code.delx.au - gnu-emacs/blob - lisp/calc/calc-comb.el

   1 ;;; calc-comb.el --- combinatoric functions for Calc
   2
   3 ;; Copyright (C) 1990, 1991, 1992, 1993, 2001, 2002, 2003, 2004,
   4 ;;   2005, 2006, 2007, 2008, 2009, 2010, 2011 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 3 of the License, or
  14 ;; (at your option) 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.  If not, see <http://www.gnu.org/licenses/>.
  23
  24 ;;; Commentary:
  25
  26 ;;; Code:
  27
  28 ;; This file is autoloaded from calc-ext.el.
  29
  30 (require 'calc-ext)
  31 (require 'calc-macs)
  32
  33 (defconst math-primes-table
  34   [2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89
  35      97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181
  36      191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277
  37      281 283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383
  38      389 397 401 409 419 421 431 433 439 443 449 457 461 463 467 479 487
  39      491 499 503 509 521 523 541 547 557 563 569 571 577 587 593 599 601
  40      607 613 617 619 631 641 643 647 653 659 661 673 677 683 691 701 709
  41      719 727 733 739 743 751 757 761 769 773 787 797 809 811 821 823 827
  42      829 839 853 857 859 863 877 881 883 887 907 911 919 929 937 941 947
  43      953 967 971 977 983 991 997 1009 1013 1019 1021 1031 1033 1039 1049
  44      1051 1061 1063 1069 1087 1091 1093 1097 1103 1109 1117 1123 1129 1151
  45      1153 1163 1171 1181 1187 1193 1201 1213 1217 1223 1229 1231 1237 1249
  46      1259 1277 1279 1283 1289 1291 1297 1301 1303 1307 1319 1321 1327 1361
  47      1367 1373 1381 1399 1409 1423 1427 1429 1433 1439 1447 1451 1453 1459
  48      1471 1481 1483 1487 1489 1493 1499 1511 1523 1531 1543 1549 1553 1559
  49      1567 1571 1579 1583 1597 1601 1607 1609 1613 1619 1621 1627 1637 1657
  50      1663 1667 1669 1693 1697 1699 1709 1721 1723 1733 1741 1747 1753 1759
  51      1777 1783 1787 1789 1801 1811 1823 1831 1847 1861 1867 1871 1873 1877
  52      1879 1889 1901 1907 1913 1931 1933 1949 1951 1973 1979 1987 1993 1997
  53      1999 2003 2011 2017 2027 2029 2039 2053 2063 2069 2081 2083 2087 2089
  54      2099 2111 2113 2129 2131 2137 2141 2143 2153 2161 2179 2203 2207 2213
  55      2221 2237 2239 2243 2251 2267 2269 2273 2281 2287 2293 2297 2309 2311
  56      2333 2339 2341 2347 2351 2357 2371 2377 2381 2383 2389 2393 2399 2411
  57      2417 2423 2437 2441 2447 2459 2467 2473 2477 2503 2521 2531 2539 2543
  58      2549 2551 2557 2579 2591 2593 2609 2617 2621 2633 2647 2657 2659 2663
  59      2671 2677 2683 2687 2689 2693 2699 2707 2711 2713 2719 2729 2731 2741
  60      2749 2753 2767 2777 2789 2791 2797 2801 2803 2819 2833 2837 2843 2851
  61      2857 2861 2879 2887 2897 2903 2909 2917 2927 2939 2953 2957 2963 2969
  62      2971 2999 3001 3011 3019 3023 3037 3041 3049 3061 3067 3079 3083 3089
  63      3109 3119 3121 3137 3163 3167 3169 3181 3187 3191 3203 3209 3217 3221
  64      3229 3251 3253 3257 3259 3271 3299 3301 3307 3313 3319 3323 3329 3331
  65      3343 3347 3359 3361 3371 3373 3389 3391 3407 3413 3433 3449 3457 3461
  66      3463 3467 3469 3491 3499 3511 3517 3527 3529 3533 3539 3541 3547 3557
  67      3559 3571 3581 3583 3593 3607 3613 3617 3623 3631 3637 3643 3659 3671
  68      3673 3677 3691 3697 3701 3709 3719 3727 3733 3739 3761 3767 3769 3779
  69      3793 3797 3803 3821 3823 3833 3847 3851 3853 3863 3877 3881 3889 3907
  70      3911 3917 3919 3923 3929 3931 3943 3947 3967 3989 4001 4003 4007 4013
  71      4019 4021 4027 4049 4051 4057 4073 4079 4091 4093 4099 4111 4127 4129
  72      4133 4139 4153 4157 4159 4177 4201 4211 4217 4219 4229 4231 4241 4243
  73      4253 4259 4261 4271 4273 4283 4289 4297 4327 4337 4339 4349 4357 4363
  74      4373 4391 4397 4409 4421 4423 4441 4447 4451 4457 4463 4481 4483 4493
  75      4507 4513 4517 4519 4523 4547 4549 4561 4567 4583 4591 4597 4603 4621
  76      4637 4639 4643 4649 4651 4657 4663 4673 4679 4691 4703 4721 4723 4729
  77      4733 4751 4759 4783 4787 4789 4793 4799 4801 4813 4817 4831 4861 4871
  78      4877 4889 4903 4909 4919 4931 4933 4937 4943 4951 4957 4967 4969 4973
  79      4987 4993 4999 5003])
  80
  81 ;; The variable math-prime-factors-finished is set by calcFunc-prfac to
  82 ;; indicate whether factoring is complete, and used by calcFunc-factors,
  83 ;; calcFunc-totient and calcFunc-moebius.
  84 (defvar math-prime-factors-finished)
  85
  86 ;;; Combinatorics
  87
  88 (defun calc-gcd (arg)
  89   (interactive "P")
  90   (calc-slow-wrapper
  91    (calc-binary-op "gcd" 'calcFunc-gcd arg)))
  92
  93 (defun calc-lcm (arg)
  94   (interactive "P")
  95   (calc-slow-wrapper
  96    (calc-binary-op "lcm" 'calcFunc-lcm arg)))
  97
  98 (defun calc-extended-gcd ()
  99   (interactive)
 100   (calc-slow-wrapper
 101    (calc-enter-result 2 "egcd" (cons 'calcFunc-egcd (calc-top-list-n 2)))))
 102
 103 (defun calc-factorial (arg)
 104   (interactive "P")
 105   (calc-slow-wrapper
 106    (calc-unary-op "fact" 'calcFunc-fact arg)))
 107
 108 (defun calc-gamma (arg)
 109   (interactive "P")
 110   (calc-slow-wrapper
 111    (calc-unary-op "gmma" 'calcFunc-gamma arg)))
 112
 113 (defun calc-double-factorial (arg)
 114   (interactive "P")
 115   (calc-slow-wrapper
 116    (calc-unary-op "dfac" 'calcFunc-dfact arg)))
 117
 118 (defun calc-choose (arg)
 119   (interactive "P")
 120   (calc-slow-wrapper
 121    (if (calc-is-hyperbolic)
 122        (calc-binary-op "perm" 'calcFunc-perm arg)
 123      (calc-binary-op "chos" 'calcFunc-choose arg))))
 124
 125 (defun calc-perm (arg)
 126   (interactive "P")
 127   (calc-hyperbolic-func)
 128   (calc-choose arg))
 129
 130 (defvar calc-last-random-limit '(float 1 0))
 131 (defun calc-random (n)
 132   (interactive "P")
 133   (calc-slow-wrapper
 134    (if n
 135        (calc-enter-result 0 "rand" (list 'calcFunc-random
 136                                          (calc-get-random-limit
 137                                           (prefix-numeric-value n))))
 138      (calc-enter-result 1 "rand" (list 'calcFunc-random
 139                                        (calc-get-random-limit
 140                                         (calc-top-n 1)))))))
 141
 142 (defun calc-get-random-limit (val)
 143   (if (eq val 0)
 144       calc-last-random-limit
 145     (setq calc-last-random-limit val)))
 146
 147 (defun calc-rrandom ()
 148   (interactive)
 149   (calc-slow-wrapper
 150    (setq calc-last-random-limit '(float 1 0))
 151    (calc-enter-result 0 "rand" (list 'calcFunc-random '(float 1 0)))))
 152
 153 (defun calc-random-again (arg)
 154   (interactive "p")
 155   (calc-slow-wrapper
 156    (while (>= (setq arg (1- arg)) 0)
 157      (calc-enter-result 0 "rand" (list 'calcFunc-random
 158                                        calc-last-random-limit)))))
 159
 160 (defun calc-shuffle (n)
 161   (interactive "P")
 162   (calc-slow-wrapper
 163    (if n
 164        (calc-enter-result 1 "shuf" (list 'calcFunc-shuffle
 165                                          (prefix-numeric-value n)
 166                                          (calc-get-random-limit
 167                                           (calc-top-n 1))))
 168      (calc-enter-result 2 "shuf" (list 'calcFunc-shuffle
 169                                        (calc-top-n 1)
 170                                        (calc-get-random-limit
 171                                         (calc-top-n 2)))))))
 172
 173 (defun calc-report-prime-test (res)
 174   (cond ((eq (car res) t)
 175          (calc-record-message "prim" "Prime (guaranteed)"))
 176         ((eq (car res) nil)
 177          (if (cdr res)
 178              (if (eq (nth 1 res) 'unknown)
 179                  (calc-record-message
 180                   "prim" "Non-prime (factors unknown)")
 181                (calc-record-message
 182                 "prim" "Non-prime (%s is a factor)"
 183                 (math-format-number (nth 1 res))))
 184            (calc-record-message "prim" "Non-prime")))
 185         (t
 186          (calc-record-message
 187           "prim" "Probably prime (%d iters; %s%% chance of error)"
 188           (nth 1 res)
 189           (let ((calc-float-format '(fix 2)))
 190             (math-format-number (nth 2 res)))))))
 191
 192 (defun calc-prime-test (iters)
 193   (interactive "p")
 194   (calc-slow-wrapper
 195    (let* ((n (calc-top-n 1))
 196           (res (math-prime-test n iters)))
 197      (calc-report-prime-test res))))
 198
 199 (defvar calc-verbose-nextprime nil)
 200
 201 (defun calc-next-prime (iters)
 202   (interactive "p")
 203   (calc-slow-wrapper
 204    (let ((calc-verbose-nextprime t))
 205      (if (calc-is-inverse)
 206          (calc-enter-result 1 "prvp" (list 'calcFunc-prevprime
 207                                            (calc-top-n 1) (math-abs iters)))
 208        (calc-enter-result 1 "nxtp" (list 'calcFunc-nextprime
 209                                          (calc-top-n 1) (math-abs iters)))))))
 210
 211 (defun calc-prev-prime (iters)
 212   (interactive "p")
 213   (calc-invert-func)
 214   (calc-next-prime iters))
 215
 216 (defun calc-prime-factors (iters)
 217   (interactive "p")
 218   (calc-slow-wrapper
 219    (let ((res (calcFunc-prfac (calc-top-n 1))))
 220      (if (not math-prime-factors-finished)
 221          (calc-record-message "pfac" "Warning:  May not be fully factored"))
 222      (calc-enter-result 1 "pfac" res))))
 223
 224 (defun calc-totient (arg)
 225   (interactive "P")
 226   (calc-slow-wrapper
 227    (calc-unary-op "phi" 'calcFunc-totient arg)))
 228
 229 (defun calc-moebius (arg)
 230   (interactive "P")
 231   (calc-slow-wrapper
 232    (calc-unary-op "mu" 'calcFunc-moebius arg)))
 233
 234
 235 (defun calcFunc-gcd (a b)
 236   (if (Math-messy-integerp a)
 237       (setq a (math-trunc a)))
 238   (if (Math-messy-integerp b)
 239       (setq b (math-trunc b)))
 240   (cond ((and (Math-integerp a) (Math-integerp b))
 241          (math-gcd a b))
 242         ((Math-looks-negp a)
 243          (calcFunc-gcd (math-neg a) b))
 244         ((Math-looks-negp b)
 245          (calcFunc-gcd a (math-neg b)))
 246         ((Math-zerop a) b)
 247         ((Math-zerop b) a)
 248         ((and (Math-ratp a)
 249               (Math-ratp b))
 250          (math-make-frac (math-gcd (if (eq (car-safe a) 'frac) (nth 1 a) a)
 251                                    (if (eq (car-safe b) 'frac) (nth 1 b) b))
 252                          (calcFunc-lcm
 253                           (if (eq (car-safe a) 'frac) (nth 2 a) 1)
 254                           (if (eq (car-safe b) 'frac) (nth 2 b) 1))))
 255         ((not (Math-integerp a))
 256          (calc-record-why 'integerp a)
 257          (list 'calcFunc-gcd a b))
 258         (t
 259          (calc-record-why 'integerp b)
 260          (list 'calcFunc-gcd a b))))
 261
 262 (defun calcFunc-lcm (a b)
 263   (let ((g (calcFunc-gcd a b)))
 264     (if (Math-numberp g)
 265         (math-div (math-mul a b) g)
 266       (list 'calcFunc-lcm a b))))
 267
 268 (defun calcFunc-egcd (a b)   ; Knuth section 4.5.2
 269   (cond
 270    ((not (Math-integerp a))
 271     (if (Math-messy-integerp a)
 272         (calcFunc-egcd (math-trunc a) b)
 273       (calc-record-why 'integerp a)
 274       (list 'calcFunc-egcd a b)))
 275    ((not (Math-integerp b))
 276     (if (Math-messy-integerp b)
 277         (calcFunc-egcd a (math-trunc b))
 278       (calc-record-why 'integerp b)
 279       (list 'calcFunc-egcd a b)))
 280    (t
 281     (let ((u1 1) (u2 0) (u3 a)
 282           (v1 0) (v2 1) (v3 b)
 283           t1 t2 q)
 284       (while (not (eq v3 0))
 285         (setq q (math-idivmod u3 v3)
 286               t1 (math-sub u1 (math-mul v1 (car q)))
 287               t2 (math-sub u2 (math-mul v2 (car q)))
 288               u1 v1  u2 v2  u3 v3
 289               v1 t1  v2 t2  v3 (cdr q)))
 290       (list 'vec u3 u1 u2)))))
 291
 292
 293 ;;; Factorial and related functions.
 294
 295 (defconst math-small-factorial-table
 296   (vector 1 1 2 6 24 120 720 5040 40320 362880 3628800 39916800
 297           (math-read-number-simple "479001600")
 298           (math-read-number-simple "6227020800")
 299           (math-read-number-simple "87178291200")
 300           (math-read-number-simple "1307674368000")
 301           (math-read-number-simple "20922789888000")
 302           (math-read-number-simple "355687428096000")
 303           (math-read-number-simple "6402373705728000")
 304           (math-read-number-simple "121645100408832000")
 305           (math-read-number-simple "2432902008176640000")))
 306
 307 (defun calcFunc-fact (n)   ; [I I] [F F] [Public]
 308   (let (temp)
 309     (cond ((Math-integer-negp n)
 310            (if calc-infinite-mode
 311                '(var uinf var-uinf)
 312              (math-reject-arg n 'range)))
 313           ((integerp n)
 314            (if (<= n 20)
 315                (aref math-small-factorial-table n)
 316              (math-factorial-iter (1- n) 2 1)))
 317           ((and (math-messy-integerp n)
 318                 (Math-lessp n 100))
 319            (math-inexact-result)
 320            (setq temp (math-trunc n))
 321            (if (>= temp 0)
 322                (if (<= temp 20)
 323                    (math-float (calcFunc-fact temp))
 324                  (math-with-extra-prec 1
 325                    (math-factorial-iter (1- temp) 2 '(float 1 0))))
 326              (math-reject-arg n 'range)))
 327           ((math-numberp n)
 328            (let* ((q (math-quarter-integer n))
 329                   (tn (and q (Math-lessp n 1000) (Math-lessp -1000 n)
 330                            (1+ (math-floor n)))))
 331              (cond ((and tn (= q 2)
 332                          (or calc-symbolic-mode (< (math-abs tn) 20)))
 333                     (let ((q (if (< tn 0)
 334                                  (math-div
 335                                   (math-pow -2 (- tn))
 336                                   (math-double-factorial-iter (* -2 tn) 3 1 2))
 337                                (math-div
 338                                 (math-double-factorial-iter (* 2 tn) 3 1 2)
 339                                 (math-pow 2 tn)))))
 340                       (math-mul q (if calc-symbolic-mode
 341                                       (list 'calcFunc-sqrt '(var pi var-pi))
 342                                     (math-sqrt-pi)))))
 343                    ((and tn (>= tn 0) (< tn 20)
 344                          (memq q '(1 3)))
 345                     (math-inexact-result)
 346                     (math-div
 347                      (math-mul (math-double-factorial-iter (* 4 tn) q 1 4)
 348                                (if (= q 1) (math-gamma-1q) (math-gamma-3q)))
 349                      (math-pow 4 tn)))
 350                    (t
 351                     (math-inexact-result)
 352                     (math-with-extra-prec 3
 353                       (math-gammap1-raw (math-float n)))))))
 354           ((equal n '(var inf var-inf)) n)
 355           (t (calc-record-why 'numberp n)
 356              (list 'calcFunc-fact n)))))
 357
 358 (math-defcache math-gamma-1q nil
 359   (math-with-extra-prec 3
 360     (math-gammap1-raw '(float -75 -2))))
 361
 362 (math-defcache math-gamma-3q nil
 363   (math-with-extra-prec 3
 364     (math-gammap1-raw '(float -25 -2))))
 365
 366 (defun math-factorial-iter (count n f)
 367   (if (= (% n 5) 1)
 368       (math-working (format "factorial(%d)" (1- n)) f))
 369   (if (> count 0)
 370       (math-factorial-iter (1- count) (1+ n) (math-mul n f))
 371     f))
 372
 373 (defun calcFunc-dfact (n)   ; [I I] [F F] [Public]
 374   (cond ((Math-integer-negp n)
 375          (if (math-oddp n)
 376              (if (eq n -1)
 377                  1
 378                (math-div (if (eq (math-mod n 4) 3) 1 -1)
 379                          (calcFunc-dfact (math-sub -2 n))))
 380            (list 'calcFunc-dfact n)))
 381         ((Math-zerop n) 1)
 382         ((integerp n) (math-double-factorial-iter n (+ 2 (% n 2)) 1 2))
 383         ((math-messy-integerp n)
 384          (let ((temp (math-trunc n)))
 385            (math-inexact-result)
 386            (if (natnump temp)
 387                (if (Math-lessp temp 200)
 388                    (math-with-extra-prec 1
 389                      (math-double-factorial-iter temp (+ 2 (% temp 2))
 390                                                  '(float 1 0) 2))
 391                  (let* ((half (math-div2 temp))
 392                         (even (math-mul (math-pow 2 half)
 393                                         (calcFunc-fact (math-float half)))))
 394                    (if (math-evenp temp)
 395                        even
 396                      (math-div (calcFunc-fact n) even))))
 397              (list 'calcFunc-dfact n))))
 398         ((equal n '(var inf var-inf)) n)
 399         (t (calc-record-why 'natnump n)
 400            (list 'calcFunc-dfact n))))
 401
 402 (defun math-double-factorial-iter (max n f step)
 403   (if (< (% n 12) step)
 404       (math-working (format "dfact(%d)" (- n step)) f))
 405   (if (<= n max)
 406       (math-double-factorial-iter max (+ n step) (math-mul n f) step)
 407     f))
 408
 409 (defun calcFunc-perm (n m)   ; [I I I] [F F F] [Public]
 410   (cond ((and (integerp n) (integerp m) (<= m n) (>= m 0))
 411          (math-factorial-iter m (1+ (- n m)) 1))
 412         ((or (not (math-num-integerp n))
 413              (and (math-messy-integerp n) (Math-lessp 100 n))
 414              (not (math-num-integerp m))
 415              (and (math-messy-integerp m) (Math-lessp 100 m)))
 416          (or (math-realp n) (equal n '(var inf var-inf))
 417              (math-reject-arg n 'realp))
 418          (or (math-realp m) (equal m '(var inf var-inf))
 419              (math-reject-arg m 'realp))
 420          (and (math-num-integerp n) (math-negp n) (math-reject-arg n 'range))
 421          (and (math-num-integerp m) (math-negp m) (math-reject-arg m 'range))
 422          (math-div (calcFunc-fact n) (calcFunc-fact (math-sub n m))))
 423         (t
 424          (let ((tn (math-trunc n))
 425                (tm (math-trunc m)))
 426            (math-inexact-result)
 427            (or (integerp tn) (math-reject-arg tn 'fixnump))
 428            (or (integerp tm) (math-reject-arg tm 'fixnump))
 429            (or (and (<= tm tn) (>= tm 0)) (math-reject-arg tm 'range))
 430            (math-with-extra-prec 1
 431              (math-factorial-iter tm (1+ (- tn tm)) '(float 1 0)))))))
 432
 433 (defun calcFunc-choose (n m)   ; [I I I] [F F F] [Public]
 434   (cond ((and (integerp n) (integerp m) (<= m n) (>= m 0))
 435          (if (> m (/ n 2))
 436              (math-choose-iter (- n m) n 1 1)
 437            (math-choose-iter m n 1 1)))
 438         ((not (math-realp n))
 439          (math-reject-arg n 'realp))
 440         ((not (math-realp m))
 441          (math-reject-arg m 'realp))
 442         ((not (math-num-integerp m))
 443          (if (and (math-num-integerp n) (math-negp n))
 444              (list 'calcFunc-choose n m)
 445            (math-div (calcFunc-fact (math-float n))
 446                      (math-mul (calcFunc-fact m)
 447                                (calcFunc-fact (math-sub n m))))))
 448         ((math-negp m) 0)
 449         ((math-negp n)
 450          (let ((val (calcFunc-choose (math-add (math-add n m) -1) m)))
 451            (if (math-evenp (math-trunc m))
 452                val
 453              (math-neg val))))
 454         ((and (math-num-integerp n)
 455               (Math-lessp n m))
 456          0)
 457         (t
 458          (math-inexact-result)
 459          (let ((tm (math-trunc m)))
 460            (or (integerp tm) (math-reject-arg tm 'fixnump))
 461            (if (> tm 100)
 462                (math-div (calcFunc-fact (math-float n))
 463                          (math-mul (calcFunc-fact (math-float m))
 464                                    (calcFunc-fact (math-float
 465                                                    (math-sub n m)))))
 466              (math-with-extra-prec 1
 467                (math-choose-float-iter tm n 1 1)))))))
 468
 469 (defun math-choose-iter (m n i c)
 470   (if (and (= (% i 5) 1) (> i 5))
 471       (math-working (format "choose(%d)" (1- i)) c))
 472   (if (<= i m)
 473       (math-choose-iter m (1- n) (1+ i)
 474                         (math-quotient (math-mul c n) i))
 475     c))
 476
 477 (defun math-choose-float-iter (count n i c)
 478   (if (= (% i 5) 1)
 479       (math-working (format "choose(%d)" (1- i)) c))
 480   (if (> count 0)
 481       (math-choose-float-iter (1- count) (math-sub n 1) (1+ i)
 482                               (math-div (math-mul c n) i))
 483     c))
 484
 485
 486 ;;; Stirling numbers.
 487
 488 (defun calcFunc-stir1 (n m)
 489   (math-stirling-number n m 1))
 490
 491 (defun calcFunc-stir2 (n m)
 492   (math-stirling-number n m 0))
 493
 494 (defvar math-stirling-cache (vector [[1]] [[1]]))
 495
 496 ;; The variable math-stirling-local-cache is local to
 497 ;; math-stirling-number, but is used by math-stirling-1
 498 ;; and math-stirling-2, which are called by math-stirling-number.
 499 (defvar math-stirling-local-cache)
 500
 501 (defun math-stirling-number (n m k)
 502   (or (math-num-natnump n) (math-reject-arg n 'natnump))
 503   (or (math-num-natnump m) (math-reject-arg m 'natnump))
 504   (if (consp n) (setq n (math-trunc n)))
 505   (or (integerp n) (math-reject-arg n 'fixnump))
 506   (if (consp m) (setq m (math-trunc m)))
 507   (or (integerp m) (math-reject-arg m 'fixnump))
 508   (if (< n m)
 509       0
 510     (let ((math-stirling-local-cache (aref math-stirling-cache k)))
 511       (while (<= (length math-stirling-local-cache) n)
 512         (let ((i (1- (length math-stirling-local-cache)))
 513               row)
 514           (setq math-stirling-local-cache
 515                 (vconcat math-stirling-local-cache
 516                          (make-vector (length math-stirling-local-cache) nil)))
 517           (aset math-stirling-cache k math-stirling-local-cache)
 518           (while (< (setq i (1+ i)) (length math-stirling-local-cache))
 519             (aset math-stirling-local-cache i (setq row (make-vector (1+ i) nil)))
 520             (aset row 0 0)
 521             (aset row i 1))))
 522       (if (= k 1)
 523           (math-stirling-1 n m)
 524         (math-stirling-2 n m)))))
 525
 526 (defun math-stirling-1 (n m)
 527   (or (aref (aref math-stirling-local-cache n) m)
 528       (aset (aref math-stirling-local-cache n) m
 529             (math-add (math-stirling-1 (1- n) (1- m))
 530                       (math-mul (- 1 n) (math-stirling-1 (1- n) m))))))
 531
 532 (defun math-stirling-2 (n m)
 533   (or (aref (aref math-stirling-local-cache n) m)
 534       (aset (aref math-stirling-local-cache n) m
 535             (math-add (math-stirling-2 (1- n) (1- m))
 536                       (math-mul m (math-stirling-2 (1- n) m))))))
 537
 538 (defvar math-random-table nil)
 539 (defvar math-last-RandSeed nil)
 540 (defvar math-random-ptr1 nil)
 541 (defvar math-random-ptr2 nil)
 542 (defvar math-random-shift nil)
 543
 544 ;;; Produce a random 10-bit integer, with (random) if no seed provided,
 545 ;;; or else with Numerical Recipes algorithm ran3 / Knuth 3.2.2-A.
 546
 547 (defvar var-RandSeed)
 548 (defvar math-random-cache nil)
 549 (defvar math-gaussian-cache nil)
 550
 551 (defun math-init-random-base ()
 552   (if (and (boundp 'var-RandSeed) var-RandSeed)
 553       (if (eq (car-safe var-RandSeed) 'vec)
 554           nil
 555         (if (Math-integerp var-RandSeed)
 556             (let* ((seed (math-sub 161803 var-RandSeed))
 557                    (mj (1+ (math-mod seed 1000000)))
 558                    (mk (1+ (math-mod (math-quotient seed 1000000)
 559                                      1000000)))
 560                    (i 0))
 561               (setq math-random-table (cons 'vec (make-list 55 mj)))
 562               (while (<= (setq i (1+ i)) 54)
 563                 (let* ((ii (% (* i 21) 55))
 564                        (p (nthcdr ii math-random-table)))
 565                   (setcar p mk)
 566                   (setq mk (- mj mk)
 567                         mj (car p)))))
 568           (math-reject-arg var-RandSeed "*RandSeed must be an integer"))
 569         (setq var-RandSeed (list 'vec var-RandSeed)
 570               math-random-ptr1 math-random-table
 571               math-random-cache nil
 572               math-random-ptr2 (nthcdr 31 math-random-table))
 573         (let ((i 200))
 574           (while (> (setq i (1- i)) 0)
 575             (math-random-base))))
 576     (random t)
 577     (setq var-RandSeed nil
 578           math-random-cache nil
 579           math-random-shift -4)  ; assume RAND_MAX >= 16383
 580     ;; This exercises the random number generator and also helps
 581     ;; deduce a better value for RAND_MAX.
 582     (let ((i 0))
 583       (while (< (setq i (1+ i)) 30)
 584         (if (> (lsh (math-abs (random)) math-random-shift) 4095)
 585             (setq math-random-shift (1- math-random-shift))))))
 586   (setq math-last-RandSeed var-RandSeed
 587         math-gaussian-cache nil))
 588
 589 (defun math-random-base ()
 590   (if var-RandSeed
 591       (progn
 592         (setq math-random-ptr1 (or (cdr math-random-ptr1)
 593                                    (cdr math-random-table))
 594               math-random-ptr2 (or (cdr math-random-ptr2)
 595                                    (cdr math-random-table)))
 596         (logand (lsh (setcar math-random-ptr1
 597                              (logand (- (car math-random-ptr1)
 598                                         (car math-random-ptr2)) 524287))
 599                      -6) 1023))
 600     (logand (lsh (random) math-random-shift) 1023)))
 601
 602
 603 ;;; Produce a random digit in the range 0..999.
 604 ;;; Avoid various pitfalls that may lurk in the built-in (random) function!
 605 ;;; Shuffling algorithm from Numerical Recipes, section 7.1.
 606 (defvar math-random-last)
 607 (defun math-random-three-digit-number ()
 608   "Return a random three digit number."
 609   (let (i)
 610     (or (and (boundp 'var-RandSeed) (eq var-RandSeed math-last-RandSeed))
 611         (math-init-random-base))
 612     (or math-random-cache
 613         (progn
 614           (setq math-random-last (math-random-base)
 615                 math-random-cache (make-vector 13 nil)
 616                 i -1)
 617           (while (< (setq i (1+ i)) 13)
 618             (aset math-random-cache i (math-random-base)))))
 619     (while (progn
 620              (setq i (/ math-random-last 79)   ; 0 <= i < 13
 621                    math-random-last (aref math-random-cache i))
 622              (aset math-random-cache i (math-random-base))
 623              (>= math-random-last 1000)))
 624     math-random-last))
 625
 626 ;;; Produce an N-digit random integer.
 627 (defun math-random-digits (n)
 628   "Produce a random N digit integer."
 629   (let* ((slop (% (- 3 (% n 3)) 3))
 630          (i (/ (+ n slop) 3))
 631          (rnum 0))
 632     (while (> i 0)
 633       (setq rnum
 634             (math-add
 635              (math-random-three-digit-number)
 636              (math-mul rnum 1000)))
 637       (setq i (1- i)))
 638     (math-normalize (math-scale-right rnum slop))))
 639
 640 ;;; Produce a uniformly-distributed random float 0 <= N < 1.
 641 (defun math-random-float ()
 642   (math-make-float (math-random-digits calc-internal-prec)
 643                    (- calc-internal-prec)))
 644
 645 ;;; Produce a Gaussian-distributed random float with mean=0, sigma=1.
 646 (defun math-gaussian-float ()
 647   (math-with-extra-prec 2
 648     (if (and math-gaussian-cache
 649              (= (car math-gaussian-cache) calc-internal-prec))
 650         (prog1
 651             (cdr math-gaussian-cache)
 652           (setq math-gaussian-cache nil))
 653       (let* ((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         (while (or (not (Math-lessp r 1)) (math-zerop r))
 657           (setq v1 (math-add (math-mul (math-random-float) 2) -1)
 658                 v2 (math-add (math-mul (math-random-float) 2) -1)
 659                 r (math-add (math-sqr v1) (math-sqr v2))))
 660         (let ((fac (math-sqrt (math-mul (math-div (calcFunc-ln r) r) -2))))
 661           (setq math-gaussian-cache (cons calc-internal-prec
 662                                           (math-mul v1 fac)))
 663           (math-mul v2 fac))))))
 664
 665 ;;; Produce a random integer or real 0 <= N < MAX.
 666 (defun calcFunc-random (max)
 667   (cond ((Math-zerop max)
 668          (math-gaussian-float))
 669         ((Math-integerp max)
 670          (let* ((digs (math-numdigs max))
 671                 (r (math-random-digits (+ digs 3))))
 672            (math-mod r max)))
 673         ((Math-realp max)
 674          (math-mul (math-random-float) max))
 675         ((and (eq (car max) 'intv) (math-constp max)
 676               (Math-lessp (nth 2 max) (nth 3 max)))
 677          (if (math-floatp max)
 678              (let ((val (math-add (math-mul (math-random-float)
 679                                             (math-sub (nth 3 max) (nth 2 max)))
 680                                   (nth 2 max))))
 681                (if (or (and (memq (nth 1 max) '(0 1))      ; almost not worth
 682                             (Math-equal val (nth 2 max)))  ;   checking!
 683                        (and (memq (nth 1 max) '(0 2))
 684                             (Math-equal val (nth 3 max))))
 685                    (calcFunc-random max)
 686                  val))
 687            (let ((lo (if (memq (nth 1 max) '(0 1))
 688                          (math-add (nth 2 max) 1) (nth 2 max)))
 689                  (hi (if (memq (nth 1 max) '(1 3))
 690                          (math-add (nth 3 max) 1) (nth 3 max))))
 691              (if (Math-lessp lo hi)
 692                  (math-add (calcFunc-random (math-sub hi lo)) lo)
 693                (math-reject-arg max "*Empty interval")))))
 694         ((eq (car max) 'vec)
 695          (if (cdr max)
 696              (nth (1+ (calcFunc-random (1- (length max)))) max)
 697            (math-reject-arg max "*Empty list")))
 698         ((and (eq (car max) 'sdev) (math-constp max) (Math-realp (nth 1 max)))
 699          (math-add (math-mul (math-gaussian-float) (nth 2 max)) (nth 1 max)))
 700         (t (math-reject-arg max 'realp))))
 701
 702 ;;; Choose N objects at random from the set MAX without duplicates.
 703 (defun calcFunc-shuffle (n &optional max)
 704   (or max (setq max n n -1))
 705   (or (and (Math-num-integerp n)
 706            (or (natnump (setq n (math-trunc n))) (eq n -1)))
 707       (math-reject-arg n 'integerp))
 708   (cond ((or (math-zerop max)
 709              (math-floatp max)
 710              (eq (car-safe max) 'sdev))
 711          (if (< n 0)
 712              (math-reject-arg n 'natnump)
 713            (math-simple-shuffle n max)))
 714         ((and (<= n 1) (>= n 0))
 715          (math-simple-shuffle n max))
 716         ((and (eq (car-safe max) 'intv) (math-constp max))
 717          (let ((num (math-add (math-sub (nth 3 max) (nth 2 max))
 718                               (cdr (assq (nth 1 max)
 719                                          '((0 . -1) (1 . 0)
 720                                            (2 . 0) (3 . 1))))))
 721                (min (math-add (nth 2 max) (if (memq (nth 1 max) '(0 1))
 722                                               1 0))))
 723            (if (< n 0) (setq n num))
 724            (or (math-posp num) (math-reject-arg max 'range))
 725            (and (Math-lessp num n) (math-reject-arg n 'range))
 726            (if (Math-lessp n (math-quotient num 3))
 727                (math-simple-shuffle n max)
 728              (if (> (* n 4) (* num 3))
 729                  (math-add (math-sub min 1)
 730                            (math-shuffle-list n num (calcFunc-index num)))
 731                (let ((tot 0)
 732                      (m 0)
 733                      (vec nil))
 734                  (while (< m n)
 735                    (if (< (calcFunc-random (- num tot)) (- n m))
 736                        (setq vec (cons (math-add min tot) vec)
 737                              m (1+ m)))
 738                    (setq tot (1+ tot)))
 739                  (math-shuffle-list n n (cons 'vec vec)))))))
 740         ((eq (car-safe max) 'vec)
 741          (let ((size (1- (length max))))
 742            (if (< n 0) (setq n size))
 743            (if (and (> n (/ size 2)) (<= n size))
 744                (math-shuffle-list n size (copy-sequence max))
 745              (let* ((vals (calcFunc-shuffle
 746                            n (list 'intv 3 1 (1- (length max)))))
 747                     (p vals))
 748                (while (setq p (cdr p))
 749                  (setcar p (nth (car p) max)))
 750                vals))))
 751         ((math-integerp max)
 752          (if (math-posp max)
 753              (calcFunc-shuffle n (list 'intv 2 0 max))
 754            (calcFunc-shuffle n (list 'intv 1 max 0))))
 755         (t (math-reject-arg max 'realp))))
 756
 757 (defun math-simple-shuffle (n max)
 758   (let ((vec nil)
 759         val)
 760     (while (>= (setq n (1- n)) 0)
 761       (while (math-member (setq val (calcFunc-random max)) vec))
 762       (setq vec (cons val vec)))
 763     (cons 'vec vec)))
 764
 765 (defun math-shuffle-list (n size vec)
 766   (let ((j size)
 767         k temp
 768         (p vec))
 769     (while (cdr (setq p (cdr p)))
 770       (setq k (calcFunc-random j)
 771             j (1- j)
 772             temp (nth k p))
 773       (setcar (nthcdr k p) (car p))
 774       (setcar p temp))
 775     (cons 'vec (nthcdr (- size n -1) vec))))
 776
 777 (defun math-member (x list)
 778   (while (and list (not (equal x (car list))))
 779     (setq list (cdr list)))
 780   list)
 781
 782
 783 ;;; Check if the integer N is prime.  [X I]
 784 ;;; Return (nil) if non-prime,
 785 ;;;        (nil N) if non-prime with known factor N,
 786 ;;;        (nil unknown) if non-prime with no known factors,
 787 ;;;        (t) if prime,
 788 ;;;        (maybe N P) if probably prime (after N iters with probability P%)
 789 (defvar math-prime-test-cache '(-1))
 790
 791 (defvar math-prime-test-cache-k)
 792 (defvar math-prime-test-cache-q)
 793 (defvar math-prime-test-cache-nm1)
 794
 795 (defun math-prime-test (n iters)
 796   (if (and (Math-vectorp n) (cdr n))
 797       (setq n (nth (1- (length n)) n)))
 798   (if (Math-messy-integerp n)
 799       (setq n (math-trunc n)))
 800   (let ((res))
 801     (while (> iters 0)
 802       (setq res
 803             (cond ((and (integerp n) (<= n 5003))
 804                    (list (= (math-next-small-prime n) n)))
 805                   ((not (Math-integerp n))
 806                    (error "Argument must be an integer"))
 807                   ((Math-integer-negp n)
 808                    '(nil))
 809                   ((Math-natnum-lessp n 8000000)
 810                    (setq n (math-fixnum n))
 811                    (let ((i -1) v)
 812                      (while (and (> (% n (setq v (aref math-primes-table
 813                                                        (setq i (1+ i)))))
 814                                     0)
 815                                  (< (* v v) n)))
 816                      (if (= (% n v) 0)
 817                          (list nil v)
 818                        '(t))))
 819                   ((not (equal n (car math-prime-test-cache)))
 820                    (cond ((= (% (nth 1 n) 2) 0) '(nil 2))
 821                          ((= (% (nth 1 n) 5) 0) '(nil 5))
 822                          (t (let ((q n) (sum 0))
 823                               (while (not (eq q 0))
 824                                 (setq sum (%
 825                                            (+
 826                                             sum
 827                                             (calcFunc-mod
 828                                              q 1000000))
 829                                            111111))
 830                                 (setq q
 831                                       (math-quotient
 832                                        q 1000000)))
 833                               (cond ((= (% sum 3) 0) '(nil 3))
 834                                     ((= (% sum 7) 0) '(nil 7))
 835                                     ((= (% sum 11) 0) '(nil 11))
 836                                     ((= (% sum 13) 0) '(nil 13))
 837                                     ((= (% sum 37) 0) '(nil 37))
 838                                     (t
 839                                      (setq math-prime-test-cache-k 1
 840                                            math-prime-test-cache-q
 841                                            (math-div2 n)
 842                                            math-prime-test-cache-nm1
 843                                            (math-add n -1))
 844                                      (while (math-evenp
 845                                              math-prime-test-cache-q)
 846                                        (setq math-prime-test-cache-k
 847                                              (1+ math-prime-test-cache-k)
 848                                              math-prime-test-cache-q
 849                                              (math-div2
 850                                               math-prime-test-cache-q)))
 851                                      (setq iters (1+ iters))
 852                                      (list 'maybe
 853                                            0
 854                                            (math-sub
 855                                             100
 856                                             (math-div
 857                                              '(float 232 0)
 858                                              (math-numdigs n))))))))))
 859                   ((not (eq (car (nth 1 math-prime-test-cache)) 'maybe))
 860                    (nth 1 math-prime-test-cache))
 861                   (t   ; Fermat step
 862                    (let* ((x (math-add (calcFunc-random (math-add n -2)) 2))
 863                           (y (math-pow-mod x math-prime-test-cache-q n))
 864                           (j 0))
 865                      (while (and (not (eq y 1))
 866                                  (not (equal y math-prime-test-cache-nm1))
 867                                  (< (setq j (1+ j)) math-prime-test-cache-k))
 868                        (setq y (math-mod (math-mul y y) n)))
 869                      (if (or (equal y math-prime-test-cache-nm1)
 870                              (and (eq y 1) (eq j 0)))
 871                          (list 'maybe
 872                                (1+ (nth 1 (nth 1 math-prime-test-cache)))
 873                                (math-mul (nth 2 (nth 1 math-prime-test-cache))
 874                                          '(float 25 -2)))
 875                        '(nil unknown))))))
 876       (setq math-prime-test-cache (list n res)
 877             iters (if (eq (car res) 'maybe)
 878                       (1- iters)
 879                     0)))
 880     res))
 881
 882 (defun calcFunc-prime (n &optional iters)
 883   (or (math-num-integerp n) (math-reject-arg n 'integerp))
 884   (or (not iters) (math-num-integerp iters) (math-reject-arg iters 'integerp))
 885   (if (car (math-prime-test (math-trunc n) (math-trunc (or iters 1))))
 886       1
 887     0))
 888
 889 ;;; Theory: summing base-10^6 digits modulo 111111 is "casting out 999999s".
 890 ;;; Initial probability that N is prime is 1/ln(N) = log10(e)/log10(N).
 891 ;;; After culling [2,3,5,7,11,13,37], probability of primality is 5.36 x more.
 892 ;;; Initial reported probability of non-primality is thus 100% - this.
 893 ;;; Each Fermat step multiplies this probability by 25%.
 894 ;;; The Fermat step is algorithm P from Knuth section 4.5.4.
 895
 896
 897 (defun calcFunc-prfac (n)
 898   (setq math-prime-factors-finished t)
 899   (if (Math-messy-integerp n)
 900       (setq n (math-trunc n)))
 901   (if (Math-natnump n)
 902       (if (Math-natnum-lessp 2 n)
 903           (let (factors res p (i 0))
 904             (while (and (not (eq n 1))
 905                         (< i (length math-primes-table)))
 906               (setq p (aref math-primes-table i))
 907               (while (eq (cdr (setq res (cond ((eq n p) (cons 1 0))
 908                                               ((eq n 1) (cons 0 1))
 909                                               ((consp n) (math-idivmod n p))
 910                                               (t (cons (/ n p) (% n p))))))
 911                          0)
 912                 (math-working "factor" p)
 913                 (setq factors (nconc factors (list p))
 914                       n (car res)))
 915               (or (eq n 1)
 916                   (Math-natnum-lessp p (car res))
 917                   (setq factors (nconc factors (list n))
 918                         n 1))
 919               (setq i (1+ i)))
 920             (or (setq math-prime-factors-finished (eq n 1))
 921                 (setq factors (nconc factors (list n))))
 922             (cons 'vec factors))
 923         (list 'vec n))
 924     (if (Math-integerp n)
 925         (if (eq n -1)
 926             (list 'vec n)
 927           (cons 'vec (cons -1 (cdr (calcFunc-prfac (math-neg n))))))
 928       (calc-record-why 'integerp n)
 929       (list 'calcFunc-prfac n))))
 930
 931 (defun calcFunc-totient (n)
 932   (if (Math-messy-integerp n)
 933       (setq n (math-trunc n)))
 934   (if (Math-natnump n)
 935       (if (Math-natnum-lessp n 2)
 936           (if (Math-negp n)
 937               (calcFunc-totient (math-abs n))
 938             n)
 939         (let ((factors (cdr (calcFunc-prfac n)))
 940               p)
 941           (if math-prime-factors-finished
 942               (progn
 943                 (while factors
 944                   (setq p (car factors)
 945                         n (math-mul (math-div n p) (math-add p -1)))
 946                   (while (equal p (car factors))
 947                     (setq factors (cdr factors))))
 948                 n)
 949             (calc-record-why "*Number too big to factor" n)
 950             (list 'calcFunc-totient n))))
 951     (calc-record-why 'natnump n)
 952     (list 'calcFunc-totient n)))
 953
 954 (defun calcFunc-moebius (n)
 955   (if (Math-messy-integerp n)
 956       (setq n (math-trunc n)))
 957   (if (and (Math-natnump n) (not (eq n 0)))
 958       (if (Math-natnum-lessp n 2)
 959           (if (Math-negp n)
 960               (calcFunc-moebius (math-abs n))
 961             1)
 962         (let ((factors (cdr (calcFunc-prfac n)))
 963               (mu 1))
 964           (if math-prime-factors-finished
 965               (progn
 966                 (while factors
 967                   (setq mu (if (equal (car factors) (nth 1 factors))
 968                                0 (math-neg mu))
 969                         factors (cdr factors)))
 970                 mu)
 971             (calc-record-why "Number too big to factor" n)
 972             (list 'calcFunc-moebius n))))
 973     (calc-record-why 'posintp n)
 974     (list 'calcFunc-moebius n)))
 975
 976
 977 (defun calcFunc-nextprime (n &optional iters)
 978   (if (Math-integerp n)
 979       (if (Math-integer-negp n)
 980           2
 981         (if (and (integerp n) (< n 5003))
 982             (math-next-small-prime (1+ n))
 983           (if (math-evenp n)
 984               (setq n (math-add n -1)))
 985           (let (res)
 986             (while (not (car (setq res (math-prime-test
 987                                         (setq n (math-add n 2))
 988                                         (or iters 1))))))
 989             (if (and calc-verbose-nextprime
 990                      (eq (car res) 'maybe))
 991                 (calc-report-prime-test res)))
 992           n))
 993     (if (Math-realp n)
 994         (calcFunc-nextprime (math-trunc n) iters)
 995       (math-reject-arg n 'integerp))))
 996
 997 (defun calcFunc-prevprime (n &optional iters)
 998   (if (Math-integerp n)
 999       (if (Math-lessp n 4)
1000           2
1001         (if (math-evenp n)
1002             (setq n (math-add n 1)))
1003         (let (res)
1004           (while (not (car (setq res (math-prime-test
1005                                       (setq n (math-add n -2))
1006                                       (or iters 1))))))
1007           (if (and calc-verbose-nextprime
1008                    (eq (car res) 'maybe))
1009               (calc-report-prime-test res)))
1010         n)
1011     (if (Math-realp n)
1012         (calcFunc-prevprime (math-ceiling n) iters)
1013       (math-reject-arg n 'integerp))))
1014
1015 (defun math-next-small-prime (n)
1016   (if (and (integerp n) (> n 2))
1017       (let ((lo -1)
1018             (hi (length math-primes-table))
1019             mid)
1020         (while (> (- hi lo) 1)
1021           (if (> n (aref math-primes-table
1022                          (setq mid (ash (+ lo hi) -1))))
1023               (setq lo mid)
1024             (setq hi mid)))
1025         (aref math-primes-table hi))
1026     2))
1027
1028 (provide 'calc-comb)
1029
1030 ;;; calc-comb.el ends here