1;;; calc-comb.el --- combinatoric functions for Calc 2 3;; Copyright (C) 1990, 1991, 1992, 1993, 2001, 2002, 2003, 2004, 4;; 2005, 2006, 2007 Free Software Foundation, Inc. 5 6;; Author: David Gillespie <daveg@synaptics.com> 7;; Maintainer: Jay Belanger <jay.p.belanger@gmail.com> 8 9;; This file is part of GNU Emacs. 10 11;; GNU Emacs is free software; you can redistribute it and/or modify 12;; it under the terms of the GNU General Public License as published by 13;; the Free Software Foundation; either version 2, or (at your option) 14;; any later version. 15 16;; GNU Emacs is distributed in the hope that it will be useful, 17;; but WITHOUT ANY WARRANTY; without even the implied warranty of 18;; MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 19;; GNU General Public License for more details. 20 21;; You should have received a copy of the GNU General Public License 22;; along with GNU Emacs; see the file COPYING. If not, write to the 23;; Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, 24;; Boston, MA 02110-1301, USA. 25 26;;; Commentary: 27 28;;; Code: 29 30;; This file is autoloaded from calc-ext.el. 31 32(require 'calc-ext) 33(require 'calc-macs) 34 35(defconst math-primes-table 36 [2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 37 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 38 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 39 281 283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383 40 389 397 401 409 419 421 431 433 439 443 449 457 461 463 467 479 487 41 491 499 503 509 521 523 541 547 557 563 569 571 577 587 593 599 601 42 607 613 617 619 631 641 643 647 653 659 661 673 677 683 691 701 709 43 719 727 733 739 743 751 757 761 769 773 787 797 809 811 821 823 827 44 829 839 853 857 859 863 877 881 883 887 907 911 919 929 937 941 947 45 953 967 971 977 983 991 997 1009 1013 1019 1021 1031 1033 1039 1049 46 1051 1061 1063 1069 1087 1091 1093 1097 1103 1109 1117 1123 1129 1151 47 1153 1163 1171 1181 1187 1193 1201 1213 1217 1223 1229 1231 1237 1249 48 1259 1277 1279 1283 1289 1291 1297 1301 1303 1307 1319 1321 1327 1361 49 1367 1373 1381 1399 1409 1423 1427 1429 1433 1439 1447 1451 1453 1459 50 1471 1481 1483 1487 1489 1493 1499 1511 1523 1531 1543 1549 1553 1559 51 1567 1571 1579 1583 1597 1601 1607 1609 1613 1619 1621 1627 1637 1657 52 1663 1667 1669 1693 1697 1699 1709 1721 1723 1733 1741 1747 1753 1759 53 1777 1783 1787 1789 1801 1811 1823 1831 1847 1861 1867 1871 1873 1877 54 1879 1889 1901 1907 1913 1931 1933 1949 1951 1973 1979 1987 1993 1997 55 1999 2003 2011 2017 2027 2029 2039 2053 2063 2069 2081 2083 2087 2089 56 2099 2111 2113 2129 2131 2137 2141 2143 2153 2161 2179 2203 2207 2213 57 2221 2237 2239 2243 2251 2267 2269 2273 2281 2287 2293 2297 2309 2311 58 2333 2339 2341 2347 2351 2357 2371 2377 2381 2383 2389 2393 2399 2411 59 2417 2423 2437 2441 2447 2459 2467 2473 2477 2503 2521 2531 2539 2543 60 2549 2551 2557 2579 2591 2593 2609 2617 2621 2633 2647 2657 2659 2663 61 2671 2677 2683 2687 2689 2693 2699 2707 2711 2713 2719 2729 2731 2741 62 2749 2753 2767 2777 2789 2791 2797 2801 2803 2819 2833 2837 2843 2851 63 2857 2861 2879 2887 2897 2903 2909 2917 2927 2939 2953 2957 2963 2969 64 2971 2999 3001 3011 3019 3023 3037 3041 3049 3061 3067 3079 3083 3089 65 3109 3119 3121 3137 3163 3167 3169 3181 3187 3191 3203 3209 3217 3221 66 3229 3251 3253 3257 3259 3271 3299 3301 3307 3313 3319 3323 3329 3331 67 3343 3347 3359 3361 3371 3373 3389 3391 3407 3413 3433 3449 3457 3461 68 3463 3467 3469 3491 3499 3511 3517 3527 3529 3533 3539 3541 3547 3557 69 3559 3571 3581 3583 3593 3607 3613 3617 3623 3631 3637 3643 3659 3671 70 3673 3677 3691 3697 3701 3709 3719 3727 3733 3739 3761 3767 3769 3779 71 3793 3797 3803 3821 3823 3833 3847 3851 3853 3863 3877 3881 3889 3907 72 3911 3917 3919 3923 3929 3931 3943 3947 3967 3989 4001 4003 4007 4013 73 4019 4021 4027 4049 4051 4057 4073 4079 4091 4093 4099 4111 4127 4129 74 4133 4139 4153 4157 4159 4177 4201 4211 4217 4219 4229 4231 4241 4243 75 4253 4259 4261 4271 4273 4283 4289 4297 4327 4337 4339 4349 4357 4363 76 4373 4391 4397 4409 4421 4423 4441 4447 4451 4457 4463 4481 4483 4493 77 4507 4513 4517 4519 4523 4547 4549 4561 4567 4583 4591 4597 4603 4621 78 4637 4639 4643 4649 4651 4657 4663 4673 4679 4691 4703 4721 4723 4729 79 4733 4751 4759 4783 4787 4789 4793 4799 4801 4813 4817 4831 4861 4871 80 4877 4889 4903 4909 4919 4931 4933 4937 4943 4951 4957 4967 4969 4973 81 4987 4993 4999 5003]) 82 83;; The variable math-prime-factors-finished is set by calcFunc-prfac to 84;; indicate whether factoring is complete, and used by calcFunc-factors, 85;; calcFunc-totient and calcFunc-moebius. 86(defvar math-prime-factors-finished) 87 88;;; Combinatorics 89 90(defun calc-gcd (arg) 91 (interactive "P") 92 (calc-slow-wrapper 93 (calc-binary-op "gcd" 'calcFunc-gcd arg))) 94 95(defun calc-lcm (arg) 96 (interactive "P") 97 (calc-slow-wrapper 98 (calc-binary-op "lcm" 'calcFunc-lcm arg))) 99 100(defun calc-extended-gcd () 101 (interactive) 102 (calc-slow-wrapper 103 (calc-enter-result 2 "egcd" (cons 'calcFunc-egcd (calc-top-list-n 2))))) 104 105(defun calc-factorial (arg) 106 (interactive "P") 107 (calc-slow-wrapper 108 (calc-unary-op "fact" 'calcFunc-fact arg))) 109 110(defun calc-gamma (arg) 111 (interactive "P") 112 (calc-slow-wrapper 113 (calc-unary-op "gmma" 'calcFunc-gamma arg))) 114 115(defun calc-double-factorial (arg) 116 (interactive "P") 117 (calc-slow-wrapper 118 (calc-unary-op "dfac" 'calcFunc-dfact arg))) 119 120(defun calc-choose (arg) 121 (interactive "P") 122 (calc-slow-wrapper 123 (if (calc-is-hyperbolic) 124 (calc-binary-op "perm" 'calcFunc-perm arg) 125 (calc-binary-op "chos" 'calcFunc-choose arg)))) 126 127(defun calc-perm (arg) 128 (interactive "P") 129 (calc-hyperbolic-func) 130 (calc-choose arg)) 131 132(defvar calc-last-random-limit '(float 1 0)) 133(defun calc-random (n) 134 (interactive "P") 135 (calc-slow-wrapper 136 (if n 137 (calc-enter-result 0 "rand" (list 'calcFunc-random 138 (calc-get-random-limit 139 (prefix-numeric-value n)))) 140 (calc-enter-result 1 "rand" (list 'calcFunc-random 141 (calc-get-random-limit 142 (calc-top-n 1))))))) 143 144(defun calc-get-random-limit (val) 145 (if (eq val 0) 146 calc-last-random-limit 147 (setq calc-last-random-limit val))) 148 149(defun calc-rrandom () 150 (interactive) 151 (calc-slow-wrapper 152 (setq calc-last-random-limit '(float 1 0)) 153 (calc-enter-result 0 "rand" (list 'calcFunc-random '(float 1 0))))) 154 155(defun calc-random-again (arg) 156 (interactive "p") 157 (calc-slow-wrapper 158 (while (>= (setq arg (1- arg)) 0) 159 (calc-enter-result 0 "rand" (list 'calcFunc-random 160 calc-last-random-limit))))) 161 162(defun calc-shuffle (n) 163 (interactive "P") 164 (calc-slow-wrapper 165 (if n 166 (calc-enter-result 1 "shuf" (list 'calcFunc-shuffle 167 (prefix-numeric-value n) 168 (calc-get-random-limit 169 (calc-top-n 1)))) 170 (calc-enter-result 2 "shuf" (list 'calcFunc-shuffle 171 (calc-top-n 1) 172 (calc-get-random-limit 173 (calc-top-n 2))))))) 174 175(defun calc-report-prime-test (res) 176 (cond ((eq (car res) t) 177 (calc-record-message "prim" "Prime (guaranteed)")) 178 ((eq (car res) nil) 179 (if (cdr res) 180 (if (eq (nth 1 res) 'unknown) 181 (calc-record-message 182 "prim" "Non-prime (factors unknown)") 183 (calc-record-message 184 "prim" "Non-prime (%s is a factor)" 185 (math-format-number (nth 1 res)))) 186 (calc-record-message "prim" "Non-prime"))) 187 (t 188 (calc-record-message 189 "prim" "Probably prime (%d iters; %s%% chance of error)" 190 (nth 1 res) 191 (let ((calc-float-format '(fix 2))) 192 (math-format-number (nth 2 res))))))) 193 194(defun calc-prime-test (iters) 195 (interactive "p") 196 (calc-slow-wrapper 197 (let* ((n (calc-top-n 1)) 198 (res (math-prime-test n iters))) 199 (calc-report-prime-test res)))) 200 201(defvar calc-verbose-nextprime nil) 202 203(defun calc-next-prime (iters) 204 (interactive "p") 205 (calc-slow-wrapper 206 (let ((calc-verbose-nextprime t)) 207 (if (calc-is-inverse) 208 (calc-enter-result 1 "prvp" (list 'calcFunc-prevprime 209 (calc-top-n 1) (math-abs iters))) 210 (calc-enter-result 1 "nxtp" (list 'calcFunc-nextprime 211 (calc-top-n 1) (math-abs iters))))))) 212 213(defun calc-prev-prime (iters) 214 (interactive "p") 215 (calc-invert-func) 216 (calc-next-prime iters)) 217 218(defun calc-prime-factors (iters) 219 (interactive "p") 220 (calc-slow-wrapper 221 (let ((res (calcFunc-prfac (calc-top-n 1)))) 222 (if (not math-prime-factors-finished) 223 (calc-record-message "pfac" "Warning: May not be fully factored")) 224 (calc-enter-result 1 "pfac" res)))) 225 226(defun calc-totient (arg) 227 (interactive "P") 228 (calc-slow-wrapper 229 (calc-unary-op "phi" 'calcFunc-totient arg))) 230 231(defun calc-moebius (arg) 232 (interactive "P") 233 (calc-slow-wrapper 234 (calc-unary-op "mu" 'calcFunc-moebius arg))) 235 236 237(defun calcFunc-gcd (a b) 238 (if (Math-messy-integerp a) 239 (setq a (math-trunc a))) 240 (if (Math-messy-integerp b) 241 (setq b (math-trunc b))) 242 (cond ((and (Math-integerp a) (Math-integerp b)) 243 (math-gcd a b)) 244 ((Math-looks-negp a) 245 (calcFunc-gcd (math-neg a) b)) 246 ((Math-looks-negp b) 247 (calcFunc-gcd a (math-neg b))) 248 ((Math-zerop a) b) 249 ((Math-zerop b) a) 250 ((and (Math-ratp a) 251 (Math-ratp b)) 252 (math-make-frac (math-gcd (if (eq (car-safe a) 'frac) (nth 1 a) a) 253 (if (eq (car-safe b) 'frac) (nth 1 b) b)) 254 (calcFunc-lcm 255 (if (eq (car-safe a) 'frac) (nth 2 a) 1) 256 (if (eq (car-safe b) 'frac) (nth 2 b) 1)))) 257 ((not (Math-integerp a)) 258 (calc-record-why 'integerp a) 259 (list 'calcFunc-gcd a b)) 260 (t 261 (calc-record-why 'integerp b) 262 (list 'calcFunc-gcd a b)))) 263 264(defun calcFunc-lcm (a b) 265 (let ((g (calcFunc-gcd a b))) 266 (if (Math-numberp g) 267 (math-div (math-mul a b) g) 268 (list 'calcFunc-lcm a b)))) 269 270(defun calcFunc-egcd (a b) ; Knuth section 4.5.2 271 (cond 272 ((not (Math-integerp a)) 273 (if (Math-messy-integerp a) 274 (calcFunc-egcd (math-trunc a) b) 275 (calc-record-why 'integerp a) 276 (list 'calcFunc-egcd a b))) 277 ((not (Math-integerp b)) 278 (if (Math-messy-integerp b) 279 (calcFunc-egcd a (math-trunc b)) 280 (calc-record-why 'integerp b) 281 (list 'calcFunc-egcd a b))) 282 (t 283 (let ((u1 1) (u2 0) (u3 a) 284 (v1 0) (v2 1) (v3 b) 285 t1 t2 q) 286 (while (not (eq v3 0)) 287 (setq q (math-idivmod u3 v3) 288 t1 (math-sub u1 (math-mul v1 (car q))) 289 t2 (math-sub u2 (math-mul v2 (car q))) 290 u1 v1 u2 v2 u3 v3 291 v1 t1 v2 t2 v3 (cdr q))) 292 (list 'vec u3 u1 u2))))) 293 294 295;;; Factorial and related functions. 296 297(defun calcFunc-fact (n) ; [I I] [F F] [Public] 298 (let (temp) 299 (cond ((Math-integer-negp n) 300 (if calc-infinite-mode 301 '(var uinf var-uinf) 302 (math-reject-arg n 'range))) 303 ((integerp n) 304 (if (<= n 20) 305 (aref '[1 1 2 6 24 120 720 5040 40320 362880 306 (bigpos 800 628 3) (bigpos 800 916 39) 307 (bigpos 600 1 479) (bigpos 800 20 227 6) 308 (bigpos 200 291 178 87) (bigpos 0 368 674 307 1) 309 (bigpos 0 888 789 922 20) (bigpos 0 96 428 687 355) 310 (bigpos 0 728 705 373 402 6) 311 (bigpos 0 832 408 100 645 121) 312 (bigpos 0 640 176 8 902 432 2)] n) 313 (math-factorial-iter (1- n) 2 1))) 314 ((and (math-messy-integerp n) 315 (Math-lessp n 100)) 316 (math-inexact-result) 317 (setq temp (math-trunc n)) 318 (if (>= temp 0) 319 (if (<= temp 20) 320 (math-float (calcFunc-fact temp)) 321 (math-with-extra-prec 1 322 (math-factorial-iter (1- temp) 2 '(float 1 0)))) 323 (math-reject-arg n 'range))) 324 ((math-numberp n) 325 (let* ((q (math-quarter-integer n)) 326 (tn (and q (Math-lessp n 1000) (Math-lessp -1000 n) 327 (1+ (math-floor n))))) 328 (cond ((and tn (= q 2) 329 (or calc-symbolic-mode (< (math-abs tn) 20))) 330 (let ((q (if (< tn 0) 331 (math-div 332 (math-pow -2 (- tn)) 333 (math-double-factorial-iter (* -2 tn) 3 1 2)) 334 (math-div 335 (math-double-factorial-iter (* 2 tn) 3 1 2) 336 (math-pow 2 tn))))) 337 (math-mul q (if calc-symbolic-mode 338 (list 'calcFunc-sqrt '(var pi var-pi)) 339 (math-sqrt-pi))))) 340 ((and tn (>= tn 0) (< tn 20) 341 (memq q '(1 3))) 342 (math-inexact-result) 343 (math-div 344 (math-mul (math-double-factorial-iter (* 4 tn) q 1 4) 345 (if (= q 1) (math-gamma-1q) (math-gamma-3q))) 346 (math-pow 4 tn))) 347 (t 348 (math-inexact-result) 349 (math-with-extra-prec 3 350 (math-gammap1-raw (math-float n))))))) 351 ((equal n '(var inf var-inf)) n) 352 (t (calc-record-why 'numberp n) 353 (list 'calcFunc-fact n))))) 354 355(math-defcache math-gamma-1q nil 356 (math-with-extra-prec 3 357 (math-gammap1-raw '(float -75 -2)))) 358 359(math-defcache math-gamma-3q nil 360 (math-with-extra-prec 3 361 (math-gammap1-raw '(float -25 -2)))) 362 363(defun math-factorial-iter (count n f) 364 (if (= (% n 5) 1) 365 (math-working (format "factorial(%d)" (1- n)) f)) 366 (if (> count 0) 367 (math-factorial-iter (1- count) (1+ n) (math-mul n f)) 368 f)) 369 370(defun calcFunc-dfact (n) ; [I I] [F F] [Public] 371 (cond ((Math-integer-negp n) 372 (if (math-oddp n) 373 (if (eq n -1) 374 1 375 (math-div (if (eq (math-mod n 4) 3) 1 -1) 376 (calcFunc-dfact (math-sub -2 n)))) 377 (list 'calcFunc-dfact n))) 378 ((Math-zerop n) 1) 379 ((integerp n) (math-double-factorial-iter n (+ 2 (% n 2)) 1 2)) 380 ((math-messy-integerp n) 381 (let ((temp (math-trunc n))) 382 (math-inexact-result) 383 (if (natnump temp) 384 (if (Math-lessp temp 200) 385 (math-with-extra-prec 1 386 (math-double-factorial-iter temp (+ 2 (% temp 2)) 387 '(float 1 0) 2)) 388 (let* ((half (math-div2 temp)) 389 (even (math-mul (math-pow 2 half) 390 (calcFunc-fact (math-float half))))) 391 (if (math-evenp temp) 392 even 393 (math-div (calcFunc-fact n) even)))) 394 (list 'calcFunc-dfact n)))) 395 ((equal n '(var inf var-inf)) n) 396 (t (calc-record-why 'natnump n) 397 (list 'calcFunc-dfact n)))) 398 399(defun math-double-factorial-iter (max n f step) 400 (if (< (% n 12) step) 401 (math-working (format "dfact(%d)" (- n step)) f)) 402 (if (<= n max) 403 (math-double-factorial-iter max (+ n step) (math-mul n f) step) 404 f)) 405 406(defun calcFunc-perm (n m) ; [I I I] [F F F] [Public] 407 (cond ((and (integerp n) (integerp m) (<= m n) (>= m 0)) 408 (math-factorial-iter m (1+ (- n m)) 1)) 409 ((or (not (math-num-integerp n)) 410 (and (math-messy-integerp n) (Math-lessp 100 n)) 411 (not (math-num-integerp m)) 412 (and (math-messy-integerp m) (Math-lessp 100 m))) 413 (or (math-realp n) (equal n '(var inf var-inf)) 414 (math-reject-arg n 'realp)) 415 (or (math-realp m) (equal m '(var inf var-inf)) 416 (math-reject-arg m 'realp)) 417 (and (math-num-integerp n) (math-negp n) (math-reject-arg n 'range)) 418 (and (math-num-integerp m) (math-negp m) (math-reject-arg m 'range)) 419 (math-div (calcFunc-fact n) (calcFunc-fact (math-sub n m)))) 420 (t 421 (let ((tn (math-trunc n)) 422 (tm (math-trunc m))) 423 (math-inexact-result) 424 (or (integerp tn) (math-reject-arg tn 'fixnump)) 425 (or (integerp tm) (math-reject-arg tm 'fixnump)) 426 (or (and (<= tm tn) (>= tm 0)) (math-reject-arg tm 'range)) 427 (math-with-extra-prec 1 428 (math-factorial-iter tm (1+ (- tn tm)) '(float 1 0))))))) 429 430(defun calcFunc-choose (n m) ; [I I I] [F F F] [Public] 431 (cond ((and (integerp n) (integerp m) (<= m n) (>= m 0)) 432 (if (> m (/ n 2)) 433 (math-choose-iter (- n m) n 1 1) 434 (math-choose-iter m n 1 1))) 435 ((not (math-realp n)) 436 (math-reject-arg n 'realp)) 437 ((not (math-realp m)) 438 (math-reject-arg m 'realp)) 439 ((not (math-num-integerp m)) 440 (if (and (math-num-integerp n) (math-negp n)) 441 (list 'calcFunc-choose n m) 442 (math-div (calcFunc-fact (math-float n)) 443 (math-mul (calcFunc-fact m) 444 (calcFunc-fact (math-sub n m)))))) 445 ((math-negp m) 0) 446 ((math-negp n) 447 (let ((val (calcFunc-choose (math-add (math-add n m) -1) m))) 448 (if (math-evenp (math-trunc m)) 449 val 450 (math-neg val)))) 451 ((and (math-num-integerp n) 452 (Math-lessp n m)) 453 0) 454 (t 455 (math-inexact-result) 456 (let ((tm (math-trunc m))) 457 (or (integerp tm) (math-reject-arg tm 'fixnump)) 458 (if (> tm 100) 459 (math-div (calcFunc-fact (math-float n)) 460 (math-mul (calcFunc-fact (math-float m)) 461 (calcFunc-fact (math-float 462 (math-sub n m))))) 463 (math-with-extra-prec 1 464 (math-choose-float-iter tm n 1 1))))))) 465 466(defun math-choose-iter (m n i c) 467 (if (and (= (% i 5) 1) (> i 5)) 468 (math-working (format "choose(%d)" (1- i)) c)) 469 (if (<= i m) 470 (math-choose-iter m (1- n) (1+ i) 471 (math-quotient (math-mul c n) i)) 472 c)) 473 474(defun math-choose-float-iter (count n i c) 475 (if (= (% i 5) 1) 476 (math-working (format "choose(%d)" (1- i)) c)) 477 (if (> count 0) 478 (math-choose-float-iter (1- count) (math-sub n 1) (1+ i) 479 (math-div (math-mul c n) i)) 480 c)) 481 482 483;;; Stirling numbers. 484 485(defun calcFunc-stir1 (n m) 486 (math-stirling-number n m 1)) 487 488(defun calcFunc-stir2 (n m) 489 (math-stirling-number n m 0)) 490 491(defvar math-stirling-cache (vector [[1]] [[1]])) 492 493;; The variable math-stirling-local-cache is local to 494;; math-stirling-number, but is used by math-stirling-1 495;; and math-stirling-2, which are called by math-stirling-number. 496(defvar math-stirling-local-cache) 497 498(defun math-stirling-number (n m k) 499 (or (math-num-natnump n) (math-reject-arg n 'natnump)) 500 (or (math-num-natnump m) (math-reject-arg m 'natnump)) 501 (if (consp n) (setq n (math-trunc n))) 502 (or (integerp n) (math-reject-arg n 'fixnump)) 503 (if (consp m) (setq m (math-trunc m))) 504 (or (integerp m) (math-reject-arg m 'fixnump)) 505 (if (< n m) 506 0 507 (let ((math-stirling-local-cache (aref math-stirling-cache k))) 508 (while (<= (length math-stirling-local-cache) n) 509 (let ((i (1- (length math-stirling-local-cache))) 510 row) 511 (setq math-stirling-local-cache 512 (vconcat math-stirling-local-cache 513 (make-vector (length math-stirling-local-cache) nil))) 514 (aset math-stirling-cache k math-stirling-local-cache) 515 (while (< (setq i (1+ i)) (length math-stirling-local-cache)) 516 (aset math-stirling-local-cache i (setq row (make-vector (1+ i) nil))) 517 (aset row 0 0) 518 (aset row i 1)))) 519 (if (= k 1) 520 (math-stirling-1 n m) 521 (math-stirling-2 n m))))) 522 523(defun math-stirling-1 (n m) 524 (or (aref (aref math-stirling-local-cache n) m) 525 (aset (aref math-stirling-local-cache n) m 526 (math-add (math-stirling-1 (1- n) (1- m)) 527 (math-mul (- 1 n) (math-stirling-1 (1- n) m)))))) 528 529(defun math-stirling-2 (n m) 530 (or (aref (aref math-stirling-local-cache n) m) 531 (aset (aref math-stirling-local-cache n) m 532 (math-add (math-stirling-2 (1- n) (1- m)) 533 (math-mul m (math-stirling-2 (1- n) m)))))) 534 535(defvar math-random-table nil) 536(defvar math-last-RandSeed nil) 537(defvar math-random-ptr1 nil) 538(defvar math-random-ptr2 nil) 539(defvar math-random-shift nil) 540 541;;; Produce a random 10-bit integer, with (random) if no seed provided, 542;;; or else with Numerical Recipes algorithm ran3 / Knuth 3.2.2-A. 543 544(defvar var-RandSeed) 545(defvar math-random-cache nil) 546(defvar math-gaussian-cache nil) 547 548(defun math-init-random-base () 549 (if (and (boundp 'var-RandSeed) var-RandSeed) 550 (if (eq (car-safe var-RandSeed) 'vec) 551 nil 552 (if (Math-integerp var-RandSeed) 553 (let* ((seed (math-sub 161803 var-RandSeed)) 554 (mj (1+ (math-mod seed '(bigpos 0 0 1)))) 555 (mk (1+ (math-mod (math-quotient seed '(bigpos 0 0 1)) 556 '(bigpos 0 0 1)))) 557 (i 0)) 558 (setq math-random-table (cons 'vec (make-list 55 mj))) 559 (while (<= (setq i (1+ i)) 54) 560 (let* ((ii (% (* i 21) 55)) 561 (p (nthcdr ii math-random-table))) 562 (setcar p mk) 563 (setq mk (- mj mk) 564 mj (car p))))) 565 (math-reject-arg var-RandSeed "*RandSeed must be an integer")) 566 (setq var-RandSeed (list 'vec var-RandSeed) 567 math-random-ptr1 math-random-table 568 math-random-cache nil 569 math-random-ptr2 (nthcdr 31 math-random-table)) 570 (let ((i 200)) 571 (while (> (setq i (1- i)) 0) 572 (math-random-base)))) 573 (random t) 574 (setq var-RandSeed nil 575 math-random-cache nil 576 math-random-shift -4) ; assume RAND_MAX >= 16383 577 ;; This exercises the random number generator and also helps 578 ;; deduce a better value for RAND_MAX. 579 (let ((i 0)) 580 (while (< (setq i (1+ i)) 30) 581 (if (> (lsh (math-abs (random)) math-random-shift) 4095) 582 (setq math-random-shift (1- math-random-shift)))))) 583 (setq math-last-RandSeed var-RandSeed 584 math-gaussian-cache nil)) 585 586(defun math-random-base () 587 (if var-RandSeed 588 (progn 589 (setq math-random-ptr1 (or (cdr math-random-ptr1) 590 (cdr math-random-table)) 591 math-random-ptr2 (or (cdr math-random-ptr2) 592 (cdr math-random-table))) 593 (logand (lsh (setcar math-random-ptr1 594 (logand (- (car math-random-ptr1) 595 (car math-random-ptr2)) 524287)) 596 -6) 1023)) 597 (logand (lsh (random) math-random-shift) 1023))) 598 599 600;;; Produce a random digit in the range 0..999. 601;;; Avoid various pitfalls that may lurk in the built-in (random) function! 602;;; Shuffling algorithm from Numerical Recipes, section 7.1. 603(defvar math-random-last) 604(defun math-random-digit () 605 (let (i) 606 (or (and (boundp 'var-RandSeed) (eq var-RandSeed math-last-RandSeed)) 607 (math-init-random-base)) 608 (or math-random-cache 609 (progn 610 (setq math-random-last (math-random-base) 611 math-random-cache (make-vector 13 nil) 612 i -1) 613 (while (< (setq i (1+ i)) 13) 614 (aset math-random-cache i (math-random-base))))) 615 (while (progn 616 (setq i (/ math-random-last 79) ; 0 <= i < 13 617 math-random-last (aref math-random-cache i)) 618 (aset math-random-cache i (math-random-base)) 619 (>= math-random-last 1000))) 620 math-random-last)) 621 622;;; Produce an N-digit random integer. 623(defun math-random-digits (n) 624 (cond ((<= n 6) 625 (math-scale-right (+ (* (math-random-digit) 1000) (math-random-digit)) 626 (- 6 n))) 627 (t (let* ((slop (% (- 900003 n) 3)) 628 (i (/ (+ n slop) 3)) 629 (digs nil)) 630 (while (> i 0) 631 (setq digs (cons (math-random-digit) digs) 632 i (1- i))) 633 (math-normalize (math-scale-right (cons 'bigpos digs) 634 slop)))))) 635 636;;; Produce a uniformly-distributed random float 0 <= N < 1. 637(defun math-random-float () 638 (math-make-float (math-random-digits calc-internal-prec) 639 (- calc-internal-prec))) 640 641;;; Produce a Gaussian-distributed random float with mean=0, sigma=1. 642(defun math-gaussian-float () 643 (math-with-extra-prec 2 644 (if (and math-gaussian-cache 645 (= (car math-gaussian-cache) calc-internal-prec)) 646 (prog1 647 (cdr math-gaussian-cache) 648 (setq math-gaussian-cache nil)) 649 (let* ((v1 (math-add (math-mul (math-random-float) 2) -1)) 650 (v2 (math-add (math-mul (math-random-float) 2) -1)) 651 (r (math-add (math-sqr v1) (math-sqr v2)))) 652 (while (or (not (Math-lessp r 1)) (math-zerop r)) 653 (setq v1 (math-add (math-mul (math-random-float) 2) -1) 654 v2 (math-add (math-mul (math-random-float) 2) -1) 655 r (math-add (math-sqr v1) (math-sqr v2)))) 656 (let ((fac (math-sqrt (math-mul (math-div (calcFunc-ln r) r) -2)))) 657 (setq math-gaussian-cache (cons calc-internal-prec 658 (math-mul v1 fac))) 659 (math-mul v2 fac)))))) 660 661;;; Produce a random integer or real 0 <= N < MAX. 662(defun calcFunc-random (max) 663 (cond ((Math-zerop max) 664 (math-gaussian-float)) 665 ((Math-integerp max) 666 (let* ((digs (math-numdigs max)) 667 (r (math-random-digits (+ digs 3)))) 668 (math-mod r max))) 669 ((Math-realp max) 670 (math-mul (math-random-float) max)) 671 ((and (eq (car max) 'intv) (math-constp max) 672 (Math-lessp (nth 2 max) (nth 3 max))) 673 (if (math-floatp max) 674 (let ((val (math-add (math-mul (math-random-float) 675 (math-sub (nth 3 max) (nth 2 max))) 676 (nth 2 max)))) 677 (if (or (and (memq (nth 1 max) '(0 1)) ; almost not worth 678 (Math-equal val (nth 2 max))) ; checking! 679 (and (memq (nth 1 max) '(0 2)) 680 (Math-equal val (nth 3 max)))) 681 (calcFunc-random max) 682 val)) 683 (let ((lo (if (memq (nth 1 max) '(0 1)) 684 (math-add (nth 2 max) 1) (nth 2 max))) 685 (hi (if (memq (nth 1 max) '(1 3)) 686 (math-add (nth 3 max) 1) (nth 3 max)))) 687 (if (Math-lessp lo hi) 688 (math-add (calcFunc-random (math-sub hi lo)) lo) 689 (math-reject-arg max "*Empty interval"))))) 690 ((eq (car max) 'vec) 691 (if (cdr max) 692 (nth (1+ (calcFunc-random (1- (length max)))) max) 693 (math-reject-arg max "*Empty list"))) 694 ((and (eq (car max) 'sdev) (math-constp max) (Math-realp (nth 1 max))) 695 (math-add (math-mul (math-gaussian-float) (nth 2 max)) (nth 1 max))) 696 (t (math-reject-arg max 'realp)))) 697 698;;; Choose N objects at random from the set MAX without duplicates. 699(defun calcFunc-shuffle (n &optional max) 700 (or max (setq max n n -1)) 701 (or (and (Math-num-integerp n) 702 (or (natnump (setq n (math-trunc n))) (eq n -1))) 703 (math-reject-arg n 'integerp)) 704 (cond ((or (math-zerop max) 705 (math-floatp max) 706 (eq (car-safe max) 'sdev)) 707 (if (< n 0) 708 (math-reject-arg n 'natnump) 709 (math-simple-shuffle n max))) 710 ((and (<= n 1) (>= n 0)) 711 (math-simple-shuffle n max)) 712 ((and (eq (car-safe max) 'intv) (math-constp max)) 713 (let ((num (math-add (math-sub (nth 3 max) (nth 2 max)) 714 (cdr (assq (nth 1 max) 715 '((0 . -1) (1 . 0) 716 (2 . 0) (3 . 1)))))) 717 (min (math-add (nth 2 max) (if (memq (nth 1 max) '(0 1)) 718 1 0)))) 719 (if (< n 0) (setq n num)) 720 (or (math-posp num) (math-reject-arg max 'range)) 721 (and (Math-lessp num n) (math-reject-arg n 'range)) 722 (if (Math-lessp n (math-quotient num 3)) 723 (math-simple-shuffle n max) 724 (if (> (* n 4) (* num 3)) 725 (math-add (math-sub min 1) 726 (math-shuffle-list n num (calcFunc-index num))) 727 (let ((tot 0) 728 (m 0) 729 (vec nil)) 730 (while (< m n) 731 (if (< (calcFunc-random (- num tot)) (- n m)) 732 (setq vec (cons (math-add min tot) vec) 733 m (1+ m))) 734 (setq tot (1+ tot))) 735 (math-shuffle-list n n (cons 'vec vec))))))) 736 ((eq (car-safe max) 'vec) 737 (let ((size (1- (length max)))) 738 (if (< n 0) (setq n size)) 739 (if (and (> n (/ size 2)) (<= n size)) 740 (math-shuffle-list n size (copy-sequence max)) 741 (let* ((vals (calcFunc-shuffle 742 n (list 'intv 3 1 (1- (length max))))) 743 (p vals)) 744 (while (setq p (cdr p)) 745 (setcar p (nth (car p) max))) 746 vals)))) 747 ((math-integerp max) 748 (if (math-posp max) 749 (calcFunc-shuffle n (list 'intv 2 0 max)) 750 (calcFunc-shuffle n (list 'intv 1 max 0)))) 751 (t (math-reject-arg max 'realp)))) 752 753(defun math-simple-shuffle (n max) 754 (let ((vec nil) 755 val) 756 (while (>= (setq n (1- n)) 0) 757 (while (math-member (setq val (calcFunc-random max)) vec)) 758 (setq vec (cons val vec))) 759 (cons 'vec vec))) 760 761(defun math-shuffle-list (n size vec) 762 (let ((j size) 763 k temp 764 (p vec)) 765 (while (cdr (setq p (cdr p))) 766 (setq k (calcFunc-random j) 767 j (1- j) 768 temp (nth k p)) 769 (setcar (nthcdr k p) (car p)) 770 (setcar p temp)) 771 (cons 'vec (nthcdr (- size n -1) vec)))) 772 773(defun math-member (x list) 774 (while (and list (not (equal x (car list)))) 775 (setq list (cdr list))) 776 list) 777 778 779;;; Check if the integer N is prime. [X I] 780;;; Return (nil) if non-prime, 781;;; (nil N) if non-prime with known factor N, 782;;; (nil unknown) if non-prime with no known factors, 783;;; (t) if prime, 784;;; (maybe N P) if probably prime (after N iters with probability P%) 785(defvar math-prime-test-cache '(-1)) 786 787(defvar math-prime-test-cache-k) 788(defvar math-prime-test-cache-q) 789(defvar math-prime-test-cache-nm1) 790 791(defun math-prime-test (n iters) 792 (if (and (Math-vectorp n) (cdr n)) 793 (setq n (nth (1- (length n)) n))) 794 (if (Math-messy-integerp n) 795 (setq n (math-trunc n))) 796 (let ((res)) 797 (while (> iters 0) 798 (setq res 799 (cond ((and (integerp n) (<= n 5003)) 800 (list (= (math-next-small-prime n) n))) 801 ((not (Math-integerp n)) 802 (error "Argument must be an integer")) 803 ((Math-integer-negp n) 804 '(nil)) 805 ((Math-natnum-lessp n '(bigpos 0 0 8)) 806 (setq n (math-fixnum n)) 807 (let ((i -1) v) 808 (while (and (> (% n (setq v (aref math-primes-table 809 (setq i (1+ i))))) 810 0) 811 (< (* v v) n))) 812 (if (= (% n v) 0) 813 (list nil v) 814 '(t)))) 815 ((not (equal n (car math-prime-test-cache))) 816 (cond ((= (% (nth 1 n) 2) 0) '(nil 2)) 817 ((= (% (nth 1 n) 5) 0) '(nil 5)) 818 (t (let ((dig (cdr n)) (sum 0)) 819 (while dig 820 (if (cdr dig) 821 (setq sum (% (+ (+ sum (car dig)) 822 (* (nth 1 dig) 1000)) 823 111111) 824 dig (cdr (cdr dig))) 825 (setq sum (% (+ sum (car dig)) 111111) 826 dig nil))) 827 (cond ((= (% sum 3) 0) '(nil 3)) 828 ((= (% sum 7) 0) '(nil 7)) 829 ((= (% sum 11) 0) '(nil 11)) 830 ((= (% sum 13) 0) '(nil 13)) 831 ((= (% sum 37) 0) '(nil 37)) 832 (t 833 (setq math-prime-test-cache-k 1 834 math-prime-test-cache-q 835 (math-div2 n) 836 math-prime-test-cache-nm1 837 (math-add n -1)) 838 (while (math-evenp 839 math-prime-test-cache-q) 840 (setq math-prime-test-cache-k 841 (1+ math-prime-test-cache-k) 842 math-prime-test-cache-q 843 (math-div2 844 math-prime-test-cache-q))) 845 (setq iters (1+ iters)) 846 (list 'maybe 847 0 848 (math-sub 849 100 850 (math-div 851 '(float 232 0) 852 (math-numdigs n)))))))))) 853 ((not (eq (car (nth 1 math-prime-test-cache)) 'maybe)) 854 (nth 1 math-prime-test-cache)) 855 (t ; Fermat step 856 (let* ((x (math-add (calcFunc-random (math-add n -2)) 2)) 857 (y (math-pow-mod x math-prime-test-cache-q n)) 858 (j 0)) 859 (while (and (not (eq y 1)) 860 (not (equal y math-prime-test-cache-nm1)) 861 (< (setq j (1+ j)) math-prime-test-cache-k)) 862 (setq y (math-mod (math-mul y y) n))) 863 (if (or (equal y math-prime-test-cache-nm1) 864 (and (eq y 1) (eq j 0))) 865 (list 'maybe 866 (1+ (nth 1 (nth 1 math-prime-test-cache))) 867 (math-mul (nth 2 (nth 1 math-prime-test-cache)) 868 '(float 25 -2))) 869 '(nil unknown)))))) 870 (setq math-prime-test-cache (list n res) 871 iters (if (eq (car res) 'maybe) 872 (1- iters) 873 0))) 874 res)) 875 876(defun calcFunc-prime (n &optional iters) 877 (or (math-num-integerp n) (math-reject-arg n 'integerp)) 878 (or (not iters) (math-num-integerp iters) (math-reject-arg iters 'integerp)) 879 (if (car (math-prime-test (math-trunc n) (math-trunc (or iters 1)))) 880 1 881 0)) 882 883;;; Theory: summing base-10^6 digits modulo 111111 is "casting out 999999s". 884;;; Initial probability that N is prime is 1/ln(N) = log10(e)/log10(N). 885;;; After culling [2,3,5,7,11,13,37], probability of primality is 5.36 x more. 886;;; Initial reported probability of non-primality is thus 100% - this. 887;;; Each Fermat step multiplies this probability by 25%. 888;;; The Fermat step is algorithm P from Knuth section 4.5.4. 889 890 891(defun calcFunc-prfac (n) 892 (setq math-prime-factors-finished t) 893 (if (Math-messy-integerp n) 894 (setq n (math-trunc n))) 895 (if (Math-natnump n) 896 (if (Math-natnum-lessp 2 n) 897 (let (factors res p (i 0)) 898 (while (and (not (eq n 1)) 899 (< i (length math-primes-table))) 900 (setq p (aref math-primes-table i)) 901 (while (eq (cdr (setq res (cond ((eq n p) (cons 1 0)) 902 ((eq n 1) (cons 0 1)) 903 ((consp n) (math-idivmod n p)) 904 (t (cons (/ n p) (% n p)))))) 905 0) 906 (math-working "factor" p) 907 (setq factors (nconc factors (list p)) 908 n (car res))) 909 (or (eq n 1) 910 (Math-natnum-lessp p (car res)) 911 (setq factors (nconc factors (list n)) 912 n 1)) 913 (setq i (1+ i))) 914 (or (setq math-prime-factors-finished (eq n 1)) 915 (setq factors (nconc factors (list n)))) 916 (cons 'vec factors)) 917 (list 'vec n)) 918 (if (Math-integerp n) 919 (if (eq n -1) 920 (list 'vec n) 921 (cons 'vec (cons -1 (cdr (calcFunc-prfac (math-neg n)))))) 922 (calc-record-why 'integerp n) 923 (list 'calcFunc-prfac n)))) 924 925(defun calcFunc-totient (n) 926 (if (Math-messy-integerp n) 927 (setq n (math-trunc n))) 928 (if (Math-natnump n) 929 (if (Math-natnum-lessp n 2) 930 (if (Math-negp n) 931 (calcFunc-totient (math-abs n)) 932 n) 933 (let ((factors (cdr (calcFunc-prfac n))) 934 p) 935 (if math-prime-factors-finished 936 (progn 937 (while factors 938 (setq p (car factors) 939 n (math-mul (math-div n p) (math-add p -1))) 940 (while (equal p (car factors)) 941 (setq factors (cdr factors)))) 942 n) 943 (calc-record-why "*Number too big to factor" n) 944 (list 'calcFunc-totient n)))) 945 (calc-record-why 'natnump n) 946 (list 'calcFunc-totient n))) 947 948(defun calcFunc-moebius (n) 949 (if (Math-messy-integerp n) 950 (setq n (math-trunc n))) 951 (if (and (Math-natnump n) (not (eq n 0))) 952 (if (Math-natnum-lessp n 2) 953 (if (Math-negp n) 954 (calcFunc-moebius (math-abs n)) 955 1) 956 (let ((factors (cdr (calcFunc-prfac n))) 957 (mu 1)) 958 (if math-prime-factors-finished 959 (progn 960 (while factors 961 (setq mu (if (equal (car factors) (nth 1 factors)) 962 0 (math-neg mu)) 963 factors (cdr factors))) 964 mu) 965 (calc-record-why "Number too big to factor" n) 966 (list 'calcFunc-moebius n)))) 967 (calc-record-why 'posintp n) 968 (list 'calcFunc-moebius n))) 969 970 971(defun calcFunc-nextprime (n &optional iters) 972 (if (Math-integerp n) 973 (if (Math-integer-negp n) 974 2 975 (if (and (integerp n) (< n 5003)) 976 (math-next-small-prime (1+ n)) 977 (if (math-evenp n) 978 (setq n (math-add n -1))) 979 (let (res) 980 (while (not (car (setq res (math-prime-test 981 (setq n (math-add n 2)) 982 (or iters 1)))))) 983 (if (and calc-verbose-nextprime 984 (eq (car res) 'maybe)) 985 (calc-report-prime-test res))) 986 n)) 987 (if (Math-realp n) 988 (calcFunc-nextprime (math-trunc n) iters) 989 (math-reject-arg n 'integerp)))) 990 991(defun calcFunc-prevprime (n &optional iters) 992 (if (Math-integerp n) 993 (if (Math-lessp n 4) 994 2 995 (if (math-evenp n) 996 (setq n (math-add n 1))) 997 (let (res) 998 (while (not (car (setq res (math-prime-test 999 (setq n (math-add n -2)) 1000 (or iters 1)))))) 1001 (if (and calc-verbose-nextprime 1002 (eq (car res) 'maybe)) 1003 (calc-report-prime-test res))) 1004 n) 1005 (if (Math-realp n) 1006 (calcFunc-prevprime (math-ceiling n) iters) 1007 (math-reject-arg n 'integerp)))) 1008 1009(defun math-next-small-prime (n) 1010 (if (and (integerp n) (> n 2)) 1011 (let ((lo -1) 1012 (hi (length math-primes-table)) 1013 mid) 1014 (while (> (- hi lo) 1) 1015 (if (> n (aref math-primes-table 1016 (setq mid (ash (+ lo hi) -1)))) 1017 (setq lo mid) 1018 (setq hi mid))) 1019 (aref math-primes-table hi)) 1020 2)) 1021 1022(provide 'calc-comb) 1023 1024;;; arch-tag: 1d75ee9b-0815-42bd-a321-bb3dc001cc02 1025;;; calc-comb.el ends here 1026