HCoop / bpt / emacs.git / blame
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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> |
| 136211a9 EZ |
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 | |
| 075969b4 | 13 | ;; the Free Software Foundation; either version 3, or (at your option) |
| 7c671b23 GM |
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))) |
| 136211a9 | 149 | |
| 3bf97114 JB |
150 | (defun math-gammap1-raw (x &optional fprec nfprec) |
| 151 | "Compute gamma(1+X) to the appropriate precision." | |
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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) | |
| bf77c646 | 182 | 2))))))) |
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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) | |
| bf77c646 | 200 | (math-gamma-series next (math-mul x xinvsqr) xinvsqr term (+ n 2)))))) |
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201 | |
| 202 | ||
| 203 | ;;; Incomplete gamma function. | |
| 204 | ||
| 12f1951a | 205 | (defvar math-current-gamma-value nil) |
| 136211a9 EZ |
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))) | |
| bf77c646 | 217 | (math-div (calcFunc-gammag a x) math-current-gamma-value))))) |
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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))) | |
| bf77c646 | 238 | (math-div (calcFunc-gammaG a x) math-current-gamma-value))))) |
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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)) | |
| bf77c646 | 255 | (math-inc-gamma-cfrac a x)))))) |
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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)) | |
| bf77c646 | 272 | (math-inc-gamma-cfrac a x))))) |
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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))) | |
| bf77c646 | 280 | (math-inc-gamma-series-step start start a x)))))) |
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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 | |
| bf77c646 | 289 | (math-inc-gamma-series-step next term a x)))) |
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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) | |
| bf77c646 | 298 | a x)))) |
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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 | |
| bf77c646 | 315 | (math-inc-gamma-cfrac-step a0 a1 b0 b1 n fac next a x)))))) |
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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) | |
| bf77c646 | 332 | x)))))) |
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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)) | |
| bf77c646 | 341 | (math-sub 1 (calcFunc-erf x))))) |
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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))) | |
| bf77c646 | 347 | x)) |
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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)) | |
| bf77c646 | 360 | x))) |
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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)) | |
| bf77c646 | 373 | (calcFunc-gamma (math-add a b)))))) |
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374 | |
| 375 | ||
| 376 | ;;; Incomplete beta function. | |
| 377 | ||
| 12f1951a | 378 | (defvar math-current-beta-value nil) |
| 136211a9 EZ |
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))) | |
| bf77c646 | 400 | (math-div (calcFunc-betaB x a b) math-current-beta-value))))) |
| 136211a9 EZ |
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))) | |
| bf77c646 | 452 | b)))))))) |
| 136211a9 EZ |
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) | |
| bf77c646 | 463 | qab qap qam a b x))) |
| 136211a9 EZ |
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)) | |
| bf77c646 | 483 | qab qap qam a b x)))) |
| 136211a9 EZ |
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))) | |
| bf77c646 | 553 | (math-div ans (math-sub (math-mul 2 sum) bj)))))))) |
| 136211a9 EZ |
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 | |
| bf77c646 | 563 | (math-besJ-series next term k zz vk)))) |
| 136211a9 EZ |
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)) | |
| a8775b6d JB |
571 | (xx (math-add x |
| 572 | (eval-when-compile | |
| f83b9601 | 573 | (math-read-number-simple "-0.785398164")))) |
| 136211a9 | 574 | (a1 (math-poly-eval y |
| a8775b6d JB |
575 | (eval-when-compile |
| 576 | (list | |
| f83b9601 JB |
577 | (math-read-number-simple "0.0000002093887211") |
| 578 | (math-read-number-simple "-0.000002073370639") | |
| 579 | (math-read-number-simple "0.00002734510407") | |
| 580 | (math-read-number-simple "-0.001098628627") | |
| a8775b6d | 581 | '(float 1 0))))) |
| 136211a9 | 582 | (a2 (math-poly-eval y |
| a8775b6d JB |
583 | (eval-when-compile |
| 584 | (list | |
| f83b9601 JB |
585 | (math-read-number-simple "-0.0000000934935152") |
| 586 | (math-read-number-simple "0.0000007621095161") | |
| 587 | (math-read-number-simple "-0.000006911147651") | |
| 588 | (math-read-number-simple "0.0001430488765") | |
| 589 | (math-read-number-simple "-0.01562499995"))))) | |
| 136211a9 EZ |
590 | (sc (math-sin-cos-raw xx))) |
| 591 | (if yflag | |
| 592 | (setq sc (cons (math-neg (cdr sc)) (car sc)))) | |
| 593 | (math-mul (math-sqrt | |
| a8775b6d | 594 | (math-div (eval-when-compile |
| f83b9601 | 595 | (math-read-number-simple "0.636619722")) |
| a8775b6d | 596 | x)) |
| 136211a9 EZ |
597 | (math-sub (math-mul (cdr sc) a1) |
| 598 | (math-mul (car sc) (math-mul z a2)))))) | |
| 599 | (t | |
| 600 | (let ((y (math-sqr x))) | |
| 601 | (math-div (math-poly-eval y | |
| a8775b6d JB |
602 | (eval-when-compile |
| 603 | (list | |
| f83b9601 JB |
604 | (math-read-number-simple "-184.9052456") |
| 605 | (math-read-number-simple "77392.33017") | |
| 606 | (math-read-number-simple "-11214424.18") | |
| 607 | (math-read-number-simple "651619640.7") | |
| 608 | (math-read-number-simple "-13362590354.0") | |
| 609 | (math-read-number-simple "57568490574.0")))) | |
| 136211a9 | 610 | (math-poly-eval y |
| a8775b6d JB |
611 | (eval-when-compile |
| 612 | (list | |
| 613 | '(float 1 0) | |
| f83b9601 JB |
614 | (math-read-number-simple "267.8532712") |
| 615 | (math-read-number-simple "59272.64853") | |
| 616 | (math-read-number-simple "9494680.718") | |
| 617 | (math-read-number-simple "1029532985.0") | |
| 618 | (math-read-number-simple "57568490411.0"))))))))) | |
| 136211a9 EZ |
619 | |
| 620 | (defun math-besJ1 (x &optional yflag) | |
| 621 | (cond ((and (math-negp (calcFunc-re x)) (not yflag)) | |
| 622 | (math-neg (math-besJ1 (math-neg x)))) | |
| 623 | ((Math-lessp '(float 8 0) (math-abs-approx x)) | |
| 624 | (let* ((z (math-div '(float 8 0) x)) | |
| 625 | (y (math-sqr z)) | |
| a8775b6d | 626 | (xx (math-add x (eval-when-compile |
| f83b9601 | 627 | (math-read-number-simple "-2.356194491")))) |
| 136211a9 | 628 | (a1 (math-poly-eval y |
| a8775b6d JB |
629 | (eval-when-compile |
| 630 | (list | |
| f83b9601 JB |
631 | (math-read-number-simple "-0.000000240337019") |
| 632 | (math-read-number-simple "0.000002457520174") | |
| 633 | (math-read-number-simple "-0.00003516396496") | |
| a8775b6d JB |
634 | '(float 183105 -8) |
| 635 | '(float 1 0))))) | |
| 136211a9 | 636 | (a2 (math-poly-eval y |
| a8775b6d JB |
637 | (eval-when-compile |
| 638 | (list | |
| f83b9601 JB |
639 | (math-read-number-simple "0.000000105787412") |
| 640 | (math-read-number-simple "-0.00000088228987") | |
| 641 | (math-read-number-simple "0.000008449199096") | |
| 642 | (math-read-number-simple "-0.0002002690873") | |
| 643 | (math-read-number-simple "0.04687499995"))))) | |
| 136211a9 EZ |
644 | (sc (math-sin-cos-raw xx))) |
| 645 | (if yflag | |
| 646 | (setq sc (cons (math-neg (cdr sc)) (car sc))) | |
| 647 | (if (math-negp x) | |
| 648 | (setq sc (cons (math-neg (car sc)) (math-neg (cdr sc)))))) | |
| a8775b6d JB |
649 | (math-mul (math-sqrt (math-div |
| 650 | (eval-when-compile | |
| f83b9601 | 651 | (math-read-number-simple "0.636619722")) |
| a8775b6d | 652 | x)) |
| 136211a9 EZ |
653 | (math-sub (math-mul (cdr sc) a1) |
| 654 | (math-mul (car sc) (math-mul z a2)))))) | |
| 655 | (t | |
| 656 | (let ((y (math-sqr x))) | |
| 657 | (math-mul | |
| 658 | x | |
| 659 | (math-div (math-poly-eval y | |
| a8775b6d JB |
660 | (eval-when-compile |
| 661 | (list | |
| f83b9601 JB |
662 | (math-read-number-simple "-30.16036606") |
| 663 | (math-read-number-simple "15704.4826") | |
| 664 | (math-read-number-simple "-2972611.439") | |
| 665 | (math-read-number-simple "242396853.1") | |
| 666 | (math-read-number-simple "-7895059235.0") | |
| 667 | (math-read-number-simple "72362614232.0")))) | |
| 136211a9 | 668 | (math-poly-eval y |
| a8775b6d JB |
669 | (eval-when-compile |
| 670 | (list | |
| 671 | '(float 1 0) | |
| f83b9601 JB |
672 | (math-read-number-simple "376.9991397") |
| 673 | (math-read-number-simple "99447.43394") | |
| 674 | (math-read-number-simple "18583304.74") | |
| 675 | (math-read-number-simple "2300535178.0") | |
| 676 | (math-read-number-simple "144725228442.0")))))))))) | |
| 136211a9 EZ |
677 | |
| 678 | (defun calcFunc-besY (v x) | |
| 679 | (math-inexact-result) | |
| 680 | (or (math-numberp v) (math-reject-arg v 'numberp)) | |
| 681 | (or (math-numberp x) (math-reject-arg x 'numberp)) | |
| 682 | (let ((calc-internal-prec (min 8 calc-internal-prec))) | |
| 683 | (math-with-extra-prec 3 | |
| 684 | (setq x (math-float (math-normalize x))) | |
| 685 | (setq v (math-float (math-normalize v))) | |
| 686 | (cond ((not (math-num-integerp v)) | |
| 687 | (let ((sc (math-sin-cos-raw (math-mul v (math-pi))))) | |
| 688 | (math-div (math-sub (math-mul (calcFunc-besJ v x) (cdr sc)) | |
| 689 | (calcFunc-besJ (math-neg v) x)) | |
| 690 | (car sc)))) | |
| 691 | ((math-negp (setq v (math-trunc v))) | |
| 692 | (if (math-oddp v) | |
| 693 | (math-neg (calcFunc-besY (math-neg v) x)) | |
| 694 | (calcFunc-besY (math-neg v) x))) | |
| 695 | ((eq v 0) | |
| 696 | (math-besY0 x)) | |
| 697 | ((eq v 1) | |
| 698 | (math-besY1 x)) | |
| 699 | (t | |
| 700 | (let ((j 0) | |
| 701 | (bym (math-besY0 x)) | |
| 702 | (by (math-besY1 x)) | |
| 703 | (two-over-x (math-div 2 x)) | |
| 704 | byp) | |
| 705 | (while (< (setq j (1+ j)) v) | |
| 706 | (setq byp (math-sub (math-mul (math-mul j two-over-x) by) | |
| 707 | bym) | |
| 708 | bym by | |
| 709 | by byp)) | |
| bf77c646 | 710 | by)))))) |
| 136211a9 EZ |
711 | |
| 712 | (defun math-besY0 (x) | |
| 713 | (cond ((Math-lessp (math-abs-approx x) '(float 8 0)) | |
| 714 | (let ((y (math-sqr x))) | |
| 715 | (math-add | |
| 716 | (math-div (math-poly-eval y | |
| a8775b6d JB |
717 | (eval-when-compile |
| 718 | (list | |
| f83b9601 JB |
719 | (math-read-number-simple "228.4622733") |
| 720 | (math-read-number-simple "-86327.92757") | |
| 721 | (math-read-number-simple "10879881.29") | |
| 722 | (math-read-number-simple "-512359803.6") | |
| 723 | (math-read-number-simple "7062834065.0") | |
| 724 | (math-read-number-simple "-2957821389.0")))) | |
| 136211a9 | 725 | (math-poly-eval y |
| a8775b6d JB |
726 | (eval-when-compile |
| 727 | (list | |
| 728 | '(float 1 0) | |
| f83b9601 JB |
729 | (math-read-number-simple "226.1030244") |
| 730 | (math-read-number-simple "47447.2647") | |
| 731 | (math-read-number-simple "7189466.438") | |
| 732 | (math-read-number-simple "745249964.8") | |
| 733 | (math-read-number-simple "40076544269.0"))))) | |
| a8775b6d | 734 | (math-mul (eval-when-compile |
| f83b9601 | 735 | (math-read-number-simple "0.636619772")) |
| 136211a9 EZ |
736 | (math-mul (math-besJ0 x) (math-ln-raw x)))))) |
| 737 | ((math-negp (calcFunc-re x)) | |
| 738 | (math-add (math-besJ0 (math-neg x) t) | |
| 739 | (math-mul '(cplx 0 2) | |
| 740 | (math-besJ0 (math-neg x))))) | |
| 741 | (t | |
| bf77c646 | 742 | (math-besJ0 x t)))) |
| 136211a9 EZ |
743 | |
| 744 | (defun math-besY1 (x) | |
| 745 | (cond ((Math-lessp (math-abs-approx x) '(float 8 0)) | |
| 746 | (let ((y (math-sqr x))) | |
| 747 | (math-add | |
| 748 | (math-mul | |
| 749 | x | |
| 750 | (math-div (math-poly-eval y | |
| a8775b6d JB |
751 | (eval-when-compile |
| 752 | (list | |
| f83b9601 JB |
753 | (math-read-number-simple "8511.937935") |
| 754 | (math-read-number-simple "-4237922.726") | |
| 755 | (math-read-number-simple "734926455.1") | |
| 756 | (math-read-number-simple "-51534381390.0") | |
| 757 | (math-read-number-simple "1275274390000.0") | |
| 758 | (math-read-number-simple "-4900604943000.0")))) | |
| 136211a9 | 759 | (math-poly-eval y |
| a8775b6d JB |
760 | (eval-when-compile |
| 761 | (list | |
| 762 | '(float 1 0) | |
| f83b9601 JB |
763 | (math-read-number-simple "354.9632885") |
| 764 | (math-read-number-simple "102042.605") | |
| 765 | (math-read-number-simple "22459040.02") | |
| 766 | (math-read-number-simple "3733650367.0") | |
| 767 | (math-read-number-simple "424441966400.0") | |
| 768 | (math-read-number-simple "24995805700000.0")))))) | |
| 769 | (math-mul (eval-when-compile (math-read-number-simple "0.636619772")) | |
| a8775b6d | 770 | (math-sub (math-mul (math-besJ1 x) (math-ln-raw x)) |
| 136211a9 EZ |
771 | (math-div 1 x)))))) |
| 772 | ((math-negp (calcFunc-re x)) | |
| 773 | (math-neg | |
| 774 | (math-add (math-besJ1 (math-neg x) t) | |
| 775 | (math-mul '(cplx 0 2) | |
| 776 | (math-besJ1 (math-neg x)))))) | |
| 777 | (t | |
| bf77c646 | 778 | (math-besJ1 x t)))) |
| 136211a9 EZ |
779 | |
| 780 | (defun math-poly-eval (x coefs) | |
| 781 | (let ((accum (car coefs))) | |
| 782 | (while (setq coefs (cdr coefs)) | |
| 783 | (setq accum (math-add (car coefs) (math-mul accum x)))) | |
| bf77c646 | 784 | accum)) |
| 136211a9 EZ |
785 | |
| 786 | ||
| 787 | ;;;; Bernoulli and Euler polynomials and numbers. | |
| 788 | ||
| 789 | (defun calcFunc-bern (n &optional x) | |
| 790 | (if (and x (not (math-zerop x))) | |
| 791 | (if (and calc-symbolic-mode (math-floatp x)) | |
| 792 | (math-inexact-result) | |
| 793 | (math-build-polynomial-expr (math-bernoulli-coefs n) x)) | |
| 794 | (or (math-num-natnump n) (math-reject-arg n 'natnump)) | |
| 795 | (if (consp n) | |
| 796 | (progn | |
| 797 | (math-inexact-result) | |
| 798 | (math-float (math-bernoulli-number (math-trunc n)))) | |
| bf77c646 | 799 | (math-bernoulli-number n)))) |
| 136211a9 EZ |
800 | |
| 801 | (defun calcFunc-euler (n &optional x) | |
| 802 | (or (math-num-natnump n) (math-reject-arg n 'natnump)) | |
| 803 | (if x | |
| 804 | (let* ((n1 (math-add n 1)) | |
| 805 | (coefs (math-bernoulli-coefs n1)) | |
| 806 | (fac (math-div (math-pow 2 n1) n1)) | |
| 807 | (k -1) | |
| 808 | (x1 (math-div (math-add x 1) 2)) | |
| 809 | (x2 (math-div x 2))) | |
| 810 | (if (math-numberp x) | |
| 811 | (if (and calc-symbolic-mode (math-floatp x)) | |
| 812 | (math-inexact-result) | |
| 813 | (math-mul fac | |
| 814 | (math-sub (math-build-polynomial-expr coefs x1) | |
| 815 | (math-build-polynomial-expr coefs x2)))) | |
| 816 | (calcFunc-collect | |
| 817 | (math-reduce-vec | |
| 818 | 'math-add | |
| 819 | (cons 'vec | |
| 820 | (mapcar (function | |
| 821 | (lambda (c) | |
| 822 | (setq k (1+ k)) | |
| 823 | (math-mul (math-mul fac c) | |
| 824 | (math-sub (math-pow x1 k) | |
| 825 | (math-pow x2 k))))) | |
| 826 | coefs))) | |
| 827 | x))) | |
| 828 | (math-mul (math-pow 2 n) | |
| 829 | (if (consp n) | |
| 830 | (progn | |
| 831 | (math-inexact-result) | |
| 832 | (calcFunc-euler n '(float 5 -1))) | |
| bf77c646 | 833 | (calcFunc-euler n '(frac 1 2)))))) |
| 136211a9 | 834 | |
| a8775b6d JB |
835 | (defvar math-bernoulli-b-cache |
| 836 | (eval-when-compile | |
| 837 | (list | |
| 838 | (list 'frac | |
| 839 | -174611 | |
| f83b9601 | 840 | (math-read-number-simple "802857662698291200000")) |
| a8775b6d JB |
841 | (list 'frac |
| 842 | 43867 | |
| f83b9601 | 843 | (math-read-number-simple "5109094217170944000")) |
| a8775b6d JB |
844 | (list 'frac |
| 845 | -3617 | |
| f83b9601 | 846 | (math-read-number-simple "10670622842880000")) |
| a8775b6d JB |
847 | (list 'frac |
| 848 | 1 | |
| f83b9601 | 849 | (math-read-number-simple "74724249600")) |
| a8775b6d JB |
850 | (list 'frac |
| 851 | -691 | |
| f83b9601 | 852 | (math-read-number-simple "1307674368000")) |
| a8775b6d JB |
853 | (list 'frac |
| 854 | 1 | |
| f83b9601 | 855 | (math-read-number-simple "47900160")) |
| a8775b6d JB |
856 | (list 'frac |
| 857 | -1 | |
| f83b9601 | 858 | (math-read-number-simple "1209600")) |
| a8775b6d JB |
859 | (list 'frac |
| 860 | 1 | |
| 861 | 30240) | |
| 862 | (list 'frac | |
| 863 | -1 | |
| 864 | 720) | |
| 865 | (list 'frac | |
| 866 | 1 | |
| 867 | 12) | |
| 868 | 1 ))) | |
| 12f1951a CW |
869 | |
| 870 | (defvar math-bernoulli-B-cache '((frac -174611 330) (frac 43867 798) | |
| 871 | (frac -3617 510) (frac 7 6) (frac -691 2730) | |
| 872 | (frac 5 66) (frac -1 30) (frac 1 42) | |
| 873 | (frac -1 30) (frac 1 6) 1 )) | |
| 874 | ||
| 875 | (defvar math-bernoulli-cache-size 11) | |
| 136211a9 EZ |
876 | (defun math-bernoulli-coefs (n) |
| 877 | (let* ((coefs (list (calcFunc-bern n))) | |
| 878 | (nn (math-trunc n)) | |
| 879 | (k nn) | |
| 880 | (term nn) | |
| 881 | coef | |
| 882 | (calc-prefer-frac (or (integerp n) calc-prefer-frac))) | |
| 883 | (while (>= (setq k (1- k)) 0) | |
| 884 | (setq term (math-div term (- nn k)) | |
| 885 | coef (math-mul term (math-bernoulli-number k)) | |
| 886 | coefs (cons (if (consp n) (math-float coef) coef) coefs) | |
| 887 | term (math-mul term k))) | |
| bf77c646 | 888 | (nreverse coefs))) |
| 136211a9 EZ |
889 | |
| 890 | (defun math-bernoulli-number (n) | |
| 891 | (if (= (% n 2) 1) | |
| 892 | (if (= n 1) | |
| 893 | '(frac -1 2) | |
| 894 | 0) | |
| 895 | (setq n (/ n 2)) | |
| 896 | (while (>= n math-bernoulli-cache-size) | |
| 897 | (let* ((sum 0) | |
| 898 | (nk 1) ; nk = n-k+1 | |
| 899 | (fact 1) ; fact = (n-k+1)! | |
| 900 | ofact | |
| 901 | (p math-bernoulli-b-cache) | |
| 902 | (calc-prefer-frac t)) | |
| 903 | (math-working "bernoulli B" (* 2 math-bernoulli-cache-size)) | |
| 904 | (while p | |
| 905 | (setq nk (+ nk 2) | |
| 906 | ofact fact | |
| 907 | fact (math-mul fact (* nk (1- nk))) | |
| 908 | sum (math-add sum (math-div (car p) fact)) | |
| 909 | p (cdr p))) | |
| 910 | (setq ofact (math-mul ofact (1- nk)) | |
| 911 | sum (math-sub (math-div '(frac 1 2) ofact) sum) | |
| 912 | math-bernoulli-b-cache (cons sum math-bernoulli-b-cache) | |
| 913 | math-bernoulli-B-cache (cons (math-mul sum ofact) | |
| 914 | math-bernoulli-B-cache) | |
| 915 | math-bernoulli-cache-size (1+ math-bernoulli-cache-size)))) | |
| bf77c646 | 916 | (nth (- math-bernoulli-cache-size n 1) math-bernoulli-B-cache))) |
| 136211a9 EZ |
917 | |
| 918 | ;;; Bn = n! bn | |
| 919 | ;;; bn = - sum_k=0^n-1 bk / (n-k+1)! | |
| 920 | ||
| 921 | ;;; A faster method would be to use "tangent numbers", c.f., Concrete | |
| 922 | ;;; Mathematics pg. 273. | |
| 923 | ||
| 136211a9 EZ |
924 | |
| 925 | ;;; Probability distributions. | |
| 926 | ||
| 927 | ;;; Binomial. | |
| 928 | (defun calcFunc-utpb (x n p) | |
| 929 | (if math-expand-formulas | |
| 930 | (math-normalize (list 'calcFunc-betaI p x (list '+ (list '- n x) 1))) | |
| bf77c646 | 931 | (calcFunc-betaI p x (math-add (math-sub n x) 1)))) |
| 136211a9 EZ |
932 | (put 'calcFunc-utpb 'math-expandable t) |
| 933 | ||
| 934 | (defun calcFunc-ltpb (x n p) | |
| bf77c646 | 935 | (math-sub 1 (calcFunc-utpb x n p))) |
| 136211a9 EZ |
936 | (put 'calcFunc-ltpb 'math-expandable t) |
| 937 | ||
| 938 | ;;; Chi-square. | |
| 939 | (defun calcFunc-utpc (chisq v) | |
| 940 | (if math-expand-formulas | |
| 941 | (math-normalize (list 'calcFunc-gammaQ (list '/ v 2) (list '/ chisq 2))) | |
| bf77c646 | 942 | (calcFunc-gammaQ (math-div v 2) (math-div chisq 2)))) |
| 136211a9 EZ |
943 | (put 'calcFunc-utpc 'math-expandable t) |
| 944 | ||
| 945 | (defun calcFunc-ltpc (chisq v) | |
| 946 | (if math-expand-formulas | |
| 947 | (math-normalize (list 'calcFunc-gammaP (list '/ v 2) (list '/ chisq 2))) | |
| bf77c646 | 948 | (calcFunc-gammaP (math-div v 2) (math-div chisq 2)))) |
| 136211a9 EZ |
949 | (put 'calcFunc-ltpc 'math-expandable t) |
| 950 | ||
| 951 | ;;; F-distribution. | |
| 952 | (defun calcFunc-utpf (f v1 v2) | |
| 953 | (if math-expand-formulas | |
| 954 | (math-normalize (list 'calcFunc-betaI | |
| 955 | (list '/ v2 (list '+ v2 (list '* v1 f))) | |
| 956 | (list '/ v2 2) | |
| 957 | (list '/ v1 2))) | |
| 958 | (calcFunc-betaI (math-div v2 (math-add v2 (math-mul v1 f))) | |
| 959 | (math-div v2 2) | |
| bf77c646 | 960 | (math-div v1 2)))) |
| 136211a9 EZ |
961 | (put 'calcFunc-utpf 'math-expandable t) |
| 962 | ||
| 963 | (defun calcFunc-ltpf (f v1 v2) | |
| bf77c646 | 964 | (math-sub 1 (calcFunc-utpf f v1 v2))) |
| 136211a9 EZ |
965 | (put 'calcFunc-ltpf 'math-expandable t) |
| 966 | ||
| 967 | ;;; Normal. | |
| 968 | (defun calcFunc-utpn (x mean sdev) | |
| 969 | (if math-expand-formulas | |
| 970 | (math-normalize | |
| 971 | (list '/ | |
| 972 | (list '+ 1 | |
| 973 | (list 'calcFunc-erf | |
| 974 | (list '/ (list '- mean x) | |
| 975 | (list '* sdev (list 'calcFunc-sqrt 2))))) | |
| 976 | 2)) | |
| 977 | (math-mul (math-add '(float 1 0) | |
| 978 | (calcFunc-erf | |
| 979 | (math-div (math-sub mean x) | |
| 980 | (math-mul sdev (math-sqrt-2))))) | |
| bf77c646 | 981 | '(float 5 -1)))) |
| 136211a9 EZ |
982 | (put 'calcFunc-utpn 'math-expandable t) |
| 983 | ||
| 984 | (defun calcFunc-ltpn (x mean sdev) | |
| 985 | (if math-expand-formulas | |
| 986 | (math-normalize | |
| 987 | (list '/ | |
| 988 | (list '+ 1 | |
| 989 | (list 'calcFunc-erf | |
| 990 | (list '/ (list '- x mean) | |
| 991 | (list '* sdev (list 'calcFunc-sqrt 2))))) | |
| 992 | 2)) | |
| 993 | (math-mul (math-add '(float 1 0) | |
| 994 | (calcFunc-erf | |
| 995 | (math-div (math-sub x mean) | |
| 996 | (math-mul sdev (math-sqrt-2))))) | |
| bf77c646 | 997 | '(float 5 -1)))) |
| 136211a9 EZ |
998 | (put 'calcFunc-ltpn 'math-expandable t) |
| 999 | ||
| 1000 | ;;; Poisson. | |
| 1001 | (defun calcFunc-utpp (n x) | |
| 1002 | (if math-expand-formulas | |
| 1003 | (math-normalize (list 'calcFunc-gammaP x n)) | |
| bf77c646 | 1004 | (calcFunc-gammaP x n))) |
| 136211a9 EZ |
1005 | (put 'calcFunc-utpp 'math-expandable t) |
| 1006 | ||
| 1007 | (defun calcFunc-ltpp (n x) | |
| 1008 | (if math-expand-formulas | |
| 1009 | (math-normalize (list 'calcFunc-gammaQ x n)) | |
| bf77c646 | 1010 | (calcFunc-gammaQ x n))) |
| 136211a9 EZ |
1011 | (put 'calcFunc-ltpp 'math-expandable t) |
| 1012 | ||
| 1013 | ;;; Student's t. (As defined in Abramowitz & Stegun and Numerical Recipes.) | |
| 1014 | (defun calcFunc-utpt (tt v) | |
| 1015 | (if math-expand-formulas | |
| 1016 | (math-normalize (list 'calcFunc-betaI | |
| 1017 | (list '/ v (list '+ v (list '^ tt 2))) | |
| 1018 | (list '/ v 2) | |
| 1019 | '(float 5 -1))) | |
| 1020 | (calcFunc-betaI (math-div v (math-add v (math-sqr tt))) | |
| 1021 | (math-div v 2) | |
| bf77c646 | 1022 | '(float 5 -1)))) |
| 136211a9 EZ |
1023 | (put 'calcFunc-utpt 'math-expandable t) |
| 1024 | ||
| 1025 | (defun calcFunc-ltpt (tt v) | |
| bf77c646 | 1026 | (math-sub 1 (calcFunc-utpt tt v))) |
| 136211a9 EZ |
1027 | (put 'calcFunc-ltpt 'math-expandable t) |
| 1028 | ||
| 300f28d3 | 1029 | (provide 'calc-funcs) |
| 136211a9 | 1030 | |
| ab5796a9 | 1031 | ;;; arch-tag: 421ddb7a-550f-4dda-a31c-06638ebfc43a |
| bf77c646 | 1032 | ;;; calc-funcs.el ends here |