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*> \brief \b DSVDCH
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition:
* ===========
*
* SUBROUTINE DSVDCH( N, S, E, SVD, TOL, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, N
* DOUBLE PRECISION TOL
* ..
* .. Array Arguments ..
* DOUBLE PRECISION E( * ), S( * ), SVD( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSVDCH checks to see if SVD(1) ,..., SVD(N) are accurate singular
*> values of the bidiagonal matrix B with diagonal entries
*> S(1) ,..., S(N) and superdiagonal entries E(1) ,..., E(N-1)).
*> It does this by expanding each SVD(I) into an interval
*> [SVD(I) * (1-EPS) , SVD(I) * (1+EPS)], merging overlapping intervals
*> if any, and using Sturm sequences to count and verify whether each
*> resulting interval has the correct number of singular values (using
*> DSVDCT). Here EPS=TOL*MAX(N/10,1)*MAZHEP, where MACHEP is the
*> machine precision. The routine assumes the singular values are sorted
*> with SVD(1) the largest and SVD(N) smallest. If each interval
*> contains the correct number of singular values, INFO = 0 is returned,
*> otherwise INFO is the index of the first singular value in the first
*> bad interval.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The dimension of the bidiagonal matrix B.
*> \endverbatim
*>
*> \param[in] S
*> \verbatim
*> S is DOUBLE PRECISION array, dimension (N)
*> The diagonal entries of the bidiagonal matrix B.
*> \endverbatim
*>
*> \param[in] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (N-1)
*> The superdiagonal entries of the bidiagonal matrix B.
*> \endverbatim
*>
*> \param[in] SVD
*> \verbatim
*> SVD is DOUBLE PRECISION array, dimension (N)
*> The computed singular values to be checked.
*> \endverbatim
*>
*> \param[in] TOL
*> \verbatim
*> TOL is DOUBLE PRECISION
*> Error tolerance for checking, a multiplier of the
*> machine precision.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> =0 if the singular values are all correct (to within
*> 1 +- TOL*MAZHEPS)
*> >0 if the interval containing the INFO-th singular value
*> contains the incorrect number of singular values.
*> \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 DSVDCH( N, S, E, SVD, TOL, INFO )
*
* -- 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 ..
INTEGER INFO, N
DOUBLE PRECISION TOL
* ..
* .. Array Arguments ..
DOUBLE PRECISION E( * ), S( * ), SVD( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D0 )
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D0 )
* ..
* .. Local Scalars ..
INTEGER BPNT, COUNT, NUML, NUMU, TPNT
DOUBLE PRECISION EPS, LOWER, OVFL, TUPPR, UNFL, UNFLEP, UPPER
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. External Subroutines ..
EXTERNAL DSVDCT
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, SQRT
* ..
* .. Executable Statements ..
*
* Get machine constants
*
INFO = 0
IF( N.LE.0 )
$ RETURN
UNFL = DLAMCH( 'Safe minimum' )
OVFL = DLAMCH( 'Overflow' )
EPS = DLAMCH( 'Epsilon' )*DLAMCH( 'Base' )
*
* UNFLEP is chosen so that when an eigenvalue is multiplied by the
* scale factor sqrt(OVFL)*sqrt(sqrt(UNFL))/MX in DSVDCT, it exceeds
* sqrt(UNFL), which is the lower limit for DSVDCT.
*
UNFLEP = ( SQRT( SQRT( UNFL ) ) / SQRT( OVFL ) )*SVD( 1 ) +
$ UNFL / EPS
*
* The value of EPS works best when TOL .GE. 10.
*
EPS = TOL*MAX( N / 10, 1 )*EPS
*
* TPNT points to singular value at right endpoint of interval
* BPNT points to singular value at left endpoint of interval
*
TPNT = 1
BPNT = 1
*
* Begin loop over all intervals
*
10 CONTINUE
UPPER = ( ONE+EPS )*SVD( TPNT ) + UNFLEP
LOWER = ( ONE-EPS )*SVD( BPNT ) - UNFLEP
IF( LOWER.LE.UNFLEP )
$ LOWER = -UPPER
*
* Begin loop merging overlapping intervals
*
20 CONTINUE
IF( BPNT.EQ.N )
$ GO TO 30
TUPPR = ( ONE+EPS )*SVD( BPNT+1 ) + UNFLEP
IF( TUPPR.LT.LOWER )
$ GO TO 30
*
* Merge
*
BPNT = BPNT + 1
LOWER = ( ONE-EPS )*SVD( BPNT ) - UNFLEP
IF( LOWER.LE.UNFLEP )
$ LOWER = -UPPER
GO TO 20
30 CONTINUE
*
* Count singular values in interval [ LOWER, UPPER ]
*
CALL DSVDCT( N, S, E, LOWER, NUML )
CALL DSVDCT( N, S, E, UPPER, NUMU )
COUNT = NUMU - NUML
IF( LOWER.LT.ZERO )
$ COUNT = COUNT / 2
IF( COUNT.NE.BPNT-TPNT+1 ) THEN
*
* Wrong number of singular values in interval
*
INFO = TPNT
GO TO 40
END IF
TPNT = BPNT + 1
BPNT = TPNT
IF( TPNT.LE.N )
$ GO TO 10
40 CONTINUE
RETURN
*
* End of DSVDCH
*
END
|
