1*> \brief \b SQRT01P 2* 3* =========== DOCUMENTATION =========== 4* 5* Online html documentation available at 6* http://www.netlib.org/lapack/explore-html/ 7* 8* Definition: 9* =========== 10* 11* SUBROUTINE SQRT01P( M, N, A, AF, Q, R, LDA, TAU, WORK, LWORK, 12* RWORK, RESULT ) 13* 14* .. Scalar Arguments .. 15* INTEGER LDA, LWORK, M, N 16* .. 17* .. Array Arguments .. 18* REAL A( LDA, * ), AF( LDA, * ), Q( LDA, * ), 19* $ R( LDA, * ), RESULT( * ), RWORK( * ), TAU( * ), 20* $ WORK( LWORK ) 21* .. 22* 23* 24*> \par Purpose: 25* ============= 26*> 27*> \verbatim 28*> 29*> SQRT01P tests SGEQRFP, which computes the QR factorization of an m-by-n 30*> matrix A, and partially tests SORGQR which forms the m-by-m 31*> orthogonal matrix Q. 32*> 33*> SQRT01P compares R with Q'*A, and checks that Q is orthogonal. 34*> \endverbatim 35* 36* Arguments: 37* ========== 38* 39*> \param[in] M 40*> \verbatim 41*> M is INTEGER 42*> The number of rows of the matrix A. M >= 0. 43*> \endverbatim 44*> 45*> \param[in] N 46*> \verbatim 47*> N is INTEGER 48*> The number of columns of the matrix A. N >= 0. 49*> \endverbatim 50*> 51*> \param[in] A 52*> \verbatim 53*> A is REAL array, dimension (LDA,N) 54*> The m-by-n matrix A. 55*> \endverbatim 56*> 57*> \param[out] AF 58*> \verbatim 59*> AF is REAL array, dimension (LDA,N) 60*> Details of the QR factorization of A, as returned by SGEQRFP. 61*> See SGEQRFP for further details. 62*> \endverbatim 63*> 64*> \param[out] Q 65*> \verbatim 66*> Q is REAL array, dimension (LDA,M) 67*> The m-by-m orthogonal matrix Q. 68*> \endverbatim 69*> 70*> \param[out] R 71*> \verbatim 72*> R is REAL array, dimension (LDA,max(M,N)) 73*> \endverbatim 74*> 75*> \param[in] LDA 76*> \verbatim 77*> LDA is INTEGER 78*> The leading dimension of the arrays A, AF, Q and R. 79*> LDA >= max(M,N). 80*> \endverbatim 81*> 82*> \param[out] TAU 83*> \verbatim 84*> TAU is REAL array, dimension (min(M,N)) 85*> The scalar factors of the elementary reflectors, as returned 86*> by SGEQRFP. 87*> \endverbatim 88*> 89*> \param[out] WORK 90*> \verbatim 91*> WORK is REAL array, dimension (LWORK) 92*> \endverbatim 93*> 94*> \param[in] LWORK 95*> \verbatim 96*> LWORK is INTEGER 97*> The dimension of the array WORK. 98*> \endverbatim 99*> 100*> \param[out] RWORK 101*> \verbatim 102*> RWORK is REAL array, dimension (M) 103*> \endverbatim 104*> 105*> \param[out] RESULT 106*> \verbatim 107*> RESULT is REAL array, dimension (2) 108*> The test ratios: 109*> RESULT(1) = norm( R - Q'*A ) / ( M * norm(A) * EPS ) 110*> RESULT(2) = norm( I - Q'*Q ) / ( M * EPS ) 111*> \endverbatim 112* 113* Authors: 114* ======== 115* 116*> \author Univ. of Tennessee 117*> \author Univ. of California Berkeley 118*> \author Univ. of Colorado Denver 119*> \author NAG Ltd. 120* 121*> \ingroup single_lin 122* 123* ===================================================================== 124 SUBROUTINE SQRT01P( M, N, A, AF, Q, R, LDA, TAU, WORK, LWORK, 125 $ RWORK, RESULT ) 126* 127* -- LAPACK test routine -- 128* -- LAPACK is a software package provided by Univ. of Tennessee, -- 129* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 130* 131* .. Scalar Arguments .. 132 INTEGER LDA, LWORK, M, N 133* .. 134* .. Array Arguments .. 135 REAL A( LDA, * ), AF( LDA, * ), Q( LDA, * ), 136 $ R( LDA, * ), RESULT( * ), RWORK( * ), TAU( * ), 137 $ WORK( LWORK ) 138* .. 139* 140* ===================================================================== 141* 142* .. Parameters .. 143 REAL ZERO, ONE 144 PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) 145 REAL ROGUE 146 PARAMETER ( ROGUE = -1.0E+10 ) 147* .. 148* .. Local Scalars .. 149 INTEGER INFO, MINMN 150 REAL ANORM, EPS, RESID 151* .. 152* .. External Functions .. 153 REAL SLAMCH, SLANGE, SLANSY 154 EXTERNAL SLAMCH, SLANGE, SLANSY 155* .. 156* .. External Subroutines .. 157 EXTERNAL SGEMM, SGEQRFP, SLACPY, SLASET, SORGQR, SSYRK 158* .. 159* .. Intrinsic Functions .. 160 INTRINSIC MAX, MIN, REAL 161* .. 162* .. Scalars in Common .. 163 CHARACTER*32 SRNAMT 164* .. 165* .. Common blocks .. 166 COMMON / SRNAMC / SRNAMT 167* .. 168* .. Executable Statements .. 169* 170 MINMN = MIN( M, N ) 171 EPS = SLAMCH( 'Epsilon' ) 172* 173* Copy the matrix A to the array AF. 174* 175 CALL SLACPY( 'Full', M, N, A, LDA, AF, LDA ) 176* 177* Factorize the matrix A in the array AF. 178* 179 SRNAMT = 'SGEQRFP' 180 CALL SGEQRFP( M, N, AF, LDA, TAU, WORK, LWORK, INFO ) 181* 182* Copy details of Q 183* 184 CALL SLASET( 'Full', M, M, ROGUE, ROGUE, Q, LDA ) 185 CALL SLACPY( 'Lower', M-1, N, AF( 2, 1 ), LDA, Q( 2, 1 ), LDA ) 186* 187* Generate the m-by-m matrix Q 188* 189 SRNAMT = 'SORGQR' 190 CALL SORGQR( M, M, MINMN, Q, LDA, TAU, WORK, LWORK, INFO ) 191* 192* Copy R 193* 194 CALL SLASET( 'Full', M, N, ZERO, ZERO, R, LDA ) 195 CALL SLACPY( 'Upper', M, N, AF, LDA, R, LDA ) 196* 197* Compute R - Q'*A 198* 199 CALL SGEMM( 'Transpose', 'No transpose', M, N, M, -ONE, Q, LDA, A, 200 $ LDA, ONE, R, LDA ) 201* 202* Compute norm( R - Q'*A ) / ( M * norm(A) * EPS ) . 203* 204 ANORM = SLANGE( '1', M, N, A, LDA, RWORK ) 205 RESID = SLANGE( '1', M, N, R, LDA, RWORK ) 206 IF( ANORM.GT.ZERO ) THEN 207 RESULT( 1 ) = ( ( RESID / REAL( MAX( 1, M ) ) ) / ANORM ) / EPS 208 ELSE 209 RESULT( 1 ) = ZERO 210 END IF 211* 212* Compute I - Q'*Q 213* 214 CALL SLASET( 'Full', M, M, ZERO, ONE, R, LDA ) 215 CALL SSYRK( 'Upper', 'Transpose', M, M, -ONE, Q, LDA, ONE, R, 216 $ LDA ) 217* 218* Compute norm( I - Q'*Q ) / ( M * EPS ) . 219* 220 RESID = SLANSY( '1', 'Upper', M, R, LDA, RWORK ) 221* 222 RESULT( 2 ) = ( RESID / REAL( MAX( 1, M ) ) ) / EPS 223* 224 RETURN 225* 226* End of SQRT01P 227* 228 END 229