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1 SUBROUTINE ZGEBRD( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK, |
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2 $ INFO ) |
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3 * |
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4 * -- LAPACK routine (version 2.0) -- |
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5 * Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., |
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6 * Courant Institute, Argonne National Lab, and Rice University |
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7 * September 30, 1994 |
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8 * |
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9 * .. Scalar Arguments .. |
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10 INTEGER INFO, LDA, LWORK, M, N |
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11 * .. |
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12 * .. Array Arguments .. |
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13 DOUBLE PRECISION D( * ), E( * ) |
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14 COMPLEX*16 A( LDA, * ), TAUP( * ), TAUQ( * ), |
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15 $ WORK( LWORK ) |
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16 * .. |
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17 * |
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18 * Purpose |
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19 * ======= |
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20 * |
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21 * ZGEBRD reduces a general complex M-by-N matrix A to upper or lower |
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22 * bidiagonal form B by a unitary transformation: Q**H * A * P = B. |
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23 * |
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24 * If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal. |
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25 * |
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26 * Arguments |
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27 * ========= |
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28 * |
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29 * M (input) INTEGER |
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30 * The number of rows in the matrix A. M >= 0. |
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31 * |
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32 * N (input) INTEGER |
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33 * The number of columns in the matrix A. N >= 0. |
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34 * |
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35 * A (input/output) COMPLEX*16 array, dimension (LDA,N) |
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36 * On entry, the M-by-N general matrix to be reduced. |
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37 * On exit, |
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38 * if m >= n, the diagonal and the first superdiagonal are |
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39 * overwritten with the upper bidiagonal matrix B; the |
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40 * elements below the diagonal, with the array TAUQ, represent |
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41 * the unitary matrix Q as a product of elementary |
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42 * reflectors, and the elements above the first superdiagonal, |
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43 * with the array TAUP, represent the unitary matrix P as |
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44 * a product of elementary reflectors; |
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45 * if m < n, the diagonal and the first subdiagonal are |
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46 * overwritten with the lower bidiagonal matrix B; the |
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47 * elements below the first subdiagonal, with the array TAUQ, |
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48 * represent the unitary matrix Q as a product of |
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49 * elementary reflectors, and the elements above the diagonal, |
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50 * with the array TAUP, represent the unitary matrix P as |
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51 * a product of elementary reflectors. |
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52 * See Further Details. |
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53 * |
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54 * LDA (input) INTEGER |
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55 * The leading dimension of the array A. LDA >= max(1,M). |
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56 * |
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57 * D (output) DOUBLE PRECISION array, dimension (min(M,N)) |
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58 * The diagonal elements of the bidiagonal matrix B: |
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59 * D(i) = A(i,i). |
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60 * |
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61 * E (output) DOUBLE PRECISION array, dimension (min(M,N)-1) |
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62 * The off-diagonal elements of the bidiagonal matrix B: |
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63 * if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1; |
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64 * if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1. |
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65 * |
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66 * TAUQ (output) COMPLEX*16 array dimension (min(M,N)) |
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67 * The scalar factors of the elementary reflectors which |
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68 * represent the unitary matrix Q. See Further Details. |
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69 * |
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70 * TAUP (output) COMPLEX*16 array, dimension (min(M,N)) |
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71 * The scalar factors of the elementary reflectors which |
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72 * represent the unitary matrix P. See Further Details. |
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73 * |
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74 * WORK (workspace/output) COMPLEX*16 array, dimension (LWORK) |
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75 * On exit, if INFO = 0, WORK(1) returns the optimal LWORK. |
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76 * |
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77 * LWORK (input) INTEGER |
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78 * The length of the array WORK. LWORK >= max(1,M,N). |
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79 * For optimum performance LWORK >= (M+N)*NB, where NB |
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80 * is the optimal blocksize. |
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81 * |
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82 * INFO (output) INTEGER |
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83 * = 0: successful exit. |
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84 * < 0: if INFO = -i, the i-th argument had an illegal value. |
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85 * |
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86 * Further Details |
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87 * =============== |
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88 * |
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89 * The matrices Q and P are represented as products of elementary |
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90 * reflectors: |
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91 * |
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92 * If m >= n, |
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93 * |
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94 * Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1) |
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95 * |
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96 * Each H(i) and G(i) has the form: |
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97 * |
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98 * H(i) = I - tauq * v * v' and G(i) = I - taup * u * u' |
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99 * |
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100 * where tauq and taup are complex scalars, and v and u are complex |
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101 * vectors; v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in |
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102 * 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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103 * A(i,i+2:n); tauq is stored in TAUQ(i) and taup in TAUP(i). |
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104 * |
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105 * If m < n, |
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106 * |
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107 * Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m) |
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108 * |
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109 * Each H(i) and G(i) has the form: |
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110 * |
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111 * H(i) = I - tauq * v * v' and G(i) = I - taup * u * u' |
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112 * |
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113 * where tauq and taup are complex scalars, and v and u are complex |
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114 * vectors; v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in |
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115 * A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in |
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116 * A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i). |
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117 * |
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118 * The contents of A on exit are illustrated by the following examples: |
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119 * |
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120 * m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n): |
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121 * |
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122 * ( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 ) |
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123 * ( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 ) |
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124 * ( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 ) |
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125 * ( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 ) |
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126 * ( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 ) |
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127 * ( v1 v2 v3 v4 v5 ) |
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128 * |
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129 * where d and e denote diagonal and off-diagonal elements of B, vi |
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130 * denotes an element of the vector defining H(i), and ui an element of |
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131 * the vector defining G(i). |
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132 * |
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133 * ===================================================================== |
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134 * |
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135 * .. Parameters .. |
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136 COMPLEX*16 ONE |
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137 PARAMETER ( ONE = ( 1.0D+0, 0.0D+0 ) ) |
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138 * .. |
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139 * .. Local Scalars .. |
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140 INTEGER I, IINFO, J, LDWRKX, LDWRKY, MINMN, NB, NBMIN, |
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141 $ NX |
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142 DOUBLE PRECISION WS |
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143 * .. |
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144 * .. External Subroutines .. |
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145 EXTERNAL XERBLA, ZGEBD2, ZGEMM, ZLABRD |
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146 * .. |
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147 * .. Intrinsic Functions .. |
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148 INTRINSIC MAX, MIN |
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149 * .. |
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150 * .. External Functions .. |
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151 INTEGER ILAENV |
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152 EXTERNAL ILAENV |
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153 * .. |
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154 * .. Executable Statements .. |
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155 * |
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156 * Test the input parameters |
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157 * |
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158 INFO = 0 |
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159 IF( M.LT.0 ) THEN |
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160 INFO = -1 |
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161 ELSE IF( N.LT.0 ) THEN |
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162 INFO = -2 |
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163 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN |
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164 INFO = -4 |
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165 ELSE IF( LWORK.LT.MAX( 1, M, N ) ) THEN |
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166 INFO = -10 |
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167 END IF |
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168 IF( INFO.LT.0 ) THEN |
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169 CALL XERBLA( 'ZGEBRD', -INFO ) |
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170 RETURN |
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171 END IF |
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172 * |
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173 * Quick return if possible |
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174 * |
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175 MINMN = MIN( M, N ) |
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176 IF( MINMN.EQ.0 ) THEN |
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177 WORK( 1 ) = 1 |
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178 RETURN |
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179 END IF |
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180 * |
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181 WS = MAX( M, N ) |
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182 LDWRKX = M |
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183 LDWRKY = N |
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184 * |
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185 * Set the block size NB and the crossover point NX. |
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186 * |
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187 NB = MAX( 1, ILAENV( 1, 'ZGEBRD', ' ', M, N, -1, -1 ) ) |
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188 * |
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189 IF( NB.GT.1 .AND. NB.LT.MINMN ) THEN |
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190 * |
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191 * Determine when to switch from blocked to unblocked code. |
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192 * |
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193 NX = MAX( NB, ILAENV( 3, 'ZGEBRD', ' ', M, N, -1, -1 ) ) |
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194 IF( NX.LT.MINMN ) THEN |
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195 WS = ( M+N )*NB |
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196 IF( LWORK.LT.WS ) THEN |
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197 * |
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198 * Not enough work space for the optimal NB, consider using |
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199 * a smaller block size. |
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200 * |
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201 NBMIN = ILAENV( 2, 'ZGEBRD', ' ', M, N, -1, -1 ) |
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202 IF( LWORK.GE.( M+N )*NBMIN ) THEN |
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203 NB = LWORK / ( M+N ) |
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204 ELSE |
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205 NB = 1 |
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206 NX = MINMN |
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207 END IF |
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208 END IF |
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209 END IF |
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210 ELSE |
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211 NX = MINMN |
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212 END IF |
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213 * |
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214 DO 30 I = 1, MINMN - NX, NB |
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215 * |
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216 * Reduce rows and columns i:i+ib-1 to bidiagonal form and return |
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217 * the matrices X and Y which are needed to update the unreduced |
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218 * part of the matrix |
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219 * |
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220 CALL ZLABRD( M-I+1, N-I+1, NB, A( I, I ), LDA, D( I ), E( I ), |
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221 $ TAUQ( I ), TAUP( I ), WORK, LDWRKX, |
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222 $ WORK( LDWRKX*NB+1 ), LDWRKY ) |
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223 * |
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224 * Update the trailing submatrix A(i+ib:m,i+ib:n), using |
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225 * an update of the form A := A - V*Y' - X*U' |
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226 * |
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227 CALL ZGEMM( 'No transpose', 'Conjugate transpose', M-I-NB+1, |
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228 $ N-I-NB+1, NB, -ONE, A( I+NB, I ), LDA, |
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229 $ WORK( LDWRKX*NB+NB+1 ), LDWRKY, ONE, |
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230 $ A( I+NB, I+NB ), LDA ) |
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231 CALL ZGEMM( 'No transpose', 'No transpose', M-I-NB+1, N-I-NB+1, |
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232 $ NB, -ONE, WORK( NB+1 ), LDWRKX, A( I, I+NB ), LDA, |
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233 $ ONE, A( I+NB, I+NB ), LDA ) |
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234 * |
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235 * Copy diagonal and off-diagonal elements of B back into A |
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236 * |
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237 IF( M.GE.N ) THEN |
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238 DO 10 J = I, I + NB - 1 |
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239 A( J, J ) = D( J ) |
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240 A( J, J+1 ) = E( J ) |
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241 10 CONTINUE |
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242 ELSE |
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243 DO 20 J = I, I + NB - 1 |
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244 A( J, J ) = D( J ) |
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245 A( J+1, J ) = E( J ) |
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246 20 CONTINUE |
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247 END IF |
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248 30 CONTINUE |
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249 * |
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250 * Use unblocked code to reduce the remainder of the matrix |
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251 * |
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252 CALL ZGEBD2( M-I+1, N-I+1, A( I, I ), LDA, D( I ), E( I ), |
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253 $ TAUQ( I ), TAUP( I ), WORK, IINFO ) |
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254 WORK( 1 ) = WS |
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255 RETURN |
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256 * |
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257 * End of ZGEBRD |
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258 * |
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259 END |
