/* glplib01.c (bignum arithmetic) */

/***********************************************************************

*  This code is part of GLPK (GNU Linear Programming Kit).

*

*  Copyright (C) 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008,

*  2009, 2010 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 "glpenv.h"

#include "glplib.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 "glplib.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 "glplib.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;

}

/**********************************************************************/

#if 0

#include <assert.h>

#include <stdio.h>

#include <stdlib.h>

#include "glprng.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 */