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 Free Software Foundation, Inc.
   4
   5 ;; Author: David Gillespie <daveg@synaptics.com>
   6 ;; Maintainers: D. Goel <deego@gnufans.org>
   7 ;;              Colin Walters <walters@debian.org>
   8
   9 ;; This file is part of GNU Emacs.
  10
  11 ;; GNU Emacs is distributed in the hope that it will be useful,
  12 ;; but WITHOUT ANY WARRANTY.  No author or distributor
  13 ;; accepts responsibility to anyone for the consequences of using it
  14 ;; or for whether it serves any particular purpose or works at all,
  15 ;; unless he says so in writing.  Refer to the GNU Emacs General Public
  16 ;; License for full details.
  17
  18 ;; Everyone is granted permission to copy, modify and redistribute
  19 ;; GNU Emacs, but only under the conditions described in the
  20 ;; GNU Emacs General Public License.   A copy of this license is
  21 ;; supposed to have been given to you along with GNU Emacs so you
  22 ;; can know your rights and responsibilities.  It should be in a
  23 ;; file named COPYING.  Among other things, the copyright notice
  24 ;; and this notice must be preserved on all copies.
  25
  26 ;;; Commentary:
  27
  28 ;;; Code:
  29
  30 ;; This file is autoloaded from calc-ext.el.
  31 (require 'calc-ext)
  32
  33 (require 'calc-macs)
  34
  35 (defun calc-Need-calc-comb () nil)
  36
  37 (defconst math-primes-table
  38   [2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89
  39      97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181
  40      191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277
  41      281 283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383
  42      389 397 401 409 419 421 431 433 439 443 449 457 461 463 467 479 487
  43      491 499 503 509 521 523 541 547 557 563 569 571 577 587 593 599 601
  44      607 613 617 619 631 641 643 647 653 659 661 673 677 683 691 701 709
  45      719 727 733 739 743 751 757 761 769 773 787 797 809 811 821 823 827
  46      829 839 853 857 859 863 877 881 883 887 907 911 919 929 937 941 947
  47      953 967 971 977 983 991 997 1009 1013 1019 1021 1031 1033 1039 1049
  48      1051 1061 1063 1069 1087 1091 1093 1097 1103 1109 1117 1123 1129 1151
  49      1153 1163 1171 1181 1187 1193 1201 1213 1217 1223 1229 1231 1237 1249
  50      1259 1277 1279 1283 1289 1291 1297 1301 1303 1307 1319 1321 1327 1361
  51      1367 1373 1381 1399 1409 1423 1427 1429 1433 1439 1447 1451 1453 1459
  52      1471 1481 1483 1487 1489 1493 1499 1511 1523 1531 1543 1549 1553 1559
  53      1567 1571 1579 1583 1597 1601 1607 1609 1613 1619 1621 1627 1637 1657
  54      1663 1667 1669 1693 1697 1699 1709 1721 1723 1733 1741 1747 1753 1759
  55      1777 1783 1787 1789 1801 1811 1823 1831 1847 1861 1867 1871 1873 1877
  56      1879 1889 1901 1907 1913 1931 1933 1949 1951 1973 1979 1987 1993 1997
  57      1999 2003 2011 2017 2027 2029 2039 2053 2063 2069 2081 2083 2087 2089
  58      2099 2111 2113 2129 2131 2137 2141 2143 2153 2161 2179 2203 2207 2213
  59      2221 2237 2239 2243 2251 2267 2269 2273 2281 2287 2293 2297 2309 2311
  60      2333 2339 2341 2347 2351 2357 2371 2377 2381 2383 2389 2393 2399 2411
  61      2417 2423 2437 2441 2447 2459 2467 2473 2477 2503 2521 2531 2539 2543
  62      2549 2551 2557 2579 2591 2593 2609 2617 2621 2633 2647 2657 2659 2663
  63      2671 2677 2683 2687 2689 2693 2699 2707 2711 2713 2719 2729 2731 2741
  64      2749 2753 2767 2777 2789 2791 2797 2801 2803 2819 2833 2837 2843 2851
  65      2857 2861 2879 2887 2897 2903 2909 2917 2927 2939 2953 2957 2963 2969
  66      2971 2999 3001 3011 3019 3023 3037 3041 3049 3061 3067 3079 3083 3089
  67      3109 3119 3121 3137 3163 3167 3169 3181 3187 3191 3203 3209 3217 3221
  68      3229 3251 3253 3257 3259 3271 3299 3301 3307 3313 3319 3323 3329 3331
  69      3343 3347 3359 3361 3371 3373 3389 3391 3407 3413 3433 3449 3457 3461
  70      3463 3467 3469 3491 3499 3511 3517 3527 3529 3533 3539 3541 3547 3557
  71      3559 3571 3581 3583 3593 3607 3613 3617 3623 3631 3637 3643 3659 3671
  72      3673 3677 3691 3697 3701 3709 3719 3727 3733 3739 3761 3767 3769 3779
  73      3793 3797 3803 3821 3823 3833 3847 3851 3853 3863 3877 3881 3889 3907
  74      3911 3917 3919 3923 3929 3931 3943 3947 3967 3989 4001 4003 4007 4013
  75      4019 4021 4027 4049 4051 4057 4073 4079 4091 4093 4099 4111 4127 4129
  76      4133 4139 4153 4157 4159 4177 4201 4211 4217 4219 4229 4231 4241 4243
  77      4253 4259 4261 4271 4273 4283 4289 4297 4327 4337 4339 4349 4357 4363
  78      4373 4391 4397 4409 4421 4423 4441 4447 4451 4457 4463 4481 4483 4493
  79      4507 4513 4517 4519 4523 4547 4549 4561 4567 4583 4591 4597 4603 4621
  80      4637 4639 4643 4649 4651 4657 4663 4673 4679 4691 4703 4721 4723 4729
  81      4733 4751 4759 4783 4787 4789 4793 4799 4801 4813 4817 4831 4861 4871
  82      4877 4889 4903 4909 4919 4931 4933 4937 4943 4951 4957 4967 4969 4973
  83      4987 4993 4999 5003])
  84
  85 ;; The variable math-prime-factors-finished is set by calcFunc-prfac to
  86 ;; indicate whether factoring is complete, and used by calcFunc-factors,
  87 ;; calcFunc-totient and calcFunc-moebius.
  88 (defvar math-prime-factors-finished)
  89
  90 ;;; Combinatorics
  91
  92 (defun calc-gcd (arg)
  93   (interactive "P")
  94   (calc-slow-wrapper
  95    (calc-binary-op "gcd" 'calcFunc-gcd arg)))
  96
  97 (defun calc-lcm (arg)
  98   (interactive "P")
  99   (calc-slow-wrapper
 100    (calc-binary-op "lcm" 'calcFunc-lcm arg)))
 101
 102 (defun calc-extended-gcd ()
 103   (interactive)
 104   (calc-slow-wrapper
 105    (calc-enter-result 2 "egcd" (cons 'calcFunc-egcd (calc-top-list-n 2)))))
 106
 107 (defun calc-factorial (arg)
 108   (interactive "P")
 109   (calc-slow-wrapper
 110    (calc-unary-op "fact" 'calcFunc-fact arg)))
 111
 112 (defun calc-gamma (arg)
 113   (interactive "P")
 114   (calc-slow-wrapper
 115    (calc-unary-op "gmma" 'calcFunc-gamma arg)))
 116
 117 (defun calc-double-factorial (arg)
 118   (interactive "P")
 119   (calc-slow-wrapper
 120    (calc-unary-op "dfac" 'calcFunc-dfact arg)))
 121
 122 (defun calc-choose (arg)
 123   (interactive "P")
 124   (calc-slow-wrapper
 125    (if (calc-is-hyperbolic)
 126        (calc-binary-op "perm" 'calcFunc-perm arg)
 127      (calc-binary-op "chos" 'calcFunc-choose arg))))
 128
 129 (defun calc-perm (arg)
 130   (interactive "P")
 131   (calc-hyperbolic-func)
 132   (calc-choose arg))
 133
 134 (defvar calc-last-random-limit '(float 1 0))
 135 (defun calc-random (n)
 136   (interactive "P")
 137   (calc-slow-wrapper
 138    (if n
 139        (calc-enter-result 0 "rand" (list 'calcFunc-random
 140                                          (calc-get-random-limit
 141                                           (prefix-numeric-value n))))
 142      (calc-enter-result 1 "rand" (list 'calcFunc-random
 143                                        (calc-get-random-limit
 144                                         (calc-top-n 1)))))))
 145
 146 (defun calc-get-random-limit (val)
 147   (if (eq val 0)
 148       calc-last-random-limit
 149     (setq calc-last-random-limit val)))
 150
 151 (defun calc-rrandom ()
 152   (interactive)
 153   (calc-slow-wrapper
 154    (setq calc-last-random-limit '(float 1 0))
 155    (calc-enter-result 0 "rand" (list 'calcFunc-random '(float 1 0)))))
 156
 157 (defun calc-random-again (arg)
 158   (interactive "p")
 159   (calc-slow-wrapper
 160    (while (>= (setq arg (1- arg)) 0)
 161      (calc-enter-result 0 "rand" (list 'calcFunc-random
 162                                        calc-last-random-limit)))))
 163
 164 (defun calc-shuffle (n)
 165   (interactive "P")
 166   (calc-slow-wrapper
 167    (if n
 168        (calc-enter-result 1 "shuf" (list 'calcFunc-shuffle
 169                                          (prefix-numeric-value n)
 170                                          (calc-get-random-limit
 171                                           (calc-top-n 1))))
 172      (calc-enter-result 2 "shuf" (list 'calcFunc-shuffle
 173                                        (calc-top-n 1)
 174                                        (calc-get-random-limit
 175                                         (calc-top-n 2)))))))
 176
 177 (defun calc-report-prime-test (res)
 178   (cond ((eq (car res) t)
 179          (calc-record-message "prim" "Prime (guaranteed)"))
 180         ((eq (car res) nil)
 181          (if (cdr res)
 182              (if (eq (nth 1 res) 'unknown)
 183                  (calc-record-message
 184                   "prim" "Non-prime (factors unknown)")
 185                (calc-record-message
 186                 "prim" "Non-prime (%s is a factor)"
 187                 (math-format-number (nth 1 res))))
 188            (calc-record-message "prim" "Non-prime")))
 189         (t
 190          (calc-record-message
 191           "prim" "Probably prime (%d iters; %s%% chance of error)"
 192           (nth 1 res)
 193           (let ((calc-float-format '(fix 2)))
 194             (math-format-number (nth 2 res)))))))
 195
 196 (defun calc-prime-test (iters)
 197   (interactive "p")
 198   (calc-slow-wrapper
 199    (let* ((n (calc-top-n 1))
 200           (res (math-prime-test n iters)))
 201      (calc-report-prime-test res))))
 202
 203 (defvar calc-verbose-nextprime nil)
 204
 205 (defun calc-next-prime (iters)
 206   (interactive "p")
 207   (calc-slow-wrapper
 208    (let ((calc-verbose-nextprime t))
 209      (if (calc-is-inverse)
 210          (calc-enter-result 1 "prvp" (list 'calcFunc-prevprime
 211                                            (calc-top-n 1) (math-abs iters)))
 212        (calc-enter-result 1 "nxtp" (list 'calcFunc-nextprime
 213                                          (calc-top-n 1) (math-abs iters)))))))
 214
 215 (defun calc-prev-prime (iters)
 216   (interactive "p")
 217   (calc-invert-func)
 218   (calc-next-prime iters))
 219
 220 (defun calc-prime-factors (iters)
 221   (interactive "p")
 222   (calc-slow-wrapper
 223    (let ((res (calcFunc-prfac (calc-top-n 1))))
 224      (if (not math-prime-factors-finished)
 225          (calc-record-message "pfac" "Warning:  May not be fully factored"))
 226      (calc-enter-result 1 "pfac" res))))
 227
 228 (defun calc-totient (arg)
 229   (interactive "P")
 230   (calc-slow-wrapper
 231    (calc-unary-op "phi" 'calcFunc-totient arg)))
 232
 233 (defun calc-moebius (arg)
 234   (interactive "P")
 235   (calc-slow-wrapper
 236    (calc-unary-op "mu" 'calcFunc-moebius arg)))
 237
 238
 239 (defun calcFunc-gcd (a b)
 240   (if (Math-messy-integerp a)
 241       (setq a (math-trunc a)))
 242   (if (Math-messy-integerp b)
 243       (setq b (math-trunc b)))
 244   (cond ((and (Math-integerp a) (Math-integerp b))
 245          (math-gcd a b))
 246         ((Math-looks-negp a)
 247          (calcFunc-gcd (math-neg a) b))
 248         ((Math-looks-negp b)
 249          (calcFunc-gcd a (math-neg b)))
 250         ((Math-zerop a) b)
 251         ((Math-zerop b) a)
 252         ((and (Math-ratp a)
 253               (Math-ratp b))
 254          (math-make-frac (math-gcd (if (eq (car-safe a) 'frac) (nth 1 a) a)
 255                                    (if (eq (car-safe b) 'frac) (nth 1 b) b))
 256                          (calcFunc-lcm
 257                           (if (eq (car-safe a) 'frac) (nth 2 a) 1)
 258                           (if (eq (car-safe b) 'frac) (nth 2 b) 1))))
 259         ((not (Math-integerp a))
 260          (calc-record-why 'integerp a)
 261          (list 'calcFunc-gcd a b))
 262         (t
 263          (calc-record-why 'integerp b)
 264          (list 'calcFunc-gcd a b))))
 265
 266 (defun calcFunc-lcm (a b)
 267   (let ((g (calcFunc-gcd a b)))
 268     (if (Math-numberp g)
 269         (math-div (math-mul a b) g)
 270       (list 'calcFunc-lcm a b))))
 271
 272 (defun calcFunc-egcd (a b)   ; Knuth section 4.5.2
 273   (cond
 274    ((not (Math-integerp a))
 275     (if (Math-messy-integerp a)
 276         (calcFunc-egcd (math-trunc a) b)
 277       (calc-record-why 'integerp a)
 278       (list 'calcFunc-egcd a b)))
 279    ((not (Math-integerp b))
 280     (if (Math-messy-integerp b)
 281         (calcFunc-egcd a (math-trunc b))
 282       (calc-record-why 'integerp b)
 283       (list 'calcFunc-egcd a b)))
 284    (t
 285     (let ((u1 1) (u2 0) (u3 a)
 286           (v1 0) (v2 1) (v3 b)
 287           t1 t2 q)
 288       (while (not (eq v3 0))
 289         (setq q (math-idivmod u3 v3)
 290               t1 (math-sub u1 (math-mul v1 (car q)))
 291               t2 (math-sub u2 (math-mul v2 (car q)))
 292               u1 v1  u2 v2  u3 v3
 293               v1 t1  v2 t2  v3 (cdr q)))
 294       (list 'vec u3 u1 u2)))))
 295
 296
 297 ;;; Factorial and related functions.
 298
 299 (defun calcFunc-fact (n)   ; [I I] [F F] [Public]
 300   (let (temp)
 301     (cond ((Math-integer-negp n)
 302            (if calc-infinite-mode
 303                '(var uinf var-uinf)
 304              (math-reject-arg n 'range)))
 305           ((integerp n)
 306            (if (<= n 20)
 307                (aref '[1 1 2 6 24 120 720 5040 40320 362880
 308                          (bigpos 800 628 3) (bigpos 800 916 39)
 309                          (bigpos 600 1 479) (bigpos 800 20 227 6)
 310                          (bigpos 200 291 178 87) (bigpos 0 368 674 307 1)
 311                          (bigpos 0 888 789 922 20) (bigpos 0 96 428 687 355)
 312                          (bigpos 0 728 705 373 402 6)
 313                          (bigpos 0 832 408 100 645 121)
 314                          (bigpos 0 640 176 8 902 432 2)] n)
 315              (math-factorial-iter (1- n) 2 1)))
 316           ((and (math-messy-integerp n)
 317                 (Math-lessp n 100))
 318            (math-inexact-result)
 319            (setq temp (math-trunc n))
 320            (if (>= temp 0)
 321                (if (<= temp 20)
 322                    (math-float (calcFunc-fact temp))
 323                  (math-with-extra-prec 1
 324                    (math-factorial-iter (1- temp) 2 '(float 1 0))))
 325              (math-reject-arg n 'range)))
 326           ((math-numberp n)
 327            (let* ((q (math-quarter-integer n))
 328                   (tn (and q (Math-lessp n 1000) (Math-lessp -1000 n)
 329                            (1+ (math-floor n)))))
 330              (cond ((and tn (= q 2)
 331                          (or calc-symbolic-mode (< (math-abs tn) 20)))
 332                     (let ((q (if (< tn 0)
 333                                  (math-div
 334                                   (math-pow -2 (- tn))
 335                                   (math-double-factorial-iter (* -2 tn) 3 1 2))
 336                                (math-div
 337                                 (math-double-factorial-iter (* 2 tn) 3 1 2)
 338                                 (math-pow 2 tn)))))
 339                       (math-mul q (if calc-symbolic-mode
 340                                       (list 'calcFunc-sqrt '(var pi var-pi))
 341                                     (math-sqrt-pi)))))
 342                    ((and tn (>= tn 0) (< tn 20)
 343                          (memq q '(1 3)))
 344                     (math-inexact-result)
 345                     (math-div
 346                      (math-mul (math-double-factorial-iter (* 4 tn) q 1 4)
 347                                (if (= q 1) (math-gamma-1q) (math-gamma-3q)))
 348                      (math-pow 4 tn)))
 349                    (t
 350                     (math-inexact-result)
 351                     (math-with-extra-prec 3
 352                       (math-gammap1-raw (math-float n)))))))
 353           ((equal n '(var inf var-inf)) n)
 354           (t (calc-record-why 'numberp n)
 355              (list 'calcFunc-fact n)))))
 356
 357 (math-defcache math-gamma-1q nil
 358   (math-with-extra-prec 3
 359     (math-gammap1-raw '(float -75 -2))))
 360
 361 (math-defcache math-gamma-3q nil
 362   (math-with-extra-prec 3
 363     (math-gammap1-raw '(float -25 -2))))
 364
 365 (defun math-factorial-iter (count n f)
 366   (if (= (% n 5) 1)
 367       (math-working (format "factorial(%d)" (1- n)) f))
 368   (if (> count 0)
 369       (math-factorial-iter (1- count) (1+ n) (math-mul n f))
 370     f))
 371
 372 (defun calcFunc-dfact (n)   ; [I I] [F F] [Public]
 373   (cond ((Math-integer-negp n)
 374          (if (math-oddp n)
 375              (if (eq n -1)
 376                  1
 377                (math-div (if (eq (math-mod n 4) 3) 1 -1)
 378                          (calcFunc-dfact (math-sub -2 n))))
 379            (list 'calcFunc-dfact n)))
 380         ((Math-zerop n) 1)
 381         ((integerp n) (math-double-factorial-iter n (+ 2 (% n 2)) 1 2))
 382         ((math-messy-integerp n)
 383          (let ((temp (math-trunc n)))
 384            (math-inexact-result)
 385            (if (natnump temp)
 386                (if (Math-lessp temp 200)
 387                    (math-with-extra-prec 1
 388                      (math-double-factorial-iter temp (+ 2 (% temp 2))
 389                                                  '(float 1 0) 2))
 390                  (let* ((half (math-div2 temp))
 391                         (even (math-mul (math-pow 2 half)
 392                                         (calcFunc-fact (math-float half)))))
 393                    (if (math-evenp temp)
 394                        even
 395                      (math-div (calcFunc-fact n) even))))
 396              (list 'calcFunc-dfact n))))
 397         ((equal n '(var inf var-inf)) n)
 398         (t (calc-record-why 'natnump n)
 399            (list 'calcFunc-dfact n))))
 400
 401 (defun math-double-factorial-iter (max n f step)
 402   (if (< (% n 12) step)
 403       (math-working (format "dfact(%d)" (- n step)) f))
 404   (if (<= n max)
 405       (math-double-factorial-iter max (+ n step) (math-mul n f) step)
 406     f))
 407
 408 (defun calcFunc-perm (n m)   ; [I I I] [F F F] [Public]
 409   (cond ((and (integerp n) (integerp m) (<= m n) (>= m 0))
 410          (math-factorial-iter m (1+ (- n m)) 1))
 411         ((or (not (math-num-integerp n))
 412              (and (math-messy-integerp n) (Math-lessp 100 n))
 413              (not (math-num-integerp m))
 414              (and (math-messy-integerp m) (Math-lessp 100 m)))
 415          (or (math-realp n) (equal n '(var inf var-inf))
 416              (math-reject-arg n 'realp))
 417          (or (math-realp m) (equal m '(var inf var-inf))
 418              (math-reject-arg m 'realp))
 419          (and (math-num-integerp n) (math-negp n) (math-reject-arg n 'range))
 420          (and (math-num-integerp m) (math-negp m) (math-reject-arg m 'range))
 421          (math-div (calcFunc-fact n) (calcFunc-fact (math-sub n m))))
 422         (t
 423          (let ((tn (math-trunc n))
 424                (tm (math-trunc m)))
 425            (math-inexact-result)
 426            (or (integerp tn) (math-reject-arg tn 'fixnump))
 427            (or (integerp tm) (math-reject-arg tm 'fixnump))
 428            (or (and (<= tm tn) (>= tm 0)) (math-reject-arg tm 'range))
 429            (math-with-extra-prec 1
 430              (math-factorial-iter tm (1+ (- tn tm)) '(float 1 0)))))))
 431
 432 (defun calcFunc-choose (n m)   ; [I I I] [F F F] [Public]
 433   (cond ((and (integerp n) (integerp m) (<= m n) (>= m 0))
 434          (if (> m (/ n 2))
 435              (math-choose-iter (- n m) n 1 1)
 436            (math-choose-iter m n 1 1)))
 437         ((not (math-realp n))
 438          (math-reject-arg n 'realp))
 439         ((not (math-realp m))
 440          (math-reject-arg m 'realp))
 441         ((not (math-num-integerp m))
 442          (if (and (math-num-integerp n) (math-negp n))
 443              (list 'calcFunc-choose n m)
 444            (math-div (calcFunc-fact (math-float n))
 445                      (math-mul (calcFunc-fact m)
 446                                (calcFunc-fact (math-sub n m))))))
 447         ((math-negp m) 0)
 448         ((math-negp n)
 449          (let ((val (calcFunc-choose (math-add (math-add n m) -1) m)))
 450            (if (math-evenp (math-trunc m))
 451                val
 452              (math-neg val))))
 453         ((and (math-num-integerp n)
 454               (Math-lessp n m))
 455          0)
 456         (t
 457          (math-inexact-result)
 458          (let ((tm (math-trunc m)))
 459            (or (integerp tm) (math-reject-arg tm 'fixnump))
 460            (if (> tm 100)
 461                (math-div (calcFunc-fact (math-float n))
 462                          (math-mul (calcFunc-fact (math-float m))
 463                                    (calcFunc-fact (math-float
 464                                                    (math-sub n m)))))
 465              (math-with-extra-prec 1
 466                (math-choose-float-iter tm n 1 1)))))))
 467
 468 (defun math-choose-iter (m n i c)
 469   (if (and (= (% i 5) 1) (> i 5))
 470       (math-working (format "choose(%d)" (1- i)) c))
 471   (if (<= i m)
 472       (math-choose-iter m (1- n) (1+ i)
 473                         (math-quotient (math-mul c n) i))
 474     c))
 475
 476 (defun math-choose-float-iter (count n i c)
 477   (if (= (% i 5) 1)
 478       (math-working (format "choose(%d)" (1- i)) c))
 479   (if (> count 0)
 480       (math-choose-float-iter (1- count) (math-sub n 1) (1+ i)
 481                               (math-div (math-mul c n) i))
 482     c))
 483
 484
 485 ;;; Stirling numbers.
 486
 487 (defun calcFunc-stir1 (n m)
 488   (math-stirling-number n m 1))
 489
 490 (defun calcFunc-stir2 (n m)
 491   (math-stirling-number n m 0))
 492
 493 (defvar math-stirling-cache (vector [[1]] [[1]]))
 494
 495 ;; The variable math-stirling-local-cache is local to
 496 ;; math-stirling-number, but is used by math-stirling-1
 497 ;; and math-stirling-2, which are called by math-stirling-number.
 498 (defvar math-stirling-local-cache)
 499
 500 (defun math-stirling-number (n m k)
 501   (or (math-num-natnump n) (math-reject-arg n 'natnump))
 502   (or (math-num-natnump m) (math-reject-arg m 'natnump))
 503   (if (consp n) (setq n (math-trunc n)))
 504   (or (integerp n) (math-reject-arg n 'fixnump))
 505   (if (consp m) (setq m (math-trunc m)))
 506   (or (integerp m) (math-reject-arg m 'fixnump))
 507   (if (< n m)
 508       0
 509     (let ((math-stirling-local-cache (aref math-stirling-cache k)))
 510       (while (<= (length math-stirling-local-cache) n)
 511         (let ((i (1- (length math-stirling-local-cache)))
 512               row)
 513           (setq math-stirling-local-cache
 514                 (vconcat math-stirling-local-cache
 515                          (make-vector (length math-stirling-local-cache) nil)))
 516           (aset math-stirling-cache k math-stirling-local-cache)
 517           (while (< (setq i (1+ i)) (length math-stirling-local-cache))
 518             (aset math-stirling-local-cache i (setq row (make-vector (1+ i) nil)))
 519             (aset row 0 0)
 520             (aset row i 1))))
 521       (if (= k 1)
 522           (math-stirling-1 n m)
 523         (math-stirling-2 n m)))))
 524
 525 (defun math-stirling-1 (n m)
 526   (or (aref (aref math-stirling-local-cache n) m)
 527       (aset (aref math-stirling-local-cache n) m
 528             (math-add (math-stirling-1 (1- n) (1- m))
 529                       (math-mul (- 1 n) (math-stirling-1 (1- n) m))))))
 530
 531 (defun math-stirling-2 (n m)
 532   (or (aref (aref math-stirling-local-cache n) m)
 533       (aset (aref math-stirling-local-cache n) m
 534             (math-add (math-stirling-2 (1- n) (1- m))
 535                       (math-mul m (math-stirling-2 (1- n) m))))))
 536
 537 (defvar math-random-table nil)
 538 (defvar math-last-RandSeed nil)
 539 (defvar math-random-ptr1 nil)
 540 (defvar math-random-ptr2 nil)
 541 (defvar math-random-shift nil)
 542
 543 ;;; Produce a random 10-bit integer, with (random) if no seed provided,
 544 ;;; or else with Numerical Recipes algorithm ran3 / Knuth 3.2.2-A.
 545
 546 (defvar var-RandSeed nil)
 547 (defvar math-random-cache nil)
 548 (defvar math-gaussian-cache nil)
 549
 550 (defun math-init-random-base ()
 551   (if var-RandSeed
 552       (if (eq (car-safe var-RandSeed) 'vec)
 553           nil
 554         (if (Math-integerp var-RandSeed)
 555             (let* ((seed (math-sub 161803 var-RandSeed))
 556                    (mj (1+ (math-mod seed '(bigpos 0 0 1))))
 557                    (mk (1+ (math-mod (math-quotient seed '(bigpos 0 0 1))
 558                                      '(bigpos 0 0 1))))
 559                    (i 0))
 560               (setq math-random-table (cons 'vec (make-list 55 mj)))
 561               (while (<= (setq i (1+ i)) 54)
 562                 (let* ((ii (% (* i 21) 55))
 563                        (p (nthcdr ii math-random-table)))
 564                   (setcar p mk)
 565                   (setq mk (- mj mk)
 566                         mj (car p)))))
 567           (math-reject-arg var-RandSeed "*RandSeed must be an integer"))
 568         (setq var-RandSeed (list 'vec var-RandSeed)
 569               math-random-ptr1 math-random-table
 570               math-random-cache nil
 571               math-random-ptr2 (nthcdr 31 math-random-table))
 572         (let ((i 200))
 573           (while (> (setq i (1- i)) 0)
 574             (math-random-base))))
 575     (random t)
 576     (setq var-RandSeed nil
 577           math-random-cache nil
 578           math-random-shift -4)  ; assume RAND_MAX >= 16383
 579     ;; This exercises the random number generator and also helps
 580     ;; deduce a better value for RAND_MAX.
 581     (let ((i 0))
 582       (while (< (setq i (1+ i)) 30)
 583         (if (> (lsh (math-abs (random)) math-random-shift) 4095)
 584             (setq math-random-shift (1- math-random-shift))))))
 585   (setq math-last-RandSeed var-RandSeed
 586         math-gaussian-cache nil))
 587
 588 (defun math-random-base ()
 589   (if var-RandSeed
 590       (progn
 591         (setq math-random-ptr1 (or (cdr math-random-ptr1)
 592                                    (cdr math-random-table))
 593               math-random-ptr2 (or (cdr math-random-ptr2)
 594                                    (cdr math-random-table)))
 595         (logand (lsh (setcar math-random-ptr1
 596                              (logand (- (car math-random-ptr1)
 597                                         (car math-random-ptr2)) 524287))
 598                      -6) 1023))
 599     (logand (lsh (random) math-random-shift) 1023)))
 600
 601
 602 ;;; Produce a random digit in the range 0..999.
 603 ;;; Avoid various pitfalls that may lurk in the built-in (random) function!
 604 ;;; Shuffling algorithm from Numerical Recipes, section 7.1.
 605 (defun math-random-digit ()
 606   (let (i math-random-last)
 607     (or (eq var-RandSeed math-last-RandSeed)
 608         (math-init-random-base))
 609     (or math-random-cache
 610         (progn
 611           (setq math-random-last (math-random-base)
 612                 math-random-cache (make-vector 13 nil)
 613                 i -1)
 614           (while (< (setq i (1+ i)) 13)
 615             (aset math-random-cache i (math-random-base)))))
 616     (while (progn
 617              (setq i (/ math-random-last 79)   ; 0 <= i < 13
 618                    math-random-last (aref math-random-cache i))
 619              (aset math-random-cache i (math-random-base))
 620              (>= math-random-last 1000)))
 621     math-random-last))
 622
 623 ;;; Produce an N-digit random integer.
 624 (defun math-random-digits (n)
 625   (cond ((<= n 6)
 626          (math-scale-right (+ (* (math-random-digit) 1000) (math-random-digit))
 627                            (- 6 n)))
 628         (t (let* ((slop (% (- 900003 n) 3))
 629                   (i (/ (+ n slop) 3))
 630                   (digs nil))
 631              (while (> i 0)
 632                (setq digs (cons (math-random-digit) digs)
 633                      i (1- i)))
 634              (math-normalize (math-scale-right (cons 'bigpos digs)
 635                                                slop))))))
 636
 637 ;;; Produce a uniformly-distributed random float 0 <= N < 1.
 638 (defun math-random-float ()
 639   (math-make-float (math-random-digits calc-internal-prec)
 640                    (- calc-internal-prec)))
 641
 642 ;;; Produce a Gaussian-distributed random float with mean=0, sigma=1.
 643 (defun math-gaussian-float ()
 644   (math-with-extra-prec 2
 645     (if (and math-gaussian-cache
 646              (= (car math-gaussian-cache) calc-internal-prec))
 647         (prog1
 648             (cdr math-gaussian-cache)
 649           (setq math-gaussian-cache nil))
 650       (let* ((v1 (math-add (math-mul (math-random-float) 2) -1))
 651              (v2 (math-add (math-mul (math-random-float) 2) -1))
 652              (r (math-add (math-sqr v1) (math-sqr v2))))
 653         (while (or (not (Math-lessp r 1)) (math-zerop r))
 654           (setq v1 (math-add (math-mul (math-random-float) 2) -1)
 655                 v2 (math-add (math-mul (math-random-float) 2) -1)
 656                 r (math-add (math-sqr v1) (math-sqr v2))))
 657         (let ((fac (math-sqrt (math-mul (math-div (calcFunc-ln r) r) -2))))
 658           (setq math-gaussian-cache (cons calc-internal-prec
 659                                           (math-mul v1 fac)))
 660           (math-mul v2 fac))))))
 661
 662 ;;; Produce a random integer or real 0 <= N < MAX.
 663 (defun calcFunc-random (max)
 664   (cond ((Math-zerop max)
 665          (math-gaussian-float))
 666         ((Math-integerp max)
 667          (let* ((digs (math-numdigs max))
 668                 (r (math-random-digits (+ digs 3))))
 669            (math-mod r max)))
 670         ((Math-realp max)
 671          (math-mul (math-random-float) max))
 672         ((and (eq (car max) 'intv) (math-constp max)
 673               (Math-lessp (nth 2 max) (nth 3 max)))
 674          (if (math-floatp max)
 675              (let ((val (math-add (math-mul (math-random-float)
 676                                             (math-sub (nth 3 max) (nth 2 max)))
 677                                   (nth 2 max))))
 678                (if (or (and (memq (nth 1 max) '(0 1))      ; almost not worth
 679                             (Math-equal val (nth 2 max)))  ;   checking!
 680                        (and (memq (nth 1 max) '(0 2))
 681                             (Math-equal val (nth 3 max))))
 682                    (calcFunc-random max)
 683                  val))
 684            (let ((lo (if (memq (nth 1 max) '(0 1))
 685                          (math-add (nth 2 max) 1) (nth 2 max)))
 686                  (hi (if (memq (nth 1 max) '(1 3))
 687                          (math-add (nth 3 max) 1) (nth 3 max))))
 688              (if (Math-lessp lo hi)
 689                  (math-add (calcFunc-random (math-sub hi lo)) lo)
 690                (math-reject-arg max "*Empty interval")))))
 691         ((eq (car max) 'vec)
 692          (if (cdr max)
 693              (nth (1+ (calcFunc-random (1- (length max)))) max)
 694            (math-reject-arg max "*Empty list")))
 695         ((and (eq (car max) 'sdev) (math-constp max) (Math-realp (nth 1 max)))
 696          (math-add (math-mul (math-gaussian-float) (nth 2 max)) (nth 1 max)))
 697         (t (math-reject-arg max 'realp))))
 698
 699 ;;; Choose N objects at random from the set MAX without duplicates.
 700 (defun calcFunc-shuffle (n &optional max)
 701   (or max (setq max n n -1))
 702   (or (and (Math-num-integerp n)
 703            (or (natnump (setq n (math-trunc n))) (eq n -1)))
 704       (math-reject-arg n 'integerp))
 705   (cond ((or (math-zerop max)
 706              (math-floatp max)
 707              (eq (car-safe max) 'sdev))
 708          (if (< n 0)
 709              (math-reject-arg n 'natnump)
 710            (math-simple-shuffle n max)))
 711         ((and (<= n 1) (>= n 0))
 712          (math-simple-shuffle n max))
 713         ((and (eq (car-safe max) 'intv) (math-constp max))
 714          (let ((num (math-add (math-sub (nth 3 max) (nth 2 max))
 715                               (cdr (assq (nth 1 max)
 716                                          '((0 . -1) (1 . 0)
 717                                            (2 . 0) (3 . 1))))))
 718                (min (math-add (nth 2 max) (if (memq (nth 1 max) '(0 1))
 719                                               1 0))))
 720            (if (< n 0) (setq n num))
 721            (or (math-posp num) (math-reject-arg max 'range))
 722            (and (Math-lessp num n) (math-reject-arg n 'range))
 723            (if (Math-lessp n (math-quotient num 3))
 724                (math-simple-shuffle n max)
 725              (if (> (* n 4) (* num 3))
 726                  (math-add (math-sub min 1)
 727                            (math-shuffle-list n num (calcFunc-index num)))
 728                (let ((tot 0)
 729                      (m 0)
 730                      (vec nil))
 731                  (while (< m n)
 732                    (if (< (calcFunc-random (- num tot)) (- n m))
 733                        (setq vec (cons (math-add min tot) vec)
 734                              m (1+ m)))
 735                    (setq tot (1+ tot)))
 736                  (math-shuffle-list n n (cons 'vec vec)))))))
 737         ((eq (car-safe max) 'vec)
 738          (let ((size (1- (length max))))
 739            (if (< n 0) (setq n size))
 740            (if (and (> n (/ size 2)) (<= n size))
 741                (math-shuffle-list n size (copy-sequence max))
 742              (let* ((vals (calcFunc-shuffle
 743                            n (list 'intv 3 1 (1- (length max)))))
 744                     (p vals))
 745                (while (setq p (cdr p))
 746                  (setcar p (nth (car p) max)))
 747                vals))))
 748         ((math-integerp max)
 749          (if (math-posp max)
 750              (calcFunc-shuffle n (list 'intv 2 0 max))
 751            (calcFunc-shuffle n (list 'intv 1 max 0))))
 752         (t (math-reject-arg max 'realp))))
 753
 754 (defun math-simple-shuffle (n max)
 755   (let ((vec nil)
 756         val)
 757     (while (>= (setq n (1- n)) 0)
 758       (while (math-member (setq val (calcFunc-random max)) vec))
 759       (setq vec (cons val vec)))
 760     (cons 'vec vec)))
 761
 762 (defun math-shuffle-list (n size vec)
 763   (let ((j size)
 764         k temp
 765         (p vec))
 766     (while (cdr (setq p (cdr p)))
 767       (setq k (calcFunc-random j)
 768             j (1- j)
 769             temp (nth k p))
 770       (setcar (nthcdr k p) (car p))
 771       (setcar p temp))
 772     (cons 'vec (nthcdr (- size n -1) vec))))
 773
 774 (defun math-member (x list)
 775   (while (and list (not (equal x (car list))))
 776     (setq list (cdr list)))
 777   list)
 778
 779
 780 ;;; Check if the integer N is prime.  [X I]
 781 ;;; Return (nil) if non-prime,
 782 ;;;        (nil N) if non-prime with known factor N,
 783 ;;;        (nil unknown) if non-prime with no known factors,
 784 ;;;        (t) if prime,
 785 ;;;        (maybe N P) if probably prime (after N iters with probability P%)
 786 (defvar math-prime-test-cache '(-1))
 787
 788 (defvar math-prime-test-cache-k)
 789 (defvar math-prime-test-cache-q)
 790 (defvar math-prime-test-cache-nm1)
 791
 792 (defun math-prime-test (n iters)
 793   (if (and (Math-vectorp n) (cdr n))
 794       (setq n (nth (1- (length n)) n)))
 795   (if (Math-messy-integerp n)
 796       (setq n (math-trunc n)))
 797   (let ((res))
 798     (while (> iters 0)
 799       (setq res
 800             (cond ((and (integerp n) (<= n 5003))
 801                    (list (= (math-next-small-prime n) n)))
 802                   ((not (Math-integerp n))
 803                    (error "Argument must be an integer"))
 804                   ((Math-integer-negp n)
 805                    '(nil))
 806                   ((Math-natnum-lessp n '(bigpos 0 0 8))
 807                    (setq n (math-fixnum n))
 808                    (let ((i -1) v)
 809                      (while (and (> (% n (setq v (aref math-primes-table
 810                                                        (setq i (1+ i)))))
 811                                     0)
 812                                  (< (* v v) n)))
 813                      (if (= (% n v) 0)
 814                          (list nil v)
 815                        '(t))))
 816                   ((not (equal n (car math-prime-test-cache)))
 817                    (cond ((= (% (nth 1 n) 2) 0) '(nil 2))
 818                          ((= (% (nth 1 n) 5) 0) '(nil 5))
 819                          (t (let ((dig (cdr n)) (sum 0))
 820                               (while dig
 821                                 (if (cdr dig)
 822                                     (setq sum (% (+ (+ sum (car dig))
 823                                                     (* (nth 1 dig) 1000))
 824                                                  111111)
 825                                           dig (cdr (cdr dig)))
 826                                   (setq sum (% (+ sum (car dig)) 111111)
 827                                         dig nil)))
 828                               (cond ((= (% sum 3) 0) '(nil 3))
 829                                     ((= (% sum 7) 0) '(nil 7))
 830                                     ((= (% sum 11) 0) '(nil 11))
 831                                     ((= (% sum 13) 0) '(nil 13))
 832                                     ((= (% sum 37) 0) '(nil 37))
 833                                     (t
 834                                      (setq math-prime-test-cache-k 1
 835                                            math-prime-test-cache-q
 836                                            (math-div2 n)
 837                                            math-prime-test-cache-nm1
 838                                            (math-add n -1))
 839                                      (while (math-evenp
 840                                              math-prime-test-cache-q)
 841                                        (setq math-prime-test-cache-k
 842                                              (1+ math-prime-test-cache-k)
 843                                              math-prime-test-cache-q
 844                                              (math-div2
 845                                               math-prime-test-cache-q)))
 846                                      (setq iters (1+ iters))
 847                                      (list 'maybe
 848                                            0
 849                                            (math-sub
 850                                             100
 851                                             (math-div
 852                                              '(float 232 0)
 853                                              (math-numdigs n))))))))))
 854                   ((not (eq (car (nth 1 math-prime-test-cache)) 'maybe))
 855                    (nth 1 math-prime-test-cache))
 856                   (t   ; Fermat step
 857                    (let* ((x (math-add (calcFunc-random (math-add n -2)) 2))
 858                           (y (math-pow-mod x math-prime-test-cache-q n))
 859                           (j 0))
 860                      (while (and (not (eq y 1))
 861                                  (not (equal y math-prime-test-cache-nm1))
 862                                  (< (setq j (1+ j)) math-prime-test-cache-k))
 863                        (setq y (math-mod (math-mul y y) n)))
 864                      (if (or (equal y math-prime-test-cache-nm1)
 865                              (and (eq y 1) (eq j 0)))
 866                          (list 'maybe
 867                                (1+ (nth 1 (nth 1 math-prime-test-cache)))
 868                                (math-mul (nth 2 (nth 1 math-prime-test-cache))
 869                                          '(float 25 -2)))
 870                        '(nil unknown))))))
 871       (setq math-prime-test-cache (list n res)
 872             iters (if (eq (car res) 'maybe)
 873                       (1- iters)
 874                     0)))
 875     res))
 876
 877 (defun calcFunc-prime (n &optional iters)
 878   (or (math-num-integerp n) (math-reject-arg n 'integerp))
 879   (or (not iters) (math-num-integerp iters) (math-reject-arg iters 'integerp))
 880   (if (car (math-prime-test (math-trunc n) (math-trunc (or iters 1))))
 881       1
 882     0))
 883
 884 ;;; Theory: summing base-10^6 digits modulo 111111 is "casting out 999999s".
 885 ;;; Initial probability that N is prime is 1/ln(N) = log10(e)/log10(N).
 886 ;;; After culling [2,3,5,7,11,13,37], probability of primality is 5.36 x more.
 887 ;;; Initial reported probability of non-primality is thus 100% - this.
 888 ;;; Each Fermat step multiplies this probability by 25%.
 889 ;;; The Fermat step is algorithm P from Knuth section 4.5.4.
 890
 891
 892 (defun calcFunc-prfac (n)
 893   (setq math-prime-factors-finished t)
 894   (if (Math-messy-integerp n)
 895       (setq n (math-trunc n)))
 896   (if (Math-natnump n)
 897       (if (Math-natnum-lessp 2 n)
 898           (let (factors res p (i 0))
 899             (while (and (not (eq n 1))
 900                         (< i (length math-primes-table)))
 901               (setq p (aref math-primes-table i))
 902               (while (eq (cdr (setq res (cond ((eq n p) (cons 1 0))
 903                                               ((eq n 1) (cons 0 1))
 904                                               ((consp n) (math-idivmod n p))
 905                                               (t (cons (/ n p) (% n p))))))
 906                          0)
 907                 (math-working "factor" p)
 908                 (setq factors (nconc factors (list p))
 909                       n (car res)))
 910               (or (eq n 1)
 911                   (Math-natnum-lessp p (car res))
 912                   (setq factors (nconc factors (list n))
 913                         n 1))
 914               (setq i (1+ i)))
 915             (or (setq math-prime-factors-finished (eq n 1))
 916                 (setq factors (nconc factors (list n))))
 917             (cons 'vec factors))
 918         (list 'vec n))
 919     (if (Math-integerp n)
 920         (if (eq n -1)
 921             (list 'vec n)
 922           (cons 'vec (cons -1 (cdr (calcFunc-prfac (math-neg n))))))
 923       (calc-record-why 'integerp n)
 924       (list 'calcFunc-prfac n))))
 925
 926 (defun calcFunc-totient (n)
 927   (if (Math-messy-integerp n)
 928       (setq n (math-trunc n)))
 929   (if (Math-natnump n)
 930       (if (Math-natnum-lessp n 2)
 931           (if (Math-negp n)
 932               (calcFunc-totient (math-abs n))
 933             n)
 934         (let ((factors (cdr (calcFunc-prfac n)))
 935               p)
 936           (if math-prime-factors-finished
 937               (progn
 938                 (while factors
 939                   (setq p (car factors)
 940                         n (math-mul (math-div n p) (math-add p -1)))
 941                   (while (equal p (car factors))
 942                     (setq factors (cdr factors))))
 943                 n)
 944             (calc-record-why "*Number too big to factor" n)
 945             (list 'calcFunc-totient n))))
 946     (calc-record-why 'natnump n)
 947     (list 'calcFunc-totient n)))
 948
 949 (defun calcFunc-moebius (n)
 950   (if (Math-messy-integerp n)
 951       (setq n (math-trunc n)))
 952   (if (and (Math-natnump n) (not (eq n 0)))
 953       (if (Math-natnum-lessp n 2)
 954           (if (Math-negp n)
 955               (calcFunc-moebius (math-abs n))
 956             1)
 957         (let ((factors (cdr (calcFunc-prfac n)))
 958               (mu 1))
 959           (if math-prime-factors-finished
 960               (progn
 961                 (while factors
 962                   (setq mu (if (equal (car factors) (nth 1 factors))
 963                                0 (math-neg mu))
 964                         factors (cdr factors)))
 965                 mu)
 966             (calc-record-why "Number too big to factor" n)
 967             (list 'calcFunc-moebius n))))
 968     (calc-record-why 'posintp n)
 969     (list 'calcFunc-moebius n)))
 970
 971
 972 (defun calcFunc-nextprime (n &optional iters)
 973   (if (Math-integerp n)
 974       (if (Math-integer-negp n)
 975           2
 976         (if (and (integerp n) (< n 5003))
 977             (math-next-small-prime (1+ n))
 978           (if (math-evenp n)
 979               (setq n (math-add n -1)))
 980           (let (res)
 981             (while (not (car (setq res (math-prime-test
 982                                         (setq n (math-add n 2))
 983                                         (or iters 1))))))
 984             (if (and calc-verbose-nextprime
 985                      (eq (car res) 'maybe))
 986                 (calc-report-prime-test res)))
 987           n))
 988     (if (Math-realp n)
 989         (calcFunc-nextprime (math-trunc n) iters)
 990       (math-reject-arg n 'integerp))))
 991
 992 (defun calcFunc-prevprime (n &optional iters)
 993   (if (Math-integerp n)
 994       (if (Math-lessp n 4)
 995           2
 996         (if (math-evenp n)
 997             (setq n (math-add n 1)))
 998         (let (res)
 999           (while (not (car (setq res (math-prime-test
1000                                       (setq n (math-add n -2))
1001                                       (or iters 1))))))
1002           (if (and calc-verbose-nextprime
1003                    (eq (car res) 'maybe))
1004               (calc-report-prime-test res)))
1005         n)
1006     (if (Math-realp n)
1007         (calcFunc-prevprime (math-ceiling n) iters)
1008       (math-reject-arg n 'integerp))))
1009
1010 (defun math-next-small-prime (n)
1011   (if (and (integerp n) (> n 2))
1012       (let ((lo -1)
1013             (hi (length math-primes-table))
1014             mid)
1015         (while (> (- hi lo) 1)
1016           (if (> n (aref math-primes-table
1017                          (setq mid (ash (+ lo hi) -1))))
1018               (setq lo mid)
1019             (setq hi mid)))
1020         (aref math-primes-table hi))
1021     2))
1022
1023
1024
1025
1026 ;;; arch-tag: 1d75ee9b-0815-42bd-a321-bb3dc001cc02
1027 ;;; calc-comb.el ends here