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annotate libcruft/lapack/sgelsy.f @ 7789:82be108cc558

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First attempt at single precision tyeps * * * corrections to qrupdate single precision routines * * * prefer demotion to single over promotion to double * * * Add single precision support to log2 function * * * Trivial PROJECT file update * * * Cache optimized hermitian/transpose methods * * * Add tests for tranpose/hermitian and ChangeLog entry for new transpose code
author David Bateman <dbateman@free.fr>
date Sun, 27 Apr 2008 22:34:17 +0200
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7789
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1 SUBROUTINE SGELSY( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK,
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2 $ WORK, LWORK, INFO )
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3 *
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4 * -- LAPACK driver 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 INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
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10 REAL RCOND
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11 * ..
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12 * .. Array Arguments ..
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13 INTEGER JPVT( * )
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14 REAL A( LDA, * ), B( LDB, * ), WORK( * )
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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 * SGELSY computes the minimum-norm solution to a real linear least
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21 * squares problem:
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22 * minimize || A * X - B ||
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23 * using a complete orthogonal factorization of A. A is an M-by-N
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24 * matrix which may be rank-deficient.
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25 *
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26 * Several right hand side vectors b and solution vectors x can be
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27 * handled in a single call; they are stored as the columns of the
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28 * M-by-NRHS right hand side matrix B and the N-by-NRHS solution
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29 * matrix X.
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30 *
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31 * The routine first computes a QR factorization with column pivoting:
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32 * A * P = Q * [ R11 R12 ]
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33 * [ 0 R22 ]
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34 * with R11 defined as the largest leading submatrix whose estimated
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35 * condition number is less than 1/RCOND. The order of R11, RANK,
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36 * is the effective rank of A.
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37 *
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38 * Then, R22 is considered to be negligible, and R12 is annihilated
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39 * by orthogonal transformations from the right, arriving at the
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40 * complete orthogonal factorization:
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41 * A * P = Q * [ T11 0 ] * Z
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42 * [ 0 0 ]
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43 * The minimum-norm solution is then
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44 * X = P * Z' [ inv(T11)*Q1'*B ]
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45 * [ 0 ]
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46 * where Q1 consists of the first RANK columns of Q.
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47 *
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48 * This routine is basically identical to the original xGELSX except
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49 * three differences:
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50 * o The call to the subroutine xGEQPF has been substituted by the
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51 * the call to the subroutine xGEQP3. This subroutine is a Blas-3
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52 * version of the QR factorization with column pivoting.
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53 * o Matrix B (the right hand side) is updated with Blas-3.
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54 * o The permutation of matrix B (the right hand side) is faster and
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55 * more simple.
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56 *
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57 * Arguments
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58 * =========
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59 *
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60 * M (input) INTEGER
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61 * The number of rows of the matrix A. M >= 0.
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62 *
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63 * N (input) INTEGER
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64 * The number of columns of the matrix A. N >= 0.
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65 *
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66 * NRHS (input) INTEGER
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67 * The number of right hand sides, i.e., the number of
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68 * columns of matrices B and X. NRHS >= 0.
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69 *
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70 * A (input/output) REAL array, dimension (LDA,N)
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71 * On entry, the M-by-N matrix A.
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72 * On exit, A has been overwritten by details of its
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73 * complete orthogonal factorization.
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74 *
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75 * LDA (input) INTEGER
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76 * The leading dimension of the array A. LDA >= max(1,M).
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77 *
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78 * B (input/output) REAL array, dimension (LDB,NRHS)
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79 * On entry, the M-by-NRHS right hand side matrix B.
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80 * On exit, the N-by-NRHS solution matrix X.
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81 *
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82 * LDB (input) INTEGER
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83 * The leading dimension of the array B. LDB >= max(1,M,N).
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84 *
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85 * JPVT (input/output) INTEGER array, dimension (N)
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86 * On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted
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87 * to the front of AP, otherwise column i is a free column.
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88 * On exit, if JPVT(i) = k, then the i-th column of AP
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89 * was the k-th column of A.
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90 *
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91 * RCOND (input) REAL
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92 * RCOND is used to determine the effective rank of A, which
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93 * is defined as the order of the largest leading triangular
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94 * submatrix R11 in the QR factorization with pivoting of A,
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95 * whose estimated condition number < 1/RCOND.
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96 *
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97 * RANK (output) INTEGER
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98 * The effective rank of A, i.e., the order of the submatrix
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99 * R11. This is the same as the order of the submatrix T11
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100 * in the complete orthogonal factorization of A.
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101 *
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102 * WORK (workspace/output) REAL array, dimension (MAX(1,LWORK))
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103 * On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
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104 *
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105 * LWORK (input) INTEGER
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106 * The dimension of the array WORK.
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107 * The unblocked strategy requires that:
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108 * LWORK >= MAX( MN+3*N+1, 2*MN+NRHS ),
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109 * where MN = min( M, N ).
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110 * The block algorithm requires that:
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111 * LWORK >= MAX( MN+2*N+NB*(N+1), 2*MN+NB*NRHS ),
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112 * where NB is an upper bound on the blocksize returned
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113 * by ILAENV for the routines SGEQP3, STZRZF, STZRQF, SORMQR,
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114 * and SORMRZ.
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115 *
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116 * If LWORK = -1, then a workspace query is assumed; the routine
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117 * only calculates the optimal size of the WORK array, returns
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118 * this value as the first entry of the WORK array, and no error
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119 * message related to LWORK is issued by XERBLA.
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120 *
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121 * INFO (output) INTEGER
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122 * = 0: successful exit
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123 * < 0: If INFO = -i, the i-th argument had an illegal value.
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124 *
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125 * Further Details
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126 * ===============
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127 *
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128 * Based on contributions by
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129 * A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
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130 * E. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
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131 * G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
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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 INTEGER IMAX, IMIN
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137 PARAMETER ( IMAX = 1, IMIN = 2 )
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138 REAL ZERO, ONE
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139 PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
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140 * ..
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141 * .. Local Scalars ..
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142 LOGICAL LQUERY
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143 INTEGER I, IASCL, IBSCL, ISMAX, ISMIN, J, LWKMIN,
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144 $ LWKOPT, MN, NB, NB1, NB2, NB3, NB4
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145 REAL ANRM, BIGNUM, BNRM, C1, C2, S1, S2, SMAX,
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146 $ SMAXPR, SMIN, SMINPR, SMLNUM, WSIZE
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147 * ..
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148 * .. External Functions ..
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149 INTEGER ILAENV
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150 REAL SLAMCH, SLANGE
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151 EXTERNAL ILAENV, SLAMCH, SLANGE
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152 * ..
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153 * .. External Subroutines ..
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154 EXTERNAL SCOPY, SGEQP3, SLABAD, SLAIC1, SLASCL, SLASET,
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155 $ SORMQR, SORMRZ, STRSM, STZRZF, XERBLA
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156 * ..
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157 * .. Intrinsic Functions ..
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158 INTRINSIC ABS, MAX, MIN
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159 * ..
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160 * .. Executable Statements ..
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161 *
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162 MN = MIN( M, N )
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163 ISMIN = MN + 1
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164 ISMAX = 2*MN + 1
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165 *
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166 * Test the input arguments.
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167 *
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168 INFO = 0
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169 LQUERY = ( LWORK.EQ.-1 )
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170 IF( M.LT.0 ) THEN
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171 INFO = -1
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172 ELSE IF( N.LT.0 ) THEN
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173 INFO = -2
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174 ELSE IF( NRHS.LT.0 ) THEN
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175 INFO = -3
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176 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
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177 INFO = -5
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178 ELSE IF( LDB.LT.MAX( 1, M, N ) ) THEN
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179 INFO = -7
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180 END IF
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181 *
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182 * Figure out optimal block size
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183 *
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184 IF( INFO.EQ.0 ) THEN
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185 IF( MN.EQ.0 .OR. NRHS.EQ.0 ) THEN
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186 LWKMIN = 1
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187 LWKOPT = 1
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188 ELSE
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189 NB1 = ILAENV( 1, 'SGEQRF', ' ', M, N, -1, -1 )
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190 NB2 = ILAENV( 1, 'SGERQF', ' ', M, N, -1, -1 )
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191 NB3 = ILAENV( 1, 'SORMQR', ' ', M, N, NRHS, -1 )
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192 NB4 = ILAENV( 1, 'SORMRQ', ' ', M, N, NRHS, -1 )
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193 NB = MAX( NB1, NB2, NB3, NB4 )
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194 LWKMIN = MN + MAX( 2*MN, N + 1, MN + NRHS )
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195 LWKOPT = MAX( LWKMIN,
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196 $ MN + 2*N + NB*( N + 1 ), 2*MN + NB*NRHS )
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197 END IF
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198 WORK( 1 ) = LWKOPT
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199 *
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200 IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
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201 INFO = -12
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202 END IF
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203 END IF
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204 *
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205 IF( INFO.NE.0 ) THEN
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206 CALL XERBLA( 'SGELSY', -INFO )
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207 RETURN
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208 ELSE IF( LQUERY ) THEN
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209 RETURN
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210 END IF
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211 *
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212 * Quick return if possible
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213 *
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214 IF( MN.EQ.0 .OR. NRHS.EQ.0 ) THEN
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215 RANK = 0
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216 RETURN
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217 END IF
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218 *
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219 * Get machine parameters
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220 *
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221 SMLNUM = SLAMCH( 'S' ) / SLAMCH( 'P' )
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222 BIGNUM = ONE / SMLNUM
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223 CALL SLABAD( SMLNUM, BIGNUM )
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224 *
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225 * Scale A, B if max entries outside range [SMLNUM,BIGNUM]
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226 *
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227 ANRM = SLANGE( 'M', M, N, A, LDA, WORK )
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228 IASCL = 0
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229 IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
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230 *
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231 * Scale matrix norm up to SMLNUM
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232 *
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233 CALL SLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, INFO )
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234 IASCL = 1
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235 ELSE IF( ANRM.GT.BIGNUM ) THEN
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236 *
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237 * Scale matrix norm down to BIGNUM
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238 *
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239 CALL SLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, INFO )
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240 IASCL = 2
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241 ELSE IF( ANRM.EQ.ZERO ) THEN
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242 *
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243 * Matrix all zero. Return zero solution.
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244 *
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245 CALL SLASET( 'F', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB )
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246 RANK = 0
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247 GO TO 70
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248 END IF
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249 *
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250 BNRM = SLANGE( 'M', M, NRHS, B, LDB, WORK )
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251 IBSCL = 0
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252 IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
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253 *
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254 * Scale matrix norm up to SMLNUM
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255 *
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256 CALL SLASCL( 'G', 0, 0, BNRM, SMLNUM, M, NRHS, B, LDB, INFO )
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257 IBSCL = 1
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258 ELSE IF( BNRM.GT.BIGNUM ) THEN
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259 *
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260 * Scale matrix norm down to BIGNUM
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261 *
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262 CALL SLASCL( 'G', 0, 0, BNRM, BIGNUM, M, NRHS, B, LDB, INFO )
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263 IBSCL = 2
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264 END IF
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265 *
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266 * Compute QR factorization with column pivoting of A:
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267 * A * P = Q * R
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268 *
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269 CALL SGEQP3( M, N, A, LDA, JPVT, WORK( 1 ), WORK( MN+1 ),
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270 $ LWORK-MN, INFO )
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271 WSIZE = MN + WORK( MN+1 )
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272 *
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273 * workspace: MN+2*N+NB*(N+1).
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274 * Details of Householder rotations stored in WORK(1:MN).
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275 *
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276 * Determine RANK using incremental condition estimation
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277 *
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278 WORK( ISMIN ) = ONE
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279 WORK( ISMAX ) = ONE
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280 SMAX = ABS( A( 1, 1 ) )
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281 SMIN = SMAX
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282 IF( ABS( A( 1, 1 ) ).EQ.ZERO ) THEN
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283 RANK = 0
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284 CALL SLASET( 'F', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB )
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285 GO TO 70
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286 ELSE
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287 RANK = 1
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288 END IF
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289 *
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290 10 CONTINUE
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291 IF( RANK.LT.MN ) THEN
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292 I = RANK + 1
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293 CALL SLAIC1( IMIN, RANK, WORK( ISMIN ), SMIN, A( 1, I ),
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294 $ A( I, I ), SMINPR, S1, C1 )
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295 CALL SLAIC1( IMAX, RANK, WORK( ISMAX ), SMAX, A( 1, I ),
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296 $ A( I, I ), SMAXPR, S2, C2 )
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297 *
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298 IF( SMAXPR*RCOND.LE.SMINPR ) THEN
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299 DO 20 I = 1, RANK
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300 WORK( ISMIN+I-1 ) = S1*WORK( ISMIN+I-1 )
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301 WORK( ISMAX+I-1 ) = S2*WORK( ISMAX+I-1 )
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302 20 CONTINUE
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303 WORK( ISMIN+RANK ) = C1
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304 WORK( ISMAX+RANK ) = C2
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305 SMIN = SMINPR
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306 SMAX = SMAXPR
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307 RANK = RANK + 1
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308 GO TO 10
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309 END IF
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310 END IF
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311 *
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312 * workspace: 3*MN.
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313 *
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314 * Logically partition R = [ R11 R12 ]
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315 * [ 0 R22 ]
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316 * where R11 = R(1:RANK,1:RANK)
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317 *
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318 * [R11,R12] = [ T11, 0 ] * Y
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319 *
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320 IF( RANK.LT.N )
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321 $ CALL STZRZF( RANK, N, A, LDA, WORK( MN+1 ), WORK( 2*MN+1 ),
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322 $ LWORK-2*MN, INFO )
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323 *
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324 * workspace: 2*MN.
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325 * Details of Householder rotations stored in WORK(MN+1:2*MN)
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326 *
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327 * B(1:M,1:NRHS) := Q' * B(1:M,1:NRHS)
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328 *
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329 CALL SORMQR( 'Left', 'Transpose', M, NRHS, MN, A, LDA, WORK( 1 ),
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330 $ B, LDB, WORK( 2*MN+1 ), LWORK-2*MN, INFO )
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331 WSIZE = MAX( WSIZE, 2*MN+WORK( 2*MN+1 ) )
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332 *
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333 * workspace: 2*MN+NB*NRHS.
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334 *
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335 * B(1:RANK,1:NRHS) := inv(T11) * B(1:RANK,1:NRHS)
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336 *
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337 CALL STRSM( 'Left', 'Upper', 'No transpose', 'Non-unit', RANK,
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338 $ NRHS, ONE, A, LDA, B, LDB )
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339 *
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340 DO 40 J = 1, NRHS
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341 DO 30 I = RANK + 1, N
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342 B( I, J ) = ZERO
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343 30 CONTINUE
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344 40 CONTINUE
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345 *
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346 * B(1:N,1:NRHS) := Y' * B(1:N,1:NRHS)
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347 *
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348 IF( RANK.LT.N ) THEN
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349 CALL SORMRZ( 'Left', 'Transpose', N, NRHS, RANK, N-RANK, A,
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350 $ LDA, WORK( MN+1 ), B, LDB, WORK( 2*MN+1 ),
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351 $ LWORK-2*MN, INFO )
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352 END IF
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353 *
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354 * workspace: 2*MN+NRHS.
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355 *
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356 * B(1:N,1:NRHS) := P * B(1:N,1:NRHS)
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357 *
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358 DO 60 J = 1, NRHS
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359 DO 50 I = 1, N
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360 WORK( JPVT( I ) ) = B( I, J )
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361 50 CONTINUE
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362 CALL SCOPY( N, WORK( 1 ), 1, B( 1, J ), 1 )
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363 60 CONTINUE
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364 *
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365 * workspace: N.
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366 *
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367 * Undo scaling
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368 *
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369 IF( IASCL.EQ.1 ) THEN
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370 CALL SLASCL( 'G', 0, 0, ANRM, SMLNUM, N, NRHS, B, LDB, INFO )
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371 CALL SLASCL( 'U', 0, 0, SMLNUM, ANRM, RANK, RANK, A, LDA,
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372 $ INFO )
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373 ELSE IF( IASCL.EQ.2 ) THEN
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374 CALL SLASCL( 'G', 0, 0, ANRM, BIGNUM, N, NRHS, B, LDB, INFO )
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375 CALL SLASCL( 'U', 0, 0, BIGNUM, ANRM, RANK, RANK, A, LDA,
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376 $ INFO )
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377 END IF
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378 IF( IBSCL.EQ.1 ) THEN
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379 CALL SLASCL( 'G', 0, 0, SMLNUM, BNRM, N, NRHS, B, LDB, INFO )
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380 ELSE IF( IBSCL.EQ.2 ) THEN
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381 CALL SLASCL( 'G', 0, 0, BIGNUM, BNRM, N, NRHS, B, LDB, INFO )
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382 END IF
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383 *
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384 70 CONTINUE
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385 WORK( 1 ) = LWKOPT
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386 *
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387 RETURN
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388 *
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389 * End of SGELSY
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390 *
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391 END