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*> \brief \b DBDT03
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition:
* ===========
*
* SUBROUTINE DBDT03( UPLO, N, KD, D, E, U, LDU, S, VT, LDVT, WORK,
* RESID )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER KD, LDU, LDVT, N
* DOUBLE PRECISION RESID
* ..
* .. Array Arguments ..
* DOUBLE PRECISION D( * ), E( * ), S( * ), U( LDU, * ),
* $ VT( LDVT, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DBDT03 reconstructs a bidiagonal matrix B from its 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( B - U * S * VT ) / ( 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] KD
*> \verbatim
*> KD is INTEGER
*> The bandwidth of the bidiagonal matrix B. If KD = 1, the
*> matrix B is bidiagonal, and if KD = 0, B is diagonal and E is
*> not referenced. If KD is greater than 1, it is assumed to be
*> 1, and if KD is less than 0, it is assumed to be 0.
*> \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] U
*> \verbatim
*> U is DOUBLE PRECISION array, dimension (LDU,N)
*> The n by n orthogonal matrix U in the reduction B = U'*A*P.
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*> LDU is INTEGER
*> The leading dimension of the array U. LDU >= max(1,N)
*> \endverbatim
*>
*> \param[in] S
*> \verbatim
*> S is DOUBLE PRECISION array, dimension (N)
*> The singular values from the SVD of B, sorted in decreasing
*> order.
*> \endverbatim
*>
*> \param[in] VT
*> \verbatim
*> VT is DOUBLE PRECISION array, dimension (LDVT,N)
*> The n by n orthogonal matrix V' in the reduction
*> B = U * S * 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(B - U * S * V') / ( n * norm(A) * EPS )
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup double_eig
*
* =====================================================================
SUBROUTINE DBDT03( UPLO, N, KD, D, E, U, LDU, S, VT, LDVT, WORK,
$ RESID )
*
* -- LAPACK test routine (version 3.4.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER KD, LDU, LDVT, N
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
DOUBLE PRECISION BNORM, EPS
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER IDAMAX
DOUBLE PRECISION DASUM, DLAMCH
EXTERNAL LSAME, IDAMAX, DASUM, DLAMCH
* ..
* .. External Subroutines ..
EXTERNAL DGEMV
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, MAX, MIN
* ..
* .. Executable Statements ..
*
* Quick return if possible
*
RESID = ZERO
IF( N.LE.0 )
$ RETURN
*
* Compute B - U * S * V' one column at a time.
*
BNORM = ZERO
IF( KD.GE.1 ) THEN
*
* B is bidiagonal.
*
IF( LSAME( UPLO, 'U' ) ) THEN
*
* B is upper bidiagonal.
*
DO 20 J = 1, N
DO 10 I = 1, N
WORK( N+I ) = S( I )*VT( I, J )
10 CONTINUE
CALL DGEMV( 'No transpose', N, N, -ONE, U, LDU,
$ WORK( N+1 ), 1, ZERO, WORK, 1 )
WORK( J ) = WORK( J ) + D( J )
IF( J.GT.1 ) THEN
WORK( J-1 ) = WORK( J-1 ) + E( J-1 )
BNORM = MAX( BNORM, ABS( D( J ) )+ABS( E( J-1 ) ) )
ELSE
BNORM = MAX( BNORM, ABS( D( J ) ) )
END IF
RESID = MAX( RESID, DASUM( N, WORK, 1 ) )
20 CONTINUE
ELSE
*
* B is lower bidiagonal.
*
DO 40 J = 1, N
DO 30 I = 1, N
WORK( N+I ) = S( I )*VT( I, J )
30 CONTINUE
CALL DGEMV( 'No transpose', N, N, -ONE, U, LDU,
$ WORK( N+1 ), 1, ZERO, WORK, 1 )
WORK( J ) = WORK( J ) + D( J )
IF( J.LT.N ) THEN
WORK( J+1 ) = WORK( J+1 ) + E( J )
BNORM = MAX( BNORM, ABS( D( J ) )+ABS( E( J ) ) )
ELSE
BNORM = MAX( BNORM, ABS( D( J ) ) )
END IF
RESID = MAX( RESID, DASUM( N, WORK, 1 ) )
40 CONTINUE
END IF
ELSE
*
* B is diagonal.
*
DO 60 J = 1, N
DO 50 I = 1, N
WORK( N+I ) = S( I )*VT( I, J )
50 CONTINUE
CALL DGEMV( 'No transpose', N, N, -ONE, U, LDU, WORK( N+1 ),
$ 1, ZERO, WORK, 1 )
WORK( J ) = WORK( J ) + D( J )
RESID = MAX( RESID, DASUM( N, WORK, 1 ) )
60 CONTINUE
J = IDAMAX( N, D, 1 )
BNORM = ABS( D( J ) )
END IF
*
* Compute norm(B - U * S * V') / ( n * norm(B) * EPS )
*
EPS = DLAMCH( 'Precision' )
*
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 DBDT03
*
END
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