1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999 1000 1001 1002 1003 1004 1005 1006 1007 1008 1009 1010 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020 1021 1022 1023 1024 1025 1026 1027 1028 1029 1030 1031 1032 1033 1034 1035 1036 1037 1038 1039 1040 1041 1042 1043 1044 1045 1046 1047 1048 1049 1050 1051 1052 1053 1054 1055 1056
|
;; Calculator for GNU Emacs, part II [calc-comb.el]
;; Copyright (C) 1990, 1991, 1992, 1993 Free Software Foundation, Inc.
;; Written by Dave Gillespie, daveg@synaptics.com.
;; This file is part of GNU Emacs.
;; GNU Emacs is distributed in the hope that it will be useful,
;; but WITHOUT ANY WARRANTY. No author or distributor
;; accepts responsibility to anyone for the consequences of using it
;; or for whether it serves any particular purpose or works at all,
;; unless he says so in writing. Refer to the GNU Emacs General Public
;; License for full details.
;; Everyone is granted permission to copy, modify and redistribute
;; GNU Emacs, but only under the conditions described in the
;; GNU Emacs General Public License. A copy of this license is
;; supposed to have been given to you along with GNU Emacs so you
;; can know your rights and responsibilities. It should be in a
;; file named COPYING. Among other things, the copyright notice
;; and this notice must be preserved on all copies.
;; This file is autoloaded from calc-ext.el.
(require 'calc-ext)
(require 'calc-macs)
(defun calc-Need-calc-comb () nil)
;;; Combinatorics
(defun calc-gcd (arg)
(interactive "P")
(calc-slow-wrapper
(calc-binary-op "gcd" 'calcFunc-gcd arg))
)
(defun calc-lcm (arg)
(interactive "P")
(calc-slow-wrapper
(calc-binary-op "lcm" 'calcFunc-lcm arg))
)
(defun calc-extended-gcd ()
(interactive)
(calc-slow-wrapper
(calc-enter-result 2 "egcd" (cons 'calcFunc-egcd (calc-top-list-n 2))))
)
(defun calc-factorial (arg)
(interactive "P")
(calc-slow-wrapper
(calc-unary-op "fact" 'calcFunc-fact arg))
)
(defun calc-gamma (arg)
(interactive "P")
(calc-slow-wrapper
(calc-unary-op "gmma" 'calcFunc-gamma arg))
)
(defun calc-double-factorial (arg)
(interactive "P")
(calc-slow-wrapper
(calc-unary-op "dfac" 'calcFunc-dfact arg))
)
(defun calc-choose (arg)
(interactive "P")
(calc-slow-wrapper
(if (calc-is-hyperbolic)
(calc-binary-op "perm" 'calcFunc-perm arg)
(calc-binary-op "chos" 'calcFunc-choose arg)))
)
(defun calc-perm (arg)
(interactive "P")
(calc-hyperbolic-func)
(calc-choose arg)
)
(defvar calc-last-random-limit '(float 1 0))
(defun calc-random (n)
(interactive "P")
(calc-slow-wrapper
(if n
(calc-enter-result 0 "rand" (list 'calcFunc-random
(calc-get-random-limit
(prefix-numeric-value n))))
(calc-enter-result 1 "rand" (list 'calcFunc-random
(calc-get-random-limit
(calc-top-n 1))))))
)
(defun calc-get-random-limit (val)
(if (eq val 0)
calc-last-random-limit
(setq calc-last-random-limit val))
)
(defun calc-rrandom ()
(interactive)
(calc-slow-wrapper
(setq calc-last-random-limit '(float 1 0))
(calc-enter-result 0 "rand" (list 'calcFunc-random '(float 1 0))))
)
(defun calc-random-again (arg)
(interactive "p")
(calc-slow-wrapper
(while (>= (setq arg (1- arg)) 0)
(calc-enter-result 0 "rand" (list 'calcFunc-random
calc-last-random-limit))))
)
(defun calc-shuffle (n)
(interactive "P")
(calc-slow-wrapper
(if n
(calc-enter-result 1 "shuf" (list 'calcFunc-shuffle
(prefix-numeric-value n)
(calc-get-random-limit
(calc-top-n 1))))
(calc-enter-result 2 "shuf" (list 'calcFunc-shuffle
(calc-top-n 1)
(calc-get-random-limit
(calc-top-n 2))))))
)
(defun calc-report-prime-test (res)
(cond ((eq (car res) t)
(calc-record-message "prim" "Prime (guaranteed)"))
((eq (car res) nil)
(if (cdr res)
(if (eq (nth 1 res) 'unknown)
(calc-record-message
"prim" "Non-prime (factors unknown)")
(calc-record-message
"prim" "Non-prime (%s is a factor)"
(math-format-number (nth 1 res))))
(calc-record-message "prim" "Non-prime")))
(t
(calc-record-message
"prim" "Probably prime (%d iters; %s%% chance of error)"
(nth 1 res)
(let ((calc-float-format '(fix 2)))
(math-format-number (nth 2 res))))))
)
(defun calc-prime-test (iters)
(interactive "p")
(calc-slow-wrapper
(let* ((n (calc-top-n 1))
(res (math-prime-test n iters)))
(calc-report-prime-test res)))
)
(defun calc-next-prime (iters)
(interactive "p")
(calc-slow-wrapper
(let ((calc-verbose-nextprime t))
(if (calc-is-inverse)
(calc-enter-result 1 "prvp" (list 'calcFunc-prevprime
(calc-top-n 1) (math-abs iters)))
(calc-enter-result 1 "nxtp" (list 'calcFunc-nextprime
(calc-top-n 1) (math-abs iters))))))
)
(defun calc-prev-prime (iters)
(interactive "p")
(calc-invert-func)
(calc-next-prime iters)
)
(defun calc-prime-factors (iters)
(interactive "p")
(calc-slow-wrapper
(let ((res (calcFunc-prfac (calc-top-n 1))))
(if (not math-prime-factors-finished)
(calc-record-message "pfac" "Warning: May not be fully factored"))
(calc-enter-result 1 "pfac" res)))
)
(defun calc-totient (arg)
(interactive "P")
(calc-slow-wrapper
(calc-unary-op "phi" 'calcFunc-totient arg))
)
(defun calc-moebius (arg)
(interactive "P")
(calc-slow-wrapper
(calc-unary-op "mu" 'calcFunc-moebius arg))
)
(defun calcFunc-gcd (a b)
(if (Math-messy-integerp a)
(setq a (math-trunc a)))
(if (Math-messy-integerp b)
(setq b (math-trunc b)))
(cond ((and (Math-integerp a) (Math-integerp b))
(math-gcd a b))
((Math-looks-negp a)
(calcFunc-gcd (math-neg a) b))
((Math-looks-negp b)
(calcFunc-gcd a (math-neg b)))
((Math-zerop a) b)
((Math-zerop b) a)
((and (Math-ratp a)
(Math-ratp b))
(math-make-frac (math-gcd (if (eq (car-safe a) 'frac) (nth 1 a) a)
(if (eq (car-safe b) 'frac) (nth 1 b) b))
(calcFunc-lcm
(if (eq (car-safe a) 'frac) (nth 2 a) 1)
(if (eq (car-safe b) 'frac) (nth 2 b) 1))))
((not (Math-integerp a))
(calc-record-why 'integerp a)
(list 'calcFunc-gcd a b))
(t
(calc-record-why 'integerp b)
(list 'calcFunc-gcd a b)))
)
(defun calcFunc-lcm (a b)
(let ((g (calcFunc-gcd a b)))
(if (Math-numberp g)
(math-div (math-mul a b) g)
(list 'calcFunc-lcm a b)))
)
(defun calcFunc-egcd (a b) ; Knuth section 4.5.2
(cond
((not (Math-integerp a))
(if (Math-messy-integerp a)
(calcFunc-egcd (math-trunc a) b)
(calc-record-why 'integerp a)
(list 'calcFunc-egcd a b)))
((not (Math-integerp b))
(if (Math-messy-integerp b)
(calcFunc-egcd a (math-trunc b))
(calc-record-why 'integerp b)
(list 'calcFunc-egcd a b)))
(t
(let ((u1 1) (u2 0) (u3 a)
(v1 0) (v2 1) (v3 b)
t1 t2 q)
(while (not (eq v3 0))
(setq q (math-idivmod u3 v3)
t1 (math-sub u1 (math-mul v1 (car q)))
t2 (math-sub u2 (math-mul v2 (car q)))
u1 v1 u2 v2 u3 v3
v1 t1 v2 t2 v3 (cdr q)))
(list 'vec u3 u1 u2))))
)
;;; Factorial and related functions.
(defun calcFunc-fact (n) ; [I I] [F F] [Public]
(let (temp)
(cond ((Math-integer-negp n)
(if calc-infinite-mode
'(var uinf var-uinf)
(math-reject-arg n 'range)))
((integerp n)
(if (<= n 20)
(aref '[1 1 2 6 24 120 720 5040 40320 362880
(bigpos 800 628 3) (bigpos 800 916 39)
(bigpos 600 1 479) (bigpos 800 20 227 6)
(bigpos 200 291 178 87) (bigpos 0 368 674 307 1)
(bigpos 0 888 789 922 20) (bigpos 0 96 428 687 355)
(bigpos 0 728 705 373 402 6)
(bigpos 0 832 408 100 645 121)
(bigpos 0 640 176 8 902 432 2)] n)
(math-factorial-iter (1- n) 2 1)))
((and (math-messy-integerp n)
(Math-lessp n 100))
(math-inexact-result)
(setq temp (math-trunc n))
(if (>= temp 0)
(if (<= temp 20)
(math-float (calcFunc-fact temp))
(math-with-extra-prec 1
(math-factorial-iter (1- temp) 2 '(float 1 0))))
(math-reject-arg n 'range)))
((math-numberp n)
(let* ((q (math-quarter-integer n))
(tn (and q (Math-lessp n 1000) (Math-lessp -1000 n)
(1+ (math-floor n)))))
(cond ((and tn (= q 2)
(or calc-symbolic-mode (< (math-abs tn) 20)))
(let ((q (if (< tn 0)
(math-div
(math-pow -2 (- tn))
(math-double-factorial-iter (* -2 tn) 3 1 2))
(math-div
(math-double-factorial-iter (* 2 tn) 3 1 2)
(math-pow 2 tn)))))
(math-mul q (if calc-symbolic-mode
(list 'calcFunc-sqrt '(var pi var-pi))
(math-sqrt-pi)))))
((and tn (>= tn 0) (< tn 20)
(memq q '(1 3)))
(math-inexact-result)
(math-div
(math-mul (math-double-factorial-iter (* 4 tn) q 1 4)
(if (= q 1) (math-gamma-1q) (math-gamma-3q)))
(math-pow 4 tn)))
(t
(math-inexact-result)
(math-with-extra-prec 3
(math-gammap1-raw (math-float n)))))))
((equal n '(var inf var-inf)) n)
(t (calc-record-why 'numberp n)
(list 'calcFunc-fact n))))
)
(math-defcache math-gamma-1q nil
(math-with-extra-prec 3
(math-gammap1-raw '(float -75 -2))))
(math-defcache math-gamma-3q nil
(math-with-extra-prec 3
(math-gammap1-raw '(float -25 -2))))
(defun math-factorial-iter (count n f)
(if (= (% n 5) 1)
(math-working (format "factorial(%d)" (1- n)) f))
(if (> count 0)
(math-factorial-iter (1- count) (1+ n) (math-mul n f))
f)
)
(defun calcFunc-dfact (n) ; [I I] [F F] [Public]
(cond ((Math-integer-negp n)
(if (math-oddp n)
(if (eq n -1)
1
(math-div (if (eq (math-mod n 4) 3) 1 -1)
(calcFunc-dfact (math-sub -2 n))))
(list 'calcFunc-dfact n)))
((Math-zerop n) 1)
((integerp n) (math-double-factorial-iter n (+ 2 (% n 2)) 1 2))
((math-messy-integerp n)
(let ((temp (math-trunc n)))
(math-inexact-result)
(if (natnump temp)
(if (Math-lessp temp 200)
(math-with-extra-prec 1
(math-double-factorial-iter temp (+ 2 (% temp 2))
'(float 1 0) 2))
(let* ((half (math-div2 temp))
(even (math-mul (math-pow 2 half)
(calcFunc-fact (math-float half)))))
(if (math-evenp temp)
even
(math-div (calcFunc-fact n) even))))
(list 'calcFunc-dfact max))))
((equal n '(var inf var-inf)) n)
(t (calc-record-why 'natnump n)
(list 'calcFunc-dfact n)))
)
(defun math-double-factorial-iter (max n f step)
(if (< (% n 12) step)
(math-working (format "dfact(%d)" (- n step)) f))
(if (<= n max)
(math-double-factorial-iter max (+ n step) (math-mul n f) step)
f)
)
(defun calcFunc-perm (n m) ; [I I I] [F F F] [Public]
(cond ((and (integerp n) (integerp m) (<= m n) (>= m 0))
(math-factorial-iter m (1+ (- n m)) 1))
((or (not (math-num-integerp n))
(and (math-messy-integerp n) (Math-lessp 100 n))
(not (math-num-integerp m))
(and (math-messy-integerp m) (Math-lessp 100 m)))
(or (math-realp n) (equal n '(var inf var-inf))
(math-reject-arg n 'realp))
(or (math-realp m) (equal m '(var inf var-inf))
(math-reject-arg m 'realp))
(and (math-num-integerp n) (math-negp n) (math-reject-arg n 'range))
(and (math-num-integerp m) (math-negp m) (math-reject-arg m 'range))
(math-div (calcFunc-fact n) (calcFunc-fact (math-sub n m))))
(t
(let ((tn (math-trunc n))
(tm (math-trunc m)))
(math-inexact-result)
(or (integerp tn) (math-reject-arg tn 'fixnump))
(or (integerp tm) (math-reject-arg tm 'fixnump))
(or (and (<= tm tn) (>= tm 0)) (math-reject-arg tm 'range))
(math-with-extra-prec 1
(math-factorial-iter tm (1+ (- tn tm)) '(float 1 0))))))
)
(defun calcFunc-choose (n m) ; [I I I] [F F F] [Public]
(cond ((and (integerp n) (integerp m) (<= m n) (>= m 0))
(if (> m (/ n 2))
(math-choose-iter (- n m) n 1 1)
(math-choose-iter m n 1 1)))
((not (math-realp n))
(math-reject-arg n 'realp))
((not (math-realp m))
(math-reject-arg m 'realp))
((not (math-num-integerp m))
(if (and (math-num-integerp n) (math-negp n))
(list 'calcFunc-choose n m)
(math-div (calcFunc-fact (math-float n))
(math-mul (calcFunc-fact m)
(calcFunc-fact (math-sub n m))))))
((math-negp m) 0)
((math-negp n)
(let ((val (calcFunc-choose (math-add (math-add n m) -1) m)))
(if (math-evenp (math-trunc m))
val
(math-neg val))))
((and (math-num-integerp n)
(Math-lessp n m))
0)
(t
(math-inexact-result)
(let ((tm (math-trunc m)))
(or (integerp tm) (math-reject-arg tm 'fixnump))
(if (> tm 100)
(math-div (calcFunc-fact (math-float n))
(math-mul (calcFunc-fact (math-float m))
(calcFunc-fact (math-float
(math-sub n m)))))
(math-with-extra-prec 1
(math-choose-float-iter tm n 1 1))))))
)
(defun math-choose-iter (m n i c)
(if (and (= (% i 5) 1) (> i 5))
(math-working (format "choose(%d)" (1- i)) c))
(if (<= i m)
(math-choose-iter m (1- n) (1+ i)
(math-quotient (math-mul c n) i))
c)
)
(defun math-choose-float-iter (count n i c)
(if (= (% i 5) 1)
(math-working (format "choose(%d)" (1- i)) c))
(if (> count 0)
(math-choose-float-iter (1- count) (math-sub n 1) (1+ i)
(math-div (math-mul c n) i))
c)
)
;;; Stirling numbers.
(defun calcFunc-stir1 (n m)
(math-stirling-number n m 1)
)
(defun calcFunc-stir2 (n m)
(math-stirling-number n m 0)
)
(defun math-stirling-number (n m k)
(or (math-num-natnump n) (math-reject-arg n 'natnump))
(or (math-num-natnump m) (math-reject-arg m 'natnump))
(if (consp n) (setq n (math-trunc n)))
(or (integerp n) (math-reject-arg n 'fixnump))
(if (consp m) (setq m (math-trunc m)))
(or (integerp m) (math-reject-arg m 'fixnump))
(if (< n m)
0
(let ((cache (aref math-stirling-cache k)))
(while (<= (length cache) n)
(let ((i (1- (length cache)))
row)
(setq cache (vconcat cache (make-vector (length cache) nil)))
(aset math-stirling-cache k cache)
(while (< (setq i (1+ i)) (length cache))
(aset cache i (setq row (make-vector (1+ i) nil)))
(aset row 0 0)
(aset row i 1))))
(if (= k 1)
(math-stirling-1 n m)
(math-stirling-2 n m))))
)
(setq math-stirling-cache (vector [[1]] [[1]]))
(defun math-stirling-1 (n m)
(or (aref (aref cache n) m)
(aset (aref cache n) m
(math-add (math-stirling-1 (1- n) (1- m))
(math-mul (- 1 n) (math-stirling-1 (1- n) m)))))
)
(defun math-stirling-2 (n m)
(or (aref (aref cache n) m)
(aset (aref cache n) m
(math-add (math-stirling-2 (1- n) (1- m))
(math-mul m (math-stirling-2 (1- n) m)))))
)
;;; Produce a random 10-bit integer, with (random) if no seed provided,
;;; or else with Numerical Recipes algorithm ran3 / Knuth 3.2.2-A.
(defun math-init-random-base ()
(if (and (boundp 'var-RandSeed) var-RandSeed)
(if (eq (car-safe var-RandSeed) 'vec)
nil
(if (Math-integerp var-RandSeed)
(let* ((seed (math-sub 161803 var-RandSeed))
(mj (1+ (math-mod seed '(bigpos 0 0 1))))
(mk (1+ (math-mod (math-quotient seed '(bigpos 0 0 1))
'(bigpos 0 0 1))))
(i 0))
(setq math-random-table (cons 'vec (make-list 55 mj)))
(while (<= (setq i (1+ i)) 54)
(let* ((ii (% (* i 21) 55))
(p (nthcdr ii math-random-table)))
(setcar p mk)
(setq mk (- mj mk)
mj (car p)))))
(math-reject-arg var-RandSeed "*RandSeed must be an integer"))
(setq var-RandSeed (list 'vec var-RandSeed)
math-random-ptr1 math-random-table
math-random-cache nil
math-random-ptr2 (nthcdr 31 math-random-table))
(let ((i 200))
(while (> (setq i (1- i)) 0)
(math-random-base))))
(random t)
(setq var-RandSeed nil
math-random-cache nil
i 0
math-random-shift -4) ; assume RAND_MAX >= 16383
;; This exercises the random number generator and also helps
;; deduce a better value for RAND_MAX.
(while (< (setq i (1+ i)) 30)
(if (> (lsh (math-abs (random)) math-random-shift) 4095)
(setq math-random-shift (1- math-random-shift)))))
(setq math-last-RandSeed var-RandSeed
math-gaussian-cache nil)
)
(defun math-random-base ()
(if var-RandSeed
(progn
(setq math-random-ptr1 (or (cdr math-random-ptr1)
(cdr math-random-table))
math-random-ptr2 (or (cdr math-random-ptr2)
(cdr math-random-table)))
(logand (lsh (setcar math-random-ptr1
(logand (- (car math-random-ptr1)
(car math-random-ptr2)) 524287))
-6) 1023))
(logand (lsh (random) math-random-shift) 1023))
)
(setq math-random-table nil)
(setq math-last-RandSeed nil)
(setq math-random-ptr1 nil)
(setq math-random-ptr2 nil)
(setq math-random-shift nil)
;;; Produce a random digit in the range 0..999.
;;; Avoid various pitfalls that may lurk in the built-in (random) function!
;;; Shuffling algorithm from Numerical Recipes, section 7.1.
(defun math-random-digit ()
(let (i)
(or (and (boundp 'var-RandSeed) (eq var-RandSeed math-last-RandSeed))
(math-init-random-base))
(or math-random-cache
(progn
(setq math-random-last (math-random-base)
math-random-cache (make-vector 13 nil)
i -1)
(while (< (setq i (1+ i)) 13)
(aset math-random-cache i (math-random-base)))))
(while (progn
(setq i (/ math-random-last 79) ; 0 <= i < 13
math-random-last (aref math-random-cache i))
(aset math-random-cache i (math-random-base))
(>= math-random-last 1000)))
math-random-last)
)
(setq math-random-cache nil)
;;; Produce an N-digit random integer.
(defun math-random-digits (n)
(cond ((<= n 6)
(math-scale-right (+ (* (math-random-digit) 1000) (math-random-digit))
(- 6 n)))
(t (let* ((slop (% (- 900003 n) 3))
(i (/ (+ n slop) 3))
(digs nil))
(while (> i 0)
(setq digs (cons (math-random-digit) digs)
i (1- i)))
(math-normalize (math-scale-right (cons 'bigpos digs)
slop)))))
)
;;; Produce a uniformly-distributed random float 0 <= N < 1.
(defun math-random-float ()
(math-make-float (math-random-digits calc-internal-prec)
(- calc-internal-prec))
)
;;; Produce a Gaussian-distributed random float with mean=0, sigma=1.
(defun math-gaussian-float ()
(math-with-extra-prec 2
(if (and math-gaussian-cache
(= (car math-gaussian-cache) calc-internal-prec))
(prog1
(cdr math-gaussian-cache)
(setq math-gaussian-cache nil))
(let* ((v1 (math-add (math-mul (math-random-float) 2) -1))
(v2 (math-add (math-mul (math-random-float) 2) -1))
(r (math-add (math-sqr v1) (math-sqr v2))))
(while (or (not (Math-lessp r 1)) (math-zerop r))
(setq v1 (math-add (math-mul (math-random-float) 2) -1)
v2 (math-add (math-mul (math-random-float) 2) -1)
r (math-add (math-sqr v1) (math-sqr v2))))
(let ((fac (math-sqrt (math-mul (math-div (calcFunc-ln r) r) -2))))
(setq math-gaussian-cache (cons calc-internal-prec
(math-mul v1 fac)))
(math-mul v2 fac)))))
)
(setq math-gaussian-cache nil)
;;; Produce a random integer or real 0 <= N < MAX.
(defun calcFunc-random (max)
(cond ((Math-zerop max)
(math-gaussian-float))
((Math-integerp max)
(let* ((digs (math-numdigs max))
(r (math-random-digits (+ digs 3))))
(math-mod r max)))
((Math-realp max)
(math-mul (math-random-float) max))
((and (eq (car max) 'intv) (math-constp max)
(Math-lessp (nth 2 max) (nth 3 max)))
(if (math-floatp max)
(let ((val (math-add (math-mul (math-random-float)
(math-sub (nth 3 max) (nth 2 max)))
(nth 2 max))))
(if (or (and (memq (nth 1 max) '(0 1)) ; almost not worth
(Math-equal val (nth 2 max))) ; checking!
(and (memq (nth 1 max) '(0 2))
(Math-equal val (nth 3 max))))
(calcFunc-random max)
val))
(let ((lo (if (memq (nth 1 max) '(0 1))
(math-add (nth 2 max) 1) (nth 2 max)))
(hi (if (memq (nth 1 max) '(1 3))
(math-add (nth 3 max) 1) (nth 3 max))))
(if (Math-lessp lo hi)
(math-add (calcFunc-random (math-sub hi lo)) lo)
(math-reject-arg max "*Empty interval")))))
((eq (car max) 'vec)
(if (cdr max)
(nth (1+ (calcFunc-random (1- (length max)))) max)
(math-reject-arg max "*Empty list")))
((and (eq (car max) 'sdev) (math-constp max) (Math-realp (nth 1 max)))
(math-add (math-mul (math-gaussian-float) (nth 2 max)) (nth 1 max)))
(t (math-reject-arg max 'realp)))
)
;;; Choose N objects at random from the set MAX without duplicates.
(defun calcFunc-shuffle (n &optional max)
(or max (setq max n n -1))
(or (and (Math-num-integerp n)
(or (natnump (setq n (math-trunc n))) (eq n -1)))
(math-reject-arg n 'integerp))
(cond ((or (math-zerop max)
(math-floatp max)
(eq (car-safe max) 'sdev))
(if (< n 0)
(math-reject-arg n 'natnump)
(math-simple-shuffle n max)))
((and (<= n 1) (>= n 0))
(math-simple-shuffle n max))
((and (eq (car-safe max) 'intv) (math-constp max))
(let ((num (math-add (math-sub (nth 3 max) (nth 2 max))
(cdr (assq (nth 1 max)
'((0 . -1) (1 . 0)
(2 . 0) (3 . 1))))))
(min (math-add (nth 2 max) (if (memq (nth 1 max) '(0 1))
1 0))))
(if (< n 0) (setq n num))
(or (math-posp num) (math-reject-arg max 'range))
(and (Math-lessp num n) (math-reject-arg n 'range))
(if (Math-lessp n (math-quotient num 3))
(math-simple-shuffle n max)
(if (> (* n 4) (* num 3))
(math-add (math-sub min 1)
(math-shuffle-list n num (calcFunc-index num)))
(let ((tot 0)
(m 0)
(vec nil))
(while (< m n)
(if (< (calcFunc-random (- num tot)) (- n m))
(setq vec (cons (math-add min tot) vec)
m (1+ m)))
(setq tot (1+ tot)))
(math-shuffle-list n n (cons 'vec vec)))))))
((eq (car-safe max) 'vec)
(let ((size (1- (length max))))
(if (< n 0) (setq n size))
(if (and (> n (/ size 2)) (<= n size))
(math-shuffle-list n size (copy-sequence max))
(let* ((vals (calcFunc-shuffle
n (list 'intv 3 1 (1- (length max)))))
(p vals))
(while (setq p (cdr p))
(setcar p (nth (car p) max)))
vals))))
((math-integerp max)
(if (math-posp max)
(calcFunc-shuffle n (list 'intv 2 0 max))
(calcFunc-shuffle n (list 'intv 1 max 0))))
(t (math-reject-arg max 'realp)))
)
(defun math-simple-shuffle (n max)
(let ((vec nil)
val)
(while (>= (setq n (1- n)) 0)
(while (math-member (setq val (calcFunc-random max)) vec))
(setq vec (cons val vec)))
(cons 'vec vec))
)
(defun math-shuffle-list (n size vec)
(let ((j size)
k temp
(p vec))
(while (cdr (setq p (cdr p)))
(setq k (calcFunc-random j)
j (1- j)
temp (nth k p))
(setcar (nthcdr k p) (car p))
(setcar p temp))
(cons 'vec (nthcdr (- size n -1) vec)))
)
(defun math-member (x list)
(while (and list (not (equal x (car list))))
(setq list (cdr list)))
list
)
;;; Check if the integer N is prime. [X I]
;;; Return (nil) if non-prime,
;;; (nil N) if non-prime with known factor N,
;;; (nil unknown) if non-prime with no known factors,
;;; (t) if prime,
;;; (maybe N P) if probably prime (after N iters with probability P%)
(defun math-prime-test (n iters)
(if (and (Math-vectorp n) (cdr n))
(setq n (nth (1- (length n)) n)))
(if (Math-messy-integerp n)
(setq n (math-trunc n)))
(let ((res))
(while (> iters 0)
(setq res
(cond ((and (integerp n) (<= n 5003))
(list (= (math-next-small-prime n) n)))
((not (Math-integerp n))
(error "Argument must be an integer"))
((Math-integer-negp n)
'(nil))
((Math-natnum-lessp n '(bigpos 0 0 8))
(setq n (math-fixnum n))
(let ((i -1) v)
(while (and (> (% n (setq v (aref math-primes-table
(setq i (1+ i)))))
0)
(< (* v v) n)))
(if (= (% n v) 0)
(list nil v)
'(t))))
((not (equal n (car math-prime-test-cache)))
(cond ((= (% (nth 1 n) 2) 0) '(nil 2))
((= (% (nth 1 n) 5) 0) '(nil 5))
(t (let ((dig (cdr n)) (sum 0))
(while dig
(if (cdr dig)
(setq sum (% (+ (+ sum (car dig))
(* (nth 1 dig) 1000))
111111)
dig (cdr (cdr dig)))
(setq sum (% (+ sum (car dig)) 111111)
dig nil)))
(cond ((= (% sum 3) 0) '(nil 3))
((= (% sum 7) 0) '(nil 7))
((= (% sum 11) 0) '(nil 11))
((= (% sum 13) 0) '(nil 13))
((= (% sum 37) 0) '(nil 37))
(t
(setq math-prime-test-cache-k 1
math-prime-test-cache-q
(math-div2 n)
math-prime-test-cache-nm1
(math-add n -1))
(while (math-evenp
math-prime-test-cache-q)
(setq math-prime-test-cache-k
(1+ math-prime-test-cache-k)
math-prime-test-cache-q
(math-div2
math-prime-test-cache-q)))
(setq iters (1+ iters))
(list 'maybe
0
(math-sub
100
(math-div
'(float 232 0)
(math-numdigs n))))))))))
((not (eq (car (nth 1 math-prime-test-cache)) 'maybe))
(nth 1 math-prime-test-cache))
(t ; Fermat step
(let* ((x (math-add (calcFunc-random (math-add n -2)) 2))
(y (math-pow-mod x math-prime-test-cache-q n))
(j 0))
(while (and (not (eq y 1))
(not (equal y math-prime-test-cache-nm1))
(< (setq j (1+ j)) math-prime-test-cache-k))
(setq y (math-mod (math-mul y y) n)))
(if (or (equal y math-prime-test-cache-nm1)
(and (eq y 1) (eq j 0)))
(list 'maybe
(1+ (nth 1 (nth 1 math-prime-test-cache)))
(math-mul (nth 2 (nth 1 math-prime-test-cache))
'(float 25 -2)))
'(nil unknown))))))
(setq math-prime-test-cache (list n res)
iters (if (eq (car res) 'maybe)
(1- iters)
0)))
res)
)
(defvar math-prime-test-cache '(-1))
(defun calcFunc-prime (n &optional iters)
(or (math-num-integerp n) (math-reject-arg n 'integerp))
(or (not iters) (math-num-integerp iters) (math-reject-arg iters 'integerp))
(if (car (math-prime-test (math-trunc n) (math-trunc (or iters 1))))
1
0)
)
;;; Theory: summing base-10^6 digits modulo 111111 is "casting out 999999s".
;;; Initial probability that N is prime is 1/ln(N) = log10(e)/log10(N).
;;; After culling [2,3,5,7,11,13,37], probability of primality is 5.36 x more.
;;; Initial reported probability of non-primality is thus 100% - this.
;;; Each Fermat step multiplies this probability by 25%.
;;; The Fermat step is algorithm P from Knuth section 4.5.4.
(defun calcFunc-prfac (n)
(setq math-prime-factors-finished t)
(if (Math-messy-integerp n)
(setq n (math-trunc n)))
(if (Math-natnump n)
(if (Math-natnum-lessp 2 n)
(let (factors res p (i 0))
(while (and (not (eq n 1))
(< i (length math-primes-table)))
(setq p (aref math-primes-table i))
(while (eq (cdr (setq res (cond ((eq n p) (cons 1 0))
((eq n 1) (cons 0 1))
((consp n) (math-idivmod n p))
(t (cons (/ n p) (% n p))))))
0)
(math-working "factor" p)
(setq factors (nconc factors (list p))
n (car res)))
(or (eq n 1)
(Math-natnum-lessp p (car res))
(setq factors (nconc factors (list n))
n 1))
(setq i (1+ i)))
(or (setq math-prime-factors-finished (eq n 1))
(setq factors (nconc factors (list n))))
(cons 'vec factors))
(list 'vec n))
(if (Math-integerp n)
(if (eq n -1)
(list 'vec n)
(cons 'vec (cons -1 (cdr (calcFunc-prfac (math-neg n))))))
(calc-record-why 'integerp n)
(list 'calcFunc-prfac n)))
)
(defun calcFunc-totient (n)
(if (Math-messy-integerp n)
(setq n (math-trunc n)))
(if (Math-natnump n)
(if (Math-natnum-lessp n 2)
(if (Math-negp n)
(calcFunc-totient (math-abs n))
n)
(let ((factors (cdr (calcFunc-prfac n)))
p)
(if math-prime-factors-finished
(progn
(while factors
(setq p (car factors)
n (math-mul (math-div n p) (math-add p -1)))
(while (equal p (car factors))
(setq factors (cdr factors))))
n)
(calc-record-why "*Number too big to factor" n)
(list 'calcFunc-totient n))))
(calc-record-why 'natnump n)
(list 'calcFunc-totient n))
)
(defun calcFunc-moebius (n)
(if (Math-messy-integerp n)
(setq n (math-trunc n)))
(if (and (Math-natnump n) (not (eq n 0)))
(if (Math-natnum-lessp n 2)
(if (Math-negp n)
(calcFunc-moebius (math-abs n))
1)
(let ((factors (cdr (calcFunc-prfac n)))
(mu 1))
(if math-prime-factors-finished
(progn
(while factors
(setq mu (if (equal (car factors) (nth 1 factors))
0 (math-neg mu))
factors (cdr factors)))
mu)
(calc-record-why "Number too big to factor" n)
(list 'calcFunc-moebius n))))
(calc-record-why 'posintp n)
(list 'calcFunc-moebius n))
)
(defun calcFunc-nextprime (n &optional iters)
(if (Math-integerp n)
(if (Math-integer-negp n)
2
(if (and (integerp n) (< n 5003))
(math-next-small-prime (1+ n))
(if (math-evenp n)
(setq n (math-add n -1)))
(let (res)
(while (not (car (setq res (math-prime-test
(setq n (math-add n 2))
(or iters 1))))))
(if (and calc-verbose-nextprime
(eq (car res) 'maybe))
(calc-report-prime-test res)))
n))
(if (Math-realp n)
(calcFunc-nextprime (math-trunc n) iters)
(math-reject-arg n 'integerp)))
)
(setq calc-verbose-nextprime nil)
(defun calcFunc-prevprime (n &optional iters)
(if (Math-integerp n)
(if (Math-lessp n 4)
2
(if (math-evenp n)
(setq n (math-add n 1)))
(let (res)
(while (not (car (setq res (math-prime-test
(setq n (math-add n -2))
(or iters 1))))))
(if (and calc-verbose-nextprime
(eq (car res) 'maybe))
(calc-report-prime-test res)))
n)
(if (Math-realp n)
(calcFunc-prevprime (math-ceiling n) iters)
(math-reject-arg n 'integerp)))
)
(defun math-next-small-prime (n)
(if (and (integerp n) (> n 2))
(let ((lo -1)
(hi (length math-primes-table))
mid)
(while (> (- hi lo) 1)
(if (> n (aref math-primes-table
(setq mid (ash (+ lo hi) -1))))
(setq lo mid)
(setq hi mid)))
(aref math-primes-table hi))
2)
)
(defconst math-primes-table
[2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89
97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181
191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277
281 283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383
389 397 401 409 419 421 431 433 439 443 449 457 461 463 467 479 487
491 499 503 509 521 523 541 547 557 563 569 571 577 587 593 599 601
607 613 617 619 631 641 643 647 653 659 661 673 677 683 691 701 709
719 727 733 739 743 751 757 761 769 773 787 797 809 811 821 823 827
829 839 853 857 859 863 877 881 883 887 907 911 919 929 937 941 947
953 967 971 977 983 991 997 1009 1013 1019 1021 1031 1033 1039 1049
1051 1061 1063 1069 1087 1091 1093 1097 1103 1109 1117 1123 1129 1151
1153 1163 1171 1181 1187 1193 1201 1213 1217 1223 1229 1231 1237 1249
1259 1277 1279 1283 1289 1291 1297 1301 1303 1307 1319 1321 1327 1361
1367 1373 1381 1399 1409 1423 1427 1429 1433 1439 1447 1451 1453 1459
1471 1481 1483 1487 1489 1493 1499 1511 1523 1531 1543 1549 1553 1559
1567 1571 1579 1583 1597 1601 1607 1609 1613 1619 1621 1627 1637 1657
1663 1667 1669 1693 1697 1699 1709 1721 1723 1733 1741 1747 1753 1759
1777 1783 1787 1789 1801 1811 1823 1831 1847 1861 1867 1871 1873 1877
1879 1889 1901 1907 1913 1931 1933 1949 1951 1973 1979 1987 1993 1997
1999 2003 2011 2017 2027 2029 2039 2053 2063 2069 2081 2083 2087 2089
2099 2111 2113 2129 2131 2137 2141 2143 2153 2161 2179 2203 2207 2213
2221 2237 2239 2243 2251 2267 2269 2273 2281 2287 2293 2297 2309 2311
2333 2339 2341 2347 2351 2357 2371 2377 2381 2383 2389 2393 2399 2411
2417 2423 2437 2441 2447 2459 2467 2473 2477 2503 2521 2531 2539 2543
2549 2551 2557 2579 2591 2593 2609 2617 2621 2633 2647 2657 2659 2663
2671 2677 2683 2687 2689 2693 2699 2707 2711 2713 2719 2729 2731 2741
2749 2753 2767 2777 2789 2791 2797 2801 2803 2819 2833 2837 2843 2851
2857 2861 2879 2887 2897 2903 2909 2917 2927 2939 2953 2957 2963 2969
2971 2999 3001 3011 3019 3023 3037 3041 3049 3061 3067 3079 3083 3089
3109 3119 3121 3137 3163 3167 3169 3181 3187 3191 3203 3209 3217 3221
3229 3251 3253 3257 3259 3271 3299 3301 3307 3313 3319 3323 3329 3331
3343 3347 3359 3361 3371 3373 3389 3391 3407 3413 3433 3449 3457 3461
3463 3467 3469 3491 3499 3511 3517 3527 3529 3533 3539 3541 3547 3557
3559 3571 3581 3583 3593 3607 3613 3617 3623 3631 3637 3643 3659 3671
3673 3677 3691 3697 3701 3709 3719 3727 3733 3739 3761 3767 3769 3779
3793 3797 3803 3821 3823 3833 3847 3851 3853 3863 3877 3881 3889 3907
3911 3917 3919 3923 3929 3931 3943 3947 3967 3989 4001 4003 4007 4013
4019 4021 4027 4049 4051 4057 4073 4079 4091 4093 4099 4111 4127 4129
4133 4139 4153 4157 4159 4177 4201 4211 4217 4219 4229 4231 4241 4243
4253 4259 4261 4271 4273 4283 4289 4297 4327 4337 4339 4349 4357 4363
4373 4391 4397 4409 4421 4423 4441 4447 4451 4457 4463 4481 4483 4493
4507 4513 4517 4519 4523 4547 4549 4561 4567 4583 4591 4597 4603 4621
4637 4639 4643 4649 4651 4657 4663 4673 4679 4691 4703 4721 4723 4729
4733 4751 4759 4783 4787 4789 4793 4799 4801 4813 4817 4831 4861 4871
4877 4889 4903 4909 4919 4931 4933 4937 4943 4951 4957 4967 4969 4973
4987 4993 4999 5003])
|