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1 SUBROUTINE ZLABRD( M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y, |
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2 $ LDY ) |
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3 * |
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4 * -- LAPACK auxiliary 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 INTEGER LDA, LDX, LDY, M, N, NB |
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10 * .. |
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11 * .. Array Arguments .. |
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12 DOUBLE PRECISION D( * ), E( * ) |
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13 COMPLEX*16 A( LDA, * ), TAUP( * ), TAUQ( * ), X( LDX, * ), |
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14 $ Y( LDY, * ) |
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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 * ZLABRD reduces the first NB rows and columns of a complex general |
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21 * m by n matrix A to upper or lower real bidiagonal form by a unitary |
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22 * transformation Q' * A * P, and returns the matrices X and Y which |
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23 * are needed to apply the transformation to the unreduced part of A. |
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24 * |
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25 * If m >= n, A is reduced to upper bidiagonal form; if m < n, to lower |
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26 * bidiagonal form. |
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27 * |
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28 * This is an auxiliary routine called by ZGEBRD |
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29 * |
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30 * Arguments |
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31 * ========= |
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32 * |
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33 * M (input) INTEGER |
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34 * The number of rows in the matrix A. |
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35 * |
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36 * N (input) INTEGER |
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37 * The number of columns in the matrix A. |
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38 * |
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39 * NB (input) INTEGER |
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40 * The number of leading rows and columns of A to be reduced. |
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41 * |
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42 * A (input/output) COMPLEX*16 array, dimension (LDA,N) |
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43 * On entry, the m by n general matrix to be reduced. |
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44 * On exit, the first NB rows and columns of the matrix are |
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45 * overwritten; the rest of the array is unchanged. |
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46 * If m >= n, elements on and below the diagonal in the first NB |
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47 * columns, with the array TAUQ, represent the unitary |
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48 * matrix Q as a product of elementary reflectors; and |
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49 * elements above the diagonal in the first NB rows, with the |
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50 * array TAUP, represent the unitary matrix P as a product |
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51 * of elementary reflectors. |
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52 * If m < n, elements below the diagonal in the first NB |
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53 * columns, with the array TAUQ, represent the unitary |
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54 * matrix Q as a product of elementary reflectors, and |
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55 * elements on and above the diagonal in the first NB rows, |
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56 * with the array TAUP, represent the unitary matrix P as |
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57 * a product of elementary reflectors. |
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58 * See Further Details. |
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59 * |
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60 * LDA (input) INTEGER |
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61 * The leading dimension of the array A. LDA >= max(1,M). |
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62 * |
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63 * D (output) DOUBLE PRECISION array, dimension (NB) |
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64 * The diagonal elements of the first NB rows and columns of |
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65 * the reduced matrix. D(i) = A(i,i). |
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66 * |
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67 * E (output) DOUBLE PRECISION array, dimension (NB) |
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68 * The off-diagonal elements of the first NB rows and columns of |
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69 * the reduced matrix. |
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70 * |
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71 * TAUQ (output) COMPLEX*16 array dimension (NB) |
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72 * The scalar factors of the elementary reflectors which |
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73 * represent the unitary matrix Q. See Further Details. |
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74 * |
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75 * TAUP (output) COMPLEX*16 array, dimension (NB) |
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76 * The scalar factors of the elementary reflectors which |
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77 * represent the unitary matrix P. See Further Details. |
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78 * |
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79 * X (output) COMPLEX*16 array, dimension (LDX,NB) |
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80 * The m-by-nb matrix X required to update the unreduced part |
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81 * of A. |
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82 * |
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83 * LDX (input) INTEGER |
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84 * The leading dimension of the array X. LDX >= max(1,M). |
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85 * |
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86 * Y (output) COMPLEX*16 array, dimension (LDY,NB) |
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87 * The n-by-nb matrix Y required to update the unreduced part |
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88 * of A. |
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89 * |
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90 * LDY (input) INTEGER |
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91 * The leading dimension of the array Y. LDY >= max(1,N). |
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92 * |
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93 * Further Details |
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94 * =============== |
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95 * |
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96 * The matrices Q and P are represented as products of elementary |
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97 * reflectors: |
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98 * |
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99 * Q = H(1) H(2) . . . H(nb) and P = G(1) G(2) . . . G(nb) |
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100 * |
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101 * Each H(i) and G(i) has the form: |
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102 * |
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103 * H(i) = I - tauq * v * v' and G(i) = I - taup * u * u' |
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104 * |
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105 * where tauq and taup are complex scalars, and v and u are complex |
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106 * vectors. |
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107 * |
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108 * If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in |
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109 * A(i:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+1:n) is stored on exit in |
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110 * A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i). |
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111 * |
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112 * If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in |
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113 * A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in |
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114 * A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i). |
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115 * |
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116 * The elements of the vectors v and u together form the m-by-nb matrix |
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117 * V and the nb-by-n matrix U' which are needed, with X and Y, to apply |
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118 * the transformation to the unreduced part of the matrix, using a block |
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119 * update of the form: A := A - V*Y' - X*U'. |
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120 * |
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121 * The contents of A on exit are illustrated by the following examples |
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122 * with nb = 2: |
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123 * |
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124 * m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n): |
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125 * |
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126 * ( 1 1 u1 u1 u1 ) ( 1 u1 u1 u1 u1 u1 ) |
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127 * ( v1 1 1 u2 u2 ) ( 1 1 u2 u2 u2 u2 ) |
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128 * ( v1 v2 a a a ) ( v1 1 a a a a ) |
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129 * ( v1 v2 a a a ) ( v1 v2 a a a a ) |
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130 * ( v1 v2 a a a ) ( v1 v2 a a a a ) |
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131 * ( v1 v2 a a a ) |
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132 * |
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133 * where a denotes an element of the original matrix which is unchanged, |
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134 * vi denotes an element of the vector defining H(i), and ui an element |
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135 * of the vector defining G(i). |
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136 * |
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137 * ===================================================================== |
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138 * |
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139 * .. Parameters .. |
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140 COMPLEX*16 ZERO, ONE |
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141 PARAMETER ( ZERO = ( 0.0D+0, 0.0D+0 ), |
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142 $ ONE = ( 1.0D+0, 0.0D+0 ) ) |
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143 * .. |
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144 * .. Local Scalars .. |
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145 INTEGER I |
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146 COMPLEX*16 ALPHA |
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147 * .. |
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148 * .. External Subroutines .. |
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149 EXTERNAL ZGEMV, ZLACGV, ZLARFG, ZSCAL |
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150 * .. |
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151 * .. Intrinsic Functions .. |
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152 INTRINSIC MIN |
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153 * .. |
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154 * .. Executable Statements .. |
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155 * |
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156 * Quick return if possible |
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157 * |
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158 IF( M.LE.0 .OR. N.LE.0 ) |
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159 $ RETURN |
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160 * |
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161 IF( M.GE.N ) THEN |
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162 * |
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163 * Reduce to upper bidiagonal form |
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164 * |
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165 DO 10 I = 1, NB |
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166 * |
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167 * Update A(i:m,i) |
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168 * |
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169 CALL ZLACGV( I-1, Y( I, 1 ), LDY ) |
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170 CALL ZGEMV( 'No transpose', M-I+1, I-1, -ONE, A( I, 1 ), |
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171 $ LDA, Y( I, 1 ), LDY, ONE, A( I, I ), 1 ) |
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172 CALL ZLACGV( I-1, Y( I, 1 ), LDY ) |
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173 CALL ZGEMV( 'No transpose', M-I+1, I-1, -ONE, X( I, 1 ), |
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174 $ LDX, A( 1, I ), 1, ONE, A( I, I ), 1 ) |
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175 * |
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176 * Generate reflection Q(i) to annihilate A(i+1:m,i) |
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177 * |
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178 ALPHA = A( I, I ) |
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179 CALL ZLARFG( M-I+1, ALPHA, A( MIN( I+1, M ), I ), 1, |
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180 $ TAUQ( I ) ) |
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181 D( I ) = ALPHA |
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182 IF( I.LT.N ) THEN |
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183 A( I, I ) = ONE |
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184 * |
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185 * Compute Y(i+1:n,i) |
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186 * |
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187 CALL ZGEMV( 'Conjugate transpose', M-I+1, N-I, ONE, |
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188 $ A( I, I+1 ), LDA, A( I, I ), 1, ZERO, |
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189 $ Y( I+1, I ), 1 ) |
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190 CALL ZGEMV( 'Conjugate transpose', M-I+1, I-1, ONE, |
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191 $ A( I, 1 ), LDA, A( I, I ), 1, ZERO, |
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192 $ Y( 1, I ), 1 ) |
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193 CALL ZGEMV( 'No transpose', N-I, I-1, -ONE, Y( I+1, 1 ), |
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194 $ LDY, Y( 1, I ), 1, ONE, Y( I+1, I ), 1 ) |
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195 CALL ZGEMV( 'Conjugate transpose', M-I+1, I-1, ONE, |
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196 $ X( I, 1 ), LDX, A( I, I ), 1, ZERO, |
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197 $ Y( 1, I ), 1 ) |
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198 CALL ZGEMV( 'Conjugate transpose', I-1, N-I, -ONE, |
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199 $ A( 1, I+1 ), LDA, Y( 1, I ), 1, ONE, |
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200 $ Y( I+1, I ), 1 ) |
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201 CALL ZSCAL( N-I, TAUQ( I ), Y( I+1, I ), 1 ) |
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202 * |
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203 * Update A(i,i+1:n) |
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204 * |
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205 CALL ZLACGV( N-I, A( I, I+1 ), LDA ) |
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206 CALL ZLACGV( I, A( I, 1 ), LDA ) |
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207 CALL ZGEMV( 'No transpose', N-I, I, -ONE, Y( I+1, 1 ), |
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208 $ LDY, A( I, 1 ), LDA, ONE, A( I, I+1 ), LDA ) |
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209 CALL ZLACGV( I, A( I, 1 ), LDA ) |
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210 CALL ZLACGV( I-1, X( I, 1 ), LDX ) |
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211 CALL ZGEMV( 'Conjugate transpose', I-1, N-I, -ONE, |
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212 $ A( 1, I+1 ), LDA, X( I, 1 ), LDX, ONE, |
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213 $ A( I, I+1 ), LDA ) |
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214 CALL ZLACGV( I-1, X( I, 1 ), LDX ) |
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215 * |
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216 * Generate reflection P(i) to annihilate A(i,i+2:n) |
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217 * |
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218 ALPHA = A( I, I+1 ) |
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219 CALL ZLARFG( N-I, ALPHA, A( I, MIN( I+2, N ) ), LDA, |
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220 $ TAUP( I ) ) |
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221 E( I ) = ALPHA |
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222 A( I, I+1 ) = ONE |
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223 * |
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224 * Compute X(i+1:m,i) |
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225 * |
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226 CALL ZGEMV( 'No transpose', M-I, N-I, ONE, A( I+1, I+1 ), |
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227 $ LDA, A( I, I+1 ), LDA, ZERO, X( I+1, I ), 1 ) |
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228 CALL ZGEMV( 'Conjugate transpose', N-I, I, ONE, |
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229 $ Y( I+1, 1 ), LDY, A( I, I+1 ), LDA, ZERO, |
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230 $ X( 1, I ), 1 ) |
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231 CALL ZGEMV( 'No transpose', M-I, I, -ONE, A( I+1, 1 ), |
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232 $ LDA, X( 1, I ), 1, ONE, X( I+1, I ), 1 ) |
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233 CALL ZGEMV( 'No transpose', I-1, N-I, ONE, A( 1, I+1 ), |
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234 $ LDA, A( I, I+1 ), LDA, ZERO, X( 1, I ), 1 ) |
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235 CALL ZGEMV( 'No transpose', M-I, I-1, -ONE, X( I+1, 1 ), |
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236 $ LDX, X( 1, I ), 1, ONE, X( I+1, I ), 1 ) |
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237 CALL ZSCAL( M-I, TAUP( I ), X( I+1, I ), 1 ) |
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238 CALL ZLACGV( N-I, A( I, I+1 ), LDA ) |
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239 END IF |
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240 10 CONTINUE |
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241 ELSE |
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242 * |
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243 * Reduce to lower bidiagonal form |
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244 * |
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245 DO 20 I = 1, NB |
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246 * |
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247 * Update A(i,i:n) |
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248 * |
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249 CALL ZLACGV( N-I+1, A( I, I ), LDA ) |
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250 CALL ZLACGV( I-1, A( I, 1 ), LDA ) |
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251 CALL ZGEMV( 'No transpose', N-I+1, I-1, -ONE, Y( I, 1 ), |
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252 $ LDY, A( I, 1 ), LDA, ONE, A( I, I ), LDA ) |
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253 CALL ZLACGV( I-1, A( I, 1 ), LDA ) |
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254 CALL ZLACGV( I-1, X( I, 1 ), LDX ) |
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255 CALL ZGEMV( 'Conjugate transpose', I-1, N-I+1, -ONE, |
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256 $ A( 1, I ), LDA, X( I, 1 ), LDX, ONE, A( I, I ), |
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257 $ LDA ) |
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258 CALL ZLACGV( I-1, X( I, 1 ), LDX ) |
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259 * |
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260 * Generate reflection P(i) to annihilate A(i,i+1:n) |
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261 * |
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262 ALPHA = A( I, I ) |
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263 CALL ZLARFG( N-I+1, ALPHA, A( I, MIN( I+1, N ) ), LDA, |
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264 $ TAUP( I ) ) |
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265 D( I ) = ALPHA |
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266 IF( I.LT.M ) THEN |
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267 A( I, I ) = ONE |
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268 * |
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269 * Compute X(i+1:m,i) |
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270 * |
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271 CALL ZGEMV( 'No transpose', M-I, N-I+1, ONE, A( I+1, I ), |
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272 $ LDA, A( I, I ), LDA, ZERO, X( I+1, I ), 1 ) |
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273 CALL ZGEMV( 'Conjugate transpose', N-I+1, I-1, ONE, |
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274 $ Y( I, 1 ), LDY, A( I, I ), LDA, ZERO, |
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275 $ X( 1, I ), 1 ) |
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276 CALL ZGEMV( 'No transpose', M-I, I-1, -ONE, A( I+1, 1 ), |
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277 $ LDA, X( 1, I ), 1, ONE, X( I+1, I ), 1 ) |
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278 CALL ZGEMV( 'No transpose', I-1, N-I+1, ONE, A( 1, I ), |
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279 $ LDA, A( I, I ), LDA, ZERO, X( 1, I ), 1 ) |
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280 CALL ZGEMV( 'No transpose', M-I, I-1, -ONE, X( I+1, 1 ), |
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281 $ LDX, X( 1, I ), 1, ONE, X( I+1, I ), 1 ) |
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282 CALL ZSCAL( M-I, TAUP( I ), X( I+1, I ), 1 ) |
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283 CALL ZLACGV( N-I+1, A( I, I ), LDA ) |
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284 * |
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285 * Update A(i+1:m,i) |
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286 * |
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287 CALL ZLACGV( I-1, Y( I, 1 ), LDY ) |
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288 CALL ZGEMV( 'No transpose', M-I, I-1, -ONE, A( I+1, 1 ), |
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289 $ LDA, Y( I, 1 ), LDY, ONE, A( I+1, I ), 1 ) |
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290 CALL ZLACGV( I-1, Y( I, 1 ), LDY ) |
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291 CALL ZGEMV( 'No transpose', M-I, I, -ONE, X( I+1, 1 ), |
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292 $ LDX, A( 1, I ), 1, ONE, A( I+1, I ), 1 ) |
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293 * |
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294 * Generate reflection Q(i) to annihilate A(i+2:m,i) |
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295 * |
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296 ALPHA = A( I+1, I ) |
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297 CALL ZLARFG( M-I, ALPHA, A( MIN( I+2, M ), I ), 1, |
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298 $ TAUQ( I ) ) |
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299 E( I ) = ALPHA |
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300 A( I+1, I ) = ONE |
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301 * |
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302 * Compute Y(i+1:n,i) |
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303 * |
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304 CALL ZGEMV( 'Conjugate transpose', M-I, N-I, ONE, |
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305 $ A( I+1, I+1 ), LDA, A( I+1, I ), 1, ZERO, |
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306 $ Y( I+1, I ), 1 ) |
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307 CALL ZGEMV( 'Conjugate transpose', M-I, I-1, ONE, |
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308 $ A( I+1, 1 ), LDA, A( I+1, I ), 1, ZERO, |
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309 $ Y( 1, I ), 1 ) |
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310 CALL ZGEMV( 'No transpose', N-I, I-1, -ONE, Y( I+1, 1 ), |
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311 $ LDY, Y( 1, I ), 1, ONE, Y( I+1, I ), 1 ) |
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312 CALL ZGEMV( 'Conjugate transpose', M-I, I, ONE, |
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313 $ X( I+1, 1 ), LDX, A( I+1, I ), 1, ZERO, |
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314 $ Y( 1, I ), 1 ) |
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315 CALL ZGEMV( 'Conjugate transpose', I, N-I, -ONE, |
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316 $ A( 1, I+1 ), LDA, Y( 1, I ), 1, ONE, |
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317 $ Y( I+1, I ), 1 ) |
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318 CALL ZSCAL( N-I, TAUQ( I ), Y( I+1, I ), 1 ) |
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319 ELSE |
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320 CALL ZLACGV( N-I+1, A( I, I ), LDA ) |
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321 END IF |
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322 20 CONTINUE |
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323 END IF |
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324 RETURN |
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325 * |
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326 * End of ZLABRD |
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327 * |
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328 END |
