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Port various comments from new VC to reduce the noise in the diff.
[emacs.git] / lisp / calc / calc-funcs.el
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1 ;;; calc-funcs.el --- well-known functions for Calc
2
3 ;; Copyright (C) 1990, 1991, 1992, 1993, 2001, 2002, 2003, 2004,
4 ;; 2005, 2006, 2007 Free Software Foundation, Inc.
5
6 ;; Author: David Gillespie <daveg@synaptics.com>
7 ;; Maintainer: Jay Belanger <jay.p.belanger@gmail.com>
8
9 ;; This file is part of GNU Emacs.
10
11 ;; GNU Emacs is free software; you can redistribute it and/or modify
12 ;; it under the terms of the GNU General Public License as published by
13 ;; the Free Software Foundation; either version 3, or (at your option)
14 ;; any later version.
15
16 ;; GNU Emacs is distributed in the hope that it will be useful,
17 ;; but WITHOUT ANY WARRANTY; without even the implied warranty of
18 ;; MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
19 ;; GNU General Public License for more details.
20
21 ;; You should have received a copy of the GNU General Public License
22 ;; along with GNU Emacs; see the file COPYING. If not, write to the
23 ;; Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor,
24 ;; Boston, MA 02110-1301, USA.
25
26 ;;; Commentary:
27
28 ;;; Code:
29
30 ;; This file is autoloaded from calc-ext.el.
31
32 (require 'calc-ext)
33 (require 'calc-macs)
34
35 (defun calc-inc-gamma (arg)
36 (interactive "P")
37 (calc-slow-wrapper
38 (if (calc-is-inverse)
39 (if (calc-is-hyperbolic)
40 (calc-binary-op "gamG" 'calcFunc-gammaG arg)
41 (calc-binary-op "gamQ" 'calcFunc-gammaQ arg))
42 (if (calc-is-hyperbolic)
43 (calc-binary-op "gamg" 'calcFunc-gammag arg)
44 (calc-binary-op "gamP" 'calcFunc-gammaP arg)))))
45
46 (defun calc-erf (arg)
47 (interactive "P")
48 (calc-slow-wrapper
49 (if (calc-is-inverse)
50 (calc-unary-op "erfc" 'calcFunc-erfc arg)
51 (calc-unary-op "erf" 'calcFunc-erf arg))))
52
53 (defun calc-erfc (arg)
54 (interactive "P")
55 (calc-invert-func)
56 (calc-erf arg))
57
58 (defun calc-beta (arg)
59 (interactive "P")
60 (calc-slow-wrapper
61 (calc-binary-op "beta" 'calcFunc-beta arg)))
62
63 (defun calc-inc-beta ()
64 (interactive)
65 (calc-slow-wrapper
66 (if (calc-is-hyperbolic)
67 (calc-enter-result 3 "betB" (cons 'calcFunc-betaB (calc-top-list-n 3)))
68 (calc-enter-result 3 "betI" (cons 'calcFunc-betaI (calc-top-list-n 3))))))
69
70 (defun calc-bessel-J (arg)
71 (interactive "P")
72 (calc-slow-wrapper
73 (calc-binary-op "besJ" 'calcFunc-besJ arg)))
74
75 (defun calc-bessel-Y (arg)
76 (interactive "P")
77 (calc-slow-wrapper
78 (calc-binary-op "besY" 'calcFunc-besY arg)))
79
80 (defun calc-bernoulli-number (arg)
81 (interactive "P")
82 (calc-slow-wrapper
83 (if (calc-is-hyperbolic)
84 (calc-binary-op "bern" 'calcFunc-bern arg)
85 (calc-unary-op "bern" 'calcFunc-bern arg))))
86
87 (defun calc-euler-number (arg)
88 (interactive "P")
89 (calc-slow-wrapper
90 (if (calc-is-hyperbolic)
91 (calc-binary-op "eulr" 'calcFunc-euler arg)
92 (calc-unary-op "eulr" 'calcFunc-euler arg))))
93
94 (defun calc-stirling-number (arg)
95 (interactive "P")
96 (calc-slow-wrapper
97 (if (calc-is-hyperbolic)
98 (calc-binary-op "str2" 'calcFunc-stir2 arg)
99 (calc-binary-op "str1" 'calcFunc-stir1 arg))))
100
101 (defun calc-utpb ()
102 (interactive)
103 (calc-prob-dist "b" 3))
104
105 (defun calc-utpc ()
106 (interactive)
107 (calc-prob-dist "c" 2))
108
109 (defun calc-utpf ()
110 (interactive)
111 (calc-prob-dist "f" 3))
112
113 (defun calc-utpn ()
114 (interactive)
115 (calc-prob-dist "n" 3))
116
117 (defun calc-utpp ()
118 (interactive)
119 (calc-prob-dist "p" 2))
120
121 (defun calc-utpt ()
122 (interactive)
123 (calc-prob-dist "t" 2))
124
125 (defun calc-prob-dist (letter nargs)
126 (calc-slow-wrapper
127 (if (calc-is-inverse)
128 (calc-enter-result nargs (concat "ltp" letter)
129 (append (list (intern (concat "calcFunc-ltp" letter))
130 (calc-top-n 1))
131 (calc-top-list-n (1- nargs) 2)))
132 (calc-enter-result nargs (concat "utp" letter)
133 (append (list (intern (concat "calcFunc-utp" letter))
134 (calc-top-n 1))
135 (calc-top-list-n (1- nargs) 2))))))
136
137
138
139
140 ;;; Sources: Numerical Recipes, Press et al;
141 ;;; Handbook of Mathematical Functions, Abramowitz & Stegun.
142
143
144 ;;; Gamma function.
145
146 (defun calcFunc-gamma (x)
147 (or (math-numberp x) (math-reject-arg x 'numberp))
148 (calcFunc-fact (math-add x -1)))
149
150 (defun math-gammap1-raw (x &optional fprec nfprec)
151 "Compute gamma(1+X) to the appropriate precision."
152 (or fprec
153 (setq fprec (math-float calc-internal-prec)
154 nfprec (math-float (- calc-internal-prec))))
155 (cond ((math-lessp-float (calcFunc-re x) fprec)
156 (if (math-lessp-float (calcFunc-re x) nfprec)
157 (math-neg (math-div
158 (math-pi)
159 (math-mul (math-gammap1-raw
160 (math-add (math-neg x)
161 '(float -1 0))
162 fprec nfprec)
163 (math-sin-raw
164 (math-mul (math-pi) x)))))
165 (let ((xplus1 (math-add x '(float 1 0))))
166 (math-div (math-gammap1-raw xplus1 fprec nfprec) xplus1))))
167 ((and (math-realp x)
168 (math-lessp-float '(float 736276 0) x))
169 (math-overflow))
170 (t ; re(x) now >= 10.0
171 (let ((xinv (math-div 1 x))
172 (lnx (math-ln-raw x)))
173 (math-mul (math-sqrt-two-pi)
174 (math-exp-raw
175 (math-gamma-series
176 (math-sub (math-mul (math-add x '(float 5 -1))
177 lnx)
178 x)
179 xinv
180 (math-sqr xinv)
181 '(float 0 0)
182 2)))))))
183
184 (defun math-gamma-series (sum x xinvsqr oterm n)
185 (math-working "gamma" sum)
186 (let* ((bn (math-bernoulli-number n))
187 (term (math-mul (math-div-float (math-float (nth 1 bn))
188 (math-float (* (nth 2 bn)
189 (* n (1- n)))))
190 x))
191 (next (math-add sum term)))
192 (if (math-nearly-equal sum next)
193 next
194 (if (> n (* 2 calc-internal-prec))
195 (progn
196 ;; Need this because series eventually diverges for large enough n.
197 (calc-record-why
198 "*Gamma computation stopped early, not all digits may be valid")
199 next)
200 (math-gamma-series next (math-mul x xinvsqr) xinvsqr term (+ n 2))))))
201
202
203 ;;; Incomplete gamma function.
204
205 (defvar math-current-gamma-value nil)
206 (defun calcFunc-gammaP (a x)
207 (if (equal x '(var inf var-inf))
208 '(float 1 0)
209 (math-inexact-result)
210 (or (Math-numberp a) (math-reject-arg a 'numberp))
211 (or (math-numberp x) (math-reject-arg x 'numberp))
212 (if (and (math-num-integerp a)
213 (integerp (setq a (math-trunc a)))
214 (> a 0) (< a 20))
215 (math-sub 1 (calcFunc-gammaQ a x))
216 (let ((math-current-gamma-value (calcFunc-gamma a)))
217 (math-div (calcFunc-gammag a x) math-current-gamma-value)))))
218
219 (defun calcFunc-gammaQ (a x)
220 (if (equal x '(var inf var-inf))
221 '(float 0 0)
222 (math-inexact-result)
223 (or (Math-numberp a) (math-reject-arg a 'numberp))
224 (or (math-numberp x) (math-reject-arg x 'numberp))
225 (if (and (math-num-integerp a)
226 (integerp (setq a (math-trunc a)))
227 (> a 0) (< a 20))
228 (let ((n 0)
229 (sum '(float 1 0))
230 (term '(float 1 0)))
231 (math-with-extra-prec 1
232 (while (< (setq n (1+ n)) a)
233 (setq term (math-div (math-mul term x) n)
234 sum (math-add sum term))
235 (math-working "gamma" sum))
236 (math-mul sum (calcFunc-exp (math-neg x)))))
237 (let ((math-current-gamma-value (calcFunc-gamma a)))
238 (math-div (calcFunc-gammaG a x) math-current-gamma-value)))))
239
240 (defun calcFunc-gammag (a x)
241 (if (equal x '(var inf var-inf))
242 (calcFunc-gamma a)
243 (math-inexact-result)
244 (or (Math-numberp a) (math-reject-arg a 'numberp))
245 (or (Math-numberp x) (math-reject-arg x 'numberp))
246 (math-with-extra-prec 2
247 (setq a (math-float a))
248 (setq x (math-float x))
249 (if (or (math-negp (calcFunc-re a))
250 (math-lessp-float (calcFunc-re x)
251 (math-add-float (calcFunc-re a)
252 '(float 1 0))))
253 (math-inc-gamma-series a x)
254 (math-sub (or math-current-gamma-value (calcFunc-gamma a))
255 (math-inc-gamma-cfrac a x))))))
256
257 (defun calcFunc-gammaG (a x)
258 (if (equal x '(var inf var-inf))
259 '(float 0 0)
260 (math-inexact-result)
261 (or (Math-numberp a) (math-reject-arg a 'numberp))
262 (or (Math-numberp x) (math-reject-arg x 'numberp))
263 (math-with-extra-prec 2
264 (setq a (math-float a))
265 (setq x (math-float x))
266 (if (or (math-negp (calcFunc-re a))
267 (math-lessp-float (calcFunc-re x)
268 (math-add-float (math-abs-approx a)
269 '(float 1 0))))
270 (math-sub (or math-current-gamma-value (calcFunc-gamma a))
271 (math-inc-gamma-series a x))
272 (math-inc-gamma-cfrac a x)))))
273
274 (defun math-inc-gamma-series (a x)
275 (if (Math-zerop x)
276 '(float 0 0)
277 (math-mul (math-exp-raw (math-sub (math-mul a (math-ln-raw x)) x))
278 (math-with-extra-prec 2
279 (let ((start (math-div '(float 1 0) a)))
280 (math-inc-gamma-series-step start start a x))))))
281
282 (defun math-inc-gamma-series-step (sum term a x)
283 (math-working "gamma" sum)
284 (setq a (math-add a '(float 1 0))
285 term (math-div (math-mul term x) a))
286 (let ((next (math-add sum term)))
287 (if (math-nearly-equal sum next)
288 next
289 (math-inc-gamma-series-step next term a x))))
290
291 (defun math-inc-gamma-cfrac (a x)
292 (if (Math-zerop x)
293 (or math-current-gamma-value (calcFunc-gamma a))
294 (math-mul (math-exp-raw (math-sub (math-mul a (math-ln-raw x)) x))
295 (math-inc-gamma-cfrac-step '(float 1 0) x
296 '(float 0 0) '(float 1 0)
297 '(float 1 0) '(float 1 0) '(float 0 0)
298 a x))))
299
300 (defun math-inc-gamma-cfrac-step (a0 a1 b0 b1 n fac g a x)
301 (let ((ana (math-sub n a))
302 (anf (math-mul n fac)))
303 (setq n (math-add n '(float 1 0))
304 a0 (math-mul (math-add a1 (math-mul a0 ana)) fac)
305 b0 (math-mul (math-add b1 (math-mul b0 ana)) fac)
306 a1 (math-add (math-mul x a0) (math-mul anf a1))
307 b1 (math-add (math-mul x b0) (math-mul anf b1)))
308 (if (math-zerop a1)
309 (math-inc-gamma-cfrac-step a0 a1 b0 b1 n fac g a x)
310 (setq fac (math-div '(float 1 0) a1))
311 (let ((next (math-mul b1 fac)))
312 (math-working "gamma" next)
313 (if (math-nearly-equal next g)
314 next
315 (math-inc-gamma-cfrac-step a0 a1 b0 b1 n fac next a x))))))
316
317
318 ;;; Error function.
319
320 (defun calcFunc-erf (x)
321 (if (equal x '(var inf var-inf))
322 '(float 1 0)
323 (if (equal x '(neg (var inf var-inf)))
324 '(float -1 0)
325 (if (Math-zerop x)
326 x
327 (let ((math-current-gamma-value (math-sqrt-pi)))
328 (math-to-same-complex-quad
329 (math-div (calcFunc-gammag '(float 5 -1)
330 (math-sqr (math-to-complex-quad-one x)))
331 math-current-gamma-value)
332 x))))))
333
334 (defun calcFunc-erfc (x)
335 (if (equal x '(var inf var-inf))
336 '(float 0 0)
337 (if (math-posp x)
338 (let ((math-current-gamma-value (math-sqrt-pi)))
339 (math-div (calcFunc-gammaG '(float 5 -1) (math-sqr x))
340 math-current-gamma-value))
341 (math-sub 1 (calcFunc-erf x)))))
342
343 (defun math-to-complex-quad-one (x)
344 (if (eq (car-safe x) 'polar) (setq x (math-complex x)))
345 (if (eq (car-safe x) 'cplx)
346 (list 'cplx (math-abs (nth 1 x)) (math-abs (nth 2 x)))
347 x))
348
349 (defun math-to-same-complex-quad (x y)
350 (if (eq (car-safe y) 'cplx)
351 (if (eq (car-safe x) 'cplx)
352 (list 'cplx
353 (if (math-negp (nth 1 y)) (math-neg (nth 1 x)) (nth 1 x))
354 (if (math-negp (nth 2 y)) (math-neg (nth 2 x)) (nth 2 x)))
355 (if (math-negp (nth 1 y)) (math-neg x) x))
356 (if (math-negp y)
357 (if (eq (car-safe x) 'cplx)
358 (list 'cplx (math-neg (nth 1 x)) (nth 2 x))
359 (math-neg x))
360 x)))
361
362
363 ;;; Beta function.
364
365 (defun calcFunc-beta (a b)
366 (if (math-num-integerp a)
367 (let ((am (math-add a -1)))
368 (or (math-numberp b) (math-reject-arg b 'numberp))
369 (math-div 1 (math-mul b (calcFunc-choose (math-add b am) am))))
370 (if (math-num-integerp b)
371 (calcFunc-beta b a)
372 (math-div (math-mul (calcFunc-gamma a) (calcFunc-gamma b))
373 (calcFunc-gamma (math-add a b))))))
374
375
376 ;;; Incomplete beta function.
377
378 (defvar math-current-beta-value nil)
379 (defun calcFunc-betaI (x a b)
380 (cond ((math-zerop x)
381 '(float 0 0))
382 ((math-equal-int x 1)
383 '(float 1 0))
384 ((or (math-zerop a)
385 (and (math-num-integerp a)
386 (math-negp a)))
387 (if (or (math-zerop b)
388 (and (math-num-integerp b)
389 (math-negp b)))
390 (math-reject-arg b 'range)
391 '(float 1 0)))
392 ((or (math-zerop b)
393 (and (math-num-integerp b)
394 (math-negp b)))
395 '(float 0 0))
396 ((not (math-numberp a)) (math-reject-arg a 'numberp))
397 ((not (math-numberp b)) (math-reject-arg b 'numberp))
398 ((math-inexact-result))
399 (t (let ((math-current-beta-value (calcFunc-beta a b)))
400 (math-div (calcFunc-betaB x a b) math-current-beta-value)))))
401
402 (defun calcFunc-betaB (x a b)
403 (cond
404 ((math-zerop x)
405 '(float 0 0))
406 ((math-equal-int x 1)
407 (calcFunc-beta a b))
408 ((not (math-numberp x)) (math-reject-arg x 'numberp))
409 ((not (math-numberp a)) (math-reject-arg a 'numberp))
410 ((not (math-numberp b)) (math-reject-arg b 'numberp))
411 ((math-zerop a) (math-reject-arg a 'nonzerop))
412 ((math-zerop b) (math-reject-arg b 'nonzerop))
413 ((and (math-num-integerp b)
414 (if (math-negp b)
415 (math-reject-arg b 'range)
416 (Math-natnum-lessp (setq b (math-trunc b)) 20)))
417 (and calc-symbolic-mode (or (math-floatp a) (math-floatp b))
418 (math-inexact-result))
419 (math-mul
420 (math-with-extra-prec 2
421 (let* ((i 0)
422 (term 1)
423 (sum (math-div term a)))
424 (while (< (setq i (1+ i)) b)
425 (setq term (math-mul (math-div (math-mul term (- i b)) i) x)
426 sum (math-add sum (math-div term (math-add a i))))
427 (math-working "beta" sum))
428 sum))
429 (math-pow x a)))
430 ((and (math-num-integerp a)
431 (if (math-negp a)
432 (math-reject-arg a 'range)
433 (Math-natnum-lessp (setq a (math-trunc a)) 20)))
434 (math-sub (or math-current-beta-value (calcFunc-beta a b))
435 (calcFunc-betaB (math-sub 1 x) b a)))
436 (t
437 (math-inexact-result)
438 (math-with-extra-prec 2
439 (setq x (math-float x))
440 (setq a (math-float a))
441 (setq b (math-float b))
442 (let ((bt (math-exp-raw (math-add (math-mul a (math-ln-raw x))
443 (math-mul b (math-ln-raw
444 (math-sub '(float 1 0)
445 x)))))))
446 (if (Math-lessp x (math-div (math-add a '(float 1 0))
447 (math-add (math-add a b) '(float 2 0))))
448 (math-div (math-mul bt (math-beta-cfrac a b x)) a)
449 (math-sub (or math-current-beta-value (calcFunc-beta a b))
450 (math-div (math-mul bt
451 (math-beta-cfrac b a (math-sub 1 x)))
452 b))))))))
453
454 (defun math-beta-cfrac (a b x)
455 (let ((qab (math-add a b))
456 (qap (math-add a '(float 1 0)))
457 (qam (math-add a '(float -1 0))))
458 (math-beta-cfrac-step '(float 1 0)
459 (math-sub '(float 1 0)
460 (math-div (math-mul qab x) qap))
461 '(float 1 0) '(float 1 0)
462 '(float 1 0)
463 qab qap qam a b x)))
464
465 (defun math-beta-cfrac-step (az bz am bm m qab qap qam a b x)
466 (let* ((two-m (math-mul m '(float 2 0)))
467 (d (math-div (math-mul (math-mul (math-sub b m) m) x)
468 (math-mul (math-add qam two-m) (math-add a two-m))))
469 (ap (math-add az (math-mul d am)))
470 (bp (math-add bz (math-mul d bm)))
471 (d2 (math-neg
472 (math-div (math-mul (math-mul (math-add a m) (math-add qab m)) x)
473 (math-mul (math-add qap two-m) (math-add a two-m)))))
474 (app (math-add ap (math-mul d2 az)))
475 (bpp (math-add bp (math-mul d2 bz)))
476 (next (math-div app bpp)))
477 (math-working "beta" next)
478 (if (math-nearly-equal next az)
479 next
480 (math-beta-cfrac-step next '(float 1 0)
481 (math-div ap bpp) (math-div bp bpp)
482 (math-add m '(float 1 0))
483 qab qap qam a b x))))
484
485
486 ;;; Bessel functions.
487
488 ;;; Should generalize this to handle arbitrary precision!
489
490 (defun calcFunc-besJ (v x)
491 (or (math-numberp v) (math-reject-arg v 'numberp))
492 (or (math-numberp x) (math-reject-arg x 'numberp))
493 (let ((calc-internal-prec (min 8 calc-internal-prec)))
494 (math-with-extra-prec 3
495 (setq x (math-float (math-normalize x)))
496 (setq v (math-float (math-normalize v)))
497 (cond ((math-zerop x)
498 (if (math-zerop v)
499 '(float 1 0)
500 '(float 0 0)))
501 ((math-inexact-result))
502 ((not (math-num-integerp v))
503 (let ((start (math-div 1 (calcFunc-fact v))))
504 (math-mul (math-besJ-series start start
505 0
506 (math-mul '(float -25 -2)
507 (math-sqr x))
508 v)
509 (math-pow (math-div x 2) v))))
510 ((math-negp (setq v (math-trunc v)))
511 (if (math-oddp v)
512 (math-neg (calcFunc-besJ (math-neg v) x))
513 (calcFunc-besJ (math-neg v) x)))
514 ((eq v 0)
515 (math-besJ0 x))
516 ((eq v 1)
517 (math-besJ1 x))
518 ((Math-lessp v (math-abs-approx x))
519 (let ((j 0)
520 (bjm (math-besJ0 x))
521 (bj (math-besJ1 x))
522 (two-over-x (math-div 2 x))
523 bjp)
524 (while (< (setq j (1+ j)) v)
525 (setq bjp (math-sub (math-mul (math-mul j two-over-x) bj)
526 bjm)
527 bjm bj
528 bj bjp))
529 bj))
530 (t
531 (if (Math-lessp 100 v) (math-reject-arg v 'range))
532 (let* ((j (logior (+ v (math-isqrt-small (* 40 v))) 1))
533 (two-over-x (math-div 2 x))
534 (jsum nil)
535 (bjp '(float 0 0))
536 (sum '(float 0 0))
537 (bj '(float 1 0))
538 bjm ans)
539 (while (> (setq j (1- j)) 0)
540 (setq bjm (math-sub (math-mul (math-mul j two-over-x) bj)
541 bjp)
542 bjp bj
543 bj bjm)
544 (if (> (nth 2 (math-abs-approx bj)) 10)
545 (setq bj (math-mul bj '(float 1 -10))
546 bjp (math-mul bjp '(float 1 -10))
547 ans (and ans (math-mul ans '(float 1 -10)))
548 sum (math-mul sum '(float 1 -10))))
549 (or (setq jsum (not jsum))
550 (setq sum (math-add sum bj)))
551 (if (= j v)
552 (setq ans bjp)))
553 (math-div ans (math-sub (math-mul 2 sum) bj))))))))
554
555 (defun math-besJ-series (sum term k zz vk)
556 (math-working "besJ" sum)
557 (setq k (1+ k)
558 vk (math-add 1 vk)
559 term (math-div (math-mul term zz) (math-mul k vk)))
560 (let ((next (math-add sum term)))
561 (if (math-nearly-equal next sum)
562 next
563 (math-besJ-series next term k zz vk))))
564
565 (defun math-besJ0 (x &optional yflag)
566 (cond ((and (not yflag) (math-negp (calcFunc-re x)))
567 (math-besJ0 (math-neg x)))
568 ((Math-lessp '(float 8 0) (math-abs-approx x))
569 (let* ((z (math-div '(float 8 0) x))
570 (y (math-sqr z))
571 (xx (math-add x
572 (math-read-number-simple "-0.785398164")))
573 (a1 (math-poly-eval y
574 (list
575 (math-read-number-simple "0.0000002093887211")
576 (math-read-number-simple "-0.000002073370639")
577 (math-read-number-simple "0.00002734510407")
578 (math-read-number-simple "-0.001098628627")
579 '(float 1 0))))
580 (a2 (math-poly-eval y
581 (list
582 (math-read-number-simple "-0.0000000934935152")
583 (math-read-number-simple "0.0000007621095161")
584 (math-read-number-simple "-0.000006911147651")
585 (math-read-number-simple "0.0001430488765")
586 (math-read-number-simple "-0.01562499995"))))
587 (sc (math-sin-cos-raw xx)))
588 (if yflag
589 (setq sc (cons (math-neg (cdr sc)) (car sc))))
590 (math-mul (math-sqrt
591 (math-div (math-read-number-simple "0.636619722")
592 x))
593 (math-sub (math-mul (cdr sc) a1)
594 (math-mul (car sc) (math-mul z a2))))))
595 (t
596 (let ((y (math-sqr x)))
597 (math-div (math-poly-eval y
598 (list
599 (math-read-number-simple "-184.9052456")
600 (math-read-number-simple "77392.33017")
601 (math-read-number-simple "-11214424.18")
602 (math-read-number-simple "651619640.7")
603 (math-read-number-simple "-13362590354.0")
604 (math-read-number-simple "57568490574.0")))
605 (math-poly-eval y
606 (list
607 '(float 1 0)
608 (math-read-number-simple "267.8532712")
609 (math-read-number-simple "59272.64853")
610 (math-read-number-simple "9494680.718")
611 (math-read-number-simple "1029532985.0")
612 (math-read-number-simple "57568490411.0"))))))))
613
614 (defun math-besJ1 (x &optional yflag)
615 (cond ((and (math-negp (calcFunc-re x)) (not yflag))
616 (math-neg (math-besJ1 (math-neg x))))
617 ((Math-lessp '(float 8 0) (math-abs-approx x))
618 (let* ((z (math-div '(float 8 0) x))
619 (y (math-sqr z))
620 (xx (math-add x (math-read-number-simple "-2.356194491")))
621 (a1 (math-poly-eval y
622 (list
623 (math-read-number-simple "-0.000000240337019")
624 (math-read-number-simple "0.000002457520174")
625 (math-read-number-simple "-0.00003516396496")
626 '(float 183105 -8)
627 '(float 1 0))))
628 (a2 (math-poly-eval y
629 (list
630 (math-read-number-simple "0.000000105787412")
631 (math-read-number-simple "-0.00000088228987")
632 (math-read-number-simple "0.000008449199096")
633 (math-read-number-simple "-0.0002002690873")
634 (math-read-number-simple "0.04687499995"))))
635 (sc (math-sin-cos-raw xx)))
636 (if yflag
637 (setq sc (cons (math-neg (cdr sc)) (car sc)))
638 (if (math-negp x)
639 (setq sc (cons (math-neg (car sc)) (math-neg (cdr sc))))))
640 (math-mul (math-sqrt (math-div
641 (math-read-number-simple "0.636619722")
642 x))
643 (math-sub (math-mul (cdr sc) a1)
644 (math-mul (car sc) (math-mul z a2))))))
645 (t
646 (let ((y (math-sqr x)))
647 (math-mul
648 x
649 (math-div (math-poly-eval y
650 (list
651 (math-read-number-simple "-30.16036606")
652 (math-read-number-simple "15704.4826")
653 (math-read-number-simple "-2972611.439")
654 (math-read-number-simple "242396853.1")
655 (math-read-number-simple "-7895059235.0")
656 (math-read-number-simple "72362614232.0")))
657 (math-poly-eval y
658 (list
659 '(float 1 0)
660 (math-read-number-simple "376.9991397")
661 (math-read-number-simple "99447.43394")
662 (math-read-number-simple "18583304.74")
663 (math-read-number-simple "2300535178.0")
664 (math-read-number-simple "144725228442.0")))))))))
665
666 (defun calcFunc-besY (v x)
667 (math-inexact-result)
668 (or (math-numberp v) (math-reject-arg v 'numberp))
669 (or (math-numberp x) (math-reject-arg x 'numberp))
670 (let ((calc-internal-prec (min 8 calc-internal-prec)))
671 (math-with-extra-prec 3
672 (setq x (math-float (math-normalize x)))
673 (setq v (math-float (math-normalize v)))
674 (cond ((not (math-num-integerp v))
675 (let ((sc (math-sin-cos-raw (math-mul v (math-pi)))))
676 (math-div (math-sub (math-mul (calcFunc-besJ v x) (cdr sc))
677 (calcFunc-besJ (math-neg v) x))
678 (car sc))))
679 ((math-negp (setq v (math-trunc v)))
680 (if (math-oddp v)
681 (math-neg (calcFunc-besY (math-neg v) x))
682 (calcFunc-besY (math-neg v) x)))
683 ((eq v 0)
684 (math-besY0 x))
685 ((eq v 1)
686 (math-besY1 x))
687 (t
688 (let ((j 0)
689 (bym (math-besY0 x))
690 (by (math-besY1 x))
691 (two-over-x (math-div 2 x))
692 byp)
693 (while (< (setq j (1+ j)) v)
694 (setq byp (math-sub (math-mul (math-mul j two-over-x) by)
695 bym)
696 bym by
697 by byp))
698 by))))))
699
700 (defun math-besY0 (x)
701 (cond ((Math-lessp (math-abs-approx x) '(float 8 0))
702 (let ((y (math-sqr x)))
703 (math-add
704 (math-div (math-poly-eval y
705 (list
706 (math-read-number-simple "228.4622733")
707 (math-read-number-simple "-86327.92757")
708 (math-read-number-simple "10879881.29")
709 (math-read-number-simple "-512359803.6")
710 (math-read-number-simple "7062834065.0")
711 (math-read-number-simple "-2957821389.0")))
712 (math-poly-eval y
713 (list
714 '(float 1 0)
715 (math-read-number-simple "226.1030244")
716 (math-read-number-simple "47447.2647")
717 (math-read-number-simple "7189466.438")
718 (math-read-number-simple "745249964.8")
719 (math-read-number-simple "40076544269.0"))))
720 (math-mul (math-read-number-simple "0.636619772")
721 (math-mul (math-besJ0 x) (math-ln-raw x))))))
722 ((math-negp (calcFunc-re x))
723 (math-add (math-besJ0 (math-neg x) t)
724 (math-mul '(cplx 0 2)
725 (math-besJ0 (math-neg x)))))
726 (t
727 (math-besJ0 x t))))
728
729 (defun math-besY1 (x)
730 (cond ((Math-lessp (math-abs-approx x) '(float 8 0))
731 (let ((y (math-sqr x)))
732 (math-add
733 (math-mul
734 x
735 (math-div (math-poly-eval y
736 (list
737 (math-read-number-simple "8511.937935")
738 (math-read-number-simple "-4237922.726")
739 (math-read-number-simple "734926455.1")
740 (math-read-number-simple "-51534381390.0")
741 (math-read-number-simple "1275274390000.0")
742 (math-read-number-simple "-4900604943000.0")))
743 (math-poly-eval y
744 (list
745 '(float 1 0)
746 (math-read-number-simple "354.9632885")
747 (math-read-number-simple "102042.605")
748 (math-read-number-simple "22459040.02")
749 (math-read-number-simple "3733650367.0")
750 (math-read-number-simple "424441966400.0")
751 (math-read-number-simple "24995805700000.0")))))
752 (math-mul (math-read-number-simple "0.636619772")
753 (math-sub (math-mul (math-besJ1 x) (math-ln-raw x))
754 (math-div 1 x))))))
755 ((math-negp (calcFunc-re x))
756 (math-neg
757 (math-add (math-besJ1 (math-neg x) t)
758 (math-mul '(cplx 0 2)
759 (math-besJ1 (math-neg x))))))
760 (t
761 (math-besJ1 x t))))
762
763 (defun math-poly-eval (x coefs)
764 (let ((accum (car coefs)))
765 (while (setq coefs (cdr coefs))
766 (setq accum (math-add (car coefs) (math-mul accum x))))
767 accum))
768
769
770 ;;;; Bernoulli and Euler polynomials and numbers.
771
772 (defun calcFunc-bern (n &optional x)
773 (if (and x (not (math-zerop x)))
774 (if (and calc-symbolic-mode (math-floatp x))
775 (math-inexact-result)
776 (math-build-polynomial-expr (math-bernoulli-coefs n) x))
777 (or (math-num-natnump n) (math-reject-arg n 'natnump))
778 (if (consp n)
779 (progn
780 (math-inexact-result)
781 (math-float (math-bernoulli-number (math-trunc n))))
782 (math-bernoulli-number n))))
783
784 (defun calcFunc-euler (n &optional x)
785 (or (math-num-natnump n) (math-reject-arg n 'natnump))
786 (if x
787 (let* ((n1 (math-add n 1))
788 (coefs (math-bernoulli-coefs n1))
789 (fac (math-div (math-pow 2 n1) n1))
790 (k -1)
791 (x1 (math-div (math-add x 1) 2))
792 (x2 (math-div x 2)))
793 (if (math-numberp x)
794 (if (and calc-symbolic-mode (math-floatp x))
795 (math-inexact-result)
796 (math-mul fac
797 (math-sub (math-build-polynomial-expr coefs x1)
798 (math-build-polynomial-expr coefs x2))))
799 (calcFunc-collect
800 (math-reduce-vec
801 'math-add
802 (cons 'vec
803 (mapcar (function
804 (lambda (c)
805 (setq k (1+ k))
806 (math-mul (math-mul fac c)
807 (math-sub (math-pow x1 k)
808 (math-pow x2 k)))))
809 coefs)))
810 x)))
811 (math-mul (math-pow 2 n)
812 (if (consp n)
813 (progn
814 (math-inexact-result)
815 (calcFunc-euler n '(float 5 -1)))
816 (calcFunc-euler n '(frac 1 2))))))
817
818 (defvar math-bernoulli-b-cache
819 (list
820 (list 'frac
821 -174611
822 (math-read-number-simple "802857662698291200000"))
823 (list 'frac
824 43867
825 (math-read-number-simple "5109094217170944000"))
826 (list 'frac
827 -3617
828 (math-read-number-simple "10670622842880000"))
829 (list 'frac
830 1
831 (math-read-number-simple "74724249600"))
832 (list 'frac
833 -691
834 (math-read-number-simple "1307674368000"))
835 (list 'frac
836 1
837 (math-read-number-simple "47900160"))
838 (list 'frac
839 -1
840 (math-read-number-simple "1209600"))
841 (list 'frac
842 1
843 30240)
844 (list 'frac
845 -1
846 720)
847 (list 'frac
848 1
849 12)
850 1 ))
851
852 (defvar math-bernoulli-B-cache
853 '((frac -174611 330) (frac 43867 798)
854 (frac -3617 510) (frac 7 6) (frac -691 2730)
855 (frac 5 66) (frac -1 30) (frac 1 42)
856 (frac -1 30) (frac 1 6) 1 ))
857
858 (defvar math-bernoulli-cache-size 11)
859 (defun math-bernoulli-coefs (n)
860 (let* ((coefs (list (calcFunc-bern n)))
861 (nn (math-trunc n))
862 (k nn)
863 (term nn)
864 coef
865 (calc-prefer-frac (or (integerp n) calc-prefer-frac)))
866 (while (>= (setq k (1- k)) 0)
867 (setq term (math-div term (- nn k))
868 coef (math-mul term (math-bernoulli-number k))
869 coefs (cons (if (consp n) (math-float coef) coef) coefs)
870 term (math-mul term k)))
871 (nreverse coefs)))
872
873 (defun math-bernoulli-number (n)
874 (if (= (% n 2) 1)
875 (if (= n 1)
876 '(frac -1 2)
877 0)
878 (setq n (/ n 2))
879 (while (>= n math-bernoulli-cache-size)
880 (let* ((sum 0)
881 (nk 1) ; nk = n-k+1
882 (fact 1) ; fact = (n-k+1)!
883 ofact
884 (p math-bernoulli-b-cache)
885 (calc-prefer-frac t))
886 (math-working "bernoulli B" (* 2 math-bernoulli-cache-size))
887 (while p
888 (setq nk (+ nk 2)
889 ofact fact
890 fact (math-mul fact (* nk (1- nk)))
891 sum (math-add sum (math-div (car p) fact))
892 p (cdr p)))
893 (setq ofact (math-mul ofact (1- nk))
894 sum (math-sub (math-div '(frac 1 2) ofact) sum)
895 math-bernoulli-b-cache (cons sum math-bernoulli-b-cache)
896 math-bernoulli-B-cache (cons (math-mul sum ofact)
897 math-bernoulli-B-cache)
898 math-bernoulli-cache-size (1+ math-bernoulli-cache-size))))
899 (nth (- math-bernoulli-cache-size n 1) math-bernoulli-B-cache)))
900
901 ;;; Bn = n! bn
902 ;;; bn = - sum_k=0^n-1 bk / (n-k+1)!
903
904 ;;; A faster method would be to use "tangent numbers", c.f., Concrete
905 ;;; Mathematics pg. 273.
906
907
908 ;;; Probability distributions.
909
910 ;;; Binomial.
911 (defun calcFunc-utpb (x n p)
912 (if math-expand-formulas
913 (math-normalize (list 'calcFunc-betaI p x (list '+ (list '- n x) 1)))
914 (calcFunc-betaI p x (math-add (math-sub n x) 1))))
915 (put 'calcFunc-utpb 'math-expandable t)
916
917 (defun calcFunc-ltpb (x n p)
918 (math-sub 1 (calcFunc-utpb x n p)))
919 (put 'calcFunc-ltpb 'math-expandable t)
920
921 ;;; Chi-square.
922 (defun calcFunc-utpc (chisq v)
923 (if math-expand-formulas
924 (math-normalize (list 'calcFunc-gammaQ (list '/ v 2) (list '/ chisq 2)))
925 (calcFunc-gammaQ (math-div v 2) (math-div chisq 2))))
926 (put 'calcFunc-utpc 'math-expandable t)
927
928 (defun calcFunc-ltpc (chisq v)
929 (if math-expand-formulas
930 (math-normalize (list 'calcFunc-gammaP (list '/ v 2) (list '/ chisq 2)))
931 (calcFunc-gammaP (math-div v 2) (math-div chisq 2))))
932 (put 'calcFunc-ltpc 'math-expandable t)
933
934 ;;; F-distribution.
935 (defun calcFunc-utpf (f v1 v2)
936 (if math-expand-formulas
937 (math-normalize (list 'calcFunc-betaI
938 (list '/ v2 (list '+ v2 (list '* v1 f)))
939 (list '/ v2 2)
940 (list '/ v1 2)))
941 (calcFunc-betaI (math-div v2 (math-add v2 (math-mul v1 f)))
942 (math-div v2 2)
943 (math-div v1 2))))
944 (put 'calcFunc-utpf 'math-expandable t)
945
946 (defun calcFunc-ltpf (f v1 v2)
947 (math-sub 1 (calcFunc-utpf f v1 v2)))
948 (put 'calcFunc-ltpf 'math-expandable t)
949
950 ;;; Normal.
951 (defun calcFunc-utpn (x mean sdev)
952 (if math-expand-formulas
953 (math-normalize
954 (list '/
955 (list '+ 1
956 (list 'calcFunc-erf
957 (list '/ (list '- mean x)
958 (list '* sdev (list 'calcFunc-sqrt 2)))))
959 2))
960 (math-mul (math-add '(float 1 0)
961 (calcFunc-erf
962 (math-div (math-sub mean x)
963 (math-mul sdev (math-sqrt-2)))))
964 '(float 5 -1))))
965 (put 'calcFunc-utpn 'math-expandable t)
966
967 (defun calcFunc-ltpn (x mean sdev)
968 (if math-expand-formulas
969 (math-normalize
970 (list '/
971 (list '+ 1
972 (list 'calcFunc-erf
973 (list '/ (list '- x mean)
974 (list '* sdev (list 'calcFunc-sqrt 2)))))
975 2))
976 (math-mul (math-add '(float 1 0)
977 (calcFunc-erf
978 (math-div (math-sub x mean)
979 (math-mul sdev (math-sqrt-2)))))
980 '(float 5 -1))))
981 (put 'calcFunc-ltpn 'math-expandable t)
982
983 ;;; Poisson.
984 (defun calcFunc-utpp (n x)
985 (if math-expand-formulas
986 (math-normalize (list 'calcFunc-gammaP x n))
987 (calcFunc-gammaP x n)))
988 (put 'calcFunc-utpp 'math-expandable t)
989
990 (defun calcFunc-ltpp (n x)
991 (if math-expand-formulas
992 (math-normalize (list 'calcFunc-gammaQ x n))
993 (calcFunc-gammaQ x n)))
994 (put 'calcFunc-ltpp 'math-expandable t)
995
996 ;;; Student's t. (As defined in Abramowitz & Stegun and Numerical Recipes.)
997 (defun calcFunc-utpt (tt v)
998 (if math-expand-formulas
999 (math-normalize (list 'calcFunc-betaI
1000 (list '/ v (list '+ v (list '^ tt 2)))
1001 (list '/ v 2)
1002 '(float 5 -1)))
1003 (calcFunc-betaI (math-div v (math-add v (math-sqr tt)))
1004 (math-div v 2)
1005 '(float 5 -1))))
1006 (put 'calcFunc-utpt 'math-expandable t)
1007
1008 (defun calcFunc-ltpt (tt v)
1009 (math-sub 1 (calcFunc-utpt tt v)))
1010 (put 'calcFunc-ltpt 'math-expandable t)
1011
1012 (provide 'calc-funcs)
1013
1014 ;;; arch-tag: 421ddb7a-550f-4dda-a31c-06638ebfc43a
1015 ;;; calc-funcs.el ends here
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