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