1.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
1.2 +++ b/src/glpscf.c Mon Dec 06 13:09:21 2010 +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 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 */