alpar@1: /* glplib01.c (bignum arithmetic) */
alpar@1: 
alpar@1: /***********************************************************************
alpar@1: *  This code is part of GLPK (GNU Linear Programming Kit).
alpar@1: *
alpar@1: *  Copyright (C) 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008,
alpar@1: *  2009, 2010 Andrew Makhorin, Department for Applied Informatics,
alpar@1: *  Moscow Aviation Institute, Moscow, Russia. All rights reserved.
alpar@1: *  E-mail: <mao@gnu.org>.
alpar@1: *
alpar@1: *  GLPK is free software: you can redistribute it and/or modify it
alpar@1: *  under the terms of the GNU General Public License as published by
alpar@1: *  the Free Software Foundation, either version 3 of the License, or
alpar@1: *  (at your option) any later version.
alpar@1: *
alpar@1: *  GLPK is distributed in the hope that it will be useful, but WITHOUT
alpar@1: *  ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
alpar@1: *  or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public
alpar@1: *  License for more details.
alpar@1: *
alpar@1: *  You should have received a copy of the GNU General Public License
alpar@1: *  along with GLPK. If not, see <http://www.gnu.org/licenses/>.
alpar@1: ***********************************************************************/
alpar@1: 
alpar@1: #include "glpenv.h"
alpar@1: #include "glplib.h"
alpar@1: 
alpar@1: /***********************************************************************
alpar@1: *  Two routines below are intended to multiply and divide unsigned
alpar@1: *  integer numbers of arbitrary precision.
alpar@1: * 
alpar@1: *  The routines assume that an unsigned integer number is represented in
alpar@1: *  the positional numeral system with the base 2^16 = 65536, i.e. each
alpar@1: *  "digit" of the number is in the range [0, 65535] and represented as
alpar@1: *  a 16-bit value of the unsigned short type. In other words, a number x
alpar@1: *  has the following representation:
alpar@1: * 
alpar@1: *         n-1
alpar@1: *     x = sum d[j] * 65536^j,
alpar@1: *         j=0
alpar@1: * 
alpar@1: *  where n is the number of places (positions), and d[j] is j-th "digit"
alpar@1: *  of x, 0 <= d[j] <= 65535.
alpar@1: ***********************************************************************/
alpar@1: 
alpar@1: /***********************************************************************
alpar@1: *  NAME
alpar@1: *
alpar@1: *  bigmul - multiply unsigned integer numbers of arbitrary precision
alpar@1: * 
alpar@1: *  SYNOPSIS
alpar@1: * 
alpar@1: *  #include "glplib.h"
alpar@1: *  void bigmul(int n, int m, unsigned short x[], unsigned short y[]);
alpar@1: * 
alpar@1: *  DESCRIPTION
alpar@1: * 
alpar@1: *  The routine bigmul multiplies unsigned integer numbers of arbitrary
alpar@1: *  precision.
alpar@1: * 
alpar@1: *  n is the number of digits of multiplicand, n >= 1;
alpar@1: * 
alpar@1: *  m is the number of digits of multiplier, m >= 1;
alpar@1: * 
alpar@1: *  x is an array containing digits of the multiplicand in elements
alpar@1: *  x[m], x[m+1], ..., x[n+m-1]. Contents of x[0], x[1], ..., x[m-1] are
alpar@1: *  ignored on entry.
alpar@1: * 
alpar@1: *  y is an array containing digits of the multiplier in elements y[0],
alpar@1: *  y[1], ..., y[m-1].
alpar@1: * 
alpar@1: *  On exit digits of the product are stored in elements x[0], x[1], ...,
alpar@1: *  x[n+m-1]. The array y is not changed. */
alpar@1: 
alpar@1: void bigmul(int n, int m, unsigned short x[], unsigned short y[])
alpar@1: {     int i, j;
alpar@1:       unsigned int t;
alpar@1:       xassert(n >= 1);
alpar@1:       xassert(m >= 1);
alpar@1:       for (j = 0; j < m; j++) x[j] = 0;
alpar@1:       for (i = 0; i < n; i++)
alpar@1:       {  if (x[i+m])
alpar@1:          {  t = 0;
alpar@1:             for (j = 0; j < m; j++)
alpar@1:             {  t += (unsigned int)x[i+m] * (unsigned int)y[j] +
alpar@1:                     (unsigned int)x[i+j];
alpar@1:                x[i+j] = (unsigned short)t;
alpar@1:                t >>= 16;
alpar@1:             }
alpar@1:             x[i+m] = (unsigned short)t;
alpar@1:          }
alpar@1:       }
alpar@1:       return;
alpar@1: }
alpar@1: 
alpar@1: /***********************************************************************
alpar@1: *  NAME
alpar@1: *
alpar@1: *  bigdiv - divide unsigned integer numbers of arbitrary precision
alpar@1: * 
alpar@1: *  SYNOPSIS
alpar@1: * 
alpar@1: *  #include "glplib.h"
alpar@1: *  void bigdiv(int n, int m, unsigned short x[], unsigned short y[]);
alpar@1: * 
alpar@1: *  DESCRIPTION
alpar@1: * 
alpar@1: *  The routine bigdiv divides one unsigned integer number of arbitrary
alpar@1: *  precision by another with the algorithm described in [1].
alpar@1: * 
alpar@1: *  n is the difference between the number of digits of dividend and the
alpar@1: *  number of digits of divisor, n >= 0.
alpar@1: * 
alpar@1: *  m is the number of digits of divisor, m >= 1.
alpar@1: * 
alpar@1: *  x is an array containing digits of the dividend in elements x[0],
alpar@1: *  x[1], ..., x[n+m-1].
alpar@1: * 
alpar@1: *  y is an array containing digits of the divisor in elements y[0],
alpar@1: *  y[1], ..., y[m-1]. The highest digit y[m-1] must be non-zero.
alpar@1: * 
alpar@1: *  On exit n+1 digits of the quotient are stored in elements x[m],
alpar@1: *  x[m+1], ..., x[n+m], and m digits of the remainder are stored in
alpar@1: *  elements x[0], x[1], ..., x[m-1]. The array y is changed but then
alpar@1: *  restored.
alpar@1: *
alpar@1: *  REFERENCES
alpar@1: *
alpar@1: *  1. D. Knuth. The Art of Computer Programming. Vol. 2: Seminumerical
alpar@1: *  Algorithms. Stanford University, 1969. */
alpar@1: 
alpar@1: void bigdiv(int n, int m, unsigned short x[], unsigned short y[])
alpar@1: {     int i, j;
alpar@1:       unsigned int t;
alpar@1:       unsigned short d, q, r;
alpar@1:       xassert(n >= 0);
alpar@1:       xassert(m >= 1);
alpar@1:       xassert(y[m-1] != 0);
alpar@1:       /* special case when divisor has the only digit */
alpar@1:       if (m == 1)
alpar@1:       {  d = 0;
alpar@1:          for (i = n; i >= 0; i--)
alpar@1:          {  t = ((unsigned int)d << 16) + (unsigned int)x[i];
alpar@1:             x[i+1] = (unsigned short)(t / y[0]);
alpar@1:             d = (unsigned short)(t % y[0]);
alpar@1:          }
alpar@1:          x[0] = d;
alpar@1:          goto done;
alpar@1:       }
alpar@1:       /* multiply dividend and divisor by a normalizing coefficient in
alpar@1:          order to provide the condition y[m-1] >= base / 2 */
alpar@1:       d = (unsigned short)(0x10000 / ((unsigned int)y[m-1] + 1));
alpar@1:       if (d == 1)
alpar@1:          x[n+m] = 0;
alpar@1:       else
alpar@1:       {  t = 0;
alpar@1:          for (i = 0; i < n+m; i++)
alpar@1:          {  t += (unsigned int)x[i] * (unsigned int)d;
alpar@1:             x[i] = (unsigned short)t;
alpar@1:             t >>= 16;
alpar@1:          }
alpar@1:          x[n+m] = (unsigned short)t;
alpar@1:          t = 0;
alpar@1:          for (j = 0; j < m; j++)
alpar@1:          {  t += (unsigned int)y[j] * (unsigned int)d;
alpar@1:             y[j] = (unsigned short)t;
alpar@1:             t >>= 16;
alpar@1:          }
alpar@1:       }
alpar@1:       /* main loop */
alpar@1:       for (i = n; i >= 0; i--)
alpar@1:       {  /* estimate and correct the current digit of quotient */
alpar@1:          if (x[i+m] < y[m-1])
alpar@1:          {  t = ((unsigned int)x[i+m] << 16) + (unsigned int)x[i+m-1];
alpar@1:             q = (unsigned short)(t / (unsigned int)y[m-1]);
alpar@1:             r = (unsigned short)(t % (unsigned int)y[m-1]);
alpar@1:             if (q == 0) goto putq; else goto test;
alpar@1:          }
alpar@1:          q = 0;
alpar@1:          r = x[i+m-1];
alpar@1: decr:    q--; /* if q = 0 then q-- = 0xFFFF */
alpar@1:          t = (unsigned int)r + (unsigned int)y[m-1];
alpar@1:          r = (unsigned short)t;
alpar@1:          if (t > 0xFFFF) goto msub;
alpar@1: test:    t = (unsigned int)y[m-2] * (unsigned int)q;
alpar@1:          if ((unsigned short)(t >> 16) > r) goto decr;
alpar@1:          if ((unsigned short)(t >> 16) < r) goto msub;
alpar@1:          if ((unsigned short)t > x[i+m-2]) goto decr;
alpar@1: msub:    /* now subtract divisor multiplied by the current digit of
alpar@1:             quotient from the current dividend */
alpar@1:          if (q == 0) goto putq;
alpar@1:          t = 0;
alpar@1:          for (j = 0; j < m; j++)
alpar@1:          {  t += (unsigned int)y[j] * (unsigned int)q;
alpar@1:             if (x[i+j] < (unsigned short)t) t += 0x10000;
alpar@1:             x[i+j] -= (unsigned short)t;
alpar@1:             t >>= 16;
alpar@1:          }
alpar@1:          if (x[i+m] >= (unsigned short)t) goto putq;
alpar@1:          /* perform correcting addition, because the current digit of
alpar@1:             quotient is greater by one than its correct value */
alpar@1:          q--;
alpar@1:          t = 0;
alpar@1:          for (j = 0; j < m; j++)
alpar@1:          {  t += (unsigned int)x[i+j] + (unsigned int)y[j];
alpar@1:             x[i+j] = (unsigned short)t;
alpar@1:             t >>= 16;
alpar@1:          }
alpar@1: putq:    /* store the current digit of quotient */
alpar@1:          x[i+m] = q;
alpar@1:       }
alpar@1:       /* divide divisor and remainder by the normalizing coefficient in
alpar@1:          order to restore their original values */
alpar@1:       if (d > 1)
alpar@1:       {  t = 0;
alpar@1:          for (i = m-1; i >= 0; i--)
alpar@1:          {  t = (t << 16) + (unsigned int)x[i];
alpar@1:             x[i] = (unsigned short)(t / (unsigned int)d);
alpar@1:             t %= (unsigned int)d;
alpar@1:          }
alpar@1:          t = 0;
alpar@1:          for (j = m-1; j >= 0; j--)
alpar@1:          {  t = (t << 16) + (unsigned int)y[j];
alpar@1:             y[j] = (unsigned short)(t / (unsigned int)d);
alpar@1:             t %= (unsigned int)d;
alpar@1:          }
alpar@1:       }
alpar@1: done: return;
alpar@1: }
alpar@1: 
alpar@1: /**********************************************************************/
alpar@1: 
alpar@1: #if 0
alpar@1: #include <assert.h>
alpar@1: #include <stdio.h>
alpar@1: #include <stdlib.h>
alpar@1: #include "glprng.h"
alpar@1: 
alpar@1: #define N_MAX 7
alpar@1: /* maximal number of digits in multiplicand */
alpar@1: 
alpar@1: #define M_MAX 5
alpar@1: /* maximal number of digits in multiplier */
alpar@1: 
alpar@1: #define N_TEST 1000000
alpar@1: /* number of tests */
alpar@1: 
alpar@1: int main(void)
alpar@1: {     RNG *rand;
alpar@1:       int d, j, n, m, test;
alpar@1:       unsigned short x[N_MAX], y[M_MAX], z[N_MAX+M_MAX];
alpar@1:       rand = rng_create_rand();
alpar@1:       for (test = 1; test <= N_TEST; test++)
alpar@1:       {  /* x[0,...,n-1] := multiplicand */
alpar@1:          n = 1 + rng_unif_rand(rand, N_MAX-1);
alpar@1:          assert(1 <= n && n <= N_MAX);
alpar@1:          for (j = 0; j < n; j++)
alpar@1:          {  d = rng_unif_rand(rand, 65536);
alpar@1:             assert(0 <= d && d <= 65535);
alpar@1:             x[j] = (unsigned short)d;
alpar@1:          }
alpar@1:          /* y[0,...,m-1] := multiplier */
alpar@1:          m = 1 + rng_unif_rand(rand, M_MAX-1);
alpar@1:          assert(1 <= m && m <= M_MAX);
alpar@1:          for (j = 0; j < m; j++)
alpar@1:          {  d = rng_unif_rand(rand, 65536);
alpar@1:             assert(0 <= d && d <= 65535);
alpar@1:             y[j] = (unsigned short)d;
alpar@1:          }
alpar@1:          if (y[m-1] == 0) y[m-1] = 1;
alpar@1:          /* z[0,...,n+m-1] := x * y */
alpar@1:          for (j = 0; j < n; j++) z[m+j] = x[j];
alpar@1:          bigmul(n, m, z, y);
alpar@1:          /* z[0,...,m-1] := z mod y, z[m,...,n+m-1] := z div y */
alpar@1:          bigdiv(n, m, z, y);
alpar@1:          /* z mod y must be 0 */
alpar@1:          for (j = 0; j < m; j++) assert(z[j] == 0);
alpar@1:          /* z div y must be x */
alpar@1:          for (j = 0; j < n; j++) assert(z[m+j] == x[j]);
alpar@1:       }
alpar@1:       fprintf(stderr, "%d tests successfully passed\n", N_TEST);
alpar@1:       rng_delete_rand(rand);
alpar@1:       return 0;
alpar@1: }
alpar@1: #endif
alpar@1: 
alpar@1: /* eof */