1 | /*************************************************************************
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2 | Copyright (c) 1992-2007 The University of Tennessee. All rights reserved.
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3 |
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4 | Contributors:
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5 | * Sergey Bochkanov (ALGLIB project). Translation from FORTRAN to
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6 | pseudocode.
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7 |
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8 | See subroutines comments for additional copyrights.
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9 |
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10 | >>> SOURCE LICENSE >>>
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11 | This program is free software; you can redistribute it and/or modify
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12 | it under the terms of the GNU General Public License as published by
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13 | the Free Software Foundation (www.fsf.org); either version 2 of the
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14 | License, or (at your option) any later version.
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15 |
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16 | This program is distributed in the hope that it will be useful,
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17 | but WITHOUT ANY WARRANTY; without even the implied warranty of
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18 | MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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19 | GNU General Public License for more details.
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20 |
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21 | A copy of the GNU General Public License is available at
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22 | http://www.fsf.org/licensing/licenses
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23 |
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24 | >>> END OF LICENSE >>>
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25 | *************************************************************************/
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26 |
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27 | using System;
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28 |
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29 | namespace alglib
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30 | {
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31 | public class srcond
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32 | {
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33 | /*************************************************************************
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34 | Condition number estimate of a symmetric matrix
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35 |
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36 | The algorithm calculates a lower bound of the condition number. In this
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37 | case, the algorithm does not return a lower bound of the condition number,
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38 | but an inverse number (to avoid an overflow in case of a singular matrix).
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39 |
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40 | It should be noted that 1-norm and inf-norm condition numbers of symmetric
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41 | matrices are equal, so the algorithm doesn't take into account the
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42 | differences between these types of norms.
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43 |
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44 | Input parameters:
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45 | A - symmetric definite matrix which is given by its upper or
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46 | lower triangle depending on IsUpper.
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47 | Array with elements [0..N-1, 0..N-1].
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48 | N - size of matrix A.
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49 | IsUpper - storage format.
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50 |
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51 | Result:
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52 | 1/LowerBound(cond(A))
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53 | *************************************************************************/
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54 | public static double smatrixrcond(ref double[,] a,
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55 | int n,
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56 | bool isupper)
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57 | {
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58 | double result = 0;
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59 | int i = 0;
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60 | int j = 0;
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61 | double[,] a1 = new double[0,0];
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62 |
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63 | a1 = new double[n+1, n+1];
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64 | for(i=1; i<=n; i++)
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65 | {
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66 | if( isupper )
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67 | {
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68 | for(j=i; j<=n; j++)
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69 | {
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70 | a1[i,j] = a[i-1,j-1];
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71 | }
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72 | }
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73 | else
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74 | {
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75 | for(j=1; j<=i; j++)
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76 | {
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77 | a1[i,j] = a[i-1,j-1];
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78 | }
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79 | }
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80 | }
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81 | result = rcondsymmetric(a1, n, isupper);
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82 | return result;
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83 | }
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84 |
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85 |
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86 | /*************************************************************************
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87 | Condition number estimate of a matrix given by LDLT-decomposition
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88 |
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89 | The algorithm calculates a lower bound of the condition number. In this
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90 | case, the algorithm does not return a lower bound of the condition number,
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91 | but an inverse number (to avoid an overflow in case of a singular matrix).
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92 |
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93 | It should be noted that 1-norm and inf-norm condition numbers of symmetric
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94 | matrices are equal, so the algorithm doesn't take into account the
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95 | differences between these types of norms.
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96 |
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97 | Input parameters:
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98 | L - LDLT-decomposition of matrix A given by the upper or lower
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99 | triangle depending on IsUpper.
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100 | Output of SMatrixLDLT subroutine.
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101 | Pivots - table of permutations which were made during LDLT-decomposition,
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102 | Output of SMatrixLDLT subroutine.
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103 | N - size of matrix A.
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104 | IsUpper - storage format.
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105 |
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106 | Result:
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107 | 1/LowerBound(cond(A))
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108 | *************************************************************************/
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109 | public static double smatrixldltrcond(ref double[,] l,
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110 | ref int[] pivots,
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111 | int n,
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112 | bool isupper)
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113 | {
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114 | double result = 0;
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115 | int i = 0;
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116 | int j = 0;
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117 | double[,] l1 = new double[0,0];
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118 | int[] p1 = new int[0];
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119 |
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120 | l1 = new double[n+1, n+1];
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121 | for(i=1; i<=n; i++)
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122 | {
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123 | if( isupper )
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124 | {
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125 | for(j=i; j<=n; j++)
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126 | {
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127 | l1[i,j] = l[i-1,j-1];
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128 | }
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129 | }
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130 | else
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131 | {
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132 | for(j=1; j<=i; j++)
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133 | {
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134 | l1[i,j] = l[i-1,j-1];
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135 | }
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136 | }
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137 | }
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138 | p1 = new int[n+1];
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139 | for(i=1; i<=n; i++)
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140 | {
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141 | if( pivots[i-1]>=0 )
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142 | {
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143 | p1[i] = pivots[i-1]+1;
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144 | }
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145 | else
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146 | {
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147 | p1[i] = -(pivots[i-1]+n+1);
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148 | }
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149 | }
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150 | result = rcondldlt(ref l1, ref p1, n, isupper);
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151 | return result;
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152 | }
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153 |
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154 |
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155 | public static double rcondsymmetric(double[,] a,
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156 | int n,
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157 | bool isupper)
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158 | {
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159 | double result = 0;
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160 | int i = 0;
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161 | int j = 0;
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162 | int im = 0;
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163 | int jm = 0;
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164 | double v = 0;
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165 | double nrm = 0;
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166 | int[] pivots = new int[0];
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167 |
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168 | a = (double[,])a.Clone();
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169 |
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170 | nrm = 0;
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171 | for(j=1; j<=n; j++)
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172 | {
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173 | v = 0;
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174 | for(i=1; i<=n; i++)
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175 | {
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176 | im = i;
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177 | jm = j;
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178 | if( isupper & j<i )
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179 | {
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180 | im = j;
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181 | jm = i;
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182 | }
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183 | if( !isupper & j>i )
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184 | {
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185 | im = j;
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186 | jm = i;
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187 | }
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188 | v = v+Math.Abs(a[im,jm]);
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189 | }
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190 | nrm = Math.Max(nrm, v);
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191 | }
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192 | ldlt.ldltdecomposition(ref a, n, isupper, ref pivots);
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193 | internalldltrcond(ref a, ref pivots, n, isupper, true, nrm, ref v);
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194 | result = v;
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195 | return result;
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196 | }
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197 |
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198 |
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199 | public static double rcondldlt(ref double[,] l,
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200 | ref int[] pivots,
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201 | int n,
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202 | bool isupper)
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203 | {
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204 | double result = 0;
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205 | double v = 0;
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206 |
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207 | internalldltrcond(ref l, ref pivots, n, isupper, false, 0, ref v);
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208 | result = v;
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209 | return result;
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210 | }
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211 |
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212 |
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213 | public static void internalldltrcond(ref double[,] l,
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214 | ref int[] pivots,
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215 | int n,
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216 | bool isupper,
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217 | bool isnormprovided,
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218 | double anorm,
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219 | ref double rcond)
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220 | {
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221 | int i = 0;
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222 | int ix = 0;
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223 | int kase = 0;
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224 | int k = 0;
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225 | int km1 = 0;
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226 | int km2 = 0;
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227 | int kp1 = 0;
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228 | int kp2 = 0;
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229 | double ainvnm = 0;
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230 | double[] work0 = new double[0];
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231 | double[] work1 = new double[0];
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232 | double[] work2 = new double[0];
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233 | int[] iwork = new int[0];
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234 | double v = 0;
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235 | int i_ = 0;
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236 |
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237 | System.Diagnostics.Debug.Assert(n>=0);
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238 |
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239 | //
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240 | // Check that the diagonal matrix D is nonsingular.
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241 | //
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242 | rcond = 0;
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243 | if( isupper )
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244 | {
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245 | for(i=n; i>=1; i--)
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246 | {
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247 | if( pivots[i]>0 & (double)(l[i,i])==(double)(0) )
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248 | {
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249 | return;
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250 | }
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251 | }
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252 | }
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253 | else
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254 | {
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255 | for(i=1; i<=n; i++)
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256 | {
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257 | if( pivots[i]>0 & (double)(l[i,i])==(double)(0) )
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258 | {
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259 | return;
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260 | }
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261 | }
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262 | }
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263 |
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264 | //
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265 | // Estimate the norm of A.
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266 | //
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267 | if( !isnormprovided )
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268 | {
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269 | kase = 0;
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270 | anorm = 0;
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271 | while( true )
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272 | {
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273 | estnorm.iterativeestimate1norm(n, ref work1, ref work0, ref iwork, ref anorm, ref kase);
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274 | if( kase==0 )
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275 | {
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276 | break;
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277 | }
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278 | if( isupper )
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279 | {
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280 |
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281 | //
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282 | // Multiply by U'
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283 | //
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284 | k = n;
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285 | while( k>=1 )
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286 | {
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287 | if( pivots[k]>0 )
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288 | {
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289 |
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290 | //
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291 | // P(k)
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292 | //
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293 | v = work0[k];
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294 | work0[k] = work0[pivots[k]];
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295 | work0[pivots[k]] = v;
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296 |
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297 | //
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298 | // U(k)
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299 | //
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300 | km1 = k-1;
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301 | v = 0.0;
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302 | for(i_=1; i_<=km1;i_++)
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303 | {
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304 | v += work0[i_]*l[i_,k];
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305 | }
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306 | work0[k] = work0[k]+v;
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307 |
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308 | //
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309 | // Next k
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310 | //
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311 | k = k-1;
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312 | }
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313 | else
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314 | {
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315 |
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316 | //
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317 | // P(k)
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318 | //
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319 | v = work0[k-1];
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320 | work0[k-1] = work0[-pivots[k-1]];
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321 | work0[-pivots[k-1]] = v;
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322 |
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323 | //
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324 | // U(k)
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325 | //
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326 | km1 = k-1;
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327 | km2 = k-2;
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328 | v = 0.0;
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329 | for(i_=1; i_<=km2;i_++)
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330 | {
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331 | v += work0[i_]*l[i_,km1];
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332 | }
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333 | work0[km1] = work0[km1]+v;
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334 | v = 0.0;
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335 | for(i_=1; i_<=km2;i_++)
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336 | {
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337 | v += work0[i_]*l[i_,k];
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338 | }
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339 | work0[k] = work0[k]+v;
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340 |
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341 | //
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342 | // Next k
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343 | //
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344 | k = k-2;
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345 | }
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346 | }
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347 |
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348 | //
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349 | // Multiply by D
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350 | //
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351 | k = n;
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352 | while( k>=1 )
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353 | {
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354 | if( pivots[k]>0 )
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355 | {
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356 | work0[k] = work0[k]*l[k,k];
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357 | k = k-1;
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358 | }
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359 | else
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360 | {
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361 | v = work0[k-1];
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362 | work0[k-1] = l[k-1,k-1]*work0[k-1]+l[k-1,k]*work0[k];
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363 | work0[k] = l[k-1,k]*v+l[k,k]*work0[k];
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364 | k = k-2;
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365 | }
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366 | }
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367 |
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368 | //
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369 | // Multiply by U
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370 | //
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371 | k = 1;
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372 | while( k<=n )
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373 | {
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374 | if( pivots[k]>0 )
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375 | {
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376 |
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377 | //
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378 | // U(k)
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379 | //
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380 | km1 = k-1;
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381 | v = work0[k];
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382 | for(i_=1; i_<=km1;i_++)
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383 | {
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384 | work0[i_] = work0[i_] + v*l[i_,k];
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385 | }
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386 |
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387 | //
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388 | // P(k)
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389 | //
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390 | v = work0[k];
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391 | work0[k] = work0[pivots[k]];
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392 | work0[pivots[k]] = v;
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393 |
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394 | //
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395 | // Next k
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396 | //
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397 | k = k+1;
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398 | }
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399 | else
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400 | {
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401 |
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402 | //
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403 | // U(k)
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404 | //
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405 | km1 = k-1;
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406 | kp1 = k+1;
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407 | v = work0[k];
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408 | for(i_=1; i_<=km1;i_++)
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409 | {
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410 | work0[i_] = work0[i_] + v*l[i_,k];
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411 | }
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412 | v = work0[kp1];
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413 | for(i_=1; i_<=km1;i_++)
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414 | {
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415 | work0[i_] = work0[i_] + v*l[i_,kp1];
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416 | }
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417 |
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418 | //
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419 | // P(k)
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420 | //
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421 | v = work0[k];
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422 | work0[k] = work0[-pivots[k]];
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423 | work0[-pivots[k]] = v;
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424 |
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425 | //
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426 | // Next k
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427 | //
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428 | k = k+2;
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429 | }
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430 | }
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431 | }
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432 | else
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433 | {
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434 |
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435 | //
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436 | // Multiply by L'
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437 | //
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438 | k = 1;
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439 | while( k<=n )
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440 | {
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441 | if( pivots[k]>0 )
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442 | {
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443 |
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444 | //
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445 | // P(k)
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446 | //
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447 | v = work0[k];
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448 | work0[k] = work0[pivots[k]];
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449 | work0[pivots[k]] = v;
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450 |
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451 | //
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452 | // L(k)
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453 | //
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454 | kp1 = k+1;
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455 | v = 0.0;
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456 | for(i_=kp1; i_<=n;i_++)
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457 | {
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458 | v += work0[i_]*l[i_,k];
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459 | }
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460 | work0[k] = work0[k]+v;
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461 |
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462 | //
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463 | // Next k
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464 | //
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465 | k = k+1;
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466 | }
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467 | else
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468 | {
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469 |
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470 | //
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471 | // P(k)
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472 | //
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473 | v = work0[k+1];
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474 | work0[k+1] = work0[-pivots[k+1]];
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475 | work0[-pivots[k+1]] = v;
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476 |
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477 | //
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478 | // L(k)
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479 | //
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480 | kp1 = k+1;
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481 | kp2 = k+2;
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482 | v = 0.0;
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483 | for(i_=kp2; i_<=n;i_++)
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484 | {
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485 | v += work0[i_]*l[i_,k];
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486 | }
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487 | work0[k] = work0[k]+v;
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488 | v = 0.0;
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489 | for(i_=kp2; i_<=n;i_++)
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490 | {
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491 | v += work0[i_]*l[i_,kp1];
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492 | }
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493 | work0[kp1] = work0[kp1]+v;
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494 |
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495 | //
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496 | // Next k
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497 | //
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498 | k = k+2;
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499 | }
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500 | }
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501 |
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502 | //
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503 | // Multiply by D
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504 | //
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505 | k = n;
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506 | while( k>=1 )
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507 | {
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508 | if( pivots[k]>0 )
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509 | {
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510 | work0[k] = work0[k]*l[k,k];
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511 | k = k-1;
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512 | }
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513 | else
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514 | {
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515 | v = work0[k-1];
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516 | work0[k-1] = l[k-1,k-1]*work0[k-1]+l[k,k-1]*work0[k];
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517 | work0[k] = l[k,k-1]*v+l[k,k]*work0[k];
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518 | k = k-2;
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519 | }
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520 | }
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521 |
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522 | //
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523 | // Multiply by L
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524 | //
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525 | k = n;
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526 | while( k>=1 )
|
---|
527 | {
|
---|
528 | if( pivots[k]>0 )
|
---|
529 | {
|
---|
530 |
|
---|
531 | //
|
---|
532 | // L(k)
|
---|
533 | //
|
---|
534 | kp1 = k+1;
|
---|
535 | v = work0[k];
|
---|
536 | for(i_=kp1; i_<=n;i_++)
|
---|
537 | {
|
---|
538 | work0[i_] = work0[i_] + v*l[i_,k];
|
---|
539 | }
|
---|
540 |
|
---|
541 | //
|
---|
542 | // P(k)
|
---|
543 | //
|
---|
544 | v = work0[k];
|
---|
545 | work0[k] = work0[pivots[k]];
|
---|
546 | work0[pivots[k]] = v;
|
---|
547 |
|
---|
548 | //
|
---|
549 | // Next k
|
---|
550 | //
|
---|
551 | k = k-1;
|
---|
552 | }
|
---|
553 | else
|
---|
554 | {
|
---|
555 |
|
---|
556 | //
|
---|
557 | // L(k)
|
---|
558 | //
|
---|
559 | kp1 = k+1;
|
---|
560 | km1 = k-1;
|
---|
561 | v = work0[k];
|
---|
562 | for(i_=kp1; i_<=n;i_++)
|
---|
563 | {
|
---|
564 | work0[i_] = work0[i_] + v*l[i_,k];
|
---|
565 | }
|
---|
566 | v = work0[km1];
|
---|
567 | for(i_=kp1; i_<=n;i_++)
|
---|
568 | {
|
---|
569 | work0[i_] = work0[i_] + v*l[i_,km1];
|
---|
570 | }
|
---|
571 |
|
---|
572 | //
|
---|
573 | // P(k)
|
---|
574 | //
|
---|
575 | v = work0[k];
|
---|
576 | work0[k] = work0[-pivots[k]];
|
---|
577 | work0[-pivots[k]] = v;
|
---|
578 |
|
---|
579 | //
|
---|
580 | // Next k
|
---|
581 | //
|
---|
582 | k = k-2;
|
---|
583 | }
|
---|
584 | }
|
---|
585 | }
|
---|
586 | }
|
---|
587 | }
|
---|
588 |
|
---|
589 | //
|
---|
590 | // Quick return if possible
|
---|
591 | //
|
---|
592 | rcond = 0;
|
---|
593 | if( n==0 )
|
---|
594 | {
|
---|
595 | rcond = 1;
|
---|
596 | return;
|
---|
597 | }
|
---|
598 | if( (double)(anorm)==(double)(0) )
|
---|
599 | {
|
---|
600 | return;
|
---|
601 | }
|
---|
602 |
|
---|
603 | //
|
---|
604 | // Estimate the 1-norm of inv(A).
|
---|
605 | //
|
---|
606 | kase = 0;
|
---|
607 | while( true )
|
---|
608 | {
|
---|
609 | estnorm.iterativeestimate1norm(n, ref work1, ref work0, ref iwork, ref ainvnm, ref kase);
|
---|
610 | if( kase==0 )
|
---|
611 | {
|
---|
612 | break;
|
---|
613 | }
|
---|
614 | ssolve.solvesystemldlt(ref l, ref pivots, work0, n, isupper, ref work2);
|
---|
615 | for(i_=1; i_<=n;i_++)
|
---|
616 | {
|
---|
617 | work0[i_] = work2[i_];
|
---|
618 | }
|
---|
619 | }
|
---|
620 |
|
---|
621 | //
|
---|
622 | // Compute the estimate of the reciprocal condition number.
|
---|
623 | //
|
---|
624 | if( (double)(ainvnm)!=(double)(0) )
|
---|
625 | {
|
---|
626 | v = 1/ainvnm;
|
---|
627 | rcond = v/anorm;
|
---|
628 | }
|
---|
629 | }
|
---|
630 | }
|
---|
631 | }
|
---|