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 */ |
---|
505 | for (i = 1; i <= n; i++) |
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506 | { /* y[i] = (i-th row of F) * b */ |
---|
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--) |
---|
514 | { t = y[i]; |
---|
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 */ |
---|
520 | for (i = 1; i <= n; i++) x[p[i]] = y[i]; |
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521 | return; |
---|
522 | } |
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523 | |
---|
524 | /*********************************************************************** |
---|
525 | * The routine tsolve solves the transposed system C' * x = b. |
---|
526 | * |
---|
527 | * From the main equation F * C = U * P it follows that: |
---|
528 | * |
---|
529 | * C' * F' = P' * U', |
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530 | * |
---|
531 | * therefore: |
---|
532 | * |
---|
533 | * C' * x = b => C' * F' * inv(F') * x = b => |
---|
534 | * |
---|
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. |
---|
538 | * |
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539 | * On entry the array x contains right-hand side vector b. On exit this |
---|
540 | * array contains solution vector x. */ |
---|
541 | |
---|
542 | static void tsolve(SCF *scf, double x[]) |
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543 | { int n = scf->n; |
---|
544 | double *f = scf->f; |
---|
545 | double *u = scf->u; |
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546 | int *p = scf->p; |
---|
547 | double *y = scf->w; |
---|
548 | int i, j, ij; |
---|
549 | double t; |
---|
550 | /* y := P * b */ |
---|
551 | for (i = 1; i <= n; i++) y[i] = x[p[i]]; |
---|
552 | /* y := inv(U') * y */ |
---|
553 | for (i = 1; i <= n; i++) |
---|
554 | { /* compute y[i] */ |
---|
555 | ij = u_loc(scf, i, i); |
---|
556 | t = (y[i] /= u[ij]); |
---|
557 | /* substitute y[i] in other equations */ |
---|
558 | for (j = i+1, ij++; j <= n; j++, ij++) |
---|
559 | y[j] -= u[ij] * t; |
---|
560 | } |
---|
561 | /* x := F' * y (computed as linear combination of rows of F) */ |
---|
562 | for (j = 1; j <= n; j++) x[j] = 0.0; |
---|
563 | for (i = 1; i <= n; i++) |
---|
564 | { t = y[i]; /* coefficient of linear combination */ |
---|
565 | for (j = 1, ij = f_loc(scf, i, 1); j <= n; j++, ij++) |
---|
566 | x[j] += f[ij] * t; |
---|
567 | } |
---|
568 | return; |
---|
569 | } |
---|
570 | |
---|
571 | /*********************************************************************** |
---|
572 | * NAME |
---|
573 | * |
---|
574 | * scf_solve_it - solve either system C * x = b or C' * x = b |
---|
575 | * |
---|
576 | * SYNOPSIS |
---|
577 | * |
---|
578 | * #include "glpscf.h" |
---|
579 | * void scf_solve_it(SCF *scf, int tr, double x[]); |
---|
580 | * |
---|
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 |
---|
585 | * to C (if tr is non-zero). C is assumed to be non-singular. |
---|
586 | * |
---|
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 */ |
---|