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1 | ;;; calc-funcs.el --- well-known functions for Calc |
2 | ||
58ba2f8f | 3 | ;; Copyright (C) 1990, 1991, 1992, 1993, 2001, 2002, 2003, 2004, |
f0fa15c5 | 4 | ;; 2005, 2006, 2007 Free Software Foundation, Inc. |
12f1951a CW |
5 | |
6 | ;; Author: David Gillespie <daveg@synaptics.com> | |
e8fff8ed | 7 | ;; Maintainer: Jay Belanger <jay.p.belanger@gmail.com> |
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8 | |
9 | ;; This file is part of GNU Emacs. | |
10 | ||
7c671b23 GM |
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 2, or (at your option) | |
14 | ;; any later version. | |
15 | ||
136211a9 | 16 | ;; GNU Emacs is distributed in the hope that it will be useful, |
7c671b23 GM |
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. | |
136211a9 | 25 | |
12f1951a | 26 | ;;; Commentary: |
136211a9 | 27 | |
12f1951a | 28 | ;;; Code: |
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29 | |
30 | ;; This file is autoloaded from calc-ext.el. | |
136211a9 | 31 | |
300f28d3 | 32 | (require 'calc-ext) |
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33 | (require 'calc-macs) |
34 | ||
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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) | |
bf77c646 | 44 | (calc-binary-op "gamP" 'calcFunc-gammaP arg))))) |
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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) | |
bf77c646 | 51 | (calc-unary-op "erf" 'calcFunc-erf arg)))) |
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52 | |
53 | (defun calc-erfc (arg) | |
54 | (interactive "P") | |
55 | (calc-invert-func) | |
bf77c646 | 56 | (calc-erf arg)) |
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57 | |
58 | (defun calc-beta (arg) | |
59 | (interactive "P") | |
60 | (calc-slow-wrapper | |
bf77c646 | 61 | (calc-binary-op "beta" 'calcFunc-beta arg))) |
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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))) | |
bf77c646 | 68 | (calc-enter-result 3 "betI" (cons 'calcFunc-betaI (calc-top-list-n 3)))))) |
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69 | |
70 | (defun calc-bessel-J (arg) | |
71 | (interactive "P") | |
72 | (calc-slow-wrapper | |
bf77c646 | 73 | (calc-binary-op "besJ" 'calcFunc-besJ arg))) |
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74 | |
75 | (defun calc-bessel-Y (arg) | |
76 | (interactive "P") | |
77 | (calc-slow-wrapper | |
bf77c646 | 78 | (calc-binary-op "besY" 'calcFunc-besY arg))) |
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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) | |
bf77c646 | 85 | (calc-unary-op "bern" 'calcFunc-bern arg)))) |
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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) | |
bf77c646 | 92 | (calc-unary-op "eulr" 'calcFunc-euler arg)))) |
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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) | |
bf77c646 | 99 | (calc-binary-op "str1" 'calcFunc-stir1 arg)))) |
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100 | |
101 | (defun calc-utpb () | |
102 | (interactive) | |
bf77c646 | 103 | (calc-prob-dist "b" 3)) |
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104 | |
105 | (defun calc-utpc () | |
106 | (interactive) | |
bf77c646 | 107 | (calc-prob-dist "c" 2)) |
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108 | |
109 | (defun calc-utpf () | |
110 | (interactive) | |
bf77c646 | 111 | (calc-prob-dist "f" 3)) |
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112 | |
113 | (defun calc-utpn () | |
114 | (interactive) | |
bf77c646 | 115 | (calc-prob-dist "n" 3)) |
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116 | |
117 | (defun calc-utpp () | |
118 | (interactive) | |
bf77c646 | 119 | (calc-prob-dist "p" 2)) |
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120 | |
121 | (defun calc-utpt () | |
122 | (interactive) | |
bf77c646 | 123 | (calc-prob-dist "t" 2)) |
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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)) | |
bf77c646 | 135 | (calc-top-list-n (1- nargs) 2)))))) |
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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)) | |
bf77c646 | 148 | (calcFunc-fact (math-add x -1))) |
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149 | |
150 | (defun math-gammap1-raw (x &optional fprec nfprec) ; compute gamma(1 + x) | |
151 | (or fprec | |
152 | (setq fprec (math-float calc-internal-prec) | |
153 | nfprec (math-float (- calc-internal-prec)))) | |
154 | (cond ((math-lessp-float (calcFunc-re x) fprec) | |
155 | (if (math-lessp-float (calcFunc-re x) nfprec) | |
156 | (math-neg (math-div | |
157 | (math-pi) | |
158 | (math-mul (math-gammap1-raw | |
159 | (math-add (math-neg x) | |
160 | '(float -1 0)) | |
161 | fprec nfprec) | |
162 | (math-sin-raw | |
163 | (math-mul (math-pi) x))))) | |
164 | (let ((xplus1 (math-add x '(float 1 0)))) | |
165 | (math-div (math-gammap1-raw xplus1 fprec nfprec) xplus1)))) | |
166 | ((and (math-realp x) | |
167 | (math-lessp-float '(float 736276 0) x)) | |
168 | (math-overflow)) | |
169 | (t ; re(x) now >= 10.0 | |
170 | (let ((xinv (math-div 1 x)) | |
171 | (lnx (math-ln-raw x))) | |
172 | (math-mul (math-sqrt-two-pi) | |
173 | (math-exp-raw | |
174 | (math-gamma-series | |
175 | (math-sub (math-mul (math-add x '(float 5 -1)) | |
176 | lnx) | |
177 | x) | |
178 | xinv | |
179 | (math-sqr xinv) | |
180 | '(float 0 0) | |
bf77c646 | 181 | 2))))))) |
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182 | |
183 | (defun math-gamma-series (sum x xinvsqr oterm n) | |
184 | (math-working "gamma" sum) | |
185 | (let* ((bn (math-bernoulli-number n)) | |
186 | (term (math-mul (math-div-float (math-float (nth 1 bn)) | |
187 | (math-float (* (nth 2 bn) | |
188 | (* n (1- n))))) | |
189 | x)) | |
190 | (next (math-add sum term))) | |
191 | (if (math-nearly-equal sum next) | |
192 | next | |
193 | (if (> n (* 2 calc-internal-prec)) | |
194 | (progn | |
195 | ;; Need this because series eventually diverges for large enough n. | |
196 | (calc-record-why | |
197 | "*Gamma computation stopped early, not all digits may be valid") | |
198 | next) | |
bf77c646 | 199 | (math-gamma-series next (math-mul x xinvsqr) xinvsqr term (+ n 2)))))) |
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200 | |
201 | ||
202 | ;;; Incomplete gamma function. | |
203 | ||
12f1951a | 204 | (defvar math-current-gamma-value nil) |
136211a9 EZ |
205 | (defun calcFunc-gammaP (a x) |
206 | (if (equal x '(var inf var-inf)) | |
207 | '(float 1 0) | |
208 | (math-inexact-result) | |
209 | (or (Math-numberp a) (math-reject-arg a 'numberp)) | |
210 | (or (math-numberp x) (math-reject-arg x 'numberp)) | |
211 | (if (and (math-num-integerp a) | |
212 | (integerp (setq a (math-trunc a))) | |
213 | (> a 0) (< a 20)) | |
214 | (math-sub 1 (calcFunc-gammaQ a x)) | |
215 | (let ((math-current-gamma-value (calcFunc-gamma a))) | |
bf77c646 | 216 | (math-div (calcFunc-gammag a x) math-current-gamma-value))))) |
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217 | |
218 | (defun calcFunc-gammaQ (a x) | |
219 | (if (equal x '(var inf var-inf)) | |
220 | '(float 0 0) | |
221 | (math-inexact-result) | |
222 | (or (Math-numberp a) (math-reject-arg a 'numberp)) | |
223 | (or (math-numberp x) (math-reject-arg x 'numberp)) | |
224 | (if (and (math-num-integerp a) | |
225 | (integerp (setq a (math-trunc a))) | |
226 | (> a 0) (< a 20)) | |
227 | (let ((n 0) | |
228 | (sum '(float 1 0)) | |
229 | (term '(float 1 0))) | |
230 | (math-with-extra-prec 1 | |
231 | (while (< (setq n (1+ n)) a) | |
232 | (setq term (math-div (math-mul term x) n) | |
233 | sum (math-add sum term)) | |
234 | (math-working "gamma" sum)) | |
235 | (math-mul sum (calcFunc-exp (math-neg x))))) | |
236 | (let ((math-current-gamma-value (calcFunc-gamma a))) | |
bf77c646 | 237 | (math-div (calcFunc-gammaG a x) math-current-gamma-value))))) |
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238 | |
239 | (defun calcFunc-gammag (a x) | |
240 | (if (equal x '(var inf var-inf)) | |
241 | (calcFunc-gamma a) | |
242 | (math-inexact-result) | |
243 | (or (Math-numberp a) (math-reject-arg a 'numberp)) | |
244 | (or (Math-numberp x) (math-reject-arg x 'numberp)) | |
245 | (math-with-extra-prec 2 | |
246 | (setq a (math-float a)) | |
247 | (setq x (math-float x)) | |
248 | (if (or (math-negp (calcFunc-re a)) | |
249 | (math-lessp-float (calcFunc-re x) | |
250 | (math-add-float (calcFunc-re a) | |
251 | '(float 1 0)))) | |
252 | (math-inc-gamma-series a x) | |
253 | (math-sub (or math-current-gamma-value (calcFunc-gamma a)) | |
bf77c646 | 254 | (math-inc-gamma-cfrac a x)))))) |
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255 | |
256 | (defun calcFunc-gammaG (a x) | |
257 | (if (equal x '(var inf var-inf)) | |
258 | '(float 0 0) | |
259 | (math-inexact-result) | |
260 | (or (Math-numberp a) (math-reject-arg a 'numberp)) | |
261 | (or (Math-numberp x) (math-reject-arg x 'numberp)) | |
262 | (math-with-extra-prec 2 | |
263 | (setq a (math-float a)) | |
264 | (setq x (math-float x)) | |
265 | (if (or (math-negp (calcFunc-re a)) | |
266 | (math-lessp-float (calcFunc-re x) | |
267 | (math-add-float (math-abs-approx a) | |
268 | '(float 1 0)))) | |
269 | (math-sub (or math-current-gamma-value (calcFunc-gamma a)) | |
270 | (math-inc-gamma-series a x)) | |
bf77c646 | 271 | (math-inc-gamma-cfrac a x))))) |
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272 | |
273 | (defun math-inc-gamma-series (a x) | |
274 | (if (Math-zerop x) | |
275 | '(float 0 0) | |
276 | (math-mul (math-exp-raw (math-sub (math-mul a (math-ln-raw x)) x)) | |
277 | (math-with-extra-prec 2 | |
278 | (let ((start (math-div '(float 1 0) a))) | |
bf77c646 | 279 | (math-inc-gamma-series-step start start a x)))))) |
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280 | |
281 | (defun math-inc-gamma-series-step (sum term a x) | |
282 | (math-working "gamma" sum) | |
283 | (setq a (math-add a '(float 1 0)) | |
284 | term (math-div (math-mul term x) a)) | |
285 | (let ((next (math-add sum term))) | |
286 | (if (math-nearly-equal sum next) | |
287 | next | |
bf77c646 | 288 | (math-inc-gamma-series-step next term a x)))) |
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289 | |
290 | (defun math-inc-gamma-cfrac (a x) | |
291 | (if (Math-zerop x) | |
292 | (or math-current-gamma-value (calcFunc-gamma a)) | |
293 | (math-mul (math-exp-raw (math-sub (math-mul a (math-ln-raw x)) x)) | |
294 | (math-inc-gamma-cfrac-step '(float 1 0) x | |
295 | '(float 0 0) '(float 1 0) | |
296 | '(float 1 0) '(float 1 0) '(float 0 0) | |
bf77c646 | 297 | a x)))) |
136211a9 EZ |
298 | |
299 | (defun math-inc-gamma-cfrac-step (a0 a1 b0 b1 n fac g a x) | |
300 | (let ((ana (math-sub n a)) | |
301 | (anf (math-mul n fac))) | |
302 | (setq n (math-add n '(float 1 0)) | |
303 | a0 (math-mul (math-add a1 (math-mul a0 ana)) fac) | |
304 | b0 (math-mul (math-add b1 (math-mul b0 ana)) fac) | |
305 | a1 (math-add (math-mul x a0) (math-mul anf a1)) | |
306 | b1 (math-add (math-mul x b0) (math-mul anf b1))) | |
307 | (if (math-zerop a1) | |
308 | (math-inc-gamma-cfrac-step a0 a1 b0 b1 n fac g a x) | |
309 | (setq fac (math-div '(float 1 0) a1)) | |
310 | (let ((next (math-mul b1 fac))) | |
311 | (math-working "gamma" next) | |
312 | (if (math-nearly-equal next g) | |
313 | next | |
bf77c646 | 314 | (math-inc-gamma-cfrac-step a0 a1 b0 b1 n fac next a x)))))) |
136211a9 EZ |
315 | |
316 | ||
317 | ;;; Error function. | |
318 | ||
319 | (defun calcFunc-erf (x) | |
320 | (if (equal x '(var inf var-inf)) | |
321 | '(float 1 0) | |
322 | (if (equal x '(neg (var inf var-inf))) | |
323 | '(float -1 0) | |
324 | (if (Math-zerop x) | |
325 | x | |
326 | (let ((math-current-gamma-value (math-sqrt-pi))) | |
327 | (math-to-same-complex-quad | |
328 | (math-div (calcFunc-gammag '(float 5 -1) | |
329 | (math-sqr (math-to-complex-quad-one x))) | |
330 | math-current-gamma-value) | |
bf77c646 | 331 | x)))))) |
136211a9 EZ |
332 | |
333 | (defun calcFunc-erfc (x) | |
334 | (if (equal x '(var inf var-inf)) | |
335 | '(float 0 0) | |
336 | (if (math-posp x) | |
337 | (let ((math-current-gamma-value (math-sqrt-pi))) | |
338 | (math-div (calcFunc-gammaG '(float 5 -1) (math-sqr x)) | |
339 | math-current-gamma-value)) | |
bf77c646 | 340 | (math-sub 1 (calcFunc-erf x))))) |
136211a9 EZ |
341 | |
342 | (defun math-to-complex-quad-one (x) | |
343 | (if (eq (car-safe x) 'polar) (setq x (math-complex x))) | |
344 | (if (eq (car-safe x) 'cplx) | |
345 | (list 'cplx (math-abs (nth 1 x)) (math-abs (nth 2 x))) | |
bf77c646 | 346 | x)) |
136211a9 EZ |
347 | |
348 | (defun math-to-same-complex-quad (x y) | |
349 | (if (eq (car-safe y) 'cplx) | |
350 | (if (eq (car-safe x) 'cplx) | |
351 | (list 'cplx | |
352 | (if (math-negp (nth 1 y)) (math-neg (nth 1 x)) (nth 1 x)) | |
353 | (if (math-negp (nth 2 y)) (math-neg (nth 2 x)) (nth 2 x))) | |
354 | (if (math-negp (nth 1 y)) (math-neg x) x)) | |
355 | (if (math-negp y) | |
356 | (if (eq (car-safe x) 'cplx) | |
357 | (list 'cplx (math-neg (nth 1 x)) (nth 2 x)) | |
358 | (math-neg x)) | |
bf77c646 | 359 | x))) |
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360 | |
361 | ||
362 | ;;; Beta function. | |
363 | ||
364 | (defun calcFunc-beta (a b) | |
365 | (if (math-num-integerp a) | |
366 | (let ((am (math-add a -1))) | |
367 | (or (math-numberp b) (math-reject-arg b 'numberp)) | |
368 | (math-div 1 (math-mul b (calcFunc-choose (math-add b am) am)))) | |
369 | (if (math-num-integerp b) | |
370 | (calcFunc-beta b a) | |
371 | (math-div (math-mul (calcFunc-gamma a) (calcFunc-gamma b)) | |
bf77c646 | 372 | (calcFunc-gamma (math-add a b)))))) |
136211a9 EZ |
373 | |
374 | ||
375 | ;;; Incomplete beta function. | |
376 | ||
12f1951a | 377 | (defvar math-current-beta-value nil) |
136211a9 EZ |
378 | (defun calcFunc-betaI (x a b) |
379 | (cond ((math-zerop x) | |
380 | '(float 0 0)) | |
381 | ((math-equal-int x 1) | |
382 | '(float 1 0)) | |
383 | ((or (math-zerop a) | |
384 | (and (math-num-integerp a) | |
385 | (math-negp a))) | |
386 | (if (or (math-zerop b) | |
387 | (and (math-num-integerp b) | |
388 | (math-negp b))) | |
389 | (math-reject-arg b 'range) | |
390 | '(float 1 0))) | |
391 | ((or (math-zerop b) | |
392 | (and (math-num-integerp b) | |
393 | (math-negp b))) | |
394 | '(float 0 0)) | |
395 | ((not (math-numberp a)) (math-reject-arg a 'numberp)) | |
396 | ((not (math-numberp b)) (math-reject-arg b 'numberp)) | |
397 | ((math-inexact-result)) | |
398 | (t (let ((math-current-beta-value (calcFunc-beta a b))) | |
bf77c646 | 399 | (math-div (calcFunc-betaB x a b) math-current-beta-value))))) |
136211a9 EZ |
400 | |
401 | (defun calcFunc-betaB (x a b) | |
402 | (cond | |
403 | ((math-zerop x) | |
404 | '(float 0 0)) | |
405 | ((math-equal-int x 1) | |
406 | (calcFunc-beta a b)) | |
407 | ((not (math-numberp x)) (math-reject-arg x 'numberp)) | |
408 | ((not (math-numberp a)) (math-reject-arg a 'numberp)) | |
409 | ((not (math-numberp b)) (math-reject-arg b 'numberp)) | |
410 | ((math-zerop a) (math-reject-arg a 'nonzerop)) | |
411 | ((math-zerop b) (math-reject-arg b 'nonzerop)) | |
412 | ((and (math-num-integerp b) | |
413 | (if (math-negp b) | |
414 | (math-reject-arg b 'range) | |
415 | (Math-natnum-lessp (setq b (math-trunc b)) 20))) | |
416 | (and calc-symbolic-mode (or (math-floatp a) (math-floatp b)) | |
417 | (math-inexact-result)) | |
418 | (math-mul | |
419 | (math-with-extra-prec 2 | |
420 | (let* ((i 0) | |
421 | (term 1) | |
422 | (sum (math-div term a))) | |
423 | (while (< (setq i (1+ i)) b) | |
424 | (setq term (math-mul (math-div (math-mul term (- i b)) i) x) | |
425 | sum (math-add sum (math-div term (math-add a i)))) | |
426 | (math-working "beta" sum)) | |
427 | sum)) | |
428 | (math-pow x a))) | |
429 | ((and (math-num-integerp a) | |
430 | (if (math-negp a) | |
431 | (math-reject-arg a 'range) | |
432 | (Math-natnum-lessp (setq a (math-trunc a)) 20))) | |
433 | (math-sub (or math-current-beta-value (calcFunc-beta a b)) | |
434 | (calcFunc-betaB (math-sub 1 x) b a))) | |
435 | (t | |
436 | (math-inexact-result) | |
437 | (math-with-extra-prec 2 | |
438 | (setq x (math-float x)) | |
439 | (setq a (math-float a)) | |
440 | (setq b (math-float b)) | |
441 | (let ((bt (math-exp-raw (math-add (math-mul a (math-ln-raw x)) | |
442 | (math-mul b (math-ln-raw | |
443 | (math-sub '(float 1 0) | |
444 | x))))))) | |
445 | (if (Math-lessp x (math-div (math-add a '(float 1 0)) | |
446 | (math-add (math-add a b) '(float 2 0)))) | |
447 | (math-div (math-mul bt (math-beta-cfrac a b x)) a) | |
448 | (math-sub (or math-current-beta-value (calcFunc-beta a b)) | |
449 | (math-div (math-mul bt | |
450 | (math-beta-cfrac b a (math-sub 1 x))) | |
bf77c646 | 451 | b)))))))) |
136211a9 EZ |
452 | |
453 | (defun math-beta-cfrac (a b x) | |
454 | (let ((qab (math-add a b)) | |
455 | (qap (math-add a '(float 1 0))) | |
456 | (qam (math-add a '(float -1 0)))) | |
457 | (math-beta-cfrac-step '(float 1 0) | |
458 | (math-sub '(float 1 0) | |
459 | (math-div (math-mul qab x) qap)) | |
460 | '(float 1 0) '(float 1 0) | |
461 | '(float 1 0) | |
bf77c646 | 462 | qab qap qam a b x))) |
136211a9 EZ |
463 | |
464 | (defun math-beta-cfrac-step (az bz am bm m qab qap qam a b x) | |
465 | (let* ((two-m (math-mul m '(float 2 0))) | |
466 | (d (math-div (math-mul (math-mul (math-sub b m) m) x) | |
467 | (math-mul (math-add qam two-m) (math-add a two-m)))) | |
468 | (ap (math-add az (math-mul d am))) | |
469 | (bp (math-add bz (math-mul d bm))) | |
470 | (d2 (math-neg | |
471 | (math-div (math-mul (math-mul (math-add a m) (math-add qab m)) x) | |
472 | (math-mul (math-add qap two-m) (math-add a two-m))))) | |
473 | (app (math-add ap (math-mul d2 az))) | |
474 | (bpp (math-add bp (math-mul d2 bz))) | |
475 | (next (math-div app bpp))) | |
476 | (math-working "beta" next) | |
477 | (if (math-nearly-equal next az) | |
478 | next | |
479 | (math-beta-cfrac-step next '(float 1 0) | |
480 | (math-div ap bpp) (math-div bp bpp) | |
481 | (math-add m '(float 1 0)) | |
bf77c646 | 482 | qab qap qam a b x)))) |
136211a9 EZ |
483 | |
484 | ||
485 | ;;; Bessel functions. | |
486 | ||
487 | ;;; Should generalize this to handle arbitrary precision! | |
488 | ||
489 | (defun calcFunc-besJ (v x) | |
490 | (or (math-numberp v) (math-reject-arg v 'numberp)) | |
491 | (or (math-numberp x) (math-reject-arg x 'numberp)) | |
492 | (let ((calc-internal-prec (min 8 calc-internal-prec))) | |
493 | (math-with-extra-prec 3 | |
494 | (setq x (math-float (math-normalize x))) | |
495 | (setq v (math-float (math-normalize v))) | |
496 | (cond ((math-zerop x) | |
497 | (if (math-zerop v) | |
498 | '(float 1 0) | |
499 | '(float 0 0))) | |
500 | ((math-inexact-result)) | |
501 | ((not (math-num-integerp v)) | |
502 | (let ((start (math-div 1 (calcFunc-fact v)))) | |
503 | (math-mul (math-besJ-series start start | |
504 | 0 | |
505 | (math-mul '(float -25 -2) | |
506 | (math-sqr x)) | |
507 | v) | |
508 | (math-pow (math-div x 2) v)))) | |
509 | ((math-negp (setq v (math-trunc v))) | |
510 | (if (math-oddp v) | |
511 | (math-neg (calcFunc-besJ (math-neg v) x)) | |
512 | (calcFunc-besJ (math-neg v) x))) | |
513 | ((eq v 0) | |
514 | (math-besJ0 x)) | |
515 | ((eq v 1) | |
516 | (math-besJ1 x)) | |
517 | ((Math-lessp v (math-abs-approx x)) | |
518 | (let ((j 0) | |
519 | (bjm (math-besJ0 x)) | |
520 | (bj (math-besJ1 x)) | |
521 | (two-over-x (math-div 2 x)) | |
522 | bjp) | |
523 | (while (< (setq j (1+ j)) v) | |
524 | (setq bjp (math-sub (math-mul (math-mul j two-over-x) bj) | |
525 | bjm) | |
526 | bjm bj | |
527 | bj bjp)) | |
528 | bj)) | |
529 | (t | |
530 | (if (Math-lessp 100 v) (math-reject-arg v 'range)) | |
531 | (let* ((j (logior (+ v (math-isqrt-small (* 40 v))) 1)) | |
532 | (two-over-x (math-div 2 x)) | |
533 | (jsum nil) | |
534 | (bjp '(float 0 0)) | |
535 | (sum '(float 0 0)) | |
536 | (bj '(float 1 0)) | |
537 | bjm ans) | |
538 | (while (> (setq j (1- j)) 0) | |
539 | (setq bjm (math-sub (math-mul (math-mul j two-over-x) bj) | |
540 | bjp) | |
541 | bjp bj | |
542 | bj bjm) | |
543 | (if (> (nth 2 (math-abs-approx bj)) 10) | |
544 | (setq bj (math-mul bj '(float 1 -10)) | |
545 | bjp (math-mul bjp '(float 1 -10)) | |
546 | ans (and ans (math-mul ans '(float 1 -10))) | |
547 | sum (math-mul sum '(float 1 -10)))) | |
548 | (or (setq jsum (not jsum)) | |
549 | (setq sum (math-add sum bj))) | |
550 | (if (= j v) | |
551 | (setq ans bjp))) | |
bf77c646 | 552 | (math-div ans (math-sub (math-mul 2 sum) bj)))))))) |
136211a9 EZ |
553 | |
554 | (defun math-besJ-series (sum term k zz vk) | |
555 | (math-working "besJ" sum) | |
556 | (setq k (1+ k) | |
557 | vk (math-add 1 vk) | |
558 | term (math-div (math-mul term zz) (math-mul k vk))) | |
559 | (let ((next (math-add sum term))) | |
560 | (if (math-nearly-equal next sum) | |
561 | next | |
bf77c646 | 562 | (math-besJ-series next term k zz vk)))) |
136211a9 EZ |
563 | |
564 | (defun math-besJ0 (x &optional yflag) | |
565 | (cond ((and (not yflag) (math-negp (calcFunc-re x))) | |
566 | (math-besJ0 (math-neg x))) | |
567 | ((Math-lessp '(float 8 0) (math-abs-approx x)) | |
568 | (let* ((z (math-div '(float 8 0) x)) | |
569 | (y (math-sqr z)) | |
a8775b6d JB |
570 | (xx (math-add x |
571 | (eval-when-compile | |
f83b9601 | 572 | (math-read-number-simple "-0.785398164")))) |
136211a9 | 573 | (a1 (math-poly-eval y |
a8775b6d JB |
574 | (eval-when-compile |
575 | (list | |
f83b9601 JB |
576 | (math-read-number-simple "0.0000002093887211") |
577 | (math-read-number-simple "-0.000002073370639") | |
578 | (math-read-number-simple "0.00002734510407") | |
579 | (math-read-number-simple "-0.001098628627") | |
a8775b6d | 580 | '(float 1 0))))) |
136211a9 | 581 | (a2 (math-poly-eval y |
a8775b6d JB |
582 | (eval-when-compile |
583 | (list | |
f83b9601 JB |
584 | (math-read-number-simple "-0.0000000934935152") |
585 | (math-read-number-simple "0.0000007621095161") | |
586 | (math-read-number-simple "-0.000006911147651") | |
587 | (math-read-number-simple "0.0001430488765") | |
588 | (math-read-number-simple "-0.01562499995"))))) | |
136211a9 EZ |
589 | (sc (math-sin-cos-raw xx))) |
590 | (if yflag | |
591 | (setq sc (cons (math-neg (cdr sc)) (car sc)))) | |
592 | (math-mul (math-sqrt | |
a8775b6d | 593 | (math-div (eval-when-compile |
f83b9601 | 594 | (math-read-number-simple "0.636619722")) |
a8775b6d | 595 | x)) |
136211a9 EZ |
596 | (math-sub (math-mul (cdr sc) a1) |
597 | (math-mul (car sc) (math-mul z a2)))))) | |
598 | (t | |
599 | (let ((y (math-sqr x))) | |
600 | (math-div (math-poly-eval y | |
a8775b6d JB |
601 | (eval-when-compile |
602 | (list | |
f83b9601 JB |
603 | (math-read-number-simple "-184.9052456") |
604 | (math-read-number-simple "77392.33017") | |
605 | (math-read-number-simple "-11214424.18") | |
606 | (math-read-number-simple "651619640.7") | |
607 | (math-read-number-simple "-13362590354.0") | |
608 | (math-read-number-simple "57568490574.0")))) | |
136211a9 | 609 | (math-poly-eval y |
a8775b6d JB |
610 | (eval-when-compile |
611 | (list | |
612 | '(float 1 0) | |
f83b9601 JB |
613 | (math-read-number-simple "267.8532712") |
614 | (math-read-number-simple "59272.64853") | |
615 | (math-read-number-simple "9494680.718") | |
616 | (math-read-number-simple "1029532985.0") | |
617 | (math-read-number-simple "57568490411.0"))))))))) | |
136211a9 EZ |
618 | |
619 | (defun math-besJ1 (x &optional yflag) | |
620 | (cond ((and (math-negp (calcFunc-re x)) (not yflag)) | |
621 | (math-neg (math-besJ1 (math-neg x)))) | |
622 | ((Math-lessp '(float 8 0) (math-abs-approx x)) | |
623 | (let* ((z (math-div '(float 8 0) x)) | |
624 | (y (math-sqr z)) | |
a8775b6d | 625 | (xx (math-add x (eval-when-compile |
f83b9601 | 626 | (math-read-number-simple "-2.356194491")))) |
136211a9 | 627 | (a1 (math-poly-eval y |
a8775b6d JB |
628 | (eval-when-compile |
629 | (list | |
f83b9601 JB |
630 | (math-read-number-simple "-0.000000240337019") |
631 | (math-read-number-simple "0.000002457520174") | |
632 | (math-read-number-simple "-0.00003516396496") | |
a8775b6d JB |
633 | '(float 183105 -8) |
634 | '(float 1 0))))) | |
136211a9 | 635 | (a2 (math-poly-eval y |
a8775b6d JB |
636 | (eval-when-compile |
637 | (list | |
f83b9601 JB |
638 | (math-read-number-simple "0.000000105787412") |
639 | (math-read-number-simple "-0.00000088228987") | |
640 | (math-read-number-simple "0.000008449199096") | |
641 | (math-read-number-simple "-0.0002002690873") | |
642 | (math-read-number-simple "0.04687499995"))))) | |
136211a9 EZ |
643 | (sc (math-sin-cos-raw xx))) |
644 | (if yflag | |
645 | (setq sc (cons (math-neg (cdr sc)) (car sc))) | |
646 | (if (math-negp x) | |
647 | (setq sc (cons (math-neg (car sc)) (math-neg (cdr sc)))))) | |
a8775b6d JB |
648 | (math-mul (math-sqrt (math-div |
649 | (eval-when-compile | |
f83b9601 | 650 | (math-read-number-simple "0.636619722")) |
a8775b6d | 651 | x)) |
136211a9 EZ |
652 | (math-sub (math-mul (cdr sc) a1) |
653 | (math-mul (car sc) (math-mul z a2)))))) | |
654 | (t | |
655 | (let ((y (math-sqr x))) | |
656 | (math-mul | |
657 | x | |
658 | (math-div (math-poly-eval y | |
a8775b6d JB |
659 | (eval-when-compile |
660 | (list | |
f83b9601 JB |
661 | (math-read-number-simple "-30.16036606") |
662 | (math-read-number-simple "15704.4826") | |
663 | (math-read-number-simple "-2972611.439") | |
664 | (math-read-number-simple "242396853.1") | |
665 | (math-read-number-simple "-7895059235.0") | |
666 | (math-read-number-simple "72362614232.0")))) | |
136211a9 | 667 | (math-poly-eval y |
a8775b6d JB |
668 | (eval-when-compile |
669 | (list | |
670 | '(float 1 0) | |
f83b9601 JB |
671 | (math-read-number-simple "376.9991397") |
672 | (math-read-number-simple "99447.43394") | |
673 | (math-read-number-simple "18583304.74") | |
674 | (math-read-number-simple "2300535178.0") | |
675 | (math-read-number-simple "144725228442.0")))))))))) | |
136211a9 EZ |
676 | |
677 | (defun calcFunc-besY (v x) | |
678 | (math-inexact-result) | |
679 | (or (math-numberp v) (math-reject-arg v 'numberp)) | |
680 | (or (math-numberp x) (math-reject-arg x 'numberp)) | |
681 | (let ((calc-internal-prec (min 8 calc-internal-prec))) | |
682 | (math-with-extra-prec 3 | |
683 | (setq x (math-float (math-normalize x))) | |
684 | (setq v (math-float (math-normalize v))) | |
685 | (cond ((not (math-num-integerp v)) | |
686 | (let ((sc (math-sin-cos-raw (math-mul v (math-pi))))) | |
687 | (math-div (math-sub (math-mul (calcFunc-besJ v x) (cdr sc)) | |
688 | (calcFunc-besJ (math-neg v) x)) | |
689 | (car sc)))) | |
690 | ((math-negp (setq v (math-trunc v))) | |
691 | (if (math-oddp v) | |
692 | (math-neg (calcFunc-besY (math-neg v) x)) | |
693 | (calcFunc-besY (math-neg v) x))) | |
694 | ((eq v 0) | |
695 | (math-besY0 x)) | |
696 | ((eq v 1) | |
697 | (math-besY1 x)) | |
698 | (t | |
699 | (let ((j 0) | |
700 | (bym (math-besY0 x)) | |
701 | (by (math-besY1 x)) | |
702 | (two-over-x (math-div 2 x)) | |
703 | byp) | |
704 | (while (< (setq j (1+ j)) v) | |
705 | (setq byp (math-sub (math-mul (math-mul j two-over-x) by) | |
706 | bym) | |
707 | bym by | |
708 | by byp)) | |
bf77c646 | 709 | by)))))) |
136211a9 EZ |
710 | |
711 | (defun math-besY0 (x) | |
712 | (cond ((Math-lessp (math-abs-approx x) '(float 8 0)) | |
713 | (let ((y (math-sqr x))) | |
714 | (math-add | |
715 | (math-div (math-poly-eval y | |
a8775b6d JB |
716 | (eval-when-compile |
717 | (list | |
f83b9601 JB |
718 | (math-read-number-simple "228.4622733") |
719 | (math-read-number-simple "-86327.92757") | |
720 | (math-read-number-simple "10879881.29") | |
721 | (math-read-number-simple "-512359803.6") | |
722 | (math-read-number-simple "7062834065.0") | |
723 | (math-read-number-simple "-2957821389.0")))) | |
136211a9 | 724 | (math-poly-eval y |
a8775b6d JB |
725 | (eval-when-compile |
726 | (list | |
727 | '(float 1 0) | |
f83b9601 JB |
728 | (math-read-number-simple "226.1030244") |
729 | (math-read-number-simple "47447.2647") | |
730 | (math-read-number-simple "7189466.438") | |
731 | (math-read-number-simple "745249964.8") | |
732 | (math-read-number-simple "40076544269.0"))))) | |
a8775b6d | 733 | (math-mul (eval-when-compile |
f83b9601 | 734 | (math-read-number-simple "0.636619772")) |
136211a9 EZ |
735 | (math-mul (math-besJ0 x) (math-ln-raw x)))))) |
736 | ((math-negp (calcFunc-re x)) | |
737 | (math-add (math-besJ0 (math-neg x) t) | |
738 | (math-mul '(cplx 0 2) | |
739 | (math-besJ0 (math-neg x))))) | |
740 | (t | |
bf77c646 | 741 | (math-besJ0 x t)))) |
136211a9 EZ |
742 | |
743 | (defun math-besY1 (x) | |
744 | (cond ((Math-lessp (math-abs-approx x) '(float 8 0)) | |
745 | (let ((y (math-sqr x))) | |
746 | (math-add | |
747 | (math-mul | |
748 | x | |
749 | (math-div (math-poly-eval y | |
a8775b6d JB |
750 | (eval-when-compile |
751 | (list | |
f83b9601 JB |
752 | (math-read-number-simple "8511.937935") |
753 | (math-read-number-simple "-4237922.726") | |
754 | (math-read-number-simple "734926455.1") | |
755 | (math-read-number-simple "-51534381390.0") | |
756 | (math-read-number-simple "1275274390000.0") | |
757 | (math-read-number-simple "-4900604943000.0")))) | |
136211a9 | 758 | (math-poly-eval y |
a8775b6d JB |
759 | (eval-when-compile |
760 | (list | |
761 | '(float 1 0) | |
f83b9601 JB |
762 | (math-read-number-simple "354.9632885") |
763 | (math-read-number-simple "102042.605") | |
764 | (math-read-number-simple "22459040.02") | |
765 | (math-read-number-simple "3733650367.0") | |
766 | (math-read-number-simple "424441966400.0") | |
767 | (math-read-number-simple "24995805700000.0")))))) | |
768 | (math-mul (eval-when-compile (math-read-number-simple "0.636619772")) | |
a8775b6d | 769 | (math-sub (math-mul (math-besJ1 x) (math-ln-raw x)) |
136211a9 EZ |
770 | (math-div 1 x)))))) |
771 | ((math-negp (calcFunc-re x)) | |
772 | (math-neg | |
773 | (math-add (math-besJ1 (math-neg x) t) | |
774 | (math-mul '(cplx 0 2) | |
775 | (math-besJ1 (math-neg x)))))) | |
776 | (t | |
bf77c646 | 777 | (math-besJ1 x t)))) |
136211a9 EZ |
778 | |
779 | (defun math-poly-eval (x coefs) | |
780 | (let ((accum (car coefs))) | |
781 | (while (setq coefs (cdr coefs)) | |
782 | (setq accum (math-add (car coefs) (math-mul accum x)))) | |
bf77c646 | 783 | accum)) |
136211a9 EZ |
784 | |
785 | ||
786 | ;;;; Bernoulli and Euler polynomials and numbers. | |
787 | ||
788 | (defun calcFunc-bern (n &optional x) | |
789 | (if (and x (not (math-zerop x))) | |
790 | (if (and calc-symbolic-mode (math-floatp x)) | |
791 | (math-inexact-result) | |
792 | (math-build-polynomial-expr (math-bernoulli-coefs n) x)) | |
793 | (or (math-num-natnump n) (math-reject-arg n 'natnump)) | |
794 | (if (consp n) | |
795 | (progn | |
796 | (math-inexact-result) | |
797 | (math-float (math-bernoulli-number (math-trunc n)))) | |
bf77c646 | 798 | (math-bernoulli-number n)))) |
136211a9 EZ |
799 | |
800 | (defun calcFunc-euler (n &optional x) | |
801 | (or (math-num-natnump n) (math-reject-arg n 'natnump)) | |
802 | (if x | |
803 | (let* ((n1 (math-add n 1)) | |
804 | (coefs (math-bernoulli-coefs n1)) | |
805 | (fac (math-div (math-pow 2 n1) n1)) | |
806 | (k -1) | |
807 | (x1 (math-div (math-add x 1) 2)) | |
808 | (x2 (math-div x 2))) | |
809 | (if (math-numberp x) | |
810 | (if (and calc-symbolic-mode (math-floatp x)) | |
811 | (math-inexact-result) | |
812 | (math-mul fac | |
813 | (math-sub (math-build-polynomial-expr coefs x1) | |
814 | (math-build-polynomial-expr coefs x2)))) | |
815 | (calcFunc-collect | |
816 | (math-reduce-vec | |
817 | 'math-add | |
818 | (cons 'vec | |
819 | (mapcar (function | |
820 | (lambda (c) | |
821 | (setq k (1+ k)) | |
822 | (math-mul (math-mul fac c) | |
823 | (math-sub (math-pow x1 k) | |
824 | (math-pow x2 k))))) | |
825 | coefs))) | |
826 | x))) | |
827 | (math-mul (math-pow 2 n) | |
828 | (if (consp n) | |
829 | (progn | |
830 | (math-inexact-result) | |
831 | (calcFunc-euler n '(float 5 -1))) | |
bf77c646 | 832 | (calcFunc-euler n '(frac 1 2)))))) |
136211a9 | 833 | |
a8775b6d JB |
834 | (defvar math-bernoulli-b-cache |
835 | (eval-when-compile | |
836 | (list | |
837 | (list 'frac | |
838 | -174611 | |
f83b9601 | 839 | (math-read-number-simple "802857662698291200000")) |
a8775b6d JB |
840 | (list 'frac |
841 | 43867 | |
f83b9601 | 842 | (math-read-number-simple "5109094217170944000")) |
a8775b6d JB |
843 | (list 'frac |
844 | -3617 | |
f83b9601 | 845 | (math-read-number-simple "10670622842880000")) |
a8775b6d JB |
846 | (list 'frac |
847 | 1 | |
f83b9601 | 848 | (math-read-number-simple "74724249600")) |
a8775b6d JB |
849 | (list 'frac |
850 | -691 | |
f83b9601 | 851 | (math-read-number-simple "1307674368000")) |
a8775b6d JB |
852 | (list 'frac |
853 | 1 | |
f83b9601 | 854 | (math-read-number-simple "47900160")) |
a8775b6d JB |
855 | (list 'frac |
856 | -1 | |
f83b9601 | 857 | (math-read-number-simple "1209600")) |
a8775b6d JB |
858 | (list 'frac |
859 | 1 | |
860 | 30240) | |
861 | (list 'frac | |
862 | -1 | |
863 | 720) | |
864 | (list 'frac | |
865 | 1 | |
866 | 12) | |
867 | 1 ))) | |
12f1951a CW |
868 | |
869 | (defvar math-bernoulli-B-cache '((frac -174611 330) (frac 43867 798) | |
870 | (frac -3617 510) (frac 7 6) (frac -691 2730) | |
871 | (frac 5 66) (frac -1 30) (frac 1 42) | |
872 | (frac -1 30) (frac 1 6) 1 )) | |
873 | ||
874 | (defvar math-bernoulli-cache-size 11) | |
136211a9 EZ |
875 | (defun math-bernoulli-coefs (n) |
876 | (let* ((coefs (list (calcFunc-bern n))) | |
877 | (nn (math-trunc n)) | |
878 | (k nn) | |
879 | (term nn) | |
880 | coef | |
881 | (calc-prefer-frac (or (integerp n) calc-prefer-frac))) | |
882 | (while (>= (setq k (1- k)) 0) | |
883 | (setq term (math-div term (- nn k)) | |
884 | coef (math-mul term (math-bernoulli-number k)) | |
885 | coefs (cons (if (consp n) (math-float coef) coef) coefs) | |
886 | term (math-mul term k))) | |
bf77c646 | 887 | (nreverse coefs))) |
136211a9 EZ |
888 | |
889 | (defun math-bernoulli-number (n) | |
890 | (if (= (% n 2) 1) | |
891 | (if (= n 1) | |
892 | '(frac -1 2) | |
893 | 0) | |
894 | (setq n (/ n 2)) | |
895 | (while (>= n math-bernoulli-cache-size) | |
896 | (let* ((sum 0) | |
897 | (nk 1) ; nk = n-k+1 | |
898 | (fact 1) ; fact = (n-k+1)! | |
899 | ofact | |
900 | (p math-bernoulli-b-cache) | |
901 | (calc-prefer-frac t)) | |
902 | (math-working "bernoulli B" (* 2 math-bernoulli-cache-size)) | |
903 | (while p | |
904 | (setq nk (+ nk 2) | |
905 | ofact fact | |
906 | fact (math-mul fact (* nk (1- nk))) | |
907 | sum (math-add sum (math-div (car p) fact)) | |
908 | p (cdr p))) | |
909 | (setq ofact (math-mul ofact (1- nk)) | |
910 | sum (math-sub (math-div '(frac 1 2) ofact) sum) | |
911 | math-bernoulli-b-cache (cons sum math-bernoulli-b-cache) | |
912 | math-bernoulli-B-cache (cons (math-mul sum ofact) | |
913 | math-bernoulli-B-cache) | |
914 | math-bernoulli-cache-size (1+ math-bernoulli-cache-size)))) | |
bf77c646 | 915 | (nth (- math-bernoulli-cache-size n 1) math-bernoulli-B-cache))) |
136211a9 EZ |
916 | |
917 | ;;; Bn = n! bn | |
918 | ;;; bn = - sum_k=0^n-1 bk / (n-k+1)! | |
919 | ||
920 | ;;; A faster method would be to use "tangent numbers", c.f., Concrete | |
921 | ;;; Mathematics pg. 273. | |
922 | ||
136211a9 EZ |
923 | |
924 | ;;; Probability distributions. | |
925 | ||
926 | ;;; Binomial. | |
927 | (defun calcFunc-utpb (x n p) | |
928 | (if math-expand-formulas | |
929 | (math-normalize (list 'calcFunc-betaI p x (list '+ (list '- n x) 1))) | |
bf77c646 | 930 | (calcFunc-betaI p x (math-add (math-sub n x) 1)))) |
136211a9 EZ |
931 | (put 'calcFunc-utpb 'math-expandable t) |
932 | ||
933 | (defun calcFunc-ltpb (x n p) | |
bf77c646 | 934 | (math-sub 1 (calcFunc-utpb x n p))) |
136211a9 EZ |
935 | (put 'calcFunc-ltpb 'math-expandable t) |
936 | ||
937 | ;;; Chi-square. | |
938 | (defun calcFunc-utpc (chisq v) | |
939 | (if math-expand-formulas | |
940 | (math-normalize (list 'calcFunc-gammaQ (list '/ v 2) (list '/ chisq 2))) | |
bf77c646 | 941 | (calcFunc-gammaQ (math-div v 2) (math-div chisq 2)))) |
136211a9 EZ |
942 | (put 'calcFunc-utpc 'math-expandable t) |
943 | ||
944 | (defun calcFunc-ltpc (chisq v) | |
945 | (if math-expand-formulas | |
946 | (math-normalize (list 'calcFunc-gammaP (list '/ v 2) (list '/ chisq 2))) | |
bf77c646 | 947 | (calcFunc-gammaP (math-div v 2) (math-div chisq 2)))) |
136211a9 EZ |
948 | (put 'calcFunc-ltpc 'math-expandable t) |
949 | ||
950 | ;;; F-distribution. | |
951 | (defun calcFunc-utpf (f v1 v2) | |
952 | (if math-expand-formulas | |
953 | (math-normalize (list 'calcFunc-betaI | |
954 | (list '/ v2 (list '+ v2 (list '* v1 f))) | |
955 | (list '/ v2 2) | |
956 | (list '/ v1 2))) | |
957 | (calcFunc-betaI (math-div v2 (math-add v2 (math-mul v1 f))) | |
958 | (math-div v2 2) | |
bf77c646 | 959 | (math-div v1 2)))) |
136211a9 EZ |
960 | (put 'calcFunc-utpf 'math-expandable t) |
961 | ||
962 | (defun calcFunc-ltpf (f v1 v2) | |
bf77c646 | 963 | (math-sub 1 (calcFunc-utpf f v1 v2))) |
136211a9 EZ |
964 | (put 'calcFunc-ltpf 'math-expandable t) |
965 | ||
966 | ;;; Normal. | |
967 | (defun calcFunc-utpn (x mean sdev) | |
968 | (if math-expand-formulas | |
969 | (math-normalize | |
970 | (list '/ | |
971 | (list '+ 1 | |
972 | (list 'calcFunc-erf | |
973 | (list '/ (list '- mean x) | |
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 mean x) | |
979 | (math-mul sdev (math-sqrt-2))))) | |
bf77c646 | 980 | '(float 5 -1)))) |
136211a9 EZ |
981 | (put 'calcFunc-utpn 'math-expandable t) |
982 | ||
983 | (defun calcFunc-ltpn (x mean sdev) | |
984 | (if math-expand-formulas | |
985 | (math-normalize | |
986 | (list '/ | |
987 | (list '+ 1 | |
988 | (list 'calcFunc-erf | |
989 | (list '/ (list '- x mean) | |
990 | (list '* sdev (list 'calcFunc-sqrt 2))))) | |
991 | 2)) | |
992 | (math-mul (math-add '(float 1 0) | |
993 | (calcFunc-erf | |
994 | (math-div (math-sub x mean) | |
995 | (math-mul sdev (math-sqrt-2))))) | |
bf77c646 | 996 | '(float 5 -1)))) |
136211a9 EZ |
997 | (put 'calcFunc-ltpn 'math-expandable t) |
998 | ||
999 | ;;; Poisson. | |
1000 | (defun calcFunc-utpp (n x) | |
1001 | (if math-expand-formulas | |
1002 | (math-normalize (list 'calcFunc-gammaP x n)) | |
bf77c646 | 1003 | (calcFunc-gammaP x n))) |
136211a9 EZ |
1004 | (put 'calcFunc-utpp 'math-expandable t) |
1005 | ||
1006 | (defun calcFunc-ltpp (n x) | |
1007 | (if math-expand-formulas | |
1008 | (math-normalize (list 'calcFunc-gammaQ x n)) | |
bf77c646 | 1009 | (calcFunc-gammaQ x n))) |
136211a9 EZ |
1010 | (put 'calcFunc-ltpp 'math-expandable t) |
1011 | ||
1012 | ;;; Student's t. (As defined in Abramowitz & Stegun and Numerical Recipes.) | |
1013 | (defun calcFunc-utpt (tt v) | |
1014 | (if math-expand-formulas | |
1015 | (math-normalize (list 'calcFunc-betaI | |
1016 | (list '/ v (list '+ v (list '^ tt 2))) | |
1017 | (list '/ v 2) | |
1018 | '(float 5 -1))) | |
1019 | (calcFunc-betaI (math-div v (math-add v (math-sqr tt))) | |
1020 | (math-div v 2) | |
bf77c646 | 1021 | '(float 5 -1)))) |
136211a9 EZ |
1022 | (put 'calcFunc-utpt 'math-expandable t) |
1023 | ||
1024 | (defun calcFunc-ltpt (tt v) | |
bf77c646 | 1025 | (math-sub 1 (calcFunc-utpt tt v))) |
136211a9 EZ |
1026 | (put 'calcFunc-ltpt 'math-expandable t) |
1027 | ||
300f28d3 | 1028 | (provide 'calc-funcs) |
136211a9 | 1029 | |
ab5796a9 | 1030 | ;;; arch-tag: 421ddb7a-550f-4dda-a31c-06638ebfc43a |
bf77c646 | 1031 | ;;; calc-funcs.el ends here |