You can not select more than 25 topics
Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
334 lines
9.6 KiB
334 lines
9.6 KiB
2 years ago
|
*> \brief \b ZGQRTS
|
||
|
|
*
|
||
|
|
* =========== DOCUMENTATION ===========
|
||
|
|
*
|
||
|
|
* Online html documentation available at
|
||
|
|
* http://www.netlib.org/lapack/explore-html/
|
||
|
|
*
|
||
|
|
* Definition:
|
||
|
|
* ===========
|
||
|
|
*
|
||
|
|
* SUBROUTINE ZGQRTS( N, M, P, A, AF, Q, R, LDA, TAUA, B, BF, Z, T,
|
||
|
|
* BWK, LDB, TAUB, WORK, LWORK, RWORK, RESULT )
|
||
|
|
*
|
||
|
|
* .. Scalar Arguments ..
|
||
|
|
* INTEGER LDA, LDB, LWORK, M, N, P
|
||
|
|
* ..
|
||
|
|
* .. Array Arguments ..
|
||
|
|
* DOUBLE PRECISION RESULT( 4 ), RWORK( * )
|
||
|
|
* COMPLEX*16 A( LDA, * ), AF( LDA, * ), B( LDB, * ),
|
||
|
|
* $ BF( LDB, * ), BWK( LDB, * ), Q( LDA, * ),
|
||
|
|
* $ R( LDA, * ), T( LDB, * ), TAUA( * ), TAUB( * ),
|
||
|
|
* $ WORK( LWORK ), Z( LDB, * )
|
||
|
|
* ..
|
||
|
|
*
|
||
|
|
*
|
||
|
|
*> \par Purpose:
|
||
|
|
* =============
|
||
|
|
*>
|
||
|
|
*> \verbatim
|
||
|
|
*>
|
||
|
|
*> ZGQRTS tests ZGGQRF, which computes the GQR factorization of an
|
||
|
|
*> N-by-M matrix A and a N-by-P matrix B: A = Q*R and B = Q*T*Z.
|
||
|
|
*> \endverbatim
|
||
|
|
*
|
||
|
|
* Arguments:
|
||
|
|
* ==========
|
||
|
|
*
|
||
|
|
*> \param[in] N
|
||
|
|
*> \verbatim
|
||
|
|
*> N is INTEGER
|
||
|
|
*> The number of rows of the matrices A and B. N >= 0.
|
||
|
|
*> \endverbatim
|
||
|
|
*>
|
||
|
|
*> \param[in] M
|
||
|
|
*> \verbatim
|
||
|
|
*> M is INTEGER
|
||
|
|
*> The number of columns of the matrix A. M >= 0.
|
||
|
|
*> \endverbatim
|
||
|
|
*>
|
||
|
|
*> \param[in] P
|
||
|
|
*> \verbatim
|
||
|
|
*> P is INTEGER
|
||
|
|
*> The number of columns of the matrix B. P >= 0.
|
||
|
|
*> \endverbatim
|
||
|
|
*>
|
||
|
|
*> \param[in] A
|
||
|
|
*> \verbatim
|
||
|
|
*> A is COMPLEX*16 array, dimension (LDA,M)
|
||
|
|
*> The N-by-M matrix A.
|
||
|
|
*> \endverbatim
|
||
|
|
*>
|
||
|
|
*> \param[out] AF
|
||
|
|
*> \verbatim
|
||
|
|
*> AF is COMPLEX*16 array, dimension (LDA,N)
|
||
|
|
*> Details of the GQR factorization of A and B, as returned
|
||
|
|
*> by ZGGQRF, see CGGQRF for further details.
|
||
|
|
*> \endverbatim
|
||
|
|
*>
|
||
|
|
*> \param[out] Q
|
||
|
|
*> \verbatim
|
||
|
|
*> Q is COMPLEX*16 array, dimension (LDA,N)
|
||
|
|
*> The M-by-M unitary matrix Q.
|
||
|
|
*> \endverbatim
|
||
|
|
*>
|
||
|
|
*> \param[out] R
|
||
|
|
*> \verbatim
|
||
|
|
*> R is COMPLEX*16 array, dimension (LDA,MAX(M,N))
|
||
|
|
*> \endverbatim
|
||
|
|
*>
|
||
|
|
*> \param[in] LDA
|
||
|
|
*> \verbatim
|
||
|
|
*> LDA is INTEGER
|
||
|
|
*> The leading dimension of the arrays A, AF, R and Q.
|
||
|
|
*> LDA >= max(M,N).
|
||
|
|
*> \endverbatim
|
||
|
|
*>
|
||
|
|
*> \param[out] TAUA
|
||
|
|
*> \verbatim
|
||
|
|
*> TAUA is COMPLEX*16 array, dimension (min(M,N))
|
||
|
|
*> The scalar factors of the elementary reflectors, as returned
|
||
|
|
*> by ZGGQRF.
|
||
|
|
*> \endverbatim
|
||
|
|
*>
|
||
|
|
*> \param[in] B
|
||
|
|
*> \verbatim
|
||
|
|
*> B is COMPLEX*16 array, dimension (LDB,P)
|
||
|
|
*> On entry, the N-by-P matrix A.
|
||
|
|
*> \endverbatim
|
||
|
|
*>
|
||
|
|
*> \param[out] BF
|
||
|
|
*> \verbatim
|
||
|
|
*> BF is COMPLEX*16 array, dimension (LDB,N)
|
||
|
|
*> Details of the GQR factorization of A and B, as returned
|
||
|
|
*> by ZGGQRF, see CGGQRF for further details.
|
||
|
|
*> \endverbatim
|
||
|
|
*>
|
||
|
|
*> \param[out] Z
|
||
|
|
*> \verbatim
|
||
|
|
*> Z is COMPLEX*16 array, dimension (LDB,P)
|
||
|
|
*> The P-by-P unitary matrix Z.
|
||
|
|
*> \endverbatim
|
||
|
|
*>
|
||
|
|
*> \param[out] T
|
||
|
|
*> \verbatim
|
||
|
|
*> T is COMPLEX*16 array, dimension (LDB,max(P,N))
|
||
|
|
*> \endverbatim
|
||
|
|
*>
|
||
|
|
*> \param[out] BWK
|
||
|
|
*> \verbatim
|
||
|
|
*> BWK is COMPLEX*16 array, dimension (LDB,N)
|
||
|
|
*> \endverbatim
|
||
|
|
*>
|
||
|
|
*> \param[in] LDB
|
||
|
|
*> \verbatim
|
||
|
|
*> LDB is INTEGER
|
||
|
|
*> The leading dimension of the arrays B, BF, Z and T.
|
||
|
|
*> LDB >= max(P,N).
|
||
|
|
*> \endverbatim
|
||
|
|
*>
|
||
|
|
*> \param[out] TAUB
|
||
|
|
*> \verbatim
|
||
|
|
*> TAUB is COMPLEX*16 array, dimension (min(P,N))
|
||
|
|
*> The scalar factors of the elementary reflectors, as returned
|
||
|
|
*> by DGGRQF.
|
||
|
|
*> \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(N,M,P)**2.
|
||
|
|
*> \endverbatim
|
||
|
|
*>
|
||
|
|
*> \param[out] RWORK
|
||
|
|
*> \verbatim
|
||
|
|
*> RWORK is DOUBLE PRECISION array, dimension (max(N,M,P))
|
||
|
|
*> \endverbatim
|
||
|
|
*>
|
||
|
|
*> \param[out] RESULT
|
||
|
|
*> \verbatim
|
||
|
|
*> RESULT is DOUBLE PRECISION array, dimension (4)
|
||
|
|
*> The test ratios:
|
||
|
|
*> RESULT(1) = norm( R - Q'*A ) / ( MAX(M,N)*norm(A)*ULP)
|
||
|
|
*> RESULT(2) = norm( T*Z - Q'*B ) / (MAX(P,N)*norm(B)*ULP)
|
||
|
|
*> RESULT(3) = norm( I - Q'*Q ) / ( M*ULP )
|
||
|
|
*> RESULT(4) = norm( I - Z'*Z ) / ( P*ULP )
|
||
|
|
*> \endverbatim
|
||
|
|
*
|
||
|
|
* Authors:
|
||
|
|
* ========
|
||
|
|
*
|
||
|
|
*> \author Univ. of Tennessee
|
||
|
|
*> \author Univ. of California Berkeley
|
||
|
|
*> \author Univ. of Colorado Denver
|
||
|
|
*> \author NAG Ltd.
|
||
|
|
*
|
||
|
|
*> \ingroup complex16_eig
|
||
|
|
*
|
||
|
|
* =====================================================================
|
||
|
|
SUBROUTINE ZGQRTS( N, M, P, A, AF, Q, R, LDA, TAUA, B, BF, Z, T,
|
||
|
|
$ BWK, LDB, TAUB, 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, LWORK, M, N, P
|
||
|
|
* ..
|
||
|
|
* .. Array Arguments ..
|
||
|
|
DOUBLE PRECISION RESULT( 4 ), RWORK( * )
|
||
|
|
COMPLEX*16 A( LDA, * ), AF( LDA, * ), B( LDB, * ),
|
||
|
|
$ BF( LDB, * ), BWK( LDB, * ), Q( LDA, * ),
|
||
|
|
$ R( LDA, * ), T( LDB, * ), TAUA( * ), TAUB( * ),
|
||
|
|
$ WORK( LWORK ), Z( LDB, * )
|
||
|
|
* ..
|
||
|
|
*
|
||
|
|
* =====================================================================
|
||
|
|
*
|
||
|
|
* .. 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 ) )
|
||
|
|
COMPLEX*16 CROGUE
|
||
|
|
PARAMETER ( CROGUE = ( -1.0D+10, 0.0D+0 ) )
|
||
|
|
* ..
|
||
|
|
* .. Local Scalars ..
|
||
|
|
INTEGER INFO
|
||
|
|
DOUBLE PRECISION ANORM, BNORM, RESID, ULP, UNFL
|
||
|
|
* ..
|
||
|
|
* .. External Functions ..
|
||
|
|
DOUBLE PRECISION DLAMCH, ZLANGE, ZLANHE
|
||
|
|
EXTERNAL DLAMCH, ZLANGE, ZLANHE
|
||
|
|
* ..
|
||
|
|
* .. External Subroutines ..
|
||
|
|
EXTERNAL ZGEMM, ZGGQRF, ZHERK, ZLACPY, ZLASET, ZUNGQR,
|
||
|
|
$ ZUNGRQ
|
||
|
|
* ..
|
||
|
|
* .. Intrinsic Functions ..
|
||
|
|
INTRINSIC DBLE, MAX, MIN
|
||
|
|
* ..
|
||
|
|
* .. Executable Statements ..
|
||
|
|
*
|
||
|
|
ULP = DLAMCH( 'Precision' )
|
||
|
|
UNFL = DLAMCH( 'Safe minimum' )
|
||
|
|
*
|
||
|
|
* Copy the matrix A to the array AF.
|
||
|
|
*
|
||
|
|
CALL ZLACPY( 'Full', N, M, A, LDA, AF, LDA )
|
||
|
|
CALL ZLACPY( 'Full', N, P, B, LDB, BF, LDB )
|
||
|
|
*
|
||
|
|
ANORM = MAX( ZLANGE( '1', N, M, A, LDA, RWORK ), UNFL )
|
||
|
|
BNORM = MAX( ZLANGE( '1', N, P, B, LDB, RWORK ), UNFL )
|
||
|
|
*
|
||
|
|
* Factorize the matrices A and B in the arrays AF and BF.
|
||
|
|
*
|
||
|
|
CALL ZGGQRF( N, M, P, AF, LDA, TAUA, BF, LDB, TAUB, WORK, LWORK,
|
||
|
|
$ INFO )
|
||
|
|
*
|
||
|
|
* Generate the N-by-N matrix Q
|
||
|
|
*
|
||
|
|
CALL ZLASET( 'Full', N, N, CROGUE, CROGUE, Q, LDA )
|
||
|
|
CALL ZLACPY( 'Lower', N-1, M, AF( 2, 1 ), LDA, Q( 2, 1 ), LDA )
|
||
|
|
CALL ZUNGQR( N, N, MIN( N, M ), Q, LDA, TAUA, WORK, LWORK, INFO )
|
||
|
|
*
|
||
|
|
* Generate the P-by-P matrix Z
|
||
|
|
*
|
||
|
|
CALL ZLASET( 'Full', P, P, CROGUE, CROGUE, Z, LDB )
|
||
|
|
IF( N.LE.P ) THEN
|
||
|
|
IF( N.GT.0 .AND. N.LT.P )
|
||
|
|
$ CALL ZLACPY( 'Full', N, P-N, BF, LDB, Z( P-N+1, 1 ), LDB )
|
||
|
|
IF( N.GT.1 )
|
||
|
|
$ CALL ZLACPY( 'Lower', N-1, N-1, BF( 2, P-N+1 ), LDB,
|
||
|
|
$ Z( P-N+2, P-N+1 ), LDB )
|
||
|
|
ELSE
|
||
|
|
IF( P.GT.1 )
|
||
|
|
$ CALL ZLACPY( 'Lower', P-1, P-1, BF( N-P+2, 1 ), LDB,
|
||
|
|
$ Z( 2, 1 ), LDB )
|
||
|
|
END IF
|
||
|
|
CALL ZUNGRQ( P, P, MIN( N, P ), Z, LDB, TAUB, WORK, LWORK, INFO )
|
||
|
|
*
|
||
|
|
* Copy R
|
||
|
|
*
|
||
|
|
CALL ZLASET( 'Full', N, M, CZERO, CZERO, R, LDA )
|
||
|
|
CALL ZLACPY( 'Upper', N, M, AF, LDA, R, LDA )
|
||
|
|
*
|
||
|
|
* Copy T
|
||
|
|
*
|
||
|
|
CALL ZLASET( 'Full', N, P, CZERO, CZERO, T, LDB )
|
||
|
|
IF( N.LE.P ) THEN
|
||
|
|
CALL ZLACPY( 'Upper', N, N, BF( 1, P-N+1 ), LDB, T( 1, P-N+1 ),
|
||
|
|
$ LDB )
|
||
|
|
ELSE
|
||
|
|
CALL ZLACPY( 'Full', N-P, P, BF, LDB, T, LDB )
|
||
|
|
CALL ZLACPY( 'Upper', P, P, BF( N-P+1, 1 ), LDB, T( N-P+1, 1 ),
|
||
|
|
$ LDB )
|
||
|
|
END IF
|
||
|
|
*
|
||
|
|
* Compute R - Q'*A
|
||
|
|
*
|
||
|
|
CALL ZGEMM( 'Conjugate transpose', 'No transpose', N, M, N, -CONE,
|
||
|
|
$ Q, LDA, A, LDA, CONE, R, LDA )
|
||
|
|
*
|
||
|
|
* Compute norm( R - Q'*A ) / ( MAX(M,N)*norm(A)*ULP ) .
|
||
|
|
*
|
||
|
|
RESID = ZLANGE( '1', N, M, R, LDA, RWORK )
|
||
|
|
IF( ANORM.GT.ZERO ) THEN
|
||
|
|
RESULT( 1 ) = ( ( RESID / DBLE( MAX( 1, M, N ) ) ) / ANORM ) /
|
||
|
|
$ ULP
|
||
|
|
ELSE
|
||
|
|
RESULT( 1 ) = ZERO
|
||
|
|
END IF
|
||
|
|
*
|
||
|
|
* Compute T*Z - Q'*B
|
||
|
|
*
|
||
|
|
CALL ZGEMM( 'No Transpose', 'No transpose', N, P, P, CONE, T, LDB,
|
||
|
|
$ Z, LDB, CZERO, BWK, LDB )
|
||
|
|
CALL ZGEMM( 'Conjugate transpose', 'No transpose', N, P, N, -CONE,
|
||
|
|
$ Q, LDA, B, LDB, CONE, BWK, LDB )
|
||
|
|
*
|
||
|
|
* Compute norm( T*Z - Q'*B ) / ( MAX(P,N)*norm(A)*ULP ) .
|
||
|
|
*
|
||
|
|
RESID = ZLANGE( '1', N, P, BWK, 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 - Q'*Q
|
||
|
|
*
|
||
|
|
CALL ZLASET( 'Full', N, N, CZERO, CONE, R, LDA )
|
||
|
|
CALL ZHERK( 'Upper', 'Conjugate transpose', N, N, -ONE, Q, LDA,
|
||
|
|
$ ONE, R, LDA )
|
||
|
|
*
|
||
|
|
* Compute norm( I - Q'*Q ) / ( N * ULP ) .
|
||
|
|
*
|
||
|
|
RESID = ZLANHE( '1', 'Upper', N, R, LDA, RWORK )
|
||
|
|
RESULT( 3 ) = ( RESID / DBLE( MAX( 1, N ) ) ) / ULP
|
||
|
|
*
|
||
|
|
* Compute I - Z'*Z
|
||
|
|
*
|
||
|
|
CALL ZLASET( 'Full', P, P, CZERO, CONE, T, LDB )
|
||
|
|
CALL ZHERK( 'Upper', 'Conjugate transpose', P, P, -ONE, Z, LDB,
|
||
|
|
$ ONE, T, LDB )
|
||
|
|
*
|
||
|
|
* Compute norm( I - Z'*Z ) / ( P*ULP ) .
|
||
|
|
*
|
||
|
|
RESID = ZLANHE( '1', 'Upper', P, T, LDB, RWORK )
|
||
|
|
RESULT( 4 ) = ( RESID / DBLE( MAX( 1, P ) ) ) / ULP
|
||
|
|
*
|
||
|
|
RETURN
|
||
|
|
*
|
||
|
|
* End of ZGQRTS
|
||
|
|
*
|
||
|
|
END
|