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*> \brief \b DSYT22
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition:
* ===========
*
* SUBROUTINE DSYT22( ITYPE, UPLO, N, M, KBAND, A, LDA, D, E, U, LDU,
* V, LDV, TAU, WORK, RESULT )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER ITYPE, KBAND, LDA, LDU, LDV, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), RESULT( 2 ),
* $ TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSYT22 generally checks a decomposition of the form
*>
*> A U = U S
*>
*> where A is symmetric, the columns of U are orthonormal, and S
*> is diagonal (if KBAND=0) or symmetric tridiagonal (if
*> KBAND=1). If ITYPE=1, then U is represented as a dense matrix,
*> otherwise the U is expressed as a product of Householder
*> transformations, whose vectors are stored in the array "V" and
*> whose scaling constants are in "TAU"; we shall use the letter
*> "V" to refer to the product of Householder transformations
*> (which should be equal to U).
*>
*> Specifically, if ITYPE=1, then:
*>
*> RESULT(1) = | U' A U - S | / ( |A| m ulp ) *andC> RESULT(2) = | I - U'U | / ( m ulp )
*> \endverbatim
*
* Arguments:
* ==========
*
*> \verbatim
*> ITYPE INTEGER
*> Specifies the type of tests to be performed.
*> 1: U expressed as a dense orthogonal matrix:
*> RESULT(1) = | A - U S U' | / ( |A| n ulp ) *andC> RESULT(2) = | I - UU' | / ( n ulp )
*>
*> UPLO CHARACTER
*> If UPLO='U', the upper triangle of A will be used and the
*> (strictly) lower triangle will not be referenced. If
*> UPLO='L', the lower triangle of A will be used and the
*> (strictly) upper triangle will not be referenced.
*> Not modified.
*>
*> N INTEGER
*> The size of the matrix. If it is zero, DSYT22 does nothing.
*> It must be at least zero.
*> Not modified.
*>
*> M INTEGER
*> The number of columns of U. If it is zero, DSYT22 does
*> nothing. It must be at least zero.
*> Not modified.
*>
*> KBAND INTEGER
*> The bandwidth of the matrix. 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.
*> Not modified.
*>
*> A 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.
*> Not modified.
*>
*> LDA INTEGER
*> The leading dimension of A. It must be at least 1
*> and at least N.
*> Not modified.
*>
*> D DOUBLE PRECISION array, dimension (N)
*> The diagonal of the (symmetric tri-) diagonal matrix.
*> Not modified.
*>
*> E DOUBLE PRECISION array, dimension (N)
*> The off-diagonal of the (symmetric tri-) diagonal matrix.
*> E(1) is ignored, E(2) is the (1,2) and (2,1) element, etc.
*> Not referenced if KBAND=0.
*> Not modified.
*>
*> U DOUBLE PRECISION array, dimension (LDU, N)
*> If ITYPE=1 or 3, this contains the orthogonal matrix in
*> the decomposition, expressed as a dense matrix. If ITYPE=2,
*> then it is not referenced.
*> Not modified.
*>
*> LDU INTEGER
*> The leading dimension of U. LDU must be at least N and
*> at least 1.
*> Not modified.
*>
*> V DOUBLE PRECISION array, dimension (LDV, N)
*> If ITYPE=2 or 3, the lower triangle of this array contains
*> the Householder vectors used to describe the orthogonal
*> matrix in the decomposition. If ITYPE=1, then it is not
*> referenced.
*> Not modified.
*>
*> LDV INTEGER
*> The leading dimension of V. LDV must be at least N and
*> at least 1.
*> Not modified.
*>
*> TAU DOUBLE PRECISION array, dimension (N)
*> If ITYPE >= 2, then TAU(j) is the scalar factor of
*> v(j) v(j)' in the Householder transformation H(j) of
*> the product U = H(1)...H(n-2)
*> If ITYPE < 2, then TAU is not referenced.
*> Not modified.
*>
*> WORK DOUBLE PRECISION array, dimension (2*N**2)
*> Workspace.
*> Modified.
*>
*> RESULT 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.
*> RESULT(1) is always modified. RESULT(2) is modified only
*> if LDU is at least N.
*> Modified.
*> \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 DSYT22( ITYPE, UPLO, N, M, KBAND, A, LDA, D, E, U, LDU,
$ V, LDV, TAU, 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 ITYPE, KBAND, LDA, LDU, LDV, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), RESULT( 2 ),
$ TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
* ..
* .. Local Scalars ..
INTEGER J, JJ, JJ1, JJ2, NN, NNP1
DOUBLE PRECISION ANORM, ULP, UNFL, WNORM
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH, DLANSY
EXTERNAL DLAMCH, DLANSY
* ..
* .. External Subroutines ..
EXTERNAL DGEMM, DORT01, DSYMM
* ..
* .. Intrinsic Functions ..
INTRINSIC DBLE, MAX, MIN
* ..
* .. Executable Statements ..
*
RESULT( 1 ) = ZERO
RESULT( 2 ) = ZERO
IF( N.LE.0 .OR. M.LE.0 )
$ RETURN
*
UNFL = DLAMCH( 'Safe minimum' )
ULP = DLAMCH( 'Precision' )
*
* Do Test 1
*
* Norm of A:
*
ANORM = MAX( DLANSY( '1', UPLO, N, A, LDA, WORK ), UNFL )
*
* Compute error matrix:
*
* ITYPE=1: error = U' A U - S
*
CALL DSYMM( 'L', UPLO, N, M, ONE, A, LDA, U, LDU, ZERO, WORK, N )
NN = N*N
NNP1 = NN + 1
CALL DGEMM( 'T', 'N', M, M, N, ONE, U, LDU, WORK, N, ZERO,
$ WORK( NNP1 ), N )
DO 10 J = 1, M
JJ = NN + ( J-1 )*N + J
WORK( JJ ) = WORK( JJ ) - D( J )
10 CONTINUE
IF( KBAND.EQ.1 .AND. N.GT.1 ) THEN
DO 20 J = 2, M
JJ1 = NN + ( J-1 )*N + J - 1
JJ2 = NN + ( J-2 )*N + J
WORK( JJ1 ) = WORK( JJ1 ) - E( J-1 )
WORK( JJ2 ) = WORK( JJ2 ) - E( J-1 )
20 CONTINUE
END IF
WNORM = DLANSY( '1', UPLO, M, WORK( NNP1 ), N, WORK( 1 ) )
*
IF( ANORM.GT.WNORM ) THEN
RESULT( 1 ) = ( WNORM / ANORM ) / ( M*ULP )
ELSE
IF( ANORM.LT.ONE ) THEN
RESULT( 1 ) = ( MIN( WNORM, M*ANORM ) / ANORM ) / ( M*ULP )
ELSE
RESULT( 1 ) = MIN( WNORM / ANORM, DBLE( M ) ) / ( M*ULP )
END IF
END IF
*
* Do Test 2
*
* Compute U'U - I
*
IF( ITYPE.EQ.1 )
$ CALL DORT01( 'Columns', N, M, U, LDU, WORK, 2*N*N,
$ RESULT( 2 ) )
*
RETURN
*
* End of DSYT22
*
END
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