1 | /////////////////////////////////////////////////////////////////////////////////
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2 | //
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3 | // Solution of linear systems involved in the Levenberg - Marquardt
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4 | // minimization algorithm
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5 | // Copyright (C) 2004 Manolis Lourakis (lourakis at ics forth gr)
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6 | // Institute of Computer Science, Foundation for Research & Technology - Hellas
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7 | // Heraklion, Crete, Greece.
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8 | //
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9 | // This program is free software; you can redistribute it and/or modify
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10 | // it under the terms of the GNU General Public License as published by
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11 | // the Free Software Foundation; either version 2 of the License, or
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12 | // (at your option) any later version.
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13 | //
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14 | // This program is distributed in the hope that it will be useful,
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15 | // but WITHOUT ANY WARRANTY; without even the implied warranty of
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16 | // MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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17 | // GNU General Public License for more details.
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18 | //
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19 | /////////////////////////////////////////////////////////////////////////////////
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20 |
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21 |
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22 | /* Solvers for the linear systems Ax=b. Solvers should NOT modify their A & B arguments! */
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23 |
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24 |
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25 | #ifndef LM_REAL // not included by Axb.c
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26 | #error This file should not be compiled directly!
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27 | #endif
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28 |
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29 |
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30 | #ifdef LINSOLVERS_RETAIN_MEMORY
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31 | #define __STATIC__ static
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32 | #else
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33 | #define __STATIC__ // empty
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34 | #endif /* LINSOLVERS_RETAIN_MEMORY */
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35 |
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36 | #ifdef HAVE_LAPACK
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37 |
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38 | /* prototypes of LAPACK routines */
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39 |
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40 | #define GEQRF LM_MK_LAPACK_NAME(geqrf)
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41 | #define ORGQR LM_MK_LAPACK_NAME(orgqr)
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42 | #define TRTRS LM_MK_LAPACK_NAME(trtrs)
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43 | #define POTF2 LM_MK_LAPACK_NAME(potf2)
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44 | #define POTRF LM_MK_LAPACK_NAME(potrf)
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45 | #define POTRS LM_MK_LAPACK_NAME(potrs)
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46 | #define GETRF LM_MK_LAPACK_NAME(getrf)
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47 | #define GETRS LM_MK_LAPACK_NAME(getrs)
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48 | #define GESVD LM_MK_LAPACK_NAME(gesvd)
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49 | #define GESDD LM_MK_LAPACK_NAME(gesdd)
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50 | #define SYTRF LM_MK_LAPACK_NAME(sytrf)
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51 | #define SYTRS LM_MK_LAPACK_NAME(sytrs)
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52 | #define PLASMA_POSV LM_CAT_(PLASMA_, LM_ADD_PREFIX(posv))
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53 |
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54 | #ifdef __cplusplus
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55 | extern "C" {
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56 | #endif
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57 | /* QR decomposition */
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58 | extern int GEQRF(int *m, int *n, LM_REAL *a, int *lda, LM_REAL *tau, LM_REAL *work, int *lwork, int *info);
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59 | extern int ORGQR(int *m, int *n, int *k, LM_REAL *a, int *lda, LM_REAL *tau, LM_REAL *work, int *lwork, int *info);
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60 |
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61 | /* solution of triangular systems */
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62 | extern int TRTRS(char *uplo, char *trans, char *diag, int *n, int *nrhs, LM_REAL *a, int *lda, LM_REAL *b, int *ldb, int *info);
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63 |
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64 | /* Cholesky decomposition and systems solution */
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65 | extern int POTF2(char *uplo, int *n, LM_REAL *a, int *lda, int *info);
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66 | extern int POTRF(char *uplo, int *n, LM_REAL *a, int *lda, int *info); /* block version of dpotf2 */
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67 | extern int POTRS(char *uplo, int *n, int *nrhs, LM_REAL *a, int *lda, LM_REAL *b, int *ldb, int *info);
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68 |
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69 | /* LU decomposition and systems solution */
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70 | extern int GETRF(int *m, int *n, LM_REAL *a, int *lda, int *ipiv, int *info);
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71 | extern int GETRS(char *trans, int *n, int *nrhs, LM_REAL *a, int *lda, int *ipiv, LM_REAL *b, int *ldb, int *info);
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72 |
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73 | /* Singular Value Decomposition (SVD) */
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74 | extern int GESVD(char *jobu, char *jobvt, int *m, int *n, LM_REAL *a, int *lda, LM_REAL *s, LM_REAL *u, int *ldu,
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75 | LM_REAL *vt, int *ldvt, LM_REAL *work, int *lwork, int *info);
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76 |
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77 | /* lapack 3.0 new SVD routine, faster than xgesvd().
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78 | * In case that your version of LAPACK does not include them, use the above two older routines
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79 | */
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80 | extern int GESDD(char *jobz, int *m, int *n, LM_REAL *a, int *lda, LM_REAL *s, LM_REAL *u, int *ldu, LM_REAL *vt, int *ldvt,
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81 | LM_REAL *work, int *lwork, int *iwork, int *info);
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82 |
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83 | /* LDLt/UDUt factorization and systems solution */
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84 | extern int SYTRF(char *uplo, int *n, LM_REAL *a, int *lda, int *ipiv, LM_REAL *work, int *lwork, int *info);
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85 | extern int SYTRS(char *uplo, int *n, int *nrhs, LM_REAL *a, int *lda, int *ipiv, LM_REAL *b, int *ldb, int *info);
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86 | #ifdef __cplusplus
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87 | }
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88 | #endif
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89 |
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90 | /* precision-specific definitions */
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91 | #define AX_EQ_B_QR LM_ADD_PREFIX(Ax_eq_b_QR)
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92 | #define AX_EQ_B_QRLS LM_ADD_PREFIX(Ax_eq_b_QRLS)
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93 | #define AX_EQ_B_CHOL LM_ADD_PREFIX(Ax_eq_b_Chol)
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94 | #define AX_EQ_B_LU LM_ADD_PREFIX(Ax_eq_b_LU)
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95 | #define AX_EQ_B_SVD LM_ADD_PREFIX(Ax_eq_b_SVD)
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96 | #define AX_EQ_B_BK LM_ADD_PREFIX(Ax_eq_b_BK)
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97 | #define AX_EQ_B_PLASMA_CHOL LM_ADD_PREFIX(Ax_eq_b_PLASMA_Chol)
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98 |
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99 | /*
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100 | * This function returns the solution of Ax = b
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101 | *
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102 | * The function is based on QR decomposition with explicit computation of Q:
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103 | * If A=Q R with Q orthogonal and R upper triangular, the linear system becomes
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104 | * Q R x = b or R x = Q^T b.
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105 | * The last equation can be solved directly.
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106 | *
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107 | * A is mxm, b is mx1
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108 | *
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109 | * The function returns 0 in case of error, 1 if successful
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110 | *
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111 | * This function is often called repetitively to solve problems of identical
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112 | * dimensions. To avoid repetitive malloc's and free's, allocated memory is
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113 | * retained between calls and free'd-malloc'ed when not of the appropriate size.
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114 | * A call with NULL as the first argument forces this memory to be released.
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115 | */
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116 | int AX_EQ_B_QR(LM_REAL *A, LM_REAL *B, LM_REAL *x, int m)
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117 | {
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118 | __STATIC__ LM_REAL *buf=NULL;
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119 | __STATIC__ int buf_sz=0;
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120 |
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121 | static int nb=0; /* no __STATIC__ decl. here! */
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122 |
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123 | LM_REAL *a, *tau, *r, *work;
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124 | int a_sz, tau_sz, r_sz, tot_sz;
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125 | register int i, j;
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126 | int info, worksz, nrhs=1;
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127 | register LM_REAL sum;
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128 |
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129 | if(!A)
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130 | #ifdef LINSOLVERS_RETAIN_MEMORY
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131 | {
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132 | if(buf) free(buf);
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133 | buf=NULL;
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134 | buf_sz=0;
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135 |
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136 | return 1;
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137 | }
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138 | #else
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139 | return 1; /* NOP */
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140 | #endif /* LINSOLVERS_RETAIN_MEMORY */
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141 |
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142 | /* calculate required memory size */
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143 | a_sz=m*m;
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144 | tau_sz=m;
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145 | r_sz=m*m; /* only the upper triangular part really needed */
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146 | if(!nb){
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147 | LM_REAL tmp;
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148 |
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149 | worksz=-1; // workspace query; optimal size is returned in tmp
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150 | GEQRF((int *)&m, (int *)&m, NULL, (int *)&m, NULL, (LM_REAL *)&tmp, (int *)&worksz, (int *)&info);
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151 | nb=((int)tmp)/m; // optimal worksize is m*nb
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152 | }
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153 | worksz=nb*m;
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154 | tot_sz=a_sz + tau_sz + r_sz + worksz;
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155 |
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156 | #ifdef LINSOLVERS_RETAIN_MEMORY
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157 | if(tot_sz>buf_sz){ /* insufficient memory, allocate a "big" memory chunk at once */
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158 | if(buf) free(buf); /* free previously allocated memory */
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159 |
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160 | buf_sz=tot_sz;
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161 | buf=(LM_REAL *)malloc(buf_sz*sizeof(LM_REAL));
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162 | if(!buf){
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163 | fprintf(stderr, RCAT("memory allocation in ", AX_EQ_B_QR) "() failed!\n");
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164 | exit(1);
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165 | }
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166 | }
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167 | #else
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168 | buf_sz=tot_sz;
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169 | buf=(LM_REAL *)malloc(buf_sz*sizeof(LM_REAL));
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170 | if(!buf){
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171 | fprintf(stderr, RCAT("memory allocation in ", AX_EQ_B_QR) "() failed!\n");
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172 | exit(1);
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173 | }
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174 | #endif /* LINSOLVERS_RETAIN_MEMORY */
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175 |
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176 | a=buf;
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177 | tau=a+a_sz;
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178 | r=tau+tau_sz;
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179 | work=r+r_sz;
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180 |
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181 | /* store A (column major!) into a */
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182 | for(i=0; i<m; i++)
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183 | for(j=0; j<m; j++)
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184 | a[i+j*m]=A[i*m+j];
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185 |
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186 | /* QR decomposition of A */
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187 | GEQRF((int *)&m, (int *)&m, a, (int *)&m, tau, work, (int *)&worksz, (int *)&info);
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188 | /* error treatment */
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189 | if(info!=0){
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190 | if(info<0){
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191 | fprintf(stderr, RCAT(RCAT("LAPACK error: illegal value for argument %d of ", GEQRF) " in ", AX_EQ_B_QR) "()\n", -info);
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192 | exit(1);
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193 | }
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194 | else{
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195 | fprintf(stderr, RCAT(RCAT("Unknown LAPACK error %d for ", GEQRF) " in ", AX_EQ_B_QR) "()\n", info);
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196 | #ifndef LINSOLVERS_RETAIN_MEMORY
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197 | free(buf);
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198 | #endif
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199 |
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200 | return 0;
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201 | }
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202 | }
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203 |
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204 | /* R is stored in the upper triangular part of a; copy it in r so that ORGQR() below won't destroy it */
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205 | memcpy(r, a, r_sz*sizeof(LM_REAL));
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206 |
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207 | /* compute Q using the elementary reflectors computed by the above decomposition */
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208 | ORGQR((int *)&m, (int *)&m, (int *)&m, a, (int *)&m, tau, work, (int *)&worksz, (int *)&info);
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209 | if(info!=0){
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210 | if(info<0){
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211 | fprintf(stderr, RCAT(RCAT("LAPACK error: illegal value for argument %d of ", ORGQR) " in ", AX_EQ_B_QR) "()\n", -info);
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212 | exit(1);
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213 | }
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214 | else{
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215 | fprintf(stderr, RCAT("Unknown LAPACK error (%d) in ", AX_EQ_B_QR) "()\n", info);
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216 | #ifndef LINSOLVERS_RETAIN_MEMORY
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217 | free(buf);
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218 | #endif
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219 |
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220 | return 0;
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221 | }
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222 | }
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223 |
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224 | /* Q is now in a; compute Q^T b in x */
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225 | for(i=0; i<m; i++){
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226 | for(j=0, sum=0.0; j<m; j++)
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227 | sum+=a[i*m+j]*B[j];
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228 | x[i]=sum;
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229 | }
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230 |
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231 | /* solve the linear system R x = Q^t b */
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232 | TRTRS("U", "N", "N", (int *)&m, (int *)&nrhs, r, (int *)&m, x, (int *)&m, &info);
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233 | /* error treatment */
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234 | if(info!=0){
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235 | if(info<0){
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236 | fprintf(stderr, RCAT(RCAT("LAPACK error: illegal value for argument %d of ", TRTRS) " in ", AX_EQ_B_QR) "()\n", -info);
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237 | exit(1);
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238 | }
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239 | else{
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240 | fprintf(stderr, RCAT("LAPACK error: the %d-th diagonal element of A is zero (singular matrix) in ", AX_EQ_B_QR) "()\n", info);
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241 | #ifndef LINSOLVERS_RETAIN_MEMORY
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242 | free(buf);
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243 | #endif
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244 |
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245 | return 0;
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246 | }
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247 | }
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248 |
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249 | #ifndef LINSOLVERS_RETAIN_MEMORY
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250 | free(buf);
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251 | #endif
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252 |
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253 | return 1;
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254 | }
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255 |
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256 | /*
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257 | * This function returns the solution of min_x ||Ax - b||
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258 | *
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259 | * || . || is the second order (i.e. L2) norm. This is a least squares technique that
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260 | * is based on QR decomposition:
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261 | * If A=Q R with Q orthogonal and R upper triangular, the normal equations become
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262 | * (A^T A) x = A^T b or (R^T Q^T Q R) x = A^T b or (R^T R) x = A^T b.
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263 | * This amounts to solving R^T y = A^T b for y and then R x = y for x
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264 | * Note that Q does not need to be explicitly computed
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265 | *
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266 | * A is mxn, b is mx1
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267 | *
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268 | * The function returns 0 in case of error, 1 if successful
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269 | *
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270 | * This function is often called repetitively to solve problems of identical
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271 | * dimensions. To avoid repetitive malloc's and free's, allocated memory is
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272 | * retained between calls and free'd-malloc'ed when not of the appropriate size.
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273 | * A call with NULL as the first argument forces this memory to be released.
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274 | */
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275 | int AX_EQ_B_QRLS(LM_REAL *A, LM_REAL *B, LM_REAL *x, int m, int n)
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276 | {
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277 | __STATIC__ LM_REAL *buf=NULL;
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278 | __STATIC__ int buf_sz=0;
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279 |
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280 | static int nb=0; /* no __STATIC__ decl. here! */
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281 |
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282 | LM_REAL *a, *tau, *r, *work;
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283 | int a_sz, tau_sz, r_sz, tot_sz;
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284 | register int i, j;
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285 | int info, worksz, nrhs=1;
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286 | register LM_REAL sum;
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287 |
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288 | if(!A)
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289 | #ifdef LINSOLVERS_RETAIN_MEMORY
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290 | {
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291 | if(buf) free(buf);
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292 | buf=NULL;
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293 | buf_sz=0;
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294 |
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295 | return 1;
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296 | }
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297 | #else
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298 | return 1; /* NOP */
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299 | #endif /* LINSOLVERS_RETAIN_MEMORY */
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300 |
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301 | if(m<n){
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302 | fprintf(stderr, RCAT("Normal equations require that the number of rows is greater than number of columns in ", AX_EQ_B_QRLS) "() [%d x %d]! -- try transposing\n", m, n);
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303 | exit(1);
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304 | }
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305 |
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306 | /* calculate required memory size */
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307 | a_sz=m*n;
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308 | tau_sz=n;
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309 | r_sz=n*n;
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310 | if(!nb){
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311 | LM_REAL tmp;
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312 |
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313 | worksz=-1; // workspace query; optimal size is returned in tmp
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314 | GEQRF((int *)&m, (int *)&m, NULL, (int *)&m, NULL, (LM_REAL *)&tmp, (int *)&worksz, (int *)&info);
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315 | nb=((int)tmp)/m; // optimal worksize is m*nb
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316 | }
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317 | worksz=nb*m;
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318 | tot_sz=a_sz + tau_sz + r_sz + worksz;
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319 |
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320 | #ifdef LINSOLVERS_RETAIN_MEMORY
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321 | if(tot_sz>buf_sz){ /* insufficient memory, allocate a "big" memory chunk at once */
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322 | if(buf) free(buf); /* free previously allocated memory */
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323 |
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324 | buf_sz=tot_sz;
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325 | buf=(LM_REAL *)malloc(buf_sz*sizeof(LM_REAL));
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326 | if(!buf){
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327 | fprintf(stderr, RCAT("memory allocation in ", AX_EQ_B_QRLS) "() failed!\n");
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328 | exit(1);
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329 | }
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330 | }
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331 | #else
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332 | buf_sz=tot_sz;
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333 | buf=(LM_REAL *)malloc(buf_sz*sizeof(LM_REAL));
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334 | if(!buf){
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335 | fprintf(stderr, RCAT("memory allocation in ", AX_EQ_B_QRLS) "() failed!\n");
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336 | exit(1);
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337 | }
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338 | #endif /* LINSOLVERS_RETAIN_MEMORY */
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339 |
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340 | a=buf;
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341 | tau=a+a_sz;
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342 | r=tau+tau_sz;
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343 | work=r+r_sz;
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344 |
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345 | /* store A (column major!) into a */
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346 | for(i=0; i<m; i++)
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347 | for(j=0; j<n; j++)
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348 | a[i+j*m]=A[i*n+j];
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349 |
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350 | /* compute A^T b in x */
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351 | for(i=0; i<n; i++){
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352 | for(j=0, sum=0.0; j<m; j++)
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353 | sum+=A[j*n+i]*B[j];
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354 | x[i]=sum;
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355 | }
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356 |
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357 | /* QR decomposition of A */
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358 | GEQRF((int *)&m, (int *)&n, a, (int *)&m, tau, work, (int *)&worksz, (int *)&info);
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359 | /* error treatment */
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360 | if(info!=0){
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361 | if(info<0){
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362 | fprintf(stderr, RCAT(RCAT("LAPACK error: illegal value for argument %d of ", GEQRF) " in ", AX_EQ_B_QRLS) "()\n", -info);
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363 | exit(1);
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364 | }
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365 | else{
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366 | fprintf(stderr, RCAT(RCAT("Unknown LAPACK error %d for ", GEQRF) " in ", AX_EQ_B_QRLS) "()\n", info);
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367 | #ifndef LINSOLVERS_RETAIN_MEMORY
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368 | free(buf);
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369 | #endif
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370 |
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371 | return 0;
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372 | }
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373 | }
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374 |
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375 | /* R is stored in the upper triangular part of a. Note that a is mxn while r nxn */
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376 | for(j=0; j<n; j++){
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377 | for(i=0; i<=j; i++)
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378 | r[i+j*n]=a[i+j*m];
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379 |
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380 | /* lower part is zero */
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381 | for(i=j+1; i<n; i++)
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382 | r[i+j*n]=0.0;
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383 | }
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384 |
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385 | /* solve the linear system R^T y = A^t b */
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386 | TRTRS("U", "T", "N", (int *)&n, (int *)&nrhs, r, (int *)&n, x, (int *)&n, &info);
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387 | /* error treatment */
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388 | if(info!=0){
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389 | if(info<0){
|
---|
390 | fprintf(stderr, RCAT(RCAT("LAPACK error: illegal value for argument %d of ", TRTRS) " in ", AX_EQ_B_QRLS) "()\n", -info);
|
---|
391 | exit(1);
|
---|
392 | }
|
---|
393 | else{
|
---|
394 | fprintf(stderr, RCAT("LAPACK error: the %d-th diagonal element of A is zero (singular matrix) in ", AX_EQ_B_QRLS) "()\n", info);
|
---|
395 | #ifndef LINSOLVERS_RETAIN_MEMORY
|
---|
396 | free(buf);
|
---|
397 | #endif
|
---|
398 |
|
---|
399 | return 0;
|
---|
400 | }
|
---|
401 | }
|
---|
402 |
|
---|
403 | /* solve the linear system R x = y */
|
---|
404 | TRTRS("U", "N", "N", (int *)&n, (int *)&nrhs, r, (int *)&n, x, (int *)&n, &info);
|
---|
405 | /* error treatment */
|
---|
406 | if(info!=0){
|
---|
407 | if(info<0){
|
---|
408 | fprintf(stderr, RCAT(RCAT("LAPACK error: illegal value for argument %d of ", TRTRS) " in ", AX_EQ_B_QRLS) "()\n", -info);
|
---|
409 | exit(1);
|
---|
410 | }
|
---|
411 | else{
|
---|
412 | fprintf(stderr, RCAT("LAPACK error: the %d-th diagonal element of A is zero (singular matrix) in ", AX_EQ_B_QRLS) "()\n", info);
|
---|
413 | #ifndef LINSOLVERS_RETAIN_MEMORY
|
---|
414 | free(buf);
|
---|
415 | #endif
|
---|
416 |
|
---|
417 | return 0;
|
---|
418 | }
|
---|
419 | }
|
---|
420 |
|
---|
421 | #ifndef LINSOLVERS_RETAIN_MEMORY
|
---|
422 | free(buf);
|
---|
423 | #endif
|
---|
424 |
|
---|
425 | return 1;
|
---|
426 | }
|
---|
427 |
|
---|
428 | /*
|
---|
429 | * This function returns the solution of Ax=b
|
---|
430 | *
|
---|
431 | * The function assumes that A is symmetric & postive definite and employs
|
---|
432 | * the Cholesky decomposition:
|
---|
433 | * If A=L L^T with L lower triangular, the system to be solved becomes
|
---|
434 | * (L L^T) x = b
|
---|
435 | * This amounts to solving L y = b for y and then L^T x = y for x
|
---|
436 | *
|
---|
437 | * A is mxm, b is mx1
|
---|
438 | *
|
---|
439 | * The function returns 0 in case of error, 1 if successful
|
---|
440 | *
|
---|
441 | * This function is often called repetitively to solve problems of identical
|
---|
442 | * dimensions. To avoid repetitive malloc's and free's, allocated memory is
|
---|
443 | * retained between calls and free'd-malloc'ed when not of the appropriate size.
|
---|
444 | * A call with NULL as the first argument forces this memory to be released.
|
---|
445 | */
|
---|
446 | int AX_EQ_B_CHOL(LM_REAL *A, LM_REAL *B, LM_REAL *x, int m)
|
---|
447 | {
|
---|
448 | __STATIC__ LM_REAL *buf=NULL;
|
---|
449 | __STATIC__ int buf_sz=0;
|
---|
450 |
|
---|
451 | LM_REAL *a;
|
---|
452 | int a_sz, tot_sz;
|
---|
453 | int info, nrhs=1;
|
---|
454 |
|
---|
455 | if(!A)
|
---|
456 | #ifdef LINSOLVERS_RETAIN_MEMORY
|
---|
457 | {
|
---|
458 | if(buf) free(buf);
|
---|
459 | buf=NULL;
|
---|
460 | buf_sz=0;
|
---|
461 |
|
---|
462 | return 1;
|
---|
463 | }
|
---|
464 | #else
|
---|
465 | return 1; /* NOP */
|
---|
466 | #endif /* LINSOLVERS_RETAIN_MEMORY */
|
---|
467 |
|
---|
468 | /* calculate required memory size */
|
---|
469 | a_sz=m*m;
|
---|
470 | tot_sz=a_sz;
|
---|
471 |
|
---|
472 | #ifdef LINSOLVERS_RETAIN_MEMORY
|
---|
473 | if(tot_sz>buf_sz){ /* insufficient memory, allocate a "big" memory chunk at once */
|
---|
474 | if(buf) free(buf); /* free previously allocated memory */
|
---|
475 |
|
---|
476 | buf_sz=tot_sz;
|
---|
477 | buf=(LM_REAL *)malloc(buf_sz*sizeof(LM_REAL));
|
---|
478 | if(!buf){
|
---|
479 | fprintf(stderr, RCAT("memory allocation in ", AX_EQ_B_CHOL) "() failed!\n");
|
---|
480 | exit(1);
|
---|
481 | }
|
---|
482 | }
|
---|
483 | #else
|
---|
484 | buf_sz=tot_sz;
|
---|
485 | buf=(LM_REAL *)malloc(buf_sz*sizeof(LM_REAL));
|
---|
486 | if(!buf){
|
---|
487 | fprintf(stderr, RCAT("memory allocation in ", AX_EQ_B_CHOL) "() failed!\n");
|
---|
488 | exit(1);
|
---|
489 | }
|
---|
490 | #endif /* LINSOLVERS_RETAIN_MEMORY */
|
---|
491 |
|
---|
492 | a=buf;
|
---|
493 |
|
---|
494 | /* store A into a and B into x. A is assumed symmetric,
|
---|
495 | * hence no transposition is needed
|
---|
496 | */
|
---|
497 | memcpy(a, A, a_sz*sizeof(LM_REAL));
|
---|
498 | memcpy(x, B, m*sizeof(LM_REAL));
|
---|
499 |
|
---|
500 | /* Cholesky decomposition of A */
|
---|
501 | //POTF2("L", (int *)&m, a, (int *)&m, (int *)&info);
|
---|
502 | POTRF("L", (int *)&m, a, (int *)&m, (int *)&info);
|
---|
503 | /* error treatment */
|
---|
504 | if(info!=0){
|
---|
505 | if(info<0){
|
---|
506 | fprintf(stderr, RCAT(RCAT(RCAT("LAPACK error: illegal value for argument %d of ", POTF2) "/", POTRF) " in ",
|
---|
507 | AX_EQ_B_CHOL) "()\n", -info);
|
---|
508 | exit(1);
|
---|
509 | }
|
---|
510 | else{
|
---|
511 | fprintf(stderr, RCAT(RCAT(RCAT("LAPACK error: the leading minor of order %d is not positive definite,\nthe factorization could not be completed for ", POTF2) "/", POTRF) " in ", AX_EQ_B_CHOL) "()\n", info);
|
---|
512 | #ifndef LINSOLVERS_RETAIN_MEMORY
|
---|
513 | free(buf);
|
---|
514 | #endif
|
---|
515 |
|
---|
516 | return 0;
|
---|
517 | }
|
---|
518 | }
|
---|
519 |
|
---|
520 | /* solve using the computed Cholesky in one lapack call */
|
---|
521 | POTRS("L", (int *)&m, (int *)&nrhs, a, (int *)&m, x, (int *)&m, &info);
|
---|
522 | if(info<0){
|
---|
523 | fprintf(stderr, RCAT(RCAT("LAPACK error: illegal value for argument %d of ", POTRS) " in ", AX_EQ_B_CHOL) "()\n", -info);
|
---|
524 | exit(1);
|
---|
525 | }
|
---|
526 |
|
---|
527 | #if 0
|
---|
528 | /* alternative: solve the linear system L y = b ... */
|
---|
529 | TRTRS("L", "N", "N", (int *)&m, (int *)&nrhs, a, (int *)&m, x, (int *)&m, &info);
|
---|
530 | /* error treatment */
|
---|
531 | if(info!=0){
|
---|
532 | if(info<0){
|
---|
533 | fprintf(stderr, RCAT(RCAT("LAPACK error: illegal value for argument %d of ", TRTRS) " in ", AX_EQ_B_CHOL) "()\n", -info);
|
---|
534 | exit(1);
|
---|
535 | }
|
---|
536 | else{
|
---|
537 | fprintf(stderr, RCAT("LAPACK error: the %d-th diagonal element of A is zero (singular matrix) in ", AX_EQ_B_CHOL) "()\n", info);
|
---|
538 | #ifndef LINSOLVERS_RETAIN_MEMORY
|
---|
539 | free(buf);
|
---|
540 | #endif
|
---|
541 |
|
---|
542 | return 0;
|
---|
543 | }
|
---|
544 | }
|
---|
545 |
|
---|
546 | /* ... solve the linear system L^T x = y */
|
---|
547 | TRTRS("L", "T", "N", (int *)&m, (int *)&nrhs, a, (int *)&m, x, (int *)&m, &info);
|
---|
548 | /* error treatment */
|
---|
549 | if(info!=0){
|
---|
550 | if(info<0){
|
---|
551 | fprintf(stderr, RCAT(RCAT("LAPACK error: illegal value for argument %d of ", TRTRS) "in ", AX_EQ_B_CHOL) "()\n", -info);
|
---|
552 | exit(1);
|
---|
553 | }
|
---|
554 | else{
|
---|
555 | fprintf(stderr, RCAT("LAPACK error: the %d-th diagonal element of A is zero (singular matrix) in ", AX_EQ_B_CHOL) "()\n", info);
|
---|
556 | #ifndef LINSOLVERS_RETAIN_MEMORY
|
---|
557 | free(buf);
|
---|
558 | #endif
|
---|
559 |
|
---|
560 | return 0;
|
---|
561 | }
|
---|
562 | }
|
---|
563 | #endif /* 0 */
|
---|
564 |
|
---|
565 | #ifndef LINSOLVERS_RETAIN_MEMORY
|
---|
566 | free(buf);
|
---|
567 | #endif
|
---|
568 |
|
---|
569 | return 1;
|
---|
570 | }
|
---|
571 |
|
---|
572 | #ifdef HAVE_PLASMA
|
---|
573 |
|
---|
574 | /* Linear algebra using PLASMA parallel library for multicore CPUs.
|
---|
575 | * http://icl.cs.utk.edu/plasma/
|
---|
576 | *
|
---|
577 | * WARNING: BLAS multithreading should be disabled, e.g. setenv MKL_NUM_THREADS 1
|
---|
578 | */
|
---|
579 |
|
---|
580 | #ifndef _LM_PLASMA_MISC_
|
---|
581 | /* avoid multiple inclusion of helper code */
|
---|
582 | #define _LM_PLASMA_MISC_
|
---|
583 |
|
---|
584 | #include <plasma.h>
|
---|
585 | #include <cblas.h>
|
---|
586 | #include <lapacke.h>
|
---|
587 | #include <plasma_tmg.h>
|
---|
588 | #include <core_blas.h>
|
---|
589 |
|
---|
590 | /* programmatically determine the number of cores on the current machine */
|
---|
591 | #ifdef _WIN32
|
---|
592 | #include <windows.h>
|
---|
593 | #elif __linux
|
---|
594 | #include <unistd.h>
|
---|
595 | #endif
|
---|
596 | static int getnbcores()
|
---|
597 | {
|
---|
598 | #ifdef _WIN32
|
---|
599 | SYSTEM_INFO sysinfo;
|
---|
600 | GetSystemInfo(&sysinfo);
|
---|
601 | return sysinfo.dwNumberOfProcessors;
|
---|
602 | #elif __linux
|
---|
603 | return sysconf(_SC_NPROCESSORS_ONLN);
|
---|
604 | #else // unknown system
|
---|
605 | return 2<<1; // will be halved by right shift below
|
---|
606 | #endif
|
---|
607 | }
|
---|
608 |
|
---|
609 | static int PLASMA_ncores=-(getnbcores()>>1); // >0 if PLASMA initialized, <0 otherwise
|
---|
610 |
|
---|
611 | /* user-specified number of cores */
|
---|
612 | void levmar_PLASMA_setnbcores(int cores)
|
---|
613 | {
|
---|
614 | PLASMA_ncores=(cores>0)? -cores : ((cores)? cores : -2);
|
---|
615 | }
|
---|
616 | #endif /* _LM_PLASMA_MISC_ */
|
---|
617 |
|
---|
618 | /*
|
---|
619 | * This function returns the solution of Ax=b
|
---|
620 | *
|
---|
621 | * The function assumes that A is symmetric & positive definite and employs the
|
---|
622 | * Cholesky decomposition implemented by PLASMA for homogeneous multicore processors.
|
---|
623 | *
|
---|
624 | * A is mxm, b is mx1
|
---|
625 | *
|
---|
626 | * The function returns 0 in case of error, 1 if successfull
|
---|
627 | *
|
---|
628 | * This function is often called repetitively to solve problems of identical
|
---|
629 | * dimensions. To avoid repetitive malloc's and free's, allocated memory is
|
---|
630 | * retained between calls and free'd-malloc'ed when not of the appropriate size.
|
---|
631 | * A call with NULL as the first argument forces this memory to be released.
|
---|
632 | */
|
---|
633 | int AX_EQ_B_PLASMA_CHOL(LM_REAL *A, LM_REAL *B, LM_REAL *x, int m)
|
---|
634 | {
|
---|
635 | __STATIC__ LM_REAL *buf=NULL;
|
---|
636 | __STATIC__ int buf_sz=0;
|
---|
637 |
|
---|
638 | LM_REAL *a;
|
---|
639 | int a_sz, tot_sz;
|
---|
640 | int info, nrhs=1;
|
---|
641 |
|
---|
642 | if(A==NULL){
|
---|
643 | #ifdef LINSOLVERS_RETAIN_MEMORY
|
---|
644 | if(buf) free(buf);
|
---|
645 | buf=NULL;
|
---|
646 | buf_sz=0;
|
---|
647 | #endif /* LINSOLVERS_RETAIN_MEMORY */
|
---|
648 |
|
---|
649 | PLASMA_Finalize();
|
---|
650 | PLASMA_ncores=-PLASMA_ncores;
|
---|
651 |
|
---|
652 | return 1;
|
---|
653 | }
|
---|
654 |
|
---|
655 | /* calculate required memory size */
|
---|
656 | a_sz=m*m;
|
---|
657 | tot_sz=a_sz;
|
---|
658 |
|
---|
659 | #ifdef LINSOLVERS_RETAIN_MEMORY
|
---|
660 | if(tot_sz>buf_sz){ /* insufficient memory, allocate a "big" memory chunk at once */
|
---|
661 | if(buf) free(buf); /* free previously allocated memory */
|
---|
662 |
|
---|
663 | buf_sz=tot_sz;
|
---|
664 | buf=(LM_REAL *)malloc(buf_sz*sizeof(LM_REAL));
|
---|
665 | if(!buf){
|
---|
666 | fprintf(stderr, RCAT("memory allocation in ", AX_EQ_B_PLASMA_CHOL) "() failed!\n");
|
---|
667 | exit(1);
|
---|
668 | }
|
---|
669 | }
|
---|
670 | #else
|
---|
671 | buf_sz=tot_sz;
|
---|
672 | buf=(LM_REAL *)malloc(buf_sz*sizeof(LM_REAL));
|
---|
673 | if(!buf){
|
---|
674 | fprintf(stderr, RCAT("memory allocation in ", AX_EQ_B_PLASMA_CHOL) "() failed!\n");
|
---|
675 | exit(1);
|
---|
676 | }
|
---|
677 | #endif /* LINSOLVERS_RETAIN_MEMORY */
|
---|
678 |
|
---|
679 | a=buf;
|
---|
680 |
|
---|
681 | /* store A into a and B into x; A is assumed to be symmetric,
|
---|
682 | * hence no transposition is needed
|
---|
683 | */
|
---|
684 | memcpy(a, A, a_sz*sizeof(LM_REAL));
|
---|
685 | memcpy(x, B, m*sizeof(LM_REAL));
|
---|
686 |
|
---|
687 | /* initialize PLASMA */
|
---|
688 | if(PLASMA_ncores<0){
|
---|
689 | PLASMA_ncores=-PLASMA_ncores;
|
---|
690 | PLASMA_Init(PLASMA_ncores);
|
---|
691 | fprintf(stderr, RCAT("\n", AX_EQ_B_PLASMA_CHOL) "(): PLASMA is running on %d cores.\n\n", PLASMA_ncores);
|
---|
692 | }
|
---|
693 |
|
---|
694 | /* Solve the linear system */
|
---|
695 | info=PLASMA_POSV(PlasmaLower, m, 1, a, m, x, m);
|
---|
696 | /* error treatment */
|
---|
697 | if(info!=0){
|
---|
698 | if(info<0){
|
---|
699 | fprintf(stderr, RCAT(RCAT("LAPACK error: illegal value for argument %d of ", PLASMA_POSV) " in ",
|
---|
700 | AX_EQ_B_PLASMA_CHOL) "()\n", -info);
|
---|
701 | exit(1);
|
---|
702 | }
|
---|
703 | else{
|
---|
704 | fprintf(stderr, RCAT(RCAT("LAPACK error: the leading minor of order %d is not positive definite,\n"
|
---|
705 | "the factorization could not be completed for ", PLASMA_POSV) " in ", AX_EQ_B_CHOL) "()\n", info);
|
---|
706 | #ifndef LINSOLVERS_RETAIN_MEMORY
|
---|
707 | free(buf);
|
---|
708 | #endif
|
---|
709 | return 0;
|
---|
710 | }
|
---|
711 | }
|
---|
712 |
|
---|
713 | #ifndef LINSOLVERS_RETAIN_MEMORY
|
---|
714 | free(buf);
|
---|
715 | #endif
|
---|
716 |
|
---|
717 | return 1;
|
---|
718 | }
|
---|
719 | #endif /* HAVE_PLASMA */
|
---|
720 |
|
---|
721 | /*
|
---|
722 | * This function returns the solution of Ax = b
|
---|
723 | *
|
---|
724 | * The function employs LU decomposition:
|
---|
725 | * If A=L U with L lower and U upper triangular, then the original system
|
---|
726 | * amounts to solving
|
---|
727 | * L y = b, U x = y
|
---|
728 | *
|
---|
729 | * A is mxm, b is mx1
|
---|
730 | *
|
---|
731 | * The function returns 0 in case of error, 1 if successful
|
---|
732 | *
|
---|
733 | * This function is often called repetitively to solve problems of identical
|
---|
734 | * dimensions. To avoid repetitive malloc's and free's, allocated memory is
|
---|
735 | * retained between calls and free'd-malloc'ed when not of the appropriate size.
|
---|
736 | * A call with NULL as the first argument forces this memory to be released.
|
---|
737 | */
|
---|
738 | int AX_EQ_B_LU(LM_REAL *A, LM_REAL *B, LM_REAL *x, int m)
|
---|
739 | {
|
---|
740 | __STATIC__ LM_REAL *buf=NULL;
|
---|
741 | __STATIC__ int buf_sz=0;
|
---|
742 |
|
---|
743 | int a_sz, ipiv_sz, tot_sz;
|
---|
744 | register int i, j;
|
---|
745 | int info, *ipiv, nrhs=1;
|
---|
746 | LM_REAL *a;
|
---|
747 |
|
---|
748 | if(!A)
|
---|
749 | #ifdef LINSOLVERS_RETAIN_MEMORY
|
---|
750 | {
|
---|
751 | if(buf) free(buf);
|
---|
752 | buf=NULL;
|
---|
753 | buf_sz=0;
|
---|
754 |
|
---|
755 | return 1;
|
---|
756 | }
|
---|
757 | #else
|
---|
758 | return 1; /* NOP */
|
---|
759 | #endif /* LINSOLVERS_RETAIN_MEMORY */
|
---|
760 |
|
---|
761 | /* calculate required memory size */
|
---|
762 | ipiv_sz=m;
|
---|
763 | a_sz=m*m;
|
---|
764 | tot_sz=a_sz*sizeof(LM_REAL) + ipiv_sz*sizeof(int); /* should be arranged in that order for proper doubles alignment */
|
---|
765 |
|
---|
766 | #ifdef LINSOLVERS_RETAIN_MEMORY
|
---|
767 | if(tot_sz>buf_sz){ /* insufficient memory, allocate a "big" memory chunk at once */
|
---|
768 | if(buf) free(buf); /* free previously allocated memory */
|
---|
769 |
|
---|
770 | buf_sz=tot_sz;
|
---|
771 | buf=(LM_REAL *)malloc(buf_sz);
|
---|
772 | if(!buf){
|
---|
773 | fprintf(stderr, RCAT("memory allocation in ", AX_EQ_B_LU) "() failed!\n");
|
---|
774 | exit(1);
|
---|
775 | }
|
---|
776 | }
|
---|
777 | #else
|
---|
778 | buf_sz=tot_sz;
|
---|
779 | buf=(LM_REAL *)malloc(buf_sz);
|
---|
780 | if(!buf){
|
---|
781 | fprintf(stderr, RCAT("memory allocation in ", AX_EQ_B_LU) "() failed!\n");
|
---|
782 | exit(1);
|
---|
783 | }
|
---|
784 | #endif /* LINSOLVERS_RETAIN_MEMORY */
|
---|
785 |
|
---|
786 | a=buf;
|
---|
787 | ipiv=(int *)(a+a_sz);
|
---|
788 |
|
---|
789 | /* store A (column major!) into a and B into x */
|
---|
790 | for(i=0; i<m; i++){
|
---|
791 | for(j=0; j<m; j++)
|
---|
792 | a[i+j*m]=A[i*m+j];
|
---|
793 |
|
---|
794 | x[i]=B[i];
|
---|
795 | }
|
---|
796 |
|
---|
797 | /* LU decomposition for A */
|
---|
798 | GETRF((int *)&m, (int *)&m, a, (int *)&m, ipiv, (int *)&info);
|
---|
799 | if(info!=0){
|
---|
800 | if(info<0){
|
---|
801 | fprintf(stderr, RCAT(RCAT("argument %d of ", GETRF) " illegal in ", AX_EQ_B_LU) "()\n", -info);
|
---|
802 | exit(1);
|
---|
803 | }
|
---|
804 | else{
|
---|
805 | fprintf(stderr, RCAT(RCAT("singular matrix A for ", GETRF) " in ", AX_EQ_B_LU) "()\n");
|
---|
806 | #ifndef LINSOLVERS_RETAIN_MEMORY
|
---|
807 | free(buf);
|
---|
808 | #endif
|
---|
809 |
|
---|
810 | return 0;
|
---|
811 | }
|
---|
812 | }
|
---|
813 |
|
---|
814 | /* solve the system with the computed LU */
|
---|
815 | GETRS("N", (int *)&m, (int *)&nrhs, a, (int *)&m, ipiv, x, (int *)&m, (int *)&info);
|
---|
816 | if(info!=0){
|
---|
817 | if(info<0){
|
---|
818 | fprintf(stderr, RCAT(RCAT("argument %d of ", GETRS) " illegal in ", AX_EQ_B_LU) "()\n", -info);
|
---|
819 | exit(1);
|
---|
820 | }
|
---|
821 | else{
|
---|
822 | fprintf(stderr, RCAT(RCAT("unknown error for ", GETRS) " in ", AX_EQ_B_LU) "()\n");
|
---|
823 | #ifndef LINSOLVERS_RETAIN_MEMORY
|
---|
824 | free(buf);
|
---|
825 | #endif
|
---|
826 |
|
---|
827 | return 0;
|
---|
828 | }
|
---|
829 | }
|
---|
830 |
|
---|
831 | #ifndef LINSOLVERS_RETAIN_MEMORY
|
---|
832 | free(buf);
|
---|
833 | #endif
|
---|
834 |
|
---|
835 | return 1;
|
---|
836 | }
|
---|
837 |
|
---|
838 | /*
|
---|
839 | * This function returns the solution of Ax = b
|
---|
840 | *
|
---|
841 | * The function is based on SVD decomposition:
|
---|
842 | * If A=U D V^T with U, V orthogonal and D diagonal, the linear system becomes
|
---|
843 | * (U D V^T) x = b or x=V D^{-1} U^T b
|
---|
844 | * Note that V D^{-1} U^T is the pseudoinverse A^+
|
---|
845 | *
|
---|
846 | * A is mxm, b is mx1.
|
---|
847 | *
|
---|
848 | * The function returns 0 in case of error, 1 if successful
|
---|
849 | *
|
---|
850 | * This function is often called repetitively to solve problems of identical
|
---|
851 | * dimensions. To avoid repetitive malloc's and free's, allocated memory is
|
---|
852 | * retained between calls and free'd-malloc'ed when not of the appropriate size.
|
---|
853 | * A call with NULL as the first argument forces this memory to be released.
|
---|
854 | */
|
---|
855 | int AX_EQ_B_SVD(LM_REAL *A, LM_REAL *B, LM_REAL *x, int m)
|
---|
856 | {
|
---|
857 | __STATIC__ LM_REAL *buf=NULL;
|
---|
858 | __STATIC__ int buf_sz=0;
|
---|
859 | static LM_REAL eps=LM_CNST(-1.0);
|
---|
860 |
|
---|
861 | register int i, j;
|
---|
862 | LM_REAL *a, *u, *s, *vt, *work;
|
---|
863 | int a_sz, u_sz, s_sz, vt_sz, tot_sz;
|
---|
864 | LM_REAL thresh, one_over_denom;
|
---|
865 | register LM_REAL sum;
|
---|
866 | int info, rank, worksz, *iwork, iworksz;
|
---|
867 |
|
---|
868 | if(!A)
|
---|
869 | #ifdef LINSOLVERS_RETAIN_MEMORY
|
---|
870 | {
|
---|
871 | if(buf) free(buf);
|
---|
872 | buf=NULL;
|
---|
873 | buf_sz=0;
|
---|
874 |
|
---|
875 | return 1;
|
---|
876 | }
|
---|
877 | #else
|
---|
878 | return 1; /* NOP */
|
---|
879 | #endif /* LINSOLVERS_RETAIN_MEMORY */
|
---|
880 |
|
---|
881 | /* calculate required memory size */
|
---|
882 | #if 1 /* use optimal size */
|
---|
883 | worksz=-1; // workspace query. Keep in mind that GESDD requires more memory than GESVD
|
---|
884 | /* note that optimal work size is returned in thresh */
|
---|
885 | GESVD("A", "A", (int *)&m, (int *)&m, NULL, (int *)&m, NULL, NULL, (int *)&m, NULL, (int *)&m, (LM_REAL *)&thresh, (int *)&worksz, &info);
|
---|
886 | //GESDD("A", (int *)&m, (int *)&m, NULL, (int *)&m, NULL, NULL, (int *)&m, NULL, (int *)&m, (LM_REAL *)&thresh, (int *)&worksz, NULL, &info);
|
---|
887 | worksz=(int)thresh;
|
---|
888 | #else /* use minimum size */
|
---|
889 | worksz=5*m; // min worksize for GESVD
|
---|
890 | //worksz=m*(7*m+4); // min worksize for GESDD
|
---|
891 | #endif
|
---|
892 | iworksz=8*m;
|
---|
893 | a_sz=m*m;
|
---|
894 | u_sz=m*m; s_sz=m; vt_sz=m*m;
|
---|
895 |
|
---|
896 | tot_sz=(a_sz + u_sz + s_sz + vt_sz + worksz)*sizeof(LM_REAL) + iworksz*sizeof(int); /* should be arranged in that order for proper doubles alignment */
|
---|
897 |
|
---|
898 | #ifdef LINSOLVERS_RETAIN_MEMORY
|
---|
899 | if(tot_sz>buf_sz){ /* insufficient memory, allocate a "big" memory chunk at once */
|
---|
900 | if(buf) free(buf); /* free previously allocated memory */
|
---|
901 |
|
---|
902 | buf_sz=tot_sz;
|
---|
903 | buf=(LM_REAL *)malloc(buf_sz);
|
---|
904 | if(!buf){
|
---|
905 | fprintf(stderr, RCAT("memory allocation in ", AX_EQ_B_SVD) "() failed!\n");
|
---|
906 | exit(1);
|
---|
907 | }
|
---|
908 | }
|
---|
909 | #else
|
---|
910 | buf_sz=tot_sz;
|
---|
911 | buf=(LM_REAL *)malloc(buf_sz);
|
---|
912 | if(!buf){
|
---|
913 | fprintf(stderr, RCAT("memory allocation in ", AX_EQ_B_SVD) "() failed!\n");
|
---|
914 | exit(1);
|
---|
915 | }
|
---|
916 | #endif /* LINSOLVERS_RETAIN_MEMORY */
|
---|
917 |
|
---|
918 | a=buf;
|
---|
919 | u=a+a_sz;
|
---|
920 | s=u+u_sz;
|
---|
921 | vt=s+s_sz;
|
---|
922 | work=vt+vt_sz;
|
---|
923 | iwork=(int *)(work+worksz);
|
---|
924 |
|
---|
925 | /* store A (column major!) into a */
|
---|
926 | for(i=0; i<m; i++)
|
---|
927 | for(j=0; j<m; j++)
|
---|
928 | a[i+j*m]=A[i*m+j];
|
---|
929 |
|
---|
930 | /* SVD decomposition of A */
|
---|
931 | GESVD("A", "A", (int *)&m, (int *)&m, a, (int *)&m, s, u, (int *)&m, vt, (int *)&m, work, (int *)&worksz, &info);
|
---|
932 | //GESDD("A", (int *)&m, (int *)&m, a, (int *)&m, s, u, (int *)&m, vt, (int *)&m, work, (int *)&worksz, iwork, &info);
|
---|
933 |
|
---|
934 | /* error treatment */
|
---|
935 | if(info!=0){
|
---|
936 | if(info<0){
|
---|
937 | fprintf(stderr, RCAT(RCAT(RCAT("LAPACK error: illegal value for argument %d of ", GESVD), "/" GESDD) " in ", AX_EQ_B_SVD) "()\n", -info);
|
---|
938 | exit(1);
|
---|
939 | }
|
---|
940 | else{
|
---|
941 | fprintf(stderr, RCAT("LAPACK error: dgesdd (dbdsdc)/dgesvd (dbdsqr) failed to converge in ", AX_EQ_B_SVD) "() [info=%d]\n", info);
|
---|
942 | #ifndef LINSOLVERS_RETAIN_MEMORY
|
---|
943 | free(buf);
|
---|
944 | #endif
|
---|
945 |
|
---|
946 | return 0;
|
---|
947 | }
|
---|
948 | }
|
---|
949 |
|
---|
950 | if(eps<0.0){
|
---|
951 | LM_REAL aux;
|
---|
952 |
|
---|
953 | /* compute machine epsilon */
|
---|
954 | for(eps=LM_CNST(1.0); aux=eps+LM_CNST(1.0), aux-LM_CNST(1.0)>0.0; eps*=LM_CNST(0.5))
|
---|
955 | ;
|
---|
956 | eps*=LM_CNST(2.0);
|
---|
957 | }
|
---|
958 |
|
---|
959 | /* compute the pseudoinverse in a */
|
---|
960 | for(i=0; i<a_sz; i++) a[i]=0.0; /* initialize to zero */
|
---|
961 | for(rank=0, thresh=eps*s[0]; rank<m && s[rank]>thresh; rank++){
|
---|
962 | one_over_denom=LM_CNST(1.0)/s[rank];
|
---|
963 |
|
---|
964 | for(j=0; j<m; j++)
|
---|
965 | for(i=0; i<m; i++)
|
---|
966 | a[i*m+j]+=vt[rank+i*m]*u[j+rank*m]*one_over_denom;
|
---|
967 | }
|
---|
968 |
|
---|
969 | /* compute A^+ b in x */
|
---|
970 | for(i=0; i<m; i++){
|
---|
971 | for(j=0, sum=0.0; j<m; j++)
|
---|
972 | sum+=a[i*m+j]*B[j];
|
---|
973 | x[i]=sum;
|
---|
974 | }
|
---|
975 |
|
---|
976 | #ifndef LINSOLVERS_RETAIN_MEMORY
|
---|
977 | free(buf);
|
---|
978 | #endif
|
---|
979 |
|
---|
980 | return 1;
|
---|
981 | }
|
---|
982 |
|
---|
983 | /*
|
---|
984 | * This function returns the solution of Ax = b for a real symmetric matrix A
|
---|
985 | *
|
---|
986 | * The function is based on LDLT factorization with the pivoting
|
---|
987 | * strategy of Bunch and Kaufman:
|
---|
988 | * A is factored as L*D*L^T where L is lower triangular and
|
---|
989 | * D symmetric and block diagonal (aka spectral decomposition,
|
---|
990 | * Banachiewicz factorization, modified Cholesky factorization)
|
---|
991 | *
|
---|
992 | * A is mxm, b is mx1.
|
---|
993 | *
|
---|
994 | * The function returns 0 in case of error, 1 if successfull
|
---|
995 | *
|
---|
996 | * This function is often called repetitively to solve problems of identical
|
---|
997 | * dimensions. To avoid repetitive malloc's and free's, allocated memory is
|
---|
998 | * retained between calls and free'd-malloc'ed when not of the appropriate size.
|
---|
999 | * A call with NULL as the first argument forces this memory to be released.
|
---|
1000 | */
|
---|
1001 | int AX_EQ_B_BK(LM_REAL *A, LM_REAL *B, LM_REAL *x, int m)
|
---|
1002 | {
|
---|
1003 | __STATIC__ LM_REAL *buf=NULL;
|
---|
1004 | __STATIC__ int buf_sz=0, nb=0;
|
---|
1005 |
|
---|
1006 | LM_REAL *a, *work;
|
---|
1007 | int a_sz, ipiv_sz, work_sz, tot_sz;
|
---|
1008 | int info, *ipiv, nrhs=1;
|
---|
1009 |
|
---|
1010 | if(!A)
|
---|
1011 | #ifdef LINSOLVERS_RETAIN_MEMORY
|
---|
1012 | {
|
---|
1013 | if(buf) free(buf);
|
---|
1014 | buf=NULL;
|
---|
1015 | buf_sz=0;
|
---|
1016 |
|
---|
1017 | return 1;
|
---|
1018 | }
|
---|
1019 | #else
|
---|
1020 | return 1; /* NOP */
|
---|
1021 | #endif /* LINSOLVERS_RETAIN_MEMORY */
|
---|
1022 |
|
---|
1023 | /* calculate required memory size */
|
---|
1024 | ipiv_sz=m;
|
---|
1025 | a_sz=m*m;
|
---|
1026 | if(!nb){
|
---|
1027 | LM_REAL tmp;
|
---|
1028 |
|
---|
1029 | work_sz=-1; // workspace query; optimal size is returned in tmp
|
---|
1030 | SYTRF("L", (int *)&m, NULL, (int *)&m, NULL, (LM_REAL *)&tmp, (int *)&work_sz, (int *)&info);
|
---|
1031 | nb=((int)tmp)/m; // optimal worksize is m*nb
|
---|
1032 | }
|
---|
1033 | work_sz=(nb!=-1)? nb*m : 1;
|
---|
1034 | tot_sz=(a_sz + work_sz)*sizeof(LM_REAL) + ipiv_sz*sizeof(int); /* should be arranged in that order for proper doubles alignment */
|
---|
1035 |
|
---|
1036 | #ifdef LINSOLVERS_RETAIN_MEMORY
|
---|
1037 | if(tot_sz>buf_sz){ /* insufficient memory, allocate a "big" memory chunk at once */
|
---|
1038 | if(buf) free(buf); /* free previously allocated memory */
|
---|
1039 |
|
---|
1040 | buf_sz=tot_sz;
|
---|
1041 | buf=(LM_REAL *)malloc(buf_sz);
|
---|
1042 | if(!buf){
|
---|
1043 | fprintf(stderr, RCAT("memory allocation in ", AX_EQ_B_BK) "() failed!\n");
|
---|
1044 | exit(1);
|
---|
1045 | }
|
---|
1046 | }
|
---|
1047 | #else
|
---|
1048 | buf_sz=tot_sz;
|
---|
1049 | buf=(LM_REAL *)malloc(buf_sz);
|
---|
1050 | if(!buf){
|
---|
1051 | fprintf(stderr, RCAT("memory allocation in ", AX_EQ_B_BK) "() failed!\n");
|
---|
1052 | exit(1);
|
---|
1053 | }
|
---|
1054 | #endif /* LINSOLVERS_RETAIN_MEMORY */
|
---|
1055 |
|
---|
1056 | a=buf;
|
---|
1057 | work=a+a_sz;
|
---|
1058 | ipiv=(int *)(work+work_sz);
|
---|
1059 |
|
---|
1060 | /* store A into a and B into x; A is assumed to be symmetric, hence
|
---|
1061 | * the column and row major order representations are the same
|
---|
1062 | */
|
---|
1063 | memcpy(a, A, a_sz*sizeof(LM_REAL));
|
---|
1064 | memcpy(x, B, m*sizeof(LM_REAL));
|
---|
1065 |
|
---|
1066 | /* LDLt factorization for A */
|
---|
1067 | SYTRF("L", (int *)&m, a, (int *)&m, ipiv, work, (int *)&work_sz, (int *)&info);
|
---|
1068 | if(info!=0){
|
---|
1069 | if(info<0){
|
---|
1070 | fprintf(stderr, RCAT(RCAT("LAPACK error: illegal value for argument %d of ", SYTRF) " in ", AX_EQ_B_BK) "()\n", -info);
|
---|
1071 | exit(1);
|
---|
1072 | }
|
---|
1073 | else{
|
---|
1074 | fprintf(stderr, RCAT(RCAT("LAPACK error: singular block diagonal matrix D for", SYTRF) " in ", AX_EQ_B_BK)"() [D(%d, %d) is zero]\n", info, info);
|
---|
1075 | #ifndef LINSOLVERS_RETAIN_MEMORY
|
---|
1076 | free(buf);
|
---|
1077 | #endif
|
---|
1078 |
|
---|
1079 | return 0;
|
---|
1080 | }
|
---|
1081 | }
|
---|
1082 |
|
---|
1083 | /* solve the system with the computed factorization */
|
---|
1084 | SYTRS("L", (int *)&m, (int *)&nrhs, a, (int *)&m, ipiv, x, (int *)&m, (int *)&info);
|
---|
1085 | if(info<0){
|
---|
1086 | fprintf(stderr, RCAT(RCAT("LAPACK error: illegal value for argument %d of ", SYTRS) " in ", AX_EQ_B_BK) "()\n", -info);
|
---|
1087 | exit(1);
|
---|
1088 | }
|
---|
1089 |
|
---|
1090 | #ifndef LINSOLVERS_RETAIN_MEMORY
|
---|
1091 | free(buf);
|
---|
1092 | #endif
|
---|
1093 |
|
---|
1094 | return 1;
|
---|
1095 | }
|
---|
1096 |
|
---|
1097 | /* undefine all. IT MUST REMAIN IN THIS POSITION IN FILE */
|
---|
1098 | #undef AX_EQ_B_QR
|
---|
1099 | #undef AX_EQ_B_QRLS
|
---|
1100 | #undef AX_EQ_B_CHOL
|
---|
1101 | #undef AX_EQ_B_LU
|
---|
1102 | #undef AX_EQ_B_SVD
|
---|
1103 | #undef AX_EQ_B_BK
|
---|
1104 | #undef AX_EQ_B_PLASMA_CHOL
|
---|
1105 |
|
---|
1106 | #undef GEQRF
|
---|
1107 | #undef ORGQR
|
---|
1108 | #undef TRTRS
|
---|
1109 | #undef POTF2
|
---|
1110 | #undef POTRF
|
---|
1111 | #undef POTRS
|
---|
1112 | #undef GETRF
|
---|
1113 | #undef GETRS
|
---|
1114 | #undef GESVD
|
---|
1115 | #undef GESDD
|
---|
1116 | #undef SYTRF
|
---|
1117 | #undef SYTRS
|
---|
1118 | #undef PLASMA_POSV
|
---|
1119 |
|
---|
1120 | #else // no LAPACK
|
---|
1121 |
|
---|
1122 | /* precision-specific definitions */
|
---|
1123 | #define AX_EQ_B_LU LM_ADD_PREFIX(Ax_eq_b_LU_noLapack)
|
---|
1124 |
|
---|
1125 | /*
|
---|
1126 | * This function returns the solution of Ax = b
|
---|
1127 | *
|
---|
1128 | * The function employs LU decomposition followed by forward/back substitution (see
|
---|
1129 | * also the LAPACK-based LU solver above)
|
---|
1130 | *
|
---|
1131 | * A is mxm, b is mx1
|
---|
1132 | *
|
---|
1133 | * The function returns 0 in case of error, 1 if successful
|
---|
1134 | *
|
---|
1135 | * This function is often called repetitively to solve problems of identical
|
---|
1136 | * dimensions. To avoid repetitive malloc's and free's, allocated memory is
|
---|
1137 | * retained between calls and free'd-malloc'ed when not of the appropriate size.
|
---|
1138 | * A call with NULL as the first argument forces this memory to be released.
|
---|
1139 | */
|
---|
1140 | int AX_EQ_B_LU(LM_REAL *A, LM_REAL *B, LM_REAL *x, int m)
|
---|
1141 | {
|
---|
1142 | __STATIC__ void *buf=NULL;
|
---|
1143 | __STATIC__ int buf_sz=0;
|
---|
1144 |
|
---|
1145 | register int i, j, k;
|
---|
1146 | int *idx, maxi=-1, idx_sz, a_sz, work_sz, tot_sz;
|
---|
1147 | LM_REAL *a, *work, max, sum, tmp;
|
---|
1148 |
|
---|
1149 | if(!A)
|
---|
1150 | #ifdef LINSOLVERS_RETAIN_MEMORY
|
---|
1151 | {
|
---|
1152 | if(buf) free(buf);
|
---|
1153 | buf=NULL;
|
---|
1154 | buf_sz=0;
|
---|
1155 |
|
---|
1156 | return 1;
|
---|
1157 | }
|
---|
1158 | #else
|
---|
1159 | return 1; /* NOP */
|
---|
1160 | #endif /* LINSOLVERS_RETAIN_MEMORY */
|
---|
1161 |
|
---|
1162 | /* calculate required memory size */
|
---|
1163 | idx_sz=m;
|
---|
1164 | a_sz=m*m;
|
---|
1165 | work_sz=m;
|
---|
1166 | tot_sz=(a_sz+work_sz)*sizeof(LM_REAL) + idx_sz*sizeof(int); /* should be arranged in that order for proper doubles alignment */
|
---|
1167 |
|
---|
1168 | #ifdef LINSOLVERS_RETAIN_MEMORY
|
---|
1169 | if(tot_sz>buf_sz){ /* insufficient memory, allocate a "big" memory chunk at once */
|
---|
1170 | if(buf) free(buf); /* free previously allocated memory */
|
---|
1171 |
|
---|
1172 | buf_sz=tot_sz;
|
---|
1173 | buf=(void *)malloc(tot_sz);
|
---|
1174 | if(!buf){
|
---|
1175 | fprintf(stderr, RCAT("memory allocation in ", AX_EQ_B_LU) "() failed!\n");
|
---|
1176 | exit(1);
|
---|
1177 | }
|
---|
1178 | }
|
---|
1179 | #else
|
---|
1180 | buf_sz=tot_sz;
|
---|
1181 | buf=(void *)malloc(tot_sz);
|
---|
1182 | if(!buf){
|
---|
1183 | fprintf(stderr, RCAT("memory allocation in ", AX_EQ_B_LU) "() failed!\n");
|
---|
1184 | exit(1);
|
---|
1185 | }
|
---|
1186 | #endif /* LINSOLVERS_RETAIN_MEMORY */
|
---|
1187 |
|
---|
1188 | a=buf;
|
---|
1189 | work=a+a_sz;
|
---|
1190 | idx=(int *)(work+work_sz);
|
---|
1191 |
|
---|
1192 | /* avoid destroying A, B by copying them to a, x resp. */
|
---|
1193 | memcpy(a, A, a_sz*sizeof(LM_REAL));
|
---|
1194 | memcpy(x, B, m*sizeof(LM_REAL));
|
---|
1195 |
|
---|
1196 | /* compute the LU decomposition of a row permutation of matrix a; the permutation itself is saved in idx[] */
|
---|
1197 | for(i=0; i<m; ++i){
|
---|
1198 | max=0.0;
|
---|
1199 | for(j=0; j<m; ++j)
|
---|
1200 | if((tmp=FABS(a[i*m+j]))>max)
|
---|
1201 | max=tmp;
|
---|
1202 | if(max==0.0){
|
---|
1203 | fprintf(stderr, RCAT("Singular matrix A in ", AX_EQ_B_LU) "()!\n");
|
---|
1204 | #ifndef LINSOLVERS_RETAIN_MEMORY
|
---|
1205 | free(buf);
|
---|
1206 | #endif
|
---|
1207 |
|
---|
1208 | return 0;
|
---|
1209 | }
|
---|
1210 | work[i]=LM_CNST(1.0)/max;
|
---|
1211 | }
|
---|
1212 |
|
---|
1213 | for(j=0; j<m; ++j){
|
---|
1214 | for(i=0; i<j; ++i){
|
---|
1215 | sum=a[i*m+j];
|
---|
1216 | for(k=0; k<i; ++k)
|
---|
1217 | sum-=a[i*m+k]*a[k*m+j];
|
---|
1218 | a[i*m+j]=sum;
|
---|
1219 | }
|
---|
1220 | max=0.0;
|
---|
1221 | for(i=j; i<m; ++i){
|
---|
1222 | sum=a[i*m+j];
|
---|
1223 | for(k=0; k<j; ++k)
|
---|
1224 | sum-=a[i*m+k]*a[k*m+j];
|
---|
1225 | a[i*m+j]=sum;
|
---|
1226 | if((tmp=work[i]*FABS(sum))>=max){
|
---|
1227 | max=tmp;
|
---|
1228 | maxi=i;
|
---|
1229 | }
|
---|
1230 | }
|
---|
1231 | if(j!=maxi){
|
---|
1232 | for(k=0; k<m; ++k){
|
---|
1233 | tmp=a[maxi*m+k];
|
---|
1234 | a[maxi*m+k]=a[j*m+k];
|
---|
1235 | a[j*m+k]=tmp;
|
---|
1236 | }
|
---|
1237 | work[maxi]=work[j];
|
---|
1238 | }
|
---|
1239 | idx[j]=maxi;
|
---|
1240 | if(a[j*m+j]==0.0)
|
---|
1241 | a[j*m+j]=LM_REAL_EPSILON;
|
---|
1242 | if(j!=m-1){
|
---|
1243 | tmp=LM_CNST(1.0)/(a[j*m+j]);
|
---|
1244 | for(i=j+1; i<m; ++i)
|
---|
1245 | a[i*m+j]*=tmp;
|
---|
1246 | }
|
---|
1247 | }
|
---|
1248 |
|
---|
1249 | /* The decomposition has now replaced a. Solve the linear system using
|
---|
1250 | * forward and back substitution
|
---|
1251 | */
|
---|
1252 | for(i=k=0; i<m; ++i){
|
---|
1253 | j=idx[i];
|
---|
1254 | sum=x[j];
|
---|
1255 | x[j]=x[i];
|
---|
1256 | if(k!=0)
|
---|
1257 | for(j=k-1; j<i; ++j)
|
---|
1258 | sum-=a[i*m+j]*x[j];
|
---|
1259 | else
|
---|
1260 | if(sum!=0.0)
|
---|
1261 | k=i+1;
|
---|
1262 | x[i]=sum;
|
---|
1263 | }
|
---|
1264 |
|
---|
1265 | for(i=m-1; i>=0; --i){
|
---|
1266 | sum=x[i];
|
---|
1267 | for(j=i+1; j<m; ++j)
|
---|
1268 | sum-=a[i*m+j]*x[j];
|
---|
1269 | x[i]=sum/a[i*m+i];
|
---|
1270 | }
|
---|
1271 |
|
---|
1272 | #ifndef LINSOLVERS_RETAIN_MEMORY
|
---|
1273 | free(buf);
|
---|
1274 | #endif
|
---|
1275 |
|
---|
1276 | return 1;
|
---|
1277 | }
|
---|
1278 |
|
---|
1279 | /* undefine all. IT MUST REMAIN IN THIS POSITION IN FILE */
|
---|
1280 | #undef AX_EQ_B_LU
|
---|
1281 |
|
---|
1282 | #endif /* HAVE_LAPACK */
|
---|