1 ;;; calc-funcs.el --- well-known functions for Calc
3 ;; Copyright (C) 1990, 1991, 1992, 1993, 2001, 2005 Free Software Foundation, Inc.
5 ;; Author: David Gillespie <daveg@synaptics.com>
6 ;; Maintainer: Jay Belanger <belanger@truman.edu>
8 ;; This file is part of GNU Emacs.
10 ;; GNU Emacs is distributed in the hope that it will be useful,
11 ;; but WITHOUT ANY WARRANTY. No author or distributor
12 ;; accepts responsibility to anyone for the consequences of using it
13 ;; or for whether it serves any particular purpose or works at all,
14 ;; unless he says so in writing. Refer to the GNU Emacs General Public
15 ;; License for full details.
17 ;; Everyone is granted permission to copy, modify and redistribute
18 ;; GNU Emacs, but only under the conditions described in the
19 ;; GNU Emacs General Public License. A copy of this license is
20 ;; supposed to have been given to you along with GNU Emacs so you
21 ;; can know your rights and responsibilities. It should be in a
22 ;; file named COPYING. Among other things, the copyright notice
23 ;; and this notice must be preserved on all copies.
29 ;; This file is autoloaded from calc-ext.el.
34 (defun calc-inc-gamma (arg)
38 (if (calc-is-hyperbolic)
39 (calc-binary-op "gamG" 'calcFunc-gammaG arg
)
40 (calc-binary-op "gamQ" 'calcFunc-gammaQ arg
))
41 (if (calc-is-hyperbolic)
42 (calc-binary-op "gamg" 'calcFunc-gammag arg
)
43 (calc-binary-op "gamP" 'calcFunc-gammaP arg
)))))
49 (calc-unary-op "erfc" 'calcFunc-erfc arg
)
50 (calc-unary-op "erf" 'calcFunc-erf arg
))))
52 (defun calc-erfc (arg)
57 (defun calc-beta (arg)
60 (calc-binary-op "beta" 'calcFunc-beta arg
)))
62 (defun calc-inc-beta ()
65 (if (calc-is-hyperbolic)
66 (calc-enter-result 3 "betB" (cons 'calcFunc-betaB
(calc-top-list-n 3)))
67 (calc-enter-result 3 "betI" (cons 'calcFunc-betaI
(calc-top-list-n 3))))))
69 (defun calc-bessel-J (arg)
72 (calc-binary-op "besJ" 'calcFunc-besJ arg
)))
74 (defun calc-bessel-Y (arg)
77 (calc-binary-op "besY" 'calcFunc-besY arg
)))
79 (defun calc-bernoulli-number (arg)
82 (if (calc-is-hyperbolic)
83 (calc-binary-op "bern" 'calcFunc-bern arg
)
84 (calc-unary-op "bern" 'calcFunc-bern arg
))))
86 (defun calc-euler-number (arg)
89 (if (calc-is-hyperbolic)
90 (calc-binary-op "eulr" 'calcFunc-euler arg
)
91 (calc-unary-op "eulr" 'calcFunc-euler arg
))))
93 (defun calc-stirling-number (arg)
96 (if (calc-is-hyperbolic)
97 (calc-binary-op "str2" 'calcFunc-stir2 arg
)
98 (calc-binary-op "str1" 'calcFunc-stir1 arg
))))
102 (calc-prob-dist "b" 3))
106 (calc-prob-dist "c" 2))
110 (calc-prob-dist "f" 3))
114 (calc-prob-dist "n" 3))
118 (calc-prob-dist "p" 2))
122 (calc-prob-dist "t" 2))
124 (defun calc-prob-dist (letter nargs
)
126 (if (calc-is-inverse)
127 (calc-enter-result nargs
(concat "ltp" letter
)
128 (append (list (intern (concat "calcFunc-ltp" letter
))
130 (calc-top-list-n (1- nargs
) 2)))
131 (calc-enter-result nargs
(concat "utp" letter
)
132 (append (list (intern (concat "calcFunc-utp" letter
))
134 (calc-top-list-n (1- nargs
) 2))))))
139 ;;; Sources: Numerical Recipes, Press et al;
140 ;;; Handbook of Mathematical Functions, Abramowitz & Stegun.
145 (defun calcFunc-gamma (x)
146 (or (math-numberp x
) (math-reject-arg x
'numberp
))
147 (calcFunc-fact (math-add x -
1)))
149 (defun math-gammap1-raw (x &optional fprec nfprec
) ; compute gamma(1 + x)
151 (setq fprec
(math-float calc-internal-prec
)
152 nfprec
(math-float (- calc-internal-prec
))))
153 (cond ((math-lessp-float (calcFunc-re x
) fprec
)
154 (if (math-lessp-float (calcFunc-re x
) nfprec
)
157 (math-mul (math-gammap1-raw
158 (math-add (math-neg x
)
162 (math-mul (math-pi) x
)))))
163 (let ((xplus1 (math-add x
'(float 1 0))))
164 (math-div (math-gammap1-raw xplus1 fprec nfprec
) xplus1
))))
166 (math-lessp-float '(float 736276 0) x
))
168 (t ; re(x) now >= 10.0
169 (let ((xinv (math-div 1 x
))
170 (lnx (math-ln-raw x
)))
171 (math-mul (math-sqrt-two-pi)
174 (math-sub (math-mul (math-add x
'(float 5 -
1))
182 (defun math-gamma-series (sum x xinvsqr oterm n
)
183 (math-working "gamma" sum
)
184 (let* ((bn (math-bernoulli-number n
))
185 (term (math-mul (math-div-float (math-float (nth 1 bn
))
186 (math-float (* (nth 2 bn
)
189 (next (math-add sum term
)))
190 (if (math-nearly-equal sum next
)
192 (if (> n
(* 2 calc-internal-prec
))
194 ;; Need this because series eventually diverges for large enough n.
196 "*Gamma computation stopped early, not all digits may be valid")
198 (math-gamma-series next
(math-mul x xinvsqr
) xinvsqr term
(+ n
2))))))
201 ;;; Incomplete gamma function.
203 (defvar math-current-gamma-value nil
)
204 (defun calcFunc-gammaP (a x
)
205 (if (equal x
'(var inf var-inf
))
207 (math-inexact-result)
208 (or (Math-numberp a
) (math-reject-arg a
'numberp
))
209 (or (math-numberp x
) (math-reject-arg x
'numberp
))
210 (if (and (math-num-integerp a
)
211 (integerp (setq a
(math-trunc a
)))
213 (math-sub 1 (calcFunc-gammaQ a x
))
214 (let ((math-current-gamma-value (calcFunc-gamma a
)))
215 (math-div (calcFunc-gammag a x
) math-current-gamma-value
)))))
217 (defun calcFunc-gammaQ (a x
)
218 (if (equal x
'(var inf var-inf
))
220 (math-inexact-result)
221 (or (Math-numberp a
) (math-reject-arg a
'numberp
))
222 (or (math-numberp x
) (math-reject-arg x
'numberp
))
223 (if (and (math-num-integerp a
)
224 (integerp (setq a
(math-trunc a
)))
229 (math-with-extra-prec 1
230 (while (< (setq n
(1+ n
)) a
)
231 (setq term
(math-div (math-mul term x
) n
)
232 sum
(math-add sum term
))
233 (math-working "gamma" sum
))
234 (math-mul sum
(calcFunc-exp (math-neg x
)))))
235 (let ((math-current-gamma-value (calcFunc-gamma a
)))
236 (math-div (calcFunc-gammaG a x
) math-current-gamma-value
)))))
238 (defun calcFunc-gammag (a x
)
239 (if (equal x
'(var inf var-inf
))
241 (math-inexact-result)
242 (or (Math-numberp a
) (math-reject-arg a
'numberp
))
243 (or (Math-numberp x
) (math-reject-arg x
'numberp
))
244 (math-with-extra-prec 2
245 (setq a
(math-float a
))
246 (setq x
(math-float x
))
247 (if (or (math-negp (calcFunc-re a
))
248 (math-lessp-float (calcFunc-re x
)
249 (math-add-float (calcFunc-re a
)
251 (math-inc-gamma-series a x
)
252 (math-sub (or math-current-gamma-value
(calcFunc-gamma a
))
253 (math-inc-gamma-cfrac a x
))))))
255 (defun calcFunc-gammaG (a x
)
256 (if (equal x
'(var inf var-inf
))
258 (math-inexact-result)
259 (or (Math-numberp a
) (math-reject-arg a
'numberp
))
260 (or (Math-numberp x
) (math-reject-arg x
'numberp
))
261 (math-with-extra-prec 2
262 (setq a
(math-float a
))
263 (setq x
(math-float x
))
264 (if (or (math-negp (calcFunc-re a
))
265 (math-lessp-float (calcFunc-re x
)
266 (math-add-float (math-abs-approx a
)
268 (math-sub (or math-current-gamma-value
(calcFunc-gamma a
))
269 (math-inc-gamma-series a x
))
270 (math-inc-gamma-cfrac a x
)))))
272 (defun math-inc-gamma-series (a x
)
275 (math-mul (math-exp-raw (math-sub (math-mul a
(math-ln-raw x
)) x
))
276 (math-with-extra-prec 2
277 (let ((start (math-div '(float 1 0) a
)))
278 (math-inc-gamma-series-step start start a x
))))))
280 (defun math-inc-gamma-series-step (sum term a x
)
281 (math-working "gamma" sum
)
282 (setq a
(math-add a
'(float 1 0))
283 term
(math-div (math-mul term x
) a
))
284 (let ((next (math-add sum term
)))
285 (if (math-nearly-equal sum next
)
287 (math-inc-gamma-series-step next term a x
))))
289 (defun math-inc-gamma-cfrac (a x
)
291 (or math-current-gamma-value
(calcFunc-gamma a
))
292 (math-mul (math-exp-raw (math-sub (math-mul a
(math-ln-raw x
)) x
))
293 (math-inc-gamma-cfrac-step '(float 1 0) x
294 '(float 0 0) '(float 1 0)
295 '(float 1 0) '(float 1 0) '(float 0 0)
298 (defun math-inc-gamma-cfrac-step (a0 a1 b0 b1 n fac g a x
)
299 (let ((ana (math-sub n a
))
300 (anf (math-mul n fac
)))
301 (setq n
(math-add n
'(float 1 0))
302 a0
(math-mul (math-add a1
(math-mul a0 ana
)) fac
)
303 b0
(math-mul (math-add b1
(math-mul b0 ana
)) fac
)
304 a1
(math-add (math-mul x a0
) (math-mul anf a1
))
305 b1
(math-add (math-mul x b0
) (math-mul anf b1
)))
307 (math-inc-gamma-cfrac-step a0 a1 b0 b1 n fac g a x
)
308 (setq fac
(math-div '(float 1 0) a1
))
309 (let ((next (math-mul b1 fac
)))
310 (math-working "gamma" next
)
311 (if (math-nearly-equal next g
)
313 (math-inc-gamma-cfrac-step a0 a1 b0 b1 n fac next a x
))))))
318 (defun calcFunc-erf (x)
319 (if (equal x
'(var inf var-inf
))
321 (if (equal x
'(neg (var inf var-inf
)))
325 (let ((math-current-gamma-value (math-sqrt-pi)))
326 (math-to-same-complex-quad
327 (math-div (calcFunc-gammag '(float 5 -
1)
328 (math-sqr (math-to-complex-quad-one x
)))
329 math-current-gamma-value
)
332 (defun calcFunc-erfc (x)
333 (if (equal x
'(var inf var-inf
))
336 (let ((math-current-gamma-value (math-sqrt-pi)))
337 (math-div (calcFunc-gammaG '(float 5 -
1) (math-sqr x
))
338 math-current-gamma-value
))
339 (math-sub 1 (calcFunc-erf x
)))))
341 (defun math-to-complex-quad-one (x)
342 (if (eq (car-safe x
) 'polar
) (setq x
(math-complex x
)))
343 (if (eq (car-safe x
) 'cplx
)
344 (list 'cplx
(math-abs (nth 1 x
)) (math-abs (nth 2 x
)))
347 (defun math-to-same-complex-quad (x y
)
348 (if (eq (car-safe y
) 'cplx
)
349 (if (eq (car-safe x
) 'cplx
)
351 (if (math-negp (nth 1 y
)) (math-neg (nth 1 x
)) (nth 1 x
))
352 (if (math-negp (nth 2 y
)) (math-neg (nth 2 x
)) (nth 2 x
)))
353 (if (math-negp (nth 1 y
)) (math-neg x
) x
))
355 (if (eq (car-safe x
) 'cplx
)
356 (list 'cplx
(math-neg (nth 1 x
)) (nth 2 x
))
363 (defun calcFunc-beta (a b
)
364 (if (math-num-integerp a
)
365 (let ((am (math-add a -
1)))
366 (or (math-numberp b
) (math-reject-arg b
'numberp
))
367 (math-div 1 (math-mul b
(calcFunc-choose (math-add b am
) am
))))
368 (if (math-num-integerp b
)
370 (math-div (math-mul (calcFunc-gamma a
) (calcFunc-gamma b
))
371 (calcFunc-gamma (math-add a b
))))))
374 ;;; Incomplete beta function.
376 (defvar math-current-beta-value nil
)
377 (defun calcFunc-betaI (x a b
)
378 (cond ((math-zerop x
)
380 ((math-equal-int x
1)
383 (and (math-num-integerp a
)
385 (if (or (math-zerop b
)
386 (and (math-num-integerp b
)
388 (math-reject-arg b
'range
)
391 (and (math-num-integerp b
)
394 ((not (math-numberp a
)) (math-reject-arg a
'numberp
))
395 ((not (math-numberp b
)) (math-reject-arg b
'numberp
))
396 ((math-inexact-result))
397 (t (let ((math-current-beta-value (calcFunc-beta a b
)))
398 (math-div (calcFunc-betaB x a b
) math-current-beta-value
)))))
400 (defun calcFunc-betaB (x a b
)
404 ((math-equal-int x
1)
406 ((not (math-numberp x
)) (math-reject-arg x
'numberp
))
407 ((not (math-numberp a
)) (math-reject-arg a
'numberp
))
408 ((not (math-numberp b
)) (math-reject-arg b
'numberp
))
409 ((math-zerop a
) (math-reject-arg a
'nonzerop
))
410 ((math-zerop b
) (math-reject-arg b
'nonzerop
))
411 ((and (math-num-integerp b
)
413 (math-reject-arg b
'range
)
414 (Math-natnum-lessp (setq b
(math-trunc b
)) 20)))
415 (and calc-symbolic-mode
(or (math-floatp a
) (math-floatp b
))
416 (math-inexact-result))
418 (math-with-extra-prec 2
421 (sum (math-div term a
)))
422 (while (< (setq i
(1+ i
)) b
)
423 (setq term
(math-mul (math-div (math-mul term
(- i b
)) i
) x
)
424 sum
(math-add sum
(math-div term
(math-add a i
))))
425 (math-working "beta" sum
))
428 ((and (math-num-integerp a
)
430 (math-reject-arg a
'range
)
431 (Math-natnum-lessp (setq a
(math-trunc a
)) 20)))
432 (math-sub (or math-current-beta-value
(calcFunc-beta a b
))
433 (calcFunc-betaB (math-sub 1 x
) b a
)))
435 (math-inexact-result)
436 (math-with-extra-prec 2
437 (setq x
(math-float x
))
438 (setq a
(math-float a
))
439 (setq b
(math-float b
))
440 (let ((bt (math-exp-raw (math-add (math-mul a
(math-ln-raw x
))
441 (math-mul b
(math-ln-raw
442 (math-sub '(float 1 0)
444 (if (Math-lessp x
(math-div (math-add a
'(float 1 0))
445 (math-add (math-add a b
) '(float 2 0))))
446 (math-div (math-mul bt
(math-beta-cfrac a b x
)) a
)
447 (math-sub (or math-current-beta-value
(calcFunc-beta a b
))
448 (math-div (math-mul bt
449 (math-beta-cfrac b a
(math-sub 1 x
)))
452 (defun math-beta-cfrac (a b x
)
453 (let ((qab (math-add a b
))
454 (qap (math-add a
'(float 1 0)))
455 (qam (math-add a
'(float -
1 0))))
456 (math-beta-cfrac-step '(float 1 0)
457 (math-sub '(float 1 0)
458 (math-div (math-mul qab x
) qap
))
459 '(float 1 0) '(float 1 0)
463 (defun math-beta-cfrac-step (az bz am bm m qab qap qam a b x
)
464 (let* ((two-m (math-mul m
'(float 2 0)))
465 (d (math-div (math-mul (math-mul (math-sub b m
) m
) x
)
466 (math-mul (math-add qam two-m
) (math-add a two-m
))))
467 (ap (math-add az
(math-mul d am
)))
468 (bp (math-add bz
(math-mul d bm
)))
470 (math-div (math-mul (math-mul (math-add a m
) (math-add qab m
)) x
)
471 (math-mul (math-add qap two-m
) (math-add a two-m
)))))
472 (app (math-add ap
(math-mul d2 az
)))
473 (bpp (math-add bp
(math-mul d2 bz
)))
474 (next (math-div app bpp
)))
475 (math-working "beta" next
)
476 (if (math-nearly-equal next az
)
478 (math-beta-cfrac-step next
'(float 1 0)
479 (math-div ap bpp
) (math-div bp bpp
)
480 (math-add m
'(float 1 0))
481 qab qap qam a b x
))))
484 ;;; Bessel functions.
486 ;;; Should generalize this to handle arbitrary precision!
488 (defun calcFunc-besJ (v x
)
489 (or (math-numberp v
) (math-reject-arg v
'numberp
))
490 (or (math-numberp x
) (math-reject-arg x
'numberp
))
491 (let ((calc-internal-prec (min 8 calc-internal-prec
)))
492 (math-with-extra-prec 3
493 (setq x
(math-float (math-normalize x
)))
494 (setq v
(math-float (math-normalize v
)))
495 (cond ((math-zerop x
)
499 ((math-inexact-result))
500 ((not (math-num-integerp v
))
501 (let ((start (math-div 1 (calcFunc-fact v
))))
502 (math-mul (math-besJ-series start start
504 (math-mul '(float -
25 -
2)
507 (math-pow (math-div x
2) v
))))
508 ((math-negp (setq v
(math-trunc v
)))
510 (math-neg (calcFunc-besJ (math-neg v
) x
))
511 (calcFunc-besJ (math-neg v
) x
)))
516 ((Math-lessp v
(math-abs-approx x
))
520 (two-over-x (math-div 2 x
))
522 (while (< (setq j
(1+ j
)) v
)
523 (setq bjp
(math-sub (math-mul (math-mul j two-over-x
) bj
)
529 (if (Math-lessp 100 v
) (math-reject-arg v
'range
))
530 (let* ((j (logior (+ v
(math-isqrt-small (* 40 v
))) 1))
531 (two-over-x (math-div 2 x
))
537 (while (> (setq j
(1- j
)) 0)
538 (setq bjm
(math-sub (math-mul (math-mul j two-over-x
) bj
)
542 (if (> (nth 2 (math-abs-approx bj
)) 10)
543 (setq bj
(math-mul bj
'(float 1 -
10))
544 bjp
(math-mul bjp
'(float 1 -
10))
545 ans
(and ans
(math-mul ans
'(float 1 -
10)))
546 sum
(math-mul sum
'(float 1 -
10))))
547 (or (setq jsum
(not jsum
))
548 (setq sum
(math-add sum bj
)))
551 (math-div ans
(math-sub (math-mul 2 sum
) bj
))))))))
553 (defun math-besJ-series (sum term k zz vk
)
554 (math-working "besJ" sum
)
557 term
(math-div (math-mul term zz
) (math-mul k vk
)))
558 (let ((next (math-add sum term
)))
559 (if (math-nearly-equal next sum
)
561 (math-besJ-series next term k zz vk
))))
563 (defun math-besJ0 (x &optional yflag
)
564 (cond ((and (not yflag
) (math-negp (calcFunc-re x
)))
565 (math-besJ0 (math-neg x
)))
566 ((Math-lessp '(float 8 0) (math-abs-approx x
))
567 (let* ((z (math-div '(float 8 0) x
))
569 (xx (math-add x
'(float (bigneg 164 398 785) -
9)))
570 (a1 (math-poly-eval y
571 '((float (bigpos 211 887 093 2) -
16)
572 (float (bigneg 639 370 073 2) -
15)
573 (float (bigpos 407 510 734 2) -
14)
574 (float (bigneg 627 628 098 1) -
12)
576 (a2 (math-poly-eval y
577 '((float (bigneg 152 935 934) -
16)
578 (float (bigpos 161 095 621 7) -
16)
579 (float (bigneg 651 147 911 6) -
15)
580 (float (bigpos 765 488 430 1) -
13)
581 (float (bigneg 995 499 562 1) -
11))))
582 (sc (math-sin-cos-raw xx
)))
584 (setq sc
(cons (math-neg (cdr sc
)) (car sc
))))
586 (math-div '(float (bigpos 722 619 636) -
9) x
))
587 (math-sub (math-mul (cdr sc
) a1
)
588 (math-mul (car sc
) (math-mul z a2
))))))
590 (let ((y (math-sqr x
)))
591 (math-div (math-poly-eval y
592 '((float (bigneg 456 052 849 1) -
7)
593 (float (bigpos 017 233 739 7) -
5)
594 (float (bigneg 418 442 121 1) -
2)
595 (float (bigpos 407 196 516 6) -
1)
596 (float (bigneg 354 590 362 13) 0)
597 (float (bigpos 574 490 568 57) 0)))
600 (float (bigpos 712 532 678 2) -
7)
601 (float (bigpos 853 264 927 5) -
5)
602 (float (bigpos 718 680 494 9) -
3)
603 (float (bigpos 985 532 029 1) 0)
604 (float (bigpos 411 490 568 57) 0))))))))
606 (defun math-besJ1 (x &optional yflag
)
607 (cond ((and (math-negp (calcFunc-re x
)) (not yflag
))
608 (math-neg (math-besJ1 (math-neg x
))))
609 ((Math-lessp '(float 8 0) (math-abs-approx x
))
610 (let* ((z (math-div '(float 8 0) x
))
612 (xx (math-add x
'(float (bigneg 491 194 356 2) -
9)))
613 (a1 (math-poly-eval y
614 '((float (bigneg 019 337 240) -
15)
615 (float (bigpos 174 520 457 2) -
15)
616 (float (bigneg 496 396 516 3) -
14)
619 (a2 (math-poly-eval y
620 '((float (bigpos 412 787 105) -
15)
621 (float (bigneg 987 228 88) -
14)
622 (float (bigpos 096 199 449 8) -
15)
623 (float (bigneg 873 690 002 2) -
13)
624 (float (bigpos 995 499 687 4) -
11))))
625 (sc (math-sin-cos-raw xx
)))
627 (setq sc
(cons (math-neg (cdr sc
)) (car sc
)))
629 (setq sc
(cons (math-neg (car sc
)) (math-neg (cdr sc
))))))
630 (math-mul (math-sqrt (math-div '(float (bigpos 722 619 636) -
9) x
))
631 (math-sub (math-mul (cdr sc
) a1
)
632 (math-mul (car sc
) (math-mul z a2
))))))
634 (let ((y (math-sqr x
)))
637 (math-div (math-poly-eval y
638 '((float (bigneg 606 036 016 3) -
8)
639 (float (bigpos 826 044 157) -
4)
640 (float (bigneg 439 611 972 2) -
3)
641 (float (bigpos 531 968 423 2) -
1)
642 (float (bigneg 235 059 895 7) 0)
643 (float (bigpos 232 614 362 72) 0)))
646 (float (bigpos 397 991 769 3) -
7)
647 (float (bigpos 394 743 944 9) -
5)
648 (float (bigpos 474 330 858 1) -
2)
649 (float (bigpos 178 535 300 2) 0)
650 (float (bigpos 442 228 725 144)
653 (defun calcFunc-besY (v x
)
654 (math-inexact-result)
655 (or (math-numberp v
) (math-reject-arg v
'numberp
))
656 (or (math-numberp x
) (math-reject-arg x
'numberp
))
657 (let ((calc-internal-prec (min 8 calc-internal-prec
)))
658 (math-with-extra-prec 3
659 (setq x
(math-float (math-normalize x
)))
660 (setq v
(math-float (math-normalize v
)))
661 (cond ((not (math-num-integerp v
))
662 (let ((sc (math-sin-cos-raw (math-mul v
(math-pi)))))
663 (math-div (math-sub (math-mul (calcFunc-besJ v x
) (cdr sc
))
664 (calcFunc-besJ (math-neg v
) x
))
666 ((math-negp (setq v
(math-trunc v
)))
668 (math-neg (calcFunc-besY (math-neg v
) x
))
669 (calcFunc-besY (math-neg v
) x
)))
678 (two-over-x (math-div 2 x
))
680 (while (< (setq j
(1+ j
)) v
)
681 (setq byp
(math-sub (math-mul (math-mul j two-over-x
) by
)
687 (defun math-besY0 (x)
688 (cond ((Math-lessp (math-abs-approx x
) '(float 8 0))
689 (let ((y (math-sqr x
)))
691 (math-div (math-poly-eval y
692 '((float (bigpos 733 622 284 2) -
7)
693 (float (bigneg 757 792 632 8) -
5)
694 (float (bigpos 129 988 087 1) -
2)
695 (float (bigneg 036 598 123 5) -
1)
696 (float (bigpos 065 834 062 7) 0)
697 (float (bigneg 389 821 957 2) 0)))
700 (float (bigpos 244 030 261 2) -
7)
701 (float (bigpos 647 472 474) -
4)
702 (float (bigpos 438 466 189 7) -
3)
703 (float (bigpos 648 499 452 7) -
1)
704 (float (bigpos 269 544 076 40) 0))))
705 (math-mul '(float (bigpos 772 619 636) -
9)
706 (math-mul (math-besJ0 x
) (math-ln-raw x
))))))
707 ((math-negp (calcFunc-re x
))
708 (math-add (math-besJ0 (math-neg x
) t
)
709 (math-mul '(cplx 0 2)
710 (math-besJ0 (math-neg x
)))))
714 (defun math-besY1 (x)
715 (cond ((Math-lessp (math-abs-approx x
) '(float 8 0))
716 (let ((y (math-sqr x
)))
720 (math-div (math-poly-eval y
721 '((float (bigpos 935 937 511 8) -
6)
722 (float (bigneg 726 922 237 4) -
3)
723 (float (bigpos 551 264 349 7) -
1)
724 (float (bigneg 139 438 153 5) 1)
725 (float (bigpos 439 527 127) 4)
726 (float (bigneg 943 604 900 4) 3)))
729 (float (bigpos 885 632 549 3) -
7)
730 (float (bigpos 605 042 102) -
3)
731 (float (bigpos 002 904 245 2) -
2)
732 (float (bigpos 367 650 733 3) 0)
733 (float (bigpos 664 419 244 4) 2)
734 (float (bigpos 057 958 249) 5)))))
735 (math-mul '(float (bigpos 772 619 636) -
9)
736 (math-sub (math-mul (math-besJ1 x
) (math-ln-raw x
))
738 ((math-negp (calcFunc-re x
))
740 (math-add (math-besJ1 (math-neg x
) t
)
741 (math-mul '(cplx 0 2)
742 (math-besJ1 (math-neg x
))))))
746 (defun math-poly-eval (x coefs
)
747 (let ((accum (car coefs
)))
748 (while (setq coefs
(cdr coefs
))
749 (setq accum
(math-add (car coefs
) (math-mul accum x
))))
753 ;;;; Bernoulli and Euler polynomials and numbers.
755 (defun calcFunc-bern (n &optional x
)
756 (if (and x
(not (math-zerop x
)))
757 (if (and calc-symbolic-mode
(math-floatp x
))
758 (math-inexact-result)
759 (math-build-polynomial-expr (math-bernoulli-coefs n
) x
))
760 (or (math-num-natnump n
) (math-reject-arg n
'natnump
))
763 (math-inexact-result)
764 (math-float (math-bernoulli-number (math-trunc n
))))
765 (math-bernoulli-number n
))))
767 (defun calcFunc-euler (n &optional x
)
768 (or (math-num-natnump n
) (math-reject-arg n
'natnump
))
770 (let* ((n1 (math-add n
1))
771 (coefs (math-bernoulli-coefs n1
))
772 (fac (math-div (math-pow 2 n1
) n1
))
774 (x1 (math-div (math-add x
1) 2))
777 (if (and calc-symbolic-mode
(math-floatp x
))
778 (math-inexact-result)
780 (math-sub (math-build-polynomial-expr coefs x1
)
781 (math-build-polynomial-expr coefs x2
))))
789 (math-mul (math-mul fac c
)
790 (math-sub (math-pow x1 k
)
794 (math-mul (math-pow 2 n
)
797 (math-inexact-result)
798 (calcFunc-euler n
'(float 5 -
1)))
799 (calcFunc-euler n
'(frac 1 2))))))
801 (defvar math-bernoulli-b-cache
'((frac -
174611
802 (bigpos 0 200 291 698 662 857 802))
803 (frac 43867 (bigpos 0 944 170 217 94 109 5))
804 (frac -
3617 (bigpos 0 880 842 622 670 10))
805 (frac 1 (bigpos 600 249 724 74))
806 (frac -
691 (bigpos 0 368 674 307 1))
807 (frac 1 (bigpos 160 900 47))
808 (frac -
1 (bigpos 600 209 1))
809 (frac 1 30240) (frac -
1 720)
812 (defvar math-bernoulli-B-cache
'((frac -
174611 330) (frac 43867 798)
813 (frac -
3617 510) (frac 7 6) (frac -
691 2730)
814 (frac 5 66) (frac -
1 30) (frac 1 42)
815 (frac -
1 30) (frac 1 6) 1 ))
817 (defvar math-bernoulli-cache-size
11)
818 (defun math-bernoulli-coefs (n)
819 (let* ((coefs (list (calcFunc-bern n
)))
824 (calc-prefer-frac (or (integerp n
) calc-prefer-frac
)))
825 (while (>= (setq k
(1- k
)) 0)
826 (setq term
(math-div term
(- nn k
))
827 coef
(math-mul term
(math-bernoulli-number k
))
828 coefs
(cons (if (consp n
) (math-float coef
) coef
) coefs
)
829 term
(math-mul term k
)))
832 (defun math-bernoulli-number (n)
838 (while (>= n math-bernoulli-cache-size
)
841 (fact 1) ; fact = (n-k+1)!
843 (p math-bernoulli-b-cache
)
844 (calc-prefer-frac t
))
845 (math-working "bernoulli B" (* 2 math-bernoulli-cache-size
))
849 fact
(math-mul fact
(* nk
(1- nk
)))
850 sum
(math-add sum
(math-div (car p
) fact
))
852 (setq ofact
(math-mul ofact
(1- nk
))
853 sum
(math-sub (math-div '(frac 1 2) ofact
) sum
)
854 math-bernoulli-b-cache
(cons sum math-bernoulli-b-cache
)
855 math-bernoulli-B-cache
(cons (math-mul sum ofact
)
856 math-bernoulli-B-cache
)
857 math-bernoulli-cache-size
(1+ math-bernoulli-cache-size
))))
858 (nth (- math-bernoulli-cache-size n
1) math-bernoulli-B-cache
)))
861 ;;; bn = - sum_k=0^n-1 bk / (n-k+1)!
863 ;;; A faster method would be to use "tangent numbers", c.f., Concrete
864 ;;; Mathematics pg. 273.
867 ;;; Probability distributions.
870 (defun calcFunc-utpb (x n p
)
871 (if math-expand-formulas
872 (math-normalize (list 'calcFunc-betaI p x
(list '+ (list '- n x
) 1)))
873 (calcFunc-betaI p x
(math-add (math-sub n x
) 1))))
874 (put 'calcFunc-utpb
'math-expandable t
)
876 (defun calcFunc-ltpb (x n p
)
877 (math-sub 1 (calcFunc-utpb x n p
)))
878 (put 'calcFunc-ltpb
'math-expandable t
)
881 (defun calcFunc-utpc (chisq v
)
882 (if math-expand-formulas
883 (math-normalize (list 'calcFunc-gammaQ
(list '/ v
2) (list '/ chisq
2)))
884 (calcFunc-gammaQ (math-div v
2) (math-div chisq
2))))
885 (put 'calcFunc-utpc
'math-expandable t
)
887 (defun calcFunc-ltpc (chisq v
)
888 (if math-expand-formulas
889 (math-normalize (list 'calcFunc-gammaP
(list '/ v
2) (list '/ chisq
2)))
890 (calcFunc-gammaP (math-div v
2) (math-div chisq
2))))
891 (put 'calcFunc-ltpc
'math-expandable t
)
894 (defun calcFunc-utpf (f v1 v2
)
895 (if math-expand-formulas
896 (math-normalize (list 'calcFunc-betaI
897 (list '/ v2
(list '+ v2
(list '* v1 f
)))
900 (calcFunc-betaI (math-div v2
(math-add v2
(math-mul v1 f
)))
903 (put 'calcFunc-utpf
'math-expandable t
)
905 (defun calcFunc-ltpf (f v1 v2
)
906 (math-sub 1 (calcFunc-utpf f v1 v2
)))
907 (put 'calcFunc-ltpf
'math-expandable t
)
910 (defun calcFunc-utpn (x mean sdev
)
911 (if math-expand-formulas
916 (list '/ (list '- mean x
)
917 (list '* sdev
(list 'calcFunc-sqrt
2)))))
919 (math-mul (math-add '(float 1 0)
921 (math-div (math-sub mean x
)
922 (math-mul sdev
(math-sqrt-2)))))
924 (put 'calcFunc-utpn
'math-expandable t
)
926 (defun calcFunc-ltpn (x mean sdev
)
927 (if math-expand-formulas
932 (list '/ (list '- x mean
)
933 (list '* sdev
(list 'calcFunc-sqrt
2)))))
935 (math-mul (math-add '(float 1 0)
937 (math-div (math-sub x mean
)
938 (math-mul sdev
(math-sqrt-2)))))
940 (put 'calcFunc-ltpn
'math-expandable t
)
943 (defun calcFunc-utpp (n x
)
944 (if math-expand-formulas
945 (math-normalize (list 'calcFunc-gammaP x n
))
946 (calcFunc-gammaP x n
)))
947 (put 'calcFunc-utpp
'math-expandable t
)
949 (defun calcFunc-ltpp (n x
)
950 (if math-expand-formulas
951 (math-normalize (list 'calcFunc-gammaQ x n
))
952 (calcFunc-gammaQ x n
)))
953 (put 'calcFunc-ltpp
'math-expandable t
)
955 ;;; Student's t. (As defined in Abramowitz & Stegun and Numerical Recipes.)
956 (defun calcFunc-utpt (tt v
)
957 (if math-expand-formulas
958 (math-normalize (list 'calcFunc-betaI
959 (list '/ v
(list '+ v
(list '^ tt
2)))
962 (calcFunc-betaI (math-div v
(math-add v
(math-sqr tt
)))
965 (put 'calcFunc-utpt
'math-expandable t
)
967 (defun calcFunc-ltpt (tt v
)
968 (math-sub 1 (calcFunc-utpt tt v
)))
969 (put 'calcFunc-ltpt
'math-expandable t
)
971 (provide 'calc-funcs
)
973 ;;; arch-tag: 421ddb7a-550f-4dda-a31c-06638ebfc43a
974 ;;; calc-funcs.el ends here