/* glpscf.c (Schur complement factorization) */ /*********************************************************************** * 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: . * * 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 . ***********************************************************************/ #include "glpenv.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; } #if _GLPSCF_DEBUG /*********************************************************************** * The routine check_error computes the maximal relative error between * left- and right-hand sides of the main equality F * C = U * P. (This * routine is intended only for debugging.) */ static void check_error(SCF *scf, const char *func) { int n = scf->n; double *f = scf->f; double *u = scf->u; int *p = scf->p; double *c = scf->c; int i, j, k; double d, dmax = 0.0, s, t; xassert(c != NULL); for (i = 1; i <= n; i++) { for (j = 1; j <= n; j++) { /* compute element (i,j) of product F * C */ s = 0.0; for (k = 1; k <= n; k++) s += f[f_loc(scf, i, k)] * c[f_loc(scf, k, j)]; /* compute element (i,j) of product U * P */ k = p[j]; t = (i <= k ? u[u_loc(scf, i, k)] : 0.0); /* compute the maximal relative error */ d = fabs(s - t) / (1.0 + fabs(t)); if (dmax < d) dmax = d; } } if (dmax > 1e-8) xprintf("%s: dmax = %g; relative error too large\n", func, dmax); return; } #endif /*********************************************************************** * NAME * * scf_update_exp - update factorization on expanding C * * SYNOPSIS * * #include "glpscf.h" * int scf_update_exp(SCF *scf, const double x[], const double y[], * double z); * * DESCRIPTION * * The routine scf_update_exp updates the factorization of matrix C on * expanding it by adding a new row and column as follows: * * ( C x ) * new C = ( ) * ( y' z ) * * where x[1,...,n] is a new column, y[1,...,n] is a new row, and z is * a new diagonal element. * * If on entry the factorization is empty, the parameters x and y can * be specified as NULL. * * RETURNS * * 0 The factorization has been successfully updated. * * SCF_ESING * The factorization has been successfully updated, however, new * matrix C is singular within working precision. Note that the new * factorization remains valid. * * SCF_ELIMIT * There is not enough room to expand the factorization, because * n = n_max. The factorization remains unchanged. * * ALGORITHM * * We can see that: * * ( F 0 ) ( C x ) ( FC Fx ) ( UP Fx ) * ( ) ( ) = ( ) = ( ) = * ( 0 1 ) ( y' z ) ( y' z ) ( y' z ) * * ( U Fx ) ( P 0 ) * = ( ) ( ), * ( y'P' z ) ( 0 1 ) * * therefore to keep the main equality F * C = U * P we can take: * * ( F 0 ) ( U Fx ) ( P 0 ) * new F = ( ), new U = ( ), new P = ( ), * ( 0 1 ) ( y'P' z ) ( 0 1 ) * * and eliminate the row spike y'P' in the last row of new U to restore * its upper triangular structure. */ int scf_update_exp(SCF *scf, const double x[], const double y[], double z) { int n_max = scf->n_max; int n = scf->n; double *f = scf->f; double *u = scf->u; int *p = scf->p; #if _GLPSCF_DEBUG double *c = scf->c; #endif double *un = scf->w; int i, ij, in, j, k, nj, ret = 0; double t; /* check if the factorization can be expanded */ if (n == n_max) { /* there is not enough room */ ret = SCF_ELIMIT; goto done; } /* increase the order of the factorization */ scf->n = ++n; /* fill new zero column of matrix F */ for (i = 1, in = f_loc(scf, i, n); i < n; i++, in += n_max) f[in] = 0.0; /* fill new zero row of matrix F */ for (j = 1, nj = f_loc(scf, n, j); j < n; j++, nj++) f[nj] = 0.0; /* fill new unity diagonal element of matrix F */ f[f_loc(scf, n, n)] = 1.0; /* compute new column of matrix U, which is (old F) * x */ for (i = 1; i < n; i++) { /* u[i,n] := (i-th row of old F) * x */ t = 0.0; for (j = 1, ij = f_loc(scf, i, 1); j < n; j++, ij++) t += f[ij] * x[j]; u[u_loc(scf, i, n)] = t; } /* compute new (spiked) row of matrix U, which is (old P) * y */ for (j = 1; j < n; j++) un[j] = y[p[j]]; /* store new diagonal element of matrix U, which is z */ un[n] = z; /* expand matrix P */ p[n] = n; #if _GLPSCF_DEBUG /* expand matrix C */ /* fill its new column, which is x */ for (i = 1, in = f_loc(scf, i, n); i < n; i++, in += n_max) c[in] = x[i]; /* fill its new row, which is y */ for (j = 1, nj = f_loc(scf, n, j); j < n; j++, nj++) c[nj] = y[j]; /* fill its new diagonal element, which is z */ c[f_loc(scf, n, n)] = z; #endif /* restore upper triangular structure of matrix U */ for (k = 1; k < n; k++) if (un[k] != 0.0) break; transform(scf, k, un); /* estimate the rank of matrices C and U */ scf->rank = estimate_rank(scf); if (scf->rank != n) ret = SCF_ESING; #if _GLPSCF_DEBUG /* check that the factorization is accurate enough */ check_error(scf, "scf_update_exp"); #endif done: return ret; } /*********************************************************************** * The routine solve solves the system C * x = b. * * From the main equation F * C = U * P it follows that: * * C * x = b => F * C * x = F * b => U * P * x = F * b => * * P * x = inv(U) * F * b => x = P' * inv(U) * F * b. * * On entry the array x contains right-hand side vector b. On exit this * array contains solution vector x. */ static void solve(SCF *scf, double x[]) { int n = scf->n; double *f = scf->f; double *u = scf->u; int *p = scf->p; double *y = scf->w; int i, j, ij; double t; /* y := F * b */ for (i = 1; i <= n; i++) { /* y[i] = (i-th row of F) * b */ t = 0.0; for (j = 1, ij = f_loc(scf, i, 1); j <= n; j++, ij++) t += f[ij] * x[j]; y[i] = t; } /* y := inv(U) * y */ for (i = n; i >= 1; i--) { t = y[i]; for (j = n, ij = u_loc(scf, i, n); j > i; j--, ij--) t -= u[ij] * y[j]; y[i] = t / u[ij]; } /* x := P' * y */ for (i = 1; i <= n; i++) x[p[i]] = y[i]; return; } /*********************************************************************** * The routine tsolve solves the transposed system C' * x = b. * * From the main equation F * C = U * P it follows that: * * C' * F' = P' * U', * * therefore: * * C' * x = b => C' * F' * inv(F') * x = b => * * P' * U' * inv(F') * x = b => U' * inv(F') * x = P * b => * * inv(F') * x = inv(U') * P * b => x = F' * inv(U') * P * b. * * On entry the array x contains right-hand side vector b. On exit this * array contains solution vector x. */ static void tsolve(SCF *scf, double x[]) { int n = scf->n; double *f = scf->f; double *u = scf->u; int *p = scf->p; double *y = scf->w; int i, j, ij; double t; /* y := P * b */ for (i = 1; i <= n; i++) y[i] = x[p[i]]; /* y := inv(U') * y */ for (i = 1; i <= n; i++) { /* compute y[i] */ ij = u_loc(scf, i, i); t = (y[i] /= u[ij]); /* substitute y[i] in other equations */ for (j = i+1, ij++; j <= n; j++, ij++) y[j] -= u[ij] * t; } /* x := F' * y (computed as linear combination of rows of F) */ for (j = 1; j <= n; j++) x[j] = 0.0; for (i = 1; i <= n; i++) { t = y[i]; /* coefficient of linear combination */ for (j = 1, ij = f_loc(scf, i, 1); j <= n; j++, ij++) x[j] += f[ij] * t; } return; } /*********************************************************************** * NAME * * scf_solve_it - solve either system C * x = b or C' * x = b * * SYNOPSIS * * #include "glpscf.h" * void scf_solve_it(SCF *scf, int tr, double x[]); * * DESCRIPTION * * The routine scf_solve_it solves either the system C * x = b (if tr * is zero) or the system C' * x = b, where C' is a matrix transposed * to C (if tr is non-zero). C is assumed to be non-singular. * * On entry the array x should contain the right-hand side vector b in * locations x[1], ..., x[n], where n is the order of matrix C. On exit * the array x contains the solution vector x in the same locations. */ void scf_solve_it(SCF *scf, int tr, double x[]) { if (scf->rank < scf->n) xfault("scf_solve_it: singular matrix\n"); if (!tr) solve(scf, x); else tsolve(scf, x); return; } void scf_reset_it(SCF *scf) { /* reset factorization for empty matrix C */ scf->n = scf->rank = 0; return; } /*********************************************************************** * NAME * * scf_delete_it - delete Schur complement factorization * * SYNOPSIS * * #include "glpscf.h" * void scf_delete_it(SCF *scf); * * DESCRIPTION * * The routine scf_delete_it deletes the specified factorization and * frees all the memory allocated to this object. */ void scf_delete_it(SCF *scf) { xfree(scf->f); xfree(scf->u); xfree(scf->p); #if _GLPSCF_DEBUG xfree(scf->c); #endif xfree(scf->w); xfree(scf); return; } /* eof */