[1] | 1 | /* glpscf.c (Schur complement factorization) */ |
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| 2 | |
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| 3 | /*********************************************************************** |
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| 4 | * This code is part of GLPK (GNU Linear Programming Kit). |
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| 5 | * |
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| 6 | * Copyright (C) 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, |
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| 7 | * 2009, 2010 Andrew Makhorin, Department for Applied Informatics, |
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| 8 | * Moscow Aviation Institute, Moscow, Russia. All rights reserved. |
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| 9 | * E-mail: <mao@gnu.org>. |
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| 10 | * |
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| 11 | * GLPK is free software: you can redistribute it and/or modify it |
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| 12 | * under the terms of the GNU General Public License as published by |
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| 13 | * the Free Software Foundation, either version 3 of the License, or |
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| 14 | * (at your option) any later version. |
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| 15 | * |
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| 16 | * GLPK is distributed in the hope that it will be useful, but WITHOUT |
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| 17 | * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY |
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| 18 | * or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public |
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| 19 | * License for more details. |
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| 20 | * |
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| 21 | * You should have received a copy of the GNU General Public License |
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| 22 | * along with GLPK. If not, see <http://www.gnu.org/licenses/>. |
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| 23 | ***********************************************************************/ |
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| 24 | |
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| 25 | #include "glpenv.h" |
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| 26 | #include "glpscf.h" |
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| 27 | #define xfault xerror |
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| 28 | |
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| 29 | #define _GLPSCF_DEBUG 0 |
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| 30 | |
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| 31 | #define eps 1e-10 |
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| 32 | |
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| 33 | /*********************************************************************** |
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| 34 | * NAME |
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| 35 | * |
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| 36 | * scf_create_it - create Schur complement factorization |
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| 37 | * |
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| 38 | * SYNOPSIS |
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| 39 | * |
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| 40 | * #include "glpscf.h" |
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| 41 | * SCF *scf_create_it(int n_max); |
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| 42 | * |
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| 43 | * DESCRIPTION |
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| 44 | * |
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| 45 | * The routine scf_create_it creates the factorization of matrix C, |
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| 46 | * which initially has no rows and columns. |
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| 47 | * |
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| 48 | * The parameter n_max specifies the maximal order of matrix C to be |
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| 49 | * factorized, 1 <= n_max <= 32767. |
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| 50 | * |
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| 51 | * RETURNS |
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| 52 | * |
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| 53 | * The routine scf_create_it returns a pointer to the structure SCF, |
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| 54 | * which defines the factorization. */ |
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| 55 | |
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| 56 | SCF *scf_create_it(int n_max) |
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| 57 | { SCF *scf; |
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| 58 | #if _GLPSCF_DEBUG |
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| 59 | xprintf("scf_create_it: warning: debug mode enabled\n"); |
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| 60 | #endif |
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| 61 | if (!(1 <= n_max && n_max <= 32767)) |
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| 62 | xfault("scf_create_it: n_max = %d; invalid parameter\n", |
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| 63 | n_max); |
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| 64 | scf = xmalloc(sizeof(SCF)); |
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| 65 | scf->n_max = n_max; |
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| 66 | scf->n = 0; |
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| 67 | scf->f = xcalloc(1 + n_max * n_max, sizeof(double)); |
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| 68 | scf->u = xcalloc(1 + n_max * (n_max + 1) / 2, sizeof(double)); |
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| 69 | scf->p = xcalloc(1 + n_max, sizeof(int)); |
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| 70 | scf->t_opt = SCF_TBG; |
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| 71 | scf->rank = 0; |
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| 72 | #if _GLPSCF_DEBUG |
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| 73 | scf->c = xcalloc(1 + n_max * n_max, sizeof(double)); |
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| 74 | #else |
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| 75 | scf->c = NULL; |
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| 76 | #endif |
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| 77 | scf->w = xcalloc(1 + n_max, sizeof(double)); |
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| 78 | return scf; |
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| 79 | } |
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| 80 | |
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| 81 | /*********************************************************************** |
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| 82 | * The routine f_loc determines location of matrix element F[i,j] in |
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| 83 | * the one-dimensional array f. */ |
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| 84 | |
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| 85 | static int f_loc(SCF *scf, int i, int j) |
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| 86 | { int n_max = scf->n_max; |
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| 87 | int n = scf->n; |
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| 88 | xassert(1 <= i && i <= n); |
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| 89 | xassert(1 <= j && j <= n); |
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| 90 | return (i - 1) * n_max + j; |
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| 91 | } |
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| 92 | |
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| 93 | /*********************************************************************** |
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| 94 | * The routine u_loc determines location of matrix element U[i,j] in |
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| 95 | * the one-dimensional array u. */ |
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| 96 | |
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| 97 | static int u_loc(SCF *scf, int i, int j) |
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| 98 | { int n_max = scf->n_max; |
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| 99 | int n = scf->n; |
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| 100 | xassert(1 <= i && i <= n); |
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| 101 | xassert(i <= j && j <= n); |
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| 102 | return (i - 1) * n_max + j - i * (i - 1) / 2; |
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| 103 | } |
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| 104 | |
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| 105 | /*********************************************************************** |
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| 106 | * The routine bg_transform applies Bartels-Golub version of gaussian |
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| 107 | * elimination to restore triangular structure of matrix U. |
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| 108 | * |
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| 109 | * On entry matrix U has the following structure: |
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| 110 | * |
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| 111 | * 1 k n |
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| 112 | * 1 * * * * * * * * * * |
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| 113 | * . * * * * * * * * * |
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| 114 | * . . * * * * * * * * |
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| 115 | * . . . * * * * * * * |
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| 116 | * k . . . . * * * * * * |
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| 117 | * . . . . . * * * * * |
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| 118 | * . . . . . . * * * * |
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| 119 | * . . . . . . . * * * |
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| 120 | * . . . . . . . . * * |
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| 121 | * n . . . . # # # # # # |
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| 122 | * |
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| 123 | * where '#' is a row spike to be eliminated. |
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| 124 | * |
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| 125 | * Elements of n-th row are passed separately in locations un[k], ..., |
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| 126 | * un[n]. On exit the content of the array un is destroyed. |
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| 127 | * |
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| 128 | * REFERENCES |
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| 129 | * |
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| 130 | * R.H.Bartels, G.H.Golub, "The Simplex Method of Linear Programming |
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| 131 | * Using LU-decomposition", Comm. ACM, 12, pp. 266-68, 1969. */ |
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| 132 | |
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| 133 | static void bg_transform(SCF *scf, int k, double un[]) |
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| 134 | { int n = scf->n; |
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| 135 | double *f = scf->f; |
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| 136 | double *u = scf->u; |
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| 137 | int j, k1, kj, kk, n1, nj; |
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| 138 | double t; |
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| 139 | xassert(1 <= k && k <= n); |
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| 140 | /* main elimination loop */ |
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| 141 | for (k = k; k < n; k++) |
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| 142 | { /* determine location of U[k,k] */ |
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| 143 | kk = u_loc(scf, k, k); |
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| 144 | /* determine location of F[k,1] */ |
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| 145 | k1 = f_loc(scf, k, 1); |
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| 146 | /* determine location of F[n,1] */ |
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| 147 | n1 = f_loc(scf, n, 1); |
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| 148 | /* if |U[k,k]| < |U[n,k]|, interchange k-th and n-th rows to |
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| 149 | provide |U[k,k]| >= |U[n,k]| */ |
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| 150 | if (fabs(u[kk]) < fabs(un[k])) |
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| 151 | { /* interchange k-th and n-th rows of matrix U */ |
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| 152 | for (j = k, kj = kk; j <= n; j++, kj++) |
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| 153 | t = u[kj], u[kj] = un[j], un[j] = t; |
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| 154 | /* interchange k-th and n-th rows of matrix F to keep the |
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| 155 | main equality F * C = U * P */ |
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| 156 | for (j = 1, kj = k1, nj = n1; j <= n; j++, kj++, nj++) |
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| 157 | t = f[kj], f[kj] = f[nj], f[nj] = t; |
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| 158 | } |
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| 159 | /* now |U[k,k]| >= |U[n,k]| */ |
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| 160 | /* if U[k,k] is too small in the magnitude, replace U[k,k] and |
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| 161 | U[n,k] by exact zero */ |
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| 162 | if (fabs(u[kk]) < eps) u[kk] = un[k] = 0.0; |
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| 163 | /* if U[n,k] is already zero, elimination is not needed */ |
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| 164 | if (un[k] == 0.0) continue; |
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| 165 | /* compute gaussian multiplier t = U[n,k] / U[k,k] */ |
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| 166 | t = un[k] / u[kk]; |
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| 167 | /* apply gaussian elimination to nullify U[n,k] */ |
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| 168 | /* (n-th row of U) := (n-th row of U) - t * (k-th row of U) */ |
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| 169 | for (j = k+1, kj = kk+1; j <= n; j++, kj++) |
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| 170 | un[j] -= t * u[kj]; |
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| 171 | /* (n-th row of F) := (n-th row of F) - t * (k-th row of F) |
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| 172 | to keep the main equality F * C = U * P */ |
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| 173 | for (j = 1, kj = k1, nj = n1; j <= n; j++, kj++, nj++) |
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| 174 | f[nj] -= t * f[kj]; |
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| 175 | } |
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| 176 | /* if U[n,n] is too small in the magnitude, replace it by exact |
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| 177 | zero */ |
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| 178 | if (fabs(un[n]) < eps) un[n] = 0.0; |
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| 179 | /* store U[n,n] in a proper location */ |
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| 180 | u[u_loc(scf, n, n)] = un[n]; |
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| 181 | return; |
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| 182 | } |
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| 183 | |
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| 184 | /*********************************************************************** |
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| 185 | * The routine givens computes the parameters of Givens plane rotation |
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| 186 | * c = cos(teta) and s = sin(teta) such that: |
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| 187 | * |
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| 188 | * ( c -s ) ( a ) ( r ) |
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| 189 | * ( ) ( ) = ( ) , |
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| 190 | * ( s c ) ( b ) ( 0 ) |
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| 191 | * |
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| 192 | * where a and b are given scalars. |
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| 193 | * |
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| 194 | * REFERENCES |
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| 195 | * |
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| 196 | * G.H.Golub, C.F.Van Loan, "Matrix Computations", 2nd ed. */ |
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| 197 | |
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| 198 | static void givens(double a, double b, double *c, double *s) |
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| 199 | { double t; |
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| 200 | if (b == 0.0) |
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| 201 | (*c) = 1.0, (*s) = 0.0; |
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| 202 | else if (fabs(a) <= fabs(b)) |
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| 203 | t = - a / b, (*s) = 1.0 / sqrt(1.0 + t * t), (*c) = (*s) * t; |
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| 204 | else |
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| 205 | t = - b / a, (*c) = 1.0 / sqrt(1.0 + t * t), (*s) = (*c) * t; |
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| 206 | return; |
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| 207 | } |
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| 208 | |
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| 209 | /*---------------------------------------------------------------------- |
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| 210 | * The routine gr_transform applies Givens plane rotations to restore |
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| 211 | * triangular structure of matrix U. |
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| 212 | * |
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| 213 | * On entry matrix U has the following structure: |
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| 214 | * |
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| 215 | * 1 k n |
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| 216 | * 1 * * * * * * * * * * |
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| 217 | * . * * * * * * * * * |
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| 218 | * . . * * * * * * * * |
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| 219 | * . . . * * * * * * * |
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| 220 | * k . . . . * * * * * * |
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| 221 | * . . . . . * * * * * |
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| 222 | * . . . . . . * * * * |
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| 223 | * . . . . . . . * * * |
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| 224 | * . . . . . . . . * * |
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| 225 | * n . . . . # # # # # # |
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| 226 | * |
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| 227 | * where '#' is a row spike to be eliminated. |
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| 228 | * |
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| 229 | * Elements of n-th row are passed separately in locations un[k], ..., |
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| 230 | * un[n]. On exit the content of the array un is destroyed. |
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| 231 | * |
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| 232 | * REFERENCES |
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| 233 | * |
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| 234 | * R.H.Bartels, G.H.Golub, "The Simplex Method of Linear Programming |
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| 235 | * Using LU-decomposition", Comm. ACM, 12, pp. 266-68, 1969. */ |
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| 236 | |
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| 237 | static void gr_transform(SCF *scf, int k, double un[]) |
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| 238 | { int n = scf->n; |
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| 239 | double *f = scf->f; |
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| 240 | double *u = scf->u; |
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| 241 | int j, k1, kj, kk, n1, nj; |
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| 242 | double c, s; |
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| 243 | xassert(1 <= k && k <= n); |
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| 244 | /* main elimination loop */ |
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| 245 | for (k = k; k < n; k++) |
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| 246 | { /* determine location of U[k,k] */ |
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| 247 | kk = u_loc(scf, k, k); |
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| 248 | /* determine location of F[k,1] */ |
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| 249 | k1 = f_loc(scf, k, 1); |
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| 250 | /* determine location of F[n,1] */ |
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| 251 | n1 = f_loc(scf, n, 1); |
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| 252 | /* if both U[k,k] and U[n,k] are too small in the magnitude, |
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| 253 | replace them by exact zero */ |
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| 254 | if (fabs(u[kk]) < eps && fabs(un[k]) < eps) |
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| 255 | u[kk] = un[k] = 0.0; |
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| 256 | /* if U[n,k] is already zero, elimination is not needed */ |
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| 257 | if (un[k] == 0.0) continue; |
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| 258 | /* compute the parameters of Givens plane rotation */ |
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| 259 | givens(u[kk], un[k], &c, &s); |
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| 260 | /* apply Givens rotation to k-th and n-th rows of matrix U */ |
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| 261 | for (j = k, kj = kk; j <= n; j++, kj++) |
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| 262 | { double ukj = u[kj], unj = un[j]; |
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| 263 | u[kj] = c * ukj - s * unj; |
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| 264 | un[j] = s * ukj + c * unj; |
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| 265 | } |
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| 266 | /* apply Givens rotation to k-th and n-th rows of matrix F |
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| 267 | to keep the main equality F * C = U * P */ |
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| 268 | for (j = 1, kj = k1, nj = n1; j <= n; j++, kj++, nj++) |
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| 269 | { double fkj = f[kj], fnj = f[nj]; |
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| 270 | f[kj] = c * fkj - s * fnj; |
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| 271 | f[nj] = s * fkj + c * fnj; |
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| 272 | } |
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| 273 | } |
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| 274 | /* if U[n,n] is too small in the magnitude, replace it by exact |
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| 275 | zero */ |
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| 276 | if (fabs(un[n]) < eps) un[n] = 0.0; |
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| 277 | /* store U[n,n] in a proper location */ |
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| 278 | u[u_loc(scf, n, n)] = un[n]; |
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| 279 | return; |
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| 280 | } |
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| 281 | |
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| 282 | /*********************************************************************** |
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| 283 | * The routine transform restores triangular structure of matrix U. |
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| 284 | * It is a driver to the routines bg_transform and gr_transform (see |
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| 285 | * comments to these routines above). */ |
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| 286 | |
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| 287 | static void transform(SCF *scf, int k, double un[]) |
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| 288 | { switch (scf->t_opt) |
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| 289 | { case SCF_TBG: |
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| 290 | bg_transform(scf, k, un); |
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| 291 | break; |
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| 292 | case SCF_TGR: |
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| 293 | gr_transform(scf, k, un); |
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| 294 | break; |
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| 295 | default: |
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| 296 | xassert(scf != scf); |
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| 297 | } |
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| 298 | return; |
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| 299 | } |
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| 300 | |
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| 301 | /*********************************************************************** |
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| 302 | * The routine estimate_rank estimates the rank of matrix C. |
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| 303 | * |
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| 304 | * Since all transformations applied to matrix F are non-singular, |
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| 305 | * and F is assumed to be well conditioned, from the main equaility |
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| 306 | * F * C = U * P it follows that rank(C) = rank(U), where rank(U) is |
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| 307 | * estimated as the number of non-zero diagonal elements of U. */ |
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| 308 | |
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| 309 | static int estimate_rank(SCF *scf) |
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| 310 | { int n_max = scf->n_max; |
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| 311 | int n = scf->n; |
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| 312 | double *u = scf->u; |
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| 313 | int i, ii, inc, rank = 0; |
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| 314 | for (i = 1, ii = u_loc(scf, i, i), inc = n_max; i <= n; |
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| 315 | i++, ii += inc, inc--) |
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| 316 | if (u[ii] != 0.0) rank++; |
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| 317 | return rank; |
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| 318 | } |
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| 319 | |
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| 320 | #if _GLPSCF_DEBUG |
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| 321 | /*********************************************************************** |
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| 322 | * The routine check_error computes the maximal relative error between |
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| 323 | * left- and right-hand sides of the main equality F * C = U * P. (This |
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| 324 | * routine is intended only for debugging.) */ |
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| 325 | |
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| 326 | static void check_error(SCF *scf, const char *func) |
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| 327 | { int n = scf->n; |
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| 328 | double *f = scf->f; |
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| 329 | double *u = scf->u; |
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| 330 | int *p = scf->p; |
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| 331 | double *c = scf->c; |
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| 332 | int i, j, k; |
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| 333 | double d, dmax = 0.0, s, t; |
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| 334 | xassert(c != NULL); |
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| 335 | for (i = 1; i <= n; i++) |
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| 336 | { for (j = 1; j <= n; j++) |
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| 337 | { /* compute element (i,j) of product F * C */ |
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| 338 | s = 0.0; |
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| 339 | for (k = 1; k <= n; k++) |
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| 340 | s += f[f_loc(scf, i, k)] * c[f_loc(scf, k, j)]; |
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| 341 | /* compute element (i,j) of product U * P */ |
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| 342 | k = p[j]; |
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| 343 | t = (i <= k ? u[u_loc(scf, i, k)] : 0.0); |
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| 344 | /* compute the maximal relative error */ |
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| 345 | d = fabs(s - t) / (1.0 + fabs(t)); |
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| 346 | if (dmax < d) dmax = d; |
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| 347 | } |
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| 348 | } |
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| 349 | if (dmax > 1e-8) |
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| 350 | xprintf("%s: dmax = %g; relative error too large\n", func, |
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| 351 | dmax); |
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| 352 | return; |
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| 353 | } |
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| 354 | #endif |
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| 355 | |
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| 356 | /*********************************************************************** |
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| 357 | * NAME |
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| 358 | * |
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| 359 | * scf_update_exp - update factorization on expanding C |
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| 360 | * |
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| 361 | * SYNOPSIS |
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| 362 | * |
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| 363 | * #include "glpscf.h" |
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| 364 | * int scf_update_exp(SCF *scf, const double x[], const double y[], |
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| 365 | * double z); |
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| 366 | * |
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| 367 | * DESCRIPTION |
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| 368 | * |
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| 369 | * The routine scf_update_exp updates the factorization of matrix C on |
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| 370 | * expanding it by adding a new row and column as follows: |
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| 371 | * |
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| 372 | * ( C x ) |
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| 373 | * new C = ( ) |
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| 374 | * ( y' z ) |
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| 375 | * |
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| 376 | * where x[1,...,n] is a new column, y[1,...,n] is a new row, and z is |
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| 377 | * a new diagonal element. |
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| 378 | * |
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| 379 | * If on entry the factorization is empty, the parameters x and y can |
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| 380 | * be specified as NULL. |
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| 381 | * |
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| 382 | * RETURNS |
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| 383 | * |
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| 384 | * 0 The factorization has been successfully updated. |
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| 385 | * |
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| 386 | * SCF_ESING |
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| 387 | * The factorization has been successfully updated, however, new |
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| 388 | * matrix C is singular within working precision. Note that the new |
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| 389 | * factorization remains valid. |
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| 390 | * |
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| 391 | * SCF_ELIMIT |
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| 392 | * There is not enough room to expand the factorization, because |
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| 393 | * n = n_max. The factorization remains unchanged. |
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| 394 | * |
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| 395 | * ALGORITHM |
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| 396 | * |
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| 397 | * We can see that: |
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| 398 | * |
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| 399 | * ( F 0 ) ( C x ) ( FC Fx ) ( UP Fx ) |
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| 400 | * ( ) ( ) = ( ) = ( ) = |
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| 401 | * ( 0 1 ) ( y' z ) ( y' z ) ( y' z ) |
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| 402 | * |
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| 403 | * ( U Fx ) ( P 0 ) |
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| 404 | * = ( ) ( ), |
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| 405 | * ( y'P' z ) ( 0 1 ) |
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| 406 | * |
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| 407 | * therefore to keep the main equality F * C = U * P we can take: |
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| 408 | * |
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| 409 | * ( F 0 ) ( U Fx ) ( P 0 ) |
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| 410 | * new F = ( ), new U = ( ), new P = ( ), |
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| 411 | * ( 0 1 ) ( y'P' z ) ( 0 1 ) |
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| 412 | * |
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| 413 | * and eliminate the row spike y'P' in the last row of new U to restore |
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| 414 | * its upper triangular structure. */ |
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| 415 | |
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| 416 | int scf_update_exp(SCF *scf, const double x[], const double y[], |
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| 417 | double z) |
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| 418 | { int n_max = scf->n_max; |
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| 419 | int n = scf->n; |
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| 420 | double *f = scf->f; |
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| 421 | double *u = scf->u; |
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| 422 | int *p = scf->p; |
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| 423 | #if _GLPSCF_DEBUG |
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| 424 | double *c = scf->c; |
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| 425 | #endif |
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| 426 | double *un = scf->w; |
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| 427 | int i, ij, in, j, k, nj, ret = 0; |
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| 428 | double t; |
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| 429 | /* check if the factorization can be expanded */ |
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| 430 | if (n == n_max) |
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| 431 | { /* there is not enough room */ |
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| 432 | ret = SCF_ELIMIT; |
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| 433 | goto done; |
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| 434 | } |
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| 435 | /* increase the order of the factorization */ |
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| 436 | scf->n = ++n; |
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| 437 | /* fill new zero column of matrix F */ |
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| 438 | for (i = 1, in = f_loc(scf, i, n); i < n; i++, in += n_max) |
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| 439 | f[in] = 0.0; |
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| 440 | /* fill new zero row of matrix F */ |
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| 441 | for (j = 1, nj = f_loc(scf, n, j); j < n; j++, nj++) |
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| 442 | f[nj] = 0.0; |
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| 443 | /* fill new unity diagonal element of matrix F */ |
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| 444 | f[f_loc(scf, n, n)] = 1.0; |
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| 445 | /* compute new column of matrix U, which is (old F) * x */ |
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| 446 | for (i = 1; i < n; i++) |
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| 447 | { /* u[i,n] := (i-th row of old F) * x */ |
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| 448 | t = 0.0; |
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| 449 | for (j = 1, ij = f_loc(scf, i, 1); j < n; j++, ij++) |
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| 450 | t += f[ij] * x[j]; |
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| 451 | u[u_loc(scf, i, n)] = t; |
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| 452 | } |
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| 453 | /* compute new (spiked) row of matrix U, which is (old P) * y */ |
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| 454 | for (j = 1; j < n; j++) un[j] = y[p[j]]; |
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| 455 | /* store new diagonal element of matrix U, which is z */ |
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| 456 | un[n] = z; |
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| 457 | /* expand matrix P */ |
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| 458 | p[n] = n; |
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| 459 | #if _GLPSCF_DEBUG |
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| 460 | /* expand matrix C */ |
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| 461 | /* fill its new column, which is x */ |
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| 462 | for (i = 1, in = f_loc(scf, i, n); i < n; i++, in += n_max) |
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| 463 | c[in] = x[i]; |
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| 464 | /* fill its new row, which is y */ |
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| 465 | for (j = 1, nj = f_loc(scf, n, j); j < n; j++, nj++) |
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| 466 | c[nj] = y[j]; |
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| 467 | /* fill its new diagonal element, which is z */ |
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| 468 | c[f_loc(scf, n, n)] = z; |
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| 469 | #endif |
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| 470 | /* restore upper triangular structure of matrix U */ |
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| 471 | for (k = 1; k < n; k++) |
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| 472 | if (un[k] != 0.0) break; |
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| 473 | transform(scf, k, un); |
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| 474 | /* estimate the rank of matrices C and U */ |
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| 475 | scf->rank = estimate_rank(scf); |
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| 476 | if (scf->rank != n) ret = SCF_ESING; |
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| 477 | #if _GLPSCF_DEBUG |
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| 478 | /* check that the factorization is accurate enough */ |
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| 479 | check_error(scf, "scf_update_exp"); |
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| 480 | #endif |
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| 481 | done: return ret; |
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| 482 | } |
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| 483 | |
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| 484 | /*********************************************************************** |
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| 485 | * The routine solve solves the system C * x = b. |
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| 486 | * |
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| 487 | * From the main equation F * C = U * P it follows that: |
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| 488 | * |
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| 489 | * C * x = b => F * C * x = F * b => U * P * x = F * b => |
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| 490 | * |
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| 491 | * P * x = inv(U) * F * b => x = P' * inv(U) * F * b. |
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| 492 | * |
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| 493 | * On entry the array x contains right-hand side vector b. On exit this |
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| 494 | * array contains solution vector x. */ |
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| 495 | |
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| 496 | static void solve(SCF *scf, double x[]) |
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| 497 | { int n = scf->n; |
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| 498 | double *f = scf->f; |
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| 499 | double *u = scf->u; |
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| 500 | int *p = scf->p; |
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| 501 | double *y = scf->w; |
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| 502 | int i, j, ij; |
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| 503 | double t; |
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| 504 | /* y := F * b */ |
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| 505 | for (i = 1; i <= n; i++) |
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| 506 | { /* y[i] = (i-th row of F) * b */ |
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| 507 | t = 0.0; |
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| 508 | for (j = 1, ij = f_loc(scf, i, 1); j <= n; j++, ij++) |
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| 509 | t += f[ij] * x[j]; |
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| 510 | y[i] = t; |
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| 511 | } |
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| 512 | /* y := inv(U) * y */ |
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| 513 | for (i = n; i >= 1; i--) |
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| 514 | { t = y[i]; |
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| 515 | for (j = n, ij = u_loc(scf, i, n); j > i; j--, ij--) |
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| 516 | t -= u[ij] * y[j]; |
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| 517 | y[i] = t / u[ij]; |
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| 518 | } |
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| 519 | /* x := P' * y */ |
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| 520 | for (i = 1; i <= n; i++) x[p[i]] = y[i]; |
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| 521 | return; |
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| 522 | } |
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| 523 | |
---|
| 524 | /*********************************************************************** |
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| 525 | * The routine tsolve solves the transposed system C' * x = b. |
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| 526 | * |
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| 527 | * From the main equation F * C = U * P it follows that: |
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| 528 | * |
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| 529 | * C' * F' = P' * U', |
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| 530 | * |
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| 531 | * therefore: |
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| 532 | * |
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| 533 | * C' * x = b => C' * F' * inv(F') * x = b => |
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| 534 | * |
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| 535 | * P' * U' * inv(F') * x = b => U' * inv(F') * x = P * b => |
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| 536 | * |
---|
| 537 | * inv(F') * x = inv(U') * P * b => x = F' * inv(U') * P * b. |
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| 538 | * |
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| 539 | * On entry the array x contains right-hand side vector b. On exit this |
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| 540 | * array contains solution vector x. */ |
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| 541 | |
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| 542 | static void tsolve(SCF *scf, double x[]) |
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| 543 | { int n = scf->n; |
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| 544 | double *f = scf->f; |
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| 545 | double *u = scf->u; |
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| 546 | int *p = scf->p; |
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| 547 | double *y = scf->w; |
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| 548 | int i, j, ij; |
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| 549 | double t; |
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| 550 | /* y := P * b */ |
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| 551 | for (i = 1; i <= n; i++) y[i] = x[p[i]]; |
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| 552 | /* y := inv(U') * y */ |
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| 553 | for (i = 1; i <= n; i++) |
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| 554 | { /* compute y[i] */ |
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| 555 | ij = u_loc(scf, i, i); |
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| 556 | t = (y[i] /= u[ij]); |
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| 557 | /* substitute y[i] in other equations */ |
---|
| 558 | for (j = i+1, ij++; j <= n; j++, ij++) |
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| 559 | y[j] -= u[ij] * t; |
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| 560 | } |
---|
| 561 | /* x := F' * y (computed as linear combination of rows of F) */ |
---|
| 562 | for (j = 1; j <= n; j++) x[j] = 0.0; |
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| 563 | for (i = 1; i <= n; i++) |
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| 564 | { t = y[i]; /* coefficient of linear combination */ |
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| 565 | for (j = 1, ij = f_loc(scf, i, 1); j <= n; j++, ij++) |
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| 566 | x[j] += f[ij] * t; |
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| 567 | } |
---|
| 568 | return; |
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| 569 | } |
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| 570 | |
---|
| 571 | /*********************************************************************** |
---|
| 572 | * NAME |
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| 573 | * |
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| 574 | * scf_solve_it - solve either system C * x = b or C' * x = b |
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| 575 | * |
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| 576 | * SYNOPSIS |
---|
| 577 | * |
---|
| 578 | * #include "glpscf.h" |
---|
| 579 | * void scf_solve_it(SCF *scf, int tr, double x[]); |
---|
| 580 | * |
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| 581 | * DESCRIPTION |
---|
| 582 | * |
---|
| 583 | * The routine scf_solve_it solves either the system C * x = b (if tr |
---|
| 584 | * is zero) or the system C' * x = b, where C' is a matrix transposed |
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| 585 | * to C (if tr is non-zero). C is assumed to be non-singular. |
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| 586 | * |
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| 587 | * On entry the array x should contain the right-hand side vector b in |
---|
| 588 | * locations x[1], ..., x[n], where n is the order of matrix C. On exit |
---|
| 589 | * the array x contains the solution vector x in the same locations. */ |
---|
| 590 | |
---|
| 591 | void scf_solve_it(SCF *scf, int tr, double x[]) |
---|
| 592 | { if (scf->rank < scf->n) |
---|
| 593 | xfault("scf_solve_it: singular matrix\n"); |
---|
| 594 | if (!tr) |
---|
| 595 | solve(scf, x); |
---|
| 596 | else |
---|
| 597 | tsolve(scf, x); |
---|
| 598 | return; |
---|
| 599 | } |
---|
| 600 | |
---|
| 601 | void scf_reset_it(SCF *scf) |
---|
| 602 | { /* reset factorization for empty matrix C */ |
---|
| 603 | scf->n = scf->rank = 0; |
---|
| 604 | return; |
---|
| 605 | } |
---|
| 606 | |
---|
| 607 | /*********************************************************************** |
---|
| 608 | * NAME |
---|
| 609 | * |
---|
| 610 | * scf_delete_it - delete Schur complement factorization |
---|
| 611 | * |
---|
| 612 | * SYNOPSIS |
---|
| 613 | * |
---|
| 614 | * #include "glpscf.h" |
---|
| 615 | * void scf_delete_it(SCF *scf); |
---|
| 616 | * |
---|
| 617 | * DESCRIPTION |
---|
| 618 | * |
---|
| 619 | * The routine scf_delete_it deletes the specified factorization and |
---|
| 620 | * frees all the memory allocated to this object. */ |
---|
| 621 | |
---|
| 622 | void scf_delete_it(SCF *scf) |
---|
| 623 | { xfree(scf->f); |
---|
| 624 | xfree(scf->u); |
---|
| 625 | xfree(scf->p); |
---|
| 626 | #if _GLPSCF_DEBUG |
---|
| 627 | xfree(scf->c); |
---|
| 628 | #endif |
---|
| 629 | xfree(scf->w); |
---|
| 630 | xfree(scf); |
---|
| 631 | return; |
---|
| 632 | } |
---|
| 633 | |
---|
| 634 | /* eof */ |
---|