SUBROUTINE SGRQTS( M, P, N, A, AF, Q, R, LDA, TAUA, B, BF, Z, T, \$ BWK, LDB, TAUB, WORK, LWORK, RWORK, RESULT ) * * -- LAPACK test routine (version 3.1) -- * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. * November 2006 * * .. Scalar Arguments .. INTEGER LDA, LDB, LWORK, M, P, N * .. * .. Array Arguments .. REAL A( LDA, * ), AF( LDA, * ), R( LDA, * ), \$ Q( LDA, * ), \$ B( LDB, * ), BF( LDB, * ), T( LDB, * ), \$ Z( LDB, * ), BWK( LDB, * ), \$ TAUA( * ), TAUB( * ), \$ RESULT( 4 ), RWORK( * ), WORK( LWORK ) * .. * * Purpose * ======= * * SGRQTS tests SGGRQF, which computes the GRQ factorization of an * M-by-N matrix A and a P-by-N matrix B: A = R*Q and B = Z*T*Q. * * Arguments * ========= * * M (input) INTEGER * The number of rows of the matrix A. M >= 0. * * P (input) INTEGER * The number of rows of the matrix B. P >= 0. * * N (input) INTEGER * The number of columns of the matrices A and B. N >= 0. * * A (input) REAL array, dimension (LDA,N) * The M-by-N matrix A. * * AF (output) REAL array, dimension (LDA,N) * Details of the GRQ factorization of A and B, as returned * by SGGRQF, see SGGRQF for further details. * * Q (output) REAL array, dimension (LDA,N) * The N-by-N orthogonal matrix Q. * * R (workspace) REAL array, dimension (LDA,MAX(M,N)) * * LDA (input) INTEGER * The leading dimension of the arrays A, AF, R and Q. * LDA >= max(M,N). * * TAUA (output) REAL array, dimension (min(M,N)) * The scalar factors of the elementary reflectors, as returned * by SGGQRC. * * B (input) REAL array, dimension (LDB,N) * On entry, the P-by-N matrix A. * * BF (output) REAL array, dimension (LDB,N) * Details of the GQR factorization of A and B, as returned * by SGGRQF, see SGGRQF for further details. * * Z (output) REAL array, dimension (LDB,P) * The P-by-P orthogonal matrix Z. * * T (workspace) REAL array, dimension (LDB,max(P,N)) * * BWK (workspace) REAL array, dimension (LDB,N) * * LDB (input) INTEGER * The leading dimension of the arrays B, BF, Z and T. * LDB >= max(P,N). * * TAUB (output) REAL array, dimension (min(P,N)) * The scalar factors of the elementary reflectors, as returned * by SGGRQF. * * WORK (workspace) REAL array, dimension (LWORK) * * LWORK (input) INTEGER * The dimension of the array WORK, LWORK >= max(M,P,N)**2. * * RWORK (workspace) REAL array, dimension (M) * * RESULT (output) REAL array, dimension (4) * The test ratios: * RESULT(1) = norm( R - A*Q' ) / ( MAX(M,N)*norm(A)*ULP) * RESULT(2) = norm( T*Q - Z'*B ) / (MAX(P,N)*norm(B)*ULP) * RESULT(3) = norm( I - Q'*Q ) / ( N*ULP ) * RESULT(4) = norm( I - Z'*Z ) / ( P*ULP ) * * ===================================================================== * * .. Parameters .. REAL ZERO, ONE PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) REAL ROGUE PARAMETER ( ROGUE = -1.0E+10 ) * .. * .. Local Scalars .. INTEGER INFO REAL ANORM, BNORM, ULP, UNFL, RESID * .. * .. External Functions .. REAL SLAMCH, SLANGE, SLANSY EXTERNAL SLAMCH, SLANGE, SLANSY * .. * .. External Subroutines .. EXTERNAL SGEMM, SGGRQF, SLACPY, SLASET, SORGQR, \$ SORGRQ, SSYRK * .. * .. Intrinsic Functions .. INTRINSIC MAX, MIN, REAL * .. * .. Executable Statements .. * ULP = SLAMCH( 'Precision' ) UNFL = SLAMCH( 'Safe minimum' ) * * Copy the matrix A to the array AF. * CALL SLACPY( 'Full', M, N, A, LDA, AF, LDA ) CALL SLACPY( 'Full', P, N, B, LDB, BF, LDB ) * ANORM = MAX( SLANGE( '1', M, N, A, LDA, RWORK ), UNFL ) BNORM = MAX( SLANGE( '1', P, N, B, LDB, RWORK ), UNFL ) * * Factorize the matrices A and B in the arrays AF and BF. * CALL SGGRQF( M, P, N, AF, LDA, TAUA, BF, LDB, TAUB, WORK, \$ LWORK, INFO ) * * Generate the N-by-N matrix Q * CALL SLASET( 'Full', N, N, ROGUE, ROGUE, Q, LDA ) IF( M.LE.N ) THEN IF( M.GT.0 .AND. M.LT.N ) \$ CALL SLACPY( 'Full', M, N-M, AF, LDA, Q( N-M+1, 1 ), LDA ) IF( M.GT.1 ) \$ CALL SLACPY( 'Lower', M-1, M-1, AF( 2, N-M+1 ), LDA, \$ Q( N-M+2, N-M+1 ), LDA ) ELSE IF( N.GT.1 ) \$ CALL SLACPY( 'Lower', N-1, N-1, AF( M-N+2, 1 ), LDA, \$ Q( 2, 1 ), LDA ) END IF CALL SORGRQ( N, N, MIN( M, N ), Q, LDA, TAUA, WORK, LWORK, INFO ) * * Generate the P-by-P matrix Z * CALL SLASET( 'Full', P, P, ROGUE, ROGUE, Z, LDB ) IF( P.GT.1 ) \$ CALL SLACPY( 'Lower', P-1, N, BF( 2,1 ), LDB, Z( 2,1 ), LDB ) CALL SORGQR( P, P, MIN( P,N ), Z, LDB, TAUB, WORK, LWORK, INFO ) * * Copy R * CALL SLASET( 'Full', M, N, ZERO, ZERO, R, LDA ) IF( M.LE.N )THEN CALL SLACPY( 'Upper', M, M, AF( 1, N-M+1 ), LDA, R( 1, N-M+1 ), \$ LDA ) ELSE CALL SLACPY( 'Full', M-N, N, AF, LDA, R, LDA ) CALL SLACPY( 'Upper', N, N, AF( M-N+1, 1 ), LDA, R( M-N+1, 1 ), \$ LDA ) END IF * * Copy T * CALL SLASET( 'Full', P, N, ZERO, ZERO, T, LDB ) CALL SLACPY( 'Upper', P, N, BF, LDB, T, LDB ) * * Compute R - A*Q' * CALL SGEMM( 'No transpose', 'Transpose', M, N, N, -ONE, A, LDA, Q, \$ LDA, ONE, R, LDA ) * * Compute norm( R - A*Q' ) / ( MAX(M,N)*norm(A)*ULP ) . * RESID = SLANGE( '1', M, N, R, LDA, RWORK ) IF( ANORM.GT.ZERO ) THEN RESULT( 1 ) = ( ( RESID / REAL(MAX(1,M,N) ) ) / ANORM ) / ULP ELSE RESULT( 1 ) = ZERO END IF * * Compute T*Q - Z'*B * CALL SGEMM( 'Transpose', 'No transpose', P, N, P, ONE, Z, LDB, B, \$ LDB, ZERO, BWK, LDB ) CALL SGEMM( 'No transpose', 'No transpose', P, N, N, ONE, T, LDB, \$ Q, LDA, -ONE, BWK, LDB ) * * Compute norm( T*Q - Z'*B ) / ( MAX(P,N)*norm(A)*ULP ) . * RESID = SLANGE( '1', P, N, BWK, LDB, RWORK ) IF( BNORM.GT.ZERO ) THEN RESULT( 2 ) = ( ( RESID / REAL( MAX( 1,P,M ) ) )/BNORM ) / ULP ELSE RESULT( 2 ) = ZERO END IF * * Compute I - Q*Q' * CALL SLASET( 'Full', N, N, ZERO, ONE, R, LDA ) CALL SSYRK( 'Upper', 'No Transpose', N, N, -ONE, Q, LDA, ONE, R, \$ LDA ) * * Compute norm( I - Q'*Q ) / ( N * ULP ) . * RESID = SLANSY( '1', 'Upper', N, R, LDA, RWORK ) RESULT( 3 ) = ( RESID / REAL( MAX( 1,N ) ) ) / ULP * * Compute I - Z'*Z * CALL SLASET( 'Full', P, P, ZERO, ONE, T, LDB ) CALL SSYRK( 'Upper', 'Transpose', P, P, -ONE, Z, LDB, ONE, T, \$ LDB ) * * Compute norm( I - Z'*Z ) / ( P*ULP ) . * RESID = SLANSY( '1', 'Upper', P, T, LDB, RWORK ) RESULT( 4 ) = ( RESID / REAL( MAX( 1,P ) ) ) / ULP * RETURN * * End of SGRQTS * END