added lapack 3.7.0 with latest patches from git
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Werner Saar
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*> \brief \b DSBT21
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*
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* =========== DOCUMENTATION ===========
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*
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* Online html documentation available at
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* http://www.netlib.org/lapack/explore-html/
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*
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* Definition:
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* ===========
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*
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* SUBROUTINE DSBT21( UPLO, N, KA, KS, A, LDA, D, E, U, LDU, WORK,
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* RESULT )
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*
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* .. Scalar Arguments ..
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* CHARACTER UPLO
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* INTEGER KA, KS, LDA, LDU, N
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* ..
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* .. Array Arguments ..
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* DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), RESULT( 2 ),
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* $ U( LDU, * ), WORK( * )
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* ..
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*
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*
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*> \par Purpose:
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* =============
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*>
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*> \verbatim
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*>
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*> DSBT21 generally checks a decomposition of the form
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*>
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*> A = U S U'
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*>
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*> where ' means transpose, A is symmetric banded, U is
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*> orthogonal, and S is diagonal (if KS=0) or symmetric
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*> tridiagonal (if KS=1).
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*>
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*> Specifically:
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*>
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*> RESULT(1) = | A - U S U' | / ( |A| n ulp ) *andC> RESULT(2) = | I - UU' | / ( n ulp )
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*> \endverbatim
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*
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* Arguments:
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* ==========
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*
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*> \param[in] UPLO
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*> \verbatim
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*> UPLO is CHARACTER
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*> If UPLO='U', the upper triangle of A and V will be used and
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*> the (strictly) lower triangle will not be referenced.
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*> If UPLO='L', the lower triangle of A and V will be used and
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*> the (strictly) upper triangle will not be referenced.
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*> \endverbatim
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*>
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*> \param[in] N
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*> \verbatim
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*> N is INTEGER
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*> The size of the matrix. If it is zero, DSBT21 does nothing.
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*> It must be at least zero.
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*> \endverbatim
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*>
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*> \param[in] KA
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*> \verbatim
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*> KA is INTEGER
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*> The bandwidth of the matrix A. It must be at least zero. If
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*> it is larger than N-1, then max( 0, N-1 ) will be used.
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*> \endverbatim
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*>
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*> \param[in] KS
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*> \verbatim
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*> KS is INTEGER
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*> The bandwidth of the matrix S. It may only be zero or one.
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*> If zero, then S is diagonal, and E is not referenced. If
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*> one, then S is symmetric tri-diagonal.
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*> \endverbatim
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*>
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*> \param[in] A
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*> \verbatim
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*> A is DOUBLE PRECISION array, dimension (LDA, N)
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*> The original (unfactored) matrix. It is assumed to be
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*> symmetric, and only the upper (UPLO='U') or only the lower
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*> (UPLO='L') will be referenced.
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*> \endverbatim
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*>
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*> \param[in] LDA
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*> \verbatim
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*> LDA is INTEGER
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*> The leading dimension of A. It must be at least 1
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*> and at least min( KA, N-1 ).
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*> \endverbatim
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*>
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*> \param[in] D
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*> \verbatim
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*> D is DOUBLE PRECISION array, dimension (N)
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*> The diagonal of the (symmetric tri-) diagonal matrix S.
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*> \endverbatim
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*>
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*> \param[in] E
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*> \verbatim
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*> E is DOUBLE PRECISION array, dimension (N-1)
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*> The off-diagonal of the (symmetric tri-) diagonal matrix S.
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*> E(1) is the (1,2) and (2,1) element, E(2) is the (2,3) and
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*> (3,2) element, etc.
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*> Not referenced if KS=0.
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*> \endverbatim
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*>
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*> \param[in] U
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*> \verbatim
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*> U is DOUBLE PRECISION array, dimension (LDU, N)
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*> The orthogonal matrix in the decomposition, expressed as a
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*> dense matrix (i.e., not as a product of Householder
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*> transformations, Givens transformations, etc.)
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*> \endverbatim
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*>
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*> \param[in] LDU
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*> \verbatim
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*> LDU is INTEGER
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*> The leading dimension of U. LDU must be at least N and
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*> at least 1.
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*> \endverbatim
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*>
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*> \param[out] WORK
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*> \verbatim
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*> WORK is DOUBLE PRECISION array, dimension (N**2+N)
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*> \endverbatim
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*>
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*> \param[out] RESULT
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*> \verbatim
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*> RESULT is DOUBLE PRECISION array, dimension (2)
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*> The values computed by the two tests described above. The
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*> values are currently limited to 1/ulp, to avoid overflow.
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*> \endverbatim
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*
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* Authors:
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* ========
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*
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*> \author Univ. of Tennessee
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*> \author Univ. of California Berkeley
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*> \author Univ. of Colorado Denver
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*> \author NAG Ltd.
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*
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*> \date December 2016
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*
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*> \ingroup double_eig
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*
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* =====================================================================
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SUBROUTINE DSBT21( UPLO, N, KA, KS, A, LDA, D, E, U, LDU, WORK,
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$ RESULT )
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*
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* -- LAPACK test routine (version 3.7.0) --
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* -- LAPACK is a software package provided by Univ. of Tennessee, --
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* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
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* December 2016
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*
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* .. Scalar Arguments ..
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CHARACTER UPLO
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INTEGER KA, KS, LDA, LDU, N
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* ..
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* .. Array Arguments ..
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DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), RESULT( 2 ),
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$ U( LDU, * ), WORK( * )
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* ..
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*
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* =====================================================================
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*
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* .. Parameters ..
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DOUBLE PRECISION ZERO, ONE
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PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
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* ..
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* .. Local Scalars ..
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LOGICAL LOWER
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CHARACTER CUPLO
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INTEGER IKA, J, JC, JR, LW
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DOUBLE PRECISION ANORM, ULP, UNFL, WNORM
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* ..
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* .. External Functions ..
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LOGICAL LSAME
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DOUBLE PRECISION DLAMCH, DLANGE, DLANSB, DLANSP
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EXTERNAL LSAME, DLAMCH, DLANGE, DLANSB, DLANSP
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* ..
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* .. External Subroutines ..
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EXTERNAL DGEMM, DSPR, DSPR2
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC DBLE, MAX, MIN
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* ..
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* .. Executable Statements ..
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*
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* Constants
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*
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RESULT( 1 ) = ZERO
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RESULT( 2 ) = ZERO
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IF( N.LE.0 )
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$ RETURN
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*
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IKA = MAX( 0, MIN( N-1, KA ) )
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LW = ( N*( N+1 ) ) / 2
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*
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IF( LSAME( UPLO, 'U' ) ) THEN
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LOWER = .FALSE.
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CUPLO = 'U'
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ELSE
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LOWER = .TRUE.
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CUPLO = 'L'
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END IF
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*
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UNFL = DLAMCH( 'Safe minimum' )
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ULP = DLAMCH( 'Epsilon' )*DLAMCH( 'Base' )
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*
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* Some Error Checks
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*
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* Do Test 1
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*
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* Norm of A:
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*
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ANORM = MAX( DLANSB( '1', CUPLO, N, IKA, A, LDA, WORK ), UNFL )
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*
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* Compute error matrix: Error = A - U S U'
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*
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* Copy A from SB to SP storage format.
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*
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J = 0
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DO 50 JC = 1, N
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IF( LOWER ) THEN
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DO 10 JR = 1, MIN( IKA+1, N+1-JC )
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J = J + 1
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WORK( J ) = A( JR, JC )
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10 CONTINUE
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DO 20 JR = IKA + 2, N + 1 - JC
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J = J + 1
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WORK( J ) = ZERO
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20 CONTINUE
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ELSE
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DO 30 JR = IKA + 2, JC
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J = J + 1
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WORK( J ) = ZERO
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30 CONTINUE
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DO 40 JR = MIN( IKA, JC-1 ), 0, -1
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J = J + 1
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WORK( J ) = A( IKA+1-JR, JC )
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40 CONTINUE
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END IF
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50 CONTINUE
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*
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DO 60 J = 1, N
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CALL DSPR( CUPLO, N, -D( J ), U( 1, J ), 1, WORK )
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60 CONTINUE
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*
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IF( N.GT.1 .AND. KS.EQ.1 ) THEN
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DO 70 J = 1, N - 1
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CALL DSPR2( CUPLO, N, -E( J ), U( 1, J ), 1, U( 1, J+1 ), 1,
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$ WORK )
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70 CONTINUE
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END IF
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WNORM = DLANSP( '1', CUPLO, N, WORK, WORK( LW+1 ) )
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*
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IF( ANORM.GT.WNORM ) THEN
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RESULT( 1 ) = ( WNORM / ANORM ) / ( N*ULP )
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ELSE
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IF( ANORM.LT.ONE ) THEN
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RESULT( 1 ) = ( MIN( WNORM, N*ANORM ) / ANORM ) / ( N*ULP )
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ELSE
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RESULT( 1 ) = MIN( WNORM / ANORM, DBLE( N ) ) / ( N*ULP )
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END IF
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END IF
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*
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* Do Test 2
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*
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* Compute UU' - I
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*
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CALL DGEMM( 'N', 'C', N, N, N, ONE, U, LDU, U, LDU, ZERO, WORK,
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$ N )
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*
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DO 80 J = 1, N
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WORK( ( N+1 )*( J-1 )+1 ) = WORK( ( N+1 )*( J-1 )+1 ) - ONE
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80 CONTINUE
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*
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RESULT( 2 ) = MIN( DLANGE( '1', N, N, WORK, N, WORK( N**2+1 ) ),
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$ DBLE( N ) ) / ( N*ULP )
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*
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RETURN
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*
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* End of DSBT21
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*
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END
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