| 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 ssolve
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| 32 | {
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| 33 | /*************************************************************************
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| 34 | Solving a system of linear equations with a system matrix given by its
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| 35 | LDLT decomposition
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| 36 |
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| 37 | The algorithm solves systems with a square matrix only.
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| 38 |
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| 39 | Input parameters:
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| 40 | A - LDLT decomposition of the matrix (the result of the
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| 41 | SMatrixLDLT subroutine).
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| 42 | Pivots - row permutation table (the result of the SMatrixLDLT subroutine).
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| 43 | B - right side of a system.
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| 44 | Array whose index ranges within [0..N-1].
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| 45 | N - size of matrix A.
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| 46 | IsUpper - points to the triangle of matrix A in which the LDLT
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| 47 | decomposition is stored.
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| 48 | If IsUpper=True, the decomposition has the form of U*D*U',
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| 49 | matrix U is stored in the upper triangle of matrix A (in
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| 50 | that case, the lower triangle isn't used and isn't changed
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| 51 | by the subroutine).
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| 52 | Similarly, if IsUpper=False, the decomposition has the form
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| 53 | of L*D*L' and the lower triangle stores matrix L.
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| 54 |
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| 55 | Output parameters:
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| 56 | X - solution of a system.
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| 57 | Array whose index ranges within [0..N-1].
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| 58 |
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| 59 | Result:
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| 60 | True, if the matrix is not singular. X contains the solution.
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| 61 | False, if the matrix is singular (the determinant of matrix D is equal
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| 62 | to 0). In this case, X doesn't contain a solution.
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| 63 | *************************************************************************/
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| 64 | public static bool smatrixldltsolve(ref double[,] a,
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| 65 | ref int[] pivots,
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| 66 | double[] b,
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| 67 | int n,
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| 68 | bool isupper,
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| 69 | ref double[] x)
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| 70 | {
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| 71 | bool result = new bool();
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| 72 | int i = 0;
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| 73 | int j = 0;
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| 74 | int k = 0;
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| 75 | int kp = 0;
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| 76 | double ak = 0;
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| 77 | double akm1 = 0;
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| 78 | double akm1k = 0;
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| 79 | double bk = 0;
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| 80 | double bkm1 = 0;
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| 81 | double denom = 0;
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| 82 | double v = 0;
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| 83 | int i_ = 0;
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| 84 |
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| 85 | b = (double[])b.Clone();
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| 86 |
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| 87 |
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| 88 | //
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| 89 | // Quick return if possible
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| 90 | //
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| 91 | result = true;
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| 92 | if( n==0 )
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| 93 | {
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| 94 | return result;
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| 95 | }
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| 96 |
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| 97 | //
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| 98 | // Check that the diagonal matrix D is nonsingular
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| 99 | //
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| 100 | if( isupper )
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| 101 | {
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| 102 |
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| 103 | //
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| 104 | // Upper triangular storage: examine D from bottom to top
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| 105 | //
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| 106 | for(i=n-1; i>=0; i--)
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| 107 | {
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| 108 | if( pivots[i]>=0 & (double)(a[i,i])==(double)(0) )
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| 109 | {
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| 110 | result = false;
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| 111 | return result;
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| 112 | }
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| 113 | }
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| 114 | }
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| 115 | else
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| 116 | {
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| 117 |
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| 118 | //
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| 119 | // Lower triangular storage: examine D from top to bottom.
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| 120 | //
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| 121 | for(i=0; i<=n-1; i++)
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| 122 | {
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| 123 | if( pivots[i]>=0 & (double)(a[i,i])==(double)(0) )
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| 124 | {
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| 125 | result = false;
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| 126 | return result;
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| 127 | }
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| 128 | }
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| 129 | }
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| 130 |
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| 131 | //
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| 132 | // Solve Ax = b
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| 133 | //
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| 134 | if( isupper )
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| 135 | {
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| 136 |
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| 137 | //
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| 138 | // Solve A*X = B, where A = U*D*U'.
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| 139 | //
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| 140 | // First solve U*D*X = B, overwriting B with X.
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| 141 | //
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| 142 | // K+1 is the main loop index, decreasing from N to 1 in steps of
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| 143 | // 1 or 2, depending on the size of the diagonal blocks.
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| 144 | //
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| 145 | k = n-1;
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| 146 | while( k>=0 )
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| 147 | {
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| 148 | if( pivots[k]>=0 )
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| 149 | {
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| 150 |
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| 151 | //
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| 152 | // 1 x 1 diagonal block
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| 153 | //
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| 154 | // Interchange rows K+1 and IPIV(K+1).
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| 155 | //
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| 156 | kp = pivots[k];
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| 157 | if( kp!=k )
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| 158 | {
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| 159 | v = b[k];
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| 160 | b[k] = b[kp];
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| 161 | b[kp] = v;
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| 162 | }
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| 163 |
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| 164 | //
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| 165 | // Multiply by inv(U(K+1)), where U(K+1) is the transformation
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| 166 | // stored in column K+1 of A.
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| 167 | //
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| 168 | v = b[k];
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| 169 | for(i_=0; i_<=k-1;i_++)
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| 170 | {
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| 171 | b[i_] = b[i_] - v*a[i_,k];
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| 172 | }
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| 173 |
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| 174 | //
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| 175 | // Multiply by the inverse of the diagonal block.
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| 176 | //
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| 177 | b[k] = b[k]/a[k,k];
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| 178 | k = k-1;
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| 179 | }
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| 180 | else
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| 181 | {
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| 182 |
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| 183 | //
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| 184 | // 2 x 2 diagonal block
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| 185 | //
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| 186 | // Interchange rows K+1-1 and -IPIV(K+1).
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| 187 | //
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| 188 | kp = pivots[k]+n;
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| 189 | if( kp!=k-1 )
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| 190 | {
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| 191 | v = b[k-1];
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| 192 | b[k-1] = b[kp];
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| 193 | b[kp] = v;
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| 194 | }
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| 195 |
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| 196 | //
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| 197 | // Multiply by inv(U(K+1)), where U(K+1) is the transformation
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| 198 | // stored in columns K+1-1 and K+1 of A.
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| 199 | //
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| 200 | v = b[k];
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| 201 | for(i_=0; i_<=k-2;i_++)
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| 202 | {
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| 203 | b[i_] = b[i_] - v*a[i_,k];
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| 204 | }
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| 205 | v = b[k-1];
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| 206 | for(i_=0; i_<=k-2;i_++)
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| 207 | {
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| 208 | b[i_] = b[i_] - v*a[i_,k-1];
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| 209 | }
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| 210 |
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| 211 | //
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| 212 | // Multiply by the inverse of the diagonal block.
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| 213 | //
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| 214 | akm1k = a[k-1,k];
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| 215 | akm1 = a[k-1,k-1]/akm1k;
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| 216 | ak = a[k,k]/akm1k;
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| 217 | denom = akm1*ak-1;
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| 218 | bkm1 = b[k-1]/akm1k;
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| 219 | bk = b[k]/akm1k;
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| 220 | b[k-1] = (ak*bkm1-bk)/denom;
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| 221 | b[k] = (akm1*bk-bkm1)/denom;
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| 222 | k = k-2;
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| 223 | }
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| 224 | }
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| 225 |
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| 226 | //
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| 227 | // Next solve U'*X = B, overwriting B with X.
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| 228 | //
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| 229 | // K+1 is the main loop index, increasing from 1 to N in steps of
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| 230 | // 1 or 2, depending on the size of the diagonal blocks.
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| 231 | //
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| 232 | k = 0;
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| 233 | while( k<=n-1 )
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| 234 | {
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| 235 | if( pivots[k]>=0 )
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| 236 | {
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| 237 |
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| 238 | //
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| 239 | // 1 x 1 diagonal block
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| 240 | //
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| 241 | // Multiply by inv(U'(K+1)), where U(K+1) is the transformation
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| 242 | // stored in column K+1 of A.
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| 243 | //
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| 244 | v = 0.0;
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| 245 | for(i_=0; i_<=k-1;i_++)
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| 246 | {
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| 247 | v += b[i_]*a[i_,k];
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| 248 | }
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| 249 | b[k] = b[k]-v;
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| 250 |
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| 251 | //
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| 252 | // Interchange rows K+1 and IPIV(K+1).
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| 253 | //
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| 254 | kp = pivots[k];
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| 255 | if( kp!=k )
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| 256 | {
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| 257 | v = b[k];
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| 258 | b[k] = b[kp];
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| 259 | b[kp] = v;
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| 260 | }
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| 261 | k = k+1;
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| 262 | }
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| 263 | else
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| 264 | {
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| 265 |
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| 266 | //
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| 267 | // 2 x 2 diagonal block
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| 268 | //
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| 269 | // Multiply by inv(U'(K+1+1)), where U(K+1+1) is the transformation
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| 270 | // stored in columns K+1 and K+1+1 of A.
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| 271 | //
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| 272 | v = 0.0;
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| 273 | for(i_=0; i_<=k-1;i_++)
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| 274 | {
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| 275 | v += b[i_]*a[i_,k];
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| 276 | }
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| 277 | b[k] = b[k]-v;
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| 278 | v = 0.0;
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| 279 | for(i_=0; i_<=k-1;i_++)
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| 280 | {
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| 281 | v += b[i_]*a[i_,k+1];
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| 282 | }
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| 283 | b[k+1] = b[k+1]-v;
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| 284 |
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| 285 | //
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| 286 | // Interchange rows K+1 and -IPIV(K+1).
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| 287 | //
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| 288 | kp = pivots[k]+n;
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| 289 | if( kp!=k )
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| 290 | {
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| 291 | v = b[k];
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| 292 | b[k] = b[kp];
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| 293 | b[kp] = v;
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| 294 | }
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| 295 | k = k+2;
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| 296 | }
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| 297 | }
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| 298 | }
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| 299 | else
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| 300 | {
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| 301 |
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| 302 | //
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| 303 | // Solve A*X = B, where A = L*D*L'.
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| 304 | //
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| 305 | // First solve L*D*X = B, overwriting B with X.
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| 306 | //
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| 307 | // K+1 is the main loop index, increasing from 1 to N in steps of
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| 308 | // 1 or 2, depending on the size of the diagonal blocks.
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| 309 | //
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| 310 | k = 0;
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| 311 | while( k<=n-1 )
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| 312 | {
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| 313 | if( pivots[k]>=0 )
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| 314 | {
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| 315 |
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| 316 | //
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| 317 | // 1 x 1 diagonal block
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| 318 | //
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| 319 | // Interchange rows K+1 and IPIV(K+1).
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| 320 | //
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| 321 | kp = pivots[k];
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| 322 | if( kp!=k )
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| 323 | {
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| 324 | v = b[k];
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| 325 | b[k] = b[kp];
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| 326 | b[kp] = v;
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| 327 | }
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| 328 |
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| 329 | //
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| 330 | // Multiply by inv(L(K+1)), where L(K+1) is the transformation
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| 331 | // stored in column K+1 of A.
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| 332 | //
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| 333 | if( k+1<n )
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| 334 | {
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| 335 | v = b[k];
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| 336 | for(i_=k+1; i_<=n-1;i_++)
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| 337 | {
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| 338 | b[i_] = b[i_] - v*a[i_,k];
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| 339 | }
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| 340 | }
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| 341 |
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| 342 | //
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| 343 | // Multiply by the inverse of the diagonal block.
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| 344 | //
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| 345 | b[k] = b[k]/a[k,k];
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| 346 | k = k+1;
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| 347 | }
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| 348 | else
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| 349 | {
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| 350 |
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| 351 | //
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| 352 | // 2 x 2 diagonal block
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| 353 | //
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| 354 | // Interchange rows K+1+1 and -IPIV(K+1).
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| 355 | //
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| 356 | kp = pivots[k]+n;
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| 357 | if( kp!=k+1 )
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| 358 | {
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| 359 | v = b[k+1];
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| 360 | b[k+1] = b[kp];
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| 361 | b[kp] = v;
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| 362 | }
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| 363 |
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| 364 | //
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| 365 | // Multiply by inv(L(K+1)), where L(K+1) is the transformation
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| 366 | // stored in columns K+1 and K+1+1 of A.
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| 367 | //
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| 368 | if( k+1<n-1 )
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| 369 | {
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| 370 | v = b[k];
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| 371 | for(i_=k+2; i_<=n-1;i_++)
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| 372 | {
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| 373 | b[i_] = b[i_] - v*a[i_,k];
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| 374 | }
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| 375 | v = b[k+1];
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| 376 | for(i_=k+2; i_<=n-1;i_++)
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| 377 | {
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| 378 | b[i_] = b[i_] - v*a[i_,k+1];
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| 379 | }
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| 380 | }
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| 381 |
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| 382 | //
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| 383 | // Multiply by the inverse of the diagonal block.
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| 384 | //
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| 385 | akm1k = a[k+1,k];
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| 386 | akm1 = a[k,k]/akm1k;
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| 387 | ak = a[k+1,k+1]/akm1k;
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| 388 | denom = akm1*ak-1;
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| 389 | bkm1 = b[k]/akm1k;
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| 390 | bk = b[k+1]/akm1k;
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| 391 | b[k] = (ak*bkm1-bk)/denom;
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| 392 | b[k+1] = (akm1*bk-bkm1)/denom;
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| 393 | k = k+2;
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| 394 | }
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| 395 | }
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| 396 |
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| 397 | //
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| 398 | // Next solve L'*X = B, overwriting B with X.
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| 399 | //
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| 400 | // K+1 is the main loop index, decreasing from N to 1 in steps of
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| 401 | // 1 or 2, depending on the size of the diagonal blocks.
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| 402 | //
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| 403 | k = n-1;
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| 404 | while( k>=0 )
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| 405 | {
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| 406 | if( pivots[k]>=0 )
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| 407 | {
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| 408 |
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| 409 | //
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| 410 | // 1 x 1 diagonal block
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| 411 | //
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| 412 | // Multiply by inv(L'(K+1)), where L(K+1) is the transformation
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| 413 | // stored in column K+1 of A.
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| 414 | //
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| 415 | if( k+1<n )
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| 416 | {
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| 417 | v = 0.0;
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| 418 | for(i_=k+1; i_<=n-1;i_++)
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| 419 | {
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| 420 | v += b[i_]*a[i_,k];
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| 421 | }
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| 422 | b[k] = b[k]-v;
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| 423 | }
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| 424 |
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| 425 | //
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| 426 | // Interchange rows K+1 and IPIV(K+1).
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| 427 | //
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| 428 | kp = pivots[k];
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| 429 | if( kp!=k )
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| 430 | {
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| 431 | v = b[k];
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| 432 | b[k] = b[kp];
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| 433 | b[kp] = v;
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| 434 | }
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| 435 | k = k-1;
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| 436 | }
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| 437 | else
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| 438 | {
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| 439 |
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| 440 | //
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| 441 | // 2 x 2 diagonal block
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| 442 | //
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| 443 | // Multiply by inv(L'(K+1-1)), where L(K+1-1) is the transformation
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| 444 | // stored in columns K+1-1 and K+1 of A.
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| 445 | //
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| 446 | if( k+1<n )
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| 447 | {
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|---|
| 448 | v = 0.0;
|
|---|
| 449 | for(i_=k+1; i_<=n-1;i_++)
|
|---|
| 450 | {
|
|---|
| 451 | v += b[i_]*a[i_,k];
|
|---|
| 452 | }
|
|---|
| 453 | b[k] = b[k]-v;
|
|---|
| 454 | v = 0.0;
|
|---|
| 455 | for(i_=k+1; i_<=n-1;i_++)
|
|---|
| 456 | {
|
|---|
| 457 | v += b[i_]*a[i_,k-1];
|
|---|
| 458 | }
|
|---|
| 459 | b[k-1] = b[k-1]-v;
|
|---|
| 460 | }
|
|---|
| 461 |
|
|---|
| 462 | //
|
|---|
| 463 | // Interchange rows K+1 and -IPIV(K+1).
|
|---|
| 464 | //
|
|---|
| 465 | kp = pivots[k]+n;
|
|---|
| 466 | if( kp!=k )
|
|---|
| 467 | {
|
|---|
| 468 | v = b[k];
|
|---|
| 469 | b[k] = b[kp];
|
|---|
| 470 | b[kp] = v;
|
|---|
| 471 | }
|
|---|
| 472 | k = k-2;
|
|---|
| 473 | }
|
|---|
| 474 | }
|
|---|
| 475 | }
|
|---|
| 476 | x = new double[n-1+1];
|
|---|
| 477 | for(i_=0; i_<=n-1;i_++)
|
|---|
| 478 | {
|
|---|
| 479 | x[i_] = b[i_];
|
|---|
| 480 | }
|
|---|
| 481 | return result;
|
|---|
| 482 | }
|
|---|
| 483 |
|
|---|
| 484 |
|
|---|
| 485 | /*************************************************************************
|
|---|
| 486 | Solving a system of linear equations with a symmetric system matrix
|
|---|
| 487 |
|
|---|
| 488 | Input parameters:
|
|---|
| 489 | A - system matrix (upper or lower triangle).
|
|---|
| 490 | Array whose indexes range within [0..N-1, 0..N-1].
|
|---|
| 491 | B - right side of a system.
|
|---|
| 492 | Array whose index ranges within [0..N-1].
|
|---|
| 493 | N - size of matrix A.
|
|---|
| 494 | IsUpper - If IsUpper = True, A contains the upper triangle,
|
|---|
| 495 | otherwise A contains the lower triangle.
|
|---|
| 496 |
|
|---|
| 497 | Output parameters:
|
|---|
| 498 | X - solution of a system.
|
|---|
| 499 | Array whose index ranges within [0..N-1].
|
|---|
| 500 |
|
|---|
| 501 | Result:
|
|---|
| 502 | True, if the matrix is not singular. X contains the solution.
|
|---|
| 503 | False, if the matrix is singular (the determinant of the matrix is equal
|
|---|
| 504 | to 0). In this case, X doesn't contain a solution.
|
|---|
| 505 |
|
|---|
| 506 | -- ALGLIB --
|
|---|
| 507 | Copyright 2005 by Bochkanov Sergey
|
|---|
| 508 | *************************************************************************/
|
|---|
| 509 | public static bool smatrixsolve(double[,] a,
|
|---|
| 510 | ref double[] b,
|
|---|
| 511 | int n,
|
|---|
| 512 | bool isupper,
|
|---|
| 513 | ref double[] x)
|
|---|
| 514 | {
|
|---|
| 515 | bool result = new bool();
|
|---|
| 516 | int[] pivots = new int[0];
|
|---|
| 517 |
|
|---|
| 518 | a = (double[,])a.Clone();
|
|---|
| 519 |
|
|---|
| 520 | ldlt.smatrixldlt(ref a, n, isupper, ref pivots);
|
|---|
| 521 | result = smatrixldltsolve(ref a, ref pivots, b, n, isupper, ref x);
|
|---|
| 522 | return result;
|
|---|
| 523 | }
|
|---|
| 524 |
|
|---|
| 525 |
|
|---|
| 526 | public static bool solvesystemldlt(ref double[,] a,
|
|---|
| 527 | ref int[] pivots,
|
|---|
| 528 | double[] b,
|
|---|
| 529 | int n,
|
|---|
| 530 | bool isupper,
|
|---|
| 531 | ref double[] x)
|
|---|
| 532 | {
|
|---|
| 533 | bool result = new bool();
|
|---|
| 534 | int i = 0;
|
|---|
| 535 | int j = 0;
|
|---|
| 536 | int k = 0;
|
|---|
| 537 | int kp = 0;
|
|---|
| 538 | int km1 = 0;
|
|---|
| 539 | int km2 = 0;
|
|---|
| 540 | int kp1 = 0;
|
|---|
| 541 | int kp2 = 0;
|
|---|
| 542 | double ak = 0;
|
|---|
| 543 | double akm1 = 0;
|
|---|
| 544 | double akm1k = 0;
|
|---|
| 545 | double bk = 0;
|
|---|
| 546 | double bkm1 = 0;
|
|---|
| 547 | double denom = 0;
|
|---|
| 548 | double v = 0;
|
|---|
| 549 | int i_ = 0;
|
|---|
| 550 |
|
|---|
| 551 | b = (double[])b.Clone();
|
|---|
| 552 |
|
|---|
| 553 |
|
|---|
| 554 | //
|
|---|
| 555 | // Quick return if possible
|
|---|
| 556 | //
|
|---|
| 557 | result = true;
|
|---|
| 558 | if( n==0 )
|
|---|
| 559 | {
|
|---|
| 560 | return result;
|
|---|
| 561 | }
|
|---|
| 562 |
|
|---|
| 563 | //
|
|---|
| 564 | // Check that the diagonal matrix D is nonsingular
|
|---|
| 565 | //
|
|---|
| 566 | if( isupper )
|
|---|
| 567 | {
|
|---|
| 568 |
|
|---|
| 569 | //
|
|---|
| 570 | // Upper triangular storage: examine D from bottom to top
|
|---|
| 571 | //
|
|---|
| 572 | for(i=n; i>=1; i--)
|
|---|
| 573 | {
|
|---|
| 574 | if( pivots[i]>0 & (double)(a[i,i])==(double)(0) )
|
|---|
| 575 | {
|
|---|
| 576 | result = false;
|
|---|
| 577 | return result;
|
|---|
| 578 | }
|
|---|
| 579 | }
|
|---|
| 580 | }
|
|---|
| 581 | else
|
|---|
| 582 | {
|
|---|
| 583 |
|
|---|
| 584 | //
|
|---|
| 585 | // Lower triangular storage: examine D from top to bottom.
|
|---|
| 586 | //
|
|---|
| 587 | for(i=1; i<=n; i++)
|
|---|
| 588 | {
|
|---|
| 589 | if( pivots[i]>0 & (double)(a[i,i])==(double)(0) )
|
|---|
| 590 | {
|
|---|
| 591 | result = false;
|
|---|
| 592 | return result;
|
|---|
| 593 | }
|
|---|
| 594 | }
|
|---|
| 595 | }
|
|---|
| 596 |
|
|---|
| 597 | //
|
|---|
| 598 | // Solve Ax = b
|
|---|
| 599 | //
|
|---|
| 600 | if( isupper )
|
|---|
| 601 | {
|
|---|
| 602 |
|
|---|
| 603 | //
|
|---|
| 604 | // Solve A*X = B, where A = U*D*U'.
|
|---|
| 605 | //
|
|---|
| 606 | // First solve U*D*X = B, overwriting B with X.
|
|---|
| 607 | //
|
|---|
| 608 | // K is the main loop index, decreasing from N to 1 in steps of
|
|---|
| 609 | // 1 or 2, depending on the size of the diagonal blocks.
|
|---|
| 610 | //
|
|---|
| 611 | k = n;
|
|---|
| 612 | while( k>=1 )
|
|---|
| 613 | {
|
|---|
| 614 | if( pivots[k]>0 )
|
|---|
| 615 | {
|
|---|
| 616 |
|
|---|
| 617 | //
|
|---|
| 618 | // 1 x 1 diagonal block
|
|---|
| 619 | //
|
|---|
| 620 | // Interchange rows K and IPIV(K).
|
|---|
| 621 | //
|
|---|
| 622 | kp = pivots[k];
|
|---|
| 623 | if( kp!=k )
|
|---|
| 624 | {
|
|---|
| 625 | v = b[k];
|
|---|
| 626 | b[k] = b[kp];
|
|---|
| 627 | b[kp] = v;
|
|---|
| 628 | }
|
|---|
| 629 |
|
|---|
| 630 | //
|
|---|
| 631 | // Multiply by inv(U(K)), where U(K) is the transformation
|
|---|
| 632 | // stored in column K of A.
|
|---|
| 633 | //
|
|---|
| 634 | km1 = k-1;
|
|---|
| 635 | v = b[k];
|
|---|
| 636 | for(i_=1; i_<=km1;i_++)
|
|---|
| 637 | {
|
|---|
| 638 | b[i_] = b[i_] - v*a[i_,k];
|
|---|
| 639 | }
|
|---|
| 640 |
|
|---|
| 641 | //
|
|---|
| 642 | // Multiply by the inverse of the diagonal block.
|
|---|
| 643 | //
|
|---|
| 644 | b[k] = b[k]/a[k,k];
|
|---|
| 645 | k = k-1;
|
|---|
| 646 | }
|
|---|
| 647 | else
|
|---|
| 648 | {
|
|---|
| 649 |
|
|---|
| 650 | //
|
|---|
| 651 | // 2 x 2 diagonal block
|
|---|
| 652 | //
|
|---|
| 653 | // Interchange rows K-1 and -IPIV(K).
|
|---|
| 654 | //
|
|---|
| 655 | kp = -pivots[k];
|
|---|
| 656 | if( kp!=k-1 )
|
|---|
| 657 | {
|
|---|
| 658 | v = b[k-1];
|
|---|
| 659 | b[k-1] = b[kp];
|
|---|
| 660 | b[kp] = v;
|
|---|
| 661 | }
|
|---|
| 662 |
|
|---|
| 663 | //
|
|---|
| 664 | // Multiply by inv(U(K)), where U(K) is the transformation
|
|---|
| 665 | // stored in columns K-1 and K of A.
|
|---|
| 666 | //
|
|---|
| 667 | km2 = k-2;
|
|---|
| 668 | km1 = k-1;
|
|---|
| 669 | v = b[k];
|
|---|
| 670 | for(i_=1; i_<=km2;i_++)
|
|---|
| 671 | {
|
|---|
| 672 | b[i_] = b[i_] - v*a[i_,k];
|
|---|
| 673 | }
|
|---|
| 674 | v = b[k-1];
|
|---|
| 675 | for(i_=1; i_<=km2;i_++)
|
|---|
| 676 | {
|
|---|
| 677 | b[i_] = b[i_] - v*a[i_,km1];
|
|---|
| 678 | }
|
|---|
| 679 |
|
|---|
| 680 | //
|
|---|
| 681 | // Multiply by the inverse of the diagonal block.
|
|---|
| 682 | //
|
|---|
| 683 | akm1k = a[k-1,k];
|
|---|
| 684 | akm1 = a[k-1,k-1]/akm1k;
|
|---|
| 685 | ak = a[k,k]/akm1k;
|
|---|
| 686 | denom = akm1*ak-1;
|
|---|
| 687 | bkm1 = b[k-1]/akm1k;
|
|---|
| 688 | bk = b[k]/akm1k;
|
|---|
| 689 | b[k-1] = (ak*bkm1-bk)/denom;
|
|---|
| 690 | b[k] = (akm1*bk-bkm1)/denom;
|
|---|
| 691 | k = k-2;
|
|---|
| 692 | }
|
|---|
| 693 | }
|
|---|
| 694 |
|
|---|
| 695 | //
|
|---|
| 696 | // Next solve U'*X = B, overwriting B with X.
|
|---|
| 697 | //
|
|---|
| 698 | // K is the main loop index, increasing from 1 to N in steps of
|
|---|
| 699 | // 1 or 2, depending on the size of the diagonal blocks.
|
|---|
| 700 | //
|
|---|
| 701 | k = 1;
|
|---|
| 702 | while( k<=n )
|
|---|
| 703 | {
|
|---|
| 704 | if( pivots[k]>0 )
|
|---|
| 705 | {
|
|---|
| 706 |
|
|---|
| 707 | //
|
|---|
| 708 | // 1 x 1 diagonal block
|
|---|
| 709 | //
|
|---|
| 710 | // Multiply by inv(U'(K)), where U(K) is the transformation
|
|---|
| 711 | // stored in column K of A.
|
|---|
| 712 | //
|
|---|
| 713 | km1 = k-1;
|
|---|
| 714 | v = 0.0;
|
|---|
| 715 | for(i_=1; i_<=km1;i_++)
|
|---|
| 716 | {
|
|---|
| 717 | v += b[i_]*a[i_,k];
|
|---|
| 718 | }
|
|---|
| 719 | b[k] = b[k]-v;
|
|---|
| 720 |
|
|---|
| 721 | //
|
|---|
| 722 | // Interchange rows K and IPIV(K).
|
|---|
| 723 | //
|
|---|
| 724 | kp = pivots[k];
|
|---|
| 725 | if( kp!=k )
|
|---|
| 726 | {
|
|---|
| 727 | v = b[k];
|
|---|
| 728 | b[k] = b[kp];
|
|---|
| 729 | b[kp] = v;
|
|---|
| 730 | }
|
|---|
| 731 | k = k+1;
|
|---|
| 732 | }
|
|---|
| 733 | else
|
|---|
| 734 | {
|
|---|
| 735 |
|
|---|
| 736 | //
|
|---|
| 737 | // 2 x 2 diagonal block
|
|---|
| 738 | //
|
|---|
| 739 | // Multiply by inv(U'(K+1)), where U(K+1) is the transformation
|
|---|
| 740 | // stored in columns K and K+1 of A.
|
|---|
| 741 | //
|
|---|
| 742 | km1 = k-1;
|
|---|
| 743 | kp1 = k+1;
|
|---|
| 744 | v = 0.0;
|
|---|
| 745 | for(i_=1; i_<=km1;i_++)
|
|---|
| 746 | {
|
|---|
| 747 | v += b[i_]*a[i_,k];
|
|---|
| 748 | }
|
|---|
| 749 | b[k] = b[k]-v;
|
|---|
| 750 | v = 0.0;
|
|---|
| 751 | for(i_=1; i_<=km1;i_++)
|
|---|
| 752 | {
|
|---|
| 753 | v += b[i_]*a[i_,kp1];
|
|---|
| 754 | }
|
|---|
| 755 | b[k+1] = b[k+1]-v;
|
|---|
| 756 |
|
|---|
| 757 | //
|
|---|
| 758 | // Interchange rows K and -IPIV(K).
|
|---|
| 759 | //
|
|---|
| 760 | kp = -pivots[k];
|
|---|
| 761 | if( kp!=k )
|
|---|
| 762 | {
|
|---|
| 763 | v = b[k];
|
|---|
| 764 | b[k] = b[kp];
|
|---|
| 765 | b[kp] = v;
|
|---|
| 766 | }
|
|---|
| 767 | k = k+2;
|
|---|
| 768 | }
|
|---|
| 769 | }
|
|---|
| 770 | }
|
|---|
| 771 | else
|
|---|
| 772 | {
|
|---|
| 773 |
|
|---|
| 774 | //
|
|---|
| 775 | // Solve A*X = B, where A = L*D*L'.
|
|---|
| 776 | //
|
|---|
| 777 | // First solve L*D*X = B, overwriting B with X.
|
|---|
| 778 | //
|
|---|
| 779 | // K is the main loop index, increasing from 1 to N in steps of
|
|---|
| 780 | // 1 or 2, depending on the size of the diagonal blocks.
|
|---|
| 781 | //
|
|---|
| 782 | k = 1;
|
|---|
| 783 | while( k<=n )
|
|---|
| 784 | {
|
|---|
| 785 | if( pivots[k]>0 )
|
|---|
| 786 | {
|
|---|
| 787 |
|
|---|
| 788 | //
|
|---|
| 789 | // 1 x 1 diagonal block
|
|---|
| 790 | //
|
|---|
| 791 | // Interchange rows K and IPIV(K).
|
|---|
| 792 | //
|
|---|
| 793 | kp = pivots[k];
|
|---|
| 794 | if( kp!=k )
|
|---|
| 795 | {
|
|---|
| 796 | v = b[k];
|
|---|
| 797 | b[k] = b[kp];
|
|---|
| 798 | b[kp] = v;
|
|---|
| 799 | }
|
|---|
| 800 |
|
|---|
| 801 | //
|
|---|
| 802 | // Multiply by inv(L(K)), where L(K) is the transformation
|
|---|
| 803 | // stored in column K of A.
|
|---|
| 804 | //
|
|---|
| 805 | if( k<n )
|
|---|
| 806 | {
|
|---|
| 807 | kp1 = k+1;
|
|---|
| 808 | v = b[k];
|
|---|
| 809 | for(i_=kp1; i_<=n;i_++)
|
|---|
| 810 | {
|
|---|
| 811 | b[i_] = b[i_] - v*a[i_,k];
|
|---|
| 812 | }
|
|---|
| 813 | }
|
|---|
| 814 |
|
|---|
| 815 | //
|
|---|
| 816 | // Multiply by the inverse of the diagonal block.
|
|---|
| 817 | //
|
|---|
| 818 | b[k] = b[k]/a[k,k];
|
|---|
| 819 | k = k+1;
|
|---|
| 820 | }
|
|---|
| 821 | else
|
|---|
| 822 | {
|
|---|
| 823 |
|
|---|
| 824 | //
|
|---|
| 825 | // 2 x 2 diagonal block
|
|---|
| 826 | //
|
|---|
| 827 | // Interchange rows K+1 and -IPIV(K).
|
|---|
| 828 | //
|
|---|
| 829 | kp = -pivots[k];
|
|---|
| 830 | if( kp!=k+1 )
|
|---|
| 831 | {
|
|---|
| 832 | v = b[k+1];
|
|---|
| 833 | b[k+1] = b[kp];
|
|---|
| 834 | b[kp] = v;
|
|---|
| 835 | }
|
|---|
| 836 |
|
|---|
| 837 | //
|
|---|
| 838 | // Multiply by inv(L(K)), where L(K) is the transformation
|
|---|
| 839 | // stored in columns K and K+1 of A.
|
|---|
| 840 | //
|
|---|
| 841 | if( k<n-1 )
|
|---|
| 842 | {
|
|---|
| 843 | kp1 = k+1;
|
|---|
| 844 | kp2 = k+2;
|
|---|
| 845 | v = b[k];
|
|---|
| 846 | for(i_=kp2; i_<=n;i_++)
|
|---|
| 847 | {
|
|---|
| 848 | b[i_] = b[i_] - v*a[i_,k];
|
|---|
| 849 | }
|
|---|
| 850 | v = b[k+1];
|
|---|
| 851 | for(i_=kp2; i_<=n;i_++)
|
|---|
| 852 | {
|
|---|
| 853 | b[i_] = b[i_] - v*a[i_,kp1];
|
|---|
| 854 | }
|
|---|
| 855 | }
|
|---|
| 856 |
|
|---|
| 857 | //
|
|---|
| 858 | // Multiply by the inverse of the diagonal block.
|
|---|
| 859 | //
|
|---|
| 860 | akm1k = a[k+1,k];
|
|---|
| 861 | akm1 = a[k,k]/akm1k;
|
|---|
| 862 | ak = a[k+1,k+1]/akm1k;
|
|---|
| 863 | denom = akm1*ak-1;
|
|---|
| 864 | bkm1 = b[k]/akm1k;
|
|---|
| 865 | bk = b[k+1]/akm1k;
|
|---|
| 866 | b[k] = (ak*bkm1-bk)/denom;
|
|---|
| 867 | b[k+1] = (akm1*bk-bkm1)/denom;
|
|---|
| 868 | k = k+2;
|
|---|
| 869 | }
|
|---|
| 870 | }
|
|---|
| 871 |
|
|---|
| 872 | //
|
|---|
| 873 | // Next solve L'*X = B, overwriting B with X.
|
|---|
| 874 | //
|
|---|
| 875 | // K is the main loop index, decreasing from N to 1 in steps of
|
|---|
| 876 | // 1 or 2, depending on the size of the diagonal blocks.
|
|---|
| 877 | //
|
|---|
| 878 | k = n;
|
|---|
| 879 | while( k>=1 )
|
|---|
| 880 | {
|
|---|
| 881 | if( pivots[k]>0 )
|
|---|
| 882 | {
|
|---|
| 883 |
|
|---|
| 884 | //
|
|---|
| 885 | // 1 x 1 diagonal block
|
|---|
| 886 | //
|
|---|
| 887 | // Multiply by inv(L'(K)), where L(K) is the transformation
|
|---|
| 888 | // stored in column K of A.
|
|---|
| 889 | //
|
|---|
| 890 | if( k<n )
|
|---|
| 891 | {
|
|---|
| 892 | kp1 = k+1;
|
|---|
| 893 | v = 0.0;
|
|---|
| 894 | for(i_=kp1; i_<=n;i_++)
|
|---|
| 895 | {
|
|---|
| 896 | v += b[i_]*a[i_,k];
|
|---|
| 897 | }
|
|---|
| 898 | b[k] = b[k]-v;
|
|---|
| 899 | }
|
|---|
| 900 |
|
|---|
| 901 | //
|
|---|
| 902 | // Interchange rows K and IPIV(K).
|
|---|
| 903 | //
|
|---|
| 904 | kp = pivots[k];
|
|---|
| 905 | if( kp!=k )
|
|---|
| 906 | {
|
|---|
| 907 | v = b[k];
|
|---|
| 908 | b[k] = b[kp];
|
|---|
| 909 | b[kp] = v;
|
|---|
| 910 | }
|
|---|
| 911 | k = k-1;
|
|---|
| 912 | }
|
|---|
| 913 | else
|
|---|
| 914 | {
|
|---|
| 915 |
|
|---|
| 916 | //
|
|---|
| 917 | // 2 x 2 diagonal block
|
|---|
| 918 | //
|
|---|
| 919 | // Multiply by inv(L'(K-1)), where L(K-1) is the transformation
|
|---|
| 920 | // stored in columns K-1 and K of A.
|
|---|
| 921 | //
|
|---|
| 922 | if( k<n )
|
|---|
| 923 | {
|
|---|
| 924 | kp1 = k+1;
|
|---|
| 925 | km1 = k-1;
|
|---|
| 926 | v = 0.0;
|
|---|
| 927 | for(i_=kp1; i_<=n;i_++)
|
|---|
| 928 | {
|
|---|
| 929 | v += b[i_]*a[i_,k];
|
|---|
| 930 | }
|
|---|
| 931 | b[k] = b[k]-v;
|
|---|
| 932 | v = 0.0;
|
|---|
| 933 | for(i_=kp1; i_<=n;i_++)
|
|---|
| 934 | {
|
|---|
| 935 | v += b[i_]*a[i_,km1];
|
|---|
| 936 | }
|
|---|
| 937 | b[k-1] = b[k-1]-v;
|
|---|
| 938 | }
|
|---|
| 939 |
|
|---|
| 940 | //
|
|---|
| 941 | // Interchange rows K and -IPIV(K).
|
|---|
| 942 | //
|
|---|
| 943 | kp = -pivots[k];
|
|---|
| 944 | if( kp!=k )
|
|---|
| 945 | {
|
|---|
| 946 | v = b[k];
|
|---|
| 947 | b[k] = b[kp];
|
|---|
| 948 | b[kp] = v;
|
|---|
| 949 | }
|
|---|
| 950 | k = k-2;
|
|---|
| 951 | }
|
|---|
| 952 | }
|
|---|
| 953 | }
|
|---|
| 954 | x = new double[n+1];
|
|---|
| 955 | for(i_=1; i_<=n;i_++)
|
|---|
| 956 | {
|
|---|
| 957 | x[i_] = b[i_];
|
|---|
| 958 | }
|
|---|
| 959 | return result;
|
|---|
| 960 | }
|
|---|
| 961 |
|
|---|
| 962 |
|
|---|
| 963 | public static bool solvesymmetricsystem(double[,] a,
|
|---|
| 964 | double[] b,
|
|---|
| 965 | int n,
|
|---|
| 966 | bool isupper,
|
|---|
| 967 | ref double[] x)
|
|---|
| 968 | {
|
|---|
| 969 | bool result = new bool();
|
|---|
| 970 | int[] pivots = new int[0];
|
|---|
| 971 |
|
|---|
| 972 | a = (double[,])a.Clone();
|
|---|
| 973 | b = (double[])b.Clone();
|
|---|
| 974 |
|
|---|
| 975 | ldlt.ldltdecomposition(ref a, n, isupper, ref pivots);
|
|---|
| 976 | result = solvesystemldlt(ref a, ref pivots, b, n, isupper, ref x);
|
|---|
| 977 | return result;
|
|---|
| 978 | }
|
|---|
| 979 | }
|
|---|
| 980 | }
|
|---|
