| 1 | /* -*-c-*- |
| 2 | * |
| 3 | * Multiply many small numbers together |
| 4 | * |
| 5 | * (c) 2000 Straylight/Edgeware |
| 6 | */ |
| 7 | |
| 8 | /*----- Licensing notice --------------------------------------------------* |
| 9 | * |
| 10 | * This file is part of Catacomb. |
| 11 | * |
| 12 | * Catacomb is free software; you can redistribute it and/or modify |
| 13 | * it under the terms of the GNU Library General Public License as |
| 14 | * published by the Free Software Foundation; either version 2 of the |
| 15 | * License, or (at your option) any later version. |
| 16 | * |
| 17 | * Catacomb is distributed in the hope that it will be useful, |
| 18 | * but WITHOUT ANY WARRANTY; without even the implied warranty of |
| 19 | * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the |
| 20 | * GNU Library General Public License for more details. |
| 21 | * |
| 22 | * You should have received a copy of the GNU Library General Public |
| 23 | * License along with Catacomb; if not, write to the Free |
| 24 | * Software Foundation, Inc., 59 Temple Place - Suite 330, Boston, |
| 25 | * MA 02111-1307, USA. |
| 26 | */ |
| 27 | |
| 28 | #ifndef CATACOMB_MPMUL_H |
| 29 | #define CATACOMB_MPMUL_H |
| 30 | |
| 31 | #ifdef __cplusplus |
| 32 | extern "C" { |
| 33 | #endif |
| 34 | |
| 35 | /*----- Header files ------------------------------------------------------*/ |
| 36 | |
| 37 | #ifndef CATACOMB_MP_H |
| 38 | # include "mp.h" |
| 39 | #endif |
| 40 | |
| 41 | /*----- Magic numbers -----------------------------------------------------*/ |
| 42 | |
| 43 | /* --- How the algorithm works --- * |
| 44 | * |
| 45 | * Multiplication on large integers is least wasteful when the numbers |
| 46 | * multiplied are approximately the same size. When a new multiplier is |
| 47 | * added to the system, we push it onto a stack. Then we `reduce' the stack: |
| 48 | * while the value on the top of the stack is not shorter than the value |
| 49 | * below it, replace the top two elements by their product. |
| 50 | * |
| 51 | * Let %$b$% be the radix of our multiprecision integers, and let %$Z$% be |
| 52 | * the maximum number of digits. Then the largest integer we can represent |
| 53 | * is %$M - 1 = b^Z - 1$%. We could assume that all of the integers we're |
| 54 | * given are about the same size. This would give us the same upper bound as |
| 55 | * that derived in `mptext.c'. |
| 56 | * |
| 57 | * However, we're in less control over our inputs. In particular, if a |
| 58 | * sequence of integers with strictly decreasing lengths is input then we're |
| 59 | * sunk. Suppose that the stack contains, from top to bottom, %$b^i$%, |
| 60 | * %$b^{i+1}$%, ..., %$b^n$%. The final product will therefore be |
| 61 | * %$p = b^{(n+i)(n-i+1)/2}$%. We must now find the maximum stack depth |
| 62 | * %$d = n - i$% such that %$p > M$%. |
| 63 | * |
| 64 | * Taking logs of both sides gives that %$(d + 2 i)(d + 1) > 2 Z$%. We can |
| 65 | * maximize %$d$% by taking %$i = 0$%, which gives that %$d^2 + d > 2 Z$%, so |
| 66 | * %$d$% must be approximately %$(\sqrt{8 Z + 1} - 1)/2$%, which is |
| 67 | * uncomfortably large. |
| 68 | * |
| 69 | * We compromise by choosing double the `mptext' bound and imposing high- and |
| 70 | * low-water marks for forced reduction. |
| 71 | */ |
| 72 | |
| 73 | #define MPMUL_DEPTH (2 * (CHAR_BIT * sizeof(size_t) + 10)) |
| 74 | |
| 75 | /*----- Data structures ---------------------------------------------------*/ |
| 76 | |
| 77 | typedef struct mpmul { |
| 78 | size_t i; |
| 79 | mp *v[MPMUL_DEPTH]; |
| 80 | } mpmul; |
| 81 | |
| 82 | #define MPMUL_INIT { 0 } |
| 83 | |
| 84 | /*----- Functions provided ------------------------------------------------*/ |
| 85 | |
| 86 | /* --- @mpmul_init@ --- * |
| 87 | * |
| 88 | * Arguments: @mpmul *b@ = pointer to multiplier context to initialize |
| 89 | * |
| 90 | * Returns: --- |
| 91 | * |
| 92 | * Use: Initializes a big multiplier context for use. |
| 93 | */ |
| 94 | |
| 95 | extern void mpmul_init(mpmul */*b*/); |
| 96 | |
| 97 | /* --- @mpmul_add@ --- * |
| 98 | * |
| 99 | * Arguments: @mpmul *b@ = pointer to multiplier context |
| 100 | * @mp *x@ = the next factor to multiply in |
| 101 | * |
| 102 | * Returns: --- |
| 103 | * |
| 104 | * Use: Contributes another factor to the mix. It's important that |
| 105 | * the integer lasts at least as long as the multiplication |
| 106 | * context; this sort of rules out @mp_build@ integers. |
| 107 | */ |
| 108 | |
| 109 | extern void mpmul_add(mpmul */*b*/, mp */*x*/); |
| 110 | |
| 111 | /* --- @mpmul_done@ --- * |
| 112 | * |
| 113 | * Arguments: @mpmul *b@ = pointer to big multiplication context |
| 114 | * |
| 115 | * Returns: The product of all the numbers contributed. |
| 116 | * |
| 117 | * Use: Returns a (large) product of numbers. The context is |
| 118 | * deallocated. |
| 119 | */ |
| 120 | |
| 121 | extern mp *mpmul_done(mpmul */*b*/); |
| 122 | |
| 123 | /* --- @mp_factorial@ --- * |
| 124 | * |
| 125 | * Arguments: @unsigned long i@ = number whose factorial should be |
| 126 | * computed. |
| 127 | * |
| 128 | * Returns: The requested factorial. |
| 129 | */ |
| 130 | |
| 131 | extern mp *mp_factorial(unsigned long /*i*/); |
| 132 | |
| 133 | /*----- That's all, folks -------------------------------------------------*/ |
| 134 | |
| 135 | #ifdef __cplusplus |
| 136 | } |
| 137 | #endif |
| 138 | |
| 139 | #endif |