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3132f345 CW |
1 | ;;; calc-comb.el --- combinatoric functions for Calc |
2 | ||
bf77c646 | 3 | ;; Copyright (C) 1990, 1991, 1992, 1993, 2001 Free Software Foundation, Inc. |
3132f345 CW |
4 | |
5 | ;; Author: David Gillespie <daveg@synaptics.com> | |
a1506d29 | 6 | ;; Maintainers: D. Goel <deego@gnufans.org> |
6e1c888a | 7 | ;; Colin Walters <walters@debian.org> |
136211a9 EZ |
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 | ||
3132f345 | 26 | ;;; Commentary: |
136211a9 | 27 | |
3132f345 | 28 | ;;; Code: |
136211a9 EZ |
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 | ||
3132f345 CW |
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]) | |
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84 | |
85 | ;;; Combinatorics | |
86 | ||
87 | (defun calc-gcd (arg) | |
88 | (interactive "P") | |
89 | (calc-slow-wrapper | |
bf77c646 | 90 | (calc-binary-op "gcd" 'calcFunc-gcd arg))) |
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91 | |
92 | (defun calc-lcm (arg) | |
93 | (interactive "P") | |
94 | (calc-slow-wrapper | |
bf77c646 | 95 | (calc-binary-op "lcm" 'calcFunc-lcm arg))) |
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96 | |
97 | (defun calc-extended-gcd () | |
98 | (interactive) | |
99 | (calc-slow-wrapper | |
bf77c646 | 100 | (calc-enter-result 2 "egcd" (cons 'calcFunc-egcd (calc-top-list-n 2))))) |
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101 | |
102 | (defun calc-factorial (arg) | |
103 | (interactive "P") | |
104 | (calc-slow-wrapper | |
bf77c646 | 105 | (calc-unary-op "fact" 'calcFunc-fact arg))) |
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106 | |
107 | (defun calc-gamma (arg) | |
108 | (interactive "P") | |
109 | (calc-slow-wrapper | |
bf77c646 | 110 | (calc-unary-op "gmma" 'calcFunc-gamma arg))) |
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111 | |
112 | (defun calc-double-factorial (arg) | |
113 | (interactive "P") | |
114 | (calc-slow-wrapper | |
bf77c646 | 115 | (calc-unary-op "dfac" 'calcFunc-dfact arg))) |
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116 | |
117 | (defun calc-choose (arg) | |
118 | (interactive "P") | |
119 | (calc-slow-wrapper | |
120 | (if (calc-is-hyperbolic) | |
121 | (calc-binary-op "perm" 'calcFunc-perm arg) | |
bf77c646 | 122 | (calc-binary-op "chos" 'calcFunc-choose arg)))) |
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123 | |
124 | (defun calc-perm (arg) | |
125 | (interactive "P") | |
126 | (calc-hyperbolic-func) | |
bf77c646 | 127 | (calc-choose arg)) |
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128 | |
129 | (defvar calc-last-random-limit '(float 1 0)) | |
130 | (defun calc-random (n) | |
131 | (interactive "P") | |
132 | (calc-slow-wrapper | |
133 | (if n | |
134 | (calc-enter-result 0 "rand" (list 'calcFunc-random | |
135 | (calc-get-random-limit | |
136 | (prefix-numeric-value n)))) | |
137 | (calc-enter-result 1 "rand" (list 'calcFunc-random | |
138 | (calc-get-random-limit | |
bf77c646 | 139 | (calc-top-n 1))))))) |
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140 | |
141 | (defun calc-get-random-limit (val) | |
142 | (if (eq val 0) | |
143 | calc-last-random-limit | |
bf77c646 | 144 | (setq calc-last-random-limit val))) |
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145 | |
146 | (defun calc-rrandom () | |
147 | (interactive) | |
148 | (calc-slow-wrapper | |
149 | (setq calc-last-random-limit '(float 1 0)) | |
bf77c646 | 150 | (calc-enter-result 0 "rand" (list 'calcFunc-random '(float 1 0))))) |
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151 | |
152 | (defun calc-random-again (arg) | |
153 | (interactive "p") | |
154 | (calc-slow-wrapper | |
155 | (while (>= (setq arg (1- arg)) 0) | |
156 | (calc-enter-result 0 "rand" (list 'calcFunc-random | |
bf77c646 | 157 | calc-last-random-limit))))) |
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158 | |
159 | (defun calc-shuffle (n) | |
160 | (interactive "P") | |
161 | (calc-slow-wrapper | |
162 | (if n | |
163 | (calc-enter-result 1 "shuf" (list 'calcFunc-shuffle | |
164 | (prefix-numeric-value n) | |
165 | (calc-get-random-limit | |
166 | (calc-top-n 1)))) | |
167 | (calc-enter-result 2 "shuf" (list 'calcFunc-shuffle | |
168 | (calc-top-n 1) | |
169 | (calc-get-random-limit | |
bf77c646 | 170 | (calc-top-n 2))))))) |
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171 | |
172 | (defun calc-report-prime-test (res) | |
173 | (cond ((eq (car res) t) | |
174 | (calc-record-message "prim" "Prime (guaranteed)")) | |
175 | ((eq (car res) nil) | |
176 | (if (cdr res) | |
177 | (if (eq (nth 1 res) 'unknown) | |
178 | (calc-record-message | |
179 | "prim" "Non-prime (factors unknown)") | |
180 | (calc-record-message | |
181 | "prim" "Non-prime (%s is a factor)" | |
182 | (math-format-number (nth 1 res)))) | |
183 | (calc-record-message "prim" "Non-prime"))) | |
184 | (t | |
185 | (calc-record-message | |
186 | "prim" "Probably prime (%d iters; %s%% chance of error)" | |
187 | (nth 1 res) | |
188 | (let ((calc-float-format '(fix 2))) | |
bf77c646 | 189 | (math-format-number (nth 2 res))))))) |
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190 | |
191 | (defun calc-prime-test (iters) | |
192 | (interactive "p") | |
193 | (calc-slow-wrapper | |
194 | (let* ((n (calc-top-n 1)) | |
195 | (res (math-prime-test n iters))) | |
bf77c646 | 196 | (calc-report-prime-test res)))) |
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197 | |
198 | (defun calc-next-prime (iters) | |
199 | (interactive "p") | |
200 | (calc-slow-wrapper | |
201 | (let ((calc-verbose-nextprime t)) | |
202 | (if (calc-is-inverse) | |
203 | (calc-enter-result 1 "prvp" (list 'calcFunc-prevprime | |
204 | (calc-top-n 1) (math-abs iters))) | |
205 | (calc-enter-result 1 "nxtp" (list 'calcFunc-nextprime | |
bf77c646 | 206 | (calc-top-n 1) (math-abs iters))))))) |
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207 | |
208 | (defun calc-prev-prime (iters) | |
209 | (interactive "p") | |
210 | (calc-invert-func) | |
bf77c646 | 211 | (calc-next-prime iters)) |
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212 | |
213 | (defun calc-prime-factors (iters) | |
214 | (interactive "p") | |
215 | (calc-slow-wrapper | |
216 | (let ((res (calcFunc-prfac (calc-top-n 1)))) | |
217 | (if (not math-prime-factors-finished) | |
218 | (calc-record-message "pfac" "Warning: May not be fully factored")) | |
bf77c646 | 219 | (calc-enter-result 1 "pfac" res)))) |
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220 | |
221 | (defun calc-totient (arg) | |
222 | (interactive "P") | |
223 | (calc-slow-wrapper | |
bf77c646 | 224 | (calc-unary-op "phi" 'calcFunc-totient arg))) |
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225 | |
226 | (defun calc-moebius (arg) | |
227 | (interactive "P") | |
228 | (calc-slow-wrapper | |
bf77c646 | 229 | (calc-unary-op "mu" 'calcFunc-moebius arg))) |
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230 | |
231 | ||
232 | (defun calcFunc-gcd (a b) | |
233 | (if (Math-messy-integerp a) | |
234 | (setq a (math-trunc a))) | |
235 | (if (Math-messy-integerp b) | |
236 | (setq b (math-trunc b))) | |
237 | (cond ((and (Math-integerp a) (Math-integerp b)) | |
238 | (math-gcd a b)) | |
239 | ((Math-looks-negp a) | |
240 | (calcFunc-gcd (math-neg a) b)) | |
241 | ((Math-looks-negp b) | |
242 | (calcFunc-gcd a (math-neg b))) | |
243 | ((Math-zerop a) b) | |
244 | ((Math-zerop b) a) | |
245 | ((and (Math-ratp a) | |
246 | (Math-ratp b)) | |
247 | (math-make-frac (math-gcd (if (eq (car-safe a) 'frac) (nth 1 a) a) | |
248 | (if (eq (car-safe b) 'frac) (nth 1 b) b)) | |
249 | (calcFunc-lcm | |
250 | (if (eq (car-safe a) 'frac) (nth 2 a) 1) | |
251 | (if (eq (car-safe b) 'frac) (nth 2 b) 1)))) | |
252 | ((not (Math-integerp a)) | |
253 | (calc-record-why 'integerp a) | |
254 | (list 'calcFunc-gcd a b)) | |
255 | (t | |
256 | (calc-record-why 'integerp b) | |
bf77c646 | 257 | (list 'calcFunc-gcd a b)))) |
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258 | |
259 | (defun calcFunc-lcm (a b) | |
260 | (let ((g (calcFunc-gcd a b))) | |
261 | (if (Math-numberp g) | |
262 | (math-div (math-mul a b) g) | |
bf77c646 | 263 | (list 'calcFunc-lcm a b)))) |
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264 | |
265 | (defun calcFunc-egcd (a b) ; Knuth section 4.5.2 | |
266 | (cond | |
267 | ((not (Math-integerp a)) | |
268 | (if (Math-messy-integerp a) | |
269 | (calcFunc-egcd (math-trunc a) b) | |
270 | (calc-record-why 'integerp a) | |
271 | (list 'calcFunc-egcd a b))) | |
272 | ((not (Math-integerp b)) | |
273 | (if (Math-messy-integerp b) | |
274 | (calcFunc-egcd a (math-trunc b)) | |
275 | (calc-record-why 'integerp b) | |
276 | (list 'calcFunc-egcd a b))) | |
277 | (t | |
278 | (let ((u1 1) (u2 0) (u3 a) | |
279 | (v1 0) (v2 1) (v3 b) | |
280 | t1 t2 q) | |
281 | (while (not (eq v3 0)) | |
282 | (setq q (math-idivmod u3 v3) | |
283 | t1 (math-sub u1 (math-mul v1 (car q))) | |
284 | t2 (math-sub u2 (math-mul v2 (car q))) | |
285 | u1 v1 u2 v2 u3 v3 | |
286 | v1 t1 v2 t2 v3 (cdr q))) | |
bf77c646 | 287 | (list 'vec u3 u1 u2))))) |
136211a9 EZ |
288 | |
289 | ||
290 | ;;; Factorial and related functions. | |
291 | ||
292 | (defun calcFunc-fact (n) ; [I I] [F F] [Public] | |
293 | (let (temp) | |
294 | (cond ((Math-integer-negp n) | |
295 | (if calc-infinite-mode | |
296 | '(var uinf var-uinf) | |
297 | (math-reject-arg n 'range))) | |
298 | ((integerp n) | |
299 | (if (<= n 20) | |
300 | (aref '[1 1 2 6 24 120 720 5040 40320 362880 | |
301 | (bigpos 800 628 3) (bigpos 800 916 39) | |
302 | (bigpos 600 1 479) (bigpos 800 20 227 6) | |
303 | (bigpos 200 291 178 87) (bigpos 0 368 674 307 1) | |
304 | (bigpos 0 888 789 922 20) (bigpos 0 96 428 687 355) | |
305 | (bigpos 0 728 705 373 402 6) | |
306 | (bigpos 0 832 408 100 645 121) | |
307 | (bigpos 0 640 176 8 902 432 2)] n) | |
308 | (math-factorial-iter (1- n) 2 1))) | |
309 | ((and (math-messy-integerp n) | |
310 | (Math-lessp n 100)) | |
311 | (math-inexact-result) | |
312 | (setq temp (math-trunc n)) | |
313 | (if (>= temp 0) | |
314 | (if (<= temp 20) | |
315 | (math-float (calcFunc-fact temp)) | |
316 | (math-with-extra-prec 1 | |
317 | (math-factorial-iter (1- temp) 2 '(float 1 0)))) | |
318 | (math-reject-arg n 'range))) | |
319 | ((math-numberp n) | |
320 | (let* ((q (math-quarter-integer n)) | |
321 | (tn (and q (Math-lessp n 1000) (Math-lessp -1000 n) | |
322 | (1+ (math-floor n))))) | |
323 | (cond ((and tn (= q 2) | |
324 | (or calc-symbolic-mode (< (math-abs tn) 20))) | |
325 | (let ((q (if (< tn 0) | |
326 | (math-div | |
327 | (math-pow -2 (- tn)) | |
328 | (math-double-factorial-iter (* -2 tn) 3 1 2)) | |
a1506d29 | 329 | (math-div |
136211a9 EZ |
330 | (math-double-factorial-iter (* 2 tn) 3 1 2) |
331 | (math-pow 2 tn))))) | |
332 | (math-mul q (if calc-symbolic-mode | |
333 | (list 'calcFunc-sqrt '(var pi var-pi)) | |
334 | (math-sqrt-pi))))) | |
335 | ((and tn (>= tn 0) (< tn 20) | |
336 | (memq q '(1 3))) | |
337 | (math-inexact-result) | |
338 | (math-div | |
339 | (math-mul (math-double-factorial-iter (* 4 tn) q 1 4) | |
340 | (if (= q 1) (math-gamma-1q) (math-gamma-3q))) | |
341 | (math-pow 4 tn))) | |
342 | (t | |
343 | (math-inexact-result) | |
344 | (math-with-extra-prec 3 | |
345 | (math-gammap1-raw (math-float n))))))) | |
346 | ((equal n '(var inf var-inf)) n) | |
347 | (t (calc-record-why 'numberp n) | |
bf77c646 | 348 | (list 'calcFunc-fact n))))) |
136211a9 EZ |
349 | |
350 | (math-defcache math-gamma-1q nil | |
351 | (math-with-extra-prec 3 | |
352 | (math-gammap1-raw '(float -75 -2)))) | |
353 | ||
354 | (math-defcache math-gamma-3q nil | |
355 | (math-with-extra-prec 3 | |
356 | (math-gammap1-raw '(float -25 -2)))) | |
357 | ||
358 | (defun math-factorial-iter (count n f) | |
359 | (if (= (% n 5) 1) | |
360 | (math-working (format "factorial(%d)" (1- n)) f)) | |
361 | (if (> count 0) | |
362 | (math-factorial-iter (1- count) (1+ n) (math-mul n f)) | |
bf77c646 | 363 | f)) |
136211a9 EZ |
364 | |
365 | (defun calcFunc-dfact (n) ; [I I] [F F] [Public] | |
366 | (cond ((Math-integer-negp n) | |
367 | (if (math-oddp n) | |
368 | (if (eq n -1) | |
369 | 1 | |
370 | (math-div (if (eq (math-mod n 4) 3) 1 -1) | |
371 | (calcFunc-dfact (math-sub -2 n)))) | |
372 | (list 'calcFunc-dfact n))) | |
373 | ((Math-zerop n) 1) | |
374 | ((integerp n) (math-double-factorial-iter n (+ 2 (% n 2)) 1 2)) | |
375 | ((math-messy-integerp n) | |
376 | (let ((temp (math-trunc n))) | |
377 | (math-inexact-result) | |
378 | (if (natnump temp) | |
379 | (if (Math-lessp temp 200) | |
380 | (math-with-extra-prec 1 | |
381 | (math-double-factorial-iter temp (+ 2 (% temp 2)) | |
382 | '(float 1 0) 2)) | |
383 | (let* ((half (math-div2 temp)) | |
384 | (even (math-mul (math-pow 2 half) | |
385 | (calcFunc-fact (math-float half))))) | |
386 | (if (math-evenp temp) | |
387 | even | |
388 | (math-div (calcFunc-fact n) even)))) | |
389 | (list 'calcFunc-dfact max)))) | |
390 | ((equal n '(var inf var-inf)) n) | |
391 | (t (calc-record-why 'natnump n) | |
bf77c646 | 392 | (list 'calcFunc-dfact n)))) |
136211a9 EZ |
393 | |
394 | (defun math-double-factorial-iter (max n f step) | |
395 | (if (< (% n 12) step) | |
396 | (math-working (format "dfact(%d)" (- n step)) f)) | |
397 | (if (<= n max) | |
398 | (math-double-factorial-iter max (+ n step) (math-mul n f) step) | |
bf77c646 | 399 | f)) |
136211a9 EZ |
400 | |
401 | (defun calcFunc-perm (n m) ; [I I I] [F F F] [Public] | |
402 | (cond ((and (integerp n) (integerp m) (<= m n) (>= m 0)) | |
403 | (math-factorial-iter m (1+ (- n m)) 1)) | |
404 | ((or (not (math-num-integerp n)) | |
405 | (and (math-messy-integerp n) (Math-lessp 100 n)) | |
406 | (not (math-num-integerp m)) | |
407 | (and (math-messy-integerp m) (Math-lessp 100 m))) | |
408 | (or (math-realp n) (equal n '(var inf var-inf)) | |
409 | (math-reject-arg n 'realp)) | |
410 | (or (math-realp m) (equal m '(var inf var-inf)) | |
411 | (math-reject-arg m 'realp)) | |
412 | (and (math-num-integerp n) (math-negp n) (math-reject-arg n 'range)) | |
413 | (and (math-num-integerp m) (math-negp m) (math-reject-arg m 'range)) | |
414 | (math-div (calcFunc-fact n) (calcFunc-fact (math-sub n m)))) | |
415 | (t | |
416 | (let ((tn (math-trunc n)) | |
417 | (tm (math-trunc m))) | |
418 | (math-inexact-result) | |
419 | (or (integerp tn) (math-reject-arg tn 'fixnump)) | |
420 | (or (integerp tm) (math-reject-arg tm 'fixnump)) | |
421 | (or (and (<= tm tn) (>= tm 0)) (math-reject-arg tm 'range)) | |
422 | (math-with-extra-prec 1 | |
bf77c646 | 423 | (math-factorial-iter tm (1+ (- tn tm)) '(float 1 0))))))) |
136211a9 EZ |
424 | |
425 | (defun calcFunc-choose (n m) ; [I I I] [F F F] [Public] | |
426 | (cond ((and (integerp n) (integerp m) (<= m n) (>= m 0)) | |
427 | (if (> m (/ n 2)) | |
428 | (math-choose-iter (- n m) n 1 1) | |
429 | (math-choose-iter m n 1 1))) | |
430 | ((not (math-realp n)) | |
431 | (math-reject-arg n 'realp)) | |
432 | ((not (math-realp m)) | |
433 | (math-reject-arg m 'realp)) | |
434 | ((not (math-num-integerp m)) | |
435 | (if (and (math-num-integerp n) (math-negp n)) | |
436 | (list 'calcFunc-choose n m) | |
437 | (math-div (calcFunc-fact (math-float n)) | |
438 | (math-mul (calcFunc-fact m) | |
439 | (calcFunc-fact (math-sub n m)))))) | |
440 | ((math-negp m) 0) | |
441 | ((math-negp n) | |
442 | (let ((val (calcFunc-choose (math-add (math-add n m) -1) m))) | |
443 | (if (math-evenp (math-trunc m)) | |
444 | val | |
445 | (math-neg val)))) | |
446 | ((and (math-num-integerp n) | |
447 | (Math-lessp n m)) | |
448 | 0) | |
449 | (t | |
450 | (math-inexact-result) | |
451 | (let ((tm (math-trunc m))) | |
452 | (or (integerp tm) (math-reject-arg tm 'fixnump)) | |
453 | (if (> tm 100) | |
454 | (math-div (calcFunc-fact (math-float n)) | |
455 | (math-mul (calcFunc-fact (math-float m)) | |
456 | (calcFunc-fact (math-float | |
457 | (math-sub n m))))) | |
458 | (math-with-extra-prec 1 | |
bf77c646 | 459 | (math-choose-float-iter tm n 1 1))))))) |
136211a9 EZ |
460 | |
461 | (defun math-choose-iter (m n i c) | |
462 | (if (and (= (% i 5) 1) (> i 5)) | |
463 | (math-working (format "choose(%d)" (1- i)) c)) | |
464 | (if (<= i m) | |
465 | (math-choose-iter m (1- n) (1+ i) | |
466 | (math-quotient (math-mul c n) i)) | |
bf77c646 | 467 | c)) |
136211a9 EZ |
468 | |
469 | (defun math-choose-float-iter (count n i c) | |
470 | (if (= (% i 5) 1) | |
471 | (math-working (format "choose(%d)" (1- i)) c)) | |
472 | (if (> count 0) | |
473 | (math-choose-float-iter (1- count) (math-sub n 1) (1+ i) | |
474 | (math-div (math-mul c n) i)) | |
bf77c646 | 475 | c)) |
136211a9 EZ |
476 | |
477 | ||
478 | ;;; Stirling numbers. | |
479 | ||
480 | (defun calcFunc-stir1 (n m) | |
bf77c646 | 481 | (math-stirling-number n m 1)) |
136211a9 EZ |
482 | |
483 | (defun calcFunc-stir2 (n m) | |
bf77c646 | 484 | (math-stirling-number n m 0)) |
136211a9 | 485 | |
3132f345 | 486 | (defvar math-stirling-cache (vector [[1]] [[1]])) |
136211a9 EZ |
487 | (defun math-stirling-number (n m k) |
488 | (or (math-num-natnump n) (math-reject-arg n 'natnump)) | |
489 | (or (math-num-natnump m) (math-reject-arg m 'natnump)) | |
490 | (if (consp n) (setq n (math-trunc n))) | |
491 | (or (integerp n) (math-reject-arg n 'fixnump)) | |
492 | (if (consp m) (setq m (math-trunc m))) | |
493 | (or (integerp m) (math-reject-arg m 'fixnump)) | |
494 | (if (< n m) | |
495 | 0 | |
496 | (let ((cache (aref math-stirling-cache k))) | |
497 | (while (<= (length cache) n) | |
498 | (let ((i (1- (length cache))) | |
499 | row) | |
500 | (setq cache (vconcat cache (make-vector (length cache) nil))) | |
501 | (aset math-stirling-cache k cache) | |
502 | (while (< (setq i (1+ i)) (length cache)) | |
503 | (aset cache i (setq row (make-vector (1+ i) nil))) | |
504 | (aset row 0 0) | |
505 | (aset row i 1)))) | |
506 | (if (= k 1) | |
507 | (math-stirling-1 n m) | |
bf77c646 | 508 | (math-stirling-2 n m))))) |
136211a9 EZ |
509 | |
510 | (defun math-stirling-1 (n m) | |
511 | (or (aref (aref cache n) m) | |
512 | (aset (aref cache n) m | |
513 | (math-add (math-stirling-1 (1- n) (1- m)) | |
bf77c646 | 514 | (math-mul (- 1 n) (math-stirling-1 (1- n) m)))))) |
136211a9 EZ |
515 | |
516 | (defun math-stirling-2 (n m) | |
517 | (or (aref (aref cache n) m) | |
518 | (aset (aref cache n) m | |
519 | (math-add (math-stirling-2 (1- n) (1- m)) | |
bf77c646 | 520 | (math-mul m (math-stirling-2 (1- n) m)))))) |
136211a9 | 521 | |
3132f345 CW |
522 | (defvar math-random-table nil) |
523 | (defvar math-last-RandSeed nil) | |
524 | (defvar math-random-ptr1 nil) | |
525 | (defvar math-random-ptr2 nil) | |
526 | (defvar math-random-shift nil) | |
136211a9 EZ |
527 | |
528 | ;;; Produce a random 10-bit integer, with (random) if no seed provided, | |
529 | ;;; or else with Numerical Recipes algorithm ran3 / Knuth 3.2.2-A. | |
530 | (defun math-init-random-base () | |
531 | (if (and (boundp 'var-RandSeed) var-RandSeed) | |
532 | (if (eq (car-safe var-RandSeed) 'vec) | |
533 | nil | |
534 | (if (Math-integerp var-RandSeed) | |
535 | (let* ((seed (math-sub 161803 var-RandSeed)) | |
536 | (mj (1+ (math-mod seed '(bigpos 0 0 1)))) | |
537 | (mk (1+ (math-mod (math-quotient seed '(bigpos 0 0 1)) | |
538 | '(bigpos 0 0 1)))) | |
539 | (i 0)) | |
540 | (setq math-random-table (cons 'vec (make-list 55 mj))) | |
541 | (while (<= (setq i (1+ i)) 54) | |
542 | (let* ((ii (% (* i 21) 55)) | |
543 | (p (nthcdr ii math-random-table))) | |
544 | (setcar p mk) | |
545 | (setq mk (- mj mk) | |
546 | mj (car p))))) | |
547 | (math-reject-arg var-RandSeed "*RandSeed must be an integer")) | |
548 | (setq var-RandSeed (list 'vec var-RandSeed) | |
549 | math-random-ptr1 math-random-table | |
550 | math-random-cache nil | |
551 | math-random-ptr2 (nthcdr 31 math-random-table)) | |
552 | (let ((i 200)) | |
553 | (while (> (setq i (1- i)) 0) | |
554 | (math-random-base)))) | |
555 | (random t) | |
556 | (setq var-RandSeed nil | |
557 | math-random-cache nil | |
558 | i 0 | |
559 | math-random-shift -4) ; assume RAND_MAX >= 16383 | |
560 | ;; This exercises the random number generator and also helps | |
561 | ;; deduce a better value for RAND_MAX. | |
562 | (while (< (setq i (1+ i)) 30) | |
563 | (if (> (lsh (math-abs (random)) math-random-shift) 4095) | |
564 | (setq math-random-shift (1- math-random-shift))))) | |
565 | (setq math-last-RandSeed var-RandSeed | |
bf77c646 | 566 | math-gaussian-cache nil)) |
136211a9 EZ |
567 | |
568 | (defun math-random-base () | |
569 | (if var-RandSeed | |
570 | (progn | |
571 | (setq math-random-ptr1 (or (cdr math-random-ptr1) | |
572 | (cdr math-random-table)) | |
573 | math-random-ptr2 (or (cdr math-random-ptr2) | |
574 | (cdr math-random-table))) | |
575 | (logand (lsh (setcar math-random-ptr1 | |
576 | (logand (- (car math-random-ptr1) | |
577 | (car math-random-ptr2)) 524287)) | |
578 | -6) 1023)) | |
bf77c646 | 579 | (logand (lsh (random) math-random-shift) 1023))) |
136211a9 EZ |
580 | |
581 | ||
582 | ;;; Produce a random digit in the range 0..999. | |
583 | ;;; Avoid various pitfalls that may lurk in the built-in (random) function! | |
584 | ;;; Shuffling algorithm from Numerical Recipes, section 7.1. | |
585 | (defun math-random-digit () | |
586 | (let (i) | |
587 | (or (and (boundp 'var-RandSeed) (eq var-RandSeed math-last-RandSeed)) | |
588 | (math-init-random-base)) | |
589 | (or math-random-cache | |
590 | (progn | |
591 | (setq math-random-last (math-random-base) | |
592 | math-random-cache (make-vector 13 nil) | |
593 | i -1) | |
594 | (while (< (setq i (1+ i)) 13) | |
595 | (aset math-random-cache i (math-random-base))))) | |
596 | (while (progn | |
597 | (setq i (/ math-random-last 79) ; 0 <= i < 13 | |
598 | math-random-last (aref math-random-cache i)) | |
599 | (aset math-random-cache i (math-random-base)) | |
600 | (>= math-random-last 1000))) | |
bf77c646 | 601 | math-random-last)) |
136211a9 EZ |
602 | (setq math-random-cache nil) |
603 | ||
604 | ;;; Produce an N-digit random integer. | |
605 | (defun math-random-digits (n) | |
606 | (cond ((<= n 6) | |
607 | (math-scale-right (+ (* (math-random-digit) 1000) (math-random-digit)) | |
608 | (- 6 n))) | |
609 | (t (let* ((slop (% (- 900003 n) 3)) | |
610 | (i (/ (+ n slop) 3)) | |
611 | (digs nil)) | |
612 | (while (> i 0) | |
613 | (setq digs (cons (math-random-digit) digs) | |
614 | i (1- i))) | |
615 | (math-normalize (math-scale-right (cons 'bigpos digs) | |
bf77c646 | 616 | slop)))))) |
136211a9 EZ |
617 | |
618 | ;;; Produce a uniformly-distributed random float 0 <= N < 1. | |
619 | (defun math-random-float () | |
620 | (math-make-float (math-random-digits calc-internal-prec) | |
bf77c646 | 621 | (- calc-internal-prec))) |
136211a9 EZ |
622 | |
623 | ;;; Produce a Gaussian-distributed random float with mean=0, sigma=1. | |
624 | (defun math-gaussian-float () | |
625 | (math-with-extra-prec 2 | |
626 | (if (and math-gaussian-cache | |
627 | (= (car math-gaussian-cache) calc-internal-prec)) | |
628 | (prog1 | |
629 | (cdr math-gaussian-cache) | |
630 | (setq math-gaussian-cache nil)) | |
631 | (let* ((v1 (math-add (math-mul (math-random-float) 2) -1)) | |
632 | (v2 (math-add (math-mul (math-random-float) 2) -1)) | |
633 | (r (math-add (math-sqr v1) (math-sqr v2)))) | |
634 | (while (or (not (Math-lessp r 1)) (math-zerop r)) | |
635 | (setq v1 (math-add (math-mul (math-random-float) 2) -1) | |
636 | v2 (math-add (math-mul (math-random-float) 2) -1) | |
637 | r (math-add (math-sqr v1) (math-sqr v2)))) | |
638 | (let ((fac (math-sqrt (math-mul (math-div (calcFunc-ln r) r) -2)))) | |
639 | (setq math-gaussian-cache (cons calc-internal-prec | |
640 | (math-mul v1 fac))) | |
bf77c646 | 641 | (math-mul v2 fac)))))) |
136211a9 EZ |
642 | (setq math-gaussian-cache nil) |
643 | ||
644 | ;;; Produce a random integer or real 0 <= N < MAX. | |
645 | (defun calcFunc-random (max) | |
646 | (cond ((Math-zerop max) | |
647 | (math-gaussian-float)) | |
648 | ((Math-integerp max) | |
649 | (let* ((digs (math-numdigs max)) | |
650 | (r (math-random-digits (+ digs 3)))) | |
651 | (math-mod r max))) | |
652 | ((Math-realp max) | |
653 | (math-mul (math-random-float) max)) | |
654 | ((and (eq (car max) 'intv) (math-constp max) | |
655 | (Math-lessp (nth 2 max) (nth 3 max))) | |
656 | (if (math-floatp max) | |
657 | (let ((val (math-add (math-mul (math-random-float) | |
658 | (math-sub (nth 3 max) (nth 2 max))) | |
659 | (nth 2 max)))) | |
660 | (if (or (and (memq (nth 1 max) '(0 1)) ; almost not worth | |
661 | (Math-equal val (nth 2 max))) ; checking! | |
662 | (and (memq (nth 1 max) '(0 2)) | |
663 | (Math-equal val (nth 3 max)))) | |
664 | (calcFunc-random max) | |
665 | val)) | |
666 | (let ((lo (if (memq (nth 1 max) '(0 1)) | |
667 | (math-add (nth 2 max) 1) (nth 2 max))) | |
668 | (hi (if (memq (nth 1 max) '(1 3)) | |
669 | (math-add (nth 3 max) 1) (nth 3 max)))) | |
670 | (if (Math-lessp lo hi) | |
671 | (math-add (calcFunc-random (math-sub hi lo)) lo) | |
672 | (math-reject-arg max "*Empty interval"))))) | |
673 | ((eq (car max) 'vec) | |
674 | (if (cdr max) | |
675 | (nth (1+ (calcFunc-random (1- (length max)))) max) | |
676 | (math-reject-arg max "*Empty list"))) | |
677 | ((and (eq (car max) 'sdev) (math-constp max) (Math-realp (nth 1 max))) | |
678 | (math-add (math-mul (math-gaussian-float) (nth 2 max)) (nth 1 max))) | |
bf77c646 | 679 | (t (math-reject-arg max 'realp)))) |
136211a9 EZ |
680 | |
681 | ;;; Choose N objects at random from the set MAX without duplicates. | |
682 | (defun calcFunc-shuffle (n &optional max) | |
683 | (or max (setq max n n -1)) | |
684 | (or (and (Math-num-integerp n) | |
685 | (or (natnump (setq n (math-trunc n))) (eq n -1))) | |
686 | (math-reject-arg n 'integerp)) | |
687 | (cond ((or (math-zerop max) | |
688 | (math-floatp max) | |
689 | (eq (car-safe max) 'sdev)) | |
690 | (if (< n 0) | |
691 | (math-reject-arg n 'natnump) | |
692 | (math-simple-shuffle n max))) | |
693 | ((and (<= n 1) (>= n 0)) | |
694 | (math-simple-shuffle n max)) | |
695 | ((and (eq (car-safe max) 'intv) (math-constp max)) | |
696 | (let ((num (math-add (math-sub (nth 3 max) (nth 2 max)) | |
697 | (cdr (assq (nth 1 max) | |
698 | '((0 . -1) (1 . 0) | |
699 | (2 . 0) (3 . 1)))))) | |
700 | (min (math-add (nth 2 max) (if (memq (nth 1 max) '(0 1)) | |
701 | 1 0)))) | |
702 | (if (< n 0) (setq n num)) | |
703 | (or (math-posp num) (math-reject-arg max 'range)) | |
704 | (and (Math-lessp num n) (math-reject-arg n 'range)) | |
705 | (if (Math-lessp n (math-quotient num 3)) | |
706 | (math-simple-shuffle n max) | |
707 | (if (> (* n 4) (* num 3)) | |
708 | (math-add (math-sub min 1) | |
709 | (math-shuffle-list n num (calcFunc-index num))) | |
710 | (let ((tot 0) | |
711 | (m 0) | |
712 | (vec nil)) | |
713 | (while (< m n) | |
714 | (if (< (calcFunc-random (- num tot)) (- n m)) | |
715 | (setq vec (cons (math-add min tot) vec) | |
716 | m (1+ m))) | |
717 | (setq tot (1+ tot))) | |
718 | (math-shuffle-list n n (cons 'vec vec))))))) | |
719 | ((eq (car-safe max) 'vec) | |
720 | (let ((size (1- (length max)))) | |
721 | (if (< n 0) (setq n size)) | |
722 | (if (and (> n (/ size 2)) (<= n size)) | |
723 | (math-shuffle-list n size (copy-sequence max)) | |
724 | (let* ((vals (calcFunc-shuffle | |
725 | n (list 'intv 3 1 (1- (length max))))) | |
726 | (p vals)) | |
727 | (while (setq p (cdr p)) | |
728 | (setcar p (nth (car p) max))) | |
729 | vals)))) | |
730 | ((math-integerp max) | |
731 | (if (math-posp max) | |
732 | (calcFunc-shuffle n (list 'intv 2 0 max)) | |
733 | (calcFunc-shuffle n (list 'intv 1 max 0)))) | |
bf77c646 | 734 | (t (math-reject-arg max 'realp)))) |
136211a9 EZ |
735 | |
736 | (defun math-simple-shuffle (n max) | |
737 | (let ((vec nil) | |
738 | val) | |
739 | (while (>= (setq n (1- n)) 0) | |
740 | (while (math-member (setq val (calcFunc-random max)) vec)) | |
741 | (setq vec (cons val vec))) | |
bf77c646 | 742 | (cons 'vec vec))) |
136211a9 EZ |
743 | |
744 | (defun math-shuffle-list (n size vec) | |
745 | (let ((j size) | |
746 | k temp | |
747 | (p vec)) | |
748 | (while (cdr (setq p (cdr p))) | |
749 | (setq k (calcFunc-random j) | |
750 | j (1- j) | |
751 | temp (nth k p)) | |
752 | (setcar (nthcdr k p) (car p)) | |
753 | (setcar p temp)) | |
bf77c646 | 754 | (cons 'vec (nthcdr (- size n -1) vec)))) |
136211a9 EZ |
755 | |
756 | (defun math-member (x list) | |
757 | (while (and list (not (equal x (car list)))) | |
758 | (setq list (cdr list))) | |
bf77c646 | 759 | list) |
136211a9 EZ |
760 | |
761 | ||
762 | ;;; Check if the integer N is prime. [X I] | |
763 | ;;; Return (nil) if non-prime, | |
764 | ;;; (nil N) if non-prime with known factor N, | |
765 | ;;; (nil unknown) if non-prime with no known factors, | |
766 | ;;; (t) if prime, | |
767 | ;;; (maybe N P) if probably prime (after N iters with probability P%) | |
768 | (defun math-prime-test (n iters) | |
769 | (if (and (Math-vectorp n) (cdr n)) | |
770 | (setq n (nth (1- (length n)) n))) | |
771 | (if (Math-messy-integerp n) | |
772 | (setq n (math-trunc n))) | |
773 | (let ((res)) | |
774 | (while (> iters 0) | |
775 | (setq res | |
776 | (cond ((and (integerp n) (<= n 5003)) | |
777 | (list (= (math-next-small-prime n) n))) | |
778 | ((not (Math-integerp n)) | |
779 | (error "Argument must be an integer")) | |
780 | ((Math-integer-negp n) | |
781 | '(nil)) | |
782 | ((Math-natnum-lessp n '(bigpos 0 0 8)) | |
783 | (setq n (math-fixnum n)) | |
784 | (let ((i -1) v) | |
785 | (while (and (> (% n (setq v (aref math-primes-table | |
786 | (setq i (1+ i))))) | |
787 | 0) | |
788 | (< (* v v) n))) | |
789 | (if (= (% n v) 0) | |
790 | (list nil v) | |
791 | '(t)))) | |
792 | ((not (equal n (car math-prime-test-cache))) | |
793 | (cond ((= (% (nth 1 n) 2) 0) '(nil 2)) | |
794 | ((= (% (nth 1 n) 5) 0) '(nil 5)) | |
795 | (t (let ((dig (cdr n)) (sum 0)) | |
796 | (while dig | |
797 | (if (cdr dig) | |
798 | (setq sum (% (+ (+ sum (car dig)) | |
799 | (* (nth 1 dig) 1000)) | |
800 | 111111) | |
801 | dig (cdr (cdr dig))) | |
802 | (setq sum (% (+ sum (car dig)) 111111) | |
803 | dig nil))) | |
804 | (cond ((= (% sum 3) 0) '(nil 3)) | |
805 | ((= (% sum 7) 0) '(nil 7)) | |
806 | ((= (% sum 11) 0) '(nil 11)) | |
807 | ((= (% sum 13) 0) '(nil 13)) | |
808 | ((= (% sum 37) 0) '(nil 37)) | |
809 | (t | |
810 | (setq math-prime-test-cache-k 1 | |
811 | math-prime-test-cache-q | |
812 | (math-div2 n) | |
813 | math-prime-test-cache-nm1 | |
814 | (math-add n -1)) | |
815 | (while (math-evenp | |
816 | math-prime-test-cache-q) | |
817 | (setq math-prime-test-cache-k | |
818 | (1+ math-prime-test-cache-k) | |
819 | math-prime-test-cache-q | |
820 | (math-div2 | |
821 | math-prime-test-cache-q))) | |
822 | (setq iters (1+ iters)) | |
823 | (list 'maybe | |
824 | 0 | |
825 | (math-sub | |
826 | 100 | |
827 | (math-div | |
828 | '(float 232 0) | |
829 | (math-numdigs n)))))))))) | |
830 | ((not (eq (car (nth 1 math-prime-test-cache)) 'maybe)) | |
831 | (nth 1 math-prime-test-cache)) | |
832 | (t ; Fermat step | |
833 | (let* ((x (math-add (calcFunc-random (math-add n -2)) 2)) | |
834 | (y (math-pow-mod x math-prime-test-cache-q n)) | |
835 | (j 0)) | |
836 | (while (and (not (eq y 1)) | |
837 | (not (equal y math-prime-test-cache-nm1)) | |
838 | (< (setq j (1+ j)) math-prime-test-cache-k)) | |
839 | (setq y (math-mod (math-mul y y) n))) | |
840 | (if (or (equal y math-prime-test-cache-nm1) | |
841 | (and (eq y 1) (eq j 0))) | |
842 | (list 'maybe | |
843 | (1+ (nth 1 (nth 1 math-prime-test-cache))) | |
844 | (math-mul (nth 2 (nth 1 math-prime-test-cache)) | |
845 | '(float 25 -2))) | |
846 | '(nil unknown)))))) | |
847 | (setq math-prime-test-cache (list n res) | |
848 | iters (if (eq (car res) 'maybe) | |
849 | (1- iters) | |
850 | 0))) | |
bf77c646 | 851 | res)) |
136211a9 EZ |
852 | (defvar math-prime-test-cache '(-1)) |
853 | ||
854 | (defun calcFunc-prime (n &optional iters) | |
855 | (or (math-num-integerp n) (math-reject-arg n 'integerp)) | |
856 | (or (not iters) (math-num-integerp iters) (math-reject-arg iters 'integerp)) | |
857 | (if (car (math-prime-test (math-trunc n) (math-trunc (or iters 1)))) | |
858 | 1 | |
bf77c646 | 859 | 0)) |
136211a9 EZ |
860 | |
861 | ;;; Theory: summing base-10^6 digits modulo 111111 is "casting out 999999s". | |
862 | ;;; Initial probability that N is prime is 1/ln(N) = log10(e)/log10(N). | |
863 | ;;; After culling [2,3,5,7,11,13,37], probability of primality is 5.36 x more. | |
864 | ;;; Initial reported probability of non-primality is thus 100% - this. | |
865 | ;;; Each Fermat step multiplies this probability by 25%. | |
866 | ;;; The Fermat step is algorithm P from Knuth section 4.5.4. | |
867 | ||
868 | ||
869 | (defun calcFunc-prfac (n) | |
870 | (setq math-prime-factors-finished t) | |
871 | (if (Math-messy-integerp n) | |
872 | (setq n (math-trunc n))) | |
873 | (if (Math-natnump n) | |
874 | (if (Math-natnum-lessp 2 n) | |
875 | (let (factors res p (i 0)) | |
876 | (while (and (not (eq n 1)) | |
877 | (< i (length math-primes-table))) | |
878 | (setq p (aref math-primes-table i)) | |
879 | (while (eq (cdr (setq res (cond ((eq n p) (cons 1 0)) | |
880 | ((eq n 1) (cons 0 1)) | |
881 | ((consp n) (math-idivmod n p)) | |
882 | (t (cons (/ n p) (% n p)))))) | |
883 | 0) | |
884 | (math-working "factor" p) | |
885 | (setq factors (nconc factors (list p)) | |
886 | n (car res))) | |
887 | (or (eq n 1) | |
888 | (Math-natnum-lessp p (car res)) | |
889 | (setq factors (nconc factors (list n)) | |
890 | n 1)) | |
891 | (setq i (1+ i))) | |
892 | (or (setq math-prime-factors-finished (eq n 1)) | |
893 | (setq factors (nconc factors (list n)))) | |
894 | (cons 'vec factors)) | |
895 | (list 'vec n)) | |
896 | (if (Math-integerp n) | |
897 | (if (eq n -1) | |
898 | (list 'vec n) | |
899 | (cons 'vec (cons -1 (cdr (calcFunc-prfac (math-neg n)))))) | |
900 | (calc-record-why 'integerp n) | |
bf77c646 | 901 | (list 'calcFunc-prfac n)))) |
136211a9 EZ |
902 | |
903 | (defun calcFunc-totient (n) | |
904 | (if (Math-messy-integerp n) | |
905 | (setq n (math-trunc n))) | |
906 | (if (Math-natnump n) | |
907 | (if (Math-natnum-lessp n 2) | |
908 | (if (Math-negp n) | |
909 | (calcFunc-totient (math-abs n)) | |
910 | n) | |
911 | (let ((factors (cdr (calcFunc-prfac n))) | |
912 | p) | |
913 | (if math-prime-factors-finished | |
914 | (progn | |
915 | (while factors | |
916 | (setq p (car factors) | |
917 | n (math-mul (math-div n p) (math-add p -1))) | |
918 | (while (equal p (car factors)) | |
919 | (setq factors (cdr factors)))) | |
920 | n) | |
921 | (calc-record-why "*Number too big to factor" n) | |
922 | (list 'calcFunc-totient n)))) | |
923 | (calc-record-why 'natnump n) | |
bf77c646 | 924 | (list 'calcFunc-totient n))) |
136211a9 EZ |
925 | |
926 | (defun calcFunc-moebius (n) | |
927 | (if (Math-messy-integerp n) | |
928 | (setq n (math-trunc n))) | |
929 | (if (and (Math-natnump n) (not (eq n 0))) | |
930 | (if (Math-natnum-lessp n 2) | |
931 | (if (Math-negp n) | |
932 | (calcFunc-moebius (math-abs n)) | |
933 | 1) | |
934 | (let ((factors (cdr (calcFunc-prfac n))) | |
935 | (mu 1)) | |
936 | (if math-prime-factors-finished | |
937 | (progn | |
938 | (while factors | |
939 | (setq mu (if (equal (car factors) (nth 1 factors)) | |
940 | 0 (math-neg mu)) | |
941 | factors (cdr factors))) | |
942 | mu) | |
943 | (calc-record-why "Number too big to factor" n) | |
944 | (list 'calcFunc-moebius n)))) | |
945 | (calc-record-why 'posintp n) | |
bf77c646 | 946 | (list 'calcFunc-moebius n))) |
136211a9 EZ |
947 | |
948 | ||
949 | (defun calcFunc-nextprime (n &optional iters) | |
950 | (if (Math-integerp n) | |
951 | (if (Math-integer-negp n) | |
952 | 2 | |
953 | (if (and (integerp n) (< n 5003)) | |
954 | (math-next-small-prime (1+ n)) | |
955 | (if (math-evenp n) | |
956 | (setq n (math-add n -1))) | |
957 | (let (res) | |
958 | (while (not (car (setq res (math-prime-test | |
959 | (setq n (math-add n 2)) | |
960 | (or iters 1)))))) | |
961 | (if (and calc-verbose-nextprime | |
962 | (eq (car res) 'maybe)) | |
963 | (calc-report-prime-test res))) | |
964 | n)) | |
965 | (if (Math-realp n) | |
966 | (calcFunc-nextprime (math-trunc n) iters) | |
bf77c646 | 967 | (math-reject-arg n 'integerp)))) |
136211a9 EZ |
968 | (setq calc-verbose-nextprime nil) |
969 | ||
970 | (defun calcFunc-prevprime (n &optional iters) | |
971 | (if (Math-integerp n) | |
972 | (if (Math-lessp n 4) | |
973 | 2 | |
974 | (if (math-evenp n) | |
975 | (setq n (math-add n 1))) | |
976 | (let (res) | |
977 | (while (not (car (setq res (math-prime-test | |
978 | (setq n (math-add n -2)) | |
979 | (or iters 1)))))) | |
980 | (if (and calc-verbose-nextprime | |
981 | (eq (car res) 'maybe)) | |
982 | (calc-report-prime-test res))) | |
983 | n) | |
984 | (if (Math-realp n) | |
985 | (calcFunc-prevprime (math-ceiling n) iters) | |
bf77c646 | 986 | (math-reject-arg n 'integerp)))) |
136211a9 EZ |
987 | |
988 | (defun math-next-small-prime (n) | |
989 | (if (and (integerp n) (> n 2)) | |
990 | (let ((lo -1) | |
991 | (hi (length math-primes-table)) | |
992 | mid) | |
993 | (while (> (- hi lo) 1) | |
994 | (if (> n (aref math-primes-table | |
995 | (setq mid (ash (+ lo hi) -1)))) | |
996 | (setq lo mid) | |
997 | (setq hi mid))) | |
998 | (aref math-primes-table hi)) | |
bf77c646 | 999 | 2)) |
136211a9 | 1000 | |
3132f345 | 1001 | |
136211a9 EZ |
1002 | |
1003 | ||
ab5796a9 | 1004 | ;;; arch-tag: 1d75ee9b-0815-42bd-a321-bb3dc001cc02 |
bf77c646 | 1005 | ;;; calc-comb.el ends here |