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7072
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1 SUBROUTINE DLALS0( ICOMPQ, NL, NR, SQRE, NRHS, B, LDB, BX, LDBX, |
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2 $ PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM, LDGNUM, |
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3 $ POLES, DIFL, DIFR, Z, K, C, S, WORK, INFO ) |
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4 * |
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5 * -- LAPACK routine (version 3.1) -- |
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6 * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. |
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7 * November 2006 |
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8 * |
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9 * .. Scalar Arguments .. |
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10 INTEGER GIVPTR, ICOMPQ, INFO, K, LDB, LDBX, LDGCOL, |
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11 $ LDGNUM, NL, NR, NRHS, SQRE |
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12 DOUBLE PRECISION C, S |
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13 * .. |
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14 * .. Array Arguments .. |
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15 INTEGER GIVCOL( LDGCOL, * ), PERM( * ) |
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16 DOUBLE PRECISION B( LDB, * ), BX( LDBX, * ), DIFL( * ), |
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17 $ DIFR( LDGNUM, * ), GIVNUM( LDGNUM, * ), |
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18 $ POLES( LDGNUM, * ), WORK( * ), Z( * ) |
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19 * .. |
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20 * |
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21 * Purpose |
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22 * ======= |
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23 * |
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24 * DLALS0 applies back the multiplying factors of either the left or the |
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25 * right singular vector matrix of a diagonal matrix appended by a row |
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26 * to the right hand side matrix B in solving the least squares problem |
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27 * using the divide-and-conquer SVD approach. |
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28 * |
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29 * For the left singular vector matrix, three types of orthogonal |
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30 * matrices are involved: |
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31 * |
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32 * (1L) Givens rotations: the number of such rotations is GIVPTR; the |
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33 * pairs of columns/rows they were applied to are stored in GIVCOL; |
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34 * and the C- and S-values of these rotations are stored in GIVNUM. |
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35 * |
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36 * (2L) Permutation. The (NL+1)-st row of B is to be moved to the first |
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37 * row, and for J=2:N, PERM(J)-th row of B is to be moved to the |
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38 * J-th row. |
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39 * |
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40 * (3L) The left singular vector matrix of the remaining matrix. |
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41 * |
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42 * For the right singular vector matrix, four types of orthogonal |
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43 * matrices are involved: |
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44 * |
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45 * (1R) The right singular vector matrix of the remaining matrix. |
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46 * |
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47 * (2R) If SQRE = 1, one extra Givens rotation to generate the right |
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48 * null space. |
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49 * |
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50 * (3R) The inverse transformation of (2L). |
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51 * |
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52 * (4R) The inverse transformation of (1L). |
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53 * |
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54 * Arguments |
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55 * ========= |
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56 * |
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57 * ICOMPQ (input) INTEGER |
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58 * Specifies whether singular vectors are to be computed in |
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59 * factored form: |
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60 * = 0: Left singular vector matrix. |
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61 * = 1: Right singular vector matrix. |
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62 * |
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63 * NL (input) INTEGER |
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64 * The row dimension of the upper block. NL >= 1. |
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65 * |
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66 * NR (input) INTEGER |
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67 * The row dimension of the lower block. NR >= 1. |
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68 * |
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69 * SQRE (input) INTEGER |
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70 * = 0: the lower block is an NR-by-NR square matrix. |
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71 * = 1: the lower block is an NR-by-(NR+1) rectangular matrix. |
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72 * |
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73 * The bidiagonal matrix has row dimension N = NL + NR + 1, |
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74 * and column dimension M = N + SQRE. |
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75 * |
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76 * NRHS (input) INTEGER |
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77 * The number of columns of B and BX. NRHS must be at least 1. |
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78 * |
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79 * B (input/output) DOUBLE PRECISION array, dimension ( LDB, NRHS ) |
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80 * On input, B contains the right hand sides of the least |
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81 * squares problem in rows 1 through M. On output, B contains |
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82 * the solution X in rows 1 through N. |
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83 * |
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84 * LDB (input) INTEGER |
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85 * The leading dimension of B. LDB must be at least |
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86 * max(1,MAX( M, N ) ). |
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87 * |
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88 * BX (workspace) DOUBLE PRECISION array, dimension ( LDBX, NRHS ) |
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89 * |
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90 * LDBX (input) INTEGER |
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91 * The leading dimension of BX. |
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92 * |
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93 * PERM (input) INTEGER array, dimension ( N ) |
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94 * The permutations (from deflation and sorting) applied |
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95 * to the two blocks. |
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96 * |
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97 * GIVPTR (input) INTEGER |
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98 * The number of Givens rotations which took place in this |
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99 * subproblem. |
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100 * |
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101 * GIVCOL (input) INTEGER array, dimension ( LDGCOL, 2 ) |
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102 * Each pair of numbers indicates a pair of rows/columns |
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103 * involved in a Givens rotation. |
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104 * |
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105 * LDGCOL (input) INTEGER |
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106 * The leading dimension of GIVCOL, must be at least N. |
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107 * |
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108 * GIVNUM (input) DOUBLE PRECISION array, dimension ( LDGNUM, 2 ) |
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109 * Each number indicates the C or S value used in the |
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110 * corresponding Givens rotation. |
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111 * |
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112 * LDGNUM (input) INTEGER |
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113 * The leading dimension of arrays DIFR, POLES and |
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114 * GIVNUM, must be at least K. |
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115 * |
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116 * POLES (input) DOUBLE PRECISION array, dimension ( LDGNUM, 2 ) |
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117 * On entry, POLES(1:K, 1) contains the new singular |
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118 * values obtained from solving the secular equation, and |
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119 * POLES(1:K, 2) is an array containing the poles in the secular |
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120 * equation. |
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121 * |
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122 * DIFL (input) DOUBLE PRECISION array, dimension ( K ). |
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123 * On entry, DIFL(I) is the distance between I-th updated |
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124 * (undeflated) singular value and the I-th (undeflated) old |
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125 * singular value. |
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126 * |
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127 * DIFR (input) DOUBLE PRECISION array, dimension ( LDGNUM, 2 ). |
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128 * On entry, DIFR(I, 1) contains the distances between I-th |
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129 * updated (undeflated) singular value and the I+1-th |
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130 * (undeflated) old singular value. And DIFR(I, 2) is the |
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131 * normalizing factor for the I-th right singular vector. |
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132 * |
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133 * Z (input) DOUBLE PRECISION array, dimension ( K ) |
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134 * Contain the components of the deflation-adjusted updating row |
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135 * vector. |
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136 * |
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137 * K (input) INTEGER |
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138 * Contains the dimension of the non-deflated matrix, |
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139 * This is the order of the related secular equation. 1 <= K <=N. |
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140 * |
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141 * C (input) DOUBLE PRECISION |
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142 * C contains garbage if SQRE =0 and the C-value of a Givens |
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143 * rotation related to the right null space if SQRE = 1. |
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144 * |
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145 * S (input) DOUBLE PRECISION |
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146 * S contains garbage if SQRE =0 and the S-value of a Givens |
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147 * rotation related to the right null space if SQRE = 1. |
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148 * |
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149 * WORK (workspace) DOUBLE PRECISION array, dimension ( K ) |
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150 * |
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151 * INFO (output) INTEGER |
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152 * = 0: successful exit. |
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153 * < 0: if INFO = -i, the i-th argument had an illegal value. |
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154 * |
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155 * Further Details |
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156 * =============== |
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157 * |
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158 * Based on contributions by |
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159 * Ming Gu and Ren-Cang Li, Computer Science Division, University of |
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160 * California at Berkeley, USA |
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161 * Osni Marques, LBNL/NERSC, USA |
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162 * |
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163 * ===================================================================== |
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164 * |
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165 * .. Parameters .. |
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166 DOUBLE PRECISION ONE, ZERO, NEGONE |
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167 PARAMETER ( ONE = 1.0D0, ZERO = 0.0D0, NEGONE = -1.0D0 ) |
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168 * .. |
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169 * .. Local Scalars .. |
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170 INTEGER I, J, M, N, NLP1 |
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171 DOUBLE PRECISION DIFLJ, DIFRJ, DJ, DSIGJ, DSIGJP, TEMP |
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172 * .. |
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173 * .. External Subroutines .. |
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174 EXTERNAL DCOPY, DGEMV, DLACPY, DLASCL, DROT, DSCAL, |
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175 $ XERBLA |
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176 * .. |
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177 * .. External Functions .. |
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178 DOUBLE PRECISION DLAMC3, DNRM2 |
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179 EXTERNAL DLAMC3, DNRM2 |
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180 * .. |
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181 * .. Intrinsic Functions .. |
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182 INTRINSIC MAX |
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183 * .. |
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184 * .. Executable Statements .. |
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185 * |
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186 * Test the input parameters. |
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187 * |
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188 INFO = 0 |
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189 * |
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190 IF( ( ICOMPQ.LT.0 ) .OR. ( ICOMPQ.GT.1 ) ) THEN |
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191 INFO = -1 |
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192 ELSE IF( NL.LT.1 ) THEN |
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193 INFO = -2 |
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194 ELSE IF( NR.LT.1 ) THEN |
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195 INFO = -3 |
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196 ELSE IF( ( SQRE.LT.0 ) .OR. ( SQRE.GT.1 ) ) THEN |
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197 INFO = -4 |
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198 END IF |
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199 * |
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200 N = NL + NR + 1 |
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201 * |
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202 IF( NRHS.LT.1 ) THEN |
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203 INFO = -5 |
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204 ELSE IF( LDB.LT.N ) THEN |
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205 INFO = -7 |
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206 ELSE IF( LDBX.LT.N ) THEN |
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207 INFO = -9 |
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208 ELSE IF( GIVPTR.LT.0 ) THEN |
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209 INFO = -11 |
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210 ELSE IF( LDGCOL.LT.N ) THEN |
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211 INFO = -13 |
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212 ELSE IF( LDGNUM.LT.N ) THEN |
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213 INFO = -15 |
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214 ELSE IF( K.LT.1 ) THEN |
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215 INFO = -20 |
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216 END IF |
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217 IF( INFO.NE.0 ) THEN |
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218 CALL XERBLA( 'DLALS0', -INFO ) |
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219 RETURN |
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220 END IF |
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221 * |
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222 M = N + SQRE |
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223 NLP1 = NL + 1 |
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224 * |
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225 IF( ICOMPQ.EQ.0 ) THEN |
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226 * |
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227 * Apply back orthogonal transformations from the left. |
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228 * |
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229 * Step (1L): apply back the Givens rotations performed. |
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230 * |
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231 DO 10 I = 1, GIVPTR |
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232 CALL DROT( NRHS, B( GIVCOL( I, 2 ), 1 ), LDB, |
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233 $ B( GIVCOL( I, 1 ), 1 ), LDB, GIVNUM( I, 2 ), |
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234 $ GIVNUM( I, 1 ) ) |
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235 10 CONTINUE |
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236 * |
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237 * Step (2L): permute rows of B. |
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238 * |
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239 CALL DCOPY( NRHS, B( NLP1, 1 ), LDB, BX( 1, 1 ), LDBX ) |
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240 DO 20 I = 2, N |
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241 CALL DCOPY( NRHS, B( PERM( I ), 1 ), LDB, BX( I, 1 ), LDBX ) |
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242 20 CONTINUE |
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243 * |
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244 * Step (3L): apply the inverse of the left singular vector |
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245 * matrix to BX. |
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246 * |
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247 IF( K.EQ.1 ) THEN |
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248 CALL DCOPY( NRHS, BX, LDBX, B, LDB ) |
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249 IF( Z( 1 ).LT.ZERO ) THEN |
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250 CALL DSCAL( NRHS, NEGONE, B, LDB ) |
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251 END IF |
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252 ELSE |
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253 DO 50 J = 1, K |
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254 DIFLJ = DIFL( J ) |
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255 DJ = POLES( J, 1 ) |
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256 DSIGJ = -POLES( J, 2 ) |
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257 IF( J.LT.K ) THEN |
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258 DIFRJ = -DIFR( J, 1 ) |
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259 DSIGJP = -POLES( J+1, 2 ) |
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260 END IF |
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261 IF( ( Z( J ).EQ.ZERO ) .OR. ( POLES( J, 2 ).EQ.ZERO ) ) |
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262 $ THEN |
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263 WORK( J ) = ZERO |
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264 ELSE |
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265 WORK( J ) = -POLES( J, 2 )*Z( J ) / DIFLJ / |
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266 $ ( POLES( J, 2 )+DJ ) |
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267 END IF |
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268 DO 30 I = 1, J - 1 |
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269 IF( ( Z( I ).EQ.ZERO ) .OR. |
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270 $ ( POLES( I, 2 ).EQ.ZERO ) ) THEN |
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271 WORK( I ) = ZERO |
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272 ELSE |
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273 WORK( I ) = POLES( I, 2 )*Z( I ) / |
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274 $ ( DLAMC3( POLES( I, 2 ), DSIGJ )- |
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275 $ DIFLJ ) / ( POLES( I, 2 )+DJ ) |
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276 END IF |
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277 30 CONTINUE |
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278 DO 40 I = J + 1, K |
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279 IF( ( Z( I ).EQ.ZERO ) .OR. |
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280 $ ( POLES( I, 2 ).EQ.ZERO ) ) THEN |
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281 WORK( I ) = ZERO |
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282 ELSE |
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283 WORK( I ) = POLES( I, 2 )*Z( I ) / |
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284 $ ( DLAMC3( POLES( I, 2 ), DSIGJP )+ |
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285 $ DIFRJ ) / ( POLES( I, 2 )+DJ ) |
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286 END IF |
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287 40 CONTINUE |
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288 WORK( 1 ) = NEGONE |
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289 TEMP = DNRM2( K, WORK, 1 ) |
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290 CALL DGEMV( 'T', K, NRHS, ONE, BX, LDBX, WORK, 1, ZERO, |
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291 $ B( J, 1 ), LDB ) |
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292 CALL DLASCL( 'G', 0, 0, TEMP, ONE, 1, NRHS, B( J, 1 ), |
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293 $ LDB, INFO ) |
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294 50 CONTINUE |
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295 END IF |
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296 * |
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297 * Move the deflated rows of BX to B also. |
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298 * |
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299 IF( K.LT.MAX( M, N ) ) |
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300 $ CALL DLACPY( 'A', N-K, NRHS, BX( K+1, 1 ), LDBX, |
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301 $ B( K+1, 1 ), LDB ) |
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302 ELSE |
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303 * |
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304 * Apply back the right orthogonal transformations. |
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305 * |
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306 * Step (1R): apply back the new right singular vector matrix |
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307 * to B. |
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308 * |
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309 IF( K.EQ.1 ) THEN |
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310 CALL DCOPY( NRHS, B, LDB, BX, LDBX ) |
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311 ELSE |
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312 DO 80 J = 1, K |
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313 DSIGJ = POLES( J, 2 ) |
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314 IF( Z( J ).EQ.ZERO ) THEN |
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315 WORK( J ) = ZERO |
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316 ELSE |
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317 WORK( J ) = -Z( J ) / DIFL( J ) / |
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318 $ ( DSIGJ+POLES( J, 1 ) ) / DIFR( J, 2 ) |
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319 END IF |
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320 DO 60 I = 1, J - 1 |
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321 IF( Z( J ).EQ.ZERO ) THEN |
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322 WORK( I ) = ZERO |
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323 ELSE |
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324 WORK( I ) = Z( J ) / ( DLAMC3( DSIGJ, -POLES( I+1, |
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325 $ 2 ) )-DIFR( I, 1 ) ) / |
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326 $ ( DSIGJ+POLES( I, 1 ) ) / DIFR( I, 2 ) |
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327 END IF |
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328 60 CONTINUE |
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329 DO 70 I = J + 1, K |
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330 IF( Z( J ).EQ.ZERO ) THEN |
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331 WORK( I ) = ZERO |
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332 ELSE |
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333 WORK( I ) = Z( J ) / ( DLAMC3( DSIGJ, -POLES( I, |
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334 $ 2 ) )-DIFL( I ) ) / |
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335 $ ( DSIGJ+POLES( I, 1 ) ) / DIFR( I, 2 ) |
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336 END IF |
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337 70 CONTINUE |
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338 CALL DGEMV( 'T', K, NRHS, ONE, B, LDB, WORK, 1, ZERO, |
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339 $ BX( J, 1 ), LDBX ) |
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340 80 CONTINUE |
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341 END IF |
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342 * |
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343 * Step (2R): if SQRE = 1, apply back the rotation that is |
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344 * related to the right null space of the subproblem. |
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345 * |
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346 IF( SQRE.EQ.1 ) THEN |
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347 CALL DCOPY( NRHS, B( M, 1 ), LDB, BX( M, 1 ), LDBX ) |
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348 CALL DROT( NRHS, BX( 1, 1 ), LDBX, BX( M, 1 ), LDBX, C, S ) |
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349 END IF |
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350 IF( K.LT.MAX( M, N ) ) |
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351 $ CALL DLACPY( 'A', N-K, NRHS, B( K+1, 1 ), LDB, BX( K+1, 1 ), |
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352 $ LDBX ) |
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353 * |
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354 * Step (3R): permute rows of B. |
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355 * |
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356 CALL DCOPY( NRHS, BX( 1, 1 ), LDBX, B( NLP1, 1 ), LDB ) |
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357 IF( SQRE.EQ.1 ) THEN |
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358 CALL DCOPY( NRHS, BX( M, 1 ), LDBX, B( M, 1 ), LDB ) |
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359 END IF |
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360 DO 90 I = 2, N |
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361 CALL DCOPY( NRHS, BX( I, 1 ), LDBX, B( PERM( I ), 1 ), LDB ) |
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362 90 CONTINUE |
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363 * |
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364 * Step (4R): apply back the Givens rotations performed. |
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365 * |
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366 DO 100 I = GIVPTR, 1, -1 |
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367 CALL DROT( NRHS, B( GIVCOL( I, 2 ), 1 ), LDB, |
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368 $ B( GIVCOL( I, 1 ), 1 ), LDB, GIVNUM( I, 2 ), |
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369 $ -GIVNUM( I, 1 ) ) |
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370 100 CONTINUE |
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371 END IF |
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372 * |
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373 RETURN |
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374 * |
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375 * End of DLALS0 |
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376 * |
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377 END |
