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/* bignum.c (bignum arithmetic) */
/***********************************************************************
* This code is part of GLPK (GNU Linear Programming Kit).
*
* Copyright (C) 2006-2013 Andrew Makhorin, Department for Applied
* Informatics, Moscow Aviation Institute, Moscow, Russia. All rights
* reserved. E-mail: <mao@gnu.org>.
*
* GLPK is free software: you can redistribute it and/or modify it
* under the terms of the GNU General Public License as published by
* the Free Software Foundation, either version 3 of the License, or
* (at your option) any later version.
*
* GLPK is distributed in the hope that it will be useful, but WITHOUT
* ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
* or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public
* License for more details.
*
* You should have received a copy of the GNU General Public License
* along with GLPK. If not, see <http://www.gnu.org/licenses/>.
***********************************************************************/
#include "env.h"
#include "bignum.h"
/***********************************************************************
* Two routines below are intended to multiply and divide unsigned
* integer numbers of arbitrary precision.
*
* The routines assume that an unsigned integer number is represented in
* the positional numeral system with the base 2^16 = 65536, i.e. each
* "digit" of the number is in the range [0, 65535] and represented as
* a 16-bit value of the unsigned short type. In other words, a number x
* has the following representation:
*
* n-1
* x = sum d[j] * 65536^j,
* j=0
*
* where n is the number of places (positions), and d[j] is j-th "digit"
* of x, 0 <= d[j] <= 65535.
***********************************************************************/
/***********************************************************************
* NAME
*
* bigmul - multiply unsigned integer numbers of arbitrary precision
*
* SYNOPSIS
*
* #include "bignum.h"
* void bigmul(int n, int m, unsigned short x[], unsigned short y[]);
*
* DESCRIPTION
*
* The routine bigmul multiplies unsigned integer numbers of arbitrary
* precision.
*
* n is the number of digits of multiplicand, n >= 1;
*
* m is the number of digits of multiplier, m >= 1;
*
* x is an array containing digits of the multiplicand in elements
* x[m], x[m+1], ..., x[n+m-1]. Contents of x[0], x[1], ..., x[m-1] are
* ignored on entry.
*
* y is an array containing digits of the multiplier in elements y[0],
* y[1], ..., y[m-1].
*
* On exit digits of the product are stored in elements x[0], x[1], ...,
* x[n+m-1]. The array y is not changed. */
void bigmul(int n, int m, unsigned short x[], unsigned short y[])
{ int i, j;
unsigned int t;
xassert(n >= 1);
xassert(m >= 1);
for (j = 0; j < m; j++) x[j] = 0;
for (i = 0; i < n; i++)
{ if (x[i+m])
{ t = 0;
for (j = 0; j < m; j++)
{ t += (unsigned int)x[i+m] * (unsigned int)y[j] +
(unsigned int)x[i+j];
x[i+j] = (unsigned short)t;
t >>= 16;
}
x[i+m] = (unsigned short)t;
}
}
return;
}
/***********************************************************************
* NAME
*
* bigdiv - divide unsigned integer numbers of arbitrary precision
*
* SYNOPSIS
*
* #include "bignum.h"
* void bigdiv(int n, int m, unsigned short x[], unsigned short y[]);
*
* DESCRIPTION
*
* The routine bigdiv divides one unsigned integer number of arbitrary
* precision by another with the algorithm described in [1].
*
* n is the difference between the number of digits of dividend and the
* number of digits of divisor, n >= 0.
*
* m is the number of digits of divisor, m >= 1.
*
* x is an array containing digits of the dividend in elements x[0],
* x[1], ..., x[n+m-1].
*
* y is an array containing digits of the divisor in elements y[0],
* y[1], ..., y[m-1]. The highest digit y[m-1] must be non-zero.
*
* On exit n+1 digits of the quotient are stored in elements x[m],
* x[m+1], ..., x[n+m], and m digits of the remainder are stored in
* elements x[0], x[1], ..., x[m-1]. The array y is changed but then
* restored.
*
* REFERENCES
*
* 1. D. Knuth. The Art of Computer Programming. Vol. 2: Seminumerical
* Algorithms. Stanford University, 1969. */
void bigdiv(int n, int m, unsigned short x[], unsigned short y[])
{ int i, j;
unsigned int t;
unsigned short d, q, r;
xassert(n >= 0);
xassert(m >= 1);
xassert(y[m-1] != 0);
/* special case when divisor has the only digit */
if (m == 1)
{ d = 0;
for (i = n; i >= 0; i--)
{ t = ((unsigned int)d << 16) + (unsigned int)x[i];
x[i+1] = (unsigned short)(t / y[0]);
d = (unsigned short)(t % y[0]);
}
x[0] = d;
goto done;
}
/* multiply dividend and divisor by a normalizing coefficient in
* order to provide the condition y[m-1] >= base / 2 */
d = (unsigned short)(0x10000 / ((unsigned int)y[m-1] + 1));
if (d == 1)
x[n+m] = 0;
else
{ t = 0;
for (i = 0; i < n+m; i++)
{ t += (unsigned int)x[i] * (unsigned int)d;
x[i] = (unsigned short)t;
t >>= 16;
}
x[n+m] = (unsigned short)t;
t = 0;
for (j = 0; j < m; j++)
{ t += (unsigned int)y[j] * (unsigned int)d;
y[j] = (unsigned short)t;
t >>= 16;
}
}
/* main loop */
for (i = n; i >= 0; i--)
{ /* estimate and correct the current digit of quotient */
if (x[i+m] < y[m-1])
{ t = ((unsigned int)x[i+m] << 16) + (unsigned int)x[i+m-1];
q = (unsigned short)(t / (unsigned int)y[m-1]);
r = (unsigned short)(t % (unsigned int)y[m-1]);
if (q == 0) goto putq; else goto test;
}
q = 0;
r = x[i+m-1];
decr: q--; /* if q = 0 then q-- = 0xFFFF */
t = (unsigned int)r + (unsigned int)y[m-1];
r = (unsigned short)t;
if (t > 0xFFFF) goto msub;
test: t = (unsigned int)y[m-2] * (unsigned int)q;
if ((unsigned short)(t >> 16) > r) goto decr;
if ((unsigned short)(t >> 16) < r) goto msub;
if ((unsigned short)t > x[i+m-2]) goto decr;
msub: /* now subtract divisor multiplied by the current digit of
* quotient from the current dividend */
if (q == 0) goto putq;
t = 0;
for (j = 0; j < m; j++)
{ t += (unsigned int)y[j] * (unsigned int)q;
if (x[i+j] < (unsigned short)t) t += 0x10000;
x[i+j] -= (unsigned short)t;
t >>= 16;
}
if (x[i+m] >= (unsigned short)t) goto putq;
/* perform correcting addition, because the current digit of
* quotient is greater by one than its correct value */
q--;
t = 0;
for (j = 0; j < m; j++)
{ t += (unsigned int)x[i+j] + (unsigned int)y[j];
x[i+j] = (unsigned short)t;
t >>= 16;
}
putq: /* store the current digit of quotient */
x[i+m] = q;
}
/* divide divisor and remainder by the normalizing coefficient in
* order to restore their original values */
if (d > 1)
{ t = 0;
for (i = m-1; i >= 0; i--)
{ t = (t << 16) + (unsigned int)x[i];
x[i] = (unsigned short)(t / (unsigned int)d);
t %= (unsigned int)d;
}
t = 0;
for (j = m-1; j >= 0; j--)
{ t = (t << 16) + (unsigned int)y[j];
y[j] = (unsigned short)(t / (unsigned int)d);
t %= (unsigned int)d;
}
}
done: return;
}
/**********************************************************************/
#ifdef GLP_TEST
#include <assert.h>
#include <stdio.h>
#include <stdlib.h>
#include "rng.h"
#define N_MAX 7
/* maximal number of digits in multiplicand */
#define M_MAX 5
/* maximal number of digits in multiplier */
#define N_TEST 1000000
/* number of tests */
int main(void)
{ RNG *rand;
int d, j, n, m, test;
unsigned short x[N_MAX], y[M_MAX], z[N_MAX+M_MAX];
rand = rng_create_rand();
for (test = 1; test <= N_TEST; test++)
{ /* x[0,...,n-1] := multiplicand */
n = 1 + rng_unif_rand(rand, N_MAX-1);
assert(1 <= n && n <= N_MAX);
for (j = 0; j < n; j++)
{ d = rng_unif_rand(rand, 65536);
assert(0 <= d && d <= 65535);
x[j] = (unsigned short)d;
}
/* y[0,...,m-1] := multiplier */
m = 1 + rng_unif_rand(rand, M_MAX-1);
assert(1 <= m && m <= M_MAX);
for (j = 0; j < m; j++)
{ d = rng_unif_rand(rand, 65536);
assert(0 <= d && d <= 65535);
y[j] = (unsigned short)d;
}
if (y[m-1] == 0) y[m-1] = 1;
/* z[0,...,n+m-1] := x * y */
for (j = 0; j < n; j++) z[m+j] = x[j];
bigmul(n, m, z, y);
/* z[0,...,m-1] := z mod y, z[m,...,n+m-1] := z div y */
bigdiv(n, m, z, y);
/* z mod y must be 0 */
for (j = 0; j < m; j++) assert(z[j] == 0);
/* z div y must be x */
for (j = 0; j < n; j++) assert(z[m+j] == x[j]);
}
fprintf(stderr, "%d tests successfully passed\n", N_TEST);
rng_delete_rand(rand);
return 0;
}
#endif
/* eof */
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