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12f1951a CW |
1 | ;;; calc-funcs.el --- well-known functions for Calc |
2 | ||
f8fc5256 | 3 | ;; Copyright (C) 1990, 1991, 1992, 1993, 2001, 2005 Free Software Foundation, Inc. |
12f1951a CW |
4 | |
5 | ;; Author: David Gillespie <daveg@synaptics.com> | |
300f28d3 | 6 | ;; Maintainer: Jay Belanger <belanger@truman.edu> |
136211a9 EZ |
7 | |
8 | ;; This file is part of GNU Emacs. | |
9 | ||
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. | |
16 | ||
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. | |
24 | ||
12f1951a | 25 | ;;; Commentary: |
136211a9 | 26 | |
12f1951a | 27 | ;;; Code: |
136211a9 EZ |
28 | |
29 | ;; This file is autoloaded from calc-ext.el. | |
136211a9 | 30 | |
300f28d3 | 31 | (require 'calc-ext) |
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32 | (require 'calc-macs) |
33 | ||
136211a9 EZ |
34 | (defun calc-inc-gamma (arg) |
35 | (interactive "P") | |
36 | (calc-slow-wrapper | |
37 | (if (calc-is-inverse) | |
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) | |
bf77c646 | 43 | (calc-binary-op "gamP" 'calcFunc-gammaP arg))))) |
136211a9 EZ |
44 | |
45 | (defun calc-erf (arg) | |
46 | (interactive "P") | |
47 | (calc-slow-wrapper | |
48 | (if (calc-is-inverse) | |
49 | (calc-unary-op "erfc" 'calcFunc-erfc arg) | |
bf77c646 | 50 | (calc-unary-op "erf" 'calcFunc-erf arg)))) |
136211a9 EZ |
51 | |
52 | (defun calc-erfc (arg) | |
53 | (interactive "P") | |
54 | (calc-invert-func) | |
bf77c646 | 55 | (calc-erf arg)) |
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56 | |
57 | (defun calc-beta (arg) | |
58 | (interactive "P") | |
59 | (calc-slow-wrapper | |
bf77c646 | 60 | (calc-binary-op "beta" 'calcFunc-beta arg))) |
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61 | |
62 | (defun calc-inc-beta () | |
63 | (interactive) | |
64 | (calc-slow-wrapper | |
65 | (if (calc-is-hyperbolic) | |
66 | (calc-enter-result 3 "betB" (cons 'calcFunc-betaB (calc-top-list-n 3))) | |
bf77c646 | 67 | (calc-enter-result 3 "betI" (cons 'calcFunc-betaI (calc-top-list-n 3)))))) |
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68 | |
69 | (defun calc-bessel-J (arg) | |
70 | (interactive "P") | |
71 | (calc-slow-wrapper | |
bf77c646 | 72 | (calc-binary-op "besJ" 'calcFunc-besJ arg))) |
136211a9 EZ |
73 | |
74 | (defun calc-bessel-Y (arg) | |
75 | (interactive "P") | |
76 | (calc-slow-wrapper | |
bf77c646 | 77 | (calc-binary-op "besY" 'calcFunc-besY arg))) |
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78 | |
79 | (defun calc-bernoulli-number (arg) | |
80 | (interactive "P") | |
81 | (calc-slow-wrapper | |
82 | (if (calc-is-hyperbolic) | |
83 | (calc-binary-op "bern" 'calcFunc-bern arg) | |
bf77c646 | 84 | (calc-unary-op "bern" 'calcFunc-bern arg)))) |
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85 | |
86 | (defun calc-euler-number (arg) | |
87 | (interactive "P") | |
88 | (calc-slow-wrapper | |
89 | (if (calc-is-hyperbolic) | |
90 | (calc-binary-op "eulr" 'calcFunc-euler arg) | |
bf77c646 | 91 | (calc-unary-op "eulr" 'calcFunc-euler arg)))) |
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92 | |
93 | (defun calc-stirling-number (arg) | |
94 | (interactive "P") | |
95 | (calc-slow-wrapper | |
96 | (if (calc-is-hyperbolic) | |
97 | (calc-binary-op "str2" 'calcFunc-stir2 arg) | |
bf77c646 | 98 | (calc-binary-op "str1" 'calcFunc-stir1 arg)))) |
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99 | |
100 | (defun calc-utpb () | |
101 | (interactive) | |
bf77c646 | 102 | (calc-prob-dist "b" 3)) |
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103 | |
104 | (defun calc-utpc () | |
105 | (interactive) | |
bf77c646 | 106 | (calc-prob-dist "c" 2)) |
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107 | |
108 | (defun calc-utpf () | |
109 | (interactive) | |
bf77c646 | 110 | (calc-prob-dist "f" 3)) |
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111 | |
112 | (defun calc-utpn () | |
113 | (interactive) | |
bf77c646 | 114 | (calc-prob-dist "n" 3)) |
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115 | |
116 | (defun calc-utpp () | |
117 | (interactive) | |
bf77c646 | 118 | (calc-prob-dist "p" 2)) |
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119 | |
120 | (defun calc-utpt () | |
121 | (interactive) | |
bf77c646 | 122 | (calc-prob-dist "t" 2)) |
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123 | |
124 | (defun calc-prob-dist (letter nargs) | |
125 | (calc-slow-wrapper | |
126 | (if (calc-is-inverse) | |
127 | (calc-enter-result nargs (concat "ltp" letter) | |
128 | (append (list (intern (concat "calcFunc-ltp" letter)) | |
129 | (calc-top-n 1)) | |
130 | (calc-top-list-n (1- nargs) 2))) | |
131 | (calc-enter-result nargs (concat "utp" letter) | |
132 | (append (list (intern (concat "calcFunc-utp" letter)) | |
133 | (calc-top-n 1)) | |
bf77c646 | 134 | (calc-top-list-n (1- nargs) 2)))))) |
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135 | |
136 | ||
137 | ||
138 | ||
139 | ;;; Sources: Numerical Recipes, Press et al; | |
140 | ;;; Handbook of Mathematical Functions, Abramowitz & Stegun. | |
141 | ||
142 | ||
143 | ;;; Gamma function. | |
144 | ||
145 | (defun calcFunc-gamma (x) | |
146 | (or (math-numberp x) (math-reject-arg x 'numberp)) | |
bf77c646 | 147 | (calcFunc-fact (math-add x -1))) |
136211a9 EZ |
148 | |
149 | (defun math-gammap1-raw (x &optional fprec nfprec) ; compute gamma(1 + x) | |
150 | (or fprec | |
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) | |
155 | (math-neg (math-div | |
156 | (math-pi) | |
157 | (math-mul (math-gammap1-raw | |
158 | (math-add (math-neg x) | |
159 | '(float -1 0)) | |
160 | fprec nfprec) | |
161 | (math-sin-raw | |
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)))) | |
165 | ((and (math-realp x) | |
166 | (math-lessp-float '(float 736276 0) x)) | |
167 | (math-overflow)) | |
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) | |
172 | (math-exp-raw | |
173 | (math-gamma-series | |
174 | (math-sub (math-mul (math-add x '(float 5 -1)) | |
175 | lnx) | |
176 | x) | |
177 | xinv | |
178 | (math-sqr xinv) | |
179 | '(float 0 0) | |
bf77c646 | 180 | 2))))))) |
136211a9 EZ |
181 | |
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) | |
187 | (* n (1- n))))) | |
188 | x)) | |
189 | (next (math-add sum term))) | |
190 | (if (math-nearly-equal sum next) | |
191 | next | |
192 | (if (> n (* 2 calc-internal-prec)) | |
193 | (progn | |
194 | ;; Need this because series eventually diverges for large enough n. | |
195 | (calc-record-why | |
196 | "*Gamma computation stopped early, not all digits may be valid") | |
197 | next) | |
bf77c646 | 198 | (math-gamma-series next (math-mul x xinvsqr) xinvsqr term (+ n 2)))))) |
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199 | |
200 | ||
201 | ;;; Incomplete gamma function. | |
202 | ||
12f1951a | 203 | (defvar math-current-gamma-value nil) |
136211a9 EZ |
204 | (defun calcFunc-gammaP (a x) |
205 | (if (equal x '(var inf var-inf)) | |
206 | '(float 1 0) | |
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))) | |
212 | (> a 0) (< a 20)) | |
213 | (math-sub 1 (calcFunc-gammaQ a x)) | |
214 | (let ((math-current-gamma-value (calcFunc-gamma a))) | |
bf77c646 | 215 | (math-div (calcFunc-gammag a x) math-current-gamma-value))))) |
136211a9 EZ |
216 | |
217 | (defun calcFunc-gammaQ (a x) | |
218 | (if (equal x '(var inf var-inf)) | |
219 | '(float 0 0) | |
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))) | |
225 | (> a 0) (< a 20)) | |
226 | (let ((n 0) | |
227 | (sum '(float 1 0)) | |
228 | (term '(float 1 0))) | |
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))) | |
bf77c646 | 236 | (math-div (calcFunc-gammaG a x) math-current-gamma-value))))) |
136211a9 EZ |
237 | |
238 | (defun calcFunc-gammag (a x) | |
239 | (if (equal x '(var inf var-inf)) | |
240 | (calcFunc-gamma a) | |
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) | |
250 | '(float 1 0)))) | |
251 | (math-inc-gamma-series a x) | |
252 | (math-sub (or math-current-gamma-value (calcFunc-gamma a)) | |
bf77c646 | 253 | (math-inc-gamma-cfrac a x)))))) |
136211a9 EZ |
254 | |
255 | (defun calcFunc-gammaG (a x) | |
256 | (if (equal x '(var inf var-inf)) | |
257 | '(float 0 0) | |
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) | |
267 | '(float 1 0)))) | |
268 | (math-sub (or math-current-gamma-value (calcFunc-gamma a)) | |
269 | (math-inc-gamma-series a x)) | |
bf77c646 | 270 | (math-inc-gamma-cfrac a x))))) |
136211a9 EZ |
271 | |
272 | (defun math-inc-gamma-series (a x) | |
273 | (if (Math-zerop x) | |
274 | '(float 0 0) | |
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))) | |
bf77c646 | 278 | (math-inc-gamma-series-step start start a x)))))) |
136211a9 EZ |
279 | |
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) | |
286 | next | |
bf77c646 | 287 | (math-inc-gamma-series-step next term a x)))) |
136211a9 EZ |
288 | |
289 | (defun math-inc-gamma-cfrac (a x) | |
290 | (if (Math-zerop 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) | |
bf77c646 | 296 | a x)))) |
136211a9 EZ |
297 | |
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))) | |
306 | (if (math-zerop a1) | |
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) | |
312 | next | |
bf77c646 | 313 | (math-inc-gamma-cfrac-step a0 a1 b0 b1 n fac next a x)))))) |
136211a9 EZ |
314 | |
315 | ||
316 | ;;; Error function. | |
317 | ||
318 | (defun calcFunc-erf (x) | |
319 | (if (equal x '(var inf var-inf)) | |
320 | '(float 1 0) | |
321 | (if (equal x '(neg (var inf var-inf))) | |
322 | '(float -1 0) | |
323 | (if (Math-zerop x) | |
324 | x | |
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) | |
bf77c646 | 330 | x)))))) |
136211a9 EZ |
331 | |
332 | (defun calcFunc-erfc (x) | |
333 | (if (equal x '(var inf var-inf)) | |
334 | '(float 0 0) | |
335 | (if (math-posp x) | |
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)) | |
bf77c646 | 339 | (math-sub 1 (calcFunc-erf x))))) |
136211a9 EZ |
340 | |
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))) | |
bf77c646 | 345 | x)) |
136211a9 EZ |
346 | |
347 | (defun math-to-same-complex-quad (x y) | |
348 | (if (eq (car-safe y) 'cplx) | |
349 | (if (eq (car-safe x) 'cplx) | |
350 | (list '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)) | |
354 | (if (math-negp y) | |
355 | (if (eq (car-safe x) 'cplx) | |
356 | (list 'cplx (math-neg (nth 1 x)) (nth 2 x)) | |
357 | (math-neg x)) | |
bf77c646 | 358 | x))) |
136211a9 EZ |
359 | |
360 | ||
361 | ;;; Beta function. | |
362 | ||
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) | |
369 | (calcFunc-beta b a) | |
370 | (math-div (math-mul (calcFunc-gamma a) (calcFunc-gamma b)) | |
bf77c646 | 371 | (calcFunc-gamma (math-add a b)))))) |
136211a9 EZ |
372 | |
373 | ||
374 | ;;; Incomplete beta function. | |
375 | ||
12f1951a | 376 | (defvar math-current-beta-value nil) |
136211a9 EZ |
377 | (defun calcFunc-betaI (x a b) |
378 | (cond ((math-zerop x) | |
379 | '(float 0 0)) | |
380 | ((math-equal-int x 1) | |
381 | '(float 1 0)) | |
382 | ((or (math-zerop a) | |
383 | (and (math-num-integerp a) | |
384 | (math-negp a))) | |
385 | (if (or (math-zerop b) | |
386 | (and (math-num-integerp b) | |
387 | (math-negp b))) | |
388 | (math-reject-arg b 'range) | |
389 | '(float 1 0))) | |
390 | ((or (math-zerop b) | |
391 | (and (math-num-integerp b) | |
392 | (math-negp b))) | |
393 | '(float 0 0)) | |
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))) | |
bf77c646 | 398 | (math-div (calcFunc-betaB x a b) math-current-beta-value))))) |
136211a9 EZ |
399 | |
400 | (defun calcFunc-betaB (x a b) | |
401 | (cond | |
402 | ((math-zerop x) | |
403 | '(float 0 0)) | |
404 | ((math-equal-int x 1) | |
405 | (calcFunc-beta a b)) | |
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) | |
412 | (if (math-negp 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)) | |
417 | (math-mul | |
418 | (math-with-extra-prec 2 | |
419 | (let* ((i 0) | |
420 | (term 1) | |
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)) | |
426 | sum)) | |
427 | (math-pow x a))) | |
428 | ((and (math-num-integerp a) | |
429 | (if (math-negp 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))) | |
434 | (t | |
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) | |
443 | x))))))) | |
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))) | |
bf77c646 | 450 | b)))))))) |
136211a9 EZ |
451 | |
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) | |
460 | '(float 1 0) | |
bf77c646 | 461 | qab qap qam a b x))) |
136211a9 EZ |
462 | |
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))) | |
469 | (d2 (math-neg | |
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) | |
477 | next | |
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)) | |
bf77c646 | 481 | qab qap qam a b x)))) |
136211a9 EZ |
482 | |
483 | ||
484 | ;;; Bessel functions. | |
485 | ||
486 | ;;; Should generalize this to handle arbitrary precision! | |
487 | ||
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) | |
496 | (if (math-zerop v) | |
497 | '(float 1 0) | |
498 | '(float 0 0))) | |
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 | |
503 | 0 | |
504 | (math-mul '(float -25 -2) | |
505 | (math-sqr x)) | |
506 | v) | |
507 | (math-pow (math-div x 2) v)))) | |
508 | ((math-negp (setq v (math-trunc v))) | |
509 | (if (math-oddp v) | |
510 | (math-neg (calcFunc-besJ (math-neg v) x)) | |
511 | (calcFunc-besJ (math-neg v) x))) | |
512 | ((eq v 0) | |
513 | (math-besJ0 x)) | |
514 | ((eq v 1) | |
515 | (math-besJ1 x)) | |
516 | ((Math-lessp v (math-abs-approx x)) | |
517 | (let ((j 0) | |
518 | (bjm (math-besJ0 x)) | |
519 | (bj (math-besJ1 x)) | |
520 | (two-over-x (math-div 2 x)) | |
521 | bjp) | |
522 | (while (< (setq j (1+ j)) v) | |
523 | (setq bjp (math-sub (math-mul (math-mul j two-over-x) bj) | |
524 | bjm) | |
525 | bjm bj | |
526 | bj bjp)) | |
527 | bj)) | |
528 | (t | |
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)) | |
532 | (jsum nil) | |
533 | (bjp '(float 0 0)) | |
534 | (sum '(float 0 0)) | |
535 | (bj '(float 1 0)) | |
536 | bjm ans) | |
537 | (while (> (setq j (1- j)) 0) | |
538 | (setq bjm (math-sub (math-mul (math-mul j two-over-x) bj) | |
539 | bjp) | |
540 | bjp bj | |
541 | bj bjm) | |
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))) | |
549 | (if (= j v) | |
550 | (setq ans bjp))) | |
bf77c646 | 551 | (math-div ans (math-sub (math-mul 2 sum) bj)))))))) |
136211a9 EZ |
552 | |
553 | (defun math-besJ-series (sum term k zz vk) | |
554 | (math-working "besJ" sum) | |
555 | (setq k (1+ k) | |
556 | vk (math-add 1 vk) | |
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) | |
560 | next | |
bf77c646 | 561 | (math-besJ-series next term k zz vk)))) |
136211a9 EZ |
562 | |
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)) | |
568 | (y (math-sqr z)) | |
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) | |
575 | (float 1 0)))) | |
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))) | |
583 | (if yflag | |
584 | (setq sc (cons (math-neg (cdr sc)) (car sc)))) | |
585 | (math-mul (math-sqrt | |
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)))))) | |
589 | (t | |
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))) | |
598 | (math-poly-eval y | |
599 | '((float 1 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) | |
bf77c646 | 604 | (float (bigpos 411 490 568 57) 0)))))))) |
136211a9 EZ |
605 | |
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)) | |
611 | (y (math-sqr z)) | |
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) | |
617 | (float 183105 -8) | |
618 | (float 1 0)))) | |
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))) | |
626 | (if yflag | |
627 | (setq sc (cons (math-neg (cdr sc)) (car sc))) | |
628 | (if (math-negp x) | |
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)))))) | |
633 | (t | |
634 | (let ((y (math-sqr x))) | |
635 | (math-mul | |
636 | 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))) | |
644 | (math-poly-eval y | |
645 | '((float 1 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) | |
bf77c646 | 651 | 0))))))))) |
136211a9 EZ |
652 | |
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)) | |
665 | (car sc)))) | |
666 | ((math-negp (setq v (math-trunc v))) | |
667 | (if (math-oddp v) | |
668 | (math-neg (calcFunc-besY (math-neg v) x)) | |
669 | (calcFunc-besY (math-neg v) x))) | |
670 | ((eq v 0) | |
671 | (math-besY0 x)) | |
672 | ((eq v 1) | |
673 | (math-besY1 x)) | |
674 | (t | |
675 | (let ((j 0) | |
676 | (bym (math-besY0 x)) | |
677 | (by (math-besY1 x)) | |
678 | (two-over-x (math-div 2 x)) | |
679 | byp) | |
680 | (while (< (setq j (1+ j)) v) | |
681 | (setq byp (math-sub (math-mul (math-mul j two-over-x) by) | |
682 | bym) | |
683 | bym by | |
684 | by byp)) | |
bf77c646 | 685 | by)))))) |
136211a9 EZ |
686 | |
687 | (defun math-besY0 (x) | |
688 | (cond ((Math-lessp (math-abs-approx x) '(float 8 0)) | |
689 | (let ((y (math-sqr x))) | |
690 | (math-add | |
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))) | |
698 | (math-poly-eval y | |
699 | '((float 1 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))))) | |
711 | (t | |
bf77c646 | 712 | (math-besJ0 x t)))) |
136211a9 EZ |
713 | |
714 | (defun math-besY1 (x) | |
715 | (cond ((Math-lessp (math-abs-approx x) '(float 8 0)) | |
716 | (let ((y (math-sqr x))) | |
717 | (math-add | |
718 | (math-mul | |
719 | 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))) | |
727 | (math-poly-eval y | |
728 | '((float 1 0) | |
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)) | |
737 | (math-div 1 x)))))) | |
738 | ((math-negp (calcFunc-re x)) | |
739 | (math-neg | |
740 | (math-add (math-besJ1 (math-neg x) t) | |
741 | (math-mul '(cplx 0 2) | |
742 | (math-besJ1 (math-neg x)))))) | |
743 | (t | |
bf77c646 | 744 | (math-besJ1 x t)))) |
136211a9 EZ |
745 | |
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)))) | |
bf77c646 | 750 | accum)) |
136211a9 EZ |
751 | |
752 | ||
753 | ;;;; Bernoulli and Euler polynomials and numbers. | |
754 | ||
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)) | |
761 | (if (consp n) | |
762 | (progn | |
763 | (math-inexact-result) | |
764 | (math-float (math-bernoulli-number (math-trunc n)))) | |
bf77c646 | 765 | (math-bernoulli-number n)))) |
136211a9 EZ |
766 | |
767 | (defun calcFunc-euler (n &optional x) | |
768 | (or (math-num-natnump n) (math-reject-arg n 'natnump)) | |
769 | (if x | |
770 | (let* ((n1 (math-add n 1)) | |
771 | (coefs (math-bernoulli-coefs n1)) | |
772 | (fac (math-div (math-pow 2 n1) n1)) | |
773 | (k -1) | |
774 | (x1 (math-div (math-add x 1) 2)) | |
775 | (x2 (math-div x 2))) | |
776 | (if (math-numberp x) | |
777 | (if (and calc-symbolic-mode (math-floatp x)) | |
778 | (math-inexact-result) | |
779 | (math-mul fac | |
780 | (math-sub (math-build-polynomial-expr coefs x1) | |
781 | (math-build-polynomial-expr coefs x2)))) | |
782 | (calcFunc-collect | |
783 | (math-reduce-vec | |
784 | 'math-add | |
785 | (cons 'vec | |
786 | (mapcar (function | |
787 | (lambda (c) | |
788 | (setq k (1+ k)) | |
789 | (math-mul (math-mul fac c) | |
790 | (math-sub (math-pow x1 k) | |
791 | (math-pow x2 k))))) | |
792 | coefs))) | |
793 | x))) | |
794 | (math-mul (math-pow 2 n) | |
795 | (if (consp n) | |
796 | (progn | |
797 | (math-inexact-result) | |
798 | (calcFunc-euler n '(float 5 -1))) | |
bf77c646 | 799 | (calcFunc-euler n '(frac 1 2)))))) |
136211a9 | 800 | |
12f1951a CW |
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) | |
810 | (frac 1 12) 1 )) | |
811 | ||
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 )) | |
816 | ||
817 | (defvar math-bernoulli-cache-size 11) | |
136211a9 EZ |
818 | (defun math-bernoulli-coefs (n) |
819 | (let* ((coefs (list (calcFunc-bern n))) | |
820 | (nn (math-trunc n)) | |
821 | (k nn) | |
822 | (term nn) | |
823 | coef | |
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))) | |
bf77c646 | 830 | (nreverse coefs))) |
136211a9 EZ |
831 | |
832 | (defun math-bernoulli-number (n) | |
833 | (if (= (% n 2) 1) | |
834 | (if (= n 1) | |
835 | '(frac -1 2) | |
836 | 0) | |
837 | (setq n (/ n 2)) | |
838 | (while (>= n math-bernoulli-cache-size) | |
839 | (let* ((sum 0) | |
840 | (nk 1) ; nk = n-k+1 | |
841 | (fact 1) ; fact = (n-k+1)! | |
842 | ofact | |
843 | (p math-bernoulli-b-cache) | |
844 | (calc-prefer-frac t)) | |
845 | (math-working "bernoulli B" (* 2 math-bernoulli-cache-size)) | |
846 | (while p | |
847 | (setq nk (+ nk 2) | |
848 | ofact fact | |
849 | fact (math-mul fact (* nk (1- nk))) | |
850 | sum (math-add sum (math-div (car p) fact)) | |
851 | p (cdr p))) | |
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)))) | |
bf77c646 | 858 | (nth (- math-bernoulli-cache-size n 1) math-bernoulli-B-cache))) |
136211a9 EZ |
859 | |
860 | ;;; Bn = n! bn | |
861 | ;;; bn = - sum_k=0^n-1 bk / (n-k+1)! | |
862 | ||
863 | ;;; A faster method would be to use "tangent numbers", c.f., Concrete | |
864 | ;;; Mathematics pg. 273. | |
865 | ||
136211a9 EZ |
866 | |
867 | ;;; Probability distributions. | |
868 | ||
869 | ;;; Binomial. | |
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))) | |
bf77c646 | 873 | (calcFunc-betaI p x (math-add (math-sub n x) 1)))) |
136211a9 EZ |
874 | (put 'calcFunc-utpb 'math-expandable t) |
875 | ||
876 | (defun calcFunc-ltpb (x n p) | |
bf77c646 | 877 | (math-sub 1 (calcFunc-utpb x n p))) |
136211a9 EZ |
878 | (put 'calcFunc-ltpb 'math-expandable t) |
879 | ||
880 | ;;; Chi-square. | |
881 | (defun calcFunc-utpc (chisq v) | |
882 | (if math-expand-formulas | |
883 | (math-normalize (list 'calcFunc-gammaQ (list '/ v 2) (list '/ chisq 2))) | |
bf77c646 | 884 | (calcFunc-gammaQ (math-div v 2) (math-div chisq 2)))) |
136211a9 EZ |
885 | (put 'calcFunc-utpc 'math-expandable t) |
886 | ||
887 | (defun calcFunc-ltpc (chisq v) | |
888 | (if math-expand-formulas | |
889 | (math-normalize (list 'calcFunc-gammaP (list '/ v 2) (list '/ chisq 2))) | |
bf77c646 | 890 | (calcFunc-gammaP (math-div v 2) (math-div chisq 2)))) |
136211a9 EZ |
891 | (put 'calcFunc-ltpc 'math-expandable t) |
892 | ||
893 | ;;; F-distribution. | |
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))) | |
898 | (list '/ v2 2) | |
899 | (list '/ v1 2))) | |
900 | (calcFunc-betaI (math-div v2 (math-add v2 (math-mul v1 f))) | |
901 | (math-div v2 2) | |
bf77c646 | 902 | (math-div v1 2)))) |
136211a9 EZ |
903 | (put 'calcFunc-utpf 'math-expandable t) |
904 | ||
905 | (defun calcFunc-ltpf (f v1 v2) | |
bf77c646 | 906 | (math-sub 1 (calcFunc-utpf f v1 v2))) |
136211a9 EZ |
907 | (put 'calcFunc-ltpf 'math-expandable t) |
908 | ||
909 | ;;; Normal. | |
910 | (defun calcFunc-utpn (x mean sdev) | |
911 | (if math-expand-formulas | |
912 | (math-normalize | |
913 | (list '/ | |
914 | (list '+ 1 | |
915 | (list 'calcFunc-erf | |
916 | (list '/ (list '- mean x) | |
917 | (list '* sdev (list 'calcFunc-sqrt 2))))) | |
918 | 2)) | |
919 | (math-mul (math-add '(float 1 0) | |
920 | (calcFunc-erf | |
921 | (math-div (math-sub mean x) | |
922 | (math-mul sdev (math-sqrt-2))))) | |
bf77c646 | 923 | '(float 5 -1)))) |
136211a9 EZ |
924 | (put 'calcFunc-utpn 'math-expandable t) |
925 | ||
926 | (defun calcFunc-ltpn (x mean sdev) | |
927 | (if math-expand-formulas | |
928 | (math-normalize | |
929 | (list '/ | |
930 | (list '+ 1 | |
931 | (list 'calcFunc-erf | |
932 | (list '/ (list '- x mean) | |
933 | (list '* sdev (list 'calcFunc-sqrt 2))))) | |
934 | 2)) | |
935 | (math-mul (math-add '(float 1 0) | |
936 | (calcFunc-erf | |
937 | (math-div (math-sub x mean) | |
938 | (math-mul sdev (math-sqrt-2))))) | |
bf77c646 | 939 | '(float 5 -1)))) |
136211a9 EZ |
940 | (put 'calcFunc-ltpn 'math-expandable t) |
941 | ||
942 | ;;; Poisson. | |
943 | (defun calcFunc-utpp (n x) | |
944 | (if math-expand-formulas | |
945 | (math-normalize (list 'calcFunc-gammaP x n)) | |
bf77c646 | 946 | (calcFunc-gammaP x n))) |
136211a9 EZ |
947 | (put 'calcFunc-utpp 'math-expandable t) |
948 | ||
949 | (defun calcFunc-ltpp (n x) | |
950 | (if math-expand-formulas | |
951 | (math-normalize (list 'calcFunc-gammaQ x n)) | |
bf77c646 | 952 | (calcFunc-gammaQ x n))) |
136211a9 EZ |
953 | (put 'calcFunc-ltpp 'math-expandable t) |
954 | ||
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))) | |
960 | (list '/ v 2) | |
961 | '(float 5 -1))) | |
962 | (calcFunc-betaI (math-div v (math-add v (math-sqr tt))) | |
963 | (math-div v 2) | |
bf77c646 | 964 | '(float 5 -1)))) |
136211a9 EZ |
965 | (put 'calcFunc-utpt 'math-expandable t) |
966 | ||
967 | (defun calcFunc-ltpt (tt v) | |
bf77c646 | 968 | (math-sub 1 (calcFunc-utpt tt v))) |
136211a9 EZ |
969 | (put 'calcFunc-ltpt 'math-expandable t) |
970 | ||
300f28d3 | 971 | (provide 'calc-funcs) |
136211a9 | 972 | |
ab5796a9 | 973 | ;;; arch-tag: 421ddb7a-550f-4dda-a31c-06638ebfc43a |
bf77c646 | 974 | ;;; calc-funcs.el ends here |