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1 SUBROUTINE ZGEBD2( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO ) |
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2 * |
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3 * -- LAPACK routine (version 3.1) -- |
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4 * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. |
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5 * November 2006 |
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6 * |
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7 * .. Scalar Arguments .. |
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8 INTEGER INFO, LDA, M, N |
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9 * .. |
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10 * .. Array Arguments .. |
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11 DOUBLE PRECISION D( * ), E( * ) |
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12 COMPLEX*16 A( LDA, * ), TAUP( * ), TAUQ( * ), WORK( * ) |
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13 * .. |
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14 * |
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15 * Purpose |
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16 * ======= |
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17 * |
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18 * ZGEBD2 reduces a complex general m by n matrix A to upper or lower |
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19 * real bidiagonal form B by a unitary transformation: Q' * A * P = B. |
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20 * |
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21 * If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal. |
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22 * |
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23 * Arguments |
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24 * ========= |
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25 * |
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26 * M (input) INTEGER |
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27 * The number of rows in the matrix A. M >= 0. |
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28 * |
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29 * N (input) INTEGER |
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30 * The number of columns in the matrix A. N >= 0. |
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31 * |
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32 * A (input/output) COMPLEX*16 array, dimension (LDA,N) |
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33 * On entry, the m by n general matrix to be reduced. |
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34 * On exit, |
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35 * if m >= n, the diagonal and the first superdiagonal are |
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36 * overwritten with the upper bidiagonal matrix B; the |
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37 * elements below the diagonal, with the array TAUQ, represent |
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38 * the unitary matrix Q as a product of elementary |
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39 * reflectors, and the elements above the first superdiagonal, |
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40 * with the array TAUP, represent the unitary matrix P as |
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41 * a product of elementary reflectors; |
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42 * if m < n, the diagonal and the first subdiagonal are |
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43 * overwritten with the lower bidiagonal matrix B; the |
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44 * elements below the first subdiagonal, with the array TAUQ, |
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45 * represent the unitary matrix Q as a product of |
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46 * elementary reflectors, and the elements above the diagonal, |
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47 * with the array TAUP, represent the unitary matrix P as |
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48 * a product of elementary reflectors. |
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49 * See Further Details. |
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50 * |
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51 * LDA (input) INTEGER |
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52 * The leading dimension of the array A. LDA >= max(1,M). |
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53 * |
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54 * D (output) DOUBLE PRECISION array, dimension (min(M,N)) |
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55 * The diagonal elements of the bidiagonal matrix B: |
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56 * D(i) = A(i,i). |
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57 * |
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58 * E (output) DOUBLE PRECISION array, dimension (min(M,N)-1) |
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59 * The off-diagonal elements of the bidiagonal matrix B: |
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60 * if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1; |
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61 * if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1. |
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62 * |
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63 * TAUQ (output) COMPLEX*16 array dimension (min(M,N)) |
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64 * The scalar factors of the elementary reflectors which |
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65 * represent the unitary matrix Q. See Further Details. |
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66 * |
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67 * TAUP (output) COMPLEX*16 array, dimension (min(M,N)) |
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68 * The scalar factors of the elementary reflectors which |
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69 * represent the unitary matrix P. See Further Details. |
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70 * |
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71 * WORK (workspace) COMPLEX*16 array, dimension (max(M,N)) |
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72 * |
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73 * INFO (output) INTEGER |
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74 * = 0: successful exit |
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75 * < 0: if INFO = -i, the i-th argument had an illegal value. |
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76 * |
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77 * Further Details |
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78 * =============== |
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79 * |
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80 * The matrices Q and P are represented as products of elementary |
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81 * reflectors: |
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82 * |
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83 * If m >= n, |
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84 * |
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85 * Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1) |
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86 * |
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87 * Each H(i) and G(i) has the form: |
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88 * |
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89 * H(i) = I - tauq * v * v' and G(i) = I - taup * u * u' |
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90 * |
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91 * where tauq and taup are complex scalars, and v and u are complex |
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92 * vectors; v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in |
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93 * A(i+1:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in |
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94 * A(i,i+2:n); tauq is stored in TAUQ(i) and taup in TAUP(i). |
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95 * |
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96 * If m < n, |
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97 * |
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98 * Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m) |
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99 * |
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100 * Each H(i) and G(i) has the form: |
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101 * |
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102 * H(i) = I - tauq * v * v' and G(i) = I - taup * u * u' |
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103 * |
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104 * where tauq and taup are complex scalars, v and u are complex vectors; |
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105 * v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i); |
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106 * u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n); |
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107 * tauq is stored in TAUQ(i) and taup in TAUP(i). |
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108 * |
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109 * The contents of A on exit are illustrated by the following examples: |
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110 * |
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111 * m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n): |
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112 * |
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113 * ( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 ) |
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114 * ( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 ) |
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115 * ( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 ) |
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116 * ( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 ) |
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117 * ( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 ) |
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118 * ( v1 v2 v3 v4 v5 ) |
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119 * |
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120 * where d and e denote diagonal and off-diagonal elements of B, vi |
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121 * denotes an element of the vector defining H(i), and ui an element of |
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122 * the vector defining G(i). |
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123 * |
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124 * ===================================================================== |
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125 * |
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126 * .. Parameters .. |
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127 COMPLEX*16 ZERO, ONE |
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128 PARAMETER ( ZERO = ( 0.0D+0, 0.0D+0 ), |
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129 $ ONE = ( 1.0D+0, 0.0D+0 ) ) |
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130 * .. |
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131 * .. Local Scalars .. |
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132 INTEGER I |
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133 COMPLEX*16 ALPHA |
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134 * .. |
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135 * .. External Subroutines .. |
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136 EXTERNAL XERBLA, ZLACGV, ZLARF, ZLARFG |
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137 * .. |
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138 * .. Intrinsic Functions .. |
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139 INTRINSIC DCONJG, MAX, MIN |
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140 * .. |
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141 * .. Executable Statements .. |
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142 * |
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143 * Test the input parameters |
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144 * |
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145 INFO = 0 |
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146 IF( M.LT.0 ) THEN |
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147 INFO = -1 |
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148 ELSE IF( N.LT.0 ) THEN |
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149 INFO = -2 |
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150 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN |
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151 INFO = -4 |
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152 END IF |
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153 IF( INFO.LT.0 ) THEN |
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154 CALL XERBLA( 'ZGEBD2', -INFO ) |
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155 RETURN |
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156 END IF |
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157 * |
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158 IF( M.GE.N ) THEN |
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159 * |
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160 * Reduce to upper bidiagonal form |
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161 * |
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162 DO 10 I = 1, N |
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163 * |
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164 * Generate elementary reflector H(i) to annihilate A(i+1:m,i) |
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165 * |
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166 ALPHA = A( I, I ) |
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167 CALL ZLARFG( M-I+1, ALPHA, A( MIN( I+1, M ), I ), 1, |
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168 $ TAUQ( I ) ) |
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169 D( I ) = ALPHA |
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170 A( I, I ) = ONE |
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171 * |
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172 * Apply H(i)' to A(i:m,i+1:n) from the left |
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173 * |
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174 IF( I.LT.N ) |
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175 $ CALL ZLARF( 'Left', M-I+1, N-I, A( I, I ), 1, |
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176 $ DCONJG( TAUQ( I ) ), A( I, I+1 ), LDA, WORK ) |
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177 A( I, I ) = D( I ) |
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178 * |
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179 IF( I.LT.N ) THEN |
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180 * |
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181 * Generate elementary reflector G(i) to annihilate |
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182 * A(i,i+2:n) |
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183 * |
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184 CALL ZLACGV( N-I, A( I, I+1 ), LDA ) |
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185 ALPHA = A( I, I+1 ) |
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186 CALL ZLARFG( N-I, ALPHA, A( I, MIN( I+2, N ) ), LDA, |
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187 $ TAUP( I ) ) |
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188 E( I ) = ALPHA |
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189 A( I, I+1 ) = ONE |
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190 * |
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191 * Apply G(i) to A(i+1:m,i+1:n) from the right |
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192 * |
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193 CALL ZLARF( 'Right', M-I, N-I, A( I, I+1 ), LDA, |
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194 $ TAUP( I ), A( I+1, I+1 ), LDA, WORK ) |
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195 CALL ZLACGV( N-I, A( I, I+1 ), LDA ) |
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196 A( I, I+1 ) = E( I ) |
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197 ELSE |
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198 TAUP( I ) = ZERO |
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199 END IF |
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200 10 CONTINUE |
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201 ELSE |
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202 * |
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203 * Reduce to lower bidiagonal form |
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204 * |
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205 DO 20 I = 1, M |
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206 * |
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207 * Generate elementary reflector G(i) to annihilate A(i,i+1:n) |
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208 * |
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209 CALL ZLACGV( N-I+1, A( I, I ), LDA ) |
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210 ALPHA = A( I, I ) |
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211 CALL ZLARFG( N-I+1, ALPHA, A( I, MIN( I+1, N ) ), LDA, |
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212 $ TAUP( I ) ) |
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213 D( I ) = ALPHA |
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214 A( I, I ) = ONE |
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215 * |
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216 * Apply G(i) to A(i+1:m,i:n) from the right |
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217 * |
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218 IF( I.LT.M ) |
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219 $ CALL ZLARF( 'Right', M-I, N-I+1, A( I, I ), LDA, |
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220 $ TAUP( I ), A( I+1, I ), LDA, WORK ) |
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221 CALL ZLACGV( N-I+1, A( I, I ), LDA ) |
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222 A( I, I ) = D( I ) |
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223 * |
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224 IF( I.LT.M ) THEN |
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225 * |
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226 * Generate elementary reflector H(i) to annihilate |
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227 * A(i+2:m,i) |
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228 * |
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229 ALPHA = A( I+1, I ) |
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230 CALL ZLARFG( M-I, ALPHA, A( MIN( I+2, M ), I ), 1, |
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231 $ TAUQ( I ) ) |
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232 E( I ) = ALPHA |
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233 A( I+1, I ) = ONE |
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234 * |
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235 * Apply H(i)' to A(i+1:m,i+1:n) from the left |
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236 * |
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237 CALL ZLARF( 'Left', M-I, N-I, A( I+1, I ), 1, |
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238 $ DCONJG( TAUQ( I ) ), A( I+1, I+1 ), LDA, |
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239 $ WORK ) |
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240 A( I+1, I ) = E( I ) |
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241 ELSE |
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242 TAUQ( I ) = ZERO |
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243 END IF |
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244 20 CONTINUE |
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245 END IF |
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246 RETURN |
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247 * |
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248 * End of ZGEBD2 |
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249 * |
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250 END |
