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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 3.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* September 30, 1994
*
* .. 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
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