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3132f345
CW		 1;;; calc-comb.el --- combinatoric functions for Calc
2
58ba2f8f 3;; Copyright (C) 1990, 1991, 1992, 1993, 2001, 2002, 2003, 2004,
f0fa15c5 4;; 2005, 2006, 2007 Free Software Foundation, Inc.
3132f345
CW		 5
6;; Author: David Gillespie <daveg@synaptics.com>
e8fff8ed 7;; Maintainer: Jay Belanger <jay.p.belanger@gmail.com>
136211a9
EZ		 8
9;; This file is part of GNU Emacs.
10
7c671b23
GM		 11;; GNU Emacs is free software; you can redistribute it and/or modify
12;; it under the terms of the GNU General Public License as published by
075969b4 13;; the Free Software Foundation; either version 3, or (at your option)
7c671b23
GM		 14;; any later version.
15
136211a9 16;; GNU Emacs is distributed in the hope that it will be useful,
7c671b23
GM		 17;; but WITHOUT ANY WARRANTY; without even the implied warranty of
18;; MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
19;; GNU General Public License for more details.
20
21;; You should have received a copy of the GNU General Public License
22;; along with GNU Emacs; see the file COPYING. If not, write to the
23;; Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor,
24;; Boston, MA 02110-1301, USA.
136211a9 25
3132f345 26;;; Commentary:
136211a9 27
3132f345 28;;; Code:
136211a9
EZ		 29
30;; This file is autoloaded from calc-ext.el.
136211a9 31
43f34ccc 32(require 'calc-ext)
136211a9
EZ		 33(require 'calc-macs)
34
3132f345
CW		 35(defconst math-primes-table
36 [2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89
37 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181
38 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277
39 281 283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383
40 389 397 401 409 419 421 431 433 439 443 449 457 461 463 467 479 487
41 491 499 503 509 521 523 541 547 557 563 569 571 577 587 593 599 601
42 607 613 617 619 631 641 643 647 653 659 661 673 677 683 691 701 709
43 719 727 733 739 743 751 757 761 769 773 787 797 809 811 821 823 827
44 829 839 853 857 859 863 877 881 883 887 907 911 919 929 937 941 947
45 953 967 971 977 983 991 997 1009 1013 1019 1021 1031 1033 1039 1049
46 1051 1061 1063 1069 1087 1091 1093 1097 1103 1109 1117 1123 1129 1151
47 1153 1163 1171 1181 1187 1193 1201 1213 1217 1223 1229 1231 1237 1249
48 1259 1277 1279 1283 1289 1291 1297 1301 1303 1307 1319 1321 1327 1361
49 1367 1373 1381 1399 1409 1423 1427 1429 1433 1439 1447 1451 1453 1459
50 1471 1481 1483 1487 1489 1493 1499 1511 1523 1531 1543 1549 1553 1559
51 1567 1571 1579 1583 1597 1601 1607 1609 1613 1619 1621 1627 1637 1657
52 1663 1667 1669 1693 1697 1699 1709 1721 1723 1733 1741 1747 1753 1759
53 1777 1783 1787 1789 1801 1811 1823 1831 1847 1861 1867 1871 1873 1877
54 1879 1889 1901 1907 1913 1931 1933 1949 1951 1973 1979 1987 1993 1997
55 1999 2003 2011 2017 2027 2029 2039 2053 2063 2069 2081 2083 2087 2089
56 2099 2111 2113 2129 2131 2137 2141 2143 2153 2161 2179 2203 2207 2213
57 2221 2237 2239 2243 2251 2267 2269 2273 2281 2287 2293 2297 2309 2311
58 2333 2339 2341 2347 2351 2357 2371 2377 2381 2383 2389 2393 2399 2411
59 2417 2423 2437 2441 2447 2459 2467 2473 2477 2503 2521 2531 2539 2543
60 2549 2551 2557 2579 2591 2593 2609 2617 2621 2633 2647 2657 2659 2663
61 2671 2677 2683 2687 2689 2693 2699 2707 2711 2713 2719 2729 2731 2741
62 2749 2753 2767 2777 2789 2791 2797 2801 2803 2819 2833 2837 2843 2851
63 2857 2861 2879 2887 2897 2903 2909 2917 2927 2939 2953 2957 2963 2969
64 2971 2999 3001 3011 3019 3023 3037 3041 3049 3061 3067 3079 3083 3089
65 3109 3119 3121 3137 3163 3167 3169 3181 3187 3191 3203 3209 3217 3221
66 3229 3251 3253 3257 3259 3271 3299 3301 3307 3313 3319 3323 3329 3331
67 3343 3347 3359 3361 3371 3373 3389 3391 3407 3413 3433 3449 3457 3461
68 3463 3467 3469 3491 3499 3511 3517 3527 3529 3533 3539 3541 3547 3557
69 3559 3571 3581 3583 3593 3607 3613 3617 3623 3631 3637 3643 3659 3671
70 3673 3677 3691 3697 3701 3709 3719 3727 3733 3739 3761 3767 3769 3779
71 3793 3797 3803 3821 3823 3833 3847 3851 3853 3863 3877 3881 3889 3907
72 3911 3917 3919 3923 3929 3931 3943 3947 3967 3989 4001 4003 4007 4013
73 4019 4021 4027 4049 4051 4057 4073 4079 4091 4093 4099 4111 4127 4129
74 4133 4139 4153 4157 4159 4177 4201 4211 4217 4219 4229 4231 4241 4243
75 4253 4259 4261 4271 4273 4283 4289 4297 4327 4337 4339 4349 4357 4363
76 4373 4391 4397 4409 4421 4423 4441 4447 4451 4457 4463 4481 4483 4493
77 4507 4513 4517 4519 4523 4547 4549 4561 4567 4583 4591 4597 4603 4621
78 4637 4639 4643 4649 4651 4657 4663 4673 4679 4691 4703 4721 4723 4729
79 4733 4751 4759 4783 4787 4789 4793 4799 4801 4813 4817 4831 4861 4871
80 4877 4889 4903 4909 4919 4931 4933 4937 4943 4951 4957 4967 4969 4973
81 4987 4993 4999 5003])
136211a9 82
8d7498c1
JB		 83;; The variable math-prime-factors-finished is set by calcFunc-prfac to
84;; indicate whether factoring is complete, and used by calcFunc-factors,
85;; calcFunc-totient and calcFunc-moebius.
86(defvar math-prime-factors-finished)
87
136211a9
EZ		 88;;; Combinatorics
89
90(defun calc-gcd (arg)
91 (interactive "P")
92 (calc-slow-wrapper
bf77c646 93 (calc-binary-op "gcd" 'calcFunc-gcd arg)))
136211a9
EZ		 94
95(defun calc-lcm (arg)
96 (interactive "P")
97 (calc-slow-wrapper
bf77c646 98 (calc-binary-op "lcm" 'calcFunc-lcm arg)))
136211a9
EZ		 99
100(defun calc-extended-gcd ()
101 (interactive)
102 (calc-slow-wrapper
bf77c646 103 (calc-enter-result 2 "egcd" (cons 'calcFunc-egcd (calc-top-list-n 2)))))
136211a9
EZ		 104
105(defun calc-factorial (arg)
106 (interactive "P")
107 (calc-slow-wrapper
bf77c646 108 (calc-unary-op "fact" 'calcFunc-fact arg)))
136211a9
EZ		 109
110(defun calc-gamma (arg)
111 (interactive "P")
112 (calc-slow-wrapper
bf77c646 113 (calc-unary-op "gmma" 'calcFunc-gamma arg)))
136211a9
EZ		 114
115(defun calc-double-factorial (arg)
116 (interactive "P")
117 (calc-slow-wrapper
bf77c646 118 (calc-unary-op "dfac" 'calcFunc-dfact arg)))
136211a9
EZ		 119
120(defun calc-choose (arg)
121 (interactive "P")
122 (calc-slow-wrapper
123 (if (calc-is-hyperbolic)
124 (calc-binary-op "perm" 'calcFunc-perm arg)
bf77c646 125 (calc-binary-op "chos" 'calcFunc-choose arg))))
136211a9
EZ		 126
127(defun calc-perm (arg)
128 (interactive "P")
129 (calc-hyperbolic-func)
bf77c646 130 (calc-choose arg))
136211a9
EZ		 131
132(defvar calc-last-random-limit '(float 1 0))
133(defun calc-random (n)
134 (interactive "P")
135 (calc-slow-wrapper
136 (if n
137 (calc-enter-result 0 "rand" (list 'calcFunc-random
138 (calc-get-random-limit
139 (prefix-numeric-value n))))
140 (calc-enter-result 1 "rand" (list 'calcFunc-random
141 (calc-get-random-limit
bf77c646 142 (calc-top-n 1)))))))
136211a9
EZ		 143
144(defun calc-get-random-limit (val)
145 (if (eq val 0)
146 calc-last-random-limit
bf77c646 147 (setq calc-last-random-limit val)))
136211a9
EZ		 148
149(defun calc-rrandom ()
150 (interactive)
151 (calc-slow-wrapper
152 (setq calc-last-random-limit '(float 1 0))
bf77c646 153 (calc-enter-result 0 "rand" (list 'calcFunc-random '(float 1 0)))))
136211a9
EZ		 154
155(defun calc-random-again (arg)
156 (interactive "p")
157 (calc-slow-wrapper
158 (while (>= (setq arg (1- arg)) 0)
159 (calc-enter-result 0 "rand" (list 'calcFunc-random
bf77c646 160 calc-last-random-limit)))))
136211a9
EZ		 161
162(defun calc-shuffle (n)
163 (interactive "P")
164 (calc-slow-wrapper
165 (if n
166 (calc-enter-result 1 "shuf" (list 'calcFunc-shuffle
167 (prefix-numeric-value n)
168 (calc-get-random-limit
169 (calc-top-n 1))))
170 (calc-enter-result 2 "shuf" (list 'calcFunc-shuffle
171 (calc-top-n 1)
172 (calc-get-random-limit
bf77c646 173 (calc-top-n 2)))))))
136211a9
EZ		 174
175(defun calc-report-prime-test (res)
176 (cond ((eq (car res) t)
177 (calc-record-message "prim" "Prime (guaranteed)"))
178 ((eq (car res) nil)
179 (if (cdr res)
180 (if (eq (nth 1 res) 'unknown)
181 (calc-record-message
182 "prim" "Non-prime (factors unknown)")
183 (calc-record-message
184 "prim" "Non-prime (%s is a factor)"
185 (math-format-number (nth 1 res))))
186 (calc-record-message "prim" "Non-prime")))
187 (t
188 (calc-record-message
189 "prim" "Probably prime (%d iters; %s%% chance of error)"
190 (nth 1 res)
191 (let ((calc-float-format '(fix 2)))
bf77c646 192 (math-format-number (nth 2 res)))))))
136211a9
EZ		 193
194(defun calc-prime-test (iters)
195 (interactive "p")
196 (calc-slow-wrapper
197 (let* ((n (calc-top-n 1))
198 (res (math-prime-test n iters)))
bf77c646 199 (calc-report-prime-test res))))
136211a9 200
8d7498c1
JB		 201(defvar calc-verbose-nextprime nil)
202
136211a9
EZ		 203(defun calc-next-prime (iters)
204 (interactive "p")
205 (calc-slow-wrapper
206 (let ((calc-verbose-nextprime t))
207 (if (calc-is-inverse)
208 (calc-enter-result 1 "prvp" (list 'calcFunc-prevprime
209 (calc-top-n 1) (math-abs iters)))
210 (calc-enter-result 1 "nxtp" (list 'calcFunc-nextprime
bf77c646 211 (calc-top-n 1) (math-abs iters)))))))
136211a9
EZ		 212
213(defun calc-prev-prime (iters)
214 (interactive "p")
215 (calc-invert-func)
bf77c646 216 (calc-next-prime iters))
136211a9
EZ		 217
218(defun calc-prime-factors (iters)
219 (interactive "p")
220 (calc-slow-wrapper
221 (let ((res (calcFunc-prfac (calc-top-n 1))))
222 (if (not math-prime-factors-finished)
223 (calc-record-message "pfac" "Warning: May not be fully factored"))
bf77c646 224 (calc-enter-result 1 "pfac" res))))
136211a9
EZ		 225
226(defun calc-totient (arg)
227 (interactive "P")
228 (calc-slow-wrapper
bf77c646 229 (calc-unary-op "phi" 'calcFunc-totient arg)))
136211a9
EZ		 230
231(defun calc-moebius (arg)
232 (interactive "P")
233 (calc-slow-wrapper
bf77c646 234 (calc-unary-op "mu" 'calcFunc-moebius arg)))
136211a9
EZ		 235
236
237(defun calcFunc-gcd (a b)
238 (if (Math-messy-integerp a)
239 (setq a (math-trunc a)))
240 (if (Math-messy-integerp b)
241 (setq b (math-trunc b)))
242 (cond ((and (Math-integerp a) (Math-integerp b))
243 (math-gcd a b))
244 ((Math-looks-negp a)
245 (calcFunc-gcd (math-neg a) b))
246 ((Math-looks-negp b)
247 (calcFunc-gcd a (math-neg b)))
248 ((Math-zerop a) b)
249 ((Math-zerop b) a)
250 ((and (Math-ratp a)
251 (Math-ratp b))
252 (math-make-frac (math-gcd (if (eq (car-safe a) 'frac) (nth 1 a) a)
253 (if (eq (car-safe b) 'frac) (nth 1 b) b))
254 (calcFunc-lcm
255 (if (eq (car-safe a) 'frac) (nth 2 a) 1)
256 (if (eq (car-safe b) 'frac) (nth 2 b) 1))))
257 ((not (Math-integerp a))
258 (calc-record-why 'integerp a)
259 (list 'calcFunc-gcd a b))
260 (t
261 (calc-record-why 'integerp b)
bf77c646 262 (list 'calcFunc-gcd a b))))
136211a9
EZ		 263
264(defun calcFunc-lcm (a b)
265 (let ((g (calcFunc-gcd a b)))
266 (if (Math-numberp g)
267 (math-div (math-mul a b) g)
bf77c646 268 (list 'calcFunc-lcm a b))))
136211a9
EZ		 269
270(defun calcFunc-egcd (a b) ; Knuth section 4.5.2
271 (cond
272 ((not (Math-integerp a))
273 (if (Math-messy-integerp a)
274 (calcFunc-egcd (math-trunc a) b)
275 (calc-record-why 'integerp a)
276 (list 'calcFunc-egcd a b)))
277 ((not (Math-integerp b))
278 (if (Math-messy-integerp b)
279 (calcFunc-egcd a (math-trunc b))
280 (calc-record-why 'integerp b)
281 (list 'calcFunc-egcd a b)))
282 (t
283 (let ((u1 1) (u2 0) (u3 a)
284 (v1 0) (v2 1) (v3 b)
285 t1 t2 q)
286 (while (not (eq v3 0))
287 (setq q (math-idivmod u3 v3)
288 t1 (math-sub u1 (math-mul v1 (car q)))
289 t2 (math-sub u2 (math-mul v2 (car q)))
290 u1 v1 u2 v2 u3 v3
291 v1 t1 v2 t2 v3 (cdr q)))
bf77c646 292 (list 'vec u3 u1 u2)))))
136211a9
EZ		 293
294
295;;; Factorial and related functions.
296
887fc3b8 297(defconst math-small-factorial-table
2aad2a84
JB		 298 (vector 1 1 2 6 24 120 720 5040 40320 362880 3628800 39916800
299 (math-read-number-simple "479001600")
300 (math-read-number-simple "6227020800")
301 (math-read-number-simple "87178291200")
302 (math-read-number-simple "1307674368000")
303 (math-read-number-simple "20922789888000")
304 (math-read-number-simple "355687428096000")
305 (math-read-number-simple "6402373705728000")
306 (math-read-number-simple "121645100408832000")
307 (math-read-number-simple "2432902008176640000")))
887fc3b8 308
136211a9
EZ		 309(defun calcFunc-fact (n) ; [I I] [F F] [Public]
310 (let (temp)
311 (cond ((Math-integer-negp n)
312 (if calc-infinite-mode
313 '(var uinf var-uinf)
314 (math-reject-arg n 'range)))
315 ((integerp n)
316 (if (<= n 20)
887fc3b8 317 (aref math-small-factorial-table n)
136211a9
EZ		 318 (math-factorial-iter (1- n) 2 1)))
319 ((and (math-messy-integerp n)
320 (Math-lessp n 100))
321 (math-inexact-result)
322 (setq temp (math-trunc n))
323 (if (>= temp 0)
324 (if (<= temp 20)
325 (math-float (calcFunc-fact temp))
326 (math-with-extra-prec 1
327 (math-factorial-iter (1- temp) 2 '(float 1 0))))
328 (math-reject-arg n 'range)))
329 ((math-numberp n)
330 (let* ((q (math-quarter-integer n))
331 (tn (and q (Math-lessp n 1000) (Math-lessp -1000 n)
332 (1+ (math-floor n)))))
333 (cond ((and tn (= q 2)
334 (or calc-symbolic-mode (< (math-abs tn) 20)))
335 (let ((q (if (< tn 0)
336 (math-div
337 (math-pow -2 (- tn))
338 (math-double-factorial-iter (* -2 tn) 3 1 2))
a1506d29 339 (math-div
136211a9
EZ		 340 (math-double-factorial-iter (* 2 tn) 3 1 2)
341 (math-pow 2 tn)))))
342 (math-mul q (if calc-symbolic-mode
343 (list 'calcFunc-sqrt '(var pi var-pi))
344 (math-sqrt-pi)))))
345 ((and tn (>= tn 0) (< tn 20)
346 (memq q '(1 3)))
347 (math-inexact-result)
348 (math-div
349 (math-mul (math-double-factorial-iter (* 4 tn) q 1 4)
350 (if (= q 1) (math-gamma-1q) (math-gamma-3q)))
351 (math-pow 4 tn)))
352 (t
353 (math-inexact-result)
354 (math-with-extra-prec 3
355 (math-gammap1-raw (math-float n)))))))
356 ((equal n '(var inf var-inf)) n)
357 (t (calc-record-why 'numberp n)
bf77c646 358 (list 'calcFunc-fact n)))))
136211a9
EZ		 359
360(math-defcache math-gamma-1q nil
361 (math-with-extra-prec 3
362 (math-gammap1-raw '(float -75 -2))))
363
364(math-defcache math-gamma-3q nil
365 (math-with-extra-prec 3
366 (math-gammap1-raw '(float -25 -2))))
367
368(defun math-factorial-iter (count n f)
369 (if (= (% n 5) 1)
370 (math-working (format "factorial(%d)" (1- n)) f))
371 (if (> count 0)
372 (math-factorial-iter (1- count) (1+ n) (math-mul n f))
bf77c646 373 f))
136211a9
EZ		 374
375(defun calcFunc-dfact (n) ; [I I] [F F] [Public]
376 (cond ((Math-integer-negp n)
377 (if (math-oddp n)
378 (if (eq n -1)
379 1
380 (math-div (if (eq (math-mod n 4) 3) 1 -1)
381 (calcFunc-dfact (math-sub -2 n))))
382 (list 'calcFunc-dfact n)))
383 ((Math-zerop n) 1)
384 ((integerp n) (math-double-factorial-iter n (+ 2 (% n 2)) 1 2))
385 ((math-messy-integerp n)
386 (let ((temp (math-trunc n)))
387 (math-inexact-result)
388 (if (natnump temp)
389 (if (Math-lessp temp 200)
390 (math-with-extra-prec 1
391 (math-double-factorial-iter temp (+ 2 (% temp 2))
392 '(float 1 0) 2))
393 (let* ((half (math-div2 temp))
394 (even (math-mul (math-pow 2 half)
395 (calcFunc-fact (math-float half)))))
396 (if (math-evenp temp)
397 even
398 (math-div (calcFunc-fact n) even))))
8d7498c1 399 (list 'calcFunc-dfact n))))
136211a9
EZ		 400 ((equal n '(var inf var-inf)) n)
401 (t (calc-record-why 'natnump n)
bf77c646 402 (list 'calcFunc-dfact n))))
136211a9
EZ		 403
404(defun math-double-factorial-iter (max n f step)
405 (if (< (% n 12) step)
406 (math-working (format "dfact(%d)" (- n step)) f))
407 (if (<= n max)
408 (math-double-factorial-iter max (+ n step) (math-mul n f) step)
bf77c646 409 f))
136211a9
EZ		 410
411(defun calcFunc-perm (n m) ; [I I I] [F F F] [Public]
412 (cond ((and (integerp n) (integerp m) (<= m n) (>= m 0))
413 (math-factorial-iter m (1+ (- n m)) 1))
414 ((or (not (math-num-integerp n))
415 (and (math-messy-integerp n) (Math-lessp 100 n))
416 (not (math-num-integerp m))
417 (and (math-messy-integerp m) (Math-lessp 100 m)))
418 (or (math-realp n) (equal n '(var inf var-inf))
419 (math-reject-arg n 'realp))
420 (or (math-realp m) (equal m '(var inf var-inf))
421 (math-reject-arg m 'realp))
422 (and (math-num-integerp n) (math-negp n) (math-reject-arg n 'range))
423 (and (math-num-integerp m) (math-negp m) (math-reject-arg m 'range))
424 (math-div (calcFunc-fact n) (calcFunc-fact (math-sub n m))))
425 (t
426 (let ((tn (math-trunc n))
427 (tm (math-trunc m)))
428 (math-inexact-result)
429 (or (integerp tn) (math-reject-arg tn 'fixnump))
430 (or (integerp tm) (math-reject-arg tm 'fixnump))
431 (or (and (<= tm tn) (>= tm 0)) (math-reject-arg tm 'range))
432 (math-with-extra-prec 1
bf77c646 433 (math-factorial-iter tm (1+ (- tn tm)) '(float 1 0)))))))
136211a9
EZ		 434
435(defun calcFunc-choose (n m) ; [I I I] [F F F] [Public]
436 (cond ((and (integerp n) (integerp m) (<= m n) (>= m 0))
437 (if (> m (/ n 2))
438 (math-choose-iter (- n m) n 1 1)
439 (math-choose-iter m n 1 1)))
440 ((not (math-realp n))
441 (math-reject-arg n 'realp))
442 ((not (math-realp m))
443 (math-reject-arg m 'realp))
444 ((not (math-num-integerp m))
445 (if (and (math-num-integerp n) (math-negp n))
446 (list 'calcFunc-choose n m)
447 (math-div (calcFunc-fact (math-float n))
448 (math-mul (calcFunc-fact m)
449 (calcFunc-fact (math-sub n m))))))
450 ((math-negp m) 0)
451 ((math-negp n)
452 (let ((val (calcFunc-choose (math-add (math-add n m) -1) m)))
453 (if (math-evenp (math-trunc m))
454 val
455 (math-neg val))))
456 ((and (math-num-integerp n)
457 (Math-lessp n m))
458 0)
459 (t
460 (math-inexact-result)
461 (let ((tm (math-trunc m)))
462 (or (integerp tm) (math-reject-arg tm 'fixnump))
463 (if (> tm 100)
464 (math-div (calcFunc-fact (math-float n))
465 (math-mul (calcFunc-fact (math-float m))
466 (calcFunc-fact (math-float
467 (math-sub n m)))))
468 (math-with-extra-prec 1
bf77c646 469 (math-choose-float-iter tm n 1 1)))))))
136211a9
EZ		 470
471(defun math-choose-iter (m n i c)
472 (if (and (= (% i 5) 1) (> i 5))
473 (math-working (format "choose(%d)" (1- i)) c))
474 (if (<= i m)
475 (math-choose-iter m (1- n) (1+ i)
476 (math-quotient (math-mul c n) i))
bf77c646 477 c))
136211a9
EZ		 478
479(defun math-choose-float-iter (count n i c)
480 (if (= (% i 5) 1)
481 (math-working (format "choose(%d)" (1- i)) c))
482 (if (> count 0)
483 (math-choose-float-iter (1- count) (math-sub n 1) (1+ i)
484 (math-div (math-mul c n) i))
bf77c646 485 c))
136211a9
EZ		 486
487
488;;; Stirling numbers.
489
490(defun calcFunc-stir1 (n m)
bf77c646 491 (math-stirling-number n m 1))
136211a9
EZ		 492
493(defun calcFunc-stir2 (n m)
bf77c646 494 (math-stirling-number n m 0))
136211a9 495
3132f345 496(defvar math-stirling-cache (vector [[1]] [[1]]))
8d7498c1
JB		 497
498;; The variable math-stirling-local-cache is local to
499;; math-stirling-number, but is used by math-stirling-1
500;; and math-stirling-2, which are called by math-stirling-number.
501(defvar math-stirling-local-cache)
502
136211a9
EZ		 503(defun math-stirling-number (n m k)
504 (or (math-num-natnump n) (math-reject-arg n 'natnump))
505 (or (math-num-natnump m) (math-reject-arg m 'natnump))
506 (if (consp n) (setq n (math-trunc n)))
507 (or (integerp n) (math-reject-arg n 'fixnump))
508 (if (consp m) (setq m (math-trunc m)))
509 (or (integerp m) (math-reject-arg m 'fixnump))
510 (if (< n m)
511 0
8d7498c1
JB		 512 (let ((math-stirling-local-cache (aref math-stirling-cache k)))
513 (while (<= (length math-stirling-local-cache) n)
514 (let ((i (1- (length math-stirling-local-cache)))
136211a9 515 row)
8d7498c1
JB		 516 (setq math-stirling-local-cache
517 (vconcat math-stirling-local-cache
518 (make-vector (length math-stirling-local-cache) nil)))
519 (aset math-stirling-cache k math-stirling-local-cache)
520 (while (< (setq i (1+ i)) (length math-stirling-local-cache))
521 (aset math-stirling-local-cache i (setq row (make-vector (1+ i) nil)))
136211a9
EZ		 522 (aset row 0 0)
523 (aset row i 1))))
524 (if (= k 1)
525 (math-stirling-1 n m)
bf77c646 526 (math-stirling-2 n m)))))
136211a9
EZ		 527
528(defun math-stirling-1 (n m)
8d7498c1
JB		 529 (or (aref (aref math-stirling-local-cache n) m)
530 (aset (aref math-stirling-local-cache n) m
136211a9 531 (math-add (math-stirling-1 (1- n) (1- m))
bf77c646 532 (math-mul (- 1 n) (math-stirling-1 (1- n) m))))))
136211a9
EZ		 533
534(defun math-stirling-2 (n m)
8d7498c1
JB		 535 (or (aref (aref math-stirling-local-cache n) m)
536 (aset (aref math-stirling-local-cache n) m
136211a9 537 (math-add (math-stirling-2 (1- n) (1- m))
bf77c646 538 (math-mul m (math-stirling-2 (1- n) m))))))
136211a9 539
3132f345
CW		 540(defvar math-random-table nil)
541(defvar math-last-RandSeed nil)
542(defvar math-random-ptr1 nil)
543(defvar math-random-ptr2 nil)
544(defvar math-random-shift nil)
136211a9
EZ		 545
546;;; Produce a random 10-bit integer, with (random) if no seed provided,
547;;; or else with Numerical Recipes algorithm ran3 / Knuth 3.2.2-A.
8d7498c1 548
a59a1687 549(defvar var-RandSeed)
8d7498c1
JB		 550(defvar math-random-cache nil)
551(defvar math-gaussian-cache nil)
552
136211a9 553(defun math-init-random-base ()
a59a1687 554 (if (and (boundp 'var-RandSeed) var-RandSeed)
136211a9
EZ		 555 (if (eq (car-safe var-RandSeed) 'vec)
556 nil
557 (if (Math-integerp var-RandSeed)
558 (let* ((seed (math-sub 161803 var-RandSeed))
4b4b19bd
JB		 559 (mj (1+ (math-mod seed 1000000)))
560 (mk (1+ (math-mod (math-quotient seed 1000000)
561 1000000)))
136211a9
EZ		 562 (i 0))
563 (setq math-random-table (cons 'vec (make-list 55 mj)))
564 (while (<= (setq i (1+ i)) 54)
565 (let* ((ii (% (* i 21) 55))
566 (p (nthcdr ii math-random-table)))
567 (setcar p mk)
568 (setq mk (- mj mk)
569 mj (car p)))))
570 (math-reject-arg var-RandSeed "*RandSeed must be an integer"))
571 (setq var-RandSeed (list 'vec var-RandSeed)
572 math-random-ptr1 math-random-table
573 math-random-cache nil
574 math-random-ptr2 (nthcdr 31 math-random-table))
575 (let ((i 200))
576 (while (> (setq i (1- i)) 0)
577 (math-random-base))))
578 (random t)
579 (setq var-RandSeed nil
580 math-random-cache nil
136211a9
EZ		 581 math-random-shift -4) ; assume RAND_MAX >= 16383
582 ;; This exercises the random number generator and also helps
583 ;; deduce a better value for RAND_MAX.
8d7498c1
JB		 584 (let ((i 0))
585 (while (< (setq i (1+ i)) 30)
586 (if (> (lsh (math-abs (random)) math-random-shift) 4095)
587 (setq math-random-shift (1- math-random-shift))))))
136211a9 588 (setq math-last-RandSeed var-RandSeed
bf77c646 589 math-gaussian-cache nil))
136211a9
EZ		 590
591(defun math-random-base ()
592 (if var-RandSeed
593 (progn
594 (setq math-random-ptr1 (or (cdr math-random-ptr1)
595 (cdr math-random-table))
596 math-random-ptr2 (or (cdr math-random-ptr2)
597 (cdr math-random-table)))
598 (logand (lsh (setcar math-random-ptr1
599 (logand (- (car math-random-ptr1)
600 (car math-random-ptr2)) 524287))
601 -6) 1023))
bf77c646 602 (logand (lsh (random) math-random-shift) 1023)))
136211a9
EZ		 603
604
605;;; Produce a random digit in the range 0..999.
606;;; Avoid various pitfalls that may lurk in the built-in (random) function!
607;;; Shuffling algorithm from Numerical Recipes, section 7.1.
a59a1687 608(defvar math-random-last)
e4ebbf49
JB		 609(defun math-random-three-digit-number ()
610 "Return a random three digit number."
a59a1687
JB		 611 (let (i)
612 (or (and (boundp 'var-RandSeed) (eq var-RandSeed math-last-RandSeed))
136211a9
EZ		 613 (math-init-random-base))
614 (or math-random-cache
615 (progn
616 (setq math-random-last (math-random-base)
617 math-random-cache (make-vector 13 nil)
618 i -1)
619 (while (< (setq i (1+ i)) 13)
620 (aset math-random-cache i (math-random-base)))))
621 (while (progn
622 (setq i (/ math-random-last 79) ; 0 <= i < 13
623 math-random-last (aref math-random-cache i))
624 (aset math-random-cache i (math-random-base))
625 (>= math-random-last 1000)))
bf77c646 626 math-random-last))
136211a9
EZ		 627
628;;; Produce an N-digit random integer.
629(defun math-random-digits (n)
e4ebbf49
JB		 630 "Produce a random N digit integer."
631 (let* ((slop (% (- 3 (% n 3)) 3))
632 (i (/ (+ n slop) 3))
633 (rnum 0))
634 (while (> i 0)
635 (setq rnum
636 (math-add
637 (math-random-three-digit-number)
638 (math-mul rnum 1000)))
639 (setq i (1- i)))
640 (math-normalize (math-scale-right rnum slop))))
136211a9
EZ		 641
642;;; Produce a uniformly-distributed random float 0 <= N < 1.
643(defun math-random-float ()
644 (math-make-float (math-random-digits calc-internal-prec)
bf77c646 645 (- calc-internal-prec)))
136211a9
EZ		 646
647;;; Produce a Gaussian-distributed random float with mean=0, sigma=1.
648(defun math-gaussian-float ()
649 (math-with-extra-prec 2
650 (if (and math-gaussian-cache
651 (= (car math-gaussian-cache) calc-internal-prec))
652 (prog1
653 (cdr math-gaussian-cache)
654 (setq math-gaussian-cache nil))
655 (let* ((v1 (math-add (math-mul (math-random-float) 2) -1))
656 (v2 (math-add (math-mul (math-random-float) 2) -1))
657 (r (math-add (math-sqr v1) (math-sqr v2))))
658 (while (or (not (Math-lessp r 1)) (math-zerop r))
659 (setq v1 (math-add (math-mul (math-random-float) 2) -1)
660 v2 (math-add (math-mul (math-random-float) 2) -1)
661 r (math-add (math-sqr v1) (math-sqr v2))))
662 (let ((fac (math-sqrt (math-mul (math-div (calcFunc-ln r) r) -2))))
663 (setq math-gaussian-cache (cons calc-internal-prec
664 (math-mul v1 fac)))
bf77c646 665 (math-mul v2 fac))))))
136211a9
EZ		 666
667;;; Produce a random integer or real 0 <= N < MAX.
668(defun calcFunc-random (max)
669 (cond ((Math-zerop max)
670 (math-gaussian-float))
671 ((Math-integerp max)
672 (let* ((digs (math-numdigs max))
673 (r (math-random-digits (+ digs 3))))
674 (math-mod r max)))
675 ((Math-realp max)
676 (math-mul (math-random-float) max))
677 ((and (eq (car max) 'intv) (math-constp max)
678 (Math-lessp (nth 2 max) (nth 3 max)))
679 (if (math-floatp max)
680 (let ((val (math-add (math-mul (math-random-float)
681 (math-sub (nth 3 max) (nth 2 max)))
682 (nth 2 max))))
683 (if (or (and (memq (nth 1 max) '(0 1)) ; almost not worth
684 (Math-equal val (nth 2 max))) ; checking!
685 (and (memq (nth 1 max) '(0 2))
686 (Math-equal val (nth 3 max))))
687 (calcFunc-random max)
688 val))
689 (let ((lo (if (memq (nth 1 max) '(0 1))
690 (math-add (nth 2 max) 1) (nth 2 max)))
691 (hi (if (memq (nth 1 max) '(1 3))
692 (math-add (nth 3 max) 1) (nth 3 max))))
693 (if (Math-lessp lo hi)
694 (math-add (calcFunc-random (math-sub hi lo)) lo)
695 (math-reject-arg max "*Empty interval")))))
696 ((eq (car max) 'vec)
697 (if (cdr max)
698 (nth (1+ (calcFunc-random (1- (length max)))) max)
699 (math-reject-arg max "*Empty list")))
700 ((and (eq (car max) 'sdev) (math-constp max) (Math-realp (nth 1 max)))
701 (math-add (math-mul (math-gaussian-float) (nth 2 max)) (nth 1 max)))
bf77c646 702 (t (math-reject-arg max 'realp))))
136211a9
EZ		 703
704;;; Choose N objects at random from the set MAX without duplicates.
705(defun calcFunc-shuffle (n &optional max)
706 (or max (setq max n n -1))
707 (or (and (Math-num-integerp n)
708 (or (natnump (setq n (math-trunc n))) (eq n -1)))
709 (math-reject-arg n 'integerp))
710 (cond ((or (math-zerop max)
711 (math-floatp max)
712 (eq (car-safe max) 'sdev))
713 (if (< n 0)
714 (math-reject-arg n 'natnump)
715 (math-simple-shuffle n max)))
716 ((and (<= n 1) (>= n 0))
717 (math-simple-shuffle n max))
718 ((and (eq (car-safe max) 'intv) (math-constp max))
719 (let ((num (math-add (math-sub (nth 3 max) (nth 2 max))
720 (cdr (assq (nth 1 max)
721 '((0 . -1) (1 . 0)
722 (2 . 0) (3 . 1))))))
723 (min (math-add (nth 2 max) (if (memq (nth 1 max) '(0 1))
724 1 0))))
725 (if (< n 0) (setq n num))
726 (or (math-posp num) (math-reject-arg max 'range))
727 (and (Math-lessp num n) (math-reject-arg n 'range))
728 (if (Math-lessp n (math-quotient num 3))
729 (math-simple-shuffle n max)
730 (if (> (* n 4) (* num 3))
731 (math-add (math-sub min 1)
732 (math-shuffle-list n num (calcFunc-index num)))
733 (let ((tot 0)
734 (m 0)
735 (vec nil))
736 (while (< m n)
737 (if (< (calcFunc-random (- num tot)) (- n m))
738 (setq vec (cons (math-add min tot) vec)
739 m (1+ m)))
740 (setq tot (1+ tot)))
741 (math-shuffle-list n n (cons 'vec vec)))))))
742 ((eq (car-safe max) 'vec)
743 (let ((size (1- (length max))))
744 (if (< n 0) (setq n size))
745 (if (and (> n (/ size 2)) (<= n size))
746 (math-shuffle-list n size (copy-sequence max))
747 (let* ((vals (calcFunc-shuffle
748 n (list 'intv 3 1 (1- (length max)))))
749 (p vals))
750 (while (setq p (cdr p))
751 (setcar p (nth (car p) max)))
752 vals))))
753 ((math-integerp max)
754 (if (math-posp max)
755 (calcFunc-shuffle n (list 'intv 2 0 max))
756 (calcFunc-shuffle n (list 'intv 1 max 0))))
bf77c646 757 (t (math-reject-arg max 'realp))))
136211a9
EZ		 758
759(defun math-simple-shuffle (n max)
760 (let ((vec nil)
761 val)
762 (while (>= (setq n (1- n)) 0)
763 (while (math-member (setq val (calcFunc-random max)) vec))
764 (setq vec (cons val vec)))
bf77c646 765 (cons 'vec vec)))
136211a9
EZ		 766
767(defun math-shuffle-list (n size vec)
768 (let ((j size)
769 k temp
770 (p vec))
771 (while (cdr (setq p (cdr p)))
772 (setq k (calcFunc-random j)
773 j (1- j)
774 temp (nth k p))
775 (setcar (nthcdr k p) (car p))
776 (setcar p temp))
bf77c646 777 (cons 'vec (nthcdr (- size n -1) vec))))
136211a9
EZ		 778
779(defun math-member (x list)
780 (while (and list (not (equal x (car list))))
781 (setq list (cdr list)))
bf77c646 782 list)
136211a9
EZ		 783
784
785;;; Check if the integer N is prime. [X I]
786;;; Return (nil) if non-prime,
787;;; (nil N) if non-prime with known factor N,
788;;; (nil unknown) if non-prime with no known factors,
789;;; (t) if prime,
790;;; (maybe N P) if probably prime (after N iters with probability P%)
8d7498c1
JB		 791(defvar math-prime-test-cache '(-1))
792
793(defvar math-prime-test-cache-k)
794(defvar math-prime-test-cache-q)
795(defvar math-prime-test-cache-nm1)
796
136211a9
EZ		 797(defun math-prime-test (n iters)
798 (if (and (Math-vectorp n) (cdr n))
799 (setq n (nth (1- (length n)) n)))
800 (if (Math-messy-integerp n)
801 (setq n (math-trunc n)))
802 (let ((res))
803 (while (> iters 0)
804 (setq res
805 (cond ((and (integerp n) (<= n 5003))
806 (list (= (math-next-small-prime n) n)))
807 ((not (Math-integerp n))
808 (error "Argument must be an integer"))
809 ((Math-integer-negp n)
810 '(nil))
4b4b19bd 811 ((Math-natnum-lessp n 8000000)
136211a9
EZ		 812 (setq n (math-fixnum n))
813 (let ((i -1) v)
814 (while (and (> (% n (setq v (aref math-primes-table
815 (setq i (1+ i)))))
816 0)
817 (< (* v v) n)))
818 (if (= (% n v) 0)
819 (list nil v)
820 '(t))))
821 ((not (equal n (car math-prime-test-cache)))
822 (cond ((= (% (nth 1 n) 2) 0) '(nil 2))
823 ((= (% (nth 1 n) 5) 0) '(nil 5))
40cddce8
JB		 824 (t (let ((q n) (sum 0))
825 (while (not (eq q 0))
826 (setq sum (%
827 (+
828 sum
829 (calcFunc-mod
4b4b19bd 830 q 1000000))
40cddce8
JB		 831 111111))
832 (setq q
833 (math-quotient
4b4b19bd 834 q 1000000)))
136211a9
EZ		 835 (cond ((= (% sum 3) 0) '(nil 3))
836 ((= (% sum 7) 0) '(nil 7))
837 ((= (% sum 11) 0) '(nil 11))
838 ((= (% sum 13) 0) '(nil 13))
839 ((= (% sum 37) 0) '(nil 37))
840 (t
841 (setq math-prime-test-cache-k 1
842 math-prime-test-cache-q
843 (math-div2 n)
844 math-prime-test-cache-nm1
845 (math-add n -1))
846 (while (math-evenp
847 math-prime-test-cache-q)
848 (setq math-prime-test-cache-k
849 (1+ math-prime-test-cache-k)
850 math-prime-test-cache-q
851 (math-div2
852 math-prime-test-cache-q)))
853 (setq iters (1+ iters))
854 (list 'maybe
855 0
856 (math-sub
857 100
858 (math-div
859 '(float 232 0)
860 (math-numdigs n))))))))))
861 ((not (eq (car (nth 1 math-prime-test-cache)) 'maybe))
862 (nth 1 math-prime-test-cache))
863 (t ; Fermat step
864 (let* ((x (math-add (calcFunc-random (math-add n -2)) 2))
865 (y (math-pow-mod x math-prime-test-cache-q n))
866 (j 0))
867 (while (and (not (eq y 1))
868 (not (equal y math-prime-test-cache-nm1))
869 (< (setq j (1+ j)) math-prime-test-cache-k))
870 (setq y (math-mod (math-mul y y) n)))
871 (if (or (equal y math-prime-test-cache-nm1)
872 (and (eq y 1) (eq j 0)))
873 (list 'maybe
874 (1+ (nth 1 (nth 1 math-prime-test-cache)))
875 (math-mul (nth 2 (nth 1 math-prime-test-cache))
876 '(float 25 -2)))
877 '(nil unknown))))))
878 (setq math-prime-test-cache (list n res)
879 iters (if (eq (car res) 'maybe)
880 (1- iters)
881 0)))
bf77c646 882 res))
136211a9
EZ		 883
884(defun calcFunc-prime (n &optional iters)
885 (or (math-num-integerp n) (math-reject-arg n 'integerp))
886 (or (not iters) (math-num-integerp iters) (math-reject-arg iters 'integerp))
887 (if (car (math-prime-test (math-trunc n) (math-trunc (or iters 1))))
888 1
bf77c646 889 0))
136211a9
EZ		 890
891;;; Theory: summing base-10^6 digits modulo 111111 is "casting out 999999s".
892;;; Initial probability that N is prime is 1/ln(N) = log10(e)/log10(N).
893;;; After culling [2,3,5,7,11,13,37], probability of primality is 5.36 x more.
894;;; Initial reported probability of non-primality is thus 100% - this.
895;;; Each Fermat step multiplies this probability by 25%.
896;;; The Fermat step is algorithm P from Knuth section 4.5.4.
897
898
899(defun calcFunc-prfac (n)
900 (setq math-prime-factors-finished t)
901 (if (Math-messy-integerp n)
902 (setq n (math-trunc n)))
903 (if (Math-natnump n)
904 (if (Math-natnum-lessp 2 n)
905 (let (factors res p (i 0))
906 (while (and (not (eq n 1))
907 (< i (length math-primes-table)))
908 (setq p (aref math-primes-table i))
909 (while (eq (cdr (setq res (cond ((eq n p) (cons 1 0))
910 ((eq n 1) (cons 0 1))
911 ((consp n) (math-idivmod n p))
912 (t (cons (/ n p) (% n p))))))
913 0)
914 (math-working "factor" p)
915 (setq factors (nconc factors (list p))
916 n (car res)))
917 (or (eq n 1)
918 (Math-natnum-lessp p (car res))
919 (setq factors (nconc factors (list n))
920 n 1))
921 (setq i (1+ i)))
922 (or (setq math-prime-factors-finished (eq n 1))
923 (setq factors (nconc factors (list n))))
924 (cons 'vec factors))
925 (list 'vec n))
926 (if (Math-integerp n)
927 (if (eq n -1)
928 (list 'vec n)
929 (cons 'vec (cons -1 (cdr (calcFunc-prfac (math-neg n))))))
930 (calc-record-why 'integerp n)
bf77c646 931 (list 'calcFunc-prfac n))))
136211a9
EZ		 932
933(defun calcFunc-totient (n)
934 (if (Math-messy-integerp n)
935 (setq n (math-trunc n)))
936 (if (Math-natnump n)
937 (if (Math-natnum-lessp n 2)
938 (if (Math-negp n)
939 (calcFunc-totient (math-abs n))
940 n)
941 (let ((factors (cdr (calcFunc-prfac n)))
942 p)
943 (if math-prime-factors-finished
944 (progn
945 (while factors
946 (setq p (car factors)
947 n (math-mul (math-div n p) (math-add p -1)))
948 (while (equal p (car factors))
949 (setq factors (cdr factors))))
950 n)
951 (calc-record-why "*Number too big to factor" n)
952 (list 'calcFunc-totient n))))
953 (calc-record-why 'natnump n)
bf77c646 954 (list 'calcFunc-totient n)))
136211a9
EZ		 955
956(defun calcFunc-moebius (n)
957 (if (Math-messy-integerp n)
958 (setq n (math-trunc n)))
959 (if (and (Math-natnump n) (not (eq n 0)))
960 (if (Math-natnum-lessp n 2)
961 (if (Math-negp n)
962 (calcFunc-moebius (math-abs n))
963 1)
964 (let ((factors (cdr (calcFunc-prfac n)))
965 (mu 1))
966 (if math-prime-factors-finished
967 (progn
968 (while factors
969 (setq mu (if (equal (car factors) (nth 1 factors))
970 0 (math-neg mu))
971 factors (cdr factors)))
972 mu)
973 (calc-record-why "Number too big to factor" n)
974 (list 'calcFunc-moebius n))))
975 (calc-record-why 'posintp n)
bf77c646 976 (list 'calcFunc-moebius n)))
136211a9
EZ		 977
978
979(defun calcFunc-nextprime (n &optional iters)
980 (if (Math-integerp n)
981 (if (Math-integer-negp n)
982 2
983 (if (and (integerp n) (< n 5003))
984 (math-next-small-prime (1+ n))
985 (if (math-evenp n)
986 (setq n (math-add n -1)))
987 (let (res)
988 (while (not (car (setq res (math-prime-test
989 (setq n (math-add n 2))
990 (or iters 1))))))
991 (if (and calc-verbose-nextprime
992 (eq (car res) 'maybe))
993 (calc-report-prime-test res)))
994 n))
995 (if (Math-realp n)
996 (calcFunc-nextprime (math-trunc n) iters)
bf77c646 997 (math-reject-arg n 'integerp))))
136211a9
EZ		 998
999(defun calcFunc-prevprime (n &optional iters)
1000 (if (Math-integerp n)
1001 (if (Math-lessp n 4)
1002 2
1003 (if (math-evenp n)
1004 (setq n (math-add n 1)))
1005 (let (res)
1006 (while (not (car (setq res (math-prime-test
1007 (setq n (math-add n -2))
1008 (or iters 1))))))
1009 (if (and calc-verbose-nextprime
1010 (eq (car res) 'maybe))
1011 (calc-report-prime-test res)))
1012 n)
1013 (if (Math-realp n)
1014 (calcFunc-prevprime (math-ceiling n) iters)
bf77c646 1015 (math-reject-arg n 'integerp))))
136211a9
EZ		 1016
1017(defun math-next-small-prime (n)
1018 (if (and (integerp n) (> n 2))
1019 (let ((lo -1)
1020 (hi (length math-primes-table))
1021 mid)
1022 (while (> (- hi lo) 1)
1023 (if (> n (aref math-primes-table
1024 (setq mid (ash (+ lo hi) -1))))
1025 (setq lo mid)
1026 (setq hi mid)))
1027 (aref math-primes-table hi))
bf77c646 1028 2))
136211a9 1029
43f34ccc 1030(provide 'calc-comb)
136211a9 1031
ab5796a9 1032;;; arch-tag: 1d75ee9b-0815-42bd-a321-bb3dc001cc02
bf77c646 1033;;; calc-comb.el ends here
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