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|
*> \brief \b ZGSVTS3
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition:
* ===========
*
* SUBROUTINE ZGSVTS3( 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( * )
* DOUBLE PRECISION ALPHA( * ), BETA( * ), RESULT( 6 ), RWORK( * )
* COMPLEX*16 A( LDA, * ), AF( LDA, * ), B( LDB, * ),
* $ BF( LDB, * ), Q( LDQ, * ), R( LDR, * ),
* $ U( LDU, * ), V( LDV, * ), WORK( LWORK )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> ZGSVTS3 tests ZGGSVD3, 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*16 array, dimension (LDA,M)
*> The M-by-N matrix A.
*> \endverbatim
*>
*> \param[out] AF
*> \verbatim
*> AF is COMPLEX*16 array, dimension (LDA,N)
*> Details of the GSVD of A and B, as returned by ZGGSVD3,
*> see ZGGSVD3 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*16 array, dimension (LDB,P)
*> On entry, the P-by-N matrix B.
*> \endverbatim
*>
*> \param[out] BF
*> \verbatim
*> BF is COMPLEX*16 array, dimension (LDB,N)
*> Details of the GSVD of A and B, as returned by ZGGSVD3,
*> see ZGGSVD3 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*16 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*16 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*16 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 DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] BETA
*> \verbatim
*> BETA is DOUBLE PRECISION 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 ZGGSVD3 for details.
*> \endverbatim
*>
*> \param[out] R
*> \verbatim
*> R is COMPLEX*16 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*16 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 DOUBLE PRECISION array, dimension (max(M,P,N))
*> \endverbatim
*>
*> \param[out] RESULT
*> \verbatim
*> RESULT is DOUBLE PRECISION 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.
*
*> \date August 2015
*
*> \ingroup complex16_eig
*
* =====================================================================
SUBROUTINE ZGSVTS3( 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 (version 3.6.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* August 2015
*
* .. Scalar Arguments ..
INTEGER LDA, LDB, LDQ, LDR, LDU, LDV, LWORK, M, N, P
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION ALPHA( * ), BETA( * ), RESULT( 6 ), RWORK( * )
COMPLEX*16 A( LDA, * ), AF( LDA, * ), B( LDB, * ),
$ BF( LDB, * ), Q( LDQ, * ), R( LDR, * ),
$ U( LDU, * ), V( LDV, * ), WORK( LWORK )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
COMPLEX*16 CZERO, CONE
PARAMETER ( CZERO = ( 0.0D+0, 0.0D+0 ),
$ CONE = ( 1.0D+0, 0.0D+0 ) )
* ..
* .. Local Scalars ..
INTEGER I, INFO, J, K, L
DOUBLE PRECISION ANORM, BNORM, RESID, TEMP, ULP, ULPINV, UNFL
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH, ZLANGE, ZLANHE
EXTERNAL DLAMCH, ZLANGE, ZLANHE
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, ZGEMM, ZGGSVD3, ZHERK, ZLACPY, ZLASET
* ..
* .. Intrinsic Functions ..
INTRINSIC DBLE, MAX, MIN
* ..
* .. Executable Statements ..
*
ULP = DLAMCH( 'Precision' )
ULPINV = ONE / ULP
UNFL = DLAMCH( 'Safe minimum' )
*
* Copy the matrix A to the array AF.
*
CALL ZLACPY( 'Full', M, N, A, LDA, AF, LDA )
CALL ZLACPY( 'Full', P, N, B, LDB, BF, LDB )
*
ANORM = MAX( ZLANGE( '1', M, N, A, LDA, RWORK ), UNFL )
BNORM = MAX( ZLANGE( '1', P, N, B, LDB, RWORK ), UNFL )
*
* Factorize the matrices A and B in the arrays AF and BF.
*
CALL ZGGSVD3( '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 ZGEMM( 'No transpose', 'No transpose', M, N, N, CONE, A, LDA,
$ Q, LDQ, CZERO, WORK, LDA )
*
CALL ZGEMM( '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 = ZLANGE( '1', M, N, A, LDA, RWORK )
IF( ANORM.GT.ZERO ) THEN
RESULT( 1 ) = ( ( RESID / DBLE( MAX( 1, M, N ) ) ) / ANORM ) /
$ ULP
ELSE
RESULT( 1 ) = ZERO
END IF
*
* Compute B := V'*B*Q - D2*R
*
CALL ZGEMM( 'No transpose', 'No transpose', P, N, N, CONE, B, LDB,
$ Q, LDQ, CZERO, WORK, LDB )
*
CALL ZGEMM( '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 = ZLANGE( '1', P, N, B, LDB, RWORK )
IF( BNORM.GT.ZERO ) THEN
RESULT( 2 ) = ( ( RESID / DBLE( MAX( 1, P, N ) ) ) / BNORM ) /
$ ULP
ELSE
RESULT( 2 ) = ZERO
END IF
*
* Compute I - U'*U
*
CALL ZLASET( 'Full', M, M, CZERO, CONE, WORK, LDQ )
CALL ZHERK( 'Upper', 'Conjugate transpose', M, M, -ONE, U, LDU,
$ ONE, WORK, LDU )
*
* Compute norm( I - U'*U ) / ( M * ULP ) .
*
RESID = ZLANHE( '1', 'Upper', M, WORK, LDU, RWORK )
RESULT( 3 ) = ( RESID / DBLE( MAX( 1, M ) ) ) / ULP
*
* Compute I - V'*V
*
CALL ZLASET( 'Full', P, P, CZERO, CONE, WORK, LDV )
CALL ZHERK( 'Upper', 'Conjugate transpose', P, P, -ONE, V, LDV,
$ ONE, WORK, LDV )
*
* Compute norm( I - V'*V ) / ( P * ULP ) .
*
RESID = ZLANHE( '1', 'Upper', P, WORK, LDV, RWORK )
RESULT( 4 ) = ( RESID / DBLE( MAX( 1, P ) ) ) / ULP
*
* Compute I - Q'*Q
*
CALL ZLASET( 'Full', N, N, CZERO, CONE, WORK, LDQ )
CALL ZHERK( 'Upper', 'Conjugate transpose', N, N, -ONE, Q, LDQ,
$ ONE, WORK, LDQ )
*
* Compute norm( I - Q'*Q ) / ( N * ULP ) .
*
RESID = ZLANHE( '1', 'Upper', N, WORK, LDQ, RWORK )
RESULT( 5 ) = ( RESID / DBLE( MAX( 1, N ) ) ) / ULP
*
* Check sorting
*
CALL DCOPY( 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 ZGSVTS3
*
END
|
