diff -r d59bea55db9b -r c445c931472f src/glpscf.c --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/src/glpscf.c Mon Dec 06 13:09:21 2010 +0100 @@ -0,0 +1,634 @@ +/* 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 */