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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, |
8b72699e | 4 | ;; 2005, 2006, 2007, 2008 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 | ||
662c9c64 | 11 | ;; GNU Emacs is free software: you can redistribute it and/or modify |
7c671b23 | 12 | ;; it under the terms of the GNU General Public License as published by |
662c9c64 GM |
13 | ;; the Free Software Foundation, either version 3 of the License, or |
14 | ;; (at your option) any later version. | |
7c671b23 | 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 | |
662c9c64 | 22 | ;; along with GNU Emacs. If not, see <http://www.gnu.org/licenses/>. |
136211a9 | 23 | |
3132f345 | 24 | ;;; Commentary: |
136211a9 | 25 | |
3132f345 | 26 | ;;; Code: |
136211a9 EZ |
27 | |
28 | ;; This file is autoloaded from calc-ext.el. | |
136211a9 | 29 | |
43f34ccc | 30 | (require 'calc-ext) |
136211a9 EZ |
31 | (require 'calc-macs) |
32 | ||
3132f345 CW |
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]) | |
136211a9 | 80 | |
8d7498c1 JB |
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 | ||
136211a9 EZ |
86 | ;;; Combinatorics |
87 | ||
88 | (defun calc-gcd (arg) | |
89 | (interactive "P") | |
90 | (calc-slow-wrapper | |
bf77c646 | 91 | (calc-binary-op "gcd" 'calcFunc-gcd arg))) |
136211a9 EZ |
92 | |
93 | (defun calc-lcm (arg) | |
94 | (interactive "P") | |
95 | (calc-slow-wrapper | |
bf77c646 | 96 | (calc-binary-op "lcm" 'calcFunc-lcm arg))) |
136211a9 EZ |
97 | |
98 | (defun calc-extended-gcd () | |
99 | (interactive) | |
100 | (calc-slow-wrapper | |
bf77c646 | 101 | (calc-enter-result 2 "egcd" (cons 'calcFunc-egcd (calc-top-list-n 2))))) |
136211a9 EZ |
102 | |
103 | (defun calc-factorial (arg) | |
104 | (interactive "P") | |
105 | (calc-slow-wrapper | |
bf77c646 | 106 | (calc-unary-op "fact" 'calcFunc-fact arg))) |
136211a9 EZ |
107 | |
108 | (defun calc-gamma (arg) | |
109 | (interactive "P") | |
110 | (calc-slow-wrapper | |
bf77c646 | 111 | (calc-unary-op "gmma" 'calcFunc-gamma arg))) |
136211a9 EZ |
112 | |
113 | (defun calc-double-factorial (arg) | |
114 | (interactive "P") | |
115 | (calc-slow-wrapper | |
bf77c646 | 116 | (calc-unary-op "dfac" 'calcFunc-dfact arg))) |
136211a9 EZ |
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) | |
bf77c646 | 123 | (calc-binary-op "chos" 'calcFunc-choose arg)))) |
136211a9 EZ |
124 | |
125 | (defun calc-perm (arg) | |
126 | (interactive "P") | |
127 | (calc-hyperbolic-func) | |
bf77c646 | 128 | (calc-choose arg)) |
136211a9 EZ |
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 | |
bf77c646 | 140 | (calc-top-n 1))))))) |
136211a9 EZ |
141 | |
142 | (defun calc-get-random-limit (val) | |
143 | (if (eq val 0) | |
144 | calc-last-random-limit | |
bf77c646 | 145 | (setq calc-last-random-limit val))) |
136211a9 EZ |
146 | |
147 | (defun calc-rrandom () | |
148 | (interactive) | |
149 | (calc-slow-wrapper | |
150 | (setq calc-last-random-limit '(float 1 0)) | |
bf77c646 | 151 | (calc-enter-result 0 "rand" (list 'calcFunc-random '(float 1 0))))) |
136211a9 EZ |
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 | |
bf77c646 | 158 | calc-last-random-limit))))) |
136211a9 EZ |
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 | |
bf77c646 | 171 | (calc-top-n 2))))))) |
136211a9 EZ |
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))) | |
bf77c646 | 190 | (math-format-number (nth 2 res))))))) |
136211a9 EZ |
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))) | |
bf77c646 | 197 | (calc-report-prime-test res)))) |
136211a9 | 198 | |
8d7498c1 JB |
199 | (defvar calc-verbose-nextprime nil) |
200 | ||
136211a9 EZ |
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 | |
bf77c646 | 209 | (calc-top-n 1) (math-abs iters))))))) |
136211a9 EZ |
210 | |
211 | (defun calc-prev-prime (iters) | |
212 | (interactive "p") | |
213 | (calc-invert-func) | |
bf77c646 | 214 | (calc-next-prime iters)) |
136211a9 EZ |
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")) | |
bf77c646 | 222 | (calc-enter-result 1 "pfac" res)))) |
136211a9 EZ |
223 | |
224 | (defun calc-totient (arg) | |
225 | (interactive "P") | |
226 | (calc-slow-wrapper | |
bf77c646 | 227 | (calc-unary-op "phi" 'calcFunc-totient arg))) |
136211a9 EZ |
228 | |
229 | (defun calc-moebius (arg) | |
230 | (interactive "P") | |
231 | (calc-slow-wrapper | |
bf77c646 | 232 | (calc-unary-op "mu" 'calcFunc-moebius arg))) |
136211a9 EZ |
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) | |
bf77c646 | 260 | (list 'calcFunc-gcd a b)))) |
136211a9 EZ |
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) | |
bf77c646 | 266 | (list 'calcFunc-lcm a b)))) |
136211a9 EZ |
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))) | |
bf77c646 | 290 | (list 'vec u3 u1 u2))))) |
136211a9 EZ |
291 | |
292 | ||
293 | ;;; Factorial and related functions. | |
294 | ||
887fc3b8 | 295 | (defconst math-small-factorial-table |
2aad2a84 JB |
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"))) | |
887fc3b8 | 306 | |
136211a9 EZ |
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) | |
887fc3b8 | 315 | (aref math-small-factorial-table n) |
136211a9 EZ |
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)) | |
a1506d29 | 337 | (math-div |
136211a9 EZ |
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) | |
bf77c646 | 356 | (list 'calcFunc-fact n))))) |
136211a9 EZ |
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)) | |
bf77c646 | 371 | f)) |
136211a9 EZ |
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)))) | |
8d7498c1 | 397 | (list 'calcFunc-dfact n)))) |
136211a9 EZ |
398 | ((equal n '(var inf var-inf)) n) |
399 | (t (calc-record-why 'natnump n) | |
bf77c646 | 400 | (list 'calcFunc-dfact n)))) |
136211a9 EZ |
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) | |
bf77c646 | 407 | f)) |
136211a9 EZ |
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 | |
bf77c646 | 431 | (math-factorial-iter tm (1+ (- tn tm)) '(float 1 0))))))) |
136211a9 EZ |
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 | |
bf77c646 | 467 | (math-choose-float-iter tm n 1 1))))))) |
136211a9 EZ |
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)) | |
bf77c646 | 475 | c)) |
136211a9 EZ |
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)) | |
bf77c646 | 483 | c)) |
136211a9 EZ |
484 | |
485 | ||
486 | ;;; Stirling numbers. | |
487 | ||
488 | (defun calcFunc-stir1 (n m) | |
bf77c646 | 489 | (math-stirling-number n m 1)) |
136211a9 EZ |
490 | |
491 | (defun calcFunc-stir2 (n m) | |
bf77c646 | 492 | (math-stirling-number n m 0)) |
136211a9 | 493 | |
3132f345 | 494 | (defvar math-stirling-cache (vector [[1]] [[1]])) |
8d7498c1 JB |
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 | ||
136211a9 EZ |
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 | |
8d7498c1 JB |
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))) | |
136211a9 | 513 | row) |
8d7498c1 JB |
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))) | |
136211a9 EZ |
520 | (aset row 0 0) |
521 | (aset row i 1)))) | |
522 | (if (= k 1) | |
523 | (math-stirling-1 n m) | |
bf77c646 | 524 | (math-stirling-2 n m))))) |
136211a9 EZ |
525 | |
526 | (defun math-stirling-1 (n m) | |
8d7498c1 JB |
527 | (or (aref (aref math-stirling-local-cache n) m) |
528 | (aset (aref math-stirling-local-cache n) m | |
136211a9 | 529 | (math-add (math-stirling-1 (1- n) (1- m)) |
bf77c646 | 530 | (math-mul (- 1 n) (math-stirling-1 (1- n) m)))))) |
136211a9 EZ |
531 | |
532 | (defun math-stirling-2 (n m) | |
8d7498c1 JB |
533 | (or (aref (aref math-stirling-local-cache n) m) |
534 | (aset (aref math-stirling-local-cache n) m | |
136211a9 | 535 | (math-add (math-stirling-2 (1- n) (1- m)) |
bf77c646 | 536 | (math-mul m (math-stirling-2 (1- n) m)))))) |
136211a9 | 537 | |
3132f345 CW |
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) | |
136211a9 EZ |
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. | |
8d7498c1 | 546 | |
a59a1687 | 547 | (defvar var-RandSeed) |
8d7498c1 JB |
548 | (defvar math-random-cache nil) |
549 | (defvar math-gaussian-cache nil) | |
550 | ||
136211a9 | 551 | (defun math-init-random-base () |
a59a1687 | 552 | (if (and (boundp 'var-RandSeed) var-RandSeed) |
136211a9 EZ |
553 | (if (eq (car-safe var-RandSeed) 'vec) |
554 | nil | |
555 | (if (Math-integerp var-RandSeed) | |
556 | (let* ((seed (math-sub 161803 var-RandSeed)) | |
4b4b19bd JB |
557 | (mj (1+ (math-mod seed 1000000))) |
558 | (mk (1+ (math-mod (math-quotient seed 1000000) | |
559 | 1000000))) | |
136211a9 EZ |
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 | |
136211a9 EZ |
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. | |
8d7498c1 JB |
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)))))) | |
136211a9 | 586 | (setq math-last-RandSeed var-RandSeed |
bf77c646 | 587 | math-gaussian-cache nil)) |
136211a9 EZ |
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)) | |
bf77c646 | 600 | (logand (lsh (random) math-random-shift) 1023))) |
136211a9 EZ |
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. | |
a59a1687 | 606 | (defvar math-random-last) |
e4ebbf49 JB |
607 | (defun math-random-three-digit-number () |
608 | "Return a random three digit number." | |
a59a1687 JB |
609 | (let (i) |
610 | (or (and (boundp 'var-RandSeed) (eq var-RandSeed math-last-RandSeed)) | |
136211a9 EZ |
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))) | |
bf77c646 | 624 | math-random-last)) |
136211a9 EZ |
625 | |
626 | ;;; Produce an N-digit random integer. | |
627 | (defun math-random-digits (n) | |
e4ebbf49 JB |
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)))) | |
136211a9 EZ |
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) | |
bf77c646 | 643 | (- calc-internal-prec))) |
136211a9 EZ |
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))) | |
bf77c646 | 663 | (math-mul v2 fac)))))) |
136211a9 EZ |
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))) | |
bf77c646 | 700 | (t (math-reject-arg max 'realp)))) |
136211a9 EZ |
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)))) | |
bf77c646 | 755 | (t (math-reject-arg max 'realp)))) |
136211a9 EZ |
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))) | |
bf77c646 | 763 | (cons 'vec vec))) |
136211a9 EZ |
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)) | |
bf77c646 | 775 | (cons 'vec (nthcdr (- size n -1) vec)))) |
136211a9 EZ |
776 | |
777 | (defun math-member (x list) | |
778 | (while (and list (not (equal x (car list)))) | |
779 | (setq list (cdr list))) | |
bf77c646 | 780 | list) |
136211a9 EZ |
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%) | |
8d7498c1 JB |
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 | ||
136211a9 EZ |
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)) | |
4b4b19bd | 809 | ((Math-natnum-lessp n 8000000) |
136211a9 EZ |
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)) | |
40cddce8 JB |
822 | (t (let ((q n) (sum 0)) |
823 | (while (not (eq q 0)) | |
824 | (setq sum (% | |
825 | (+ | |
826 | sum | |
827 | (calcFunc-mod | |
4b4b19bd | 828 | q 1000000)) |
40cddce8 JB |
829 | 111111)) |
830 | (setq q | |
831 | (math-quotient | |
4b4b19bd | 832 | q 1000000))) |
136211a9 EZ |
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))) | |
bf77c646 | 880 | res)) |
136211a9 EZ |
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 | |
bf77c646 | 887 | 0)) |
136211a9 EZ |
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) | |
bf77c646 | 929 | (list 'calcFunc-prfac n)))) |
136211a9 EZ |
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) | |
bf77c646 | 952 | (list 'calcFunc-totient n))) |
136211a9 EZ |
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) | |
bf77c646 | 974 | (list 'calcFunc-moebius n))) |
136211a9 EZ |
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) | |
bf77c646 | 995 | (math-reject-arg n 'integerp)))) |
136211a9 EZ |
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) | |
bf77c646 | 1013 | (math-reject-arg n 'integerp)))) |
136211a9 EZ |
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)) | |
bf77c646 | 1026 | 2)) |
136211a9 | 1027 | |
43f34ccc | 1028 | (provide 'calc-comb) |
136211a9 | 1029 | |
cbee283d | 1030 | ;; arch-tag: 1d75ee9b-0815-42bd-a321-bb3dc001cc02 |
bf77c646 | 1031 | ;;; calc-comb.el ends here |