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|
*> \brief \b DBDT04
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition:
* ===========
*
* SUBROUTINE DBDT04( UPLO, N, D, E, S, NS, U, LDU, VT, LDVT,
* WORK, RESID )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER LDU, LDVT, N, NS
* DOUBLE PRECISION RESID
* ..
* .. Array Arguments ..
* DOUBLE PRECISION D( * ), E( * ), S( * ), U( LDU, * ),
* $ VT( LDVT, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DBDT04 reconstructs a bidiagonal matrix B from its (partial) SVD:
*> S = U' * B * V
*> where U and V are orthogonal matrices and S is diagonal.
*>
*> The test ratio to test the singular value decomposition is
*> RESID = norm( S - U' * B * V ) / ( n * norm(B) * EPS )
*> where VT = V' and EPS is the machine precision.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> Specifies whether the matrix B is upper or lower bidiagonal.
*> = 'U': Upper bidiagonal
*> = 'L': Lower bidiagonal
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix B.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> The n diagonal elements of the bidiagonal matrix B.
*> \endverbatim
*>
*> \param[in] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (N-1)
*> The (n-1) superdiagonal elements of the bidiagonal matrix B
*> if UPLO = 'U', or the (n-1) subdiagonal elements of B if
*> UPLO = 'L'.
*> \endverbatim
*>
*> \param[in] S
*> \verbatim
*> S is DOUBLE PRECISION array, dimension (NS)
*> The singular values from the (partial) SVD of B, sorted in
*> decreasing order.
*> \endverbatim
*>
*> \param[in] NS
*> \verbatim
*> NS is INTEGER
*> The number of singular values/vectors from the (partial)
*> SVD of B.
*> \endverbatim
*>
*> \param[in] U
*> \verbatim
*> U is DOUBLE PRECISION array, dimension (LDU,NS)
*> The n by ns orthogonal matrix U in S = U' * B * V.
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*> LDU is INTEGER
*> The leading dimension of the array U. LDU >= max(1,N)
*> \endverbatim
*>
*> \param[in] VT
*> \verbatim
*> VT is DOUBLE PRECISION array, dimension (LDVT,N)
*> The n by ns orthogonal matrix V in S = U' * B * V.
*> \endverbatim
*>
*> \param[in] LDVT
*> \verbatim
*> LDVT is INTEGER
*> The leading dimension of the array VT.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (2*N)
*> \endverbatim
*>
*> \param[out] RESID
*> \verbatim
*> RESID is DOUBLE PRECISION
*> The test ratio: norm(S - U' * B * V) / ( n * norm(B) * EPS )
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date December 2016
*
*> \ingroup double_eig
*
* =====================================================================
SUBROUTINE DBDT04( UPLO, N, D, E, S, NS, U, LDU, VT, LDVT, WORK,
$ RESID )
*
* -- LAPACK test routine (version 3.7.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* December 2016
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER LDU, LDVT, N, NS
DOUBLE PRECISION RESID
* ..
* .. Array Arguments ..
DOUBLE PRECISION D( * ), E( * ), S( * ), U( LDU, * ),
$ VT( LDVT, * ), WORK( * )
* ..
*
* ======================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, J, K
DOUBLE PRECISION BNORM, EPS
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER IDAMAX
DOUBLE PRECISION DASUM, DLAMCH
EXTERNAL LSAME, IDAMAX, DASUM, DLAMCH
* ..
* .. External Subroutines ..
EXTERNAL DGEMM
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, MAX, MIN
* ..
* .. Executable Statements ..
*
* Quick return if possible.
*
RESID = ZERO
IF( N.LE.0 .OR. NS.LE.0 )
$ RETURN
*
EPS = DLAMCH( 'Precision' )
*
* Compute S - U' * B * V.
*
BNORM = ZERO
*
IF( LSAME( UPLO, 'U' ) ) THEN
*
* B is upper bidiagonal.
*
K = 0
DO 20 I = 1, NS
DO 10 J = 1, N-1
K = K + 1
WORK( K ) = D( J )*VT( I, J ) + E( J )*VT( I, J+1 )
10 CONTINUE
K = K + 1
WORK( K ) = D( N )*VT( I, N )
20 CONTINUE
BNORM = ABS( D( 1 ) )
DO 30 I = 2, N
BNORM = MAX( BNORM, ABS( D( I ) )+ABS( E( I-1 ) ) )
30 CONTINUE
ELSE
*
* B is lower bidiagonal.
*
K = 0
DO 50 I = 1, NS
K = K + 1
WORK( K ) = D( 1 )*VT( I, 1 )
DO 40 J = 1, N-1
K = K + 1
WORK( K ) = E( J )*VT( I, J ) + D( J+1 )*VT( I, J+1 )
40 CONTINUE
50 CONTINUE
BNORM = ABS( D( N ) )
DO 60 I = 1, N-1
BNORM = MAX( BNORM, ABS( D( I ) )+ABS( E( I ) ) )
60 CONTINUE
END IF
*
CALL DGEMM( 'T', 'N', NS, NS, N, -ONE, U, LDU, WORK( 1 ),
$ N, ZERO, WORK( 1+N*NS ), NS )
*
* norm(S - U' * B * V)
*
K = N*NS
DO 70 I = 1, NS
WORK( K+I ) = WORK( K+I ) + S( I )
RESID = MAX( RESID, DASUM( NS, WORK( K+1 ), 1 ) )
K = K + NS
70 CONTINUE
*
IF( BNORM.LE.ZERO ) THEN
IF( RESID.NE.ZERO )
$ RESID = ONE / EPS
ELSE
IF( BNORM.GE.RESID ) THEN
RESID = ( RESID / BNORM ) / ( DBLE( N )*EPS )
ELSE
IF( BNORM.LT.ONE ) THEN
RESID = ( MIN( RESID, DBLE( N )*BNORM ) / BNORM ) /
$ ( DBLE( N )*EPS )
ELSE
RESID = MIN( RESID / BNORM, DBLE( N ) ) /
$ ( DBLE( N )*EPS )
END IF
END IF
END IF
*
RETURN
*
* End of DBDT04
*
END
|
