode, the
30680 closest one before the formula is used. Annotations after the
30681 formula are never used (except for global annotations, described
30684 The scan does not look for the leading @samp{% }, only for the
30685 square brackets and the text they enclose. You can edit the mode
30686 annotations to a style that works better in context if you wish.
30687 @xref{Customizing Embedded Mode}, to see how to change the style
30688 that Calc uses when it generates the annotations. You can write
30689 mode annotations into the file yourself if you know the syntax;
30690 the easiest way to find the syntax for a given mode is to let
30691 Calc write the annotation for it once and see what it does.
30693 If you give a mode-changing command for a mode that already has
30694 a suitable annotation just above the current formula, Calc will
30695 modify that annotation rather than generating a new, conflicting
30698 Mode annotations have three parts, separated by colons. (Spaces
30699 after the colons are optional.) The first identifies the kind
30700 of mode setting, the second is a name for the mode itself, and
30701 the third is the value in the form of a Lisp symbol, number,
30702 or list. Annotations with unrecognizable text in the first or
30703 second parts are ignored. The third part is not checked to make
30704 sure the value is of a valid type or range; if you write an
30705 annotation by hand, be sure to give a proper value or results
30706 will be unpredictable. Mode-setting annotations are case-sensitive.
30708 While Embedded mode is enabled, the word @code{Local} appears in
30709 the mode line. This is to show that mode setting commands generate
30710 annotations that are ``local'' to the current formula or set of
30711 formulas. The @kbd{m R} (@code{calc-mode-record-mode}) command
30712 causes Calc to generate different kinds of annotations. Pressing
30713 @kbd{m R} repeatedly cycles through the possible modes.
30715 @code{LocEdit} and @code{LocPerm} modes generate annotations
30716 that look like this, respectively:
30719 % [calc-edit-mode: float-format: (sci 0)]
30720 % [calc-perm-mode: float-format: (sci 5)]
30723 The first kind of annotation will be used only while a formula
30724 is enabled in Embedded mode. The second kind will be used only
30725 when the formula is @emph{not} enabled. (Whether the formula
30726 is ``active'' or not, i.e., whether Calc has seen this formula
30727 yet, is not relevant here.)
30729 @code{Global} mode generates an annotation like this at the end
30733 % [calc-global-mode: fractions t]
30736 Global mode annotations affect all formulas throughout the file,
30737 and may appear anywhere in the file. This allows you to tuck your
30738 mode annotations somewhere out of the way, say, on a new page of
30739 the file, as long as those mode settings are suitable for all
30740 formulas in the file.
30742 Enabling a formula with @kbd{M-# e} causes a fresh scan for local
30743 mode annotations; you will have to use this after adding annotations
30744 above a formula by hand to get the formula to notice them. Updating
30745 a formula with @kbd{M-# u} will also re-scan the local modes, but
30746 global modes are only re-scanned by @kbd{M-# a}.
30748 Another way that modes can get out of date is if you add a local
30749 mode annotation to a formula that has another formula after it.
30750 In this example, we have used the @kbd{d s} command while the
30751 first of the two embedded formulas is active. But the second
30752 formula has not changed its style to match, even though by the
30753 rules of reading annotations the @samp{(sci 0)} applies to it, too.
30756 % [calc-mode: float-format: (sci 0)]
30762 We would have to go down to the other formula and press @kbd{M-# u}
30763 on it in order to get it to notice the new annotation.
30765 Two more mode-recording modes selectable by @kbd{m R} are available
30766 which are also available outside of Embedded mode.
30767 (@pxref{General Mode Commands}.) They are @code{Save}, in which mode
30768 settings are recorded permanently in your Calc init file (the file given
30769 by the variable @code{calc-settings-file}, typically @file{~/.calc.el})
30770 rather than by annotating the current document, and no-recording
30771 mode (where there is no symbol like @code{Save} or @code{Local} in
30772 the mode line), in which mode-changing commands do not leave any
30773 annotations at all.
30775 When Embedded mode is not enabled, mode-recording modes except
30776 for @code{Save} have no effect.
30778 @node Customizing Embedded Mode, , Mode Settings in Embedded Mode, Embedded Mode
30779 @section Customizing Embedded Mode
30782 You can modify Embedded mode's behavior by setting various Lisp
30783 variables described here. These variables are customizable
30784 (@pxref{Customizable Variables}), or you can use @kbd{M-x set-variable}
30785 or @kbd{M-x edit-options} to adjust a variable on the fly.
30786 (Another possibility would
30787 be to use a file-local variable annotation at the end of the
30788 file; @pxref{File Variables, , Local Variables in Files, emacs, the
30791 While none of these variables will be buffer-local by default, you
30792 can make any of them local to any Embedded mode buffer. (Their
30793 values in the @samp{*Calculator*} buffer are never used.)
30795 @vindex calc-embedded-open-formula
30796 The @code{calc-embedded-open-formula} variable holds a regular
30797 expression for the opening delimiter of a formula. @xref{Regexp Search,
30798 , Regular Expression Search, emacs, the Emacs manual}, to see
30799 how regular expressions work. Basically, a regular expression is a
30800 pattern that Calc can search for. A regular expression that considers
30801 blank lines, @samp{$}, and @samp{$$} to be opening delimiters is
30802 @code{"\\`\\|^\n\\|\\$\\$?"}. Just in case the meaning of this
30803 regular expression is not completely plain, let's go through it
30806 The surrounding @samp{" "} marks quote the text between them as a
30807 Lisp string. If you left them off, @code{set-variable} or
30808 @code{edit-options} would try to read the regular expression as a
30811 The most obvious property of this regular expression is that it
30812 contains indecently many backslashes. There are actually two levels
30813 of backslash usage going on here. First, when Lisp reads a quoted
30814 string, all pairs of characters beginning with a backslash are
30815 interpreted as special characters. Here, @code{\n} changes to a
30816 new-line character, and @code{\\} changes to a single backslash.
30817 So the actual regular expression seen by Calc is
30818 @samp{\`\|^ @r{(newline)} \|\$\$?}.
30820 Regular expressions also consider pairs beginning with backslash
30821 to have special meanings. Sometimes the backslash is used to quote
30822 a character that otherwise would have a special meaning in a regular
30823 expression, like @samp{$}, which normally means ``end-of-line,''
30824 or @samp{?}, which means that the preceding item is optional. So
30825 @samp{\$\$?} matches either one or two dollar signs.
30827 The other codes in this regular expression are @samp{^}, which matches
30828 ``beginning-of-line,'' @samp{\|}, which means ``or,'' and @samp{\`},
30829 which matches ``beginning-of-buffer.'' So the whole pattern means
30830 that a formula begins at the beginning of the buffer, or on a newline
30831 that occurs at the beginning of a line (i.e., a blank line), or at
30832 one or two dollar signs.
30834 The default value of @code{calc-embedded-open-formula} looks just
30835 like this example, with several more alternatives added on to
30836 recognize various other common kinds of delimiters.
30838 By the way, the reason to use @samp{^\n} rather than @samp{^$}
30839 or @samp{\n\n}, which also would appear to match blank lines,
30840 is that the former expression actually ``consumes'' only one
30841 newline character as @emph{part of} the delimiter, whereas the
30842 latter expressions consume zero or two newlines, respectively.
30843 The former choice gives the most natural behavior when Calc
30844 must operate on a whole formula including its delimiters.
30846 See the Emacs manual for complete details on regular expressions.
30847 But just for your convenience, here is a list of all characters
30848 which must be quoted with backslash (like @samp{\$}) to avoid
30849 some special interpretation: @samp{. * + ? [ ] ^ $ \}. (Note
30850 the backslash in this list; for example, to match @samp{\[} you
30851 must use @code{"\\\\\\["}. An exercise for the reader is to
30852 account for each of these six backslashes!)
30854 @vindex calc-embedded-close-formula
30855 The @code{calc-embedded-close-formula} variable holds a regular
30856 expression for the closing delimiter of a formula. A closing
30857 regular expression to match the above example would be
30858 @code{"\\'\\|\n$\\|\\$\\$?"}. This is almost the same as the
30859 other one, except it now uses @samp{\'} (``end-of-buffer'') and
30860 @samp{\n$} (newline occurring at end of line, yet another way
30861 of describing a blank line that is more appropriate for this
30864 @vindex calc-embedded-open-word
30865 @vindex calc-embedded-close-word
30866 The @code{calc-embedded-open-word} and @code{calc-embedded-close-word}
30867 variables are similar expressions used when you type @kbd{M-# w}
30868 instead of @kbd{M-# e} to enable Embedded mode.
30870 @vindex calc-embedded-open-plain
30871 The @code{calc-embedded-open-plain} variable is a string which
30872 begins a ``plain'' formula written in front of the formatted
30873 formula when @kbd{d p} mode is turned on. Note that this is an
30874 actual string, not a regular expression, because Calc must be able
30875 to write this string into a buffer as well as to recognize it.
30876 The default string is @code{"%%% "} (note the trailing space).
30878 @vindex calc-embedded-close-plain
30879 The @code{calc-embedded-close-plain} variable is a string which
30880 ends a ``plain'' formula. The default is @code{" %%%\n"}. Without
30881 the trailing newline here, the first line of a Big mode formula
30882 that followed might be shifted over with respect to the other lines.
30884 @vindex calc-embedded-open-new-formula
30885 The @code{calc-embedded-open-new-formula} variable is a string
30886 which is inserted at the front of a new formula when you type
30887 @kbd{M-# f}. Its default value is @code{"\n\n"}. If this
30888 string begins with a newline character and the @kbd{M-# f} is
30889 typed at the beginning of a line, @kbd{M-# f} will skip this
30890 first newline to avoid introducing unnecessary blank lines in
30893 @vindex calc-embedded-close-new-formula
30894 The @code{calc-embedded-close-new-formula} variable is the corresponding
30895 string which is inserted at the end of a new formula. Its default
30896 value is also @code{"\n\n"}. The final newline is omitted by
30897 @w{@kbd{M-# f}} if typed at the end of a line. (It follows that if
30898 @kbd{M-# f} is typed on a blank line, both a leading opening
30899 newline and a trailing closing newline are omitted.)
30901 @vindex calc-embedded-announce-formula
30902 The @code{calc-embedded-announce-formula} variable is a regular
30903 expression which is sure to be followed by an embedded formula.
30904 The @kbd{M-# a} command searches for this pattern as well as for
30905 @samp{=>} and @samp{:=} operators. Note that @kbd{M-# a} will
30906 not activate just anything surrounded by formula delimiters; after
30907 all, blank lines are considered formula delimiters by default!
30908 But if your language includes a delimiter which can only occur
30909 actually in front of a formula, you can take advantage of it here.
30910 The default pattern is @code{"%Embed\n\\(% .*\n\\)*"}, which
30911 checks for @samp{%Embed} followed by any number of lines beginning
30912 with @samp{%} and a space. This last is important to make Calc
30913 consider mode annotations part of the pattern, so that the formula's
30914 opening delimiter really is sure to follow the pattern.
30916 @vindex calc-embedded-open-mode
30917 The @code{calc-embedded-open-mode} variable is a string (not a
30918 regular expression) which should precede a mode annotation.
30919 Calc never scans for this string; Calc always looks for the
30920 annotation itself. But this is the string that is inserted before
30921 the opening bracket when Calc adds an annotation on its own.
30922 The default is @code{"% "}.
30924 @vindex calc-embedded-close-mode
30925 The @code{calc-embedded-close-mode} variable is a string which
30926 follows a mode annotation written by Calc. Its default value
30927 is simply a newline, @code{"\n"}. If you change this, it is a
30928 good idea still to end with a newline so that mode annotations
30929 will appear on lines by themselves.
30931 @node Programming, Customizable Variables, Embedded Mode, Top
30932 @chapter Programming
30935 There are several ways to ``program'' the Emacs Calculator, depending
30936 on the nature of the problem you need to solve.
30940 @dfn{Keyboard macros} allow you to record a sequence of keystrokes
30941 and play them back at a later time. This is just the standard Emacs
30942 keyboard macro mechanism, dressed up with a few more features such
30943 as loops and conditionals.
30946 @dfn{Algebraic definitions} allow you to use any formula to define a
30947 new function. This function can then be used in algebraic formulas or
30948 as an interactive command.
30951 @dfn{Rewrite rules} are discussed in the section on algebra commands.
30952 @xref{Rewrite Rules}. If you put your rewrite rules in the variable
30953 @code{EvalRules}, they will be applied automatically to all Calc
30954 results in just the same way as an internal ``rule'' is applied to
30955 evaluate @samp{sqrt(9)} to 3 and so on. @xref{Automatic Rewrites}.
30958 @dfn{Lisp} is the programming language that Calc (and most of Emacs)
30959 is written in. If the above techniques aren't powerful enough, you
30960 can write Lisp functions to do anything that built-in Calc commands
30961 can do. Lisp code is also somewhat faster than keyboard macros or
30966 Programming features are available through the @kbd{z} and @kbd{Z}
30967 prefix keys. New commands that you define are two-key sequences
30968 beginning with @kbd{z}. Commands for managing these definitions
30969 use the shift-@kbd{Z} prefix. (The @kbd{Z T} (@code{calc-timing})
30970 command is described elsewhere; @pxref{Troubleshooting Commands}.
30971 The @kbd{Z C} (@code{calc-user-define-composition}) command is also
30972 described elsewhere; @pxref{User-Defined Compositions}.)
30975 * Creating User Keys::
30976 * Keyboard Macros::
30977 * Invocation Macros::
30978 * Algebraic Definitions::
30979 * Lisp Definitions::
30982 @node Creating User Keys, Keyboard Macros, Programming, Programming
30983 @section Creating User Keys
30987 @pindex calc-user-define
30988 Any Calculator command may be bound to a key using the @kbd{Z D}
30989 (@code{calc-user-define}) command. Actually, it is bound to a two-key
30990 sequence beginning with the lower-case @kbd{z} prefix.
30992 The @kbd{Z D} command first prompts for the key to define. For example,
30993 press @kbd{Z D a} to define the new key sequence @kbd{z a}. You are then
30994 prompted for the name of the Calculator command that this key should
30995 run. For example, the @code{calc-sincos} command is not normally
30996 available on a key. Typing @kbd{Z D s sincos @key{RET}} programs the
30997 @kbd{z s} key sequence to run @code{calc-sincos}. This definition will remain
30998 in effect for the rest of this Emacs session, or until you redefine
30999 @kbd{z s} to be something else.
31001 You can actually bind any Emacs command to a @kbd{z} key sequence by
31002 backspacing over the @samp{calc-} when you are prompted for the command name.
31004 As with any other prefix key, you can type @kbd{z ?} to see a list of
31005 all the two-key sequences you have defined that start with @kbd{z}.
31006 Initially, no @kbd{z} sequences (except @kbd{z ?} itself) are defined.
31008 User keys are typically letters, but may in fact be any key.
31009 (@key{META}-keys are not permitted, nor are a terminal's special
31010 function keys which generate multi-character sequences when pressed.)
31011 You can define different commands on the shifted and unshifted versions
31012 of a letter if you wish.
31015 @pindex calc-user-undefine
31016 The @kbd{Z U} (@code{calc-user-undefine}) command unbinds a user key.
31017 For example, the key sequence @kbd{Z U s} will undefine the @code{sincos}
31018 key we defined above.
31021 @pindex calc-user-define-permanent
31022 @cindex Storing user definitions
31023 @cindex Permanent user definitions
31024 @cindex Calc init file, user-defined commands
31025 The @kbd{Z P} (@code{calc-user-define-permanent}) command makes a key
31026 binding permanent so that it will remain in effect even in future Emacs
31027 sessions. (It does this by adding a suitable bit of Lisp code into
31028 your Calc init file; that is, the file given by the variable
31029 @code{calc-settings-file}, typically @file{~/.calc.el}.) For example,
31030 @kbd{Z P s} would register our @code{sincos} command permanently. If
31031 you later wish to unregister this command you must edit your Calc init
31032 file by hand. (@xref{General Mode Commands}, for a way to tell Calc to
31033 use a different file for the Calc init file.)
31035 The @kbd{Z P} command also saves the user definition, if any, for the
31036 command bound to the key. After @kbd{Z F} and @kbd{Z C}, a given user
31037 key could invoke a command, which in turn calls an algebraic function,
31038 which might have one or more special display formats. A single @kbd{Z P}
31039 command will save all of these definitions.
31040 To save an algebraic function, type @kbd{'} (the apostrophe)
31041 when prompted for a key, and type the function name. To save a command
31042 without its key binding, type @kbd{M-x} and enter a function name. (The
31043 @samp{calc-} prefix will automatically be inserted for you.)
31044 (If the command you give implies a function, the function will be saved,
31045 and if the function has any display formats, those will be saved, but
31046 not the other way around: Saving a function will not save any commands
31047 or key bindings associated with the function.)
31050 @pindex calc-user-define-edit
31051 @cindex Editing user definitions
31052 The @kbd{Z E} (@code{calc-user-define-edit}) command edits the definition
31053 of a user key. This works for keys that have been defined by either
31054 keyboard macros or formulas; further details are contained in the relevant
31055 following sections.
31057 @node Keyboard Macros, Invocation Macros, Creating User Keys, Programming
31058 @section Programming with Keyboard Macros
31062 @cindex Programming with keyboard macros
31063 @cindex Keyboard macros
31064 The easiest way to ``program'' the Emacs Calculator is to use standard
31065 keyboard macros. Press @w{@kbd{C-x (}} to begin recording a macro. From
31066 this point on, keystrokes you type will be saved away as well as
31067 performing their usual functions. Press @kbd{C-x )} to end recording.
31068 Press shift-@kbd{X} (or the standard Emacs key sequence @kbd{C-x e}) to
31069 execute your keyboard macro by replaying the recorded keystrokes.
31070 @xref{Keyboard Macros, , , emacs, the Emacs Manual}, for further
31073 When you use @kbd{X} to invoke a keyboard macro, the entire macro is
31074 treated as a single command by the undo and trail features. The stack
31075 display buffer is not updated during macro execution, but is instead
31076 fixed up once the macro completes. Thus, commands defined with keyboard
31077 macros are convenient and efficient. The @kbd{C-x e} command, on the
31078 other hand, invokes the keyboard macro with no special treatment: Each
31079 command in the macro will record its own undo information and trail entry,
31080 and update the stack buffer accordingly. If your macro uses features
31081 outside of Calc's control to operate on the contents of the Calc stack
31082 buffer, or if it includes Undo, Redo, or last-arguments commands, you
31083 must use @kbd{C-x e} to make sure the buffer and undo list are up-to-date
31084 at all times. You could also consider using @kbd{K} (@code{calc-keep-args})
31085 instead of @kbd{M-@key{RET}} (@code{calc-last-args}).
31087 Calc extends the standard Emacs keyboard macros in several ways.
31088 Keyboard macros can be used to create user-defined commands. Keyboard
31089 macros can include conditional and iteration structures, somewhat
31090 analogous to those provided by a traditional programmable calculator.
31093 * Naming Keyboard Macros::
31094 * Conditionals in Macros::
31095 * Loops in Macros::
31096 * Local Values in Macros::
31097 * Queries in Macros::
31100 @node Naming Keyboard Macros, Conditionals in Macros, Keyboard Macros, Keyboard Macros
31101 @subsection Naming Keyboard Macros
31105 @pindex calc-user-define-kbd-macro
31106 Once you have defined a keyboard macro, you can bind it to a @kbd{z}
31107 key sequence with the @kbd{Z K} (@code{calc-user-define-kbd-macro}) command.
31108 This command prompts first for a key, then for a command name. For
31109 example, if you type @kbd{C-x ( n @key{TAB} n @key{TAB} C-x )} you will
31110 define a keyboard macro which negates the top two numbers on the stack
31111 (@key{TAB} swaps the top two stack elements). Now you can type
31112 @kbd{Z K n @key{RET}} to define this keyboard macro onto the @kbd{z n} key
31113 sequence. The default command name (if you answer the second prompt with
31114 just the @key{RET} key as in this example) will be something like
31115 @samp{calc-User-n}. The keyboard macro will now be available as both
31116 @kbd{z n} and @kbd{M-x calc-User-n}. You can backspace and enter a more
31117 descriptive command name if you wish.
31119 Macros defined by @kbd{Z K} act like single commands; they are executed
31120 in the same way as by the @kbd{X} key. If you wish to define the macro
31121 as a standard no-frills Emacs macro (to be executed as if by @kbd{C-x e}),
31122 give a negative prefix argument to @kbd{Z K}.
31124 Once you have bound your keyboard macro to a key, you can use
31125 @kbd{Z P} to register it permanently with Emacs. @xref{Creating User Keys}.
31127 @cindex Keyboard macros, editing
31128 The @kbd{Z E} (@code{calc-user-define-edit}) command on a key that has
31129 been defined by a keyboard macro tries to use the @code{edmacro} package
31130 edit the macro. Type @kbd{C-c C-c} to finish editing and update
31131 the definition stored on the key, or, to cancel the edit, kill the
31132 buffer with @kbd{C-x k}.
31133 The special characters @code{RET}, @code{LFD}, @code{TAB}, @code{SPC},
31134 @code{DEL}, and @code{NUL} must be entered as these three character
31135 sequences, written in all uppercase, as must the prefixes @code{C-} and
31136 @code{M-}. Spaces and line breaks are ignored. Other characters are
31137 copied verbatim into the keyboard macro. Basically, the notation is the
31138 same as is used in all of this manual's examples, except that the manual
31139 takes some liberties with spaces: When we say @kbd{' [1 2 3] @key{RET}},
31140 we take it for granted that it is clear we really mean
31141 @kbd{' [1 @key{SPC} 2 @key{SPC} 3] @key{RET}}.
31144 @pindex read-kbd-macro
31145 The @kbd{M-# m} (@code{read-kbd-macro}) command reads an Emacs ``region''
31146 of spelled-out keystrokes and defines it as the current keyboard macro.
31147 It is a convenient way to define a keyboard macro that has been stored
31148 in a file, or to define a macro without executing it at the same time.
31150 @node Conditionals in Macros, Loops in Macros, Naming Keyboard Macros, Keyboard Macros
31151 @subsection Conditionals in Keyboard Macros
31156 @pindex calc-kbd-if
31157 @pindex calc-kbd-else
31158 @pindex calc-kbd-else-if
31159 @pindex calc-kbd-end-if
31160 @cindex Conditional structures
31161 The @kbd{Z [} (@code{calc-kbd-if}) and @kbd{Z ]} (@code{calc-kbd-end-if})
31162 commands allow you to put simple tests in a keyboard macro. When Calc
31163 sees the @kbd{Z [}, it pops an object from the stack and, if the object is
31164 a non-zero value, continues executing keystrokes. But if the object is
31165 zero, or if it is not provably nonzero, Calc skips ahead to the matching
31166 @kbd{Z ]} keystroke. @xref{Logical Operations}, for a set of commands for
31167 performing tests which conveniently produce 1 for true and 0 for false.
31169 For example, @kbd{@key{RET} 0 a < Z [ n Z ]} implements an absolute-value
31170 function in the form of a keyboard macro. This macro duplicates the
31171 number on the top of the stack, pushes zero and compares using @kbd{a <}
31172 (@code{calc-less-than}), then, if the number was less than zero,
31173 executes @kbd{n} (@code{calc-change-sign}). Otherwise, the change-sign
31174 command is skipped.
31176 To program this macro, type @kbd{C-x (}, type the above sequence of
31177 keystrokes, then type @kbd{C-x )}. Note that the keystrokes will be
31178 executed while you are making the definition as well as when you later
31179 re-execute the macro by typing @kbd{X}. Thus you should make sure a
31180 suitable number is on the stack before defining the macro so that you
31181 don't get a stack-underflow error during the definition process.
31183 Conditionals can be nested arbitrarily. However, there should be exactly
31184 one @kbd{Z ]} for each @kbd{Z [} in a keyboard macro.
31187 The @kbd{Z :} (@code{calc-kbd-else}) command allows you to choose between
31188 two keystroke sequences. The general format is @kbd{@var{cond} Z [
31189 @var{then-part} Z : @var{else-part} Z ]}. If @var{cond} is true
31190 (i.e., if the top of stack contains a non-zero number after @var{cond}
31191 has been executed), the @var{then-part} will be executed and the
31192 @var{else-part} will be skipped. Otherwise, the @var{then-part} will
31193 be skipped and the @var{else-part} will be executed.
31196 The @kbd{Z |} (@code{calc-kbd-else-if}) command allows you to choose
31197 between any number of alternatives. For example,
31198 @kbd{@var{cond1} Z [ @var{part1} Z : @var{cond2} Z | @var{part2} Z :
31199 @var{part3} Z ]} will execute @var{part1} if @var{cond1} is true,
31200 otherwise it will execute @var{part2} if @var{cond2} is true, otherwise
31201 it will execute @var{part3}.
31203 More precisely, @kbd{Z [} pops a number and conditionally skips to the
31204 next matching @kbd{Z :} or @kbd{Z ]} key. @w{@kbd{Z ]}} has no effect when
31205 actually executed. @kbd{Z :} skips to the next matching @kbd{Z ]}.
31206 @kbd{Z |} pops a number and conditionally skips to the next matching
31207 @kbd{Z :} or @kbd{Z ]}; thus, @kbd{Z [} and @kbd{Z |} are functionally
31208 equivalent except that @kbd{Z [} participates in nesting but @kbd{Z |}
31211 Calc's conditional and looping constructs work by scanning the
31212 keyboard macro for occurrences of character sequences like @samp{Z:}
31213 and @samp{Z]}. One side-effect of this is that if you use these
31214 constructs you must be careful that these character pairs do not
31215 occur by accident in other parts of the macros. Since Calc rarely
31216 uses shift-@kbd{Z} for any purpose except as a prefix character, this
31217 is not likely to be a problem. Another side-effect is that it will
31218 not work to define your own custom key bindings for these commands.
31219 Only the standard shift-@kbd{Z} bindings will work correctly.
31222 If Calc gets stuck while skipping characters during the definition of a
31223 macro, type @kbd{Z C-g} to cancel the definition. (Typing plain @kbd{C-g}
31224 actually adds a @kbd{C-g} keystroke to the macro.)
31226 @node Loops in Macros, Local Values in Macros, Conditionals in Macros, Keyboard Macros
31227 @subsection Loops in Keyboard Macros
31232 @pindex calc-kbd-repeat
31233 @pindex calc-kbd-end-repeat
31234 @cindex Looping structures
31235 @cindex Iterative structures
31236 The @kbd{Z <} (@code{calc-kbd-repeat}) and @kbd{Z >}
31237 (@code{calc-kbd-end-repeat}) commands pop a number from the stack,
31238 which must be an integer, then repeat the keystrokes between the brackets
31239 the specified number of times. If the integer is zero or negative, the
31240 body is skipped altogether. For example, @kbd{1 @key{TAB} Z < 2 * Z >}
31241 computes two to a nonnegative integer power. First, we push 1 on the
31242 stack and then swap the integer argument back to the top. The @kbd{Z <}
31243 pops that argument leaving the 1 back on top of the stack. Then, we
31244 repeat a multiply-by-two step however many times.
31246 Once again, the keyboard macro is executed as it is being entered.
31247 In this case it is especially important to set up reasonable initial
31248 conditions before making the definition: Suppose the integer 1000 just
31249 happened to be sitting on the stack before we typed the above definition!
31250 Another approach is to enter a harmless dummy definition for the macro,
31251 then go back and edit in the real one with a @kbd{Z E} command. Yet
31252 another approach is to type the macro as written-out keystroke names
31253 in a buffer, then use @kbd{M-# m} (@code{read-kbd-macro}) to read the
31258 The @kbd{Z /} (@code{calc-kbd-break}) command allows you to break out
31259 of a keyboard macro loop prematurely. It pops an object from the stack;
31260 if that object is true (a non-zero number), control jumps out of the
31261 innermost enclosing @kbd{Z <} @dots{} @kbd{Z >} loop and continues
31262 after the @kbd{Z >}. If the object is false, the @kbd{Z /} has no
31263 effect. Thus @kbd{@var{cond} Z /} is similar to @samp{if (@var{cond}) break;}
31268 @pindex calc-kbd-for
31269 @pindex calc-kbd-end-for
31270 The @kbd{Z (} (@code{calc-kbd-for}) and @kbd{Z )} (@code{calc-kbd-end-for})
31271 commands are similar to @kbd{Z <} and @kbd{Z >}, except that they make the
31272 value of the counter available inside the loop. The general layout is
31273 @kbd{@var{init} @var{final} Z ( @var{body} @var{step} Z )}. The @kbd{Z (}
31274 command pops initial and final values from the stack. It then creates
31275 a temporary internal counter and initializes it with the value @var{init}.
31276 The @kbd{Z (} command then repeatedly pushes the counter value onto the
31277 stack and executes @var{body} and @var{step}, adding @var{step} to the
31278 counter each time until the loop finishes.
31280 @cindex Summations (by keyboard macros)
31281 By default, the loop finishes when the counter becomes greater than (or
31282 less than) @var{final}, assuming @var{initial} is less than (greater
31283 than) @var{final}. If @var{initial} is equal to @var{final}, the body
31284 executes exactly once. The body of the loop always executes at least
31285 once. For example, @kbd{0 1 10 Z ( 2 ^ + 1 Z )} computes the sum of the
31286 squares of the integers from 1 to 10, in steps of 1.
31288 If you give a numeric prefix argument of 1 to @kbd{Z (}, the loop is
31289 forced to use upward-counting conventions. In this case, if @var{initial}
31290 is greater than @var{final} the body will not be executed at all.
31291 Note that @var{step} may still be negative in this loop; the prefix
31292 argument merely constrains the loop-finished test. Likewise, a prefix
31293 argument of @mathit{-1} forces downward-counting conventions.
31297 @pindex calc-kbd-loop
31298 @pindex calc-kbd-end-loop
31299 The @kbd{Z @{} (@code{calc-kbd-loop}) and @kbd{Z @}}
31300 (@code{calc-kbd-end-loop}) commands are similar to @kbd{Z <} and
31301 @kbd{Z >}, except that they do not pop a count from the stack---they
31302 effectively create an infinite loop. Every @kbd{Z @{} @dots{} @kbd{Z @}}
31303 loop ought to include at least one @kbd{Z /} to make sure the loop
31304 doesn't run forever. (If any error message occurs which causes Emacs
31305 to beep, the keyboard macro will also be halted; this is a standard
31306 feature of Emacs. You can also generally press @kbd{C-g} to halt a
31307 running keyboard macro, although not all versions of Unix support
31310 The conditional and looping constructs are not actually tied to
31311 keyboard macros, but they are most often used in that context.
31312 For example, the keystrokes @kbd{10 Z < 23 @key{RET} Z >} push
31313 ten copies of 23 onto the stack. This can be typed ``live'' just
31314 as easily as in a macro definition.
31316 @xref{Conditionals in Macros}, for some additional notes about
31317 conditional and looping commands.
31319 @node Local Values in Macros, Queries in Macros, Loops in Macros, Keyboard Macros
31320 @subsection Local Values in Macros
31323 @cindex Local variables
31324 @cindex Restoring saved modes
31325 Keyboard macros sometimes want to operate under known conditions
31326 without affecting surrounding conditions. For example, a keyboard
31327 macro may wish to turn on Fraction mode, or set a particular
31328 precision, independent of the user's normal setting for those
31333 @pindex calc-kbd-push
31334 @pindex calc-kbd-pop
31335 Macros also sometimes need to use local variables. Assignments to
31336 local variables inside the macro should not affect any variables
31337 outside the macro. The @kbd{Z `} (@code{calc-kbd-push}) and @kbd{Z '}
31338 (@code{calc-kbd-pop}) commands give you both of these capabilities.
31340 When you type @kbd{Z `} (with a backquote or accent grave character),
31341 the values of various mode settings are saved away. The ten ``quick''
31342 variables @code{q0} through @code{q9} are also saved. When
31343 you type @w{@kbd{Z '}} (with an apostrophe), these values are restored.
31344 Pairs of @kbd{Z `} and @kbd{Z '} commands may be nested.
31346 If a keyboard macro halts due to an error in between a @kbd{Z `} and
31347 a @kbd{Z '}, the saved values will be restored correctly even though
31348 the macro never reaches the @kbd{Z '} command. Thus you can use
31349 @kbd{Z `} and @kbd{Z '} without having to worry about what happens
31350 in exceptional conditions.
31352 If you type @kbd{Z `} ``live'' (not in a keyboard macro), Calc puts
31353 you into a ``recursive edit.'' You can tell you are in a recursive
31354 edit because there will be extra square brackets in the mode line,
31355 as in @samp{[(Calculator)]}. These brackets will go away when you
31356 type the matching @kbd{Z '} command. The modes and quick variables
31357 will be saved and restored in just the same way as if actual keyboard
31358 macros were involved.
31360 The modes saved by @kbd{Z `} and @kbd{Z '} are the current precision
31361 and binary word size, the angular mode (Deg, Rad, or HMS), the
31362 simplification mode, Algebraic mode, Symbolic mode, Infinite mode,
31363 Matrix or Scalar mode, Fraction mode, and the current complex mode
31364 (Polar or Rectangular). The ten ``quick'' variables' values (or lack
31365 thereof) are also saved.
31367 Most mode-setting commands act as toggles, but with a numeric prefix
31368 they force the mode either on (positive prefix) or off (negative
31369 or zero prefix). Since you don't know what the environment might
31370 be when you invoke your macro, it's best to use prefix arguments
31371 for all mode-setting commands inside the macro.
31373 In fact, @kbd{C-u Z `} is like @kbd{Z `} except that it sets the modes
31374 listed above to their default values. As usual, the matching @kbd{Z '}
31375 will restore the modes to their settings from before the @kbd{C-u Z `}.
31376 Also, @w{@kbd{Z `}} with a negative prefix argument resets the algebraic mode
31377 to its default (off) but leaves the other modes the same as they were
31378 outside the construct.
31380 The contents of the stack and trail, values of non-quick variables, and
31381 other settings such as the language mode and the various display modes,
31382 are @emph{not} affected by @kbd{Z `} and @kbd{Z '}.
31384 @node Queries in Macros, , Local Values in Macros, Keyboard Macros
31385 @subsection Queries in Keyboard Macros
31389 @pindex calc-kbd-report
31390 The @kbd{Z =} (@code{calc-kbd-report}) command displays an informative
31391 message including the value on the top of the stack. You are prompted
31392 to enter a string. That string, along with the top-of-stack value,
31393 is displayed unless @kbd{m w} (@code{calc-working}) has been used
31394 to turn such messages off.
31397 @pindex calc-kbd-query
31398 The @kbd{Z #} (@code{calc-kbd-query}) command displays a prompt message
31399 (which you enter during macro definition), then does an algebraic entry
31400 which takes its input from the keyboard, even during macro execution.
31401 This command allows your keyboard macros to accept numbers or formulas
31402 as interactive input. All the normal conventions of algebraic input,
31403 including the use of @kbd{$} characters, are supported.
31405 @xref{Keyboard Macro Query, , , emacs, the Emacs Manual}, for a description of
31406 @kbd{C-x q} (@code{kbd-macro-query}), the standard Emacs way to accept
31407 keyboard input during a keyboard macro. In particular, you can use
31408 @kbd{C-x q} to enter a recursive edit, which allows the user to perform
31409 any Calculator operations interactively before pressing @kbd{C-M-c} to
31410 return control to the keyboard macro.
31412 @node Invocation Macros, Algebraic Definitions, Keyboard Macros, Programming
31413 @section Invocation Macros
31417 @pindex calc-user-invocation
31418 @pindex calc-user-define-invocation
31419 Calc provides one special keyboard macro, called up by @kbd{M-# z}
31420 (@code{calc-user-invocation}), that is intended to allow you to define
31421 your own special way of starting Calc. To define this ``invocation
31422 macro,'' create the macro in the usual way with @kbd{C-x (} and
31423 @kbd{C-x )}, then type @kbd{Z I} (@code{calc-user-define-invocation}).
31424 There is only one invocation macro, so you don't need to type any
31425 additional letters after @kbd{Z I}. From now on, you can type
31426 @kbd{M-# z} at any time to execute your invocation macro.
31428 For example, suppose you find yourself often grabbing rectangles of
31429 numbers into Calc and multiplying their columns. You can do this
31430 by typing @kbd{M-# r} to grab, and @kbd{V R : *} to multiply columns.
31431 To make this into an invocation macro, just type @kbd{C-x ( M-# r
31432 V R : * C-x )}, then @kbd{Z I}. Then, to multiply a rectangle of data,
31433 just mark the data in its buffer in the usual way and type @kbd{M-# z}.
31435 Invocation macros are treated like regular Emacs keyboard macros;
31436 all the special features described above for @kbd{Z K}-style macros
31437 do not apply. @kbd{M-# z} is just like @kbd{C-x e}, except that it
31438 uses the macro that was last stored by @kbd{Z I}. (In fact, the
31439 macro does not even have to have anything to do with Calc!)
31441 The @kbd{m m} command saves the last invocation macro defined by
31442 @kbd{Z I} along with all the other Calc mode settings.
31443 @xref{General Mode Commands}.
31445 @node Algebraic Definitions, Lisp Definitions, Invocation Macros, Programming
31446 @section Programming with Formulas
31450 @pindex calc-user-define-formula
31451 @cindex Programming with algebraic formulas
31452 Another way to create a new Calculator command uses algebraic formulas.
31453 The @kbd{Z F} (@code{calc-user-define-formula}) command stores the
31454 formula at the top of the stack as the definition for a key. This
31455 command prompts for five things: The key, the command name, the function
31456 name, the argument list, and the behavior of the command when given
31457 non-numeric arguments.
31459 For example, suppose we type @kbd{' a+2b @key{RET}} to push the formula
31460 @samp{a + 2*b} onto the stack. We now type @kbd{Z F m} to define this
31461 formula on the @kbd{z m} key sequence. The next prompt is for a command
31462 name, beginning with @samp{calc-}, which should be the long (@kbd{M-x}) form
31463 for the new command. If you simply press @key{RET}, a default name like
31464 @code{calc-User-m} will be constructed. In our example, suppose we enter
31465 @kbd{spam @key{RET}} to define the new command as @code{calc-spam}.
31467 If you want to give the formula a long-style name only, you can press
31468 @key{SPC} or @key{RET} when asked which single key to use. For example
31469 @kbd{Z F @key{RET} spam @key{RET}} defines the new command as
31470 @kbd{M-x calc-spam}, with no keyboard equivalent.
31472 The third prompt is for an algebraic function name. The default is to
31473 use the same name as the command name but without the @samp{calc-}
31474 prefix. (If this is of the form @samp{User-m}, the hyphen is removed so
31475 it won't be taken for a minus sign in algebraic formulas.)
31476 This is the name you will use if you want to enter your
31477 new function in an algebraic formula. Suppose we enter @kbd{yow @key{RET}}.
31478 Then the new function can be invoked by pushing two numbers on the
31479 stack and typing @kbd{z m} or @kbd{x spam}, or by entering the algebraic
31480 formula @samp{yow(x,y)}.
31482 The fourth prompt is for the function's argument list. This is used to
31483 associate values on the stack with the variables that appear in the formula.
31484 The default is a list of all variables which appear in the formula, sorted
31485 into alphabetical order. In our case, the default would be @samp{(a b)}.
31486 This means that, when the user types @kbd{z m}, the Calculator will remove
31487 two numbers from the stack, substitute these numbers for @samp{a} and
31488 @samp{b} (respectively) in the formula, then simplify the formula and
31489 push the result on the stack. In other words, @kbd{10 @key{RET} 100 z m}
31490 would replace the 10 and 100 on the stack with the number 210, which is
31491 @expr{a + 2 b} with @expr{a=10} and @expr{b=100}. Likewise, the formula
31492 @samp{yow(10, 100)} will be evaluated by substituting @expr{a=10} and
31493 @expr{b=100} in the definition.
31495 You can rearrange the order of the names before pressing @key{RET} to
31496 control which stack positions go to which variables in the formula. If
31497 you remove a variable from the argument list, that variable will be left
31498 in symbolic form by the command. Thus using an argument list of @samp{(b)}
31499 for our function would cause @kbd{10 z m} to replace the 10 on the stack
31500 with the formula @samp{a + 20}. If we had used an argument list of
31501 @samp{(b a)}, the result with inputs 10 and 100 would have been 120.
31503 You can also put a nameless function on the stack instead of just a
31504 formula, as in @samp{<a, b : a + 2 b>}. @xref{Specifying Operators}.
31505 In this example, the command will be defined by the formula @samp{a + 2 b}
31506 using the argument list @samp{(a b)}.
31508 The final prompt is a y-or-n question concerning what to do if symbolic
31509 arguments are given to your function. If you answer @kbd{y}, then
31510 executing @kbd{z m} (using the original argument list @samp{(a b)}) with
31511 arguments @expr{10} and @expr{x} will leave the function in symbolic
31512 form, i.e., @samp{yow(10,x)}. On the other hand, if you answer @kbd{n},
31513 then the formula will always be expanded, even for non-constant
31514 arguments: @samp{10 + 2 x}. If you never plan to feed algebraic
31515 formulas to your new function, it doesn't matter how you answer this
31518 If you answered @kbd{y} to this question you can still cause a function
31519 call to be expanded by typing @kbd{a "} (@code{calc-expand-formula}).
31520 Also, Calc will expand the function if necessary when you take a
31521 derivative or integral or solve an equation involving the function.
31524 @pindex calc-get-user-defn
31525 Once you have defined a formula on a key, you can retrieve this formula
31526 with the @kbd{Z G} (@code{calc-user-define-get-defn}) command. Press a
31527 key, and this command pushes the formula that was used to define that
31528 key onto the stack. Actually, it pushes a nameless function that
31529 specifies both the argument list and the defining formula. You will get
31530 an error message if the key is undefined, or if the key was not defined
31531 by a @kbd{Z F} command.
31533 The @kbd{Z E} (@code{calc-user-define-edit}) command on a key that has
31534 been defined by a formula uses a variant of the @code{calc-edit} command
31535 to edit the defining formula. Press @kbd{C-c C-c} to finish editing and
31536 store the new formula back in the definition, or kill the buffer with
31538 cancel the edit. (The argument list and other properties of the
31539 definition are unchanged; to adjust the argument list, you can use
31540 @kbd{Z G} to grab the function onto the stack, edit with @kbd{`}, and
31541 then re-execute the @kbd{Z F} command.)
31543 As usual, the @kbd{Z P} command records your definition permanently.
31544 In this case it will permanently record all three of the relevant
31545 definitions: the key, the command, and the function.
31547 You may find it useful to turn off the default simplifications with
31548 @kbd{m O} (@code{calc-no-simplify-mode}) when entering a formula to be
31549 used as a function definition. For example, the formula @samp{deriv(a^2,v)}
31550 which might be used to define a new function @samp{dsqr(a,v)} will be
31551 ``simplified'' to 0 immediately upon entry since @code{deriv} considers
31552 @expr{a} to be constant with respect to @expr{v}. Turning off
31553 default simplifications cures this problem: The definition will be stored
31554 in symbolic form without ever activating the @code{deriv} function. Press
31555 @kbd{m D} to turn the default simplifications back on afterwards.
31557 @node Lisp Definitions, , Algebraic Definitions, Programming
31558 @section Programming with Lisp
31561 The Calculator can be programmed quite extensively in Lisp. All you
31562 do is write a normal Lisp function definition, but with @code{defmath}
31563 in place of @code{defun}. This has the same form as @code{defun}, but it
31564 automagically replaces calls to standard Lisp functions like @code{+} and
31565 @code{zerop} with calls to the corresponding functions in Calc's own library.
31566 Thus you can write natural-looking Lisp code which operates on all of the
31567 standard Calculator data types. You can then use @kbd{Z D} if you wish to
31568 bind your new command to a @kbd{z}-prefix key sequence. The @kbd{Z E} command
31569 will not edit a Lisp-based definition.
31571 Emacs Lisp is described in the GNU Emacs Lisp Reference Manual. This section
31572 assumes a familiarity with Lisp programming concepts; if you do not know
31573 Lisp, you may find keyboard macros or rewrite rules to be an easier way
31574 to program the Calculator.
31576 This section first discusses ways to write commands, functions, or
31577 small programs to be executed inside of Calc. Then it discusses how
31578 your own separate programs are able to call Calc from the outside.
31579 Finally, there is a list of internal Calc functions and data structures
31580 for the true Lisp enthusiast.
31583 * Defining Functions::
31584 * Defining Simple Commands::
31585 * Defining Stack Commands::
31586 * Argument Qualifiers::
31587 * Example Definitions::
31589 * Calling Calc from Your Programs::
31593 @node Defining Functions, Defining Simple Commands, Lisp Definitions, Lisp Definitions
31594 @subsection Defining New Functions
31598 The @code{defmath} function (actually a Lisp macro) is like @code{defun}
31599 except that code in the body of the definition can make use of the full
31600 range of Calculator data types. The prefix @samp{calcFunc-} is added
31601 to the specified name to get the actual Lisp function name. As a simple
31605 (defmath myfact (n)
31607 (* n (myfact (1- n)))
31612 This actually expands to the code,
31615 (defun calcFunc-myfact (n)
31617 (math-mul n (calcFunc-myfact (math-add n -1)))
31622 This function can be used in algebraic expressions, e.g., @samp{myfact(5)}.
31624 The @samp{myfact} function as it is defined above has the bug that an
31625 expression @samp{myfact(a+b)} will be simplified to 1 because the
31626 formula @samp{a+b} is not considered to be @code{posp}. A robust
31627 factorial function would be written along the following lines:
31630 (defmath myfact (n)
31632 (* n (myfact (1- n)))
31635 nil))) ; this could be simplified as: (and (= n 0) 1)
31638 If a function returns @code{nil}, it is left unsimplified by the Calculator
31639 (except that its arguments will be simplified). Thus, @samp{myfact(a+1+2)}
31640 will be simplified to @samp{myfact(a+3)} but no further. Beware that every
31641 time the Calculator reexamines this formula it will attempt to resimplify
31642 it, so your function ought to detect the returning-@code{nil} case as
31643 efficiently as possible.
31645 The following standard Lisp functions are treated by @code{defmath}:
31646 @code{+}, @code{-}, @code{*}, @code{/}, @code{%}, @code{^} or
31647 @code{expt}, @code{=}, @code{<}, @code{>}, @code{<=}, @code{>=},
31648 @code{/=}, @code{1+}, @code{1-}, @code{logand}, @code{logior}, @code{logxor},
31649 @code{logandc2}, @code{lognot}. Also, @code{~=} is an abbreviation for
31650 @code{math-nearly-equal}, which is useful in implementing Taylor series.
31652 For other functions @var{func}, if a function by the name
31653 @samp{calcFunc-@var{func}} exists it is used, otherwise if a function by the
31654 name @samp{math-@var{func}} exists it is used, otherwise if @var{func} itself
31655 is defined as a function it is used, otherwise @samp{calcFunc-@var{func}} is
31656 used on the assumption that this is a to-be-defined math function. Also, if
31657 the function name is quoted as in @samp{('integerp a)} the function name is
31658 always used exactly as written (but not quoted).
31660 Variable names have @samp{var-} prepended to them unless they appear in
31661 the function's argument list or in an enclosing @code{let}, @code{let*},
31662 @code{for}, or @code{foreach} form,
31663 or their names already contain a @samp{-} character. Thus a reference to
31664 @samp{foo} is the same as a reference to @samp{var-foo}.
31666 A few other Lisp extensions are available in @code{defmath} definitions:
31670 The @code{elt} function accepts any number of index variables.
31671 Note that Calc vectors are stored as Lisp lists whose first
31672 element is the symbol @code{vec}; thus, @samp{(elt v 2)} yields
31673 the second element of vector @code{v}, and @samp{(elt m i j)}
31674 yields one element of a Calc matrix.
31677 The @code{setq} function has been extended to act like the Common
31678 Lisp @code{setf} function. (The name @code{setf} is recognized as
31679 a synonym of @code{setq}.) Specifically, the first argument of
31680 @code{setq} can be an @code{nth}, @code{elt}, @code{car}, or @code{cdr} form,
31681 in which case the effect is to store into the specified
31682 element of a list. Thus, @samp{(setq (elt m i j) x)} stores @expr{x}
31683 into one element of a matrix.
31686 A @code{for} looping construct is available. For example,
31687 @samp{(for ((i 0 10)) body)} executes @code{body} once for each
31688 binding of @expr{i} from zero to 10. This is like a @code{let}
31689 form in that @expr{i} is temporarily bound to the loop count
31690 without disturbing its value outside the @code{for} construct.
31691 Nested loops, as in @samp{(for ((i 0 10) (j 0 (1- i) 2)) body)},
31692 are also available. For each value of @expr{i} from zero to 10,
31693 @expr{j} counts from 0 to @expr{i-1} in steps of two. Note that
31694 @code{for} has the same general outline as @code{let*}, except
31695 that each element of the header is a list of three or four
31696 things, not just two.
31699 The @code{foreach} construct loops over elements of a list.
31700 For example, @samp{(foreach ((x (cdr v))) body)} executes
31701 @code{body} with @expr{x} bound to each element of Calc vector
31702 @expr{v} in turn. The purpose of @code{cdr} here is to skip over
31703 the initial @code{vec} symbol in the vector.
31706 The @code{break} function breaks out of the innermost enclosing
31707 @code{while}, @code{for}, or @code{foreach} loop. If given a
31708 value, as in @samp{(break x)}, this value is returned by the
31709 loop. (Lisp loops otherwise always return @code{nil}.)
31712 The @code{return} function prematurely returns from the enclosing
31713 function. For example, @samp{(return (+ x y))} returns @expr{x+y}
31714 as the value of a function. You can use @code{return} anywhere
31715 inside the body of the function.
31718 Non-integer numbers (and extremely large integers) cannot be included
31719 directly into a @code{defmath} definition. This is because the Lisp
31720 reader will fail to parse them long before @code{defmath} ever gets control.
31721 Instead, use the notation, @samp{:"3.1415"}. In fact, any algebraic
31722 formula can go between the quotes. For example,
31725 (defmath sqexp (x) ; sqexp(x) == sqrt(exp(x)) == exp(x*0.5)
31733 (defun calcFunc-sqexp (x)
31734 (and (math-numberp x)
31735 (calcFunc-exp (math-mul x '(float 5 -1)))))
31738 Note the use of @code{numberp} as a guard to ensure that the argument is
31739 a number first, returning @code{nil} if not. The exponential function
31740 could itself have been included in the expression, if we had preferred:
31741 @samp{:"exp(x * 0.5)"}. As another example, the multiplication-and-recursion
31742 step of @code{myfact} could have been written
31748 A good place to put your @code{defmath} commands is your Calc init file
31749 (the file given by @code{calc-settings-file}, typically
31750 @file{~/.calc.el}), which will not be loaded until Calc starts.
31751 If a file named @file{.emacs} exists in your home directory, Emacs reads
31752 and executes the Lisp forms in this file as it starts up. While it may
31753 seem reasonable to put your favorite @code{defmath} commands there,
31754 this has the unfortunate side-effect that parts of the Calculator must be
31755 loaded in to process the @code{defmath} commands whether or not you will
31756 actually use the Calculator! If you want to put the @code{defmath}
31757 commands there (for example, if you redefine @code{calc-settings-file}
31758 to be @file{.emacs}), a better effect can be had by writing
31761 (put 'calc-define 'thing '(progn
31768 @vindex calc-define
31769 The @code{put} function adds a @dfn{property} to a symbol. Each Lisp
31770 symbol has a list of properties associated with it. Here we add a
31771 property with a name of @code{thing} and a @samp{(progn ...)} form as
31772 its value. When Calc starts up, and at the start of every Calc command,
31773 the property list for the symbol @code{calc-define} is checked and the
31774 values of any properties found are evaluated as Lisp forms. The
31775 properties are removed as they are evaluated. The property names
31776 (like @code{thing}) are not used; you should choose something like the
31777 name of your project so as not to conflict with other properties.
31779 The net effect is that you can put the above code in your @file{.emacs}
31780 file and it will not be executed until Calc is loaded. Or, you can put
31781 that same code in another file which you load by hand either before or
31782 after Calc itself is loaded.
31784 The properties of @code{calc-define} are evaluated in the same order
31785 that they were added. They can assume that the Calc modules @file{calc.el},
31786 @file{calc-ext.el}, and @file{calc-macs.el} have been fully loaded, and
31787 that the @samp{*Calculator*} buffer will be the current buffer.
31789 If your @code{calc-define} property only defines algebraic functions,
31790 you can be sure that it will have been evaluated before Calc tries to
31791 call your function, even if the file defining the property is loaded
31792 after Calc is loaded. But if the property defines commands or key
31793 sequences, it may not be evaluated soon enough. (Suppose it defines the
31794 new command @code{tweak-calc}; the user can load your file, then type
31795 @kbd{M-x tweak-calc} before Calc has had chance to do anything.) To
31796 protect against this situation, you can put
31799 (run-hooks 'calc-check-defines)
31802 @findex calc-check-defines
31804 at the end of your file. The @code{calc-check-defines} function is what
31805 looks for and evaluates properties on @code{calc-define}; @code{run-hooks}
31806 has the advantage that it is quietly ignored if @code{calc-check-defines}
31807 is not yet defined because Calc has not yet been loaded.
31809 Examples of things that ought to be enclosed in a @code{calc-define}
31810 property are @code{defmath} calls, @code{define-key} calls that modify
31811 the Calc key map, and any calls that redefine things defined inside Calc.
31812 Ordinary @code{defun}s need not be enclosed with @code{calc-define}.
31814 @node Defining Simple Commands, Defining Stack Commands, Defining Functions, Lisp Definitions
31815 @subsection Defining New Simple Commands
31818 @findex interactive
31819 If a @code{defmath} form contains an @code{interactive} clause, it defines
31820 a Calculator command. Actually such a @code{defmath} results in @emph{two}
31821 function definitions: One, a @samp{calcFunc-} function as was just described,
31822 with the @code{interactive} clause removed. Two, a @samp{calc-} function
31823 with a suitable @code{interactive} clause and some sort of wrapper to make
31824 the command work in the Calc environment.
31826 In the simple case, the @code{interactive} clause has the same form as
31827 for normal Emacs Lisp commands:
31830 (defmath increase-precision (delta)
31831 "Increase precision by DELTA." ; This is the "documentation string"
31832 (interactive "p") ; Register this as a M-x-able command
31833 (setq calc-internal-prec (+ calc-internal-prec delta)))
31836 This expands to the pair of definitions,
31839 (defun calc-increase-precision (delta)
31840 "Increase precision by DELTA."
31843 (setq calc-internal-prec (math-add calc-internal-prec delta))))
31845 (defun calcFunc-increase-precision (delta)
31846 "Increase precision by DELTA."
31847 (setq calc-internal-prec (math-add calc-internal-prec delta)))
31851 where in this case the latter function would never really be used! Note
31852 that since the Calculator stores small integers as plain Lisp integers,
31853 the @code{math-add} function will work just as well as the native
31854 @code{+} even when the intent is to operate on native Lisp integers.
31856 @findex calc-wrapper
31857 The @samp{calc-wrapper} call invokes a macro which surrounds the body of
31858 the function with code that looks roughly like this:
31861 (let ((calc-command-flags nil))
31864 (calc-select-buffer)
31865 @emph{body of function}
31866 @emph{renumber stack}
31867 @emph{clear} Working @emph{message})
31868 @emph{realign cursor and window}
31869 @emph{clear Inverse, Hyperbolic, and Keep Args flags}
31870 @emph{update Emacs mode line}))
31873 @findex calc-select-buffer
31874 The @code{calc-select-buffer} function selects the @samp{*Calculator*}
31875 buffer if necessary, say, because the command was invoked from inside
31876 the @samp{*Calc Trail*} window.
31878 @findex calc-set-command-flag
31879 You can call, for example, @code{(calc-set-command-flag 'no-align)} to
31880 set the above-mentioned command flags. Calc routines recognize the
31881 following command flags:
31885 Stack line numbers @samp{1:}, @samp{2:}, and so on must be renumbered
31886 after this command completes. This is set by routines like
31889 @item clear-message
31890 Calc should call @samp{(message "")} if this command completes normally
31891 (to clear a ``Working@dots{}'' message out of the echo area).
31894 Do not move the cursor back to the @samp{.} top-of-stack marker.
31896 @item position-point
31897 Use the variables @code{calc-position-point-line} and
31898 @code{calc-position-point-column} to position the cursor after
31899 this command finishes.
31902 Do not clear @code{calc-inverse-flag}, @code{calc-hyperbolic-flag},
31903 and @code{calc-keep-args-flag} at the end of this command.
31906 Switch to buffer @samp{*Calc Edit*} after this command.
31909 Do not move trail pointer to end of trail when something is recorded
31915 @vindex calc-Y-help-msgs
31916 Calc reserves a special prefix key, shift-@kbd{Y}, for user-written
31917 extensions to Calc. There are no built-in commands that work with
31918 this prefix key; you must call @code{define-key} from Lisp (probably
31919 from inside a @code{calc-define} property) to add to it. Initially only
31920 @kbd{Y ?} is defined; it takes help messages from a list of strings
31921 (initially @code{nil}) in the variable @code{calc-Y-help-msgs}. All
31922 other undefined keys except for @kbd{Y} are reserved for use by
31923 future versions of Calc.
31925 If you are writing a Calc enhancement which you expect to give to
31926 others, it is best to minimize the number of @kbd{Y}-key sequences
31927 you use. In fact, if you have more than one key sequence you should
31928 consider defining three-key sequences with a @kbd{Y}, then a key that
31929 stands for your package, then a third key for the particular command
31930 within your package.
31932 Users may wish to install several Calc enhancements, and it is possible
31933 that several enhancements will choose to use the same key. In the
31934 example below, a variable @code{inc-prec-base-key} has been defined
31935 to contain the key that identifies the @code{inc-prec} package. Its
31936 value is initially @code{"P"}, but a user can change this variable
31937 if necessary without having to modify the file.
31939 Here is a complete file, @file{inc-prec.el}, which makes a @kbd{Y P I}
31940 command that increases the precision, and a @kbd{Y P D} command that
31941 decreases the precision.
31944 ;;; Increase and decrease Calc precision. Dave Gillespie, 5/31/91.
31945 ;;; (Include copyright or copyleft stuff here.)
31947 (defvar inc-prec-base-key "P"
31948 "Base key for inc-prec.el commands.")
31950 (put 'calc-define 'inc-prec '(progn
31952 (define-key calc-mode-map (format "Y%sI" inc-prec-base-key)
31953 'increase-precision)
31954 (define-key calc-mode-map (format "Y%sD" inc-prec-base-key)
31955 'decrease-precision)
31957 (setq calc-Y-help-msgs
31958 (cons (format "%s + Inc-prec, Dec-prec" inc-prec-base-key)
31961 (defmath increase-precision (delta)
31962 "Increase precision by DELTA."
31964 (setq calc-internal-prec (+ calc-internal-prec delta)))
31966 (defmath decrease-precision (delta)
31967 "Decrease precision by DELTA."
31969 (setq calc-internal-prec (- calc-internal-prec delta)))
31971 )) ; end of calc-define property
31973 (run-hooks 'calc-check-defines)
31976 @node Defining Stack Commands, Argument Qualifiers, Defining Simple Commands, Lisp Definitions
31977 @subsection Defining New Stack-Based Commands
31980 To define a new computational command which takes and/or leaves arguments
31981 on the stack, a special form of @code{interactive} clause is used.
31984 (interactive @var{num} @var{tag})
31988 where @var{num} is an integer, and @var{tag} is a string. The effect is
31989 to pop @var{num} values off the stack, resimplify them by calling
31990 @code{calc-normalize}, and hand them to your function according to the
31991 function's argument list. Your function may include @code{&optional} and
31992 @code{&rest} parameters, so long as calling the function with @var{num}
31993 parameters is valid.
31995 Your function must return either a number or a formula in a form
31996 acceptable to Calc, or a list of such numbers or formulas. These value(s)
31997 are pushed onto the stack when the function completes. They are also
31998 recorded in the Calc Trail buffer on a line beginning with @var{tag},
31999 a string of (normally) four characters or less. If you omit @var{tag}
32000 or use @code{nil} as a tag, the result is not recorded in the trail.
32002 As an example, the definition
32005 (defmath myfact (n)
32006 "Compute the factorial of the integer at the top of the stack."
32007 (interactive 1 "fact")
32009 (* n (myfact (1- n)))
32014 is a version of the factorial function shown previously which can be used
32015 as a command as well as an algebraic function. It expands to
32018 (defun calc-myfact ()
32019 "Compute the factorial of the integer at the top of the stack."
32022 (calc-enter-result 1 "fact"
32023 (cons 'calcFunc-myfact (calc-top-list-n 1)))))
32025 (defun calcFunc-myfact (n)
32026 "Compute the factorial of the integer at the top of the stack."
32028 (math-mul n (calcFunc-myfact (math-add n -1)))
32029 (and (math-zerop n) 1)))
32032 @findex calc-slow-wrapper
32033 The @code{calc-slow-wrapper} function is a version of @code{calc-wrapper}
32034 that automatically puts up a @samp{Working...} message before the
32035 computation begins. (This message can be turned off by the user
32036 with an @kbd{m w} (@code{calc-working}) command.)
32038 @findex calc-top-list-n
32039 The @code{calc-top-list-n} function returns a list of the specified number
32040 of values from the top of the stack. It resimplifies each value by
32041 calling @code{calc-normalize}. If its argument is zero it returns an
32042 empty list. It does not actually remove these values from the stack.
32044 @findex calc-enter-result
32045 The @code{calc-enter-result} function takes an integer @var{num} and string
32046 @var{tag} as described above, plus a third argument which is either a
32047 Calculator data object or a list of such objects. These objects are
32048 resimplified and pushed onto the stack after popping the specified number
32049 of values from the stack. If @var{tag} is non-@code{nil}, the values
32050 being pushed are also recorded in the trail.
32052 Note that if @code{calcFunc-myfact} returns @code{nil} this represents
32053 ``leave the function in symbolic form.'' To return an actual empty list,
32054 in the sense that @code{calc-enter-result} will push zero elements back
32055 onto the stack, you should return the special value @samp{'(nil)}, a list
32056 containing the single symbol @code{nil}.
32058 The @code{interactive} declaration can actually contain a limited
32059 Emacs-style code string as well which comes just before @var{num} and
32060 @var{tag}. Currently the only Emacs code supported is @samp{"p"}, as in
32063 (defmath foo (a b &optional c)
32064 (interactive "p" 2 "foo")
32068 In this example, the command @code{calc-foo} will evaluate the expression
32069 @samp{foo(a,b)} if executed with no argument, or @samp{foo(a,b,n)} if
32070 executed with a numeric prefix argument of @expr{n}.
32072 The other code string allowed is @samp{"m"} (unrelated to the usual @samp{"m"}
32073 code as used with @code{defun}). It uses the numeric prefix argument as the
32074 number of objects to remove from the stack and pass to the function.
32075 In this case, the integer @var{num} serves as a default number of
32076 arguments to be used when no prefix is supplied.
32078 @node Argument Qualifiers, Example Definitions, Defining Stack Commands, Lisp Definitions
32079 @subsection Argument Qualifiers
32082 Anywhere a parameter name can appear in the parameter list you can also use
32083 an @dfn{argument qualifier}. Thus the general form of a definition is:
32086 (defmath @var{name} (@var{param} @var{param...}
32087 &optional @var{param} @var{param...}
32093 where each @var{param} is either a symbol or a list of the form
32096 (@var{qual} @var{param})
32099 The following qualifiers are recognized:
32104 The argument must not be an incomplete vector, interval, or complex number.
32105 (This is rarely needed since the Calculator itself will never call your
32106 function with an incomplete argument. But there is nothing stopping your
32107 own Lisp code from calling your function with an incomplete argument.)
32111 The argument must be an integer. If it is an integer-valued float
32112 it will be accepted but converted to integer form. Non-integers and
32113 formulas are rejected.
32117 Like @samp{integer}, but the argument must be non-negative.
32121 Like @samp{integer}, but the argument must fit into a native Lisp integer,
32122 which on most systems means less than 2^23 in absolute value. The
32123 argument is converted into Lisp-integer form if necessary.
32127 The argument is converted to floating-point format if it is a number or
32128 vector. If it is a formula it is left alone. (The argument is never
32129 actually rejected by this qualifier.)
32132 The argument must satisfy predicate @var{pred}, which is one of the
32133 standard Calculator predicates. @xref{Predicates}.
32135 @item not-@var{pred}
32136 The argument must @emph{not} satisfy predicate @var{pred}.
32142 (defmath foo (a (constp (not-matrixp b)) &optional (float c)
32151 (defun calcFunc-foo (a b &optional c &rest d)
32152 (and (math-matrixp b)
32153 (math-reject-arg b 'not-matrixp))
32154 (or (math-constp b)
32155 (math-reject-arg b 'constp))
32156 (and c (setq c (math-check-float c)))
32157 (setq d (mapcar 'math-check-integer d))
32162 which performs the necessary checks and conversions before executing the
32163 body of the function.
32165 @node Example Definitions, Calling Calc from Your Programs, Argument Qualifiers, Lisp Definitions
32166 @subsection Example Definitions
32169 This section includes some Lisp programming examples on a larger scale.
32170 These programs make use of some of the Calculator's internal functions;
32174 * Bit Counting Example::
32178 @node Bit Counting Example, Sine Example, Example Definitions, Example Definitions
32179 @subsubsection Bit-Counting
32186 Calc does not include a built-in function for counting the number of
32187 ``one'' bits in a binary integer. It's easy to invent one using @kbd{b u}
32188 to convert the integer to a set, and @kbd{V #} to count the elements of
32189 that set; let's write a function that counts the bits without having to
32190 create an intermediate set.
32193 (defmath bcount ((natnum n))
32194 (interactive 1 "bcnt")
32198 (setq count (1+ count)))
32199 (setq n (lsh n -1)))
32204 When this is expanded by @code{defmath}, it will become the following
32205 Emacs Lisp function:
32208 (defun calcFunc-bcount (n)
32209 (setq n (math-check-natnum n))
32211 (while (math-posp n)
32213 (setq count (math-add count 1)))
32214 (setq n (calcFunc-lsh n -1)))
32218 If the input numbers are large, this function involves a fair amount
32219 of arithmetic. A binary right shift is essentially a division by two;
32220 recall that Calc stores integers in decimal form so bit shifts must
32221 involve actual division.
32223 To gain a bit more efficiency, we could divide the integer into
32224 @var{n}-bit chunks, each of which can be handled quickly because
32225 they fit into Lisp integers. It turns out that Calc's arithmetic
32226 routines are especially fast when dividing by an integer less than
32227 1000, so we can set @var{n = 9} bits and use repeated division by 512:
32230 (defmath bcount ((natnum n))
32231 (interactive 1 "bcnt")
32233 (while (not (fixnump n))
32234 (let ((qr (idivmod n 512)))
32235 (setq count (+ count (bcount-fixnum (cdr qr)))
32237 (+ count (bcount-fixnum n))))
32239 (defun bcount-fixnum (n)
32242 (setq count (+ count (logand n 1))
32248 Note that the second function uses @code{defun}, not @code{defmath}.
32249 Because this function deals only with native Lisp integers (``fixnums''),
32250 it can use the actual Emacs @code{+} and related functions rather
32251 than the slower but more general Calc equivalents which @code{defmath}
32254 The @code{idivmod} function does an integer division, returning both
32255 the quotient and the remainder at once. Again, note that while it
32256 might seem that @samp{(logand n 511)} and @samp{(lsh n -9)} are
32257 more efficient ways to split off the bottom nine bits of @code{n},
32258 actually they are less efficient because each operation is really
32259 a division by 512 in disguise; @code{idivmod} allows us to do the
32260 same thing with a single division by 512.
32262 @node Sine Example, , Bit Counting Example, Example Definitions
32263 @subsubsection The Sine Function
32270 A somewhat limited sine function could be defined as follows, using the
32271 well-known Taylor series expansion for
32272 @texline @math{\sin x}:
32273 @infoline @samp{sin(x)}:
32276 (defmath mysin ((float (anglep x)))
32277 (interactive 1 "mysn")
32278 (setq x (to-radians x)) ; Convert from current angular mode.
32279 (let ((sum x) ; Initial term of Taylor expansion of sin.
32281 (nfact 1) ; "nfact" equals "n" factorial at all times.
32282 (xnegsqr :"-(x^2)")) ; "xnegsqr" equals -x^2.
32283 (for ((n 3 100 2)) ; Upper limit of 100 is a good precaution.
32284 (working "mysin" sum) ; Display "Working" message, if enabled.
32285 (setq nfact (* nfact (1- n) n)
32287 newsum (+ sum (/ x nfact)))
32288 (if (~= newsum sum) ; If newsum is "nearly equal to" sum,
32289 (break)) ; then we are done.
32294 The actual @code{sin} function in Calc works by first reducing the problem
32295 to a sine or cosine of a nonnegative number less than @cpiover{4}. This
32296 ensures that the Taylor series will converge quickly. Also, the calculation
32297 is carried out with two extra digits of precision to guard against cumulative
32298 round-off in @samp{sum}. Finally, complex arguments are allowed and handled
32299 by a separate algorithm.
32302 (defmath mysin ((float (scalarp x)))
32303 (interactive 1 "mysn")
32304 (setq x (to-radians x)) ; Convert from current angular mode.
32305 (with-extra-prec 2 ; Evaluate with extra precision.
32306 (cond ((complexp x)
32309 (- (mysin-raw (- x))) ; Always call mysin-raw with x >= 0.
32310 (t (mysin-raw x))))))
32312 (defmath mysin-raw (x)
32314 (mysin-raw (% x (two-pi)))) ; Now x < 7.
32316 (- (mysin-raw (- x (pi))))) ; Now -pi/2 <= x <= pi/2.
32318 (mycos-raw (- x (pi-over-2)))) ; Now -pi/2 <= x <= pi/4.
32319 ((< x (- (pi-over-4)))
32320 (- (mycos-raw (+ x (pi-over-2))))) ; Now -pi/4 <= x <= pi/4,
32321 (t (mysin-series x)))) ; so the series will be efficient.
32325 where @code{mysin-complex} is an appropriate function to handle complex
32326 numbers, @code{mysin-series} is the routine to compute the sine Taylor
32327 series as before, and @code{mycos-raw} is a function analogous to
32328 @code{mysin-raw} for cosines.
32330 The strategy is to ensure that @expr{x} is nonnegative before calling
32331 @code{mysin-raw}. This function then recursively reduces its argument
32332 to a suitable range, namely, plus-or-minus @cpiover{4}. Note that each
32333 test, and particularly the first comparison against 7, is designed so
32334 that small roundoff errors cannot produce an infinite loop. (Suppose
32335 we compared with @samp{(two-pi)} instead; if due to roundoff problems
32336 the modulo operator ever returned @samp{(two-pi)} exactly, an infinite
32337 recursion could result!) We use modulo only for arguments that will
32338 clearly get reduced, knowing that the next rule will catch any reductions
32339 that this rule misses.
32341 If a program is being written for general use, it is important to code
32342 it carefully as shown in this second example. For quick-and-dirty programs,
32343 when you know that your own use of the sine function will never encounter
32344 a large argument, a simpler program like the first one shown is fine.
32346 @node Calling Calc from Your Programs, Internals, Example Definitions, Lisp Definitions
32347 @subsection Calling Calc from Your Lisp Programs
32350 A later section (@pxref{Internals}) gives a full description of
32351 Calc's internal Lisp functions. It's not hard to call Calc from
32352 inside your programs, but the number of these functions can be daunting.
32353 So Calc provides one special ``programmer-friendly'' function called
32354 @code{calc-eval} that can be made to do just about everything you
32355 need. It's not as fast as the low-level Calc functions, but it's
32356 much simpler to use!
32358 It may seem that @code{calc-eval} itself has a daunting number of
32359 options, but they all stem from one simple operation.
32361 In its simplest manifestation, @samp{(calc-eval "1+2")} parses the
32362 string @code{"1+2"} as if it were a Calc algebraic entry and returns
32363 the result formatted as a string: @code{"3"}.
32365 Since @code{calc-eval} is on the list of recommended @code{autoload}
32366 functions, you don't need to make any special preparations to load
32367 Calc before calling @code{calc-eval} the first time. Calc will be
32368 loaded and initialized for you.
32370 All the Calc modes that are currently in effect will be used when
32371 evaluating the expression and formatting the result.
32378 @subsubsection Additional Arguments to @code{calc-eval}
32381 If the input string parses to a list of expressions, Calc returns
32382 the results separated by @code{", "}. You can specify a different
32383 separator by giving a second string argument to @code{calc-eval}:
32384 @samp{(calc-eval "1+2,3+4" ";")} returns @code{"3;7"}.
32386 The ``separator'' can also be any of several Lisp symbols which
32387 request other behaviors from @code{calc-eval}. These are discussed
32390 You can give additional arguments to be substituted for
32391 @samp{$}, @samp{$$}, and so on in the main expression. For
32392 example, @samp{(calc-eval "$/$$" nil "7" "1+1")} evaluates the
32393 expression @code{"7/(1+1)"} to yield the result @code{"3.5"}
32394 (assuming Fraction mode is not in effect). Note the @code{nil}
32395 used as a placeholder for the item-separator argument.
32402 @subsubsection Error Handling
32405 If @code{calc-eval} encounters an error, it returns a list containing
32406 the character position of the error, plus a suitable message as a
32407 string. Note that @samp{1 / 0} is @emph{not} an error by Calc's
32408 standards; it simply returns the string @code{"1 / 0"} which is the
32409 division left in symbolic form. But @samp{(calc-eval "1/")} will
32410 return the list @samp{(2 "Expected a number")}.
32412 If you bind the variable @code{calc-eval-error} to @code{t}
32413 using a @code{let} form surrounding the call to @code{calc-eval},
32414 errors instead call the Emacs @code{error} function which aborts
32415 to the Emacs command loop with a beep and an error message.
32417 If you bind this variable to the symbol @code{string}, error messages
32418 are returned as strings instead of lists. The character position is
32421 As a courtesy to other Lisp code which may be using Calc, be sure
32422 to bind @code{calc-eval-error} using @code{let} rather than changing
32423 it permanently with @code{setq}.
32430 @subsubsection Numbers Only
32433 Sometimes it is preferable to treat @samp{1 / 0} as an error
32434 rather than returning a symbolic result. If you pass the symbol
32435 @code{num} as the second argument to @code{calc-eval}, results
32436 that are not constants are treated as errors. The error message
32437 reported is the first @code{calc-why} message if there is one,
32438 or otherwise ``Number expected.''
32440 A result is ``constant'' if it is a number, vector, or other
32441 object that does not include variables or function calls. If it
32442 is a vector, the components must themselves be constants.
32449 @subsubsection Default Modes
32452 If the first argument to @code{calc-eval} is a list whose first
32453 element is a formula string, then @code{calc-eval} sets all the
32454 various Calc modes to their default values while the formula is
32455 evaluated and formatted. For example, the precision is set to 12
32456 digits, digit grouping is turned off, and the Normal language
32459 This same principle applies to the other options discussed below.
32460 If the first argument would normally be @var{x}, then it can also
32461 be the list @samp{(@var{x})} to use the default mode settings.
32463 If there are other elements in the list, they are taken as
32464 variable-name/value pairs which override the default mode
32465 settings. Look at the documentation at the front of the
32466 @file{calc.el} file to find the names of the Lisp variables for
32467 the various modes. The mode settings are restored to their
32468 original values when @code{calc-eval} is done.
32470 For example, @samp{(calc-eval '("$+$$" calc-internal-prec 8) 'num a b)}
32471 computes the sum of two numbers, requiring a numeric result, and
32472 using default mode settings except that the precision is 8 instead
32473 of the default of 12.
32475 It's usually best to use this form of @code{calc-eval} unless your
32476 program actually considers the interaction with Calc's mode settings
32477 to be a feature. This will avoid all sorts of potential ``gotchas'';
32478 consider what happens with @samp{(calc-eval "sqrt(2)" 'num)}
32479 when the user has left Calc in Symbolic mode or No-Simplify mode.
32481 As another example, @samp{(equal (calc-eval '("$<$$") nil a b) "1")}
32482 checks if the number in string @expr{a} is less than the one in
32483 string @expr{b}. Without using a list, the integer 1 might
32484 come out in a variety of formats which would be hard to test for
32485 conveniently: @code{"1"}, @code{"8#1"}, @code{"00001"}. (But
32486 see ``Predicates'' mode, below.)
32493 @subsubsection Raw Numbers
32496 Normally all input and output for @code{calc-eval} is done with strings.
32497 You can do arithmetic with, say, @samp{(calc-eval "$+$$" nil a b)}
32498 in place of @samp{(+ a b)}, but this is very inefficient since the
32499 numbers must be converted to and from string format as they are passed
32500 from one @code{calc-eval} to the next.
32502 If the separator is the symbol @code{raw}, the result will be returned
32503 as a raw Calc data structure rather than a string. You can read about
32504 how these objects look in the following sections, but usually you can
32505 treat them as ``black box'' objects with no important internal
32508 There is also a @code{rawnum} symbol, which is a combination of
32509 @code{raw} (returning a raw Calc object) and @code{num} (signaling
32510 an error if that object is not a constant).
32512 You can pass a raw Calc object to @code{calc-eval} in place of a
32513 string, either as the formula itself or as one of the @samp{$}
32514 arguments. Thus @samp{(calc-eval "$+$$" 'raw a b)} is an
32515 addition function that operates on raw Calc objects. Of course
32516 in this case it would be easier to call the low-level @code{math-add}
32517 function in Calc, if you can remember its name.
32519 In particular, note that a plain Lisp integer is acceptable to Calc
32520 as a raw object. (All Lisp integers are accepted on input, but
32521 integers of more than six decimal digits are converted to ``big-integer''
32522 form for output. @xref{Data Type Formats}.)
32524 When it comes time to display the object, just use @samp{(calc-eval a)}
32525 to format it as a string.
32527 It is an error if the input expression evaluates to a list of
32528 values. The separator symbol @code{list} is like @code{raw}
32529 except that it returns a list of one or more raw Calc objects.
32531 Note that a Lisp string is not a valid Calc object, nor is a list
32532 containing a string. Thus you can still safely distinguish all the
32533 various kinds of error returns discussed above.
32540 @subsubsection Predicates
32543 If the separator symbol is @code{pred}, the result of the formula is
32544 treated as a true/false value; @code{calc-eval} returns @code{t} or
32545 @code{nil}, respectively. A value is considered ``true'' if it is a
32546 non-zero number, or false if it is zero or if it is not a number.
32548 For example, @samp{(calc-eval "$<$$" 'pred a b)} tests whether
32549 one value is less than another.
32551 As usual, it is also possible for @code{calc-eval} to return one of
32552 the error indicators described above. Lisp will interpret such an
32553 indicator as ``true'' if you don't check for it explicitly. If you
32554 wish to have an error register as ``false'', use something like
32555 @samp{(eq (calc-eval ...) t)}.
32562 @subsubsection Variable Values
32565 Variables in the formula passed to @code{calc-eval} are not normally
32566 replaced by their values. If you wish this, you can use the
32567 @code{evalv} function (@pxref{Algebraic Manipulation}). For example,
32568 if 4 is stored in Calc variable @code{a} (i.e., in Lisp variable
32569 @code{var-a}), then @samp{(calc-eval "a+pi")} will return the
32570 formula @code{"a + pi"}, but @samp{(calc-eval "evalv(a+pi)")}
32571 will return @code{"7.14159265359"}.
32573 To store in a Calc variable, just use @code{setq} to store in the
32574 corresponding Lisp variable. (This is obtained by prepending
32575 @samp{var-} to the Calc variable name.) Calc routines will
32576 understand either string or raw form values stored in variables,
32577 although raw data objects are much more efficient. For example,
32578 to increment the Calc variable @code{a}:
32581 (setq var-a (calc-eval "evalv(a+1)" 'raw))
32589 @subsubsection Stack Access
32592 If the separator symbol is @code{push}, the formula argument is
32593 evaluated (with possible @samp{$} expansions, as usual). The
32594 result is pushed onto the Calc stack. The return value is @code{nil}
32595 (unless there is an error from evaluating the formula, in which
32596 case the return value depends on @code{calc-eval-error} in the
32599 If the separator symbol is @code{pop}, the first argument to
32600 @code{calc-eval} must be an integer instead of a string. That
32601 many values are popped from the stack and thrown away. A negative
32602 argument deletes the entry at that stack level. The return value
32603 is the number of elements remaining in the stack after popping;
32604 @samp{(calc-eval 0 'pop)} is a good way to measure the size of
32607 If the separator symbol is @code{top}, the first argument to
32608 @code{calc-eval} must again be an integer. The value at that
32609 stack level is formatted as a string and returned. Thus
32610 @samp{(calc-eval 1 'top)} returns the top-of-stack value. If the
32611 integer is out of range, @code{nil} is returned.
32613 The separator symbol @code{rawtop} is just like @code{top} except
32614 that the stack entry is returned as a raw Calc object instead of
32617 In all of these cases the first argument can be made a list in
32618 order to force the default mode settings, as described above.
32619 Thus @samp{(calc-eval '(2 calc-number-radix 16) 'top)} returns the
32620 second-to-top stack entry, formatted as a string using the default
32621 instead of current display modes, except that the radix is
32622 hexadecimal instead of decimal.
32624 It is, of course, polite to put the Calc stack back the way you
32625 found it when you are done, unless the user of your program is
32626 actually expecting it to affect the stack.
32628 Note that you do not actually have to switch into the @samp{*Calculator*}
32629 buffer in order to use @code{calc-eval}; it temporarily switches into
32630 the stack buffer if necessary.
32637 @subsubsection Keyboard Macros
32640 If the separator symbol is @code{macro}, the first argument must be a
32641 string of characters which Calc can execute as a sequence of keystrokes.
32642 This switches into the Calc buffer for the duration of the macro.
32643 For example, @samp{(calc-eval "vx5\rVR+" 'macro)} pushes the
32644 vector @samp{[1,2,3,4,5]} on the stack and then replaces it
32645 with the sum of those numbers. Note that @samp{\r} is the Lisp
32646 notation for the carriage-return, @key{RET}, character.
32648 If your keyboard macro wishes to pop the stack, @samp{\C-d} is
32649 safer than @samp{\177} (the @key{DEL} character) because some
32650 installations may have switched the meanings of @key{DEL} and
32651 @kbd{C-h}. Calc always interprets @kbd{C-d} as a synonym for
32652 ``pop-stack'' regardless of key mapping.
32654 If you provide a third argument to @code{calc-eval}, evaluation
32655 of the keyboard macro will leave a record in the Trail using
32656 that argument as a tag string. Normally the Trail is unaffected.
32658 The return value in this case is always @code{nil}.
32665 @subsubsection Lisp Evaluation
32668 Finally, if the separator symbol is @code{eval}, then the Lisp
32669 @code{eval} function is called on the first argument, which must
32670 be a Lisp expression rather than a Calc formula. Remember to
32671 quote the expression so that it is not evaluated until inside
32674 The difference from plain @code{eval} is that @code{calc-eval}
32675 switches to the Calc buffer before evaluating the expression.
32676 For example, @samp{(calc-eval '(setq calc-internal-prec 17) 'eval)}
32677 will correctly affect the buffer-local Calc precision variable.
32679 An alternative would be @samp{(calc-eval '(calc-precision 17) 'eval)}.
32680 This is evaluating a call to the function that is normally invoked
32681 by the @kbd{p} key, giving it 17 as its ``numeric prefix argument.''
32682 Note that this function will leave a message in the echo area as
32683 a side effect. Also, all Calc functions switch to the Calc buffer
32684 automatically if not invoked from there, so the above call is
32685 also equivalent to @samp{(calc-precision 17)} by itself.
32686 In all cases, Calc uses @code{save-excursion} to switch back to
32687 your original buffer when it is done.
32689 As usual the first argument can be a list that begins with a Lisp
32690 expression to use default instead of current mode settings.
32692 The result of @code{calc-eval} in this usage is just the result
32693 returned by the evaluated Lisp expression.
32700 @subsubsection Example
32703 @findex convert-temp
32704 Here is a sample Emacs command that uses @code{calc-eval}. Suppose
32705 you have a document with lots of references to temperatures on the
32706 Fahrenheit scale, say ``98.6 F'', and you wish to convert these
32707 references to Centigrade. The following command does this conversion.
32708 Place the Emacs cursor right after the letter ``F'' and invoke the
32709 command to change ``98.6 F'' to ``37 C''. Or, if the temperature is
32710 already in Centigrade form, the command changes it back to Fahrenheit.
32713 (defun convert-temp ()
32716 (re-search-backward "[^-.0-9]\\([-.0-9]+\\) *\\([FC]\\)")
32717 (let* ((top1 (match-beginning 1))
32718 (bot1 (match-end 1))
32719 (number (buffer-substring top1 bot1))
32720 (top2 (match-beginning 2))
32721 (bot2 (match-end 2))
32722 (type (buffer-substring top2 bot2)))
32723 (if (equal type "F")
32725 number (calc-eval "($ - 32)*5/9" nil number))
32727 number (calc-eval "$*9/5 + 32" nil number)))
32729 (delete-region top2 bot2)
32730 (insert-before-markers type)
32732 (delete-region top1 bot1)
32733 (if (string-match "\\.$" number) ; change "37." to "37"
32734 (setq number (substring number 0 -1)))
32738 Note the use of @code{insert-before-markers} when changing between
32739 ``F'' and ``C'', so that the character winds up before the cursor
32740 instead of after it.
32742 @node Internals, , Calling Calc from Your Programs, Lisp Definitions
32743 @subsection Calculator Internals
32746 This section describes the Lisp functions defined by the Calculator that
32747 may be of use to user-written Calculator programs (as described in the
32748 rest of this chapter). These functions are shown by their names as they
32749 conventionally appear in @code{defmath}. Their full Lisp names are
32750 generally gotten by prepending @samp{calcFunc-} or @samp{math-} to their
32751 apparent names. (Names that begin with @samp{calc-} are already in
32752 their full Lisp form.) You can use the actual full names instead if you
32753 prefer them, or if you are calling these functions from regular Lisp.
32755 The functions described here are scattered throughout the various
32756 Calc component files. Note that @file{calc.el} includes @code{autoload}s
32757 for only a few component files; when Calc wants to call an advanced
32758 function it calls @samp{(calc-extensions)} first; this function
32759 autoloads @file{calc-ext.el}, which in turn autoloads all the functions
32760 in the remaining component files.
32762 Because @code{defmath} itself uses the extensions, user-written code
32763 generally always executes with the extensions already loaded, so
32764 normally you can use any Calc function and be confident that it will
32765 be autoloaded for you when necessary. If you are doing something
32766 special, check carefully to make sure each function you are using is
32767 from @file{calc.el} or its components, and call @samp{(calc-extensions)}
32768 before using any function based in @file{calc-ext.el} if you can't
32769 prove this file will already be loaded.
32772 * Data Type Formats::
32773 * Interactive Lisp Functions::
32774 * Stack Lisp Functions::
32776 * Computational Lisp Functions::
32777 * Vector Lisp Functions::
32778 * Symbolic Lisp Functions::
32779 * Formatting Lisp Functions::
32783 @node Data Type Formats, Interactive Lisp Functions, Internals, Internals
32784 @subsubsection Data Type Formats
32787 Integers are stored in either of two ways, depending on their magnitude.
32788 Integers less than one million in absolute value are stored as standard
32789 Lisp integers. This is the only storage format for Calc data objects
32790 which is not a Lisp list.
32792 Large integers are stored as lists of the form @samp{(bigpos @var{d0}
32793 @var{d1} @var{d2} @dots{})} for positive integers 1000000 or more, or
32794 @samp{(bigneg @var{d0} @var{d1} @var{d2} @dots{})} for negative integers
32795 @mathit{-1000000} or less. Each @var{d} is a base-1000 ``digit,'' a Lisp integer
32796 from 0 to 999. The least significant digit is @var{d0}; the last digit,
32797 @var{dn}, which is always nonzero, is the most significant digit. For
32798 example, the integer @mathit{-12345678} is stored as @samp{(bigneg 678 345 12)}.
32800 The distinction between small and large integers is entirely hidden from
32801 the user. In @code{defmath} definitions, the Lisp predicate @code{integerp}
32802 returns true for either kind of integer, and in general both big and small
32803 integers are accepted anywhere the word ``integer'' is used in this manual.
32804 If the distinction must be made, native Lisp integers are called @dfn{fixnums}
32805 and large integers are called @dfn{bignums}.
32807 Fractions are stored as a list of the form, @samp{(frac @var{n} @var{d})}
32808 where @var{n} is an integer (big or small) numerator, @var{d} is an
32809 integer denominator greater than one, and @var{n} and @var{d} are relatively
32810 prime. Note that fractions where @var{d} is one are automatically converted
32811 to plain integers by all math routines; fractions where @var{d} is negative
32812 are normalized by negating the numerator and denominator.
32814 Floating-point numbers are stored in the form, @samp{(float @var{mant}
32815 @var{exp})}, where @var{mant} (the ``mantissa'') is an integer less than
32816 @samp{10^@var{p}} in absolute value (@var{p} represents the current
32817 precision), and @var{exp} (the ``exponent'') is a fixnum. The value of
32818 the float is @samp{@var{mant} * 10^@var{exp}}. For example, the number
32819 @mathit{-3.14} is stored as @samp{(float -314 -2) = -314*10^-2}. Other constraints
32820 are that the number 0.0 is always stored as @samp{(float 0 0)}, and,
32821 except for the 0.0 case, the rightmost base-10 digit of @var{mant} is
32822 always nonzero. (If the rightmost digit is zero, the number is
32823 rearranged by dividing @var{mant} by ten and incrementing @var{exp}.)
32825 Rectangular complex numbers are stored in the form @samp{(cplx @var{re}
32826 @var{im})}, where @var{re} and @var{im} are each real numbers, either
32827 integers, fractions, or floats. The value is @samp{@var{re} + @var{im}i}.
32828 The @var{im} part is nonzero; complex numbers with zero imaginary
32829 components are converted to real numbers automatically.
32831 Polar complex numbers are stored in the form @samp{(polar @var{r}
32832 @var{theta})}, where @var{r} is a positive real value and @var{theta}
32833 is a real value or HMS form representing an angle. This angle is
32834 usually normalized to lie in the interval @samp{(-180 ..@: 180)} degrees,
32835 or @samp{(-pi ..@: pi)} radians, according to the current angular mode.
32836 If the angle is 0 the value is converted to a real number automatically.
32837 (If the angle is 180 degrees, the value is usually also converted to a
32838 negative real number.)
32840 Hours-minutes-seconds forms are stored as @samp{(hms @var{h} @var{m}
32841 @var{s})}, where @var{h} is an integer or an integer-valued float (i.e.,
32842 a float with @samp{@var{exp} >= 0}), @var{m} is an integer or integer-valued
32843 float in the range @w{@samp{[0 ..@: 60)}}, and @var{s} is any real number
32844 in the range @samp{[0 ..@: 60)}.
32846 Date forms are stored as @samp{(date @var{n})}, where @var{n} is
32847 a real number that counts days since midnight on the morning of
32848 January 1, 1 AD. If @var{n} is an integer, this is a pure date
32849 form. If @var{n} is a fraction or float, this is a date/time form.
32851 Modulo forms are stored as @samp{(mod @var{n} @var{m})}, where @var{m} is a
32852 positive real number or HMS form, and @var{n} is a real number or HMS
32853 form in the range @samp{[0 ..@: @var{m})}.
32855 Error forms are stored as @samp{(sdev @var{x} @var{sigma})}, where @var{x}
32856 is the mean value and @var{sigma} is the standard deviation. Each
32857 component is either a number, an HMS form, or a symbolic object
32858 (a variable or function call). If @var{sigma} is zero, the value is
32859 converted to a plain real number. If @var{sigma} is negative or
32860 complex, it is automatically normalized to be a positive real.
32862 Interval forms are stored as @samp{(intv @var{mask} @var{lo} @var{hi})},
32863 where @var{mask} is one of the integers 0, 1, 2, or 3, and @var{lo} and
32864 @var{hi} are real numbers, HMS forms, or symbolic objects. The @var{mask}
32865 is a binary integer where 1 represents the fact that the interval is
32866 closed on the high end, and 2 represents the fact that it is closed on
32867 the low end. (Thus 3 represents a fully closed interval.) The interval
32868 @w{@samp{(intv 3 @var{x} @var{x})}} is converted to the plain number @var{x};
32869 intervals @samp{(intv @var{mask} @var{x} @var{x})} for any other @var{mask}
32870 represent empty intervals. If @var{hi} is less than @var{lo}, the interval
32871 is converted to a standard empty interval by replacing @var{hi} with @var{lo}.
32873 Vectors are stored as @samp{(vec @var{v1} @var{v2} @dots{})}, where @var{v1}
32874 is the first element of the vector, @var{v2} is the second, and so on.
32875 An empty vector is stored as @samp{(vec)}. A matrix is simply a vector
32876 where all @var{v}'s are themselves vectors of equal lengths. Note that
32877 Calc vectors are unrelated to the Emacs Lisp ``vector'' type, which is
32878 generally unused by Calc data structures.
32880 Variables are stored as @samp{(var @var{name} @var{sym})}, where
32881 @var{name} is a Lisp symbol whose print name is used as the visible name
32882 of the variable, and @var{sym} is a Lisp symbol in which the variable's
32883 value is actually stored. Thus, @samp{(var pi var-pi)} represents the
32884 special constant @samp{pi}. Almost always, the form is @samp{(var
32885 @var{v} var-@var{v})}. If the variable name was entered with @code{#}
32886 signs (which are converted to hyphens internally), the form is
32887 @samp{(var @var{u} @var{v})}, where @var{u} is a symbol whose name
32888 contains @code{#} characters, and @var{v} is a symbol that contains
32889 @code{-} characters instead. The value of a variable is the Calc
32890 object stored in its @var{sym} symbol's value cell. If the symbol's
32891 value cell is void or if it contains @code{nil}, the variable has no
32892 value. Special constants have the form @samp{(special-const
32893 @var{value})} stored in their value cell, where @var{value} is a formula
32894 which is evaluated when the constant's value is requested. Variables
32895 which represent units are not stored in any special way; they are units
32896 only because their names appear in the units table. If the value
32897 cell contains a string, it is parsed to get the variable's value when
32898 the variable is used.
32900 A Lisp list with any other symbol as the first element is a function call.
32901 The symbols @code{+}, @code{-}, @code{*}, @code{/}, @code{%}, @code{^},
32902 and @code{|} represent special binary operators; these lists are always
32903 of the form @samp{(@var{op} @var{lhs} @var{rhs})} where @var{lhs} is the
32904 sub-formula on the lefthand side and @var{rhs} is the sub-formula on the
32905 right. The symbol @code{neg} represents unary negation; this list is always
32906 of the form @samp{(neg @var{arg})}. Any other symbol @var{func} represents a
32907 function that would be displayed in function-call notation; the symbol
32908 @var{func} is in general always of the form @samp{calcFunc-@var{name}}.
32909 The function cell of the symbol @var{func} should contain a Lisp function
32910 for evaluating a call to @var{func}. This function is passed the remaining
32911 elements of the list (themselves already evaluated) as arguments; such
32912 functions should return @code{nil} or call @code{reject-arg} to signify
32913 that they should be left in symbolic form, or they should return a Calc
32914 object which represents their value, or a list of such objects if they
32915 wish to return multiple values. (The latter case is allowed only for
32916 functions which are the outer-level call in an expression whose value is
32917 about to be pushed on the stack; this feature is considered obsolete
32918 and is not used by any built-in Calc functions.)
32920 @node Interactive Lisp Functions, Stack Lisp Functions, Data Type Formats, Internals
32921 @subsubsection Interactive Functions
32924 The functions described here are used in implementing interactive Calc
32925 commands. Note that this list is not exhaustive! If there is an
32926 existing command that behaves similarly to the one you want to define,
32927 you may find helpful tricks by checking the source code for that command.
32929 @defun calc-set-command-flag flag
32930 Set the command flag @var{flag}. This is generally a Lisp symbol, but
32931 may in fact be anything. The effect is to add @var{flag} to the list
32932 stored in the variable @code{calc-command-flags}, unless it is already
32933 there. @xref{Defining Simple Commands}.
32936 @defun calc-clear-command-flag flag
32937 If @var{flag} appears among the list of currently-set command flags,
32938 remove it from that list.
32941 @defun calc-record-undo rec
32942 Add the ``undo record'' @var{rec} to the list of steps to take if the
32943 current operation should need to be undone. Stack push and pop functions
32944 automatically call @code{calc-record-undo}, so the kinds of undo records
32945 you might need to create take the form @samp{(set @var{sym} @var{value})},
32946 which says that the Lisp variable @var{sym} was changed and had previously
32947 contained @var{value}; @samp{(store @var{var} @var{value})} which says that
32948 the Calc variable @var{var} (a string which is the name of the symbol that
32949 contains the variable's value) was stored and its previous value was
32950 @var{value} (either a Calc data object, or @code{nil} if the variable was
32951 previously void); or @samp{(eval @var{undo} @var{redo} @var{args} @dots{})},
32952 which means that to undo requires calling the function @samp{(@var{undo}
32953 @var{args} @dots{})} and, if the undo is later redone, calling
32954 @samp{(@var{redo} @var{args} @dots{})}.
32957 @defun calc-record-why msg args
32958 Record the error or warning message @var{msg}, which is normally a string.
32959 This message will be replayed if the user types @kbd{w} (@code{calc-why});
32960 if the message string begins with a @samp{*}, it is considered important
32961 enough to display even if the user doesn't type @kbd{w}. If one or more
32962 @var{args} are present, the displayed message will be of the form,
32963 @samp{@var{msg}: @var{arg1}, @var{arg2}, @dots{}}, where the arguments are
32964 formatted on the assumption that they are either strings or Calc objects of
32965 some sort. If @var{msg} is a symbol, it is the name of a Calc predicate
32966 (such as @code{integerp} or @code{numvecp}) which the arguments did not
32967 satisfy; it is expanded to a suitable string such as ``Expected an
32968 integer.'' The @code{reject-arg} function calls @code{calc-record-why}
32969 automatically; @pxref{Predicates}.
32972 @defun calc-is-inverse
32973 This predicate returns true if the current command is inverse,
32974 i.e., if the Inverse (@kbd{I} key) flag was set.
32977 @defun calc-is-hyperbolic
32978 This predicate is the analogous function for the @kbd{H} key.
32981 @node Stack Lisp Functions, Predicates, Interactive Lisp Functions, Internals
32982 @subsubsection Stack-Oriented Functions
32985 The functions described here perform various operations on the Calc
32986 stack and trail. They are to be used in interactive Calc commands.
32988 @defun calc-push-list vals n
32989 Push the Calc objects in list @var{vals} onto the stack at stack level
32990 @var{n}. If @var{n} is omitted it defaults to 1, so that the elements
32991 are pushed at the top of the stack. If @var{n} is greater than 1, the
32992 elements will be inserted into the stack so that the last element will
32993 end up at level @var{n}, the next-to-last at level @var{n}+1, etc.
32994 The elements of @var{vals} are assumed to be valid Calc objects, and
32995 are not evaluated, rounded, or renormalized in any way. If @var{vals}
32996 is an empty list, nothing happens.
32998 The stack elements are pushed without any sub-formula selections.
32999 You can give an optional third argument to this function, which must
33000 be a list the same size as @var{vals} of selections. Each selection
33001 must be @code{eq} to some sub-formula of the corresponding formula
33002 in @var{vals}, or @code{nil} if that formula should have no selection.
33005 @defun calc-top-list n m
33006 Return a list of the @var{n} objects starting at level @var{m} of the
33007 stack. If @var{m} is omitted it defaults to 1, so that the elements are
33008 taken from the top of the stack. If @var{n} is omitted, it also
33009 defaults to 1, so that the top stack element (in the form of a
33010 one-element list) is returned. If @var{m} is greater than 1, the
33011 @var{m}th stack element will be at the end of the list, the @var{m}+1st
33012 element will be next-to-last, etc. If @var{n} or @var{m} are out of
33013 range, the command is aborted with a suitable error message. If @var{n}
33014 is zero, the function returns an empty list. The stack elements are not
33015 evaluated, rounded, or renormalized.
33017 If any stack elements contain selections, and selections have not
33018 been disabled by the @kbd{j e} (@code{calc-enable-selections}) command,
33019 this function returns the selected portions rather than the entire
33020 stack elements. It can be given a third ``selection-mode'' argument
33021 which selects other behaviors. If it is the symbol @code{t}, then
33022 a selection in any of the requested stack elements produces an
33023 ``invalid operation on selections'' error. If it is the symbol @code{full},
33024 the whole stack entry is always returned regardless of selections.
33025 If it is the symbol @code{sel}, the selected portion is always returned,
33026 or @code{nil} if there is no selection. (This mode ignores the @kbd{j e}
33027 command.) If the symbol is @code{entry}, the complete stack entry in
33028 list form is returned; the first element of this list will be the whole
33029 formula, and the third element will be the selection (or @code{nil}).
33032 @defun calc-pop-stack n m
33033 Remove the specified elements from the stack. The parameters @var{n}
33034 and @var{m} are defined the same as for @code{calc-top-list}. The return
33035 value of @code{calc-pop-stack} is uninteresting.
33037 If there are any selected sub-formulas among the popped elements, and
33038 @kbd{j e} has not been used to disable selections, this produces an
33039 error without changing the stack. If you supply an optional third
33040 argument of @code{t}, the stack elements are popped even if they
33041 contain selections.
33044 @defun calc-record-list vals tag
33045 This function records one or more results in the trail. The @var{vals}
33046 are a list of strings or Calc objects. The @var{tag} is the four-character
33047 tag string to identify the values. If @var{tag} is omitted, a blank tag
33051 @defun calc-normalize n
33052 This function takes a Calc object and ``normalizes'' it. At the very
33053 least this involves re-rounding floating-point values according to the
33054 current precision and other similar jobs. Also, unless the user has
33055 selected No-Simplify mode (@pxref{Simplification Modes}), this involves
33056 actually evaluating a formula object by executing the function calls
33057 it contains, and possibly also doing algebraic simplification, etc.
33060 @defun calc-top-list-n n m
33061 This function is identical to @code{calc-top-list}, except that it calls
33062 @code{calc-normalize} on the values that it takes from the stack. They
33063 are also passed through @code{check-complete}, so that incomplete
33064 objects will be rejected with an error message. All computational
33065 commands should use this in preference to @code{calc-top-list}; the only
33066 standard Calc commands that operate on the stack without normalizing
33067 are stack management commands like @code{calc-enter} and @code{calc-roll-up}.
33068 This function accepts the same optional selection-mode argument as
33069 @code{calc-top-list}.
33072 @defun calc-top-n m
33073 This function is a convenient form of @code{calc-top-list-n} in which only
33074 a single element of the stack is taken and returned, rather than a list
33075 of elements. This also accepts an optional selection-mode argument.
33078 @defun calc-enter-result n tag vals
33079 This function is a convenient interface to most of the above functions.
33080 The @var{vals} argument should be either a single Calc object, or a list
33081 of Calc objects; the object or objects are normalized, and the top @var{n}
33082 stack entries are replaced by the normalized objects. If @var{tag} is
33083 non-@code{nil}, the normalized objects are also recorded in the trail.
33084 A typical stack-based computational command would take the form,
33087 (calc-enter-result @var{n} @var{tag} (cons 'calcFunc-@var{func}
33088 (calc-top-list-n @var{n})))
33091 If any of the @var{n} stack elements replaced contain sub-formula
33092 selections, and selections have not been disabled by @kbd{j e},
33093 this function takes one of two courses of action. If @var{n} is
33094 equal to the number of elements in @var{vals}, then each element of
33095 @var{vals} is spliced into the corresponding selection; this is what
33096 happens when you use the @key{TAB} key, or when you use a unary
33097 arithmetic operation like @code{sqrt}. If @var{vals} has only one
33098 element but @var{n} is greater than one, there must be only one
33099 selection among the top @var{n} stack elements; the element from
33100 @var{vals} is spliced into that selection. This is what happens when
33101 you use a binary arithmetic operation like @kbd{+}. Any other
33102 combination of @var{n} and @var{vals} is an error when selections
33106 @defun calc-unary-op tag func arg
33107 This function implements a unary operator that allows a numeric prefix
33108 argument to apply the operator over many stack entries. If the prefix
33109 argument @var{arg} is @code{nil}, this uses @code{calc-enter-result}
33110 as outlined above. Otherwise, it maps the function over several stack
33111 elements; @pxref{Prefix Arguments}. For example,
33114 (defun calc-zeta (arg)
33116 (calc-unary-op "zeta" 'calcFunc-zeta arg))
33120 @defun calc-binary-op tag func arg ident unary
33121 This function implements a binary operator, analogously to
33122 @code{calc-unary-op}. The optional @var{ident} and @var{unary}
33123 arguments specify the behavior when the prefix argument is zero or
33124 one, respectively. If the prefix is zero, the value @var{ident}
33125 is pushed onto the stack, if specified, otherwise an error message
33126 is displayed. If the prefix is one, the unary function @var{unary}
33127 is applied to the top stack element, or, if @var{unary} is not
33128 specified, nothing happens. When the argument is two or more,
33129 the binary function @var{func} is reduced across the top @var{arg}
33130 stack elements; when the argument is negative, the function is
33131 mapped between the next-to-top @mathit{-@var{arg}} stack elements and the
33135 @defun calc-stack-size
33136 Return the number of elements on the stack as an integer. This count
33137 does not include elements that have been temporarily hidden by stack
33138 truncation; @pxref{Truncating the Stack}.
33141 @defun calc-cursor-stack-index n
33142 Move the point to the @var{n}th stack entry. If @var{n} is zero, this
33143 will be the @samp{.} line. If @var{n} is from 1 to the current stack size,
33144 this will be the beginning of the first line of that stack entry's display.
33145 If line numbers are enabled, this will move to the first character of the
33146 line number, not the stack entry itself.
33149 @defun calc-substack-height n
33150 Return the number of lines between the beginning of the @var{n}th stack
33151 entry and the bottom of the buffer. If @var{n} is zero, this
33152 will be one (assuming no stack truncation). If all stack entries are
33153 one line long (i.e., no matrices are displayed), the return value will
33154 be equal @var{n}+1 as long as @var{n} is in range. (Note that in Big
33155 mode, the return value includes the blank lines that separate stack
33159 @defun calc-refresh
33160 Erase the @code{*Calculator*} buffer and reformat its contents from memory.
33161 This must be called after changing any parameter, such as the current
33162 display radix, which might change the appearance of existing stack
33163 entries. (During a keyboard macro invoked by the @kbd{X} key, refreshing
33164 is suppressed, but a flag is set so that the entire stack will be refreshed
33165 rather than just the top few elements when the macro finishes.)
33168 @node Predicates, Computational Lisp Functions, Stack Lisp Functions, Internals
33169 @subsubsection Predicates
33172 The functions described here are predicates, that is, they return a
33173 true/false value where @code{nil} means false and anything else means
33174 true. These predicates are expanded by @code{defmath}, for example,
33175 from @code{zerop} to @code{math-zerop}. In many cases they correspond
33176 to native Lisp functions by the same name, but are extended to cover
33177 the full range of Calc data types.
33180 Returns true if @var{x} is numerically zero, in any of the Calc data
33181 types. (Note that for some types, such as error forms and intervals,
33182 it never makes sense to return true.) In @code{defmath}, the expression
33183 @samp{(= x 0)} will automatically be converted to @samp{(math-zerop x)},
33184 and @samp{(/= x 0)} will be converted to @samp{(not (math-zerop x))}.
33188 Returns true if @var{x} is negative. This accepts negative real numbers
33189 of various types, negative HMS and date forms, and intervals in which
33190 all included values are negative. In @code{defmath}, the expression
33191 @samp{(< x 0)} will automatically be converted to @samp{(math-negp x)},
33192 and @samp{(>= x 0)} will be converted to @samp{(not (math-negp x))}.
33196 Returns true if @var{x} is positive (and non-zero). For complex
33197 numbers, none of these three predicates will return true.
33200 @defun looks-negp x
33201 Returns true if @var{x} is ``negative-looking.'' This returns true if
33202 @var{x} is a negative number, or a formula with a leading minus sign
33203 such as @samp{-a/b}. In other words, this is an object which can be
33204 made simpler by calling @code{(- @var{x})}.
33208 Returns true if @var{x} is an integer of any size.
33212 Returns true if @var{x} is a native Lisp integer.
33216 Returns true if @var{x} is a nonnegative integer of any size.
33219 @defun fixnatnump x
33220 Returns true if @var{x} is a nonnegative Lisp integer.
33223 @defun num-integerp x
33224 Returns true if @var{x} is numerically an integer, i.e., either a
33225 true integer or a float with no significant digits to the right of
33229 @defun messy-integerp x
33230 Returns true if @var{x} is numerically, but not literally, an integer.
33231 A value is @code{num-integerp} if it is @code{integerp} or
33232 @code{messy-integerp} (but it is never both at once).
33235 @defun num-natnump x
33236 Returns true if @var{x} is numerically a nonnegative integer.
33240 Returns true if @var{x} is an even integer.
33243 @defun looks-evenp x
33244 Returns true if @var{x} is an even integer, or a formula with a leading
33245 multiplicative coefficient which is an even integer.
33249 Returns true if @var{x} is an odd integer.
33253 Returns true if @var{x} is a rational number, i.e., an integer or a
33258 Returns true if @var{x} is a real number, i.e., an integer, fraction,
33259 or floating-point number.
33263 Returns true if @var{x} is a real number or HMS form.
33267 Returns true if @var{x} is a float, or a complex number, error form,
33268 interval, date form, or modulo form in which at least one component
33273 Returns true if @var{x} is a rectangular or polar complex number
33274 (but not a real number).
33277 @defun rect-complexp x
33278 Returns true if @var{x} is a rectangular complex number.
33281 @defun polar-complexp x
33282 Returns true if @var{x} is a polar complex number.
33286 Returns true if @var{x} is a real number or a complex number.
33290 Returns true if @var{x} is a real or complex number or an HMS form.
33294 Returns true if @var{x} is a vector (this simply checks if its argument
33295 is a list whose first element is the symbol @code{vec}).
33299 Returns true if @var{x} is a number or vector.
33303 Returns true if @var{x} is a matrix, i.e., a vector of one or more vectors,
33304 all of the same size.
33307 @defun square-matrixp x
33308 Returns true if @var{x} is a square matrix.
33312 Returns true if @var{x} is any numeric Calc object, including real and
33313 complex numbers, HMS forms, date forms, error forms, intervals, and
33314 modulo forms. (Note that error forms and intervals may include formulas
33315 as their components; see @code{constp} below.)
33319 Returns true if @var{x} is an object or a vector. This also accepts
33320 incomplete objects, but it rejects variables and formulas (except as
33321 mentioned above for @code{objectp}).
33325 Returns true if @var{x} is a ``primitive'' or ``atomic'' Calc object,
33326 i.e., one whose components cannot be regarded as sub-formulas. This
33327 includes variables, and all @code{objectp} types except error forms
33332 Returns true if @var{x} is constant, i.e., a real or complex number,
33333 HMS form, date form, or error form, interval, or vector all of whose
33334 components are @code{constp}.
33338 Returns true if @var{x} is numerically less than @var{y}. Returns false
33339 if @var{x} is greater than or equal to @var{y}, or if the order is
33340 undefined or cannot be determined. Generally speaking, this works
33341 by checking whether @samp{@var{x} - @var{y}} is @code{negp}. In
33342 @code{defmath}, the expression @samp{(< x y)} will automatically be
33343 converted to @samp{(lessp x y)}; expressions involving @code{>}, @code{<=},
33344 and @code{>=} are similarly converted in terms of @code{lessp}.
33348 Returns true if @var{x} comes before @var{y} in a canonical ordering
33349 of Calc objects. If @var{x} and @var{y} are both real numbers, this
33350 will be the same as @code{lessp}. But whereas @code{lessp} considers
33351 other types of objects to be unordered, @code{beforep} puts any two
33352 objects into a definite, consistent order. The @code{beforep}
33353 function is used by the @kbd{V S} vector-sorting command, and also
33354 by @kbd{a s} to put the terms of a product into canonical order:
33355 This allows @samp{x y + y x} to be simplified easily to @samp{2 x y}.
33359 This is the standard Lisp @code{equal} predicate; it returns true if
33360 @var{x} and @var{y} are structurally identical. This is the usual way
33361 to compare numbers for equality, but note that @code{equal} will treat
33362 0 and 0.0 as different.
33365 @defun math-equal x y
33366 Returns true if @var{x} and @var{y} are numerically equal, either because
33367 they are @code{equal}, or because their difference is @code{zerop}. In
33368 @code{defmath}, the expression @samp{(= x y)} will automatically be
33369 converted to @samp{(math-equal x y)}.
33372 @defun equal-int x n
33373 Returns true if @var{x} and @var{n} are numerically equal, where @var{n}
33374 is a fixnum which is not a multiple of 10. This will automatically be
33375 used by @code{defmath} in place of the more general @code{math-equal}
33379 @defun nearly-equal x y
33380 Returns true if @var{x} and @var{y}, as floating-point numbers, are
33381 equal except possibly in the last decimal place. For example,
33382 314.159 and 314.166 are considered nearly equal if the current
33383 precision is 6 (since they differ by 7 units), but not if the current
33384 precision is 7 (since they differ by 70 units). Most functions which
33385 use series expansions use @code{with-extra-prec} to evaluate the
33386 series with 2 extra digits of precision, then use @code{nearly-equal}
33387 to decide when the series has converged; this guards against cumulative
33388 error in the series evaluation without doing extra work which would be
33389 lost when the result is rounded back down to the current precision.
33390 In @code{defmath}, this can be written @samp{(~= @var{x} @var{y})}.
33391 The @var{x} and @var{y} can be numbers of any kind, including complex.
33394 @defun nearly-zerop x y
33395 Returns true if @var{x} is nearly zero, compared to @var{y}. This
33396 checks whether @var{x} plus @var{y} would by be @code{nearly-equal}
33397 to @var{y} itself, to within the current precision, in other words,
33398 if adding @var{x} to @var{y} would have a negligible effect on @var{y}
33399 due to roundoff error. @var{X} may be a real or complex number, but
33400 @var{y} must be real.
33404 Return true if the formula @var{x} represents a true value in
33405 Calc, not Lisp, terms. It tests if @var{x} is a non-zero number
33406 or a provably non-zero formula.
33409 @defun reject-arg val pred
33410 Abort the current function evaluation due to unacceptable argument values.
33411 This calls @samp{(calc-record-why @var{pred} @var{val})}, then signals a
33412 Lisp error which @code{normalize} will trap. The net effect is that the
33413 function call which led here will be left in symbolic form.
33416 @defun inexact-value
33417 If Symbolic mode is enabled, this will signal an error that causes
33418 @code{normalize} to leave the formula in symbolic form, with the message
33419 ``Inexact result.'' (This function has no effect when not in Symbolic mode.)
33420 Note that if your function calls @samp{(sin 5)} in Symbolic mode, the
33421 @code{sin} function will call @code{inexact-value}, which will cause your
33422 function to be left unsimplified. You may instead wish to call
33423 @samp{(normalize (list 'calcFunc-sin 5))}, which in Symbolic mode will
33424 return the formula @samp{sin(5)} to your function.
33428 This signals an error that will be reported as a floating-point overflow.
33432 This signals a floating-point underflow.
33435 @node Computational Lisp Functions, Vector Lisp Functions, Predicates, Internals
33436 @subsubsection Computational Functions
33439 The functions described here do the actual computational work of the
33440 Calculator. In addition to these, note that any function described in
33441 the main body of this manual may be called from Lisp; for example, if
33442 the documentation refers to the @code{calc-sqrt} [@code{sqrt}] command,
33443 this means @code{calc-sqrt} is an interactive stack-based square-root
33444 command and @code{sqrt} (which @code{defmath} expands to @code{calcFunc-sqrt})
33445 is the actual Lisp function for taking square roots.
33447 The functions @code{math-add}, @code{math-sub}, @code{math-mul},
33448 @code{math-div}, @code{math-mod}, and @code{math-neg} are not included
33449 in this list, since @code{defmath} allows you to write native Lisp
33450 @code{+}, @code{-}, @code{*}, @code{/}, @code{%}, and unary @code{-},
33451 respectively, instead.
33453 @defun normalize val
33454 (Full form: @code{math-normalize}.)
33455 Reduce the value @var{val} to standard form. For example, if @var{val}
33456 is a fixnum, it will be converted to a bignum if it is too large, and
33457 if @var{val} is a bignum it will be normalized by clipping off trailing
33458 (i.e., most-significant) zero digits and converting to a fixnum if it is
33459 small. All the various data types are similarly converted to their standard
33460 forms. Variables are left alone, but function calls are actually evaluated
33461 in formulas. For example, normalizing @samp{(+ 2 (calcFunc-abs -4))} will
33464 If a function call fails, because the function is void or has the wrong
33465 number of parameters, or because it returns @code{nil} or calls
33466 @code{reject-arg} or @code{inexact-result}, @code{normalize} returns
33467 the formula still in symbolic form.
33469 If the current simplification mode is ``none'' or ``numeric arguments
33470 only,'' @code{normalize} will act appropriately. However, the more
33471 powerful simplification modes (like Algebraic Simplification) are
33472 not handled by @code{normalize}. They are handled by @code{calc-normalize},
33473 which calls @code{normalize} and possibly some other routines, such
33474 as @code{simplify} or @code{simplify-units}. Programs generally will
33475 never call @code{calc-normalize} except when popping or pushing values
33479 @defun evaluate-expr expr
33480 Replace all variables in @var{expr} that have values with their values,
33481 then use @code{normalize} to simplify the result. This is what happens
33482 when you press the @kbd{=} key interactively.
33485 @defmac with-extra-prec n body
33486 Evaluate the Lisp forms in @var{body} with precision increased by @var{n}
33487 digits. This is a macro which expands to
33491 (let ((calc-internal-prec (+ calc-internal-prec @var{n})))
33495 The surrounding call to @code{math-normalize} causes a floating-point
33496 result to be rounded down to the original precision afterwards. This
33497 is important because some arithmetic operations assume a number's
33498 mantissa contains no more digits than the current precision allows.
33501 @defun make-frac n d
33502 Build a fraction @samp{@var{n}:@var{d}}. This is equivalent to calling
33503 @samp{(normalize (list 'frac @var{n} @var{d}))}, but more efficient.
33506 @defun make-float mant exp
33507 Build a floating-point value out of @var{mant} and @var{exp}, both
33508 of which are arbitrary integers. This function will return a
33509 properly normalized float value, or signal an overflow or underflow
33510 if @var{exp} is out of range.
33513 @defun make-sdev x sigma
33514 Build an error form out of @var{x} and the absolute value of @var{sigma}.
33515 If @var{sigma} is zero, the result is the number @var{x} directly.
33516 If @var{sigma} is negative or complex, its absolute value is used.
33517 If @var{x} or @var{sigma} is not a valid type of object for use in
33518 error forms, this calls @code{reject-arg}.
33521 @defun make-intv mask lo hi
33522 Build an interval form out of @var{mask} (which is assumed to be an
33523 integer from 0 to 3), and the limits @var{lo} and @var{hi}. If
33524 @var{lo} is greater than @var{hi}, an empty interval form is returned.
33525 This calls @code{reject-arg} if @var{lo} or @var{hi} is unsuitable.
33528 @defun sort-intv mask lo hi
33529 Build an interval form, similar to @code{make-intv}, except that if
33530 @var{lo} is less than @var{hi} they are simply exchanged, and the
33531 bits of @var{mask} are swapped accordingly.
33534 @defun make-mod n m
33535 Build a modulo form out of @var{n} and the modulus @var{m}. Since modulo
33536 forms do not allow formulas as their components, if @var{n} or @var{m}
33537 is not a real number or HMS form the result will be a formula which
33538 is a call to @code{makemod}, the algebraic version of this function.
33542 Convert @var{x} to floating-point form. Integers and fractions are
33543 converted to numerically equivalent floats; components of complex
33544 numbers, vectors, HMS forms, date forms, error forms, intervals, and
33545 modulo forms are recursively floated. If the argument is a variable
33546 or formula, this calls @code{reject-arg}.
33550 Compare the numbers @var{x} and @var{y}, and return @mathit{-1} if
33551 @samp{(lessp @var{x} @var{y})}, 1 if @samp{(lessp @var{y} @var{x})},
33552 0 if @samp{(math-equal @var{x} @var{y})}, or 2 if the order is
33553 undefined or cannot be determined.
33557 Return the number of digits of integer @var{n}, effectively
33558 @samp{ceil(log10(@var{n}))}, but much more efficient. Zero is
33559 considered to have zero digits.
33562 @defun scale-int x n
33563 Shift integer @var{x} left @var{n} decimal digits, or right @mathit{-@var{n}}
33564 digits with truncation toward zero.
33567 @defun scale-rounding x n
33568 Like @code{scale-int}, except that a right shift rounds to the nearest
33569 integer rather than truncating.
33573 Return the integer @var{n} as a fixnum, i.e., a native Lisp integer.
33574 If @var{n} is outside the permissible range for Lisp integers (usually
33575 24 binary bits) the result is undefined.
33579 Compute the square of @var{x}; short for @samp{(* @var{x} @var{x})}.
33582 @defun quotient x y
33583 Divide integer @var{x} by integer @var{y}; return an integer quotient
33584 and discard the remainder. If @var{x} or @var{y} is negative, the
33585 direction of rounding is undefined.
33589 Perform an integer division; if @var{x} and @var{y} are both nonnegative
33590 integers, this uses the @code{quotient} function, otherwise it computes
33591 @samp{floor(@var{x}/@var{y})}. Thus the result is well-defined but
33592 slower than for @code{quotient}.
33596 Divide integer @var{x} by integer @var{y}; return the integer remainder
33597 and discard the quotient. Like @code{quotient}, this works only for
33598 integer arguments and is not well-defined for negative arguments.
33599 For a more well-defined result, use @samp{(% @var{x} @var{y})}.
33603 Divide integer @var{x} by integer @var{y}; return a cons cell whose
33604 @code{car} is @samp{(quotient @var{x} @var{y})} and whose @code{cdr}
33605 is @samp{(imod @var{x} @var{y})}.
33609 Compute @var{x} to the power @var{y}. In @code{defmath} code, this can
33610 also be written @samp{(^ @var{x} @var{y})} or
33611 @w{@samp{(expt @var{x} @var{y})}}.
33614 @defun abs-approx x
33615 Compute a fast approximation to the absolute value of @var{x}. For
33616 example, for a rectangular complex number the result is the sum of
33617 the absolute values of the components.
33623 @findex pi-over-180
33624 @findex sqrt-two-pi
33630 The function @samp{(pi)} computes @samp{pi} to the current precision.
33631 Other related constant-generating functions are @code{two-pi},
33632 @code{pi-over-2}, @code{pi-over-4}, @code{pi-over-180}, @code{sqrt-two-pi},
33633 @code{e}, @code{sqrt-e}, @code{ln-2}, and @code{ln-10}. Each function
33634 returns a floating-point value in the current precision, and each uses
33635 caching so that all calls after the first are essentially free.
33638 @defmac math-defcache @var{func} @var{initial} @var{form}
33639 This macro, usually used as a top-level call like @code{defun} or
33640 @code{defvar}, defines a new cached constant analogous to @code{pi}, etc.
33641 It defines a function @code{func} which returns the requested value;
33642 if @var{initial} is non-@code{nil} it must be a @samp{(float @dots{})}
33643 form which serves as an initial value for the cache. If @var{func}
33644 is called when the cache is empty or does not have enough digits to
33645 satisfy the current precision, the Lisp expression @var{form} is evaluated
33646 with the current precision increased by four, and the result minus its
33647 two least significant digits is stored in the cache. For example,
33648 calling @samp{(pi)} with a precision of 30 computes @samp{pi} to 34
33649 digits, rounds it down to 32 digits for future use, then rounds it
33650 again to 30 digits for use in the present request.
33653 @findex half-circle
33654 @findex quarter-circle
33655 @defun full-circle symb
33656 If the current angular mode is Degrees or HMS, this function returns the
33657 integer 360. In Radians mode, this function returns either the
33658 corresponding value in radians to the current precision, or the formula
33659 @samp{2*pi}, depending on the Symbolic mode. There are also similar
33660 function @code{half-circle} and @code{quarter-circle}.
33663 @defun power-of-2 n
33664 Compute two to the integer power @var{n}, as a (potentially very large)
33665 integer. Powers of two are cached, so only the first call for a
33666 particular @var{n} is expensive.
33669 @defun integer-log2 n
33670 Compute the base-2 logarithm of @var{n}, which must be an integer which
33671 is a power of two. If @var{n} is not a power of two, this function will
33675 @defun div-mod a b m
33676 Divide @var{a} by @var{b}, modulo @var{m}. This returns @code{nil} if
33677 there is no solution, or if any of the arguments are not integers.
33680 @defun pow-mod a b m
33681 Compute @var{a} to the power @var{b}, modulo @var{m}. If @var{a},
33682 @var{b}, and @var{m} are integers, this uses an especially efficient
33683 algorithm. Otherwise, it simply computes @samp{(% (^ a b) m)}.
33687 Compute the integer square root of @var{n}. This is the square root
33688 of @var{n} rounded down toward zero, i.e., @samp{floor(sqrt(@var{n}))}.
33689 If @var{n} is itself an integer, the computation is especially efficient.
33692 @defun to-hms a ang
33693 Convert the argument @var{a} into an HMS form. If @var{ang} is specified,
33694 it is the angular mode in which to interpret @var{a}, either @code{deg}
33695 or @code{rad}. Otherwise, the current angular mode is used. If @var{a}
33696 is already an HMS form it is returned as-is.
33699 @defun from-hms a ang
33700 Convert the HMS form @var{a} into a real number. If @var{ang} is specified,
33701 it is the angular mode in which to express the result, otherwise the
33702 current angular mode is used. If @var{a} is already a real number, it
33706 @defun to-radians a
33707 Convert the number or HMS form @var{a} to radians from the current
33711 @defun from-radians a
33712 Convert the number @var{a} from radians to the current angular mode.
33713 If @var{a} is a formula, this returns the formula @samp{deg(@var{a})}.
33716 @defun to-radians-2 a
33717 Like @code{to-radians}, except that in Symbolic mode a degrees to
33718 radians conversion yields a formula like @samp{@var{a}*pi/180}.
33721 @defun from-radians-2 a
33722 Like @code{from-radians}, except that in Symbolic mode a radians to
33723 degrees conversion yields a formula like @samp{@var{a}*180/pi}.
33726 @defun random-digit
33727 Produce a random base-1000 digit in the range 0 to 999.
33730 @defun random-digits n
33731 Produce a random @var{n}-digit integer; this will be an integer
33732 in the interval @samp{[0, 10^@var{n})}.
33735 @defun random-float
33736 Produce a random float in the interval @samp{[0, 1)}.
33739 @defun prime-test n iters
33740 Determine whether the integer @var{n} is prime. Return a list which has
33741 one of these forms: @samp{(nil @var{f})} means the number is non-prime
33742 because it was found to be divisible by @var{f}; @samp{(nil)} means it
33743 was found to be non-prime by table look-up (so no factors are known);
33744 @samp{(nil unknown)} means it is definitely non-prime but no factors
33745 are known because @var{n} was large enough that Fermat's probabilistic
33746 test had to be used; @samp{(t)} means the number is definitely prime;
33747 and @samp{(maybe @var{i} @var{p})} means that Fermat's test, after @var{i}
33748 iterations, is @var{p} percent sure that the number is prime. The
33749 @var{iters} parameter is the number of Fermat iterations to use, in the
33750 case that this is necessary. If @code{prime-test} returns ``maybe,''
33751 you can call it again with the same @var{n} to get a greater certainty;
33752 @code{prime-test} remembers where it left off.
33755 @defun to-simple-fraction f
33756 If @var{f} is a floating-point number which can be represented exactly
33757 as a small rational number. return that number, else return @var{f}.
33758 For example, 0.75 would be converted to 3:4. This function is very
33762 @defun to-fraction f tol
33763 Find a rational approximation to floating-point number @var{f} to within
33764 a specified tolerance @var{tol}; this corresponds to the algebraic
33765 function @code{frac}, and can be rather slow.
33768 @defun quarter-integer n
33769 If @var{n} is an integer or integer-valued float, this function
33770 returns zero. If @var{n} is a half-integer (i.e., an integer plus
33771 @mathit{1:2} or 0.5), it returns 2. If @var{n} is a quarter-integer,
33772 it returns 1 or 3. If @var{n} is anything else, this function
33773 returns @code{nil}.
33776 @node Vector Lisp Functions, Symbolic Lisp Functions, Computational Lisp Functions, Internals
33777 @subsubsection Vector Functions
33780 The functions described here perform various operations on vectors and
33783 @defun math-concat x y
33784 Do a vector concatenation; this operation is written @samp{@var{x} | @var{y}}
33785 in a symbolic formula. @xref{Building Vectors}.
33788 @defun vec-length v
33789 Return the length of vector @var{v}. If @var{v} is not a vector, the
33790 result is zero. If @var{v} is a matrix, this returns the number of
33791 rows in the matrix.
33794 @defun mat-dimens m
33795 Determine the dimensions of vector or matrix @var{m}. If @var{m} is not
33796 a vector, the result is an empty list. If @var{m} is a plain vector
33797 but not a matrix, the result is a one-element list containing the length
33798 of the vector. If @var{m} is a matrix with @var{r} rows and @var{c} columns,
33799 the result is the list @samp{(@var{r} @var{c})}. Higher-order tensors
33800 produce lists of more than two dimensions. Note that the object
33801 @samp{[[1, 2, 3], [4, 5]]} is a vector of vectors not all the same size,
33802 and is treated by this and other Calc routines as a plain vector of two
33806 @defun dimension-error
33807 Abort the current function with a message of ``Dimension error.''
33808 The Calculator will leave the function being evaluated in symbolic
33809 form; this is really just a special case of @code{reject-arg}.
33812 @defun build-vector args
33813 Return a Calc vector with @var{args} as elements.
33814 For example, @samp{(build-vector 1 2 3)} returns the Calc vector
33815 @samp{[1, 2, 3]}, stored internally as the list @samp{(vec 1 2 3)}.
33818 @defun make-vec obj dims
33819 Return a Calc vector or matrix all of whose elements are equal to
33820 @var{obj}. For example, @samp{(make-vec 27 3 4)} returns a 3x4 matrix
33824 @defun row-matrix v
33825 If @var{v} is a plain vector, convert it into a row matrix, i.e.,
33826 a matrix whose single row is @var{v}. If @var{v} is already a matrix,
33830 @defun col-matrix v
33831 If @var{v} is a plain vector, convert it into a column matrix, i.e., a
33832 matrix with each element of @var{v} as a separate row. If @var{v} is
33833 already a matrix, leave it alone.
33837 Map the Lisp function @var{f} over the Calc vector @var{v}. For example,
33838 @samp{(map-vec 'math-floor v)} returns a vector of the floored components
33842 @defun map-vec-2 f a b
33843 Map the Lisp function @var{f} over the two vectors @var{a} and @var{b}.
33844 If @var{a} and @var{b} are vectors of equal length, the result is a
33845 vector of the results of calling @samp{(@var{f} @var{ai} @var{bi})}
33846 for each pair of elements @var{ai} and @var{bi}. If either @var{a} or
33847 @var{b} is a scalar, it is matched with each value of the other vector.
33848 For example, @samp{(map-vec-2 'math-add v 1)} returns the vector @var{v}
33849 with each element increased by one. Note that using @samp{'+} would not
33850 work here, since @code{defmath} does not expand function names everywhere,
33851 just where they are in the function position of a Lisp expression.
33854 @defun reduce-vec f v
33855 Reduce the function @var{f} over the vector @var{v}. For example, if
33856 @var{v} is @samp{[10, 20, 30, 40]}, this calls @samp{(f (f (f 10 20) 30) 40)}.
33857 If @var{v} is a matrix, this reduces over the rows of @var{v}.
33860 @defun reduce-cols f m
33861 Reduce the function @var{f} over the columns of matrix @var{m}. For
33862 example, if @var{m} is @samp{[[1, 2], [3, 4], [5, 6]]}, the result
33863 is a vector of the two elements @samp{(f (f 1 3) 5)} and @samp{(f (f 2 4) 6)}.
33867 Return the @var{n}th row of matrix @var{m}. This is equivalent to
33868 @samp{(elt m n)}. For a slower but safer version, use @code{mrow}.
33869 (@xref{Extracting Elements}.)
33873 Return the @var{n}th column of matrix @var{m}, in the form of a vector.
33874 The arguments are not checked for correctness.
33877 @defun mat-less-row m n
33878 Return a copy of matrix @var{m} with its @var{n}th row deleted. The
33879 number @var{n} must be in range from 1 to the number of rows in @var{m}.
33882 @defun mat-less-col m n
33883 Return a copy of matrix @var{m} with its @var{n}th column deleted.
33887 Return the transpose of matrix @var{m}.
33890 @defun flatten-vector v
33891 Flatten nested vector @var{v} into a vector of scalars. For example,
33892 if @var{v} is @samp{[[1, 2, 3], [4, 5]]} the result is @samp{[1, 2, 3, 4, 5]}.
33895 @defun copy-matrix m
33896 If @var{m} is a matrix, return a copy of @var{m}. This maps
33897 @code{copy-sequence} over the rows of @var{m}; in Lisp terms, each
33898 element of the result matrix will be @code{eq} to the corresponding
33899 element of @var{m}, but none of the @code{cons} cells that make up
33900 the structure of the matrix will be @code{eq}. If @var{m} is a plain
33901 vector, this is the same as @code{copy-sequence}.
33904 @defun swap-rows m r1 r2
33905 Exchange rows @var{r1} and @var{r2} of matrix @var{m} in-place. In
33906 other words, unlike most of the other functions described here, this
33907 function changes @var{m} itself rather than building up a new result
33908 matrix. The return value is @var{m}, i.e., @samp{(eq (swap-rows m 1 2) m)}
33909 is true, with the side effect of exchanging the first two rows of
33913 @node Symbolic Lisp Functions, Formatting Lisp Functions, Vector Lisp Functions, Internals
33914 @subsubsection Symbolic Functions
33917 The functions described here operate on symbolic formulas in the
33920 @defun calc-prepare-selection num
33921 Prepare a stack entry for selection operations. If @var{num} is
33922 omitted, the stack entry containing the cursor is used; otherwise,
33923 it is the number of the stack entry to use. This function stores
33924 useful information about the current stack entry into a set of
33925 variables. @code{calc-selection-cache-num} contains the number of
33926 the stack entry involved (equal to @var{num} if you specified it);
33927 @code{calc-selection-cache-entry} contains the stack entry as a
33928 list (such as @code{calc-top-list} would return with @code{entry}
33929 as the selection mode); and @code{calc-selection-cache-comp} contains
33930 a special ``tagged'' composition (@pxref{Formatting Lisp Functions})
33931 which allows Calc to relate cursor positions in the buffer with
33932 their corresponding sub-formulas.
33934 A slight complication arises in the selection mechanism because
33935 formulas may contain small integers. For example, in the vector
33936 @samp{[1, 2, 1]} the first and last elements are @code{eq} to each
33937 other; selections are recorded as the actual Lisp object that
33938 appears somewhere in the tree of the whole formula, but storing
33939 @code{1} would falsely select both @code{1}'s in the vector. So
33940 @code{calc-prepare-selection} also checks the stack entry and
33941 replaces any plain integers with ``complex number'' lists of the form
33942 @samp{(cplx @var{n} 0)}. This list will be displayed the same as a
33943 plain @var{n} and the change will be completely invisible to the
33944 user, but it will guarantee that no two sub-formulas of the stack
33945 entry will be @code{eq} to each other. Next time the stack entry
33946 is involved in a computation, @code{calc-normalize} will replace
33947 these lists with plain numbers again, again invisibly to the user.
33950 @defun calc-encase-atoms x
33951 This modifies the formula @var{x} to ensure that each part of the
33952 formula is a unique atom, using the @samp{(cplx @var{n} 0)} trick
33953 described above. This function may use @code{setcar} to modify
33954 the formula in-place.
33957 @defun calc-find-selected-part
33958 Find the smallest sub-formula of the current formula that contains
33959 the cursor. This assumes @code{calc-prepare-selection} has been
33960 called already. If the cursor is not actually on any part of the
33961 formula, this returns @code{nil}.
33964 @defun calc-change-current-selection selection
33965 Change the currently prepared stack element's selection to
33966 @var{selection}, which should be @code{eq} to some sub-formula
33967 of the stack element, or @code{nil} to unselect the formula.
33968 The stack element's appearance in the Calc buffer is adjusted
33969 to reflect the new selection.
33972 @defun calc-find-nth-part expr n
33973 Return the @var{n}th sub-formula of @var{expr}. This function is used
33974 by the selection commands, and (unless @kbd{j b} has been used) treats
33975 sums and products as flat many-element formulas. Thus if @var{expr}
33976 is @samp{((a + b) - c) + d}, calling @code{calc-find-nth-part} with
33977 @var{n} equal to four will return @samp{d}.
33980 @defun calc-find-parent-formula expr part
33981 Return the sub-formula of @var{expr} which immediately contains
33982 @var{part}. If @var{expr} is @samp{a*b + (c+1)*d} and @var{part}
33983 is @code{eq} to the @samp{c+1} term of @var{expr}, then this function
33984 will return @samp{(c+1)*d}. If @var{part} turns out not to be a
33985 sub-formula of @var{expr}, the function returns @code{nil}. If
33986 @var{part} is @code{eq} to @var{expr}, the function returns @code{t}.
33987 This function does not take associativity into account.
33990 @defun calc-find-assoc-parent-formula expr part
33991 This is the same as @code{calc-find-parent-formula}, except that
33992 (unless @kbd{j b} has been used) it continues widening the selection
33993 to contain a complete level of the formula. Given @samp{a} from
33994 @samp{((a + b) - c) + d}, @code{calc-find-parent-formula} will
33995 return @samp{a + b} but @code{calc-find-assoc-parent-formula} will
33996 return the whole expression.
33999 @defun calc-grow-assoc-formula expr part
34000 This expands sub-formula @var{part} of @var{expr} to encompass a
34001 complete level of the formula. If @var{part} and its immediate
34002 parent are not compatible associative operators, or if @kbd{j b}
34003 has been used, this simply returns @var{part}.
34006 @defun calc-find-sub-formula expr part
34007 This finds the immediate sub-formula of @var{expr} which contains
34008 @var{part}. It returns an index @var{n} such that
34009 @samp{(calc-find-nth-part @var{expr} @var{n})} would return @var{part}.
34010 If @var{part} is not a sub-formula of @var{expr}, it returns @code{nil}.
34011 If @var{part} is @code{eq} to @var{expr}, it returns @code{t}. This
34012 function does not take associativity into account.
34015 @defun calc-replace-sub-formula expr old new
34016 This function returns a copy of formula @var{expr}, with the
34017 sub-formula that is @code{eq} to @var{old} replaced by @var{new}.
34020 @defun simplify expr
34021 Simplify the expression @var{expr} by applying various algebraic rules.
34022 This is what the @w{@kbd{a s}} (@code{calc-simplify}) command uses. This
34023 always returns a copy of the expression; the structure @var{expr} points
34024 to remains unchanged in memory.
34026 More precisely, here is what @code{simplify} does: The expression is
34027 first normalized and evaluated by calling @code{normalize}. If any
34028 @code{AlgSimpRules} have been defined, they are then applied. Then
34029 the expression is traversed in a depth-first, bottom-up fashion; at
34030 each level, any simplifications that can be made are made until no
34031 further changes are possible. Once the entire formula has been
34032 traversed in this way, it is compared with the original formula (from
34033 before the call to @code{normalize}) and, if it has changed,
34034 the entire procedure is repeated (starting with @code{normalize})
34035 until no further changes occur. Usually only two iterations are
34036 needed:@: one to simplify the formula, and another to verify that no
34037 further simplifications were possible.
34040 @defun simplify-extended expr
34041 Simplify the expression @var{expr}, with additional rules enabled that
34042 help do a more thorough job, while not being entirely ``safe'' in all
34043 circumstances. (For example, this mode will simplify @samp{sqrt(x^2)}
34044 to @samp{x}, which is only valid when @var{x} is positive.) This is
34045 implemented by temporarily binding the variable @code{math-living-dangerously}
34046 to @code{t} (using a @code{let} form) and calling @code{simplify}.
34047 Dangerous simplification rules are written to check this variable
34048 before taking any action.
34051 @defun simplify-units expr
34052 Simplify the expression @var{expr}, treating variable names as units
34053 whenever possible. This works by binding the variable
34054 @code{math-simplifying-units} to @code{t} while calling @code{simplify}.
34057 @defmac math-defsimplify funcs body
34058 Register a new simplification rule; this is normally called as a top-level
34059 form, like @code{defun} or @code{defmath}. If @var{funcs} is a symbol
34060 (like @code{+} or @code{calcFunc-sqrt}), this simplification rule is
34061 applied to the formulas which are calls to the specified function. Or,
34062 @var{funcs} can be a list of such symbols; the rule applies to all
34063 functions on the list. The @var{body} is written like the body of a
34064 function with a single argument called @code{expr}. The body will be
34065 executed with @code{expr} bound to a formula which is a call to one of
34066 the functions @var{funcs}. If the function body returns @code{nil}, or
34067 if it returns a result @code{equal} to the original @code{expr}, it is
34068 ignored and Calc goes on to try the next simplification rule that applies.
34069 If the function body returns something different, that new formula is
34070 substituted for @var{expr} in the original formula.
34072 At each point in the formula, rules are tried in the order of the
34073 original calls to @code{math-defsimplify}; the search stops after the
34074 first rule that makes a change. Thus later rules for that same
34075 function will not have a chance to trigger until the next iteration
34076 of the main @code{simplify} loop.
34078 Note that, since @code{defmath} is not being used here, @var{body} must
34079 be written in true Lisp code without the conveniences that @code{defmath}
34080 provides. If you prefer, you can have @var{body} simply call another
34081 function (defined with @code{defmath}) which does the real work.
34083 The arguments of a function call will already have been simplified
34084 before any rules for the call itself are invoked. Since a new argument
34085 list is consed up when this happens, this means that the rule's body is
34086 allowed to rearrange the function's arguments destructively if that is
34087 convenient. Here is a typical example of a simplification rule:
34090 (math-defsimplify calcFunc-arcsinh
34091 (or (and (math-looks-negp (nth 1 expr))
34092 (math-neg (list 'calcFunc-arcsinh
34093 (math-neg (nth 1 expr)))))
34094 (and (eq (car-safe (nth 1 expr)) 'calcFunc-sinh)
34095 (or math-living-dangerously
34096 (math-known-realp (nth 1 (nth 1 expr))))
34097 (nth 1 (nth 1 expr)))))
34100 This is really a pair of rules written with one @code{math-defsimplify}
34101 for convenience; the first replaces @samp{arcsinh(-x)} with
34102 @samp{-arcsinh(x)}, and the second, which is safe only for real @samp{x},
34103 replaces @samp{arcsinh(sinh(x))} with @samp{x}.
34106 @defun common-constant-factor expr
34107 Check @var{expr} to see if it is a sum of terms all multiplied by the
34108 same rational value. If so, return this value. If not, return @code{nil}.
34109 For example, if called on @samp{6x + 9y + 12z}, it would return 3, since
34110 3 is a common factor of all the terms.
34113 @defun cancel-common-factor expr factor
34114 Assuming @var{expr} is a sum with @var{factor} as a common factor,
34115 divide each term of the sum by @var{factor}. This is done by
34116 destructively modifying parts of @var{expr}, on the assumption that
34117 it is being used by a simplification rule (where such things are
34118 allowed; see above). For example, consider this built-in rule for
34122 (math-defsimplify calcFunc-sqrt
34123 (let ((fac (math-common-constant-factor (nth 1 expr))))
34124 (and fac (not (eq fac 1))
34125 (math-mul (math-normalize (list 'calcFunc-sqrt fac))
34127 (list 'calcFunc-sqrt
34128 (math-cancel-common-factor
34129 (nth 1 expr) fac)))))))
34133 @defun frac-gcd a b
34134 Compute a ``rational GCD'' of @var{a} and @var{b}, which must both be
34135 rational numbers. This is the fraction composed of the GCD of the
34136 numerators of @var{a} and @var{b}, over the GCD of the denominators.
34137 It is used by @code{common-constant-factor}. Note that the standard
34138 @code{gcd} function uses the LCM to combine the denominators.
34141 @defun map-tree func expr many
34142 Try applying Lisp function @var{func} to various sub-expressions of
34143 @var{expr}. Initially, call @var{func} with @var{expr} itself as an
34144 argument. If this returns an expression which is not @code{equal} to
34145 @var{expr}, apply @var{func} again until eventually it does return
34146 @var{expr} with no changes. Then, if @var{expr} is a function call,
34147 recursively apply @var{func} to each of the arguments. This keeps going
34148 until no changes occur anywhere in the expression; this final expression
34149 is returned by @code{map-tree}. Note that, unlike simplification rules,
34150 @var{func} functions may @emph{not} make destructive changes to
34151 @var{expr}. If a third argument @var{many} is provided, it is an
34152 integer which says how many times @var{func} may be applied; the
34153 default, as described above, is infinitely many times.
34156 @defun compile-rewrites rules
34157 Compile the rewrite rule set specified by @var{rules}, which should
34158 be a formula that is either a vector or a variable name. If the latter,
34159 the compiled rules are saved so that later @code{compile-rules} calls
34160 for that same variable can return immediately. If there are problems
34161 with the rules, this function calls @code{error} with a suitable
34165 @defun apply-rewrites expr crules heads
34166 Apply the compiled rewrite rule set @var{crules} to the expression
34167 @var{expr}. This will make only one rewrite and only checks at the
34168 top level of the expression. The result @code{nil} if no rules
34169 matched, or if the only rules that matched did not actually change
34170 the expression. The @var{heads} argument is optional; if is given,
34171 it should be a list of all function names that (may) appear in
34172 @var{expr}. The rewrite compiler tags each rule with the
34173 rarest-looking function name in the rule; if you specify @var{heads},
34174 @code{apply-rewrites} can use this information to narrow its search
34175 down to just a few rules in the rule set.
34178 @defun rewrite-heads expr
34179 Compute a @var{heads} list for @var{expr} suitable for use with
34180 @code{apply-rewrites}, as discussed above.
34183 @defun rewrite expr rules many
34184 This is an all-in-one rewrite function. It compiles the rule set
34185 specified by @var{rules}, then uses @code{map-tree} to apply the
34186 rules throughout @var{expr} up to @var{many} (default infinity)
34190 @defun match-patterns pat vec not-flag
34191 Given a Calc vector @var{vec} and an uncompiled pattern set or
34192 pattern set variable @var{pat}, this function returns a new vector
34193 of all elements of @var{vec} which do (or don't, if @var{not-flag} is
34194 non-@code{nil}) match any of the patterns in @var{pat}.
34197 @defun deriv expr var value symb
34198 Compute the derivative of @var{expr} with respect to variable @var{var}
34199 (which may actually be any sub-expression). If @var{value} is specified,
34200 the derivative is evaluated at the value of @var{var}; otherwise, the
34201 derivative is left in terms of @var{var}. If the expression contains
34202 functions for which no derivative formula is known, new derivative
34203 functions are invented by adding primes to the names; @pxref{Calculus}.
34204 However, if @var{symb} is non-@code{nil}, the presence of undifferentiable
34205 functions in @var{expr} instead cancels the whole differentiation, and
34206 @code{deriv} returns @code{nil} instead.
34208 Derivatives of an @var{n}-argument function can be defined by
34209 adding a @code{math-derivative-@var{n}} property to the property list
34210 of the symbol for the function's derivative, which will be the
34211 function name followed by an apostrophe. The value of the property
34212 should be a Lisp function; it is called with the same arguments as the
34213 original function call that is being differentiated. It should return
34214 a formula for the derivative. For example, the derivative of @code{ln}
34218 (put 'calcFunc-ln\' 'math-derivative-1
34219 (function (lambda (u) (math-div 1 u))))
34222 The two-argument @code{log} function has two derivatives,
34224 (put 'calcFunc-log\' 'math-derivative-2 ; d(log(x,b)) / dx
34225 (function (lambda (x b) ... )))
34226 (put 'calcFunc-log\'2 'math-derivative-2 ; d(log(x,b)) / db
34227 (function (lambda (x b) ... )))
34231 @defun tderiv expr var value symb
34232 Compute the total derivative of @var{expr}. This is the same as
34233 @code{deriv}, except that variables other than @var{var} are not
34234 assumed to be constant with respect to @var{var}.
34237 @defun integ expr var low high
34238 Compute the integral of @var{expr} with respect to @var{var}.
34239 @xref{Calculus}, for further details.
34242 @defmac math-defintegral funcs body
34243 Define a rule for integrating a function or functions of one argument;
34244 this macro is very similar in format to @code{math-defsimplify}.
34245 The main difference is that here @var{body} is the body of a function
34246 with a single argument @code{u} which is bound to the argument to the
34247 function being integrated, not the function call itself. Also, the
34248 variable of integration is available as @code{math-integ-var}. If
34249 evaluation of the integral requires doing further integrals, the body
34250 should call @samp{(math-integral @var{x})} to find the integral of
34251 @var{x} with respect to @code{math-integ-var}; this function returns
34252 @code{nil} if the integral could not be done. Some examples:
34255 (math-defintegral calcFunc-conj
34256 (let ((int (math-integral u)))
34258 (list 'calcFunc-conj int))))
34260 (math-defintegral calcFunc-cos
34261 (and (equal u math-integ-var)
34262 (math-from-radians-2 (list 'calcFunc-sin u))))
34265 In the @code{cos} example, we define only the integral of @samp{cos(x) dx},
34266 relying on the general integration-by-substitution facility to handle
34267 cosines of more complicated arguments. An integration rule should return
34268 @code{nil} if it can't do the integral; if several rules are defined for
34269 the same function, they are tried in order until one returns a non-@code{nil}
34273 @defmac math-defintegral-2 funcs body
34274 Define a rule for integrating a function or functions of two arguments.
34275 This is exactly analogous to @code{math-defintegral}, except that @var{body}
34276 is written as the body of a function with two arguments, @var{u} and
34280 @defun solve-for lhs rhs var full
34281 Attempt to solve the equation @samp{@var{lhs} = @var{rhs}} by isolating
34282 the variable @var{var} on the lefthand side; return the resulting righthand
34283 side, or @code{nil} if the equation cannot be solved. The variable
34284 @var{var} must appear at least once in @var{lhs} or @var{rhs}. Note that
34285 the return value is a formula which does not contain @var{var}; this is
34286 different from the user-level @code{solve} and @code{finv} functions,
34287 which return a rearranged equation or a functional inverse, respectively.
34288 If @var{full} is non-@code{nil}, a full solution including dummy signs
34289 and dummy integers will be produced. User-defined inverses are provided
34290 as properties in a manner similar to derivatives:
34293 (put 'calcFunc-ln 'math-inverse
34294 (function (lambda (x) (list 'calcFunc-exp x))))
34297 This function can call @samp{(math-solve-get-sign @var{x})} to create
34298 a new arbitrary sign variable, returning @var{x} times that sign, and
34299 @samp{(math-solve-get-int @var{x})} to create a new arbitrary integer
34300 variable multiplied by @var{x}. These functions simply return @var{x}
34301 if the caller requested a non-``full'' solution.
34304 @defun solve-eqn expr var full
34305 This version of @code{solve-for} takes an expression which will
34306 typically be an equation or inequality. (If it is not, it will be
34307 interpreted as the equation @samp{@var{expr} = 0}.) It returns an
34308 equation or inequality, or @code{nil} if no solution could be found.
34311 @defun solve-system exprs vars full
34312 This function solves a system of equations. Generally, @var{exprs}
34313 and @var{vars} will be vectors of equal length.
34314 @xref{Solving Systems of Equations}, for other options.
34317 @defun expr-contains expr var
34318 Returns a non-@code{nil} value if @var{var} occurs as a subexpression
34321 This function might seem at first to be identical to
34322 @code{calc-find-sub-formula}. The key difference is that
34323 @code{expr-contains} uses @code{equal} to test for matches, whereas
34324 @code{calc-find-sub-formula} uses @code{eq}. In the formula
34325 @samp{f(a, a)}, the two @samp{a}s will be @code{equal} but not
34326 @code{eq} to each other.
34329 @defun expr-contains-count expr var
34330 Returns the number of occurrences of @var{var} as a subexpression
34331 of @var{expr}, or @code{nil} if there are no occurrences.
34334 @defun expr-depends expr var
34335 Returns true if @var{expr} refers to any variable the occurs in @var{var}.
34336 In other words, it checks if @var{expr} and @var{var} have any variables
34340 @defun expr-contains-vars expr
34341 Return true if @var{expr} contains any variables, or @code{nil} if @var{expr}
34342 contains only constants and functions with constant arguments.
34345 @defun expr-subst expr old new
34346 Returns a copy of @var{expr}, with all occurrences of @var{old} replaced
34347 by @var{new}. This treats @code{lambda} forms specially with respect
34348 to the dummy argument variables, so that the effect is always to return
34349 @var{expr} evaluated at @var{old} = @var{new}.
34352 @defun multi-subst expr old new
34353 This is like @code{expr-subst}, except that @var{old} and @var{new}
34354 are lists of expressions to be substituted simultaneously. If one
34355 list is shorter than the other, trailing elements of the longer list
34359 @defun expr-weight expr
34360 Returns the ``weight'' of @var{expr}, basically a count of the total
34361 number of objects and function calls that appear in @var{expr}. For
34362 ``primitive'' objects, this will be one.
34365 @defun expr-height expr
34366 Returns the ``height'' of @var{expr}, which is the deepest level to
34367 which function calls are nested. (Note that @samp{@var{a} + @var{b}}
34368 counts as a function call.) For primitive objects, this returns zero.
34371 @defun polynomial-p expr var
34372 Check if @var{expr} is a polynomial in variable (or sub-expression)
34373 @var{var}. If so, return the degree of the polynomial, that is, the
34374 highest power of @var{var} that appears in @var{expr}. For example,
34375 for @samp{(x^2 + 3)^3 + 4} this would return 6. This function returns
34376 @code{nil} unless @var{expr}, when expanded out by @kbd{a x}
34377 (@code{calc-expand}), would consist of a sum of terms in which @var{var}
34378 appears only raised to nonnegative integer powers. Note that if
34379 @var{var} does not occur in @var{expr}, then @var{expr} is considered
34380 a polynomial of degree 0.
34383 @defun is-polynomial expr var degree loose
34384 Check if @var{expr} is a polynomial in variable or sub-expression
34385 @var{var}, and, if so, return a list representation of the polynomial
34386 where the elements of the list are coefficients of successive powers of
34387 @var{var}: @samp{@var{a} + @var{b} x + @var{c} x^3} would produce the
34388 list @samp{(@var{a} @var{b} 0 @var{c})}, and @samp{(x + 1)^2} would
34389 produce the list @samp{(1 2 1)}. The highest element of the list will
34390 be non-zero, with the special exception that if @var{expr} is the
34391 constant zero, the returned value will be @samp{(0)}. Return @code{nil}
34392 if @var{expr} is not a polynomial in @var{var}. If @var{degree} is
34393 specified, this will not consider polynomials of degree higher than that
34394 value. This is a good precaution because otherwise an input of
34395 @samp{(x+1)^1000} will cause a huge coefficient list to be built. If
34396 @var{loose} is non-@code{nil}, then a looser definition of a polynomial
34397 is used in which coefficients are no longer required not to depend on
34398 @var{var}, but are only required not to take the form of polynomials
34399 themselves. For example, @samp{sin(x) x^2 + cos(x)} is a loose
34400 polynomial with coefficients @samp{((calcFunc-cos x) 0 (calcFunc-sin
34401 x))}. The result will never be @code{nil} in loose mode, since any
34402 expression can be interpreted as a ``constant'' loose polynomial.
34405 @defun polynomial-base expr pred
34406 Check if @var{expr} is a polynomial in any variable that occurs in it;
34407 if so, return that variable. (If @var{expr} is a multivariate polynomial,
34408 this chooses one variable arbitrarily.) If @var{pred} is specified, it should
34409 be a Lisp function which is called as @samp{(@var{pred} @var{subexpr})},
34410 and which should return true if @code{mpb-top-expr} (a global name for
34411 the original @var{expr}) is a suitable polynomial in @var{subexpr}.
34412 The default predicate uses @samp{(polynomial-p mpb-top-expr @var{subexpr})};
34413 you can use @var{pred} to specify additional conditions. Or, you could
34414 have @var{pred} build up a list of every suitable @var{subexpr} that
34418 @defun poly-simplify poly
34419 Simplify polynomial coefficient list @var{poly} by (destructively)
34420 clipping off trailing zeros.
34423 @defun poly-mix a ac b bc
34424 Mix two polynomial lists @var{a} and @var{b} (in the form returned by
34425 @code{is-polynomial}) in a linear combination with coefficient expressions
34426 @var{ac} and @var{bc}. The result is a (not necessarily simplified)
34427 polynomial list representing @samp{@var{ac} @var{a} + @var{bc} @var{b}}.
34430 @defun poly-mul a b
34431 Multiply two polynomial coefficient lists @var{a} and @var{b}. The
34432 result will be in simplified form if the inputs were simplified.
34435 @defun build-polynomial-expr poly var
34436 Construct a Calc formula which represents the polynomial coefficient
34437 list @var{poly} applied to variable @var{var}. The @kbd{a c}
34438 (@code{calc-collect}) command uses @code{is-polynomial} to turn an
34439 expression into a coefficient list, then @code{build-polynomial-expr}
34440 to turn the list back into an expression in regular form.
34443 @defun check-unit-name var
34444 Check if @var{var} is a variable which can be interpreted as a unit
34445 name. If so, return the units table entry for that unit. This
34446 will be a list whose first element is the unit name (not counting
34447 prefix characters) as a symbol and whose second element is the
34448 Calc expression which defines the unit. (Refer to the Calc sources
34449 for details on the remaining elements of this list.) If @var{var}
34450 is not a variable or is not a unit name, return @code{nil}.
34453 @defun units-in-expr-p expr sub-exprs
34454 Return true if @var{expr} contains any variables which can be
34455 interpreted as units. If @var{sub-exprs} is @code{t}, the entire
34456 expression is searched. If @var{sub-exprs} is @code{nil}, this
34457 checks whether @var{expr} is directly a units expression.
34460 @defun single-units-in-expr-p expr
34461 Check whether @var{expr} contains exactly one units variable. If so,
34462 return the units table entry for the variable. If @var{expr} does
34463 not contain any units, return @code{nil}. If @var{expr} contains
34464 two or more units, return the symbol @code{wrong}.
34467 @defun to-standard-units expr which
34468 Convert units expression @var{expr} to base units. If @var{which}
34469 is @code{nil}, use Calc's native base units. Otherwise, @var{which}
34470 can specify a units system, which is a list of two-element lists,
34471 where the first element is a Calc base symbol name and the second
34472 is an expression to substitute for it.
34475 @defun remove-units expr
34476 Return a copy of @var{expr} with all units variables replaced by ones.
34477 This expression is generally normalized before use.
34480 @defun extract-units expr
34481 Return a copy of @var{expr} with everything but units variables replaced
34485 @node Formatting Lisp Functions, Hooks, Symbolic Lisp Functions, Internals
34486 @subsubsection I/O and Formatting Functions
34489 The functions described here are responsible for parsing and formatting
34490 Calc numbers and formulas.
34492 @defun calc-eval str sep arg1 arg2 @dots{}
34493 This is the simplest interface to the Calculator from another Lisp program.
34494 @xref{Calling Calc from Your Programs}.
34497 @defun read-number str
34498 If string @var{str} contains a valid Calc number, either integer,
34499 fraction, float, or HMS form, this function parses and returns that
34500 number. Otherwise, it returns @code{nil}.
34503 @defun read-expr str
34504 Read an algebraic expression from string @var{str}. If @var{str} does
34505 not have the form of a valid expression, return a list of the form
34506 @samp{(error @var{pos} @var{msg})} where @var{pos} is an integer index
34507 into @var{str} of the general location of the error, and @var{msg} is
34508 a string describing the problem.
34511 @defun read-exprs str
34512 Read a list of expressions separated by commas, and return it as a
34513 Lisp list. If an error occurs in any expressions, an error list as
34514 shown above is returned instead.
34517 @defun calc-do-alg-entry initial prompt no-norm
34518 Read an algebraic formula or formulas using the minibuffer. All
34519 conventions of regular algebraic entry are observed. The return value
34520 is a list of Calc formulas; there will be more than one if the user
34521 entered a list of values separated by commas. The result is @code{nil}
34522 if the user presses Return with a blank line. If @var{initial} is
34523 given, it is a string which the minibuffer will initially contain.
34524 If @var{prompt} is given, it is the prompt string to use; the default
34525 is ``Algebraic:''. If @var{no-norm} is @code{t}, the formulas will
34526 be returned exactly as parsed; otherwise, they will be passed through
34527 @code{calc-normalize} first.
34529 To support the use of @kbd{$} characters in the algebraic entry, use
34530 @code{let} to bind @code{calc-dollar-values} to a list of the values
34531 to be substituted for @kbd{$}, @kbd{$$}, and so on, and bind
34532 @code{calc-dollar-used} to 0. Upon return, @code{calc-dollar-used}
34533 will have been changed to the highest number of consecutive @kbd{$}s
34534 that actually appeared in the input.
34537 @defun format-number a
34538 Convert the real or complex number or HMS form @var{a} to string form.
34541 @defun format-flat-expr a prec
34542 Convert the arbitrary Calc number or formula @var{a} to string form,
34543 in the style used by the trail buffer and the @code{calc-edit} command.
34544 This is a simple format designed
34545 mostly to guarantee the string is of a form that can be re-parsed by
34546 @code{read-expr}. Most formatting modes, such as digit grouping,
34547 complex number format, and point character, are ignored to ensure the
34548 result will be re-readable. The @var{prec} parameter is normally 0; if
34549 you pass a large integer like 1000 instead, the expression will be
34550 surrounded by parentheses unless it is a plain number or variable name.
34553 @defun format-nice-expr a width
34554 This is like @code{format-flat-expr} (with @var{prec} equal to 0),
34555 except that newlines will be inserted to keep lines down to the
34556 specified @var{width}, and vectors that look like matrices or rewrite
34557 rules are written in a pseudo-matrix format. The @code{calc-edit}
34558 command uses this when only one stack entry is being edited.
34561 @defun format-value a width
34562 Convert the Calc number or formula @var{a} to string form, using the
34563 format seen in the stack buffer. Beware the string returned may
34564 not be re-readable by @code{read-expr}, for example, because of digit
34565 grouping. Multi-line objects like matrices produce strings that
34566 contain newline characters to separate the lines. The @var{w}
34567 parameter, if given, is the target window size for which to format
34568 the expressions. If @var{w} is omitted, the width of the Calculator
34572 @defun compose-expr a prec
34573 Format the Calc number or formula @var{a} according to the current
34574 language mode, returning a ``composition.'' To learn about the
34575 structure of compositions, see the comments in the Calc source code.
34576 You can specify the format of a given type of function call by putting
34577 a @code{math-compose-@var{lang}} property on the function's symbol,
34578 whose value is a Lisp function that takes @var{a} and @var{prec} as
34579 arguments and returns a composition. Here @var{lang} is a language
34580 mode name, one of @code{normal}, @code{big}, @code{c}, @code{pascal},
34581 @code{fortran}, @code{tex}, @code{eqn}, @code{math}, or @code{maple}.
34582 In Big mode, Calc actually tries @code{math-compose-big} first, then
34583 tries @code{math-compose-normal}. If this property does not exist,
34584 or if the function returns @code{nil}, the function is written in the
34585 normal function-call notation for that language.
34588 @defun composition-to-string c w
34589 Convert a composition structure returned by @code{compose-expr} into
34590 a string. Multi-line compositions convert to strings containing
34591 newline characters. The target window size is given by @var{w}.
34592 The @code{format-value} function basically calls @code{compose-expr}
34593 followed by @code{composition-to-string}.
34596 @defun comp-width c
34597 Compute the width in characters of composition @var{c}.
34600 @defun comp-height c
34601 Compute the height in lines of composition @var{c}.
34604 @defun comp-ascent c
34605 Compute the portion of the height of composition @var{c} which is on or
34606 above the baseline. For a one-line composition, this will be one.
34609 @defun comp-descent c
34610 Compute the portion of the height of composition @var{c} which is below
34611 the baseline. For a one-line composition, this will be zero.
34614 @defun comp-first-char c
34615 If composition @var{c} is a ``flat'' composition, return the first
34616 (leftmost) character of the composition as an integer. Otherwise,
34620 @defun comp-last-char c
34621 If composition @var{c} is a ``flat'' composition, return the last
34622 (rightmost) character, otherwise return @code{nil}.
34625 @comment @node Lisp Variables, Hooks, Formatting Lisp Functions, Internals
34626 @comment @subsubsection Lisp Variables
34629 @comment (This section is currently unfinished.)
34631 @node Hooks, , Formatting Lisp Functions, Internals
34632 @subsubsection Hooks
34635 Hooks are variables which contain Lisp functions (or lists of functions)
34636 which are called at various times. Calc defines a number of hooks
34637 that help you to customize it in various ways. Calc uses the Lisp
34638 function @code{run-hooks} to invoke the hooks shown below. Several
34639 other customization-related variables are also described here.
34641 @defvar calc-load-hook
34642 This hook is called at the end of @file{calc.el}, after the file has
34643 been loaded, before any functions in it have been called, but after
34644 @code{calc-mode-map} and similar variables have been set up.
34647 @defvar calc-ext-load-hook
34648 This hook is called at the end of @file{calc-ext.el}.
34651 @defvar calc-start-hook
34652 This hook is called as the last step in a @kbd{M-x calc} command.
34653 At this point, the Calc buffer has been created and initialized if
34654 necessary, the Calc window and trail window have been created,
34655 and the ``Welcome to Calc'' message has been displayed.
34658 @defvar calc-mode-hook
34659 This hook is called when the Calc buffer is being created. Usually
34660 this will only happen once per Emacs session. The hook is called
34661 after Emacs has switched to the new buffer, the mode-settings file
34662 has been read if necessary, and all other buffer-local variables
34663 have been set up. After this hook returns, Calc will perform a
34664 @code{calc-refresh} operation, set up the mode line display, then
34665 evaluate any deferred @code{calc-define} properties that have not
34666 been evaluated yet.
34669 @defvar calc-trail-mode-hook
34670 This hook is called when the Calc Trail buffer is being created.
34671 It is called as the very last step of setting up the Trail buffer.
34672 Like @code{calc-mode-hook}, this will normally happen only once
34676 @defvar calc-end-hook
34677 This hook is called by @code{calc-quit}, generally because the user
34678 presses @kbd{q} or @kbd{M-# c} while in Calc. The Calc buffer will
34679 be the current buffer. The hook is called as the very first
34680 step, before the Calc window is destroyed.
34683 @defvar calc-window-hook
34684 If this hook exists, it is called to create the Calc window.
34685 Upon return, this new Calc window should be the current window.
34686 (The Calc buffer will already be the current buffer when the
34687 hook is called.) If the hook is not defined, Calc will
34688 generally use @code{split-window}, @code{set-window-buffer},
34689 and @code{select-window} to create the Calc window.
34692 @defvar calc-trail-window-hook
34693 If this hook exists, it is called to create the Calc Trail window.
34694 The variable @code{calc-trail-buffer} will contain the buffer
34695 which the window should use. Unlike @code{calc-window-hook},
34696 this hook must @emph{not} switch into the new window.
34699 @defvar calc-edit-mode-hook
34700 This hook is called by @code{calc-edit} (and the other ``edit''
34701 commands) when the temporary editing buffer is being created.
34702 The buffer will have been selected and set up to be in
34703 @code{calc-edit-mode}, but will not yet have been filled with
34704 text. (In fact it may still have leftover text from a previous
34705 @code{calc-edit} command.)
34708 @defvar calc-mode-save-hook
34709 This hook is called by the @code{calc-save-modes} command,
34710 after Calc's own mode features have been inserted into the
34711 Calc init file and just before the ``End of mode settings''
34712 message is inserted.
34715 @defvar calc-reset-hook
34716 This hook is called after @kbd{M-# 0} (@code{calc-reset}) has
34717 reset all modes. The Calc buffer will be the current buffer.
34720 @defvar calc-other-modes
34721 This variable contains a list of strings. The strings are
34722 concatenated at the end of the modes portion of the Calc
34723 mode line (after standard modes such as ``Deg'', ``Inv'' and
34724 ``Hyp''). Each string should be a short, single word followed
34725 by a space. The variable is @code{nil} by default.
34728 @defvar calc-mode-map
34729 This is the keymap that is used by Calc mode. The best time
34730 to adjust it is probably in a @code{calc-mode-hook}. If the
34731 Calc extensions package (@file{calc-ext.el}) has not yet been
34732 loaded, many of these keys will be bound to @code{calc-missing-key},
34733 which is a command that loads the extensions package and
34734 ``retypes'' the key. If your @code{calc-mode-hook} rebinds
34735 one of these keys, it will probably be overridden when the
34736 extensions are loaded.
34739 @defvar calc-digit-map
34740 This is the keymap that is used during numeric entry. Numeric
34741 entry uses the minibuffer, but this map binds every non-numeric
34742 key to @code{calcDigit-nondigit} which generally calls
34743 @code{exit-minibuffer} and ``retypes'' the key.
34746 @defvar calc-alg-ent-map
34747 This is the keymap that is used during algebraic entry. This is
34748 mostly a copy of @code{minibuffer-local-map}.
34751 @defvar calc-store-var-map
34752 This is the keymap that is used during entry of variable names for
34753 commands like @code{calc-store} and @code{calc-recall}. This is
34754 mostly a copy of @code{minibuffer-local-completion-map}.
34757 @defvar calc-edit-mode-map
34758 This is the (sparse) keymap used by @code{calc-edit} and other
34759 temporary editing commands. It binds @key{RET}, @key{LFD},
34760 and @kbd{C-c C-c} to @code{calc-edit-finish}.
34763 @defvar calc-mode-var-list
34764 This is a list of variables which are saved by @code{calc-save-modes}.
34765 Each entry is a list of two items, the variable (as a Lisp symbol)
34766 and its default value. When modes are being saved, each variable
34767 is compared with its default value (using @code{equal}) and any
34768 non-default variables are written out.
34771 @defvar calc-local-var-list
34772 This is a list of variables which should be buffer-local to the
34773 Calc buffer. Each entry is a variable name (as a Lisp symbol).
34774 These variables also have their default values manipulated by
34775 the @code{calc} and @code{calc-quit} commands; @pxref{Multiple Calculators}.
34776 Since @code{calc-mode-hook} is called after this list has been
34777 used the first time, your hook should add a variable to the
34778 list and also call @code{make-local-variable} itself.
34781 @node Customizable Variables, Reporting Bugs, Programming, Top
34782 @appendix Customizable Variables
34784 GNU Calc is controlled by many variables, most of which can be reset
34785 from within Calc. Some variables are less involved with actual
34786 calculation, and can be set outside of Calc using Emacs's
34787 customization facilities. These variables are listed below.
34788 Typing @kbd{M-x customize-variable RET @var{variable-name} RET}
34789 will bring up a buffer in which the variable's value can be redefined.
34790 Typing @kbd{M-x customize-group RET calc RET} will bring up a buffer which
34791 contains all of Calc's customizable variables. (These variables can
34792 also be reset by putting the appropriate lines in your .emacs file;
34793 @xref{Init File, ,Init File, emacs, The GNU Emacs Manual}.)
34795 Some of the customizable variables are regular expressions. A regular
34796 expression is basically a pattern that Calc can search for.
34797 See @ref{Regexp Search,, Regular Expression Search, emacs, The GNU Emacs Manual}
34798 to see how regular expressions work.
34802 @item calc-settings-file
34804 @vindex calc-settings-file
34805 The variable @code{calc-settings-file} holds the file name in
34806 which commands like @kbd{m m} and @kbd{Z P} store ``permanent''
34808 If @code{calc-settings-file} is not your user init file (typically
34809 @file{~/.emacs}) and if the variable @code{calc-loaded-settings-file} is
34810 @code{nil}, then Calc will automatically load your settings file (if it
34811 exists) the first time Calc is invoked.
34813 The default value for this variable is @code{"~/.calc.el"}.
34815 @item calc-gnuplot-name
34817 See @ref{Graphics}.@*
34818 The variable @code{calc-gnuplot-name} should be the name of the
34819 GNUPLOT program (a string). If you have GNUPLOT installed on your
34820 system but Calc is unable to find it, you may need to set this
34821 variable. (@pxref{Customizable Variables})
34822 You may also need to set some Lisp variables to show Calc how to run
34823 GNUPLOT on your system, see @ref{Devices, ,Graphical Devices} . The default value
34824 of @code{calc-gnuplot-name} is @code{"gnuplot"}.
34826 @item calc-gnuplot-plot-command
34827 @itemx calc-gnuplot-print-command
34829 See @ref{Devices, ,Graphical Devices}.@*
34830 The variables @code{calc-gnuplot-plot-command} and
34831 @code{calc-gnuplot-print-command} represent system commands to
34832 display and print the output of GNUPLOT, respectively. These may be
34833 @code{nil} if no command is necessary, or strings which can include
34834 @samp{%s} to signify the name of the file to be displayed or printed.
34835 Or, these variables may contain Lisp expressions which are evaluated
34836 to display or print the output.
34838 The default value of @code{calc-gnuplot-plot-command} is @code{nil},
34839 and the default value of @code{calc-gnuplot-print-command} is
34842 @item calc-language-alist
34844 See @ref{Basic Embedded Mode}.@*
34845 The variable @code{calc-language-alist} controls the languages that
34846 Calc will associate with major modes. When Calc embedded mode is
34847 enabled, it will try to use the current major mode to
34848 determine what language should be used. (This can be overridden using
34849 Calc's mode changing commands, @xref{Mode Settings in Embedded Mode}.)
34850 The variable @code{calc-language-alist} consists of a list of pairs of
34851 the form @code{(@var{MAJOR-MODE} . @var{LANGUAGE})}; for example,
34852 @code{(latex-mode . latex)} is one such pair. If Calc embedded is
34853 activated in a buffer whose major mode is @var{MAJOR-MODE}, it will set itself
34854 to use the language @var{LANGUAGE}.
34856 The default value of @code{calc-language-alist} is
34858 ((latex-mode . latex)
34860 (plain-tex-mode . tex)
34861 (context-mode . tex)
34863 (pascal-mode . pascal)
34866 (fortran-mode . fortran)
34867 (f90-mode . fortran))
34870 @item calc-embedded-announce-formula
34872 See @ref{Customizing Embedded Mode}.@*
34873 The variable @code{calc-embedded-announce-formula} helps determine
34874 what formulas @kbd{M-# a} will activate in a buffer. It is a
34875 regular expression, and when activating embedded formulas with
34876 @kbd{M-# a}, it will tell Calc that what follows is a formula to be
34877 activated. (Calc also uses other patterns to find formulas, such as
34878 @samp{=>} and @samp{:=}.)
34880 The default pattern is @code{"%Embed\n\\(% .*\n\\)*"}, which checks
34881 for @samp{%Embed} followed by any number of lines beginning with
34882 @samp{%} and a space.
34884 @item calc-embedded-open-formula
34885 @itemx calc-embedded-close-formula
34887 See @ref{Customizing Embedded Mode}.@*
34888 The variables @code{calc-embedded-open-formula} and
34889 @code{calc-embedded-open-formula} control the region that Calc will
34890 activate as a formula when Embedded mode is entered with @kbd{M-# e}.
34891 They are regular expressions;
34892 Calc normally scans backward and forward in the buffer for the
34893 nearest text matching these regular expressions to be the ``formula
34896 The simplest delimiters are blank lines. Other delimiters that
34897 Embedded mode understands by default are:
34900 The @TeX{} and La@TeX{} math delimiters @samp{$ $}, @samp{$$ $$},
34901 @samp{\[ \]}, and @samp{\( \)};
34903 Lines beginning with @samp{\begin} and @samp{\end} (except matrix delimiters);
34905 Lines beginning with @samp{@@} (Texinfo delimiters).
34907 Lines beginning with @samp{.EQ} and @samp{.EN} (@dfn{eqn} delimiters);
34909 Lines containing a single @samp{%} or @samp{.\"} symbol and nothing else.
34912 @item calc-embedded-open-word
34913 @itemx calc-embedded-close-word
34915 See @ref{Customizing Embedded Mode}.@*
34916 The variables @code{calc-embedded-open-word} and
34917 @code{calc-embedded-close-word} control the region that Calc will
34918 activate when Embedded mode is entered with @kbd{M-# w}. They are
34919 regular expressions.
34921 The default values of @code{calc-embedded-open-word} and
34922 @code{calc-embedded-close-word} are @code{"^\\|[^-+0-9.eE]"} and
34923 @code{"$\\|[^-+0-9.eE]"} respectively.
34925 @item calc-embedded-open-plain
34926 @itemx calc-embedded-close-plain
34928 See @ref{Customizing Embedded Mode}.@*
34929 The variables @code{calc-embedded-open-plain} and
34930 @code{calc-embedded-open-plain} are used to delimit ``plain''
34931 formulas. Note that these are actual strings, not regular
34932 expressions, because Calc must be able to write these string into a
34933 buffer as well as to recognize them.
34935 The default string for @code{calc-embedded-open-plain} is
34936 @code{"%%% "}, note the trailing space. The default string for
34937 @code{calc-embedded-close-plain} is @code{" %%%\n"}, without
34938 the trailing newline here, the first line of a Big mode formula
34939 that followed might be shifted over with respect to the other lines.
34941 @item calc-embedded-open-new-formula
34942 @itemx calc-embedded-close-new-formula
34944 See @ref{Customizing Embedded Mode}.@*
34945 The variables @code{calc-embedded-open-new-formula} and
34946 @code{calc-embedded-close-new-formula} are strings which are
34947 inserted before and after a new formula when you type @kbd{M-# f}.
34949 The default value of @code{calc-embedded-open-new-formula} is
34950 @code{"\n\n"}. If this string begins with a newline character and the
34951 @kbd{M-# f} is typed at the beginning of a line, @kbd{M-# f} will skip
34952 this first newline to avoid introducing unnecessary blank lines in the
34953 file. The default value of @code{calc-embedded-close-new-formula} is
34954 also @code{"\n\n"}. The final newline is omitted by @w{@kbd{M-# f}}
34955 if typed at the end of a line. (It follows that if @kbd{M-# f} is
34956 typed on a blank line, both a leading opening newline and a trailing
34957 closing newline are omitted.)
34959 @item calc-embedded-open-mode
34960 @itemx calc-embedded-close-mode
34962 See @ref{Customizing Embedded Mode}.@*
34963 The variables @code{calc-embedded-open-mode} and
34964 @code{calc-embedded-close-mode} are strings which Calc will place before
34965 and after any mode annotations that it inserts. Calc never scans for
34966 these strings; Calc always looks for the annotation itself, so it is not
34967 necessary to add them to user-written annotations.
34969 The default value of @code{calc-embedded-open-mode} is @code{"% "}
34970 and the default value of @code{calc-embedded-close-mode} is
34972 If you change the value of @code{calc-embedded-close-mode}, it is a good
34973 idea still to end with a newline so that mode annotations will appear on
34974 lines by themselves.
34978 @node Reporting Bugs, Summary, Customizable Variables, Top
34979 @appendix Reporting Bugs
34982 If you find a bug in Calc, send e-mail to Jay Belanger,
34985 belanger@@truman.edu
34989 There is an automatic command @kbd{M-x report-calc-bug} which helps
34990 you to report bugs. This command prompts you for a brief subject
34991 line, then leaves you in a mail editing buffer. Type @kbd{C-c C-c} to
34992 send your mail. Make sure your subject line indicates that you are
34993 reporting a Calc bug; this command sends mail to the maintainer's
34996 If you have suggestions for additional features for Calc, please send
34997 them. Some have dared to suggest that Calc is already top-heavy with
34998 features; this obviously cannot be the case, so if you have ideas, send
35001 At the front of the source file, @file{calc.el}, is a list of ideas for
35002 future work. If any enthusiastic souls wish to take it upon themselves
35003 to work on these, please send a message (using @kbd{M-x report-calc-bug})
35004 so any efforts can be coordinated.
35006 The latest version of Calc is available from Savannah, in the Emacs
35007 CVS tree. See @uref{http://savannah.gnu.org/projects/emacs}.
35010 @node Summary, Key Index, Reporting Bugs, Top
35011 @appendix Calc Summary
35014 This section includes a complete list of Calc 2.02 keystroke commands.
35015 Each line lists the stack entries used by the command (top-of-stack
35016 last), the keystrokes themselves, the prompts asked by the command,
35017 and the result of the command (also with top-of-stack last).
35018 The result is expressed using the equivalent algebraic function.
35019 Commands which put no results on the stack show the full @kbd{M-x}
35020 command name in that position. Numbers preceding the result or
35021 command name refer to notes at the end.
35023 Algebraic functions and @kbd{M-x} commands that don't have corresponding
35024 keystrokes are not listed in this summary.
35025 @xref{Command Index}. @xref{Function Index}.
35030 \vskip-2\baselineskip \null
35031 \gdef\sumrow#1{\sumrowx#1\relax}%
35032 \gdef\sumrowx#1\:#2\:#3\:#4\:#5\:#6\relax{%
35035 \hbox to5em{\sl\hss#1}%
35036 \hbox to5em{\tt#2\hss}%
35037 \hbox to4em{\sl#3\hss}%
35038 \hbox to5em{\rm\hss#4}%
35043 \gdef\sumlpar{{\rm(}}%
35044 \gdef\sumrpar{{\rm)}}%
35045 \gdef\sumcomma{{\rm,\thinspace}}%
35046 \gdef\sumexcl{{\rm!}}%
35047 \gdef\sumbreak{\vskip-2.5\baselineskip\goodbreak}%
35048 \gdef\minus#1{{\tt-}}%
35052 @catcode`@(=@active @let(=@sumlpar
35053 @catcode`@)=@active @let)=@sumrpar
35054 @catcode`@,=@active @let,=@sumcomma
35055 @catcode`@!=@active @let!=@sumexcl
35059 @advance@baselineskip-2.5pt
35062 @r{ @: M-# a @: @: 33 @:calc-embedded-activate@:}
35063 @r{ @: M-# b @: @: @:calc-big-or-small@:}
35064 @r{ @: M-# c @: @: @:calc@:}
35065 @r{ @: M-# d @: @: @:calc-embedded-duplicate@:}
35066 @r{ @: M-# e @: @: 34 @:calc-embedded@:}
35067 @r{ @: M-# f @:formula @: @:calc-embedded-new-formula@:}
35068 @r{ @: M-# g @: @: 35 @:calc-grab-region@:}
35069 @r{ @: M-# i @: @: @:calc-info@:}
35070 @r{ @: M-# j @: @: @:calc-embedded-select@:}
35071 @r{ @: M-# k @: @: @:calc-keypad@:}
35072 @r{ @: M-# l @: @: @:calc-load-everything@:}
35073 @r{ @: M-# m @: @: @:read-kbd-macro@:}
35074 @r{ @: M-# n @: @: 4 @:calc-embedded-next@:}
35075 @r{ @: M-# o @: @: @:calc-other-window@:}
35076 @r{ @: M-# p @: @: 4 @:calc-embedded-previous@:}
35077 @r{ @: M-# q @:formula @: @:quick-calc@:}
35078 @r{ @: M-# r @: @: 36 @:calc-grab-rectangle@:}
35079 @r{ @: M-# s @: @: @:calc-info-summary@:}
35080 @r{ @: M-# t @: @: @:calc-tutorial@:}
35081 @r{ @: M-# u @: @: @:calc-embedded-update@:}
35082 @r{ @: M-# w @: @: @:calc-embedded-word@:}
35083 @r{ @: M-# x @: @: @:calc-quit@:}
35084 @r{ @: M-# y @: @:1,28,49 @:calc-copy-to-buffer@:}
35085 @r{ @: M-# z @: @: @:calc-user-invocation@:}
35086 @r{ @: M-# : @: @: 36 @:calc-grab-sum-down@:}
35087 @r{ @: M-# _ @: @: 36 @:calc-grab-sum-across@:}
35088 @r{ @: M-# ` @:editing @: 30 @:calc-embedded-edit@:}
35089 @r{ @: M-# 0 @:(zero) @: @:calc-reset@:}
35092 @r{ @: 0-9 @:number @: @:@:number}
35093 @r{ @: . @:number @: @:@:0.number}
35094 @r{ @: _ @:number @: @:-@:number}
35095 @r{ @: e @:number @: @:@:1e number}
35096 @r{ @: # @:number @: @:@:current-radix@tfn{#}number}
35097 @r{ @: P @:(in number) @: @:+/-@:}
35098 @r{ @: M @:(in number) @: @:mod@:}
35099 @r{ @: @@ ' " @: (in number)@: @:@:HMS form}
35100 @r{ @: h m s @: (in number)@: @:@:HMS form}
35103 @r{ @: ' @:formula @: 37,46 @:@:formula}
35104 @r{ @: $ @:formula @: 37,46 @:$@:formula}
35105 @r{ @: " @:string @: 37,46 @:@:string}
35108 @r{ a b@: + @: @: 2 @:add@:(a,b) a+b}
35109 @r{ a b@: - @: @: 2 @:sub@:(a,b) a@minus{}b}
35110 @r{ a b@: * @: @: 2 @:mul@:(a,b) a b, a*b}
35111 @r{ a b@: / @: @: 2 @:div@:(a,b) a/b}
35112 @r{ a b@: ^ @: @: 2 @:pow@:(a,b) a^b}
35113 @r{ a b@: I ^ @: @: 2 @:nroot@:(a,b) a^(1/b)}
35114 @r{ a b@: % @: @: 2 @:mod@:(a,b) a%b}
35115 @r{ a b@: \ @: @: 2 @:idiv@:(a,b) a\b}
35116 @r{ a b@: : @: @: 2 @:fdiv@:(a,b)}
35117 @r{ a b@: | @: @: 2 @:vconcat@:(a,b) a|b}
35118 @r{ a b@: I | @: @: @:vconcat@:(b,a) b|a}
35119 @r{ a b@: H | @: @: 2 @:append@:(a,b)}
35120 @r{ a b@: I H | @: @: @:append@:(b,a)}
35121 @r{ a@: & @: @: 1 @:inv@:(a) 1/a}
35122 @r{ a@: ! @: @: 1 @:fact@:(a) a!}
35123 @r{ a@: = @: @: 1 @:evalv@:(a)}
35124 @r{ a@: M-% @: @: @:percent@:(a) a%}
35127 @r{ ... a@: @key{RET} @: @: 1 @:@:... a a}
35128 @r{ ... a@: @key{SPC} @: @: 1 @:@:... a a}
35129 @r{... a b@: @key{TAB} @: @: 3 @:@:... b a}
35130 @r{. a b c@: M-@key{TAB} @: @: 3 @:@:... b c a}
35131 @r{... a b@: @key{LFD} @: @: 1 @:@:... a b a}
35132 @r{ ... a@: @key{DEL} @: @: 1 @:@:...}
35133 @r{... a b@: M-@key{DEL} @: @: 1 @:@:... b}
35134 @r{ @: M-@key{RET} @: @: 4 @:calc-last-args@:}
35135 @r{ a@: ` @:editing @: 1,30 @:calc-edit@:}
35138 @r{ ... a@: C-d @: @: 1 @:@:...}
35139 @r{ @: C-k @: @: 27 @:calc-kill@:}
35140 @r{ @: C-w @: @: 27 @:calc-kill-region@:}
35141 @r{ @: C-y @: @: @:calc-yank@:}
35142 @r{ @: C-_ @: @: 4 @:calc-undo@:}
35143 @r{ @: M-k @: @: 27 @:calc-copy-as-kill@:}
35144 @r{ @: M-w @: @: 27 @:calc-copy-region-as-kill@:}
35147 @r{ @: [ @: @: @:@:[...}
35148 @r{[.. a b@: ] @: @: @:@:[a,b]}
35149 @r{ @: ( @: @: @:@:(...}
35150 @r{(.. a b@: ) @: @: @:@:(a,b)}
35151 @r{ @: , @: @: @:@:vector or rect complex}
35152 @r{ @: ; @: @: @:@:matrix or polar complex}
35153 @r{ @: .. @: @: @:@:interval}
35156 @r{ @: ~ @: @: @:calc-num-prefix@:}
35157 @r{ @: < @: @: 4 @:calc-scroll-left@:}
35158 @r{ @: > @: @: 4 @:calc-scroll-right@:}
35159 @r{ @: @{ @: @: 4 @:calc-scroll-down@:}
35160 @r{ @: @} @: @: 4 @:calc-scroll-up@:}
35161 @r{ @: ? @: @: @:calc-help@:}
35164 @r{ a@: n @: @: 1 @:neg@:(a) @minus{}a}
35165 @r{ @: o @: @: 4 @:calc-realign@:}
35166 @r{ @: p @:precision @: 31 @:calc-precision@:}
35167 @r{ @: q @: @: @:calc-quit@:}
35168 @r{ @: w @: @: @:calc-why@:}
35169 @r{ @: x @:command @: @:M-x calc-@:command}
35170 @r{ a@: y @: @:1,28,49 @:calc-copy-to-buffer@:}
35173 @r{ a@: A @: @: 1 @:abs@:(a)}
35174 @r{ a b@: B @: @: 2 @:log@:(a,b)}
35175 @r{ a b@: I B @: @: 2 @:alog@:(a,b) b^a}
35176 @r{ a@: C @: @: 1 @:cos@:(a)}
35177 @r{ a@: I C @: @: 1 @:arccos@:(a)}
35178 @r{ a@: H C @: @: 1 @:cosh@:(a)}
35179 @r{ a@: I H C @: @: 1 @:arccosh@:(a)}
35180 @r{ @: D @: @: 4 @:calc-redo@:}
35181 @r{ a@: E @: @: 1 @:exp@:(a)}
35182 @r{ a@: H E @: @: 1 @:exp10@:(a) 10.^a}
35183 @r{ a@: F @: @: 1,11 @:floor@:(a,d)}
35184 @r{ a@: I F @: @: 1,11 @:ceil@:(a,d)}
35185 @r{ a@: H F @: @: 1,11 @:ffloor@:(a,d)}
35186 @r{ a@: I H F @: @: 1,11 @:fceil@:(a,d)}
35187 @r{ a@: G @: @: 1 @:arg@:(a)}
35188 @r{ @: H @:command @: 32 @:@:Hyperbolic}
35189 @r{ @: I @:command @: 32 @:@:Inverse}
35190 @r{ a@: J @: @: 1 @:conj@:(a)}
35191 @r{ @: K @:command @: 32 @:@:Keep-args}
35192 @r{ a@: L @: @: 1 @:ln@:(a)}
35193 @r{ a@: H L @: @: 1 @:log10@:(a)}
35194 @r{ @: M @: @: @:calc-more-recursion-depth@:}
35195 @r{ @: I M @: @: @:calc-less-recursion-depth@:}
35196 @r{ a@: N @: @: 5 @:evalvn@:(a)}
35197 @r{ @: P @: @: @:@:pi}
35198 @r{ @: I P @: @: @:@:gamma}
35199 @r{ @: H P @: @: @:@:e}
35200 @r{ @: I H P @: @: @:@:phi}
35201 @r{ a@: Q @: @: 1 @:sqrt@:(a)}
35202 @r{ a@: I Q @: @: 1 @:sqr@:(a) a^2}
35203 @r{ a@: R @: @: 1,11 @:round@:(a,d)}
35204 @r{ a@: I R @: @: 1,11 @:trunc@:(a,d)}
35205 @r{ a@: H R @: @: 1,11 @:fround@:(a,d)}
35206 @r{ a@: I H R @: @: 1,11 @:ftrunc@:(a,d)}
35207 @r{ a@: S @: @: 1 @:sin@:(a)}
35208 @r{ a@: I S @: @: 1 @:arcsin@:(a)}
35209 @r{ a@: H S @: @: 1 @:sinh@:(a)}
35210 @r{ a@: I H S @: @: 1 @:arcsinh@:(a)}
35211 @r{ a@: T @: @: 1 @:tan@:(a)}
35212 @r{ a@: I T @: @: 1 @:arctan@:(a)}
35213 @r{ a@: H T @: @: 1 @:tanh@:(a)}
35214 @r{ a@: I H T @: @: 1 @:arctanh@:(a)}
35215 @r{ @: U @: @: 4 @:calc-undo@:}
35216 @r{ @: X @: @: 4 @:calc-call-last-kbd-macro@:}
35219 @r{ a b@: a = @: @: 2 @:eq@:(a,b) a=b}
35220 @r{ a b@: a # @: @: 2 @:neq@:(a,b) a!=b}
35221 @r{ a b@: a < @: @: 2 @:lt@:(a,b) a<b}
35222 @r{ a b@: a > @: @: 2 @:gt@:(a,b) a>b}
35223 @r{ a b@: a [ @: @: 2 @:leq@:(a,b) a<=b}
35224 @r{ a b@: a ] @: @: 2 @:geq@:(a,b) a>=b}
35225 @r{ a b@: a @{ @: @: 2 @:in@:(a,b)}
35226 @r{ a b@: a & @: @: 2,45 @:land@:(a,b) a&&b}
35227 @r{ a b@: a | @: @: 2,45 @:lor@:(a,b) a||b}
35228 @r{ a@: a ! @: @: 1,45 @:lnot@:(a) !a}
35229 @r{ a b c@: a : @: @: 45 @:if@:(a,b,c) a?b:c}
35230 @r{ a@: a . @: @: 1 @:rmeq@:(a)}
35231 @r{ a@: a " @: @: 7,8 @:calc-expand-formula@:}
35234 @r{ a@: a + @:i, l, h @: 6,38 @:sum@:(a,i,l,h)}
35235 @r{ a@: a - @:i, l, h @: 6,38 @:asum@:(a,i,l,h)}
35236 @r{ a@: a * @:i, l, h @: 6,38 @:prod@:(a,i,l,h)}
35237 @r{ a b@: a _ @: @: 2 @:subscr@:(a,b) a_b}
35240 @r{ a b@: a \ @: @: 2 @:pdiv@:(a,b)}
35241 @r{ a b@: a % @: @: 2 @:prem@:(a,b)}
35242 @r{ a b@: a / @: @: 2 @:pdivrem@:(a,b) [q,r]}
35243 @r{ a b@: H a / @: @: 2 @:pdivide@:(a,b) q+r/b}
35246 @r{ a@: a a @: @: 1 @:apart@:(a)}
35247 @r{ a@: a b @:old, new @: 38 @:subst@:(a,old,new)}
35248 @r{ a@: a c @:v @: 38 @:collect@:(a,v)}
35249 @r{ a@: a d @:v @: 4,38 @:deriv@:(a,v)}
35250 @r{ a@: H a d @:v @: 4,38 @:tderiv@:(a,v)}
35251 @r{ a@: a e @: @: @:esimplify@:(a)}
35252 @r{ a@: a f @: @: 1 @:factor@:(a)}
35253 @r{ a@: H a f @: @: 1 @:factors@:(a)}
35254 @r{ a b@: a g @: @: 2 @:pgcd@:(a,b)}
35255 @r{ a@: a i @:v @: 38 @:integ@:(a,v)}
35256 @r{ a@: a m @:pats @: 38 @:match@:(a,pats)}
35257 @r{ a@: I a m @:pats @: 38 @:matchnot@:(a,pats)}
35258 @r{ data x@: a p @: @: 28 @:polint@:(data,x)}
35259 @r{ data x@: H a p @: @: 28 @:ratint@:(data,x)}
35260 @r{ a@: a n @: @: 1 @:nrat@:(a)}
35261 @r{ a@: a r @:rules @:4,8,38 @:rewrite@:(a,rules,n)}
35262 @r{ a@: a s @: @: @:simplify@:(a)}
35263 @r{ a@: a t @:v, n @: 31,39 @:taylor@:(a,v,n)}
35264 @r{ a@: a v @: @: 7,8 @:calc-alg-evaluate@:}
35265 @r{ a@: a x @: @: 4,8 @:expand@:(a)}
35268 @r{ data@: a F @:model, vars @: 48 @:fit@:(m,iv,pv,data)}
35269 @r{ data@: I a F @:model, vars @: 48 @:xfit@:(m,iv,pv,data)}
35270 @r{ data@: H a F @:model, vars @: 48 @:efit@:(m,iv,pv,data)}
35271 @r{ a@: a I @:v, l, h @: 38 @:ninteg@:(a,v,l,h)}
35272 @r{ a b@: a M @:op @: 22 @:mapeq@:(op,a,b)}
35273 @r{ a b@: I a M @:op @: 22 @:mapeqr@:(op,a,b)}
35274 @r{ a b@: H a M @:op @: 22 @:mapeqp@:(op,a,b)}
35275 @r{ a g@: a N @:v @: 38 @:minimize@:(a,v,g)}
35276 @r{ a g@: H a N @:v @: 38 @:wminimize@:(a,v,g)}
35277 @r{ a@: a P @:v @: 38 @:roots@:(a,v)}
35278 @r{ a g@: a R @:v @: 38 @:root@:(a,v,g)}
35279 @r{ a g@: H a R @:v @: 38 @:wroot@:(a,v,g)}
35280 @r{ a@: a S @:v @: 38 @:solve@:(a,v)}
35281 @r{ a@: I a S @:v @: 38 @:finv@:(a,v)}
35282 @r{ a@: H a S @:v @: 38 @:fsolve@:(a,v)}
35283 @r{ a@: I H a S @:v @: 38 @:ffinv@:(a,v)}
35284 @r{ a@: a T @:i, l, h @: 6,38 @:table@:(a,i,l,h)}
35285 @r{ a g@: a X @:v @: 38 @:maximize@:(a,v,g)}
35286 @r{ a g@: H a X @:v @: 38 @:wmaximize@:(a,v,g)}
35289 @r{ a b@: b a @: @: 9 @:and@:(a,b,w)}
35290 @r{ a@: b c @: @: 9 @:clip@:(a,w)}
35291 @r{ a b@: b d @: @: 9 @:diff@:(a,b,w)}
35292 @r{ a@: b l @: @: 10 @:lsh@:(a,n,w)}
35293 @r{ a n@: H b l @: @: 9 @:lsh@:(a,n,w)}
35294 @r{ a@: b n @: @: 9 @:not@:(a,w)}
35295 @r{ a b@: b o @: @: 9 @:or@:(a,b,w)}
35296 @r{ v@: b p @: @: 1 @:vpack@:(v)}
35297 @r{ a@: b r @: @: 10 @:rsh@:(a,n,w)}
35298 @r{ a n@: H b r @: @: 9 @:rsh@:(a,n,w)}
35299 @r{ a@: b t @: @: 10 @:rot@:(a,n,w)}
35300 @r{ a n@: H b t @: @: 9 @:rot@:(a,n,w)}
35301 @r{ a@: b u @: @: 1 @:vunpack@:(a)}
35302 @r{ @: b w @:w @: 9,50 @:calc-word-size@:}
35303 @r{ a b@: b x @: @: 9 @:xor@:(a,b,w)}
35306 @r{c s l p@: b D @: @: @:ddb@:(c,s,l,p)}
35307 @r{ r n p@: b F @: @: @:fv@:(r,n,p)}
35308 @r{ r n p@: I b F @: @: @:fvb@:(r,n,p)}
35309 @r{ r n p@: H b F @: @: @:fvl@:(r,n,p)}
35310 @r{ v@: b I @: @: 19 @:irr@:(v)}
35311 @r{ v@: I b I @: @: 19 @:irrb@:(v)}
35312 @r{ a@: b L @: @: 10 @:ash@:(a,n,w)}
35313 @r{ a n@: H b L @: @: 9 @:ash@:(a,n,w)}
35314 @r{ r n a@: b M @: @: @:pmt@:(r,n,a)}
35315 @r{ r n a@: I b M @: @: @:pmtb@:(r,n,a)}
35316 @r{ r n a@: H b M @: @: @:pmtl@:(r,n,a)}
35317 @r{ r v@: b N @: @: 19 @:npv@:(r,v)}
35318 @r{ r v@: I b N @: @: 19 @:npvb@:(r,v)}
35319 @r{ r n p@: b P @: @: @:pv@:(r,n,p)}
35320 @r{ r n p@: I b P @: @: @:pvb@:(r,n,p)}
35321 @r{ r n p@: H b P @: @: @:pvl@:(r,n,p)}
35322 @r{ a@: b R @: @: 10 @:rash@:(a,n,w)}
35323 @r{ a n@: H b R @: @: 9 @:rash@:(a,n,w)}
35324 @r{ c s l@: b S @: @: @:sln@:(c,s,l)}
35325 @r{ n p a@: b T @: @: @:rate@:(n,p,a)}
35326 @r{ n p a@: I b T @: @: @:rateb@:(n,p,a)}
35327 @r{ n p a@: H b T @: @: @:ratel@:(n,p,a)}
35328 @r{c s l p@: b Y @: @: @:syd@:(c,s,l,p)}
35330 @r{ r p a@: b # @: @: @:nper@:(r,p,a)}
35331 @r{ r p a@: I b # @: @: @:nperb@:(r,p,a)}
35332 @r{ r p a@: H b # @: @: @:nperl@:(r,p,a)}
35333 @r{ a b@: b % @: @: @:relch@:(a,b)}
35336 @r{ a@: c c @: @: 5 @:pclean@:(a,p)}
35337 @r{ a@: c 0-9 @: @: @:pclean@:(a,p)}
35338 @r{ a@: H c c @: @: 5 @:clean@:(a,p)}
35339 @r{ a@: H c 0-9 @: @: @:clean@:(a,p)}
35340 @r{ a@: c d @: @: 1 @:deg@:(a)}
35341 @r{ a@: c f @: @: 1 @:pfloat@:(a)}
35342 @r{ a@: H c f @: @: 1 @:float@:(a)}
35343 @r{ a@: c h @: @: 1 @:hms@:(a)}
35344 @r{ a@: c p @: @: @:polar@:(a)}
35345 @r{ a@: I c p @: @: @:rect@:(a)}
35346 @r{ a@: c r @: @: 1 @:rad@:(a)}
35349 @r{ a@: c F @: @: 5 @:pfrac@:(a,p)}
35350 @r{ a@: H c F @: @: 5 @:frac@:(a,p)}
35353 @r{ a@: c % @: @: @:percent@:(a*100)}
35356 @r{ @: d . @:char @: 50 @:calc-point-char@:}
35357 @r{ @: d , @:char @: 50 @:calc-group-char@:}
35358 @r{ @: d < @: @: 13,50 @:calc-left-justify@:}
35359 @r{ @: d = @: @: 13,50 @:calc-center-justify@:}
35360 @r{ @: d > @: @: 13,50 @:calc-right-justify@:}
35361 @r{ @: d @{ @:label @: 50 @:calc-left-label@:}
35362 @r{ @: d @} @:label @: 50 @:calc-right-label@:}
35363 @r{ @: d [ @: @: 4 @:calc-truncate-up@:}
35364 @r{ @: d ] @: @: 4 @:calc-truncate-down@:}
35365 @r{ @: d " @: @: 12,50 @:calc-display-strings@:}
35366 @r{ @: d @key{SPC} @: @: @:calc-refresh@:}
35367 @r{ @: d @key{RET} @: @: 1 @:calc-refresh-top@:}
35370 @r{ @: d 0 @: @: 50 @:calc-decimal-radix@:}
35371 @r{ @: d 2 @: @: 50 @:calc-binary-radix@:}
35372 @r{ @: d 6 @: @: 50 @:calc-hex-radix@:}
35373 @r{ @: d 8 @: @: 50 @:calc-octal-radix@:}
35376 @r{ @: d b @: @:12,13,50 @:calc-line-breaking@:}
35377 @r{ @: d c @: @: 50 @:calc-complex-notation@:}
35378 @r{ @: d d @:format @: 50 @:calc-date-notation@:}
35379 @r{ @: d e @: @: 5,50 @:calc-eng-notation@:}
35380 @r{ @: d f @:num @: 31,50 @:calc-fix-notation@:}
35381 @r{ @: d g @: @:12,13,50 @:calc-group-digits@:}
35382 @r{ @: d h @:format @: 50 @:calc-hms-notation@:}
35383 @r{ @: d i @: @: 50 @:calc-i-notation@:}
35384 @r{ @: d j @: @: 50 @:calc-j-notation@:}
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35684 @r{d oz nz@: t C @:$ @: @:tzconv@:(d,oz,nz)}
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35721 @r{ a@: u 0-9 @: @: @:calc-quick-units@:}
35724 @r{ v1 v2@: u C @: @: 20 @:vcov@:(v1,v2)}
35725 @r{ v1 v2@: I u C @: @: 20 @:vpcov@:(v1,v2)}
35726 @r{ v1 v2@: H u C @: @: 20 @:vcorr@:(v1,v2)}
35727 @r{ v@: u G @: @: 19 @:vgmean@:(v)}
35728 @r{ a b@: H u G @: @: 2 @:agmean@:(a,b)}
35729 @r{ v@: u M @: @: 19 @:vmean@:(v)}
35730 @r{ v@: I u M @: @: 19 @:vmeane@:(v)}
35731 @r{ v@: H u M @: @: 19 @:vmedian@:(v)}
35732 @r{ v@: I H u M @: @: 19 @:vhmean@:(v)}
35733 @r{ v@: u N @: @: 19 @:vmin@:(v)}
35734 @r{ v@: u S @: @: 19 @:vsdev@:(v)}
35735 @r{ v@: I u S @: @: 19 @:vpsdev@:(v)}
35736 @r{ v@: H u S @: @: 19 @:vvar@:(v)}
35737 @r{ v@: I H u S @: @: 19 @:vpvar@:(v)}
35738 @r{ @: u V @: @: @:calc-view-units-table@:}
35739 @r{ v@: u X @: @: 19 @:vmax@:(v)}
35742 @r{ v@: u + @: @: 19 @:vsum@:(v)}
35743 @r{ v@: u * @: @: 19 @:vprod@:(v)}
35744 @r{ v@: u # @: @: 19 @:vcount@:(v)}
35747 @r{ @: V ( @: @: 50 @:calc-vector-parens@:}
35748 @r{ @: V @{ @: @: 50 @:calc-vector-braces@:}
35749 @r{ @: V [ @: @: 50 @:calc-vector-brackets@:}
35750 @r{ @: V ] @:ROCP @: 50 @:calc-matrix-brackets@:}
35751 @r{ @: V , @: @: 50 @:calc-vector-commas@:}
35752 @r{ @: V < @: @: 50 @:calc-matrix-left-justify@:}
35753 @r{ @: V = @: @: 50 @:calc-matrix-center-justify@:}
35754 @r{ @: V > @: @: 50 @:calc-matrix-right-justify@:}
35755 @r{ @: V / @: @: 12,50 @:calc-break-vectors@:}
35756 @r{ @: V . @: @: 12,50 @:calc-full-vectors@:}
35759 @r{ s t@: V ^ @: @: 2 @:vint@:(s,t)}
35760 @r{ s t@: V - @: @: 2 @:vdiff@:(s,t)}
35761 @r{ s@: V ~ @: @: 1 @:vcompl@:(s)}
35762 @r{ s@: V # @: @: 1 @:vcard@:(s)}
35763 @r{ s@: V : @: @: 1 @:vspan@:(s)}
35764 @r{ s@: V + @: @: 1 @:rdup@:(s)}
35767 @r{ m@: V & @: @: 1 @:inv@:(m) 1/m}
35770 @r{ v@: v a @:n @: @:arrange@:(v,n)}
35771 @r{ a@: v b @:n @: @:cvec@:(a,n)}
35772 @r{ v@: v c @:n >0 @: 21,31 @:mcol@:(v,n)}
35773 @r{ v@: v c @:n <0 @: 31 @:mrcol@:(v,-n)}
35774 @r{ m@: v c @:0 @: 31 @:getdiag@:(m)}
35775 @r{ v@: v d @: @: 25 @:diag@:(v,n)}
35776 @r{ v m@: v e @: @: 2 @:vexp@:(v,m)}
35777 @r{ v m f@: H v e @: @: 2 @:vexp@:(v,m,f)}
35778 @r{ v a@: v f @: @: 26 @:find@:(v,a,n)}
35779 @r{ v@: v h @: @: 1 @:head@:(v)}
35780 @r{ v@: I v h @: @: 1 @:tail@:(v)}
35781 @r{ v@: H v h @: @: 1 @:rhead@:(v)}
35782 @r{ v@: I H v h @: @: 1 @:rtail@:(v)}
35783 @r{ @: v i @:n @: 31 @:idn@:(1,n)}
35784 @r{ @: v i @:0 @: 31 @:idn@:(1)}
35785 @r{ h t@: v k @: @: 2 @:cons@:(h,t)}
35786 @r{ h t@: H v k @: @: 2 @:rcons@:(h,t)}
35787 @r{ v@: v l @: @: 1 @:vlen@:(v)}
35788 @r{ v@: H v l @: @: 1 @:mdims@:(v)}
35789 @r{ v m@: v m @: @: 2 @:vmask@:(v,m)}
35790 @r{ v@: v n @: @: 1 @:rnorm@:(v)}
35791 @r{ a b c@: v p @: @: 24 @:calc-pack@:}
35792 @r{ v@: v r @:n >0 @: 21,31 @:mrow@:(v,n)}
35793 @r{ v@: v r @:n <0 @: 31 @:mrrow@:(v,-n)}
35794 @r{ m@: v r @:0 @: 31 @:getdiag@:(m)}
35795 @r{ v i j@: v s @: @: @:subvec@:(v,i,j)}
35796 @r{ v i j@: I v s @: @: @:rsubvec@:(v,i,j)}
35797 @r{ m@: v t @: @: 1 @:trn@:(m)}
35798 @r{ v@: v u @: @: 24 @:calc-unpack@:}
35799 @r{ v@: v v @: @: 1 @:rev@:(v)}
35800 @r{ @: v x @:n @: 31 @:index@:(n)}
35801 @r{ n s i@: C-u v x @: @: @:index@:(n,s,i)}
35804 @r{ v@: V A @:op @: 22 @:apply@:(op,v)}
35805 @r{ v1 v2@: V C @: @: 2 @:cross@:(v1,v2)}
35806 @r{ m@: V D @: @: 1 @:det@:(m)}
35807 @r{ s@: V E @: @: 1 @:venum@:(s)}
35808 @r{ s@: V F @: @: 1 @:vfloor@:(s)}
35809 @r{ v@: V G @: @: @:grade@:(v)}
35810 @r{ v@: I V G @: @: @:rgrade@:(v)}
35811 @r{ v@: V H @:n @: 31 @:histogram@:(v,n)}
35812 @r{ v w@: H V H @:n @: 31 @:histogram@:(v,w,n)}
35813 @r{ v1 v2@: V I @:mop aop @: 22 @:inner@:(mop,aop,v1,v2)}
35814 @r{ m@: V J @: @: 1 @:ctrn@:(m)}
35815 @r{ m@: V L @: @: 1 @:lud@:(m)}
35816 @r{ v@: V M @:op @: 22,23 @:map@:(op,v)}
35817 @r{ v@: V N @: @: 1 @:cnorm@:(v)}
35818 @r{ v1 v2@: V O @:op @: 22 @:outer@:(op,v1,v2)}
35819 @r{ v@: V R @:op @: 22,23 @:reduce@:(op,v)}
35820 @r{ v@: I V R @:op @: 22,23 @:rreduce@:(op,v)}
35821 @r{ a n@: H V R @:op @: 22 @:nest@:(op,a,n)}
35822 @r{ a@: I H V R @:op @: 22 @:fixp@:(op,a)}
35823 @r{ v@: V S @: @: @:sort@:(v)}
35824 @r{ v@: I V S @: @: @:rsort@:(v)}
35825 @r{ m@: V T @: @: 1 @:tr@:(m)}
35826 @r{ v@: V U @:op @: 22 @:accum@:(op,v)}
35827 @r{ v@: I V U @:op @: 22 @:raccum@:(op,v)}
35828 @r{ a n@: H V U @:op @: 22 @:anest@:(op,a,n)}
35829 @r{ a@: I H V U @:op @: 22 @:afixp@:(op,a)}
35830 @r{ s t@: V V @: @: 2 @:vunion@:(s,t)}
35831 @r{ s t@: V X @: @: 2 @:vxor@:(s,t)}
35834 @r{ @: Y @: @: @:@:user commands}
35837 @r{ @: z @: @: @:@:user commands}
35840 @r{ c@: Z [ @: @: 45 @:calc-kbd-if@:}
35841 @r{ c@: Z | @: @: 45 @:calc-kbd-else-if@:}
35842 @r{ @: Z : @: @: @:calc-kbd-else@:}
35843 @r{ @: Z ] @: @: @:calc-kbd-end-if@:}
35846 @r{ @: Z @{ @: @: 4 @:calc-kbd-loop@:}
35847 @r{ c@: Z / @: @: 45 @:calc-kbd-break@:}
35848 @r{ @: Z @} @: @: @:calc-kbd-end-loop@:}
35849 @r{ n@: Z < @: @: @:calc-kbd-repeat@:}
35850 @r{ @: Z > @: @: @:calc-kbd-end-repeat@:}
35851 @r{ n m@: Z ( @: @: @:calc-kbd-for@:}
35852 @r{ s@: Z ) @: @: @:calc-kbd-end-for@:}
35855 @r{ @: Z C-g @: @: @:@:cancel if/loop command}
35858 @r{ @: Z ` @: @: @:calc-kbd-push@:}
35859 @r{ @: Z ' @: @: @:calc-kbd-pop@:}
35860 @r{ a@: Z = @:message @: 28 @:calc-kbd-report@:}
35861 @r{ @: Z # @:prompt @: @:calc-kbd-query@:}
35864 @r{ comp@: Z C @:func, args @: 50 @:calc-user-define-composition@:}
35865 @r{ @: Z D @:key, command @: @:calc-user-define@:}
35866 @r{ @: Z E @:key, editing @: 30 @:calc-user-define-edit@:}
35867 @r{ defn@: Z F @:k, c, f, a, n@: 28 @:calc-user-define-formula@:}
35868 @r{ @: Z G @:key @: @:calc-get-user-defn@:}
35869 @r{ @: Z I @: @: @:calc-user-define-invocation@:}
35870 @r{ @: Z K @:key, command @: @:calc-user-define-kbd-macro@:}
35871 @r{ @: Z P @:key @: @:calc-user-define-permanent@:}
35872 @r{ @: Z S @: @: 30 @:calc-edit-user-syntax@:}
35873 @r{ @: Z T @: @: 12 @:calc-timing@:}
35874 @r{ @: Z U @:key @: @:calc-user-undefine@:}
35884 Positive prefix arguments apply to @expr{n} stack entries.
35885 Negative prefix arguments apply to the @expr{-n}th stack entry.
35886 A prefix of zero applies to the entire stack. (For @key{LFD} and
35887 @kbd{M-@key{DEL}}, the meaning of the sign is reversed.)
35891 Positive prefix arguments apply to @expr{n} stack entries.
35892 Negative prefix arguments apply to the top stack entry
35893 and the next @expr{-n} stack entries.
35897 Positive prefix arguments rotate top @expr{n} stack entries by one.
35898 Negative prefix arguments rotate the entire stack by @expr{-n}.
35899 A prefix of zero reverses the entire stack.
35903 Prefix argument specifies a repeat count or distance.
35907 Positive prefix arguments specify a precision @expr{p}.
35908 Negative prefix arguments reduce the current precision by @expr{-p}.
35912 A prefix argument is interpreted as an additional step-size parameter.
35913 A plain @kbd{C-u} prefix means to prompt for the step size.
35917 A prefix argument specifies simplification level and depth.
35918 1=Default, 2=like @kbd{a s}, 3=like @kbd{a e}.
35922 A negative prefix operates only on the top level of the input formula.
35926 Positive prefix arguments specify a word size of @expr{w} bits, unsigned.
35927 Negative prefix arguments specify a word size of @expr{w} bits, signed.
35931 Prefix arguments specify the shift amount @expr{n}. The @expr{w} argument
35932 cannot be specified in the keyboard version of this command.
35936 From the keyboard, @expr{d} is omitted and defaults to zero.
35940 Mode is toggled; a positive prefix always sets the mode, and a negative
35941 prefix always clears the mode.
35945 Some prefix argument values provide special variations of the mode.
35949 A prefix argument, if any, is used for @expr{m} instead of taking
35950 @expr{m} from the stack. @expr{M} may take any of these values:
35952 {@advance@tableindent10pt
35956 Random integer in the interval @expr{[0 .. m)}.
35958 Random floating-point number in the interval @expr{[0 .. m)}.
35960 Gaussian with mean 1 and standard deviation 0.
35962 Gaussian with specified mean and standard deviation.
35964 Random integer or floating-point number in that interval.
35966 Random element from the vector.
35974 A prefix argument from 1 to 6 specifies number of date components
35975 to remove from the stack. @xref{Date Conversions}.
35979 A prefix argument specifies a time zone; @kbd{C-u} says to take the
35980 time zone number or name from the top of the stack. @xref{Time Zones}.
35984 A prefix argument specifies a day number (0-6, 0-31, or 0-366).
35988 If the input has no units, you will be prompted for both the old and
35993 With a prefix argument, collect that many stack entries to form the
35994 input data set. Each entry may be a single value or a vector of values.
35998 With a prefix argument of 1, take a single
35999 @texline @var{n}@math{\times2}
36000 @infoline @mathit{@var{N}x2}
36001 matrix from the stack instead of two separate data vectors.
36005 The row or column number @expr{n} may be given as a numeric prefix
36006 argument instead. A plain @kbd{C-u} prefix says to take @expr{n}
36007 from the top of the stack. If @expr{n} is a vector or interval,
36008 a subvector/submatrix of the input is created.
36012 The @expr{op} prompt can be answered with the key sequence for the
36013 desired function, or with @kbd{x} or @kbd{z} followed by a function name,
36014 or with @kbd{$} to take a formula from the top of the stack, or with
36015 @kbd{'} and a typed formula. In the last two cases, the formula may
36016 be a nameless function like @samp{<#1+#2>} or @samp{<x, y : x+y>}, or it
36017 may include @kbd{$}, @kbd{$$}, etc. (where @kbd{$} will correspond to the
36018 last argument of the created function), or otherwise you will be
36019 prompted for an argument list. The number of vectors popped from the
36020 stack by @kbd{V M} depends on the number of arguments of the function.
36024 One of the mapping direction keys @kbd{_} (horizontal, i.e., map
36025 by rows or reduce across), @kbd{:} (vertical, i.e., map by columns or
36026 reduce down), or @kbd{=} (map or reduce by rows) may be used before
36027 entering @expr{op}; these modify the function name by adding the letter
36028 @code{r} for ``rows,'' @code{c} for ``columns,'' @code{a} for ``across,''
36029 or @code{d} for ``down.''
36033 The prefix argument specifies a packing mode. A nonnegative mode
36034 is the number of items (for @kbd{v p}) or the number of levels
36035 (for @kbd{v u}). A negative mode is as described below. With no
36036 prefix argument, the mode is taken from the top of the stack and
36037 may be an integer or a vector of integers.
36039 {@advance@tableindent-20pt
36043 (@var{2}) Rectangular complex number.
36045 (@var{2}) Polar complex number.
36047 (@var{3}) HMS form.
36049 (@var{2}) Error form.
36051 (@var{2}) Modulo form.
36053 (@var{2}) Closed interval.
36055 (@var{2}) Closed .. open interval.
36057 (@var{2}) Open .. closed interval.
36059 (@var{2}) Open interval.
36061 (@var{2}) Fraction.
36063 (@var{2}) Float with integer mantissa.
36065 (@var{2}) Float with mantissa in @expr{[1 .. 10)}.
36067 (@var{1}) Date form (using date numbers).
36069 (@var{3}) Date form (using year, month, day).
36071 (@var{6}) Date form (using year, month, day, hour, minute, second).
36079 A prefix argument specifies the size @expr{n} of the matrix. With no
36080 prefix argument, @expr{n} is omitted and the size is inferred from
36085 The prefix argument specifies the starting position @expr{n} (default 1).
36089 Cursor position within stack buffer affects this command.
36093 Arguments are not actually removed from the stack by this command.
36097 Variable name may be a single digit or a full name.
36101 Editing occurs in a separate buffer. Press @kbd{C-c C-c} (or
36102 @key{LFD}, or in some cases @key{RET}) to finish the edit, or kill the
36103 buffer with @kbd{C-x k} to cancel the edit. The @key{LFD} key prevents evaluation
36104 of the result of the edit.
36108 The number prompted for can also be provided as a prefix argument.
36112 Press this key a second time to cancel the prefix.
36116 With a negative prefix, deactivate all formulas. With a positive
36117 prefix, deactivate and then reactivate from scratch.
36121 Default is to scan for nearest formula delimiter symbols. With a
36122 prefix of zero, formula is delimited by mark and point. With a
36123 non-zero prefix, formula is delimited by scanning forward or
36124 backward by that many lines.
36128 Parse the region between point and mark as a vector. A nonzero prefix
36129 parses @var{n} lines before or after point as a vector. A zero prefix
36130 parses the current line as a vector. A @kbd{C-u} prefix parses the
36131 region between point and mark as a single formula.
36135 Parse the rectangle defined by point and mark as a matrix. A positive
36136 prefix @var{n} divides the rectangle into columns of width @var{n}.
36137 A zero or @kbd{C-u} prefix parses each line as one formula. A negative
36138 prefix suppresses special treatment of bracketed portions of a line.
36142 A numeric prefix causes the current language mode to be ignored.
36146 Responding to a prompt with a blank line answers that and all
36147 later prompts by popping additional stack entries.
36151 Answer for @expr{v} may also be of the form @expr{v = v_0} or
36156 With a positive prefix argument, stack contains many @expr{y}'s and one
36157 common @expr{x}. With a zero prefix, stack contains a vector of
36158 @expr{y}s and a common @expr{x}. With a negative prefix, stack
36159 contains many @expr{[x,y]} vectors. (For 3D plots, substitute
36160 @expr{z} for @expr{y} and @expr{x,y} for @expr{x}.)
36164 With any prefix argument, all curves in the graph are deleted.
36168 With a positive prefix, refines an existing plot with more data points.
36169 With a negative prefix, forces recomputation of the plot data.
36173 With any prefix argument, set the default value instead of the
36174 value for this graph.
36178 With a negative prefix argument, set the value for the printer.
36182 Condition is considered ``true'' if it is a nonzero real or complex
36183 number, or a formula whose value is known to be nonzero; it is ``false''
36188 Several formulas separated by commas are pushed as multiple stack
36189 entries. Trailing @kbd{)}, @kbd{]}, @kbd{@}}, @kbd{>}, and @kbd{"}
36190 delimiters may be omitted. The notation @kbd{$$$} refers to the value
36191 in stack level three, and causes the formula to replace the top three
36192 stack levels. The notation @kbd{$3} refers to stack level three without
36193 causing that value to be removed from the stack. Use @key{LFD} in place
36194 of @key{RET} to prevent evaluation; use @kbd{M-=} in place of @key{RET}
36195 to evaluate variables.
36199 The variable is replaced by the formula shown on the right. The
36200 Inverse flag reverses the order of the operands, e.g., @kbd{I s - x}
36202 @texline @math{x \coloneq a-x}.
36203 @infoline @expr{x := a-x}.
36207 Press @kbd{?} repeatedly to see how to choose a model. Answer the
36208 variables prompt with @expr{iv} or @expr{iv;pv} to specify
36209 independent and parameter variables. A positive prefix argument
36210 takes @mathit{@var{n}+1} vectors from the stack; a zero prefix takes a matrix
36211 and a vector from the stack.
36215 With a plain @kbd{C-u} prefix, replace the current region of the
36216 destination buffer with the yanked text instead of inserting.
36220 All stack entries are reformatted; the @kbd{H} prefix inhibits this.
36221 The @kbd{I} prefix sets the mode temporarily, redraws the top stack
36222 entry, then restores the original setting of the mode.
36226 A negative prefix sets the default 3D resolution instead of the
36227 default 2D resolution.
36231 This grabs a vector of the form [@var{prec}, @var{wsize}, @var{ssize},
36232 @var{radix}, @var{flfmt}, @var{ang}, @var{frac}, @var{symb}, @var{polar},
36233 @var{matrix}, @var{simp}, @var{inf}]. A prefix argument from 1 to 12
36234 grabs the @var{n}th mode value only.
36238 (Space is provided below for you to keep your own written notes.)
36246 @node Key Index, Command Index, Summary, Top
36247 @unnumbered Index of Key Sequences
36251 @node Command Index, Function Index, Key Index, Top
36252 @unnumbered Index of Calculator Commands
36254 Since all Calculator commands begin with the prefix @samp{calc-}, the
36255 @kbd{x} key has been provided as a variant of @kbd{M-x} which automatically
36256 types @samp{calc-} for you. Thus, @kbd{x last-args} is short for
36257 @kbd{M-x calc-last-args}.
36261 @node Function Index, Concept Index, Command Index, Top
36262 @unnumbered Index of Algebraic Functions
36264 This is a list of built-in functions and operators usable in algebraic
36265 expressions. Their full Lisp names are derived by adding the prefix
36266 @samp{calcFunc-}, as in @code{calcFunc-sqrt}.
36268 All functions except those noted with ``*'' have corresponding
36269 Calc keystrokes and can also be found in the Calc Summary.
36274 @node Concept Index, Variable Index, Function Index, Top
36275 @unnumbered Concept Index
36279 @node Variable Index, Lisp Function Index, Concept Index, Top
36280 @unnumbered Index of Variables
36282 The variables in this list that do not contain dashes are accessible
36283 as Calc variables. Add a @samp{var-} prefix to get the name of the
36284 corresponding Lisp variable.
36286 The remaining variables are Lisp variables suitable for @code{setq}ing
36287 in your Calc init file or @file{.emacs} file.
36291 @node Lisp Function Index, , Variable Index, Top
36292 @unnumbered Index of Lisp Math Functions
36294 The following functions are meant to be used with @code{defmath}, not
36295 @code{defun} definitions. For names that do not start with @samp{calc-},
36296 the corresponding full Lisp name is derived by adding a prefix of
36310 arch-tag: 77a71809-fa4d-40be-b2cc-da3e8fb137c0