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1 SUBROUTINE ZBDSQR( UPLO, N, NCVT, NRU, NCC, D, E, VT, LDVT, U, |
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2 $ LDU, C, LDC, RWORK, INFO ) |
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3 * |
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4 * -- LAPACK routine (version 3.1) -- |
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5 * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. |
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6 * November 2006 |
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7 * |
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8 * .. Scalar Arguments .. |
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9 CHARACTER UPLO |
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10 INTEGER INFO, LDC, LDU, LDVT, N, NCC, NCVT, NRU |
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11 * .. |
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12 * .. Array Arguments .. |
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13 DOUBLE PRECISION D( * ), E( * ), RWORK( * ) |
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14 COMPLEX*16 C( LDC, * ), U( LDU, * ), VT( LDVT, * ) |
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15 * .. |
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16 * |
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17 * Purpose |
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18 * ======= |
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19 * |
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20 * ZBDSQR computes the singular values and, optionally, the right and/or |
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21 * left singular vectors from the singular value decomposition (SVD) of |
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22 * a real N-by-N (upper or lower) bidiagonal matrix B using the implicit |
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23 * zero-shift QR algorithm. The SVD of B has the form |
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24 * |
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25 * B = Q * S * P**H |
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26 * |
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27 * where S is the diagonal matrix of singular values, Q is an orthogonal |
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28 * matrix of left singular vectors, and P is an orthogonal matrix of |
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29 * right singular vectors. If left singular vectors are requested, this |
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30 * subroutine actually returns U*Q instead of Q, and, if right singular |
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31 * vectors are requested, this subroutine returns P**H*VT instead of |
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32 * P**H, for given complex input matrices U and VT. When U and VT are |
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33 * the unitary matrices that reduce a general matrix A to bidiagonal |
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34 * form: A = U*B*VT, as computed by ZGEBRD, then |
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35 * |
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36 * A = (U*Q) * S * (P**H*VT) |
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37 * |
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38 * is the SVD of A. Optionally, the subroutine may also compute Q**H*C |
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39 * for a given complex input matrix C. |
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40 * |
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41 * See "Computing Small Singular Values of Bidiagonal Matrices With |
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42 * Guaranteed High Relative Accuracy," by J. Demmel and W. Kahan, |
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43 * LAPACK Working Note #3 (or SIAM J. Sci. Statist. Comput. vol. 11, |
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44 * no. 5, pp. 873-912, Sept 1990) and |
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45 * "Accurate singular values and differential qd algorithms," by |
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46 * B. Parlett and V. Fernando, Technical Report CPAM-554, Mathematics |
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47 * Department, University of California at Berkeley, July 1992 |
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48 * for a detailed description of the algorithm. |
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49 * |
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50 * Arguments |
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51 * ========= |
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52 * |
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53 * UPLO (input) CHARACTER*1 |
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54 * = 'U': B is upper bidiagonal; |
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55 * = 'L': B is lower bidiagonal. |
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56 * |
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57 * N (input) INTEGER |
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58 * The order of the matrix B. N >= 0. |
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59 * |
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60 * NCVT (input) INTEGER |
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61 * The number of columns of the matrix VT. NCVT >= 0. |
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62 * |
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63 * NRU (input) INTEGER |
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64 * The number of rows of the matrix U. NRU >= 0. |
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65 * |
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66 * NCC (input) INTEGER |
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67 * The number of columns of the matrix C. NCC >= 0. |
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68 * |
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69 * D (input/output) DOUBLE PRECISION array, dimension (N) |
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70 * On entry, the n diagonal elements of the bidiagonal matrix B. |
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71 * On exit, if INFO=0, the singular values of B in decreasing |
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72 * order. |
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73 * |
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74 * E (input/output) DOUBLE PRECISION array, dimension (N-1) |
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75 * On entry, the N-1 offdiagonal elements of the bidiagonal |
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76 * matrix B. |
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77 * On exit, if INFO = 0, E is destroyed; if INFO > 0, D and E |
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78 * will contain the diagonal and superdiagonal elements of a |
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79 * bidiagonal matrix orthogonally equivalent to the one given |
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80 * as input. |
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81 * |
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82 * VT (input/output) COMPLEX*16 array, dimension (LDVT, NCVT) |
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83 * On entry, an N-by-NCVT matrix VT. |
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84 * On exit, VT is overwritten by P**H * VT. |
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85 * Not referenced if NCVT = 0. |
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86 * |
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87 * LDVT (input) INTEGER |
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88 * The leading dimension of the array VT. |
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89 * LDVT >= max(1,N) if NCVT > 0; LDVT >= 1 if NCVT = 0. |
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90 * |
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91 * U (input/output) COMPLEX*16 array, dimension (LDU, N) |
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92 * On entry, an NRU-by-N matrix U. |
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93 * On exit, U is overwritten by U * Q. |
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94 * Not referenced if NRU = 0. |
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95 * |
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96 * LDU (input) INTEGER |
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97 * The leading dimension of the array U. LDU >= max(1,NRU). |
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98 * |
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99 * C (input/output) COMPLEX*16 array, dimension (LDC, NCC) |
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100 * On entry, an N-by-NCC matrix C. |
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101 * On exit, C is overwritten by Q**H * C. |
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102 * Not referenced if NCC = 0. |
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103 * |
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104 * LDC (input) INTEGER |
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105 * The leading dimension of the array C. |
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106 * LDC >= max(1,N) if NCC > 0; LDC >=1 if NCC = 0. |
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107 * |
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108 * RWORK (workspace) DOUBLE PRECISION array, dimension (2*N) |
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109 * if NCVT = NRU = NCC = 0, (max(1, 4*N-4)) otherwise |
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110 * |
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111 * INFO (output) INTEGER |
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112 * = 0: successful exit |
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113 * < 0: If INFO = -i, the i-th argument had an illegal value |
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114 * > 0: the algorithm did not converge; D and E contain the |
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115 * elements of a bidiagonal matrix which is orthogonally |
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116 * similar to the input matrix B; if INFO = i, i |
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117 * elements of E have not converged to zero. |
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118 * |
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119 * Internal Parameters |
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120 * =================== |
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121 * |
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122 * TOLMUL DOUBLE PRECISION, default = max(10,min(100,EPS**(-1/8))) |
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123 * TOLMUL controls the convergence criterion of the QR loop. |
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124 * If it is positive, TOLMUL*EPS is the desired relative |
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125 * precision in the computed singular values. |
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126 * If it is negative, abs(TOLMUL*EPS*sigma_max) is the |
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127 * desired absolute accuracy in the computed singular |
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128 * values (corresponds to relative accuracy |
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129 * abs(TOLMUL*EPS) in the largest singular value. |
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130 * abs(TOLMUL) should be between 1 and 1/EPS, and preferably |
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131 * between 10 (for fast convergence) and .1/EPS |
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132 * (for there to be some accuracy in the results). |
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133 * Default is to lose at either one eighth or 2 of the |
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134 * available decimal digits in each computed singular value |
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135 * (whichever is smaller). |
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136 * |
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137 * MAXITR INTEGER, default = 6 |
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138 * MAXITR controls the maximum number of passes of the |
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139 * algorithm through its inner loop. The algorithms stops |
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140 * (and so fails to converge) if the number of passes |
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141 * through the inner loop exceeds MAXITR*N**2. |
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142 * |
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143 * ===================================================================== |
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144 * |
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145 * .. Parameters .. |
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146 DOUBLE PRECISION ZERO |
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147 PARAMETER ( ZERO = 0.0D0 ) |
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148 DOUBLE PRECISION ONE |
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149 PARAMETER ( ONE = 1.0D0 ) |
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150 DOUBLE PRECISION NEGONE |
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151 PARAMETER ( NEGONE = -1.0D0 ) |
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152 DOUBLE PRECISION HNDRTH |
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153 PARAMETER ( HNDRTH = 0.01D0 ) |
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154 DOUBLE PRECISION TEN |
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155 PARAMETER ( TEN = 10.0D0 ) |
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156 DOUBLE PRECISION HNDRD |
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157 PARAMETER ( HNDRD = 100.0D0 ) |
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158 DOUBLE PRECISION MEIGTH |
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159 PARAMETER ( MEIGTH = -0.125D0 ) |
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160 INTEGER MAXITR |
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161 PARAMETER ( MAXITR = 6 ) |
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162 * .. |
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163 * .. Local Scalars .. |
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164 LOGICAL LOWER, ROTATE |
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165 INTEGER I, IDIR, ISUB, ITER, J, LL, LLL, M, MAXIT, NM1, |
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166 $ NM12, NM13, OLDLL, OLDM |
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167 DOUBLE PRECISION ABSE, ABSS, COSL, COSR, CS, EPS, F, G, H, MU, |
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168 $ OLDCS, OLDSN, R, SHIFT, SIGMN, SIGMX, SINL, |
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169 $ SINR, SLL, SMAX, SMIN, SMINL, SMINOA, |
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170 $ SN, THRESH, TOL, TOLMUL, UNFL |
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171 * .. |
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172 * .. External Functions .. |
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173 LOGICAL LSAME |
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174 DOUBLE PRECISION DLAMCH |
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175 EXTERNAL LSAME, DLAMCH |
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176 * .. |
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177 * .. External Subroutines .. |
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178 EXTERNAL DLARTG, DLAS2, DLASQ1, DLASV2, XERBLA, ZDROT, |
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179 $ ZDSCAL, ZLASR, ZSWAP |
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180 * .. |
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181 * .. Intrinsic Functions .. |
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182 INTRINSIC ABS, DBLE, MAX, MIN, SIGN, SQRT |
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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 LOWER = LSAME( UPLO, 'L' ) |
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190 IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LOWER ) THEN |
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191 INFO = -1 |
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192 ELSE IF( N.LT.0 ) THEN |
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193 INFO = -2 |
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194 ELSE IF( NCVT.LT.0 ) THEN |
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195 INFO = -3 |
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196 ELSE IF( NRU.LT.0 ) THEN |
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197 INFO = -4 |
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198 ELSE IF( NCC.LT.0 ) THEN |
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199 INFO = -5 |
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200 ELSE IF( ( NCVT.EQ.0 .AND. LDVT.LT.1 ) .OR. |
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201 $ ( NCVT.GT.0 .AND. LDVT.LT.MAX( 1, N ) ) ) THEN |
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202 INFO = -9 |
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203 ELSE IF( LDU.LT.MAX( 1, NRU ) ) THEN |
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204 INFO = -11 |
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205 ELSE IF( ( NCC.EQ.0 .AND. LDC.LT.1 ) .OR. |
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206 $ ( NCC.GT.0 .AND. LDC.LT.MAX( 1, N ) ) ) THEN |
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207 INFO = -13 |
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208 END IF |
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209 IF( INFO.NE.0 ) THEN |
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210 CALL XERBLA( 'ZBDSQR', -INFO ) |
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211 RETURN |
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212 END IF |
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213 IF( N.EQ.0 ) |
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214 $ RETURN |
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215 IF( N.EQ.1 ) |
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216 $ GO TO 160 |
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217 * |
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218 * ROTATE is true if any singular vectors desired, false otherwise |
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219 * |
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220 ROTATE = ( NCVT.GT.0 ) .OR. ( NRU.GT.0 ) .OR. ( NCC.GT.0 ) |
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221 * |
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222 * If no singular vectors desired, use qd algorithm |
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223 * |
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224 IF( .NOT.ROTATE ) THEN |
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225 CALL DLASQ1( N, D, E, RWORK, INFO ) |
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226 RETURN |
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227 END IF |
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228 * |
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229 NM1 = N - 1 |
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230 NM12 = NM1 + NM1 |
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231 NM13 = NM12 + NM1 |
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232 IDIR = 0 |
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233 * |
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234 * Get machine constants |
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235 * |
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236 EPS = DLAMCH( 'Epsilon' ) |
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237 UNFL = DLAMCH( 'Safe minimum' ) |
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238 * |
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239 * If matrix lower bidiagonal, rotate to be upper bidiagonal |
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240 * by applying Givens rotations on the left |
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241 * |
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242 IF( LOWER ) THEN |
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243 DO 10 I = 1, N - 1 |
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244 CALL DLARTG( D( I ), E( I ), CS, SN, R ) |
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245 D( I ) = R |
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246 E( I ) = SN*D( I+1 ) |
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247 D( I+1 ) = CS*D( I+1 ) |
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248 RWORK( I ) = CS |
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249 RWORK( NM1+I ) = SN |
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250 10 CONTINUE |
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251 * |
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252 * Update singular vectors if desired |
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253 * |
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254 IF( NRU.GT.0 ) |
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255 $ CALL ZLASR( 'R', 'V', 'F', NRU, N, RWORK( 1 ), RWORK( N ), |
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256 $ U, LDU ) |
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257 IF( NCC.GT.0 ) |
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258 $ CALL ZLASR( 'L', 'V', 'F', N, NCC, RWORK( 1 ), RWORK( N ), |
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259 $ C, LDC ) |
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260 END IF |
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261 * |
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262 * Compute singular values to relative accuracy TOL |
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263 * (By setting TOL to be negative, algorithm will compute |
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264 * singular values to absolute accuracy ABS(TOL)*norm(input matrix)) |
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265 * |
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266 TOLMUL = MAX( TEN, MIN( HNDRD, EPS**MEIGTH ) ) |
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267 TOL = TOLMUL*EPS |
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268 * |
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269 * Compute approximate maximum, minimum singular values |
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270 * |
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271 SMAX = ZERO |
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272 DO 20 I = 1, N |
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273 SMAX = MAX( SMAX, ABS( D( I ) ) ) |
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274 20 CONTINUE |
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275 DO 30 I = 1, N - 1 |
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276 SMAX = MAX( SMAX, ABS( E( I ) ) ) |
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277 30 CONTINUE |
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278 SMINL = ZERO |
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279 IF( TOL.GE.ZERO ) THEN |
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280 * |
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281 * Relative accuracy desired |
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282 * |
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283 SMINOA = ABS( D( 1 ) ) |
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284 IF( SMINOA.EQ.ZERO ) |
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285 $ GO TO 50 |
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286 MU = SMINOA |
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287 DO 40 I = 2, N |
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288 MU = ABS( D( I ) )*( MU / ( MU+ABS( E( I-1 ) ) ) ) |
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289 SMINOA = MIN( SMINOA, MU ) |
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290 IF( SMINOA.EQ.ZERO ) |
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291 $ GO TO 50 |
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292 40 CONTINUE |
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293 50 CONTINUE |
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294 SMINOA = SMINOA / SQRT( DBLE( N ) ) |
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295 THRESH = MAX( TOL*SMINOA, MAXITR*N*N*UNFL ) |
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296 ELSE |
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297 * |
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298 * Absolute accuracy desired |
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299 * |
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300 THRESH = MAX( ABS( TOL )*SMAX, MAXITR*N*N*UNFL ) |
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301 END IF |
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302 * |
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303 * Prepare for main iteration loop for the singular values |
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304 * (MAXIT is the maximum number of passes through the inner |
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305 * loop permitted before nonconvergence signalled.) |
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306 * |
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307 MAXIT = MAXITR*N*N |
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308 ITER = 0 |
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309 OLDLL = -1 |
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310 OLDM = -1 |
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311 * |
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312 * M points to last element of unconverged part of matrix |
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313 * |
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314 M = N |
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315 * |
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316 * Begin main iteration loop |
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317 * |
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318 60 CONTINUE |
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319 * |
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320 * Check for convergence or exceeding iteration count |
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321 * |
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322 IF( M.LE.1 ) |
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323 $ GO TO 160 |
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324 IF( ITER.GT.MAXIT ) |
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325 $ GO TO 200 |
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326 * |
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327 * Find diagonal block of matrix to work on |
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328 * |
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329 IF( TOL.LT.ZERO .AND. ABS( D( M ) ).LE.THRESH ) |
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330 $ D( M ) = ZERO |
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331 SMAX = ABS( D( M ) ) |
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332 SMIN = SMAX |
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333 DO 70 LLL = 1, M - 1 |
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334 LL = M - LLL |
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335 ABSS = ABS( D( LL ) ) |
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336 ABSE = ABS( E( LL ) ) |
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337 IF( TOL.LT.ZERO .AND. ABSS.LE.THRESH ) |
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338 $ D( LL ) = ZERO |
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339 IF( ABSE.LE.THRESH ) |
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340 $ GO TO 80 |
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341 SMIN = MIN( SMIN, ABSS ) |
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342 SMAX = MAX( SMAX, ABSS, ABSE ) |
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343 70 CONTINUE |
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344 LL = 0 |
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345 GO TO 90 |
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346 80 CONTINUE |
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347 E( LL ) = ZERO |
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348 * |
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349 * Matrix splits since E(LL) = 0 |
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350 * |
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351 IF( LL.EQ.M-1 ) THEN |
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352 * |
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353 * Convergence of bottom singular value, return to top of loop |
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354 * |
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355 M = M - 1 |
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356 GO TO 60 |
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357 END IF |
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358 90 CONTINUE |
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359 LL = LL + 1 |
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360 * |
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361 * E(LL) through E(M-1) are nonzero, E(LL-1) is zero |
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362 * |
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363 IF( LL.EQ.M-1 ) THEN |
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364 * |
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365 * 2 by 2 block, handle separately |
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366 * |
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367 CALL DLASV2( D( M-1 ), E( M-1 ), D( M ), SIGMN, SIGMX, SINR, |
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368 $ COSR, SINL, COSL ) |
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369 D( M-1 ) = SIGMX |
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370 E( M-1 ) = ZERO |
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371 D( M ) = SIGMN |
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372 * |
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373 * Compute singular vectors, if desired |
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374 * |
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375 IF( NCVT.GT.0 ) |
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376 $ CALL ZDROT( NCVT, VT( M-1, 1 ), LDVT, VT( M, 1 ), LDVT, |
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377 $ COSR, SINR ) |
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378 IF( NRU.GT.0 ) |
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379 $ CALL ZDROT( NRU, U( 1, M-1 ), 1, U( 1, M ), 1, COSL, SINL ) |
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380 IF( NCC.GT.0 ) |
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381 $ CALL ZDROT( NCC, C( M-1, 1 ), LDC, C( M, 1 ), LDC, COSL, |
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382 $ SINL ) |
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383 M = M - 2 |
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384 GO TO 60 |
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385 END IF |
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386 * |
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387 * If working on new submatrix, choose shift direction |
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388 * (from larger end diagonal element towards smaller) |
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389 * |
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390 IF( LL.GT.OLDM .OR. M.LT.OLDLL ) THEN |
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391 IF( ABS( D( LL ) ).GE.ABS( D( M ) ) ) THEN |
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392 * |
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393 * Chase bulge from top (big end) to bottom (small end) |
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394 * |
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395 IDIR = 1 |
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396 ELSE |
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397 * |
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398 * Chase bulge from bottom (big end) to top (small end) |
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399 * |
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400 IDIR = 2 |
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401 END IF |
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402 END IF |
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403 * |
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404 * Apply convergence tests |
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405 * |
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406 IF( IDIR.EQ.1 ) THEN |
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407 * |
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408 * Run convergence test in forward direction |
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409 * First apply standard test to bottom of matrix |
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410 * |
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411 IF( ABS( E( M-1 ) ).LE.ABS( TOL )*ABS( D( M ) ) .OR. |
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412 $ ( TOL.LT.ZERO .AND. ABS( E( M-1 ) ).LE.THRESH ) ) THEN |
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413 E( M-1 ) = ZERO |
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414 GO TO 60 |
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415 END IF |
|
|
416 * |
|
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417 IF( TOL.GE.ZERO ) THEN |
|
|
418 * |
|
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419 * If relative accuracy desired, |
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|
420 * apply convergence criterion forward |
|
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421 * |
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422 MU = ABS( D( LL ) ) |
|
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423 SMINL = MU |
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3333
|
424 DO 100 LLL = LL, M - 1 |
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425 IF( ABS( E( LLL ) ).LE.TOL*MU ) THEN |
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426 E( LLL ) = ZERO |
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3333
|
427 GO TO 60 |
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428 END IF |
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|
429 MU = ABS( D( LLL+1 ) )*( MU / ( MU+ABS( E( LLL ) ) ) ) |
|
|
430 SMINL = MIN( SMINL, MU ) |
|
3333
|
431 100 CONTINUE |
|
2329
|
432 END IF |
|
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433 * |
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434 ELSE |
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|
435 * |
|
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436 * Run convergence test in backward direction |
|
|
437 * First apply standard test to top of matrix |
|
|
438 * |
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|
439 IF( ABS( E( LL ) ).LE.ABS( TOL )*ABS( D( LL ) ) .OR. |
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440 $ ( TOL.LT.ZERO .AND. ABS( E( LL ) ).LE.THRESH ) ) THEN |
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|
441 E( LL ) = ZERO |
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3333
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442 GO TO 60 |
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443 END IF |
|
|
444 * |
|
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445 IF( TOL.GE.ZERO ) THEN |
|
|
446 * |
|
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447 * If relative accuracy desired, |
|
|
448 * apply convergence criterion backward |
|
|
449 * |
|
|
450 MU = ABS( D( M ) ) |
|
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451 SMINL = MU |
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3333
|
452 DO 110 LLL = M - 1, LL, -1 |
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453 IF( ABS( E( LLL ) ).LE.TOL*MU ) THEN |
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|
454 E( LLL ) = ZERO |
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3333
|
455 GO TO 60 |
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456 END IF |
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457 MU = ABS( D( LLL ) )*( MU / ( MU+ABS( E( LLL ) ) ) ) |
|
|
458 SMINL = MIN( SMINL, MU ) |
|
3333
|
459 110 CONTINUE |
|
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|
460 END IF |
|
|
461 END IF |
|
|
462 OLDLL = LL |
|
|
463 OLDM = M |
|
|
464 * |
|
|
465 * Compute shift. First, test if shifting would ruin relative |
|
|
466 * accuracy, and if so set the shift to zero. |
|
|
467 * |
|
|
468 IF( TOL.GE.ZERO .AND. N*TOL*( SMINL / SMAX ).LE. |
|
|
469 $ MAX( EPS, HNDRTH*TOL ) ) THEN |
|
|
470 * |
|
|
471 * Use a zero shift to avoid loss of relative accuracy |
|
|
472 * |
|
|
473 SHIFT = ZERO |
|
|
474 ELSE |
|
|
475 * |
|
|
476 * Compute the shift from 2-by-2 block at end of matrix |
|
|
477 * |
|
|
478 IF( IDIR.EQ.1 ) THEN |
|
|
479 SLL = ABS( D( LL ) ) |
|
|
480 CALL DLAS2( D( M-1 ), E( M-1 ), D( M ), SHIFT, R ) |
|
|
481 ELSE |
|
|
482 SLL = ABS( D( M ) ) |
|
|
483 CALL DLAS2( D( LL ), E( LL ), D( LL+1 ), SHIFT, R ) |
|
|
484 END IF |
|
|
485 * |
|
|
486 * Test if shift negligible, and if so set to zero |
|
|
487 * |
|
|
488 IF( SLL.GT.ZERO ) THEN |
|
|
489 IF( ( SHIFT / SLL )**2.LT.EPS ) |
|
|
490 $ SHIFT = ZERO |
|
|
491 END IF |
|
|
492 END IF |
|
|
493 * |
|
|
494 * Increment iteration count |
|
|
495 * |
|
|
496 ITER = ITER + M - LL |
|
|
497 * |
|
|
498 * If SHIFT = 0, do simplified QR iteration |
|
|
499 * |
|
|
500 IF( SHIFT.EQ.ZERO ) THEN |
|
|
501 IF( IDIR.EQ.1 ) THEN |
|
|
502 * |
|
|
503 * Chase bulge from top to bottom |
|
|
504 * Save cosines and sines for later singular vector updates |
|
|
505 * |
|
|
506 CS = ONE |
|
|
507 OLDCS = ONE |
|
3333
|
508 DO 120 I = LL, M - 1 |
|
2329
|
509 CALL DLARTG( D( I )*CS, E( I ), CS, SN, R ) |
|
3333
|
510 IF( I.GT.LL ) |
|
|
511 $ E( I-1 ) = OLDSN*R |
|
2329
|
512 CALL DLARTG( OLDCS*R, D( I+1 )*SN, OLDCS, OLDSN, D( I ) ) |
|
3333
|
513 RWORK( I-LL+1 ) = CS |
|
|
514 RWORK( I-LL+1+NM1 ) = SN |
|
|
515 RWORK( I-LL+1+NM12 ) = OLDCS |
|
|
516 RWORK( I-LL+1+NM13 ) = OLDSN |
|
|
517 120 CONTINUE |
|
2329
|
518 H = D( M )*CS |
|
|
519 D( M ) = H*OLDCS |
|
|
520 E( M-1 ) = H*OLDSN |
|
|
521 * |
|
|
522 * Update singular vectors |
|
|
523 * |
|
|
524 IF( NCVT.GT.0 ) |
|
|
525 $ CALL ZLASR( 'L', 'V', 'F', M-LL+1, NCVT, RWORK( 1 ), |
|
|
526 $ RWORK( N ), VT( LL, 1 ), LDVT ) |
|
|
527 IF( NRU.GT.0 ) |
|
|
528 $ CALL ZLASR( 'R', 'V', 'F', NRU, M-LL+1, RWORK( NM12+1 ), |
|
|
529 $ RWORK( NM13+1 ), U( 1, LL ), LDU ) |
|
|
530 IF( NCC.GT.0 ) |
|
|
531 $ CALL ZLASR( 'L', 'V', 'F', M-LL+1, NCC, RWORK( NM12+1 ), |
|
|
532 $ RWORK( NM13+1 ), C( LL, 1 ), LDC ) |
|
|
533 * |
|
|
534 * Test convergence |
|
|
535 * |
|
|
536 IF( ABS( E( M-1 ) ).LE.THRESH ) |
|
|
537 $ E( M-1 ) = ZERO |
|
|
538 * |
|
|
539 ELSE |
|
|
540 * |
|
|
541 * Chase bulge from bottom to top |
|
|
542 * Save cosines and sines for later singular vector updates |
|
|
543 * |
|
|
544 CS = ONE |
|
|
545 OLDCS = ONE |
|
3333
|
546 DO 130 I = M, LL + 1, -1 |
|
2329
|
547 CALL DLARTG( D( I )*CS, E( I-1 ), CS, SN, R ) |
|
3333
|
548 IF( I.LT.M ) |
|
|
549 $ E( I ) = OLDSN*R |
|
2329
|
550 CALL DLARTG( OLDCS*R, D( I-1 )*SN, OLDCS, OLDSN, D( I ) ) |
|
3333
|
551 RWORK( I-LL ) = CS |
|
|
552 RWORK( I-LL+NM1 ) = -SN |
|
|
553 RWORK( I-LL+NM12 ) = OLDCS |
|
|
554 RWORK( I-LL+NM13 ) = -OLDSN |
|
|
555 130 CONTINUE |
|
2329
|
556 H = D( LL )*CS |
|
|
557 D( LL ) = H*OLDCS |
|
|
558 E( LL ) = H*OLDSN |
|
|
559 * |
|
|
560 * Update singular vectors |
|
|
561 * |
|
|
562 IF( NCVT.GT.0 ) |
|
|
563 $ CALL ZLASR( 'L', 'V', 'B', M-LL+1, NCVT, RWORK( NM12+1 ), |
|
|
564 $ RWORK( NM13+1 ), VT( LL, 1 ), LDVT ) |
|
|
565 IF( NRU.GT.0 ) |
|
|
566 $ CALL ZLASR( 'R', 'V', 'B', NRU, M-LL+1, RWORK( 1 ), |
|
|
567 $ RWORK( N ), U( 1, LL ), LDU ) |
|
|
568 IF( NCC.GT.0 ) |
|
|
569 $ CALL ZLASR( 'L', 'V', 'B', M-LL+1, NCC, RWORK( 1 ), |
|
|
570 $ RWORK( N ), C( LL, 1 ), LDC ) |
|
|
571 * |
|
|
572 * Test convergence |
|
|
573 * |
|
|
574 IF( ABS( E( LL ) ).LE.THRESH ) |
|
|
575 $ E( LL ) = ZERO |
|
|
576 END IF |
|
|
577 ELSE |
|
|
578 * |
|
|
579 * Use nonzero shift |
|
|
580 * |
|
|
581 IF( IDIR.EQ.1 ) THEN |
|
|
582 * |
|
|
583 * Chase bulge from top to bottom |
|
|
584 * Save cosines and sines for later singular vector updates |
|
|
585 * |
|
|
586 F = ( ABS( D( LL ) )-SHIFT )* |
|
|
587 $ ( SIGN( ONE, D( LL ) )+SHIFT / D( LL ) ) |
|
|
588 G = E( LL ) |
|
3333
|
589 DO 140 I = LL, M - 1 |
|
2329
|
590 CALL DLARTG( F, G, COSR, SINR, R ) |
|
3333
|
591 IF( I.GT.LL ) |
|
|
592 $ E( I-1 ) = R |
|
2329
|
593 F = COSR*D( I ) + SINR*E( I ) |
|
|
594 E( I ) = COSR*E( I ) - SINR*D( I ) |
|
|
595 G = SINR*D( I+1 ) |
|
|
596 D( I+1 ) = COSR*D( I+1 ) |
|
|
597 CALL DLARTG( F, G, COSL, SINL, R ) |
|
|
598 D( I ) = R |
|
|
599 F = COSL*E( I ) + SINL*D( I+1 ) |
|
|
600 D( I+1 ) = COSL*D( I+1 ) - SINL*E( I ) |
|
3333
|
601 IF( I.LT.M-1 ) THEN |
|
|
602 G = SINL*E( I+1 ) |
|
|
603 E( I+1 ) = COSL*E( I+1 ) |
|
|
604 END IF |
|
|
605 RWORK( I-LL+1 ) = COSR |
|
|
606 RWORK( I-LL+1+NM1 ) = SINR |
|
|
607 RWORK( I-LL+1+NM12 ) = COSL |
|
|
608 RWORK( I-LL+1+NM13 ) = SINL |
|
|
609 140 CONTINUE |
|
2329
|
610 E( M-1 ) = F |
|
|
611 * |
|
|
612 * Update singular vectors |
|
|
613 * |
|
|
614 IF( NCVT.GT.0 ) |
|
|
615 $ CALL ZLASR( 'L', 'V', 'F', M-LL+1, NCVT, RWORK( 1 ), |
|
|
616 $ RWORK( N ), VT( LL, 1 ), LDVT ) |
|
|
617 IF( NRU.GT.0 ) |
|
|
618 $ CALL ZLASR( 'R', 'V', 'F', NRU, M-LL+1, RWORK( NM12+1 ), |
|
|
619 $ RWORK( NM13+1 ), U( 1, LL ), LDU ) |
|
|
620 IF( NCC.GT.0 ) |
|
|
621 $ CALL ZLASR( 'L', 'V', 'F', M-LL+1, NCC, RWORK( NM12+1 ), |
|
|
622 $ RWORK( NM13+1 ), C( LL, 1 ), LDC ) |
|
|
623 * |
|
|
624 * Test convergence |
|
|
625 * |
|
|
626 IF( ABS( E( M-1 ) ).LE.THRESH ) |
|
|
627 $ E( M-1 ) = ZERO |
|
|
628 * |
|
|
629 ELSE |
|
|
630 * |
|
|
631 * Chase bulge from bottom to top |
|
|
632 * Save cosines and sines for later singular vector updates |
|
|
633 * |
|
|
634 F = ( ABS( D( M ) )-SHIFT )*( SIGN( ONE, D( M ) )+SHIFT / |
|
|
635 $ D( M ) ) |
|
|
636 G = E( M-1 ) |
|
3333
|
637 DO 150 I = M, LL + 1, -1 |
|
2329
|
638 CALL DLARTG( F, G, COSR, SINR, R ) |
|
3333
|
639 IF( I.LT.M ) |
|
|
640 $ E( I ) = R |
|
2329
|
641 F = COSR*D( I ) + SINR*E( I-1 ) |
|
|
642 E( I-1 ) = COSR*E( I-1 ) - SINR*D( I ) |
|
|
643 G = SINR*D( I-1 ) |
|
|
644 D( I-1 ) = COSR*D( I-1 ) |
|
|
645 CALL DLARTG( F, G, COSL, SINL, R ) |
|
|
646 D( I ) = R |
|
|
647 F = COSL*E( I-1 ) + SINL*D( I-1 ) |
|
|
648 D( I-1 ) = COSL*D( I-1 ) - SINL*E( I-1 ) |
|
3333
|
649 IF( I.GT.LL+1 ) THEN |
|
|
650 G = SINL*E( I-2 ) |
|
|
651 E( I-2 ) = COSL*E( I-2 ) |
|
|
652 END IF |
|
|
653 RWORK( I-LL ) = COSR |
|
|
654 RWORK( I-LL+NM1 ) = -SINR |
|
|
655 RWORK( I-LL+NM12 ) = COSL |
|
|
656 RWORK( I-LL+NM13 ) = -SINL |
|
|
657 150 CONTINUE |
|
2329
|
658 E( LL ) = F |
|
|
659 * |
|
|
660 * Test convergence |
|
|
661 * |
|
|
662 IF( ABS( E( LL ) ).LE.THRESH ) |
|
|
663 $ E( LL ) = ZERO |
|
|
664 * |
|
|
665 * Update singular vectors if desired |
|
|
666 * |
|
|
667 IF( NCVT.GT.0 ) |
|
|
668 $ CALL ZLASR( 'L', 'V', 'B', M-LL+1, NCVT, RWORK( NM12+1 ), |
|
|
669 $ RWORK( NM13+1 ), VT( LL, 1 ), LDVT ) |
|
|
670 IF( NRU.GT.0 ) |
|
|
671 $ CALL ZLASR( 'R', 'V', 'B', NRU, M-LL+1, RWORK( 1 ), |
|
|
672 $ RWORK( N ), U( 1, LL ), LDU ) |
|
|
673 IF( NCC.GT.0 ) |
|
|
674 $ CALL ZLASR( 'L', 'V', 'B', M-LL+1, NCC, RWORK( 1 ), |
|
|
675 $ RWORK( N ), C( LL, 1 ), LDC ) |
|
|
676 END IF |
|
|
677 END IF |
|
|
678 * |
|
|
679 * QR iteration finished, go back and check convergence |
|
|
680 * |
|
3333
|
681 GO TO 60 |
|
2329
|
682 * |
|
|
683 * All singular values converged, so make them positive |
|
|
684 * |
|
3333
|
685 160 CONTINUE |
|
|
686 DO 170 I = 1, N |
|
2329
|
687 IF( D( I ).LT.ZERO ) THEN |
|
|
688 D( I ) = -D( I ) |
|
|
689 * |
|
|
690 * Change sign of singular vectors, if desired |
|
|
691 * |
|
|
692 IF( NCVT.GT.0 ) |
|
|
693 $ CALL ZDSCAL( NCVT, NEGONE, VT( I, 1 ), LDVT ) |
|
|
694 END IF |
|
3333
|
695 170 CONTINUE |
|
2329
|
696 * |
|
|
697 * Sort the singular values into decreasing order (insertion sort on |
|
|
698 * singular values, but only one transposition per singular vector) |
|
|
699 * |
|
3333
|
700 DO 190 I = 1, N - 1 |
|
2329
|
701 * |
|
|
702 * Scan for smallest D(I) |
|
|
703 * |
|
|
704 ISUB = 1 |
|
|
705 SMIN = D( 1 ) |
|
3333
|
706 DO 180 J = 2, N + 1 - I |
|
2329
|
707 IF( D( J ).LE.SMIN ) THEN |
|
|
708 ISUB = J |
|
|
709 SMIN = D( J ) |
|
|
710 END IF |
|
3333
|
711 180 CONTINUE |
|
2329
|
712 IF( ISUB.NE.N+1-I ) THEN |
|
|
713 * |
|
|
714 * Swap singular values and vectors |
|
|
715 * |
|
|
716 D( ISUB ) = D( N+1-I ) |
|
|
717 D( N+1-I ) = SMIN |
|
|
718 IF( NCVT.GT.0 ) |
|
|
719 $ CALL ZSWAP( NCVT, VT( ISUB, 1 ), LDVT, VT( N+1-I, 1 ), |
|
|
720 $ LDVT ) |
|
|
721 IF( NRU.GT.0 ) |
|
|
722 $ CALL ZSWAP( NRU, U( 1, ISUB ), 1, U( 1, N+1-I ), 1 ) |
|
|
723 IF( NCC.GT.0 ) |
|
|
724 $ CALL ZSWAP( NCC, C( ISUB, 1 ), LDC, C( N+1-I, 1 ), LDC ) |
|
|
725 END IF |
|
3333
|
726 190 CONTINUE |
|
|
727 GO TO 220 |
|
2329
|
728 * |
|
|
729 * Maximum number of iterations exceeded, failure to converge |
|
|
730 * |
|
3333
|
731 200 CONTINUE |
|
2329
|
732 INFO = 0 |
|
3333
|
733 DO 210 I = 1, N - 1 |
|
2329
|
734 IF( E( I ).NE.ZERO ) |
|
|
735 $ INFO = INFO + 1 |
|
|
736 210 CONTINUE |
|
3333
|
737 220 CONTINUE |
|
2329
|
738 RETURN |
|
|
739 * |
|
|
740 * End of ZBDSQR |
|
|
741 * |
|
|
742 END |
