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1 SUBROUTINE DGEBD2( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO ) |
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2 * |
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3 * -- LAPACK routine (version 3.0) -- |
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4 * Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., |
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5 * Courant Institute, Argonne National Lab, and Rice University |
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6 * February 29, 1992 |
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7 * |
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8 * .. Scalar Arguments .. |
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9 INTEGER INFO, LDA, M, N |
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10 * .. |
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11 * .. Array Arguments .. |
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12 DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAUP( * ), |
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13 $ TAUQ( * ), WORK( * ) |
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14 * .. |
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15 * |
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16 * Purpose |
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17 * ======= |
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18 * |
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19 * DGEBD2 reduces a real general m by n matrix A to upper or lower |
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20 * bidiagonal form B by an orthogonal transformation: Q' * A * P = B. |
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21 * |
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22 * If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal. |
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23 * |
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24 * Arguments |
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25 * ========= |
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26 * |
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27 * M (input) INTEGER |
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28 * The number of rows in the matrix A. M >= 0. |
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29 * |
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30 * N (input) INTEGER |
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31 * The number of columns in the matrix A. N >= 0. |
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32 * |
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33 * A (input/output) DOUBLE PRECISION array, dimension (LDA,N) |
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34 * On entry, the m by n general matrix to be reduced. |
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35 * On exit, |
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36 * if m >= n, the diagonal and the first superdiagonal are |
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37 * overwritten with the upper bidiagonal matrix B; the |
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38 * elements below the diagonal, with the array TAUQ, represent |
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39 * the orthogonal matrix Q as a product of elementary |
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40 * reflectors, and the elements above the first superdiagonal, |
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41 * with the array TAUP, represent the orthogonal matrix P as |
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42 * a product of elementary reflectors; |
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43 * if m < n, the diagonal and the first subdiagonal are |
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44 * overwritten with the lower bidiagonal matrix B; the |
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45 * elements below the first subdiagonal, with the array TAUQ, |
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46 * represent the orthogonal matrix Q as a product of |
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47 * elementary reflectors, and the elements above the diagonal, |
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48 * with the array TAUP, represent the orthogonal matrix P as |
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49 * a product of elementary reflectors. |
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50 * See Further Details. |
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51 * |
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52 * LDA (input) INTEGER |
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53 * The leading dimension of the array A. LDA >= max(1,M). |
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54 * |
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55 * D (output) DOUBLE PRECISION array, dimension (min(M,N)) |
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56 * The diagonal elements of the bidiagonal matrix B: |
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57 * D(i) = A(i,i). |
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58 * |
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59 * E (output) DOUBLE PRECISION array, dimension (min(M,N)-1) |
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60 * The off-diagonal elements of the bidiagonal matrix B: |
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61 * if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1; |
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62 * if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1. |
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63 * |
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64 * TAUQ (output) DOUBLE PRECISION array dimension (min(M,N)) |
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65 * The scalar factors of the elementary reflectors which |
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66 * represent the orthogonal matrix Q. See Further Details. |
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67 * |
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68 * TAUP (output) DOUBLE PRECISION array, dimension (min(M,N)) |
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69 * The scalar factors of the elementary reflectors which |
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70 * represent the orthogonal matrix P. See Further Details. |
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71 * |
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72 * WORK (workspace) DOUBLE PRECISION array, dimension (max(M,N)) |
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73 * |
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74 * INFO (output) INTEGER |
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75 * = 0: successful exit. |
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76 * < 0: if INFO = -i, the i-th argument had an illegal value. |
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77 * |
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78 * Further Details |
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79 * =============== |
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80 * |
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81 * The matrices Q and P are represented as products of elementary |
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82 * reflectors: |
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83 * |
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84 * If m >= n, |
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85 * |
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86 * Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1) |
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87 * |
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88 * Each H(i) and G(i) has the form: |
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89 * |
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90 * H(i) = I - tauq * v * v' and G(i) = I - taup * u * u' |
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91 * |
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92 * where tauq and taup are real scalars, and v and u are real vectors; |
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93 * v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(i+1:m,i); |
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94 * u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(i,i+2:n); |
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95 * tauq is stored in TAUQ(i) and taup in TAUP(i). |
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96 * |
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97 * If m < n, |
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98 * |
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99 * Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m) |
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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 real scalars, and v and u are real vectors; |
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106 * 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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107 * 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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108 * tauq is stored in TAUQ(i) and taup in TAUP(i). |
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109 * |
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110 * The contents of A on exit are illustrated by the following examples: |
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111 * |
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112 * m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n): |
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113 * |
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114 * ( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 ) |
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115 * ( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 ) |
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116 * ( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 ) |
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117 * ( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 ) |
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118 * ( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 ) |
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119 * ( v1 v2 v3 v4 v5 ) |
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120 * |
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121 * where d and e denote diagonal and off-diagonal elements of B, vi |
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122 * denotes an element of the vector defining H(i), and ui an element of |
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123 * the vector defining G(i). |
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124 * |
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125 * ===================================================================== |
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126 * |
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127 * .. Parameters .. |
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128 DOUBLE PRECISION ZERO, ONE |
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129 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+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 * .. |
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134 * .. External Subroutines .. |
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135 EXTERNAL DLARF, DLARFG, XERBLA |
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136 * .. |
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137 * .. Intrinsic Functions .. |
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138 INTRINSIC MAX, MIN |
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139 * .. |
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140 * .. Executable Statements .. |
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141 * |
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142 * Test the input parameters |
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143 * |
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144 INFO = 0 |
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145 IF( M.LT.0 ) THEN |
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146 INFO = -1 |
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147 ELSE IF( N.LT.0 ) THEN |
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148 INFO = -2 |
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149 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN |
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150 INFO = -4 |
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151 END IF |
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152 IF( INFO.LT.0 ) THEN |
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153 CALL XERBLA( 'DGEBD2', -INFO ) |
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154 RETURN |
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155 END IF |
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156 * |
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157 IF( M.GE.N ) THEN |
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158 * |
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159 * Reduce to upper bidiagonal form |
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160 * |
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161 DO 10 I = 1, N |
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162 * |
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163 * Generate elementary reflector H(i) to annihilate A(i+1:m,i) |
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164 * |
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165 CALL DLARFG( M-I+1, A( I, I ), A( MIN( I+1, M ), I ), 1, |
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166 $ TAUQ( I ) ) |
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167 D( I ) = A( I, I ) |
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168 A( I, I ) = ONE |
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169 * |
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170 * Apply H(i) to A(i:m,i+1:n) from the left |
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171 * |
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172 CALL DLARF( 'Left', M-I+1, N-I, A( I, I ), 1, TAUQ( I ), |
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173 $ A( I, I+1 ), LDA, WORK ) |
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174 A( I, I ) = D( I ) |
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175 * |
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176 IF( I.LT.N ) THEN |
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177 * |
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178 * Generate elementary reflector G(i) to annihilate |
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179 * A(i,i+2:n) |
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180 * |
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181 CALL DLARFG( N-I, A( I, I+1 ), A( I, MIN( I+2, N ) ), |
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182 $ LDA, TAUP( I ) ) |
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183 E( I ) = A( I, I+1 ) |
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184 A( I, I+1 ) = ONE |
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185 * |
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186 * Apply G(i) to A(i+1:m,i+1:n) from the right |
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187 * |
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188 CALL DLARF( 'Right', M-I, N-I, A( I, I+1 ), LDA, |
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189 $ TAUP( I ), A( I+1, I+1 ), LDA, WORK ) |
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190 A( I, I+1 ) = E( I ) |
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191 ELSE |
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192 TAUP( I ) = ZERO |
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193 END IF |
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194 10 CONTINUE |
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195 ELSE |
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196 * |
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197 * Reduce to lower bidiagonal form |
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198 * |
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199 DO 20 I = 1, M |
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200 * |
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201 * Generate elementary reflector G(i) to annihilate A(i,i+1:n) |
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202 * |
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203 CALL DLARFG( N-I+1, A( I, I ), A( I, MIN( I+1, N ) ), LDA, |
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204 $ TAUP( I ) ) |
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205 D( I ) = A( I, I ) |
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206 A( I, I ) = ONE |
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207 * |
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208 * Apply G(i) to A(i+1:m,i:n) from the right |
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209 * |
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210 CALL DLARF( 'Right', M-I, N-I+1, A( I, I ), LDA, TAUP( I ), |
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211 $ A( MIN( I+1, M ), I ), LDA, WORK ) |
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212 A( I, I ) = D( I ) |
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213 * |
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214 IF( I.LT.M ) THEN |
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215 * |
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216 * Generate elementary reflector H(i) to annihilate |
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217 * A(i+2:m,i) |
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218 * |
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219 CALL DLARFG( M-I, A( I+1, I ), A( MIN( I+2, M ), I ), 1, |
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220 $ TAUQ( I ) ) |
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221 E( I ) = A( I+1, I ) |
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222 A( I+1, I ) = ONE |
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223 * |
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224 * Apply H(i) to A(i+1:m,i+1:n) from the left |
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225 * |
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226 CALL DLARF( 'Left', M-I, N-I, A( I+1, I ), 1, TAUQ( I ), |
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227 $ A( I+1, I+1 ), LDA, WORK ) |
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228 A( I+1, I ) = E( I ) |
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229 ELSE |
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230 TAUQ( I ) = ZERO |
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231 END IF |
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232 20 CONTINUE |
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233 END IF |
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234 RETURN |
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235 * |
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236 * End of DGEBD2 |
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237 * |
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238 END |
