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7072
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1 SUBROUTINE DLASDA( ICOMPQ, SMLSIZ, N, SQRE, D, E, U, LDU, VT, K, |
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2 $ DIFL, DIFR, Z, POLES, GIVPTR, GIVCOL, LDGCOL, |
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3 $ PERM, GIVNUM, C, S, WORK, IWORK, INFO ) |
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4 * |
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5 * -- LAPACK auxiliary 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 ICOMPQ, INFO, LDGCOL, LDU, N, SMLSIZ, SQRE |
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11 * .. |
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12 * .. Array Arguments .. |
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13 INTEGER GIVCOL( LDGCOL, * ), GIVPTR( * ), IWORK( * ), |
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14 $ K( * ), PERM( LDGCOL, * ) |
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15 DOUBLE PRECISION C( * ), D( * ), DIFL( LDU, * ), DIFR( LDU, * ), |
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16 $ E( * ), GIVNUM( LDU, * ), POLES( LDU, * ), |
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17 $ S( * ), U( LDU, * ), VT( LDU, * ), WORK( * ), |
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18 $ Z( LDU, * ) |
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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 * Using a divide and conquer approach, DLASDA computes the singular |
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25 * value decomposition (SVD) of a real upper bidiagonal N-by-M matrix |
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26 * B with diagonal D and offdiagonal E, where M = N + SQRE. The |
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27 * algorithm computes the singular values in the SVD B = U * S * VT. |
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28 * The orthogonal matrices U and VT are optionally computed in |
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29 * compact form. |
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30 * |
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31 * A related subroutine, DLASD0, computes the singular values and |
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32 * the singular vectors in explicit form. |
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33 * |
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34 * Arguments |
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35 * ========= |
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36 * |
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37 * ICOMPQ (input) INTEGER |
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38 * Specifies whether singular vectors are to be computed |
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39 * in compact form, as follows |
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40 * = 0: Compute singular values only. |
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41 * = 1: Compute singular vectors of upper bidiagonal |
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42 * matrix in compact form. |
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43 * |
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44 * SMLSIZ (input) INTEGER |
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45 * The maximum size of the subproblems at the bottom of the |
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46 * computation tree. |
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47 * |
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48 * N (input) INTEGER |
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49 * The row dimension of the upper bidiagonal matrix. This is |
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50 * also the dimension of the main diagonal array D. |
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51 * |
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52 * SQRE (input) INTEGER |
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53 * Specifies the column dimension of the bidiagonal matrix. |
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54 * = 0: The bidiagonal matrix has column dimension M = N; |
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55 * = 1: The bidiagonal matrix has column dimension M = N + 1. |
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56 * |
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57 * D (input/output) DOUBLE PRECISION array, dimension ( N ) |
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58 * On entry D contains the main diagonal of the bidiagonal |
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59 * matrix. On exit D, if INFO = 0, contains its singular values. |
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60 * |
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61 * E (input) DOUBLE PRECISION array, dimension ( M-1 ) |
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62 * Contains the subdiagonal entries of the bidiagonal matrix. |
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63 * On exit, E has been destroyed. |
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64 * |
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65 * U (output) DOUBLE PRECISION array, |
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66 * dimension ( LDU, SMLSIZ ) if ICOMPQ = 1, and not referenced |
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67 * if ICOMPQ = 0. If ICOMPQ = 1, on exit, U contains the left |
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68 * singular vector matrices of all subproblems at the bottom |
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69 * level. |
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70 * |
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71 * LDU (input) INTEGER, LDU = > N. |
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72 * The leading dimension of arrays U, VT, DIFL, DIFR, POLES, |
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73 * GIVNUM, and Z. |
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74 * |
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75 * VT (output) DOUBLE PRECISION array, |
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76 * dimension ( LDU, SMLSIZ+1 ) if ICOMPQ = 1, and not referenced |
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77 * if ICOMPQ = 0. If ICOMPQ = 1, on exit, VT' contains the right |
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78 * singular vector matrices of all subproblems at the bottom |
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79 * level. |
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80 * |
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81 * K (output) INTEGER array, |
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82 * dimension ( N ) if ICOMPQ = 1 and dimension 1 if ICOMPQ = 0. |
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83 * If ICOMPQ = 1, on exit, K(I) is the dimension of the I-th |
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84 * secular equation on the computation tree. |
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85 * |
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86 * DIFL (output) DOUBLE PRECISION array, dimension ( LDU, NLVL ), |
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87 * where NLVL = floor(log_2 (N/SMLSIZ))). |
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88 * |
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89 * DIFR (output) DOUBLE PRECISION array, |
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90 * dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1 and |
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91 * dimension ( N ) if ICOMPQ = 0. |
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92 * If ICOMPQ = 1, on exit, DIFL(1:N, I) and DIFR(1:N, 2 * I - 1) |
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93 * record distances between singular values on the I-th |
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94 * level and singular values on the (I -1)-th level, and |
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95 * DIFR(1:N, 2 * I ) contains the normalizing factors for |
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96 * the right singular vector matrix. See DLASD8 for details. |
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97 * |
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98 * Z (output) DOUBLE PRECISION array, |
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99 * dimension ( LDU, NLVL ) if ICOMPQ = 1 and |
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100 * dimension ( N ) if ICOMPQ = 0. |
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101 * The first K elements of Z(1, I) contain the components of |
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102 * the deflation-adjusted updating row vector for subproblems |
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103 * on the I-th level. |
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104 * |
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105 * POLES (output) DOUBLE PRECISION array, |
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106 * dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1, and not referenced |
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107 * if ICOMPQ = 0. If ICOMPQ = 1, on exit, POLES(1, 2*I - 1) and |
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108 * POLES(1, 2*I) contain the new and old singular values |
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109 * involved in the secular equations on the I-th level. |
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110 * |
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111 * GIVPTR (output) INTEGER array, |
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112 * dimension ( N ) if ICOMPQ = 1, and not referenced if |
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113 * ICOMPQ = 0. If ICOMPQ = 1, on exit, GIVPTR( I ) records |
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114 * the number of Givens rotations performed on the I-th |
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115 * problem on the computation tree. |
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116 * |
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117 * GIVCOL (output) INTEGER array, |
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118 * dimension ( LDGCOL, 2 * NLVL ) if ICOMPQ = 1, and not |
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119 * referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, for each I, |
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120 * GIVCOL(1, 2 *I - 1) and GIVCOL(1, 2 *I) record the locations |
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121 * of Givens rotations performed on the I-th level on the |
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122 * computation tree. |
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123 * |
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124 * LDGCOL (input) INTEGER, LDGCOL = > N. |
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125 * The leading dimension of arrays GIVCOL and PERM. |
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126 * |
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127 * PERM (output) INTEGER array, |
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128 * dimension ( LDGCOL, NLVL ) if ICOMPQ = 1, and not referenced |
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129 * if ICOMPQ = 0. If ICOMPQ = 1, on exit, PERM(1, I) records |
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130 * permutations done on the I-th level of the computation tree. |
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131 * |
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132 * GIVNUM (output) DOUBLE PRECISION array, |
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133 * dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1, and not |
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134 * referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, for each I, |
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135 * GIVNUM(1, 2 *I - 1) and GIVNUM(1, 2 *I) record the C- and S- |
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136 * values of Givens rotations performed on the I-th level on |
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137 * the computation tree. |
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138 * |
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139 * C (output) DOUBLE PRECISION array, |
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140 * dimension ( N ) if ICOMPQ = 1, and dimension 1 if ICOMPQ = 0. |
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141 * If ICOMPQ = 1 and the I-th subproblem is not square, on exit, |
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142 * C( I ) contains the C-value of a Givens rotation related to |
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143 * the right null space of the I-th subproblem. |
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144 * |
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145 * S (output) DOUBLE PRECISION array, dimension ( N ) if |
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146 * ICOMPQ = 1, and dimension 1 if ICOMPQ = 0. If ICOMPQ = 1 |
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147 * and the I-th subproblem is not square, on exit, S( I ) |
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148 * contains the S-value of a Givens rotation related to |
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149 * the right null space of the I-th subproblem. |
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150 * |
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151 * WORK (workspace) DOUBLE PRECISION array, dimension |
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152 * (6 * N + (SMLSIZ + 1)*(SMLSIZ + 1)). |
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153 * |
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154 * IWORK (workspace) INTEGER array. |
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155 * Dimension must be at least (7 * N). |
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156 * |
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157 * INFO (output) INTEGER |
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158 * = 0: successful exit. |
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159 * < 0: if INFO = -i, the i-th argument had an illegal value. |
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160 * > 0: if INFO = 1, an singular value did not converge |
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161 * |
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162 * Further Details |
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163 * =============== |
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164 * |
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165 * Based on contributions by |
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166 * Ming Gu and Huan Ren, Computer Science Division, University of |
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167 * California at Berkeley, USA |
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168 * |
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169 * ===================================================================== |
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170 * |
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171 * .. Parameters .. |
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172 DOUBLE PRECISION ZERO, ONE |
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173 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) |
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174 * .. |
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175 * .. Local Scalars .. |
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176 INTEGER I, I1, IC, IDXQ, IDXQI, IM1, INODE, ITEMP, IWK, |
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177 $ J, LF, LL, LVL, LVL2, M, NCC, ND, NDB1, NDIML, |
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178 $ NDIMR, NL, NLF, NLP1, NLVL, NR, NRF, NRP1, NRU, |
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179 $ NWORK1, NWORK2, SMLSZP, SQREI, VF, VFI, VL, VLI |
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180 DOUBLE PRECISION ALPHA, BETA |
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181 * .. |
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182 * .. External Subroutines .. |
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183 EXTERNAL DCOPY, DLASD6, DLASDQ, DLASDT, DLASET, XERBLA |
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184 * .. |
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185 * .. Executable Statements .. |
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186 * |
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187 * Test the input parameters. |
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188 * |
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189 INFO = 0 |
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190 * |
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191 IF( ( ICOMPQ.LT.0 ) .OR. ( ICOMPQ.GT.1 ) ) THEN |
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192 INFO = -1 |
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193 ELSE IF( SMLSIZ.LT.3 ) THEN |
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194 INFO = -2 |
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195 ELSE IF( N.LT.0 ) THEN |
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196 INFO = -3 |
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197 ELSE IF( ( SQRE.LT.0 ) .OR. ( SQRE.GT.1 ) ) THEN |
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198 INFO = -4 |
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199 ELSE IF( LDU.LT.( N+SQRE ) ) THEN |
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200 INFO = -8 |
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201 ELSE IF( LDGCOL.LT.N ) THEN |
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202 INFO = -17 |
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203 END IF |
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204 IF( INFO.NE.0 ) THEN |
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205 CALL XERBLA( 'DLASDA', -INFO ) |
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206 RETURN |
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207 END IF |
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208 * |
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209 M = N + SQRE |
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210 * |
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211 * If the input matrix is too small, call DLASDQ to find the SVD. |
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212 * |
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213 IF( N.LE.SMLSIZ ) THEN |
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214 IF( ICOMPQ.EQ.0 ) THEN |
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215 CALL DLASDQ( 'U', SQRE, N, 0, 0, 0, D, E, VT, LDU, U, LDU, |
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216 $ U, LDU, WORK, INFO ) |
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217 ELSE |
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218 CALL DLASDQ( 'U', SQRE, N, M, N, 0, D, E, VT, LDU, U, LDU, |
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219 $ U, LDU, WORK, INFO ) |
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220 END IF |
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221 RETURN |
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222 END IF |
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223 * |
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224 * Book-keeping and set up the computation tree. |
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225 * |
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226 INODE = 1 |
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227 NDIML = INODE + N |
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228 NDIMR = NDIML + N |
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229 IDXQ = NDIMR + N |
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230 IWK = IDXQ + N |
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231 * |
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232 NCC = 0 |
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233 NRU = 0 |
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234 * |
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235 SMLSZP = SMLSIZ + 1 |
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236 VF = 1 |
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237 VL = VF + M |
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238 NWORK1 = VL + M |
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239 NWORK2 = NWORK1 + SMLSZP*SMLSZP |
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240 * |
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241 CALL DLASDT( N, NLVL, ND, IWORK( INODE ), IWORK( NDIML ), |
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242 $ IWORK( NDIMR ), SMLSIZ ) |
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243 * |
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244 * for the nodes on bottom level of the tree, solve |
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245 * their subproblems by DLASDQ. |
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246 * |
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247 NDB1 = ( ND+1 ) / 2 |
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248 DO 30 I = NDB1, ND |
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249 * |
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250 * IC : center row of each node |
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251 * NL : number of rows of left subproblem |
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252 * NR : number of rows of right subproblem |
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253 * NLF: starting row of the left subproblem |
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254 * NRF: starting row of the right subproblem |
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255 * |
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256 I1 = I - 1 |
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257 IC = IWORK( INODE+I1 ) |
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258 NL = IWORK( NDIML+I1 ) |
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259 NLP1 = NL + 1 |
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260 NR = IWORK( NDIMR+I1 ) |
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261 NLF = IC - NL |
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262 NRF = IC + 1 |
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263 IDXQI = IDXQ + NLF - 2 |
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264 VFI = VF + NLF - 1 |
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265 VLI = VL + NLF - 1 |
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266 SQREI = 1 |
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267 IF( ICOMPQ.EQ.0 ) THEN |
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268 CALL DLASET( 'A', NLP1, NLP1, ZERO, ONE, WORK( NWORK1 ), |
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269 $ SMLSZP ) |
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270 CALL DLASDQ( 'U', SQREI, NL, NLP1, NRU, NCC, D( NLF ), |
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271 $ E( NLF ), WORK( NWORK1 ), SMLSZP, |
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272 $ WORK( NWORK2 ), NL, WORK( NWORK2 ), NL, |
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273 $ WORK( NWORK2 ), INFO ) |
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274 ITEMP = NWORK1 + NL*SMLSZP |
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275 CALL DCOPY( NLP1, WORK( NWORK1 ), 1, WORK( VFI ), 1 ) |
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276 CALL DCOPY( NLP1, WORK( ITEMP ), 1, WORK( VLI ), 1 ) |
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277 ELSE |
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278 CALL DLASET( 'A', NL, NL, ZERO, ONE, U( NLF, 1 ), LDU ) |
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279 CALL DLASET( 'A', NLP1, NLP1, ZERO, ONE, VT( NLF, 1 ), LDU ) |
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280 CALL DLASDQ( 'U', SQREI, NL, NLP1, NL, NCC, D( NLF ), |
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281 $ E( NLF ), VT( NLF, 1 ), LDU, U( NLF, 1 ), LDU, |
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282 $ U( NLF, 1 ), LDU, WORK( NWORK1 ), INFO ) |
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283 CALL DCOPY( NLP1, VT( NLF, 1 ), 1, WORK( VFI ), 1 ) |
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284 CALL DCOPY( NLP1, VT( NLF, NLP1 ), 1, WORK( VLI ), 1 ) |
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285 END IF |
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286 IF( INFO.NE.0 ) THEN |
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287 RETURN |
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288 END IF |
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289 DO 10 J = 1, NL |
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290 IWORK( IDXQI+J ) = J |
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291 10 CONTINUE |
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292 IF( ( I.EQ.ND ) .AND. ( SQRE.EQ.0 ) ) THEN |
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293 SQREI = 0 |
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294 ELSE |
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295 SQREI = 1 |
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296 END IF |
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297 IDXQI = IDXQI + NLP1 |
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298 VFI = VFI + NLP1 |
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299 VLI = VLI + NLP1 |
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300 NRP1 = NR + SQREI |
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301 IF( ICOMPQ.EQ.0 ) THEN |
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302 CALL DLASET( 'A', NRP1, NRP1, ZERO, ONE, WORK( NWORK1 ), |
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303 $ SMLSZP ) |
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304 CALL DLASDQ( 'U', SQREI, NR, NRP1, NRU, NCC, D( NRF ), |
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305 $ E( NRF ), WORK( NWORK1 ), SMLSZP, |
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306 $ WORK( NWORK2 ), NR, WORK( NWORK2 ), NR, |
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307 $ WORK( NWORK2 ), INFO ) |
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308 ITEMP = NWORK1 + ( NRP1-1 )*SMLSZP |
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309 CALL DCOPY( NRP1, WORK( NWORK1 ), 1, WORK( VFI ), 1 ) |
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310 CALL DCOPY( NRP1, WORK( ITEMP ), 1, WORK( VLI ), 1 ) |
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311 ELSE |
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312 CALL DLASET( 'A', NR, NR, ZERO, ONE, U( NRF, 1 ), LDU ) |
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313 CALL DLASET( 'A', NRP1, NRP1, ZERO, ONE, VT( NRF, 1 ), LDU ) |
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314 CALL DLASDQ( 'U', SQREI, NR, NRP1, NR, NCC, D( NRF ), |
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315 $ E( NRF ), VT( NRF, 1 ), LDU, U( NRF, 1 ), LDU, |
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316 $ U( NRF, 1 ), LDU, WORK( NWORK1 ), INFO ) |
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317 CALL DCOPY( NRP1, VT( NRF, 1 ), 1, WORK( VFI ), 1 ) |
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318 CALL DCOPY( NRP1, VT( NRF, NRP1 ), 1, WORK( VLI ), 1 ) |
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319 END IF |
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320 IF( INFO.NE.0 ) THEN |
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321 RETURN |
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322 END IF |
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323 DO 20 J = 1, NR |
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324 IWORK( IDXQI+J ) = J |
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325 20 CONTINUE |
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326 30 CONTINUE |
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327 * |
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328 * Now conquer each subproblem bottom-up. |
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329 * |
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330 J = 2**NLVL |
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331 DO 50 LVL = NLVL, 1, -1 |
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332 LVL2 = LVL*2 - 1 |
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333 * |
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334 * Find the first node LF and last node LL on |
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335 * the current level LVL. |
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336 * |
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337 IF( LVL.EQ.1 ) THEN |
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338 LF = 1 |
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339 LL = 1 |
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340 ELSE |
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341 LF = 2**( LVL-1 ) |
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342 LL = 2*LF - 1 |
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343 END IF |
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344 DO 40 I = LF, LL |
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345 IM1 = I - 1 |
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346 IC = IWORK( INODE+IM1 ) |
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347 NL = IWORK( NDIML+IM1 ) |
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348 NR = IWORK( NDIMR+IM1 ) |
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349 NLF = IC - NL |
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350 NRF = IC + 1 |
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351 IF( I.EQ.LL ) THEN |
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352 SQREI = SQRE |
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353 ELSE |
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354 SQREI = 1 |
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355 END IF |
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356 VFI = VF + NLF - 1 |
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357 VLI = VL + NLF - 1 |
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358 IDXQI = IDXQ + NLF - 1 |
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359 ALPHA = D( IC ) |
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360 BETA = E( IC ) |
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361 IF( ICOMPQ.EQ.0 ) THEN |
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362 CALL DLASD6( ICOMPQ, NL, NR, SQREI, D( NLF ), |
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363 $ WORK( VFI ), WORK( VLI ), ALPHA, BETA, |
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364 $ IWORK( IDXQI ), PERM, GIVPTR( 1 ), GIVCOL, |
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365 $ LDGCOL, GIVNUM, LDU, POLES, DIFL, DIFR, Z, |
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366 $ K( 1 ), C( 1 ), S( 1 ), WORK( NWORK1 ), |
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367 $ IWORK( IWK ), INFO ) |
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368 ELSE |
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369 J = J - 1 |
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370 CALL DLASD6( ICOMPQ, NL, NR, SQREI, D( NLF ), |
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371 $ WORK( VFI ), WORK( VLI ), ALPHA, BETA, |
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372 $ IWORK( IDXQI ), PERM( NLF, LVL ), |
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373 $ GIVPTR( J ), GIVCOL( NLF, LVL2 ), LDGCOL, |
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374 $ GIVNUM( NLF, LVL2 ), LDU, |
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375 $ POLES( NLF, LVL2 ), DIFL( NLF, LVL ), |
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376 $ DIFR( NLF, LVL2 ), Z( NLF, LVL ), K( J ), |
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377 $ C( J ), S( J ), WORK( NWORK1 ), |
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378 $ IWORK( IWK ), INFO ) |
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379 END IF |
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380 IF( INFO.NE.0 ) THEN |
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381 RETURN |
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382 END IF |
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383 40 CONTINUE |
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384 50 CONTINUE |
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385 * |
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386 RETURN |
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387 * |
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388 * End of DLASDA |
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389 * |
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390 END |
