HCoop / bpt / emacs.git / blame
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| 3132f345 CW |
1 | ;;; calc-comb.el --- combinatoric functions for Calc |
| 2 | ||
| bf77c646 | 3 | ;; Copyright (C) 1990, 1991, 1992, 1993, 2001 Free Software Foundation, Inc. |
| 3132f345 CW |
4 | |
| 5 | ;; Author: David Gillespie <daveg@synaptics.com> | |
| 43f34ccc | 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 | ||
| 3132f345 | 25 | ;;; Commentary: |
| 136211a9 | 26 | |
| 3132f345 | 27 | ;;; Code: |
| 136211a9 EZ |
28 | |
| 29 | ;; This file is autoloaded from calc-ext.el. | |
| 136211a9 | 30 | |
| 43f34ccc | 31 | (require 'calc-ext) |
| 136211a9 EZ |
32 | (require 'calc-macs) |
| 33 | ||
| 3132f345 CW |
34 | (defconst math-primes-table |
| 35 | [2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 | |
| 36 | 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 | |
| 37 | 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 | |
| 38 | 281 283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383 | |
| 39 | 389 397 401 409 419 421 431 433 439 443 449 457 461 463 467 479 487 | |
| 40 | 491 499 503 509 521 523 541 547 557 563 569 571 577 587 593 599 601 | |
| 41 | 607 613 617 619 631 641 643 647 653 659 661 673 677 683 691 701 709 | |
| 42 | 719 727 733 739 743 751 757 761 769 773 787 797 809 811 821 823 827 | |
| 43 | 829 839 853 857 859 863 877 881 883 887 907 911 919 929 937 941 947 | |
| 44 | 953 967 971 977 983 991 997 1009 1013 1019 1021 1031 1033 1039 1049 | |
| 45 | 1051 1061 1063 1069 1087 1091 1093 1097 1103 1109 1117 1123 1129 1151 | |
| 46 | 1153 1163 1171 1181 1187 1193 1201 1213 1217 1223 1229 1231 1237 1249 | |
| 47 | 1259 1277 1279 1283 1289 1291 1297 1301 1303 1307 1319 1321 1327 1361 | |
| 48 | 1367 1373 1381 1399 1409 1423 1427 1429 1433 1439 1447 1451 1453 1459 | |
| 49 | 1471 1481 1483 1487 1489 1493 1499 1511 1523 1531 1543 1549 1553 1559 | |
| 50 | 1567 1571 1579 1583 1597 1601 1607 1609 1613 1619 1621 1627 1637 1657 | |
| 51 | 1663 1667 1669 1693 1697 1699 1709 1721 1723 1733 1741 1747 1753 1759 | |
| 52 | 1777 1783 1787 1789 1801 1811 1823 1831 1847 1861 1867 1871 1873 1877 | |
| 53 | 1879 1889 1901 1907 1913 1931 1933 1949 1951 1973 1979 1987 1993 1997 | |
| 54 | 1999 2003 2011 2017 2027 2029 2039 2053 2063 2069 2081 2083 2087 2089 | |
| 55 | 2099 2111 2113 2129 2131 2137 2141 2143 2153 2161 2179 2203 2207 2213 | |
| 56 | 2221 2237 2239 2243 2251 2267 2269 2273 2281 2287 2293 2297 2309 2311 | |
| 57 | 2333 2339 2341 2347 2351 2357 2371 2377 2381 2383 2389 2393 2399 2411 | |
| 58 | 2417 2423 2437 2441 2447 2459 2467 2473 2477 2503 2521 2531 2539 2543 | |
| 59 | 2549 2551 2557 2579 2591 2593 2609 2617 2621 2633 2647 2657 2659 2663 | |
| 60 | 2671 2677 2683 2687 2689 2693 2699 2707 2711 2713 2719 2729 2731 2741 | |
| 61 | 2749 2753 2767 2777 2789 2791 2797 2801 2803 2819 2833 2837 2843 2851 | |
| 62 | 2857 2861 2879 2887 2897 2903 2909 2917 2927 2939 2953 2957 2963 2969 | |
| 63 | 2971 2999 3001 3011 3019 3023 3037 3041 3049 3061 3067 3079 3083 3089 | |
| 64 | 3109 3119 3121 3137 3163 3167 3169 3181 3187 3191 3203 3209 3217 3221 | |
| 65 | 3229 3251 3253 3257 3259 3271 3299 3301 3307 3313 3319 3323 3329 3331 | |
| 66 | 3343 3347 3359 3361 3371 3373 3389 3391 3407 3413 3433 3449 3457 3461 | |
| 67 | 3463 3467 3469 3491 3499 3511 3517 3527 3529 3533 3539 3541 3547 3557 | |
| 68 | 3559 3571 3581 3583 3593 3607 3613 3617 3623 3631 3637 3643 3659 3671 | |
| 69 | 3673 3677 3691 3697 3701 3709 3719 3727 3733 3739 3761 3767 3769 3779 | |
| 70 | 3793 3797 3803 3821 3823 3833 3847 3851 3853 3863 3877 3881 3889 3907 | |
| 71 | 3911 3917 3919 3923 3929 3931 3943 3947 3967 3989 4001 4003 4007 4013 | |
| 72 | 4019 4021 4027 4049 4051 4057 4073 4079 4091 4093 4099 4111 4127 4129 | |
| 73 | 4133 4139 4153 4157 4159 4177 4201 4211 4217 4219 4229 4231 4241 4243 | |
| 74 | 4253 4259 4261 4271 4273 4283 4289 4297 4327 4337 4339 4349 4357 4363 | |
| 75 | 4373 4391 4397 4409 4421 4423 4441 4447 4451 4457 4463 4481 4483 4493 | |
| 76 | 4507 4513 4517 4519 4523 4547 4549 4561 4567 4583 4591 4597 4603 4621 | |
| 77 | 4637 4639 4643 4649 4651 4657 4663 4673 4679 4691 4703 4721 4723 4729 | |
| 78 | 4733 4751 4759 4783 4787 4789 4793 4799 4801 4813 4817 4831 4861 4871 | |
| 79 | 4877 4889 4903 4909 4919 4931 4933 4937 4943 4951 4957 4967 4969 4973 | |
| 80 | 4987 4993 4999 5003]) | |
| 136211a9 | 81 | |
| 8d7498c1 JB |
82 | ;; The variable math-prime-factors-finished is set by calcFunc-prfac to |
| 83 | ;; indicate whether factoring is complete, and used by calcFunc-factors, | |
| 84 | ;; calcFunc-totient and calcFunc-moebius. | |
| 85 | (defvar math-prime-factors-finished) | |
| 86 | ||
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87 | ;;; Combinatorics |
| 88 | ||
| 89 | (defun calc-gcd (arg) | |
| 90 | (interactive "P") | |
| 91 | (calc-slow-wrapper | |
| bf77c646 | 92 | (calc-binary-op "gcd" 'calcFunc-gcd arg))) |
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93 | |
| 94 | (defun calc-lcm (arg) | |
| 95 | (interactive "P") | |
| 96 | (calc-slow-wrapper | |
| bf77c646 | 97 | (calc-binary-op "lcm" 'calcFunc-lcm arg))) |
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98 | |
| 99 | (defun calc-extended-gcd () | |
| 100 | (interactive) | |
| 101 | (calc-slow-wrapper | |
| bf77c646 | 102 | (calc-enter-result 2 "egcd" (cons 'calcFunc-egcd (calc-top-list-n 2))))) |
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103 | |
| 104 | (defun calc-factorial (arg) | |
| 105 | (interactive "P") | |
| 106 | (calc-slow-wrapper | |
| bf77c646 | 107 | (calc-unary-op "fact" 'calcFunc-fact arg))) |
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108 | |
| 109 | (defun calc-gamma (arg) | |
| 110 | (interactive "P") | |
| 111 | (calc-slow-wrapper | |
| bf77c646 | 112 | (calc-unary-op "gmma" 'calcFunc-gamma arg))) |
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113 | |
| 114 | (defun calc-double-factorial (arg) | |
| 115 | (interactive "P") | |
| 116 | (calc-slow-wrapper | |
| bf77c646 | 117 | (calc-unary-op "dfac" 'calcFunc-dfact arg))) |
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118 | |
| 119 | (defun calc-choose (arg) | |
| 120 | (interactive "P") | |
| 121 | (calc-slow-wrapper | |
| 122 | (if (calc-is-hyperbolic) | |
| 123 | (calc-binary-op "perm" 'calcFunc-perm arg) | |
| bf77c646 | 124 | (calc-binary-op "chos" 'calcFunc-choose arg)))) |
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125 | |
| 126 | (defun calc-perm (arg) | |
| 127 | (interactive "P") | |
| 128 | (calc-hyperbolic-func) | |
| bf77c646 | 129 | (calc-choose arg)) |
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130 | |
| 131 | (defvar calc-last-random-limit '(float 1 0)) | |
| 132 | (defun calc-random (n) | |
| 133 | (interactive "P") | |
| 134 | (calc-slow-wrapper | |
| 135 | (if n | |
| 136 | (calc-enter-result 0 "rand" (list 'calcFunc-random | |
| 137 | (calc-get-random-limit | |
| 138 | (prefix-numeric-value n)))) | |
| 139 | (calc-enter-result 1 "rand" (list 'calcFunc-random | |
| 140 | (calc-get-random-limit | |
| bf77c646 | 141 | (calc-top-n 1))))))) |
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142 | |
| 143 | (defun calc-get-random-limit (val) | |
| 144 | (if (eq val 0) | |
| 145 | calc-last-random-limit | |
| bf77c646 | 146 | (setq calc-last-random-limit val))) |
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147 | |
| 148 | (defun calc-rrandom () | |
| 149 | (interactive) | |
| 150 | (calc-slow-wrapper | |
| 151 | (setq calc-last-random-limit '(float 1 0)) | |
| bf77c646 | 152 | (calc-enter-result 0 "rand" (list 'calcFunc-random '(float 1 0))))) |
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153 | |
| 154 | (defun calc-random-again (arg) | |
| 155 | (interactive "p") | |
| 156 | (calc-slow-wrapper | |
| 157 | (while (>= (setq arg (1- arg)) 0) | |
| 158 | (calc-enter-result 0 "rand" (list 'calcFunc-random | |
| bf77c646 | 159 | calc-last-random-limit))))) |
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160 | |
| 161 | (defun calc-shuffle (n) | |
| 162 | (interactive "P") | |
| 163 | (calc-slow-wrapper | |
| 164 | (if n | |
| 165 | (calc-enter-result 1 "shuf" (list 'calcFunc-shuffle | |
| 166 | (prefix-numeric-value n) | |
| 167 | (calc-get-random-limit | |
| 168 | (calc-top-n 1)))) | |
| 169 | (calc-enter-result 2 "shuf" (list 'calcFunc-shuffle | |
| 170 | (calc-top-n 1) | |
| 171 | (calc-get-random-limit | |
| bf77c646 | 172 | (calc-top-n 2))))))) |
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173 | |
| 174 | (defun calc-report-prime-test (res) | |
| 175 | (cond ((eq (car res) t) | |
| 176 | (calc-record-message "prim" "Prime (guaranteed)")) | |
| 177 | ((eq (car res) nil) | |
| 178 | (if (cdr res) | |
| 179 | (if (eq (nth 1 res) 'unknown) | |
| 180 | (calc-record-message | |
| 181 | "prim" "Non-prime (factors unknown)") | |
| 182 | (calc-record-message | |
| 183 | "prim" "Non-prime (%s is a factor)" | |
| 184 | (math-format-number (nth 1 res)))) | |
| 185 | (calc-record-message "prim" "Non-prime"))) | |
| 186 | (t | |
| 187 | (calc-record-message | |
| 188 | "prim" "Probably prime (%d iters; %s%% chance of error)" | |
| 189 | (nth 1 res) | |
| 190 | (let ((calc-float-format '(fix 2))) | |
| bf77c646 | 191 | (math-format-number (nth 2 res))))))) |
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192 | |
| 193 | (defun calc-prime-test (iters) | |
| 194 | (interactive "p") | |
| 195 | (calc-slow-wrapper | |
| 196 | (let* ((n (calc-top-n 1)) | |
| 197 | (res (math-prime-test n iters))) | |
| bf77c646 | 198 | (calc-report-prime-test res)))) |
| 136211a9 | 199 | |
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200 | (defvar calc-verbose-nextprime nil) |
| 201 | ||
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202 | (defun calc-next-prime (iters) |
| 203 | (interactive "p") | |
| 204 | (calc-slow-wrapper | |
| 205 | (let ((calc-verbose-nextprime t)) | |
| 206 | (if (calc-is-inverse) | |
| 207 | (calc-enter-result 1 "prvp" (list 'calcFunc-prevprime | |
| 208 | (calc-top-n 1) (math-abs iters))) | |
| 209 | (calc-enter-result 1 "nxtp" (list 'calcFunc-nextprime | |
| bf77c646 | 210 | (calc-top-n 1) (math-abs iters))))))) |
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211 | |
| 212 | (defun calc-prev-prime (iters) | |
| 213 | (interactive "p") | |
| 214 | (calc-invert-func) | |
| bf77c646 | 215 | (calc-next-prime iters)) |
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216 | |
| 217 | (defun calc-prime-factors (iters) | |
| 218 | (interactive "p") | |
| 219 | (calc-slow-wrapper | |
| 220 | (let ((res (calcFunc-prfac (calc-top-n 1)))) | |
| 221 | (if (not math-prime-factors-finished) | |
| 222 | (calc-record-message "pfac" "Warning: May not be fully factored")) | |
| bf77c646 | 223 | (calc-enter-result 1 "pfac" res)))) |
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224 | |
| 225 | (defun calc-totient (arg) | |
| 226 | (interactive "P") | |
| 227 | (calc-slow-wrapper | |
| bf77c646 | 228 | (calc-unary-op "phi" 'calcFunc-totient arg))) |
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229 | |
| 230 | (defun calc-moebius (arg) | |
| 231 | (interactive "P") | |
| 232 | (calc-slow-wrapper | |
| bf77c646 | 233 | (calc-unary-op "mu" 'calcFunc-moebius arg))) |
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234 | |
| 235 | ||
| 236 | (defun calcFunc-gcd (a b) | |
| 237 | (if (Math-messy-integerp a) | |
| 238 | (setq a (math-trunc a))) | |
| 239 | (if (Math-messy-integerp b) | |
| 240 | (setq b (math-trunc b))) | |
| 241 | (cond ((and (Math-integerp a) (Math-integerp b)) | |
| 242 | (math-gcd a b)) | |
| 243 | ((Math-looks-negp a) | |
| 244 | (calcFunc-gcd (math-neg a) b)) | |
| 245 | ((Math-looks-negp b) | |
| 246 | (calcFunc-gcd a (math-neg b))) | |
| 247 | ((Math-zerop a) b) | |
| 248 | ((Math-zerop b) a) | |
| 249 | ((and (Math-ratp a) | |
| 250 | (Math-ratp b)) | |
| 251 | (math-make-frac (math-gcd (if (eq (car-safe a) 'frac) (nth 1 a) a) | |
| 252 | (if (eq (car-safe b) 'frac) (nth 1 b) b)) | |
| 253 | (calcFunc-lcm | |
| 254 | (if (eq (car-safe a) 'frac) (nth 2 a) 1) | |
| 255 | (if (eq (car-safe b) 'frac) (nth 2 b) 1)))) | |
| 256 | ((not (Math-integerp a)) | |
| 257 | (calc-record-why 'integerp a) | |
| 258 | (list 'calcFunc-gcd a b)) | |
| 259 | (t | |
| 260 | (calc-record-why 'integerp b) | |
| bf77c646 | 261 | (list 'calcFunc-gcd a b)))) |
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262 | |
| 263 | (defun calcFunc-lcm (a b) | |
| 264 | (let ((g (calcFunc-gcd a b))) | |
| 265 | (if (Math-numberp g) | |
| 266 | (math-div (math-mul a b) g) | |
| bf77c646 | 267 | (list 'calcFunc-lcm a b)))) |
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268 | |
| 269 | (defun calcFunc-egcd (a b) ; Knuth section 4.5.2 | |
| 270 | (cond | |
| 271 | ((not (Math-integerp a)) | |
| 272 | (if (Math-messy-integerp a) | |
| 273 | (calcFunc-egcd (math-trunc a) b) | |
| 274 | (calc-record-why 'integerp a) | |
| 275 | (list 'calcFunc-egcd a b))) | |
| 276 | ((not (Math-integerp b)) | |
| 277 | (if (Math-messy-integerp b) | |
| 278 | (calcFunc-egcd a (math-trunc b)) | |
| 279 | (calc-record-why 'integerp b) | |
| 280 | (list 'calcFunc-egcd a b))) | |
| 281 | (t | |
| 282 | (let ((u1 1) (u2 0) (u3 a) | |
| 283 | (v1 0) (v2 1) (v3 b) | |
| 284 | t1 t2 q) | |
| 285 | (while (not (eq v3 0)) | |
| 286 | (setq q (math-idivmod u3 v3) | |
| 287 | t1 (math-sub u1 (math-mul v1 (car q))) | |
| 288 | t2 (math-sub u2 (math-mul v2 (car q))) | |
| 289 | u1 v1 u2 v2 u3 v3 | |
| 290 | v1 t1 v2 t2 v3 (cdr q))) | |
| bf77c646 | 291 | (list 'vec u3 u1 u2))))) |
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292 | |
| 293 | ||
| 294 | ;;; Factorial and related functions. | |
| 295 | ||
| 296 | (defun calcFunc-fact (n) ; [I I] [F F] [Public] | |
| 297 | (let (temp) | |
| 298 | (cond ((Math-integer-negp n) | |
| 299 | (if calc-infinite-mode | |
| 300 | '(var uinf var-uinf) | |
| 301 | (math-reject-arg n 'range))) | |
| 302 | ((integerp n) | |
| 303 | (if (<= n 20) | |
| 304 | (aref '[1 1 2 6 24 120 720 5040 40320 362880 | |
| 305 | (bigpos 800 628 3) (bigpos 800 916 39) | |
| 306 | (bigpos 600 1 479) (bigpos 800 20 227 6) | |
| 307 | (bigpos 200 291 178 87) (bigpos 0 368 674 307 1) | |
| 308 | (bigpos 0 888 789 922 20) (bigpos 0 96 428 687 355) | |
| 309 | (bigpos 0 728 705 373 402 6) | |
| 310 | (bigpos 0 832 408 100 645 121) | |
| 311 | (bigpos 0 640 176 8 902 432 2)] n) | |
| 312 | (math-factorial-iter (1- n) 2 1))) | |
| 313 | ((and (math-messy-integerp n) | |
| 314 | (Math-lessp n 100)) | |
| 315 | (math-inexact-result) | |
| 316 | (setq temp (math-trunc n)) | |
| 317 | (if (>= temp 0) | |
| 318 | (if (<= temp 20) | |
| 319 | (math-float (calcFunc-fact temp)) | |
| 320 | (math-with-extra-prec 1 | |
| 321 | (math-factorial-iter (1- temp) 2 '(float 1 0)))) | |
| 322 | (math-reject-arg n 'range))) | |
| 323 | ((math-numberp n) | |
| 324 | (let* ((q (math-quarter-integer n)) | |
| 325 | (tn (and q (Math-lessp n 1000) (Math-lessp -1000 n) | |
| 326 | (1+ (math-floor n))))) | |
| 327 | (cond ((and tn (= q 2) | |
| 328 | (or calc-symbolic-mode (< (math-abs tn) 20))) | |
| 329 | (let ((q (if (< tn 0) | |
| 330 | (math-div | |
| 331 | (math-pow -2 (- tn)) | |
| 332 | (math-double-factorial-iter (* -2 tn) 3 1 2)) | |
| a1506d29 | 333 | (math-div |
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334 | (math-double-factorial-iter (* 2 tn) 3 1 2) |
| 335 | (math-pow 2 tn))))) | |
| 336 | (math-mul q (if calc-symbolic-mode | |
| 337 | (list 'calcFunc-sqrt '(var pi var-pi)) | |
| 338 | (math-sqrt-pi))))) | |
| 339 | ((and tn (>= tn 0) (< tn 20) | |
| 340 | (memq q '(1 3))) | |
| 341 | (math-inexact-result) | |
| 342 | (math-div | |
| 343 | (math-mul (math-double-factorial-iter (* 4 tn) q 1 4) | |
| 344 | (if (= q 1) (math-gamma-1q) (math-gamma-3q))) | |
| 345 | (math-pow 4 tn))) | |
| 346 | (t | |
| 347 | (math-inexact-result) | |
| 348 | (math-with-extra-prec 3 | |
| 349 | (math-gammap1-raw (math-float n))))))) | |
| 350 | ((equal n '(var inf var-inf)) n) | |
| 351 | (t (calc-record-why 'numberp n) | |
| bf77c646 | 352 | (list 'calcFunc-fact n))))) |
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353 | |
| 354 | (math-defcache math-gamma-1q nil | |
| 355 | (math-with-extra-prec 3 | |
| 356 | (math-gammap1-raw '(float -75 -2)))) | |
| 357 | ||
| 358 | (math-defcache math-gamma-3q nil | |
| 359 | (math-with-extra-prec 3 | |
| 360 | (math-gammap1-raw '(float -25 -2)))) | |
| 361 | ||
| 362 | (defun math-factorial-iter (count n f) | |
| 363 | (if (= (% n 5) 1) | |
| 364 | (math-working (format "factorial(%d)" (1- n)) f)) | |
| 365 | (if (> count 0) | |
| 366 | (math-factorial-iter (1- count) (1+ n) (math-mul n f)) | |
| bf77c646 | 367 | f)) |
| 136211a9 EZ |
368 | |
| 369 | (defun calcFunc-dfact (n) ; [I I] [F F] [Public] | |
| 370 | (cond ((Math-integer-negp n) | |
| 371 | (if (math-oddp n) | |
| 372 | (if (eq n -1) | |
| 373 | 1 | |
| 374 | (math-div (if (eq (math-mod n 4) 3) 1 -1) | |
| 375 | (calcFunc-dfact (math-sub -2 n)))) | |
| 376 | (list 'calcFunc-dfact n))) | |
| 377 | ((Math-zerop n) 1) | |
| 378 | ((integerp n) (math-double-factorial-iter n (+ 2 (% n 2)) 1 2)) | |
| 379 | ((math-messy-integerp n) | |
| 380 | (let ((temp (math-trunc n))) | |
| 381 | (math-inexact-result) | |
| 382 | (if (natnump temp) | |
| 383 | (if (Math-lessp temp 200) | |
| 384 | (math-with-extra-prec 1 | |
| 385 | (math-double-factorial-iter temp (+ 2 (% temp 2)) | |
| 386 | '(float 1 0) 2)) | |
| 387 | (let* ((half (math-div2 temp)) | |
| 388 | (even (math-mul (math-pow 2 half) | |
| 389 | (calcFunc-fact (math-float half))))) | |
| 390 | (if (math-evenp temp) | |
| 391 | even | |
| 392 | (math-div (calcFunc-fact n) even)))) | |
| 8d7498c1 | 393 | (list 'calcFunc-dfact n)))) |
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394 | ((equal n '(var inf var-inf)) n) |
| 395 | (t (calc-record-why 'natnump n) | |
| bf77c646 | 396 | (list 'calcFunc-dfact n)))) |
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397 | |
| 398 | (defun math-double-factorial-iter (max n f step) | |
| 399 | (if (< (% n 12) step) | |
| 400 | (math-working (format "dfact(%d)" (- n step)) f)) | |
| 401 | (if (<= n max) | |
| 402 | (math-double-factorial-iter max (+ n step) (math-mul n f) step) | |
| bf77c646 | 403 | f)) |
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404 | |
| 405 | (defun calcFunc-perm (n m) ; [I I I] [F F F] [Public] | |
| 406 | (cond ((and (integerp n) (integerp m) (<= m n) (>= m 0)) | |
| 407 | (math-factorial-iter m (1+ (- n m)) 1)) | |
| 408 | ((or (not (math-num-integerp n)) | |
| 409 | (and (math-messy-integerp n) (Math-lessp 100 n)) | |
| 410 | (not (math-num-integerp m)) | |
| 411 | (and (math-messy-integerp m) (Math-lessp 100 m))) | |
| 412 | (or (math-realp n) (equal n '(var inf var-inf)) | |
| 413 | (math-reject-arg n 'realp)) | |
| 414 | (or (math-realp m) (equal m '(var inf var-inf)) | |
| 415 | (math-reject-arg m 'realp)) | |
| 416 | (and (math-num-integerp n) (math-negp n) (math-reject-arg n 'range)) | |
| 417 | (and (math-num-integerp m) (math-negp m) (math-reject-arg m 'range)) | |
| 418 | (math-div (calcFunc-fact n) (calcFunc-fact (math-sub n m)))) | |
| 419 | (t | |
| 420 | (let ((tn (math-trunc n)) | |
| 421 | (tm (math-trunc m))) | |
| 422 | (math-inexact-result) | |
| 423 | (or (integerp tn) (math-reject-arg tn 'fixnump)) | |
| 424 | (or (integerp tm) (math-reject-arg tm 'fixnump)) | |
| 425 | (or (and (<= tm tn) (>= tm 0)) (math-reject-arg tm 'range)) | |
| 426 | (math-with-extra-prec 1 | |
| bf77c646 | 427 | (math-factorial-iter tm (1+ (- tn tm)) '(float 1 0))))))) |
| 136211a9 EZ |
428 | |
| 429 | (defun calcFunc-choose (n m) ; [I I I] [F F F] [Public] | |
| 430 | (cond ((and (integerp n) (integerp m) (<= m n) (>= m 0)) | |
| 431 | (if (> m (/ n 2)) | |
| 432 | (math-choose-iter (- n m) n 1 1) | |
| 433 | (math-choose-iter m n 1 1))) | |
| 434 | ((not (math-realp n)) | |
| 435 | (math-reject-arg n 'realp)) | |
| 436 | ((not (math-realp m)) | |
| 437 | (math-reject-arg m 'realp)) | |
| 438 | ((not (math-num-integerp m)) | |
| 439 | (if (and (math-num-integerp n) (math-negp n)) | |
| 440 | (list 'calcFunc-choose n m) | |
| 441 | (math-div (calcFunc-fact (math-float n)) | |
| 442 | (math-mul (calcFunc-fact m) | |
| 443 | (calcFunc-fact (math-sub n m)))))) | |
| 444 | ((math-negp m) 0) | |
| 445 | ((math-negp n) | |
| 446 | (let ((val (calcFunc-choose (math-add (math-add n m) -1) m))) | |
| 447 | (if (math-evenp (math-trunc m)) | |
| 448 | val | |
| 449 | (math-neg val)))) | |
| 450 | ((and (math-num-integerp n) | |
| 451 | (Math-lessp n m)) | |
| 452 | 0) | |
| 453 | (t | |
| 454 | (math-inexact-result) | |
| 455 | (let ((tm (math-trunc m))) | |
| 456 | (or (integerp tm) (math-reject-arg tm 'fixnump)) | |
| 457 | (if (> tm 100) | |
| 458 | (math-div (calcFunc-fact (math-float n)) | |
| 459 | (math-mul (calcFunc-fact (math-float m)) | |
| 460 | (calcFunc-fact (math-float | |
| 461 | (math-sub n m))))) | |
| 462 | (math-with-extra-prec 1 | |
| bf77c646 | 463 | (math-choose-float-iter tm n 1 1))))))) |
| 136211a9 EZ |
464 | |
| 465 | (defun math-choose-iter (m n i c) | |
| 466 | (if (and (= (% i 5) 1) (> i 5)) | |
| 467 | (math-working (format "choose(%d)" (1- i)) c)) | |
| 468 | (if (<= i m) | |
| 469 | (math-choose-iter m (1- n) (1+ i) | |
| 470 | (math-quotient (math-mul c n) i)) | |
| bf77c646 | 471 | c)) |
| 136211a9 EZ |
472 | |
| 473 | (defun math-choose-float-iter (count n i c) | |
| 474 | (if (= (% i 5) 1) | |
| 475 | (math-working (format "choose(%d)" (1- i)) c)) | |
| 476 | (if (> count 0) | |
| 477 | (math-choose-float-iter (1- count) (math-sub n 1) (1+ i) | |
| 478 | (math-div (math-mul c n) i)) | |
| bf77c646 | 479 | c)) |
| 136211a9 EZ |
480 | |
| 481 | ||
| 482 | ;;; Stirling numbers. | |
| 483 | ||
| 484 | (defun calcFunc-stir1 (n m) | |
| bf77c646 | 485 | (math-stirling-number n m 1)) |
| 136211a9 EZ |
486 | |
| 487 | (defun calcFunc-stir2 (n m) | |
| bf77c646 | 488 | (math-stirling-number n m 0)) |
| 136211a9 | 489 | |
| 3132f345 | 490 | (defvar math-stirling-cache (vector [[1]] [[1]])) |
| 8d7498c1 JB |
491 | |
| 492 | ;; The variable math-stirling-local-cache is local to | |
| 493 | ;; math-stirling-number, but is used by math-stirling-1 | |
| 494 | ;; and math-stirling-2, which are called by math-stirling-number. | |
| 495 | (defvar math-stirling-local-cache) | |
| 496 | ||
| 136211a9 EZ |
497 | (defun math-stirling-number (n m k) |
| 498 | (or (math-num-natnump n) (math-reject-arg n 'natnump)) | |
| 499 | (or (math-num-natnump m) (math-reject-arg m 'natnump)) | |
| 500 | (if (consp n) (setq n (math-trunc n))) | |
| 501 | (or (integerp n) (math-reject-arg n 'fixnump)) | |
| 502 | (if (consp m) (setq m (math-trunc m))) | |
| 503 | (or (integerp m) (math-reject-arg m 'fixnump)) | |
| 504 | (if (< n m) | |
| 505 | 0 | |
| 8d7498c1 JB |
506 | (let ((math-stirling-local-cache (aref math-stirling-cache k))) |
| 507 | (while (<= (length math-stirling-local-cache) n) | |
| 508 | (let ((i (1- (length math-stirling-local-cache))) | |
| 136211a9 | 509 | row) |
| 8d7498c1 JB |
510 | (setq math-stirling-local-cache |
| 511 | (vconcat math-stirling-local-cache | |
| 512 | (make-vector (length math-stirling-local-cache) nil))) | |
| 513 | (aset math-stirling-cache k math-stirling-local-cache) | |
| 514 | (while (< (setq i (1+ i)) (length math-stirling-local-cache)) | |
| 515 | (aset math-stirling-local-cache i (setq row (make-vector (1+ i) nil))) | |
| 136211a9 EZ |
516 | (aset row 0 0) |
| 517 | (aset row i 1)))) | |
| 518 | (if (= k 1) | |
| 519 | (math-stirling-1 n m) | |
| bf77c646 | 520 | (math-stirling-2 n m))))) |
| 136211a9 EZ |
521 | |
| 522 | (defun math-stirling-1 (n m) | |
| 8d7498c1 JB |
523 | (or (aref (aref math-stirling-local-cache n) m) |
| 524 | (aset (aref math-stirling-local-cache n) m | |
| 136211a9 | 525 | (math-add (math-stirling-1 (1- n) (1- m)) |
| bf77c646 | 526 | (math-mul (- 1 n) (math-stirling-1 (1- n) m)))))) |
| 136211a9 EZ |
527 | |
| 528 | (defun math-stirling-2 (n m) | |
| 8d7498c1 JB |
529 | (or (aref (aref math-stirling-local-cache n) m) |
| 530 | (aset (aref math-stirling-local-cache n) m | |
| 136211a9 | 531 | (math-add (math-stirling-2 (1- n) (1- m)) |
| bf77c646 | 532 | (math-mul m (math-stirling-2 (1- n) m)))))) |
| 136211a9 | 533 | |
| 3132f345 CW |
534 | (defvar math-random-table nil) |
| 535 | (defvar math-last-RandSeed nil) | |
| 536 | (defvar math-random-ptr1 nil) | |
| 537 | (defvar math-random-ptr2 nil) | |
| 538 | (defvar math-random-shift nil) | |
| 136211a9 EZ |
539 | |
| 540 | ;;; Produce a random 10-bit integer, with (random) if no seed provided, | |
| 541 | ;;; or else with Numerical Recipes algorithm ran3 / Knuth 3.2.2-A. | |
| 8d7498c1 JB |
542 | |
| 543 | (defvar var-RandSeed nil) | |
| 544 | (defvar math-random-cache nil) | |
| 545 | (defvar math-gaussian-cache nil) | |
| 546 | ||
| 136211a9 | 547 | (defun math-init-random-base () |
| 8d7498c1 | 548 | (if var-RandSeed |
| 136211a9 EZ |
549 | (if (eq (car-safe var-RandSeed) 'vec) |
| 550 | nil | |
| 551 | (if (Math-integerp var-RandSeed) | |
| 552 | (let* ((seed (math-sub 161803 var-RandSeed)) | |
| 553 | (mj (1+ (math-mod seed '(bigpos 0 0 1)))) | |
| 554 | (mk (1+ (math-mod (math-quotient seed '(bigpos 0 0 1)) | |
| 555 | '(bigpos 0 0 1)))) | |
| 556 | (i 0)) | |
| 557 | (setq math-random-table (cons 'vec (make-list 55 mj))) | |
| 558 | (while (<= (setq i (1+ i)) 54) | |
| 559 | (let* ((ii (% (* i 21) 55)) | |
| 560 | (p (nthcdr ii math-random-table))) | |
| 561 | (setcar p mk) | |
| 562 | (setq mk (- mj mk) | |
| 563 | mj (car p))))) | |
| 564 | (math-reject-arg var-RandSeed "*RandSeed must be an integer")) | |
| 565 | (setq var-RandSeed (list 'vec var-RandSeed) | |
| 566 | math-random-ptr1 math-random-table | |
| 567 | math-random-cache nil | |
| 568 | math-random-ptr2 (nthcdr 31 math-random-table)) | |
| 569 | (let ((i 200)) | |
| 570 | (while (> (setq i (1- i)) 0) | |
| 571 | (math-random-base)))) | |
| 572 | (random t) | |
| 573 | (setq var-RandSeed nil | |
| 574 | math-random-cache nil | |
| 136211a9 EZ |
575 | math-random-shift -4) ; assume RAND_MAX >= 16383 |
| 576 | ;; This exercises the random number generator and also helps | |
| 577 | ;; deduce a better value for RAND_MAX. | |
| 8d7498c1 JB |
578 | (let ((i 0)) |
| 579 | (while (< (setq i (1+ i)) 30) | |
| 580 | (if (> (lsh (math-abs (random)) math-random-shift) 4095) | |
| 581 | (setq math-random-shift (1- math-random-shift)))))) | |
| 136211a9 | 582 | (setq math-last-RandSeed var-RandSeed |
| bf77c646 | 583 | math-gaussian-cache nil)) |
| 136211a9 EZ |
584 | |
| 585 | (defun math-random-base () | |
| 586 | (if var-RandSeed | |
| 587 | (progn | |
| 588 | (setq math-random-ptr1 (or (cdr math-random-ptr1) | |
| 589 | (cdr math-random-table)) | |
| 590 | math-random-ptr2 (or (cdr math-random-ptr2) | |
| 591 | (cdr math-random-table))) | |
| 592 | (logand (lsh (setcar math-random-ptr1 | |
| 593 | (logand (- (car math-random-ptr1) | |
| 594 | (car math-random-ptr2)) 524287)) | |
| 595 | -6) 1023)) | |
| bf77c646 | 596 | (logand (lsh (random) math-random-shift) 1023))) |
| 136211a9 EZ |
597 | |
| 598 | ||
| 599 | ;;; Produce a random digit in the range 0..999. | |
| 600 | ;;; Avoid various pitfalls that may lurk in the built-in (random) function! | |
| 601 | ;;; Shuffling algorithm from Numerical Recipes, section 7.1. | |
| 602 | (defun math-random-digit () | |
| 8d7498c1 JB |
603 | (let (i math-random-last) |
| 604 | (or (eq var-RandSeed math-last-RandSeed) | |
| 136211a9 EZ |
605 | (math-init-random-base)) |
| 606 | (or math-random-cache | |
| 607 | (progn | |
| 608 | (setq math-random-last (math-random-base) | |
| 609 | math-random-cache (make-vector 13 nil) | |
| 610 | i -1) | |
| 611 | (while (< (setq i (1+ i)) 13) | |
| 612 | (aset math-random-cache i (math-random-base))))) | |
| 613 | (while (progn | |
| 614 | (setq i (/ math-random-last 79) ; 0 <= i < 13 | |
| 615 | math-random-last (aref math-random-cache i)) | |
| 616 | (aset math-random-cache i (math-random-base)) | |
| 617 | (>= math-random-last 1000))) | |
| bf77c646 | 618 | math-random-last)) |
| 136211a9 EZ |
619 | |
| 620 | ;;; Produce an N-digit random integer. | |
| 621 | (defun math-random-digits (n) | |
| 622 | (cond ((<= n 6) | |
| 623 | (math-scale-right (+ (* (math-random-digit) 1000) (math-random-digit)) | |
| 624 | (- 6 n))) | |
| 625 | (t (let* ((slop (% (- 900003 n) 3)) | |
| 626 | (i (/ (+ n slop) 3)) | |
| 627 | (digs nil)) | |
| 628 | (while (> i 0) | |
| 629 | (setq digs (cons (math-random-digit) digs) | |
| 630 | i (1- i))) | |
| 631 | (math-normalize (math-scale-right (cons 'bigpos digs) | |
| bf77c646 | 632 | slop)))))) |
| 136211a9 EZ |
633 | |
| 634 | ;;; Produce a uniformly-distributed random float 0 <= N < 1. | |
| 635 | (defun math-random-float () | |
| 636 | (math-make-float (math-random-digits calc-internal-prec) | |
| bf77c646 | 637 | (- calc-internal-prec))) |
| 136211a9 EZ |
638 | |
| 639 | ;;; Produce a Gaussian-distributed random float with mean=0, sigma=1. | |
| 640 | (defun math-gaussian-float () | |
| 641 | (math-with-extra-prec 2 | |
| 642 | (if (and math-gaussian-cache | |
| 643 | (= (car math-gaussian-cache) calc-internal-prec)) | |
| 644 | (prog1 | |
| 645 | (cdr math-gaussian-cache) | |
| 646 | (setq math-gaussian-cache nil)) | |
| 647 | (let* ((v1 (math-add (math-mul (math-random-float) 2) -1)) | |
| 648 | (v2 (math-add (math-mul (math-random-float) 2) -1)) | |
| 649 | (r (math-add (math-sqr v1) (math-sqr v2)))) | |
| 650 | (while (or (not (Math-lessp r 1)) (math-zerop r)) | |
| 651 | (setq v1 (math-add (math-mul (math-random-float) 2) -1) | |
| 652 | v2 (math-add (math-mul (math-random-float) 2) -1) | |
| 653 | r (math-add (math-sqr v1) (math-sqr v2)))) | |
| 654 | (let ((fac (math-sqrt (math-mul (math-div (calcFunc-ln r) r) -2)))) | |
| 655 | (setq math-gaussian-cache (cons calc-internal-prec | |
| 656 | (math-mul v1 fac))) | |
| bf77c646 | 657 | (math-mul v2 fac)))))) |
| 136211a9 EZ |
658 | |
| 659 | ;;; Produce a random integer or real 0 <= N < MAX. | |
| 660 | (defun calcFunc-random (max) | |
| 661 | (cond ((Math-zerop max) | |
| 662 | (math-gaussian-float)) | |
| 663 | ((Math-integerp max) | |
| 664 | (let* ((digs (math-numdigs max)) | |
| 665 | (r (math-random-digits (+ digs 3)))) | |
| 666 | (math-mod r max))) | |
| 667 | ((Math-realp max) | |
| 668 | (math-mul (math-random-float) max)) | |
| 669 | ((and (eq (car max) 'intv) (math-constp max) | |
| 670 | (Math-lessp (nth 2 max) (nth 3 max))) | |
| 671 | (if (math-floatp max) | |
| 672 | (let ((val (math-add (math-mul (math-random-float) | |
| 673 | (math-sub (nth 3 max) (nth 2 max))) | |
| 674 | (nth 2 max)))) | |
| 675 | (if (or (and (memq (nth 1 max) '(0 1)) ; almost not worth | |
| 676 | (Math-equal val (nth 2 max))) ; checking! | |
| 677 | (and (memq (nth 1 max) '(0 2)) | |
| 678 | (Math-equal val (nth 3 max)))) | |
| 679 | (calcFunc-random max) | |
| 680 | val)) | |
| 681 | (let ((lo (if (memq (nth 1 max) '(0 1)) | |
| 682 | (math-add (nth 2 max) 1) (nth 2 max))) | |
| 683 | (hi (if (memq (nth 1 max) '(1 3)) | |
| 684 | (math-add (nth 3 max) 1) (nth 3 max)))) | |
| 685 | (if (Math-lessp lo hi) | |
| 686 | (math-add (calcFunc-random (math-sub hi lo)) lo) | |
| 687 | (math-reject-arg max "*Empty interval"))))) | |
| 688 | ((eq (car max) 'vec) | |
| 689 | (if (cdr max) | |
| 690 | (nth (1+ (calcFunc-random (1- (length max)))) max) | |
| 691 | (math-reject-arg max "*Empty list"))) | |
| 692 | ((and (eq (car max) 'sdev) (math-constp max) (Math-realp (nth 1 max))) | |
| 693 | (math-add (math-mul (math-gaussian-float) (nth 2 max)) (nth 1 max))) | |
| bf77c646 | 694 | (t (math-reject-arg max 'realp)))) |
| 136211a9 EZ |
695 | |
| 696 | ;;; Choose N objects at random from the set MAX without duplicates. | |
| 697 | (defun calcFunc-shuffle (n &optional max) | |
| 698 | (or max (setq max n n -1)) | |
| 699 | (or (and (Math-num-integerp n) | |
| 700 | (or (natnump (setq n (math-trunc n))) (eq n -1))) | |
| 701 | (math-reject-arg n 'integerp)) | |
| 702 | (cond ((or (math-zerop max) | |
| 703 | (math-floatp max) | |
| 704 | (eq (car-safe max) 'sdev)) | |
| 705 | (if (< n 0) | |
| 706 | (math-reject-arg n 'natnump) | |
| 707 | (math-simple-shuffle n max))) | |
| 708 | ((and (<= n 1) (>= n 0)) | |
| 709 | (math-simple-shuffle n max)) | |
| 710 | ((and (eq (car-safe max) 'intv) (math-constp max)) | |
| 711 | (let ((num (math-add (math-sub (nth 3 max) (nth 2 max)) | |
| 712 | (cdr (assq (nth 1 max) | |
| 713 | '((0 . -1) (1 . 0) | |
| 714 | (2 . 0) (3 . 1)))))) | |
| 715 | (min (math-add (nth 2 max) (if (memq (nth 1 max) '(0 1)) | |
| 716 | 1 0)))) | |
| 717 | (if (< n 0) (setq n num)) | |
| 718 | (or (math-posp num) (math-reject-arg max 'range)) | |
| 719 | (and (Math-lessp num n) (math-reject-arg n 'range)) | |
| 720 | (if (Math-lessp n (math-quotient num 3)) | |
| 721 | (math-simple-shuffle n max) | |
| 722 | (if (> (* n 4) (* num 3)) | |
| 723 | (math-add (math-sub min 1) | |
| 724 | (math-shuffle-list n num (calcFunc-index num))) | |
| 725 | (let ((tot 0) | |
| 726 | (m 0) | |
| 727 | (vec nil)) | |
| 728 | (while (< m n) | |
| 729 | (if (< (calcFunc-random (- num tot)) (- n m)) | |
| 730 | (setq vec (cons (math-add min tot) vec) | |
| 731 | m (1+ m))) | |
| 732 | (setq tot (1+ tot))) | |
| 733 | (math-shuffle-list n n (cons 'vec vec))))))) | |
| 734 | ((eq (car-safe max) 'vec) | |
| 735 | (let ((size (1- (length max)))) | |
| 736 | (if (< n 0) (setq n size)) | |
| 737 | (if (and (> n (/ size 2)) (<= n size)) | |
| 738 | (math-shuffle-list n size (copy-sequence max)) | |
| 739 | (let* ((vals (calcFunc-shuffle | |
| 740 | n (list 'intv 3 1 (1- (length max))))) | |
| 741 | (p vals)) | |
| 742 | (while (setq p (cdr p)) | |
| 743 | (setcar p (nth (car p) max))) | |
| 744 | vals)))) | |
| 745 | ((math-integerp max) | |
| 746 | (if (math-posp max) | |
| 747 | (calcFunc-shuffle n (list 'intv 2 0 max)) | |
| 748 | (calcFunc-shuffle n (list 'intv 1 max 0)))) | |
| bf77c646 | 749 | (t (math-reject-arg max 'realp)))) |
| 136211a9 EZ |
750 | |
| 751 | (defun math-simple-shuffle (n max) | |
| 752 | (let ((vec nil) | |
| 753 | val) | |
| 754 | (while (>= (setq n (1- n)) 0) | |
| 755 | (while (math-member (setq val (calcFunc-random max)) vec)) | |
| 756 | (setq vec (cons val vec))) | |
| bf77c646 | 757 | (cons 'vec vec))) |
| 136211a9 EZ |
758 | |
| 759 | (defun math-shuffle-list (n size vec) | |
| 760 | (let ((j size) | |
| 761 | k temp | |
| 762 | (p vec)) | |
| 763 | (while (cdr (setq p (cdr p))) | |
| 764 | (setq k (calcFunc-random j) | |
| 765 | j (1- j) | |
| 766 | temp (nth k p)) | |
| 767 | (setcar (nthcdr k p) (car p)) | |
| 768 | (setcar p temp)) | |
| bf77c646 | 769 | (cons 'vec (nthcdr (- size n -1) vec)))) |
| 136211a9 EZ |
770 | |
| 771 | (defun math-member (x list) | |
| 772 | (while (and list (not (equal x (car list)))) | |
| 773 | (setq list (cdr list))) | |
| bf77c646 | 774 | list) |
| 136211a9 EZ |
775 | |
| 776 | ||
| 777 | ;;; Check if the integer N is prime. [X I] | |
| 778 | ;;; Return (nil) if non-prime, | |
| 779 | ;;; (nil N) if non-prime with known factor N, | |
| 780 | ;;; (nil unknown) if non-prime with no known factors, | |
| 781 | ;;; (t) if prime, | |
| 782 | ;;; (maybe N P) if probably prime (after N iters with probability P%) | |
| 8d7498c1 JB |
783 | (defvar math-prime-test-cache '(-1)) |
| 784 | ||
| 785 | (defvar math-prime-test-cache-k) | |
| 786 | (defvar math-prime-test-cache-q) | |
| 787 | (defvar math-prime-test-cache-nm1) | |
| 788 | ||
| 136211a9 EZ |
789 | (defun math-prime-test (n iters) |
| 790 | (if (and (Math-vectorp n) (cdr n)) | |
| 791 | (setq n (nth (1- (length n)) n))) | |
| 792 | (if (Math-messy-integerp n) | |
| 793 | (setq n (math-trunc n))) | |
| 794 | (let ((res)) | |
| 795 | (while (> iters 0) | |
| 796 | (setq res | |
| 797 | (cond ((and (integerp n) (<= n 5003)) | |
| 798 | (list (= (math-next-small-prime n) n))) | |
| 799 | ((not (Math-integerp n)) | |
| 800 | (error "Argument must be an integer")) | |
| 801 | ((Math-integer-negp n) | |
| 802 | '(nil)) | |
| 803 | ((Math-natnum-lessp n '(bigpos 0 0 8)) | |
| 804 | (setq n (math-fixnum n)) | |
| 805 | (let ((i -1) v) | |
| 806 | (while (and (> (% n (setq v (aref math-primes-table | |
| 807 | (setq i (1+ i))))) | |
| 808 | 0) | |
| 809 | (< (* v v) n))) | |
| 810 | (if (= (% n v) 0) | |
| 811 | (list nil v) | |
| 812 | '(t)))) | |
| 813 | ((not (equal n (car math-prime-test-cache))) | |
| 814 | (cond ((= (% (nth 1 n) 2) 0) '(nil 2)) | |
| 815 | ((= (% (nth 1 n) 5) 0) '(nil 5)) | |
| 816 | (t (let ((dig (cdr n)) (sum 0)) | |
| 817 | (while dig | |
| 818 | (if (cdr dig) | |
| 819 | (setq sum (% (+ (+ sum (car dig)) | |
| 820 | (* (nth 1 dig) 1000)) | |
| 821 | 111111) | |
| 822 | dig (cdr (cdr dig))) | |
| 823 | (setq sum (% (+ sum (car dig)) 111111) | |
| 824 | dig nil))) | |
| 825 | (cond ((= (% sum 3) 0) '(nil 3)) | |
| 826 | ((= (% sum 7) 0) '(nil 7)) | |
| 827 | ((= (% sum 11) 0) '(nil 11)) | |
| 828 | ((= (% sum 13) 0) '(nil 13)) | |
| 829 | ((= (% sum 37) 0) '(nil 37)) | |
| 830 | (t | |
| 831 | (setq math-prime-test-cache-k 1 | |
| 832 | math-prime-test-cache-q | |
| 833 | (math-div2 n) | |
| 834 | math-prime-test-cache-nm1 | |
| 835 | (math-add n -1)) | |
| 836 | (while (math-evenp | |
| 837 | math-prime-test-cache-q) | |
| 838 | (setq math-prime-test-cache-k | |
| 839 | (1+ math-prime-test-cache-k) | |
| 840 | math-prime-test-cache-q | |
| 841 | (math-div2 | |
| 842 | math-prime-test-cache-q))) | |
| 843 | (setq iters (1+ iters)) | |
| 844 | (list 'maybe | |
| 845 | 0 | |
| 846 | (math-sub | |
| 847 | 100 | |
| 848 | (math-div | |
| 849 | '(float 232 0) | |
| 850 | (math-numdigs n)))))))))) | |
| 851 | ((not (eq (car (nth 1 math-prime-test-cache)) 'maybe)) | |
| 852 | (nth 1 math-prime-test-cache)) | |
| 853 | (t ; Fermat step | |
| 854 | (let* ((x (math-add (calcFunc-random (math-add n -2)) 2)) | |
| 855 | (y (math-pow-mod x math-prime-test-cache-q n)) | |
| 856 | (j 0)) | |
| 857 | (while (and (not (eq y 1)) | |
| 858 | (not (equal y math-prime-test-cache-nm1)) | |
| 859 | (< (setq j (1+ j)) math-prime-test-cache-k)) | |
| 860 | (setq y (math-mod (math-mul y y) n))) | |
| 861 | (if (or (equal y math-prime-test-cache-nm1) | |
| 862 | (and (eq y 1) (eq j 0))) | |
| 863 | (list 'maybe | |
| 864 | (1+ (nth 1 (nth 1 math-prime-test-cache))) | |
| 865 | (math-mul (nth 2 (nth 1 math-prime-test-cache)) | |
| 866 | '(float 25 -2))) | |
| 867 | '(nil unknown)))))) | |
| 868 | (setq math-prime-test-cache (list n res) | |
| 869 | iters (if (eq (car res) 'maybe) | |
| 870 | (1- iters) | |
| 871 | 0))) | |
| bf77c646 | 872 | res)) |
| 136211a9 EZ |
873 | |
| 874 | (defun calcFunc-prime (n &optional iters) | |
| 875 | (or (math-num-integerp n) (math-reject-arg n 'integerp)) | |
| 876 | (or (not iters) (math-num-integerp iters) (math-reject-arg iters 'integerp)) | |
| 877 | (if (car (math-prime-test (math-trunc n) (math-trunc (or iters 1)))) | |
| 878 | 1 | |
| bf77c646 | 879 | 0)) |
| 136211a9 EZ |
880 | |
| 881 | ;;; Theory: summing base-10^6 digits modulo 111111 is "casting out 999999s". | |
| 882 | ;;; Initial probability that N is prime is 1/ln(N) = log10(e)/log10(N). | |
| 883 | ;;; After culling [2,3,5,7,11,13,37], probability of primality is 5.36 x more. | |
| 884 | ;;; Initial reported probability of non-primality is thus 100% - this. | |
| 885 | ;;; Each Fermat step multiplies this probability by 25%. | |
| 886 | ;;; The Fermat step is algorithm P from Knuth section 4.5.4. | |
| 887 | ||
| 888 | ||
| 889 | (defun calcFunc-prfac (n) | |
| 890 | (setq math-prime-factors-finished t) | |
| 891 | (if (Math-messy-integerp n) | |
| 892 | (setq n (math-trunc n))) | |
| 893 | (if (Math-natnump n) | |
| 894 | (if (Math-natnum-lessp 2 n) | |
| 895 | (let (factors res p (i 0)) | |
| 896 | (while (and (not (eq n 1)) | |
| 897 | (< i (length math-primes-table))) | |
| 898 | (setq p (aref math-primes-table i)) | |
| 899 | (while (eq (cdr (setq res (cond ((eq n p) (cons 1 0)) | |
| 900 | ((eq n 1) (cons 0 1)) | |
| 901 | ((consp n) (math-idivmod n p)) | |
| 902 | (t (cons (/ n p) (% n p)))))) | |
| 903 | 0) | |
| 904 | (math-working "factor" p) | |
| 905 | (setq factors (nconc factors (list p)) | |
| 906 | n (car res))) | |
| 907 | (or (eq n 1) | |
| 908 | (Math-natnum-lessp p (car res)) | |
| 909 | (setq factors (nconc factors (list n)) | |
| 910 | n 1)) | |
| 911 | (setq i (1+ i))) | |
| 912 | (or (setq math-prime-factors-finished (eq n 1)) | |
| 913 | (setq factors (nconc factors (list n)))) | |
| 914 | (cons 'vec factors)) | |
| 915 | (list 'vec n)) | |
| 916 | (if (Math-integerp n) | |
| 917 | (if (eq n -1) | |
| 918 | (list 'vec n) | |
| 919 | (cons 'vec (cons -1 (cdr (calcFunc-prfac (math-neg n)))))) | |
| 920 | (calc-record-why 'integerp n) | |
| bf77c646 | 921 | (list 'calcFunc-prfac n)))) |
| 136211a9 EZ |
922 | |
| 923 | (defun calcFunc-totient (n) | |
| 924 | (if (Math-messy-integerp n) | |
| 925 | (setq n (math-trunc n))) | |
| 926 | (if (Math-natnump n) | |
| 927 | (if (Math-natnum-lessp n 2) | |
| 928 | (if (Math-negp n) | |
| 929 | (calcFunc-totient (math-abs n)) | |
| 930 | n) | |
| 931 | (let ((factors (cdr (calcFunc-prfac n))) | |
| 932 | p) | |
| 933 | (if math-prime-factors-finished | |
| 934 | (progn | |
| 935 | (while factors | |
| 936 | (setq p (car factors) | |
| 937 | n (math-mul (math-div n p) (math-add p -1))) | |
| 938 | (while (equal p (car factors)) | |
| 939 | (setq factors (cdr factors)))) | |
| 940 | n) | |
| 941 | (calc-record-why "*Number too big to factor" n) | |
| 942 | (list 'calcFunc-totient n)))) | |
| 943 | (calc-record-why 'natnump n) | |
| bf77c646 | 944 | (list 'calcFunc-totient n))) |
| 136211a9 EZ |
945 | |
| 946 | (defun calcFunc-moebius (n) | |
| 947 | (if (Math-messy-integerp n) | |
| 948 | (setq n (math-trunc n))) | |
| 949 | (if (and (Math-natnump n) (not (eq n 0))) | |
| 950 | (if (Math-natnum-lessp n 2) | |
| 951 | (if (Math-negp n) | |
| 952 | (calcFunc-moebius (math-abs n)) | |
| 953 | 1) | |
| 954 | (let ((factors (cdr (calcFunc-prfac n))) | |
| 955 | (mu 1)) | |
| 956 | (if math-prime-factors-finished | |
| 957 | (progn | |
| 958 | (while factors | |
| 959 | (setq mu (if (equal (car factors) (nth 1 factors)) | |
| 960 | 0 (math-neg mu)) | |
| 961 | factors (cdr factors))) | |
| 962 | mu) | |
| 963 | (calc-record-why "Number too big to factor" n) | |
| 964 | (list 'calcFunc-moebius n)))) | |
| 965 | (calc-record-why 'posintp n) | |
| bf77c646 | 966 | (list 'calcFunc-moebius n))) |
| 136211a9 EZ |
967 | |
| 968 | ||
| 969 | (defun calcFunc-nextprime (n &optional iters) | |
| 970 | (if (Math-integerp n) | |
| 971 | (if (Math-integer-negp n) | |
| 972 | 2 | |
| 973 | (if (and (integerp n) (< n 5003)) | |
| 974 | (math-next-small-prime (1+ n)) | |
| 975 | (if (math-evenp n) | |
| 976 | (setq n (math-add n -1))) | |
| 977 | (let (res) | |
| 978 | (while (not (car (setq res (math-prime-test | |
| 979 | (setq n (math-add n 2)) | |
| 980 | (or iters 1)))))) | |
| 981 | (if (and calc-verbose-nextprime | |
| 982 | (eq (car res) 'maybe)) | |
| 983 | (calc-report-prime-test res))) | |
| 984 | n)) | |
| 985 | (if (Math-realp n) | |
| 986 | (calcFunc-nextprime (math-trunc n) iters) | |
| bf77c646 | 987 | (math-reject-arg n 'integerp)))) |
| 136211a9 EZ |
988 | |
| 989 | (defun calcFunc-prevprime (n &optional iters) | |
| 990 | (if (Math-integerp n) | |
| 991 | (if (Math-lessp n 4) | |
| 992 | 2 | |
| 993 | (if (math-evenp n) | |
| 994 | (setq n (math-add n 1))) | |
| 995 | (let (res) | |
| 996 | (while (not (car (setq res (math-prime-test | |
| 997 | (setq n (math-add n -2)) | |
| 998 | (or iters 1)))))) | |
| 999 | (if (and calc-verbose-nextprime | |
| 1000 | (eq (car res) 'maybe)) | |
| 1001 | (calc-report-prime-test res))) | |
| 1002 | n) | |
| 1003 | (if (Math-realp n) | |
| 1004 | (calcFunc-prevprime (math-ceiling n) iters) | |
| bf77c646 | 1005 | (math-reject-arg n 'integerp)))) |
| 136211a9 EZ |
1006 | |
| 1007 | (defun math-next-small-prime (n) | |
| 1008 | (if (and (integerp n) (> n 2)) | |
| 1009 | (let ((lo -1) | |
| 1010 | (hi (length math-primes-table)) | |
| 1011 | mid) | |
| 1012 | (while (> (- hi lo) 1) | |
| 1013 | (if (> n (aref math-primes-table | |
| 1014 | (setq mid (ash (+ lo hi) -1)))) | |
| 1015 | (setq lo mid) | |
| 1016 | (setq hi mid))) | |
| 1017 | (aref math-primes-table hi)) | |
| bf77c646 | 1018 | 2)) |
| 136211a9 | 1019 | |
| 43f34ccc | 1020 | (provide 'calc-comb) |
| 136211a9 | 1021 | |
| ab5796a9 | 1022 | ;;; arch-tag: 1d75ee9b-0815-42bd-a321-bb3dc001cc02 |
| bf77c646 | 1023 | ;;; calc-comb.el ends here |