*> \brief \b CGSVTS3 * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * * Definition: * =========== * * SUBROUTINE CGSVTS3( M, P, N, A, AF, LDA, B, BF, LDB, U, LDU, V, * LDV, Q, LDQ, ALPHA, BETA, R, LDR, IWORK, WORK, * LWORK, RWORK, RESULT ) * * .. Scalar Arguments .. * INTEGER LDA, LDB, LDQ, LDR, LDU, LDV, LWORK, M, N, P * .. * .. Array Arguments .. * INTEGER IWORK( * ) * REAL ALPHA( * ), BETA( * ), RESULT( 6 ), RWORK( * ) * COMPLEX A( LDA, * ), AF( LDA, * ), B( LDB, * ), * $ BF( LDB, * ), Q( LDQ, * ), R( LDR, * ), * $ U( LDU, * ), V( LDV, * ), WORK( LWORK ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> CGSVTS3 tests CGGSVD3, which computes the GSVD of an M-by-N matrix A *> and a P-by-N matrix B: *> U'*A*Q = D1*R and V'*B*Q = D2*R. *> \endverbatim * * Arguments: * ========== * *> \param[in] M *> \verbatim *> M is INTEGER *> The number of rows of the matrix A. M >= 0. *> \endverbatim *> *> \param[in] P *> \verbatim *> P is INTEGER *> The number of rows of the matrix B. P >= 0. *> \endverbatim *> *> \param[in] N *> \verbatim *> N is INTEGER *> The number of columns of the matrices A and B. N >= 0. *> \endverbatim *> *> \param[in] A *> \verbatim *> A is COMPLEX array, dimension (LDA,M) *> The M-by-N matrix A. *> \endverbatim *> *> \param[out] AF *> \verbatim *> AF is COMPLEX array, dimension (LDA,N) *> Details of the GSVD of A and B, as returned by CGGSVD3, *> see CGGSVD3 for further details. *> \endverbatim *> *> \param[in] LDA *> \verbatim *> LDA is INTEGER *> The leading dimension of the arrays A and AF. *> LDA >= max( 1,M ). *> \endverbatim *> *> \param[in] B *> \verbatim *> B is COMPLEX array, dimension (LDB,P) *> On entry, the P-by-N matrix B. *> \endverbatim *> *> \param[out] BF *> \verbatim *> BF is COMPLEX array, dimension (LDB,N) *> Details of the GSVD of A and B, as returned by CGGSVD3, *> see CGGSVD3 for further details. *> \endverbatim *> *> \param[in] LDB *> \verbatim *> LDB is INTEGER *> The leading dimension of the arrays B and BF. *> LDB >= max(1,P). *> \endverbatim *> *> \param[out] U *> \verbatim *> U is COMPLEX array, dimension(LDU,M) *> The M by M unitary matrix U. *> \endverbatim *> *> \param[in] LDU *> \verbatim *> LDU is INTEGER *> The leading dimension of the array U. LDU >= max(1,M). *> \endverbatim *> *> \param[out] V *> \verbatim *> V is COMPLEX array, dimension(LDV,M) *> The P by P unitary matrix V. *> \endverbatim *> *> \param[in] LDV *> \verbatim *> LDV is INTEGER *> The leading dimension of the array V. LDV >= max(1,P). *> \endverbatim *> *> \param[out] Q *> \verbatim *> Q is COMPLEX array, dimension(LDQ,N) *> The N by N unitary matrix Q. *> \endverbatim *> *> \param[in] LDQ *> \verbatim *> LDQ is INTEGER *> The leading dimension of the array Q. LDQ >= max(1,N). *> \endverbatim *> *> \param[out] ALPHA *> \verbatim *> ALPHA is REAL array, dimension (N) *> \endverbatim *> *> \param[out] BETA *> \verbatim *> BETA is REAL array, dimension (N) *> *> The generalized singular value pairs of A and B, the *> ``diagonal'' matrices D1 and D2 are constructed from *> ALPHA and BETA, see subroutine CGGSVD3 for details. *> \endverbatim *> *> \param[out] R *> \verbatim *> R is COMPLEX array, dimension(LDQ,N) *> The upper triangular matrix R. *> \endverbatim *> *> \param[in] LDR *> \verbatim *> LDR is INTEGER *> The leading dimension of the array R. LDR >= max(1,N). *> \endverbatim *> *> \param[out] IWORK *> \verbatim *> IWORK is INTEGER array, dimension (N) *> \endverbatim *> *> \param[out] WORK *> \verbatim *> WORK is COMPLEX array, dimension (LWORK) *> \endverbatim *> *> \param[in] LWORK *> \verbatim *> LWORK is INTEGER *> The dimension of the array WORK, *> LWORK >= max(M,P,N)*max(M,P,N). *> \endverbatim *> *> \param[out] RWORK *> \verbatim *> RWORK is REAL array, dimension (max(M,P,N)) *> \endverbatim *> *> \param[out] RESULT *> \verbatim *> RESULT is REAL array, dimension (6) *> The test ratios: *> RESULT(1) = norm( U'*A*Q - D1*R ) / ( MAX(M,N)*norm(A)*ULP) *> RESULT(2) = norm( V'*B*Q - D2*R ) / ( MAX(P,N)*norm(B)*ULP) *> RESULT(3) = norm( I - U'*U ) / ( M*ULP ) *> RESULT(4) = norm( I - V'*V ) / ( P*ULP ) *> RESULT(5) = norm( I - Q'*Q ) / ( N*ULP ) *> RESULT(6) = 0 if ALPHA is in decreasing order; *> = ULPINV otherwise. *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \ingroup complex_eig * * ===================================================================== SUBROUTINE CGSVTS3( M, P, N, A, AF, LDA, B, BF, LDB, U, LDU, V, $ LDV, Q, LDQ, ALPHA, BETA, R, LDR, IWORK, WORK, $ LWORK, RWORK, RESULT ) * * -- LAPACK test routine -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * * .. Scalar Arguments .. INTEGER LDA, LDB, LDQ, LDR, LDU, LDV, LWORK, M, N, P * .. * .. Array Arguments .. INTEGER IWORK( * ) REAL ALPHA( * ), BETA( * ), RESULT( 6 ), RWORK( * ) COMPLEX A( LDA, * ), AF( LDA, * ), B( LDB, * ), $ BF( LDB, * ), Q( LDQ, * ), R( LDR, * ), $ U( LDU, * ), V( LDV, * ), WORK( LWORK ) * .. * * ===================================================================== * * .. Parameters .. REAL ZERO, ONE PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) COMPLEX CZERO, CONE PARAMETER ( CZERO = ( 0.0E+0, 0.0E+0 ), $ CONE = ( 1.0E+0, 0.0E+0 ) ) * .. * .. Local Scalars .. INTEGER I, INFO, J, K, L REAL ANORM, BNORM, RESID, TEMP, ULP, ULPINV, UNFL * .. * .. External Functions .. REAL CLANGE, CLANHE, SLAMCH EXTERNAL CLANGE, CLANHE, SLAMCH * .. * .. External Subroutines .. EXTERNAL CGEMM, CGGSVD3, CHERK, CLACPY, CLASET, SCOPY * .. * .. Intrinsic Functions .. INTRINSIC MAX, MIN, REAL * .. * .. Executable Statements .. * ULP = SLAMCH( 'Precision' ) ULPINV = ONE / ULP UNFL = SLAMCH( 'Safe minimum' ) * * Copy the matrix A to the array AF. * CALL CLACPY( 'Full', M, N, A, LDA, AF, LDA ) CALL CLACPY( 'Full', P, N, B, LDB, BF, LDB ) * ANORM = MAX( CLANGE( '1', M, N, A, LDA, RWORK ), UNFL ) BNORM = MAX( CLANGE( '1', P, N, B, LDB, RWORK ), UNFL ) * * Factorize the matrices A and B in the arrays AF and BF. * CALL CGGSVD3( 'U', 'V', 'Q', M, N, P, K, L, AF, LDA, BF, LDB, $ ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK, LWORK, $ RWORK, IWORK, INFO ) * * Copy R * DO 20 I = 1, MIN( K+L, M ) DO 10 J = I, K + L R( I, J ) = AF( I, N-K-L+J ) 10 CONTINUE 20 CONTINUE * IF( M-K-L.LT.0 ) THEN DO 40 I = M + 1, K + L DO 30 J = I, K + L R( I, J ) = BF( I-K, N-K-L+J ) 30 CONTINUE 40 CONTINUE END IF * * Compute A:= U'*A*Q - D1*R * CALL CGEMM( 'No transpose', 'No transpose', M, N, N, CONE, A, LDA, $ Q, LDQ, CZERO, WORK, LDA ) * CALL CGEMM( 'Conjugate transpose', 'No transpose', M, N, M, CONE, $ U, LDU, WORK, LDA, CZERO, A, LDA ) * DO 60 I = 1, K DO 50 J = I, K + L A( I, N-K-L+J ) = A( I, N-K-L+J ) - R( I, J ) 50 CONTINUE 60 CONTINUE * DO 80 I = K + 1, MIN( K+L, M ) DO 70 J = I, K + L A( I, N-K-L+J ) = A( I, N-K-L+J ) - ALPHA( I )*R( I, J ) 70 CONTINUE 80 CONTINUE * * Compute norm( U'*A*Q - D1*R ) / ( MAX(1,M,N)*norm(A)*ULP ) . * RESID = CLANGE( '1', M, N, A, LDA, RWORK ) IF( ANORM.GT.ZERO ) THEN RESULT( 1 ) = ( ( RESID / REAL( MAX( 1, M, N ) ) ) / ANORM ) / $ ULP ELSE RESULT( 1 ) = ZERO END IF * * Compute B := V'*B*Q - D2*R * CALL CGEMM( 'No transpose', 'No transpose', P, N, N, CONE, B, LDB, $ Q, LDQ, CZERO, WORK, LDB ) * CALL CGEMM( 'Conjugate transpose', 'No transpose', P, N, P, CONE, $ V, LDV, WORK, LDB, CZERO, B, LDB ) * DO 100 I = 1, L DO 90 J = I, L B( I, N-L+J ) = B( I, N-L+J ) - BETA( K+I )*R( K+I, K+J ) 90 CONTINUE 100 CONTINUE * * Compute norm( V'*B*Q - D2*R ) / ( MAX(P,N)*norm(B)*ULP ) . * RESID = CLANGE( '1', P, N, B, LDB, RWORK ) IF( BNORM.GT.ZERO ) THEN RESULT( 2 ) = ( ( RESID / REAL( MAX( 1, P, N ) ) ) / BNORM ) / $ ULP ELSE RESULT( 2 ) = ZERO END IF * * Compute I - U'*U * CALL CLASET( 'Full', M, M, CZERO, CONE, WORK, LDQ ) CALL CHERK( 'Upper', 'Conjugate transpose', M, M, -ONE, U, LDU, $ ONE, WORK, LDU ) * * Compute norm( I - U'*U ) / ( M * ULP ) . * RESID = CLANHE( '1', 'Upper', M, WORK, LDU, RWORK ) RESULT( 3 ) = ( RESID / REAL( MAX( 1, M ) ) ) / ULP * * Compute I - V'*V * CALL CLASET( 'Full', P, P, CZERO, CONE, WORK, LDV ) CALL CHERK( 'Upper', 'Conjugate transpose', P, P, -ONE, V, LDV, $ ONE, WORK, LDV ) * * Compute norm( I - V'*V ) / ( P * ULP ) . * RESID = CLANHE( '1', 'Upper', P, WORK, LDV, RWORK ) RESULT( 4 ) = ( RESID / REAL( MAX( 1, P ) ) ) / ULP * * Compute I - Q'*Q * CALL CLASET( 'Full', N, N, CZERO, CONE, WORK, LDQ ) CALL CHERK( 'Upper', 'Conjugate transpose', N, N, -ONE, Q, LDQ, $ ONE, WORK, LDQ ) * * Compute norm( I - Q'*Q ) / ( N * ULP ) . * RESID = CLANHE( '1', 'Upper', N, WORK, LDQ, RWORK ) RESULT( 5 ) = ( RESID / REAL( MAX( 1, N ) ) ) / ULP * * Check sorting * CALL SCOPY( N, ALPHA, 1, RWORK, 1 ) DO 110 I = K + 1, MIN( K+L, M ) J = IWORK( I ) IF( I.NE.J ) THEN TEMP = RWORK( I ) RWORK( I ) = RWORK( J ) RWORK( J ) = TEMP END IF 110 CONTINUE * RESULT( 6 ) = ZERO DO 120 I = K + 1, MIN( K+L, M ) - 1 IF( RWORK( I ).LT.RWORK( I+1 ) ) $ RESULT( 6 ) = ULPINV 120 CONTINUE * RETURN * * End of CGSVTS3 * END