1.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
     1.2 +++ b/deps/glpk/src/glpscf.c	Sun Nov 06 20:59:10 2011 +0100
     1.3 @@ -0,0 +1,634 @@
     1.4 +/* glpscf.c (Schur complement factorization) */
     1.5 +
     1.6 +/***********************************************************************
     1.7 +*  This code is part of GLPK (GNU Linear Programming Kit).
     1.8 +*
     1.9 +*  Copyright (C) 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008,
    1.10 +*  2009, 2010, 2011 Andrew Makhorin, Department for Applied Informatics,
    1.11 +*  Moscow Aviation Institute, Moscow, Russia. All rights reserved.
    1.12 +*  E-mail: <mao@gnu.org>.
    1.13 +*
    1.14 +*  GLPK is free software: you can redistribute it and/or modify it
    1.15 +*  under the terms of the GNU General Public License as published by
    1.16 +*  the Free Software Foundation, either version 3 of the License, or
    1.17 +*  (at your option) any later version.
    1.18 +*
    1.19 +*  GLPK is distributed in the hope that it will be useful, but WITHOUT
    1.20 +*  ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
    1.21 +*  or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public
    1.22 +*  License for more details.
    1.23 +*
    1.24 +*  You should have received a copy of the GNU General Public License
    1.25 +*  along with GLPK. If not, see <http://www.gnu.org/licenses/>.
    1.26 +***********************************************************************/
    1.27 +
    1.28 +#include "glpenv.h"
    1.29 +#include "glpscf.h"
    1.30 +#define xfault xerror
    1.31 +
    1.32 +#define _GLPSCF_DEBUG 0
    1.33 +
    1.34 +#define eps 1e-10
    1.35 +
    1.36 +/***********************************************************************
    1.37 +*  NAME
    1.38 +*
    1.39 +*  scf_create_it - create Schur complement factorization
    1.40 +*
    1.41 +*  SYNOPSIS
    1.42 +*
    1.43 +*  #include "glpscf.h"
    1.44 +*  SCF *scf_create_it(int n_max);
    1.45 +*
    1.46 +*  DESCRIPTION
    1.47 +*
    1.48 +*  The routine scf_create_it creates the factorization of matrix C,
    1.49 +*  which initially has no rows and columns.
    1.50 +*
    1.51 +*  The parameter n_max specifies the maximal order of matrix C to be
    1.52 +*  factorized, 1 <= n_max <= 32767.
    1.53 +*
    1.54 +*  RETURNS
    1.55 +*
    1.56 +*  The routine scf_create_it returns a pointer to the structure SCF,
    1.57 +*  which defines the factorization. */
    1.58 +
    1.59 +SCF *scf_create_it(int n_max)
    1.60 +{     SCF *scf;
    1.61 +#if _GLPSCF_DEBUG
    1.62 +      xprintf("scf_create_it: warning: debug mode enabled\n");
    1.63 +#endif
    1.64 +      if (!(1 <= n_max && n_max <= 32767))
    1.65 +         xfault("scf_create_it: n_max = %d; invalid parameter\n",
    1.66 +            n_max);
    1.67 +      scf = xmalloc(sizeof(SCF));
    1.68 +      scf->n_max = n_max;
    1.69 +      scf->n = 0;
    1.70 +      scf->f = xcalloc(1 + n_max * n_max, sizeof(double));
    1.71 +      scf->u = xcalloc(1 + n_max * (n_max + 1) / 2, sizeof(double));
    1.72 +      scf->p = xcalloc(1 + n_max, sizeof(int));
    1.73 +      scf->t_opt = SCF_TBG;
    1.74 +      scf->rank = 0;
    1.75 +#if _GLPSCF_DEBUG
    1.76 +      scf->c = xcalloc(1 + n_max * n_max, sizeof(double));
    1.77 +#else
    1.78 +      scf->c = NULL;
    1.79 +#endif
    1.80 +      scf->w = xcalloc(1 + n_max, sizeof(double));
    1.81 +      return scf;
    1.82 +}
    1.83 +
    1.84 +/***********************************************************************
    1.85 +*  The routine f_loc determines location of matrix element F[i,j] in
    1.86 +*  the one-dimensional array f. */
    1.87 +
    1.88 +static int f_loc(SCF *scf, int i, int j)
    1.89 +{     int n_max = scf->n_max;
    1.90 +      int n = scf->n;
    1.91 +      xassert(1 <= i && i <= n);
    1.92 +      xassert(1 <= j && j <= n);
    1.93 +      return (i - 1) * n_max + j;
    1.94 +}
    1.95 +
    1.96 +/***********************************************************************
    1.97 +*  The routine u_loc determines location of matrix element U[i,j] in
    1.98 +*  the one-dimensional array u. */
    1.99 +
   1.100 +static int u_loc(SCF *scf, int i, int j)
   1.101 +{     int n_max = scf->n_max;
   1.102 +      int n = scf->n;
   1.103 +      xassert(1 <= i && i <= n);
   1.104 +      xassert(i <= j && j <= n);
   1.105 +      return (i - 1) * n_max + j - i * (i - 1) / 2;
   1.106 +}
   1.107 +
   1.108 +/***********************************************************************
   1.109 +*  The routine bg_transform applies Bartels-Golub version of gaussian
   1.110 +*  elimination to restore triangular structure of matrix U.
   1.111 +*
   1.112 +*  On entry matrix U has the following structure:
   1.113 +*
   1.114 +*        1       k         n
   1.115 +*     1  * * * * * * * * * *
   1.116 +*        . * * * * * * * * *
   1.117 +*        . . * * * * * * * *
   1.118 +*        . . . * * * * * * *
   1.119 +*     k  . . . . * * * * * *
   1.120 +*        . . . . . * * * * *
   1.121 +*        . . . . . . * * * *
   1.122 +*        . . . . . . . * * *
   1.123 +*        . . . . . . . . * *
   1.124 +*     n  . . . . # # # # # #
   1.125 +*
   1.126 +*  where '#' is a row spike to be eliminated.
   1.127 +*
   1.128 +*  Elements of n-th row are passed separately in locations un[k], ...,
   1.129 +*  un[n]. On exit the content of the array un is destroyed.
   1.130 +*
   1.131 +*  REFERENCES
   1.132 +*
   1.133 +*  R.H.Bartels, G.H.Golub, "The Simplex Method of Linear Programming
   1.134 +*  Using LU-decomposition", Comm. ACM, 12, pp. 266-68, 1969. */
   1.135 +
   1.136 +static void bg_transform(SCF *scf, int k, double un[])
   1.137 +{     int n = scf->n;
   1.138 +      double *f = scf->f;
   1.139 +      double *u = scf->u;
   1.140 +      int j, k1, kj, kk, n1, nj;
   1.141 +      double t;
   1.142 +      xassert(1 <= k && k <= n);
   1.143 +      /* main elimination loop */
   1.144 +      for (k = k; k < n; k++)
   1.145 +      {  /* determine location of U[k,k] */
   1.146 +         kk = u_loc(scf, k, k);
   1.147 +         /* determine location of F[k,1] */
   1.148 +         k1 = f_loc(scf, k, 1);
   1.149 +         /* determine location of F[n,1] */
   1.150 +         n1 = f_loc(scf, n, 1);
   1.151 +         /* if |U[k,k]| < |U[n,k]|, interchange k-th and n-th rows to
   1.152 +            provide |U[k,k]| >= |U[n,k]| */
   1.153 +         if (fabs(u[kk]) < fabs(un[k]))
   1.154 +         {  /* interchange k-th and n-th rows of matrix U */
   1.155 +            for (j = k, kj = kk; j <= n; j++, kj++)
   1.156 +               t = u[kj], u[kj] = un[j], un[j] = t;
   1.157 +            /* interchange k-th and n-th rows of matrix F to keep the
   1.158 +               main equality F * C = U * P */
   1.159 +            for (j = 1, kj = k1, nj = n1; j <= n; j++, kj++, nj++)
   1.160 +               t = f[kj], f[kj] = f[nj], f[nj] = t;
   1.161 +         }
   1.162 +         /* now |U[k,k]| >= |U[n,k]| */
   1.163 +         /* if U[k,k] is too small in the magnitude, replace U[k,k] and
   1.164 +            U[n,k] by exact zero */
   1.165 +         if (fabs(u[kk]) < eps) u[kk] = un[k] = 0.0;
   1.166 +         /* if U[n,k] is already zero, elimination is not needed */
   1.167 +         if (un[k] == 0.0) continue;
   1.168 +         /* compute gaussian multiplier t = U[n,k] / U[k,k] */
   1.169 +         t = un[k] / u[kk];
   1.170 +         /* apply gaussian elimination to nullify U[n,k] */
   1.171 +         /* (n-th row of U) := (n-th row of U) - t * (k-th row of U) */
   1.172 +         for (j = k+1, kj = kk+1; j <= n; j++, kj++)
   1.173 +            un[j] -= t * u[kj];
   1.174 +         /* (n-th row of F) := (n-th row of F) - t * (k-th row of F)
   1.175 +            to keep the main equality F * C = U * P */
   1.176 +         for (j = 1, kj = k1, nj = n1; j <= n; j++, kj++, nj++)
   1.177 +            f[nj] -= t * f[kj];
   1.178 +      }
   1.179 +      /* if U[n,n] is too small in the magnitude, replace it by exact
   1.180 +         zero */
   1.181 +      if (fabs(un[n]) < eps) un[n] = 0.0;
   1.182 +      /* store U[n,n] in a proper location */
   1.183 +      u[u_loc(scf, n, n)] = un[n];
   1.184 +      return;
   1.185 +}
   1.186 +
   1.187 +/***********************************************************************
   1.188 +*  The routine givens computes the parameters of Givens plane rotation
   1.189 +*  c = cos(teta) and s = sin(teta) such that:
   1.190 +*
   1.191 +*     ( c -s ) ( a )   ( r )
   1.192 +*     (      ) (   ) = (   ) ,
   1.193 +*     ( s  c ) ( b )   ( 0 )
   1.194 +*
   1.195 +*  where a and b are given scalars.
   1.196 +*
   1.197 +*  REFERENCES
   1.198 +*
   1.199 +*  G.H.Golub, C.F.Van Loan, "Matrix Computations", 2nd ed. */
   1.200 +
   1.201 +static void givens(double a, double b, double *c, double *s)
   1.202 +{     double t;
   1.203 +      if (b == 0.0)
   1.204 +         (*c) = 1.0, (*s) = 0.0;
   1.205 +      else if (fabs(a) <= fabs(b))
   1.206 +         t = - a / b, (*s) = 1.0 / sqrt(1.0 + t * t), (*c) = (*s) * t;
   1.207 +      else
   1.208 +         t = - b / a, (*c) = 1.0 / sqrt(1.0 + t * t), (*s) = (*c) * t;
   1.209 +      return;
   1.210 +}
   1.211 +
   1.212 +/*----------------------------------------------------------------------
   1.213 +*  The routine gr_transform applies Givens plane rotations to restore
   1.214 +*  triangular structure of matrix U.
   1.215 +*
   1.216 +*  On entry matrix U has the following structure:
   1.217 +*
   1.218 +*        1       k         n
   1.219 +*     1  * * * * * * * * * *
   1.220 +*        . * * * * * * * * *
   1.221 +*        . . * * * * * * * *
   1.222 +*        . . . * * * * * * *
   1.223 +*     k  . . . . * * * * * *
   1.224 +*        . . . . . * * * * *
   1.225 +*        . . . . . . * * * *
   1.226 +*        . . . . . . . * * *
   1.227 +*        . . . . . . . . * *
   1.228 +*     n  . . . . # # # # # #
   1.229 +*
   1.230 +*  where '#' is a row spike to be eliminated.
   1.231 +*
   1.232 +*  Elements of n-th row are passed separately in locations un[k], ...,
   1.233 +*  un[n]. On exit the content of the array un is destroyed.
   1.234 +*
   1.235 +*  REFERENCES
   1.236 +*
   1.237 +*  R.H.Bartels, G.H.Golub, "The Simplex Method of Linear Programming
   1.238 +*  Using LU-decomposition", Comm. ACM, 12, pp. 266-68, 1969. */
   1.239 +
   1.240 +static void gr_transform(SCF *scf, int k, double un[])
   1.241 +{     int n = scf->n;
   1.242 +      double *f = scf->f;
   1.243 +      double *u = scf->u;
   1.244 +      int j, k1, kj, kk, n1, nj;
   1.245 +      double c, s;
   1.246 +      xassert(1 <= k && k <= n);
   1.247 +      /* main elimination loop */
   1.248 +      for (k = k; k < n; k++)
   1.249 +      {  /* determine location of U[k,k] */
   1.250 +         kk = u_loc(scf, k, k);
   1.251 +         /* determine location of F[k,1] */
   1.252 +         k1 = f_loc(scf, k, 1);
   1.253 +         /* determine location of F[n,1] */
   1.254 +         n1 = f_loc(scf, n, 1);
   1.255 +         /* if both U[k,k] and U[n,k] are too small in the magnitude,
   1.256 +            replace them by exact zero */
   1.257 +         if (fabs(u[kk]) < eps && fabs(un[k]) < eps)
   1.258 +            u[kk] = un[k] = 0.0;
   1.259 +         /* if U[n,k] is already zero, elimination is not needed */
   1.260 +         if (un[k] == 0.0) continue;
   1.261 +         /* compute the parameters of Givens plane rotation */
   1.262 +         givens(u[kk], un[k], &c, &s);
   1.263 +         /* apply Givens rotation to k-th and n-th rows of matrix U */
   1.264 +         for (j = k, kj = kk; j <= n; j++, kj++)
   1.265 +         {  double ukj = u[kj], unj = un[j];
   1.266 +            u[kj] = c * ukj - s * unj;
   1.267 +            un[j] = s * ukj + c * unj;
   1.268 +         }
   1.269 +         /* apply Givens rotation to k-th and n-th rows of matrix F
   1.270 +            to keep the main equality F * C = U * P */
   1.271 +         for (j = 1, kj = k1, nj = n1; j <= n; j++, kj++, nj++)
   1.272 +         {  double fkj = f[kj], fnj = f[nj];
   1.273 +            f[kj] = c * fkj - s * fnj;
   1.274 +            f[nj] = s * fkj + c * fnj;
   1.275 +         }
   1.276 +      }
   1.277 +      /* if U[n,n] is too small in the magnitude, replace it by exact
   1.278 +         zero */
   1.279 +      if (fabs(un[n]) < eps) un[n] = 0.0;
   1.280 +      /* store U[n,n] in a proper location */
   1.281 +      u[u_loc(scf, n, n)] = un[n];
   1.282 +      return;
   1.283 +}
   1.284 +
   1.285 +/***********************************************************************
   1.286 +*  The routine transform restores triangular structure of matrix U.
   1.287 +*  It is a driver to the routines bg_transform and gr_transform (see
   1.288 +*  comments to these routines above). */
   1.289 +
   1.290 +static void transform(SCF *scf, int k, double un[])
   1.291 +{     switch (scf->t_opt)
   1.292 +      {  case SCF_TBG:
   1.293 +            bg_transform(scf, k, un);
   1.294 +            break;
   1.295 +         case SCF_TGR:
   1.296 +            gr_transform(scf, k, un);
   1.297 +            break;
   1.298 +         default:
   1.299 +            xassert(scf != scf);
   1.300 +      }
   1.301 +      return;
   1.302 +}
   1.303 +
   1.304 +/***********************************************************************
   1.305 +*  The routine estimate_rank estimates the rank of matrix C.
   1.306 +*
   1.307 +*  Since all transformations applied to matrix F are non-singular,
   1.308 +*  and F is assumed to be well conditioned, from the main equaility
   1.309 +*  F * C = U * P it follows that rank(C) = rank(U), where rank(U) is
   1.310 +*  estimated as the number of non-zero diagonal elements of U. */
   1.311 +
   1.312 +static int estimate_rank(SCF *scf)
   1.313 +{     int n_max = scf->n_max;
   1.314 +      int n = scf->n;
   1.315 +      double *u = scf->u;
   1.316 +      int i, ii, inc, rank = 0;
   1.317 +      for (i = 1, ii = u_loc(scf, i, i), inc = n_max; i <= n;
   1.318 +         i++, ii += inc, inc--)
   1.319 +         if (u[ii] != 0.0) rank++;
   1.320 +      return rank;
   1.321 +}
   1.322 +
   1.323 +#if _GLPSCF_DEBUG
   1.324 +/***********************************************************************
   1.325 +*  The routine check_error computes the maximal relative error between
   1.326 +*  left- and right-hand sides of the main equality F * C = U * P. (This
   1.327 +*  routine is intended only for debugging.) */
   1.328 +
   1.329 +static void check_error(SCF *scf, const char *func)
   1.330 +{     int n = scf->n;
   1.331 +      double *f = scf->f;
   1.332 +      double *u = scf->u;
   1.333 +      int *p = scf->p;
   1.334 +      double *c = scf->c;
   1.335 +      int i, j, k;
   1.336 +      double d, dmax = 0.0, s, t;
   1.337 +      xassert(c != NULL);
   1.338 +      for (i = 1; i <= n; i++)
   1.339 +      {  for (j = 1; j <= n; j++)
   1.340 +         {  /* compute element (i,j) of product F * C */
   1.341 +            s = 0.0;
   1.342 +            for (k = 1; k <= n; k++)
   1.343 +               s += f[f_loc(scf, i, k)] * c[f_loc(scf, k, j)];
   1.344 +            /* compute element (i,j) of product U * P */
   1.345 +            k = p[j];
   1.346 +            t = (i <= k ? u[u_loc(scf, i, k)] : 0.0);
   1.347 +            /* compute the maximal relative error */
   1.348 +            d = fabs(s - t) / (1.0 + fabs(t));
   1.349 +            if (dmax < d) dmax = d;
   1.350 +         }
   1.351 +      }
   1.352 +      if (dmax > 1e-8)
   1.353 +         xprintf("%s: dmax = %g; relative error too large\n", func,
   1.354 +            dmax);
   1.355 +      return;
   1.356 +}
   1.357 +#endif
   1.358 +
   1.359 +/***********************************************************************
   1.360 +*  NAME
   1.361 +*
   1.362 +*  scf_update_exp - update factorization on expanding C
   1.363 +*
   1.364 +*  SYNOPSIS
   1.365 +*
   1.366 +*  #include "glpscf.h"
   1.367 +*  int scf_update_exp(SCF *scf, const double x[], const double y[],
   1.368 +*     double z);
   1.369 +*
   1.370 +*  DESCRIPTION
   1.371 +*
   1.372 +*  The routine scf_update_exp updates the factorization of matrix C on
   1.373 +*  expanding it by adding a new row and column as follows:
   1.374 +*
   1.375 +*             ( C  x )
   1.376 +*     new C = (      )
   1.377 +*             ( y' z )
   1.378 +*
   1.379 +*  where x[1,...,n] is a new column, y[1,...,n] is a new row, and z is
   1.380 +*  a new diagonal element.
   1.381 +*
   1.382 +*  If on entry the factorization is empty, the parameters x and y can
   1.383 +*  be specified as NULL.
   1.384 +*
   1.385 +*  RETURNS
   1.386 +*
   1.387 +*  0  The factorization has been successfully updated.
   1.388 +*
   1.389 +*  SCF_ESING
   1.390 +*     The factorization has been successfully updated, however, new
   1.391 +*     matrix C is singular within working precision. Note that the new
   1.392 +*     factorization remains valid.
   1.393 +*
   1.394 +*  SCF_ELIMIT
   1.395 +*     There is not enough room to expand the factorization, because
   1.396 +*     n = n_max. The factorization remains unchanged.
   1.397 +*
   1.398 +*  ALGORITHM
   1.399 +*
   1.400 +*  We can see that:
   1.401 +*
   1.402 +*     ( F  0 ) ( C  x )   ( FC  Fx )   ( UP  Fx )
   1.403 +*     (      ) (      ) = (        ) = (        ) =
   1.404 +*     ( 0  1 ) ( y' z )   ( y'   z )   ( y'   z )
   1.405 +*
   1.406 +*        ( U   Fx ) ( P  0 )
   1.407 +*     =  (        ) (      ),
   1.408 +*        ( y'P' z ) ( 0  1 )
   1.409 +*
   1.410 +*  therefore to keep the main equality F * C = U * P we can take:
   1.411 +*
   1.412 +*             ( F  0 )           ( U   Fx )           ( P  0 )
   1.413 +*     new F = (      ),  new U = (        ),  new P = (      ),
   1.414 +*             ( 0  1 )           ( y'P' z )           ( 0  1 )
   1.415 +*
   1.416 +*  and eliminate the row spike y'P' in the last row of new U to restore
   1.417 +*  its upper triangular structure. */
   1.418 +
   1.419 +int scf_update_exp(SCF *scf, const double x[], const double y[],
   1.420 +      double z)
   1.421 +{     int n_max = scf->n_max;
   1.422 +      int n = scf->n;
   1.423 +      double *f = scf->f;
   1.424 +      double *u = scf->u;
   1.425 +      int *p = scf->p;
   1.426 +#if _GLPSCF_DEBUG
   1.427 +      double *c = scf->c;
   1.428 +#endif
   1.429 +      double *un = scf->w;
   1.430 +      int i, ij, in, j, k, nj, ret = 0;
   1.431 +      double t;
   1.432 +      /* check if the factorization can be expanded */
   1.433 +      if (n == n_max)
   1.434 +      {  /* there is not enough room */
   1.435 +         ret = SCF_ELIMIT;
   1.436 +         goto done;
   1.437 +      }
   1.438 +      /* increase the order of the factorization */
   1.439 +      scf->n = ++n;
   1.440 +      /* fill new zero column of matrix F */
   1.441 +      for (i = 1, in = f_loc(scf, i, n); i < n; i++, in += n_max)
   1.442 +         f[in] = 0.0;
   1.443 +      /* fill new zero row of matrix F */
   1.444 +      for (j = 1, nj = f_loc(scf, n, j); j < n; j++, nj++)
   1.445 +         f[nj] = 0.0;
   1.446 +      /* fill new unity diagonal element of matrix F */
   1.447 +      f[f_loc(scf, n, n)] = 1.0;
   1.448 +      /* compute new column of matrix U, which is (old F) * x */
   1.449 +      for (i = 1; i < n; i++)
   1.450 +      {  /* u[i,n] := (i-th row of old F) * x */
   1.451 +         t = 0.0;
   1.452 +         for (j = 1, ij = f_loc(scf, i, 1); j < n; j++, ij++)
   1.453 +            t += f[ij] * x[j];
   1.454 +         u[u_loc(scf, i, n)] = t;
   1.455 +      }
   1.456 +      /* compute new (spiked) row of matrix U, which is (old P) * y */
   1.457 +      for (j = 1; j < n; j++) un[j] = y[p[j]];
   1.458 +      /* store new diagonal element of matrix U, which is z */
   1.459 +      un[n] = z;
   1.460 +      /* expand matrix P */
   1.461 +      p[n] = n;
   1.462 +#if _GLPSCF_DEBUG
   1.463 +      /* expand matrix C */
   1.464 +      /* fill its new column, which is x */
   1.465 +      for (i = 1, in = f_loc(scf, i, n); i < n; i++, in += n_max)
   1.466 +         c[in] = x[i];
   1.467 +      /* fill its new row, which is y */
   1.468 +      for (j = 1, nj = f_loc(scf, n, j); j < n; j++, nj++)
   1.469 +         c[nj] = y[j];
   1.470 +      /* fill its new diagonal element, which is z */
   1.471 +      c[f_loc(scf, n, n)] = z;
   1.472 +#endif
   1.473 +      /* restore upper triangular structure of matrix U */
   1.474 +      for (k = 1; k < n; k++)
   1.475 +         if (un[k] != 0.0) break;
   1.476 +      transform(scf, k, un);
   1.477 +      /* estimate the rank of matrices C and U */
   1.478 +      scf->rank = estimate_rank(scf);
   1.479 +      if (scf->rank != n) ret = SCF_ESING;
   1.480 +#if _GLPSCF_DEBUG
   1.481 +      /* check that the factorization is accurate enough */
   1.482 +      check_error(scf, "scf_update_exp");
   1.483 +#endif
   1.484 +done: return ret;
   1.485 +}
   1.486 +
   1.487 +/***********************************************************************
   1.488 +*  The routine solve solves the system C * x = b.
   1.489 +*
   1.490 +*  From the main equation F * C = U * P it follows that:
   1.491 +*
   1.492 +*     C * x = b  =>  F * C * x = F * b  =>  U * P * x = F * b  =>
   1.493 +*
   1.494 +*     P * x = inv(U) * F * b  =>  x = P' * inv(U) * F * b.
   1.495 +*
   1.496 +*  On entry the array x contains right-hand side vector b. On exit this
   1.497 +*  array contains solution vector x. */
   1.498 +
   1.499 +static void solve(SCF *scf, double x[])
   1.500 +{     int n = scf->n;
   1.501 +      double *f = scf->f;
   1.502 +      double *u = scf->u;
   1.503 +      int *p = scf->p;
   1.504 +      double *y = scf->w;
   1.505 +      int i, j, ij;
   1.506 +      double t;
   1.507 +      /* y := F * b */
   1.508 +      for (i = 1; i <= n; i++)
   1.509 +      {  /* y[i] = (i-th row of F) * b */
   1.510 +         t = 0.0;
   1.511 +         for (j = 1, ij = f_loc(scf, i, 1); j <= n; j++, ij++)
   1.512 +            t += f[ij] * x[j];
   1.513 +         y[i] = t;
   1.514 +      }
   1.515 +      /* y := inv(U) * y */
   1.516 +      for (i = n; i >= 1; i--)
   1.517 +      {  t = y[i];
   1.518 +         for (j = n, ij = u_loc(scf, i, n); j > i; j--, ij--)
   1.519 +            t -= u[ij] * y[j];
   1.520 +         y[i] = t / u[ij];
   1.521 +      }
   1.522 +      /* x := P' * y */
   1.523 +      for (i = 1; i <= n; i++) x[p[i]] = y[i];
   1.524 +      return;
   1.525 +}
   1.526 +
   1.527 +/***********************************************************************
   1.528 +*  The routine tsolve solves the transposed system C' * x = b.
   1.529 +*
   1.530 +*  From the main equation F * C = U * P it follows that:
   1.531 +*
   1.532 +*     C' * F' = P' * U',
   1.533 +*
   1.534 +*  therefore:
   1.535 +*
   1.536 +*     C' * x = b  =>  C' * F' * inv(F') * x = b  =>
   1.537 +*
   1.538 +*     P' * U' * inv(F') * x = b  =>  U' * inv(F') * x = P * b  =>
   1.539 +*
   1.540 +*     inv(F') * x = inv(U') * P * b  =>  x = F' * inv(U') * P * b.
   1.541 +*
   1.542 +*  On entry the array x contains right-hand side vector b. On exit this
   1.543 +*  array contains solution vector x. */
   1.544 +
   1.545 +static void tsolve(SCF *scf, double x[])
   1.546 +{     int n = scf->n;
   1.547 +      double *f = scf->f;
   1.548 +      double *u = scf->u;
   1.549 +      int *p = scf->p;
   1.550 +      double *y = scf->w;
   1.551 +      int i, j, ij;
   1.552 +      double t;
   1.553 +      /* y := P * b */
   1.554 +      for (i = 1; i <= n; i++) y[i] = x[p[i]];
   1.555 +      /* y := inv(U') * y */
   1.556 +      for (i = 1; i <= n; i++)
   1.557 +      {  /* compute y[i] */
   1.558 +         ij = u_loc(scf, i, i);
   1.559 +         t = (y[i] /= u[ij]);
   1.560 +         /* substitute y[i] in other equations */
   1.561 +         for (j = i+1, ij++; j <= n; j++, ij++)
   1.562 +            y[j] -= u[ij] * t;
   1.563 +      }
   1.564 +      /* x := F' * y (computed as linear combination of rows of F) */
   1.565 +      for (j = 1; j <= n; j++) x[j] = 0.0;
   1.566 +      for (i = 1; i <= n; i++)
   1.567 +      {  t = y[i]; /* coefficient of linear combination */
   1.568 +         for (j = 1, ij = f_loc(scf, i, 1); j <= n; j++, ij++)
   1.569 +            x[j] += f[ij] * t;
   1.570 +      }
   1.571 +      return;
   1.572 +}
   1.573 +
   1.574 +/***********************************************************************
   1.575 +*  NAME
   1.576 +*
   1.577 +*  scf_solve_it - solve either system C * x = b or C' * x = b
   1.578 +*
   1.579 +*  SYNOPSIS
   1.580 +*
   1.581 +*  #include "glpscf.h"
   1.582 +*  void scf_solve_it(SCF *scf, int tr, double x[]);
   1.583 +*
   1.584 +*  DESCRIPTION
   1.585 +*
   1.586 +*  The routine scf_solve_it solves either the system C * x = b (if tr
   1.587 +*  is zero) or the system C' * x = b, where C' is a matrix transposed
   1.588 +*  to C (if tr is non-zero). C is assumed to be non-singular.
   1.589 +*
   1.590 +*  On entry the array x should contain the right-hand side vector b in
   1.591 +*  locations x[1], ..., x[n], where n is the order of matrix C. On exit
   1.592 +*  the array x contains the solution vector x in the same locations. */
   1.593 +
   1.594 +void scf_solve_it(SCF *scf, int tr, double x[])
   1.595 +{     if (scf->rank < scf->n)
   1.596 +         xfault("scf_solve_it: singular matrix\n");
   1.597 +      if (!tr)
   1.598 +         solve(scf, x);
   1.599 +      else
   1.600 +         tsolve(scf, x);
   1.601 +      return;
   1.602 +}
   1.603 +
   1.604 +void scf_reset_it(SCF *scf)
   1.605 +{     /* reset factorization for empty matrix C */
   1.606 +      scf->n = scf->rank = 0;
   1.607 +      return;
   1.608 +}
   1.609 +
   1.610 +/***********************************************************************
   1.611 +*  NAME
   1.612 +*
   1.613 +*  scf_delete_it - delete Schur complement factorization
   1.614 +*
   1.615 +*  SYNOPSIS
   1.616 +*
   1.617 +*  #include "glpscf.h"
   1.618 +*  void scf_delete_it(SCF *scf);
   1.619 +*
   1.620 +*  DESCRIPTION
   1.621 +*
   1.622 +*  The routine scf_delete_it deletes the specified factorization and
   1.623 +*  frees all the memory allocated to this object. */
   1.624 +
   1.625 +void scf_delete_it(SCF *scf)
   1.626 +{     xfree(scf->f);
   1.627 +      xfree(scf->u);
   1.628 +      xfree(scf->p);
   1.629 +#if _GLPSCF_DEBUG
   1.630 +      xfree(scf->c);
   1.631 +#endif
   1.632 +      xfree(scf->w);
   1.633 +      xfree(scf);
   1.634 +      return;
   1.635 +}
   1.636 +
   1.637 +/* eof */