glpscf.c

著名的大规模线性规划求解器源码GLPK.C语言版本,可以修剪.内有详细帮助文档.

/* glpscf.c (Schur complement factorization) */

/***********************************************************************
*  This code is part of GLPK (GNU Linear Programming Kit).
*
*  Copyright (C) 2000,01,02,03,04,05,06,07,08,2009 Andrew Makhorin,
*  Department for Applied Informatics, Moscow Aviation Institute,
*  Moscow, Russia. All rights reserved. E-mail: <[e-mail]>.
*
*  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 "glplib.h"
#include "glpscf.h"
#define xfault xerror

#define _GLPSCF_DEBUG 0

#define eps 1e-10

/***********************************************************************
*  NAME
*
*  scf_create_it - create Schur complement factorization
*
*  SYNOPSIS
*
*  #include "glpscf.h"
*  SCF *scf_create_it(int n_max);
*
*  DESCRIPTION
*
*  The routine scf_create_it creates the factorization of matrix C,
*  which initially has no rows and columns.
*
*  The parameter n_max specifies the maximal order of matrix C to be
*  factorized, 1 <= n_max <= 32767.
*
*  RETURNS
*
*  The routine scf_create_it returns a pointer to the structure SCF,
*  which defines the factorization. */

SCF *scf_create_it(int n_max)
{     SCF *scf;
#if _GLPSCF_DEBUG
      xprintf("scf_create_it: warning: debug mode enabled\n");
#endif
      if (!(1 <= n_max && n_max <= 32767))
         xfault("scf_create_it: n_max = %d; invalid parameter\n",
            n_max);
      scf = xmalloc(sizeof(SCF));
      scf->n_max = n_max;
      scf->n = 0;
      scf->f = xcalloc(1 + n_max * n_max, sizeof(double));
      scf->u = xcalloc(1 + n_max * (n_max + 1) / 2, sizeof(double));
      scf->p = xcalloc(1 + n_max, sizeof(int));
      scf->t_opt = SCF_TBG;
      scf->rank = 0;
#if _GLPSCF_DEBUG
      scf->c = xcalloc(1 + n_max * n_max, sizeof(double));
#else
      scf->c = NULL;
#endif
      scf->w = xcalloc(1 + n_max, sizeof(double));
      return scf;
}

/***********************************************************************
*  The routine f_loc determines location of matrix element F[i,j] in
*  the one-dimensional array f. */

static int f_loc(SCF *scf, int i, int j)
{     int n_max = scf->n_max;
      int n = scf->n;
      xassert(1 <= i && i <= n);
      xassert(1 <= j && j <= n);
      return (i - 1) * n_max + j;
}

/***********************************************************************
*  The routine u_loc determines location of matrix element U[i,j] in
*  the one-dimensional array u. */

static int u_loc(SCF *scf, int i, int j)
{     int n_max = scf->n_max;
      int n = scf->n;
      xassert(1 <= i && i <= n);
      xassert(i <= j && j <= n);
      return (i - 1) * n_max + j - i * (i - 1) / 2;
}

/***********************************************************************
*  The routine bg_transform applies Bartels-Golub version of gaussian
*  elimination to restore triangular structure of matrix U.
*
*  On entry matrix U has the following structure:
*
*        1       k         n
*     1  * * * * * * * * * *
*        . * * * * * * * * *
*        . . * * * * * * * *
*        . . . * * * * * * *
*     k  . . . . * * * * * *
*        . . . . . * * * * *
*        . . . . . . * * * *
*        . . . . . . . * * *
*        . . . . . . . . * *
*     n  . . . . # # # # # #
*
*  where '#' is a row spike to be eliminated.
*
*  Elements of n-th row are passed separately in locations un[k], ...,
*  un[n]. On exit the content of the array un is destroyed.
*
*  REFERENCES
*
*  R.H.Bartels, G.H.Golub, "The Simplex Method of Linear Programming
*  Using LU-decomposition", Comm. ACM, 12, pp. 266-68, 1969. */

static void bg_transform(SCF *scf, int k, double un[])
{     int n = scf->n;
      double *f = scf->f;
      double *u = scf->u;
      int j, k1, kj, kk, n1, nj;
      double t;
      xassert(1 <= k && k <= n);
      /* main elimination loop */
      for (k = k; k < n; k++)
      {  /* determine location of U[k,k] */
         kk = u_loc(scf, k, k);
         /* determine location of F[k,1] */
         k1 = f_loc(scf, k, 1);
         /* determine location of F[n,1] */
         n1 = f_loc(scf, n, 1);
         /* if |U[k,k]| < |U[n,k]|, interchange k-th and n-th rows to
            provide |U[k,k]| >= |U[n,k]| */
         if (fabs(u[kk]) < fabs(un[k]))
         {  /* interchange k-th and n-th rows of matrix U */
            for (j = k, kj = kk; j <= n; j++, kj++)
               t = u[kj], u[kj] = un[j], un[j] = t;
            /* interchange k-th and n-th rows of matrix F to keep the
               main equality F * C = U * P */
            for (j = 1, kj = k1, nj = n1; j <= n; j++, kj++, nj++)
               t = f[kj], f[kj] = f[nj], f[nj] = t;
         }
         /* now |U[k,k]| >= |U[n,k]| */
         /* if U[k,k] is too small in the magnitude, replace U[k,k] and
            U[n,k] by exact zero */
         if (fabs(u[kk]) < eps) u[kk] = un[k] = 0.0;
         /* if U[n,k] is already zero, elimination is not needed */
         if (un[k] == 0.0) continue;
         /* compute gaussian multiplier t = U[n,k] / U[k,k] */
         t = un[k] / u[kk];
         /* apply gaussian elimination to nullify U[n,k] */
         /* (n-th row of U) := (n-th row of U) - t * (k-th row of U) */
         for (j = k+1, kj = kk+1; j <= n; j++, kj++)
            un[j] -= t * u[kj];
         /* (n-th row of F) := (n-th row of F) - t * (k-th row of F)
            to keep the main equality F * C = U * P */
         for (j = 1, kj = k1, nj = n1; j <= n; j++, kj++, nj++)
            f[nj] -= t * f[kj];
      }
      /* if U[n,n] is too small in the magnitude, replace it by exact
         zero */
      if (fabs(un[n]) < eps) un[n] = 0.0;
      /* store U[n,n] in a proper location */
      u[u_loc(scf, n, n)] = un[n];
      return;
}

/***********************************************************************
*  The routine givens computes the parameters of Givens plane rotation
*  c = cos(teta) and s = sin(teta) such that:
*
*     ( c -s ) ( a )   ( r )
*     (      ) (   ) = (   ) ,
*     ( s  c ) ( b )   ( 0 )
*
*  where a and b are given scalars.
*
*  REFERENCES
*
*  G.H.Golub, C.F.Van Loan, "Matrix Computations", 2nd ed. */

static void givens(double a, double b, double *c, double *s)
{     double t;
      if (b == 0.0)
         (*c) = 1.0, (*s) = 0.0;
      else if (fabs(a) <= fabs(b))
         t = - a / b, (*s) = 1.0 / sqrt(1.0 + t * t), (*c) = (*s) * t;
      else
         t = - b / a, (*c) = 1.0 / sqrt(1.0 + t * t), (*s) = (*c) * t;
      return;
}

/*----------------------------------------------------------------------
*  The routine gr_transform applies Givens plane rotations to restore
*  triangular structure of matrix U.
*
*  On entry matrix U has the following structure:
*
*        1       k         n
*     1  * * * * * * * * * *
*        . * * * * * * * * *
*        . . * * * * * * * *
*        . . . * * * * * * *
*     k  . . . . * * * * * *
*        . . . . . * * * * *
*        . . . . . . * * * *
*        . . . . . . . * * *
*        . . . . . . . . * *
*     n  . . . . # # # # # #
*
*  where '#' is a row spike to be eliminated.
*
*  Elements of n-th row are passed separately in locations un[k], ...,
*  un[n]. On exit the content of the array un is destroyed.
*
*  REFERENCES
*
*  R.H.Bartels, G.H.Golub, "The Simplex Method of Linear Programming
*  Using LU-decomposition", Comm. ACM, 12, pp. 266-68, 1969. */

static void gr_transform(SCF *scf, int k, double un[])
{     int n = scf->n;
      double *f = scf->f;
      double *u = scf->u;
      int j, k1, kj, kk, n1, nj;
      double c, s;
      xassert(1 <= k && k <= n);
      /* main elimination loop */
      for (k = k; k < n; k++)
      {  /* determine location of U[k,k] */
         kk = u_loc(scf, k, k);
         /* determine location of F[k,1] */
         k1 = f_loc(scf, k, 1);
         /* determine location of F[n,1] */
         n1 = f_loc(scf, n, 1);
         /* if both U[k,k] and U[n,k] are too small in the magnitude,
            replace them by exact zero */
         if (fabs(u[kk]) < eps && fabs(un[k]) < eps)
            u[kk] = un[k] = 0.0;
         /* if U[n,k] is already zero, elimination is not needed */
         if (un[k] == 0.0) continue;
         /* compute the parameters of Givens plane rotation */
         givens(u[kk], un[k], &c, &s);
         /* apply Givens rotation to k-th and n-th rows of matrix U */
         for (j = k, kj = kk; j <= n; j++, kj++)
         {  double ukj = u[kj], unj = un[j];
            u[kj] = c * ukj - s * unj;
            un[j] = s * ukj + c * unj;
         }
         /* apply Givens rotation to k-th and n-th rows of matrix F
            to keep the main equality F * C = U * P */
         for (j = 1, kj = k1, nj = n1; j <= n; j++, kj++, nj++)
         {  double fkj = f[kj], fnj = f[nj];
            f[kj] = c * fkj - s * fnj;
            f[nj] = s * fkj + c * fnj;
         }
      }
      /* if U[n,n] is too small in the magnitude, replace it by exact
         zero */
      if (fabs(un[n]) < eps) un[n] = 0.0;
      /* store U[n,n] in a proper location */
      u[u_loc(scf, n, n)] = un[n];
      return;
}

/***********************************************************************
*  The routine transform restores triangular structure of matrix U.
*  It is a driver to the routines bg_transform and gr_transform (see
*  comments to these routines above). */

static void transform(SCF *scf, int k, double un[])
{     switch (scf->t_opt)
      {  case SCF_TBG:
            bg_transform(scf, k, un);
            break;
         case SCF_TGR:
            gr_transform(scf, k, un);
            break;
         default:
            xassert(scf != scf);
      }
      return;
}

/***********************************************************************
*  The routine estimate_rank estimates the rank of matrix C.
*
*  Since all transformations applied to matrix F are non-singular,
*  and F is assumed to be well conditioned, from the main equaility
*  F * C = U * P it follows that rank(C) = rank(U), where rank(U) is
*  estimated as the number of non-zero diagonal elements of U. */

static int estimate_rank(SCF *scf)
{     int n_max = scf->n_max;
      int n = scf->n;
      double *u = scf->u;
      int i, ii, inc, rank = 0;
      for (i = 1, ii = u_loc(scf, i, i), inc = n_max; i <= n;
         i++, ii += inc, inc--)
         if (u[ii] != 0.0) rank++;
      return rank;
}