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annotate libcruft/lapack/sgehrd.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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1 SUBROUTINE SGEHRD( N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO )
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2 *
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3 * -- LAPACK routine (version 3.1) --
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4 * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
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5 * November 2006
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6 *
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7 * .. Scalar Arguments ..
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8 INTEGER IHI, ILO, INFO, LDA, LWORK, N
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9 * ..
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10 * .. Array Arguments ..
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11 REAL A( LDA, * ), TAU( * ), WORK( * )
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12 * ..
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13 *
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14 * Purpose
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15 * =======
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16 *
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17 * SGEHRD reduces a real general matrix A to upper Hessenberg form H by
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18 * an orthogonal similarity transformation: Q' * A * Q = H .
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19 *
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20 * Arguments
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21 * =========
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22 *
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23 * N (input) INTEGER
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24 * The order of the matrix A. N >= 0.
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25 *
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26 * ILO (input) INTEGER
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27 * IHI (input) INTEGER
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28 * It is assumed that A is already upper triangular in rows
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29 * and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
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30 * set by a previous call to SGEBAL; otherwise they should be
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31 * set to 1 and N respectively. See Further Details.
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32 * 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
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33 *
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34 * A (input/output) REAL array, dimension (LDA,N)
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35 * On entry, the N-by-N general matrix to be reduced.
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36 * On exit, the upper triangle and the first subdiagonal of A
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37 * are overwritten with the upper Hessenberg matrix H, and the
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38 * elements below the first subdiagonal, with the array TAU,
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39 * represent the orthogonal matrix Q as a product of elementary
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40 * reflectors. See Further Details.
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41 *
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42 * LDA (input) INTEGER
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43 * The leading dimension of the array A. LDA >= max(1,N).
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44 *
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45 * TAU (output) REAL array, dimension (N-1)
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46 * The scalar factors of the elementary reflectors (see Further
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47 * Details). Elements 1:ILO-1 and IHI:N-1 of TAU are set to
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48 * zero.
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49 *
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50 * WORK (workspace/output) REAL array, dimension (LWORK)
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51 * On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
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52 *
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53 * LWORK (input) INTEGER
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54 * The length of the array WORK. LWORK >= max(1,N).
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55 * For optimum performance LWORK >= N*NB, where NB is the
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56 * optimal blocksize.
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57 *
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58 * If LWORK = -1, then a workspace query is assumed; the routine
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59 * only calculates the optimal size of the WORK array, returns
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60 * this value as the first entry of the WORK array, and no error
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61 * message related to LWORK is issued by XERBLA.
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62 *
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63 * INFO (output) INTEGER
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64 * = 0: successful exit
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65 * < 0: if INFO = -i, the i-th argument had an illegal value.
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66 *
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67 * Further Details
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68 * ===============
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69 *
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70 * The matrix Q is represented as a product of (ihi-ilo) elementary
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71 * reflectors
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72 *
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73 * Q = H(ilo) H(ilo+1) . . . H(ihi-1).
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74 *
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75 * Each H(i) has the form
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76 *
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77 * H(i) = I - tau * v * v'
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78 *
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79 * where tau is a real scalar, and v is a real vector with
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80 * v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on
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81 * exit in A(i+2:ihi,i), and tau in TAU(i).
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82 *
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83 * The contents of A are illustrated by the following example, with
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84 * n = 7, ilo = 2 and ihi = 6:
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85 *
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86 * on entry, on exit,
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87 *
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88 * ( a a a a a a a ) ( a a h h h h a )
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89 * ( a a a a a a ) ( a h h h h a )
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90 * ( a a a a a a ) ( h h h h h h )
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91 * ( a a a a a a ) ( v2 h h h h h )
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92 * ( a a a a a a ) ( v2 v3 h h h h )
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93 * ( a a a a a a ) ( v2 v3 v4 h h h )
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94 * ( a ) ( a )
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95 *
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96 * where a denotes an element of the original matrix A, h denotes a
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97 * modified element of the upper Hessenberg matrix H, and vi denotes an
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98 * element of the vector defining H(i).
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99 *
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100 * This file is a slight modification of LAPACK-3.0's SGEHRD
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101 * subroutine incorporating improvements proposed by Quintana-Orti and
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102 * Van de Geijn (2005).
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103 *
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104 * =====================================================================
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105 *
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106 * .. Parameters ..
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107 INTEGER NBMAX, LDT
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108 PARAMETER ( NBMAX = 64, LDT = NBMAX+1 )
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109 REAL ZERO, ONE
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110 PARAMETER ( ZERO = 0.0E+0,
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111 $ ONE = 1.0E+0 )
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112 * ..
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113 * .. Local Scalars ..
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114 LOGICAL LQUERY
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115 INTEGER I, IB, IINFO, IWS, J, LDWORK, LWKOPT, NB,
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116 $ NBMIN, NH, NX
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117 REAL EI
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118 * ..
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119 * .. Local Arrays ..
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120 REAL T( LDT, NBMAX )
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121 * ..
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122 * .. External Subroutines ..
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123 EXTERNAL SAXPY, SGEHD2, SGEMM, SLAHR2, SLARFB, STRMM,
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124 $ XERBLA
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125 * ..
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126 * .. Intrinsic Functions ..
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127 INTRINSIC MAX, MIN
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128 * ..
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129 * .. External Functions ..
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130 INTEGER ILAENV
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131 EXTERNAL ILAENV
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132 * ..
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133 * .. Executable Statements ..
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134 *
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135 * Test the input parameters
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136 *
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137 INFO = 0
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138 NB = MIN( NBMAX, ILAENV( 1, 'SGEHRD', ' ', N, ILO, IHI, -1 ) )
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139 LWKOPT = N*NB
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140 WORK( 1 ) = LWKOPT
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141 LQUERY = ( LWORK.EQ.-1 )
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142 IF( N.LT.0 ) THEN
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143 INFO = -1
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144 ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN
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145 INFO = -2
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146 ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN
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147 INFO = -3
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148 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
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149 INFO = -5
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150 ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN
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151 INFO = -8
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152 END IF
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153 IF( INFO.NE.0 ) THEN
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154 CALL XERBLA( 'SGEHRD', -INFO )
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155 RETURN
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156 ELSE IF( LQUERY ) THEN
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157 RETURN
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158 END IF
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159 *
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160 * Set elements 1:ILO-1 and IHI:N-1 of TAU to zero
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161 *
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162 DO 10 I = 1, ILO - 1
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163 TAU( I ) = ZERO
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164 10 CONTINUE
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165 DO 20 I = MAX( 1, IHI ), N - 1
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166 TAU( I ) = ZERO
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167 20 CONTINUE
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168 *
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169 * Quick return if possible
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170 *
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171 NH = IHI - ILO + 1
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172 IF( NH.LE.1 ) THEN
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173 WORK( 1 ) = 1
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174 RETURN
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175 END IF
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176 *
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177 * Determine the block size
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178 *
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179 NB = MIN( NBMAX, ILAENV( 1, 'SGEHRD', ' ', N, ILO, IHI, -1 ) )
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180 NBMIN = 2
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181 IWS = 1
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182 IF( NB.GT.1 .AND. NB.LT.NH ) THEN
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183 *
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184 * Determine when to cross over from blocked to unblocked code
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185 * (last block is always handled by unblocked code)
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186 *
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187 NX = MAX( NB, ILAENV( 3, 'SGEHRD', ' ', N, ILO, IHI, -1 ) )
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188 IF( NX.LT.NH ) THEN
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189 *
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190 * Determine if workspace is large enough for blocked code
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191 *
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192 IWS = N*NB
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193 IF( LWORK.LT.IWS ) THEN
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194 *
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195 * Not enough workspace to use optimal NB: determine the
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196 * minimum value of NB, and reduce NB or force use of
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197 * unblocked code
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198 *
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199 NBMIN = MAX( 2, ILAENV( 2, 'SGEHRD', ' ', N, ILO, IHI,
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200 $ -1 ) )
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201 IF( LWORK.GE.N*NBMIN ) THEN
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202 NB = LWORK / N
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203 ELSE
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204 NB = 1
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205 END IF
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206 END IF
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207 END IF
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208 END IF
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209 LDWORK = N
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210 *
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211 IF( NB.LT.NBMIN .OR. NB.GE.NH ) THEN
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212 *
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213 * Use unblocked code below
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214 *
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215 I = ILO
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216 *
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217 ELSE
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218 *
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219 * Use blocked code
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220 *
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221 DO 40 I = ILO, IHI - 1 - NX, NB
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222 IB = MIN( NB, IHI-I )
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223 *
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224 * Reduce columns i:i+ib-1 to Hessenberg form, returning the
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225 * matrices V and T of the block reflector H = I - V*T*V'
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226 * which performs the reduction, and also the matrix Y = A*V*T
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227 *
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228 CALL SLAHR2( IHI, I, IB, A( 1, I ), LDA, TAU( I ), T, LDT,
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229 $ WORK, LDWORK )
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230 *
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231 * Apply the block reflector H to A(1:ihi,i+ib:ihi) from the
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232 * right, computing A := A - Y * V'. V(i+ib,ib-1) must be set
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233 * to 1
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234 *
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235 EI = A( I+IB, I+IB-1 )
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236 A( I+IB, I+IB-1 ) = ONE
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237 CALL SGEMM( 'No transpose', 'Transpose',
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238 $ IHI, IHI-I-IB+1,
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239 $ IB, -ONE, WORK, LDWORK, A( I+IB, I ), LDA, ONE,
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240 $ A( 1, I+IB ), LDA )
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241 A( I+IB, I+IB-1 ) = EI
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242 *
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243 * Apply the block reflector H to A(1:i,i+1:i+ib-1) from the
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244 * right
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245 *
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246 CALL STRMM( 'Right', 'Lower', 'Transpose',
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247 $ 'Unit', I, IB-1,
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248 $ ONE, A( I+1, I ), LDA, WORK, LDWORK )
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249 DO 30 J = 0, IB-2
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250 CALL SAXPY( I, -ONE, WORK( LDWORK*J+1 ), 1,
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251 $ A( 1, I+J+1 ), 1 )
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252 30 CONTINUE
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253 *
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254 * Apply the block reflector H to A(i+1:ihi,i+ib:n) from the
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255 * left
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256 *
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257 CALL SLARFB( 'Left', 'Transpose', 'Forward',
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258 $ 'Columnwise',
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259 $ IHI-I, N-I-IB+1, IB, A( I+1, I ), LDA, T, LDT,
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260 $ A( I+1, I+IB ), LDA, WORK, LDWORK )
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261 40 CONTINUE
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262 END IF
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263 *
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264 * Use unblocked code to reduce the rest of the matrix
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265 *
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266 CALL SGEHD2( N, I, IHI, A, LDA, TAU, WORK, IINFO )
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267 WORK( 1 ) = IWS
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268 *
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269 RETURN
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270 *
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271 * End of SGEHRD
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272 *
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273 END