254 lines
7.9 KiB
Fortran
254 lines
7.9 KiB
Fortran
*> \brief \b SSYT22
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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 SSYT22( ITYPE, UPLO, N, M, KBAND, A, LDA, D, E, U, LDU,
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* V, LDV, TAU, WORK, RESULT )
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*
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* .. Scalar Arguments ..
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* CHARACTER UPLO
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* INTEGER ITYPE, KBAND, LDA, LDU, LDV, M, N
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* ..
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* .. Array Arguments ..
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* REAL A( LDA, * ), D( * ), E( * ), RESULT( 2 ),
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* $ TAU( * ), U( LDU, * ), V( LDV, * ), 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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*> SSYT22 generally checks a decomposition of the form
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*>
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*> A U = U S
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*>
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*> where A is symmetric, the columns of U are orthonormal, and S
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*> is diagonal (if KBAND=0) or symmetric tridiagonal (if
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*> KBAND=1). If ITYPE=1, then U is represented as a dense matrix,
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*> otherwise the U is expressed as a product of Householder
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*> transformations, whose vectors are stored in the array "V" and
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*> whose scaling constants are in "TAU"; we shall use the letter
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*> "V" to refer to the product of Householder transformations
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*> (which should be equal to U).
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*>
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*> Specifically, if ITYPE=1, then:
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*>
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*> RESULT(1) = | U' A U - S | / ( |A| m ulp ) *andC> RESULT(2) = | I - U'U | / ( m 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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*> \verbatim
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*> ITYPE INTEGER
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*> Specifies the type of tests to be performed.
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*> 1: U expressed as a dense orthogonal matrix:
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*> RESULT(1) = | A - U S U' | / ( |A| n ulp ) *andC> RESULT(2) = | I - UU' | / ( n ulp )
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*>
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*> UPLO CHARACTER
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*> If UPLO='U', the upper triangle of A will be used and the
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*> (strictly) lower triangle will not be referenced. If
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*> UPLO='L', the lower triangle of A will be used and the
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*> (strictly) upper triangle will not be referenced.
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*> Not modified.
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*>
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*> N INTEGER
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*> The size of the matrix. If it is zero, SSYT22 does nothing.
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*> It must be at least zero.
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*> Not modified.
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*>
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*> M INTEGER
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*> The number of columns of U. If it is zero, SSYT22 does
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*> nothing. It must be at least zero.
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*> Not modified.
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*>
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*> KBAND INTEGER
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*> The bandwidth of the matrix. 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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*> Not modified.
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*>
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*> A REAL 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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*> Not modified.
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*>
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*> LDA INTEGER
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*> The leading dimension of A. It must be at least 1
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*> and at least N.
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*> Not modified.
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*>
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*> D REAL array, dimension (N)
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*> The diagonal of the (symmetric tri-) diagonal matrix.
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*> Not modified.
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*>
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*> E REAL array, dimension (N)
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*> The off-diagonal of the (symmetric tri-) diagonal matrix.
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*> E(1) is ignored, E(2) is the (1,2) and (2,1) element, etc.
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*> Not referenced if KBAND=0.
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*> Not modified.
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*>
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*> U REAL array, dimension (LDU, N)
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*> If ITYPE=1 or 3, this contains the orthogonal matrix in
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*> the decomposition, expressed as a dense matrix. If ITYPE=2,
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*> then it is not referenced.
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*> Not modified.
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*>
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*> LDU 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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*> Not modified.
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*>
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*> V REAL array, dimension (LDV, N)
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*> If ITYPE=2 or 3, the lower triangle of this array contains
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*> the Householder vectors used to describe the orthogonal
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*> matrix in the decomposition. If ITYPE=1, then it is not
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*> referenced.
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*> Not modified.
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*>
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*> LDV INTEGER
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*> The leading dimension of V. LDV must be at least N and
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*> at least 1.
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*> Not modified.
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*>
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*> TAU REAL array, dimension (N)
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*> If ITYPE >= 2, then TAU(j) is the scalar factor of
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*> v(j) v(j)' in the Householder transformation H(j) of
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*> the product U = H(1)...H(n-2)
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*> If ITYPE < 2, then TAU is not referenced.
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*> Not modified.
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*>
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*> WORK REAL array, dimension (2*N**2)
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*> Workspace.
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*> Modified.
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*>
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*> RESULT REAL 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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*> RESULT(1) is always modified. RESULT(2) is modified only
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*> if LDU is at least N.
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*> Modified.
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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 November 2011
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*
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*> \ingroup single_eig
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*
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* =====================================================================
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SUBROUTINE SSYT22( ITYPE, UPLO, N, M, KBAND, A, LDA, D, E, U, LDU,
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$ V, LDV, TAU, WORK, RESULT )
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*
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* -- LAPACK test routine (version 3.4.0) --
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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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* November 2011
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*
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* .. Scalar Arguments ..
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CHARACTER UPLO
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INTEGER ITYPE, KBAND, LDA, LDU, LDV, M, N
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* ..
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* .. Array Arguments ..
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REAL A( LDA, * ), D( * ), E( * ), RESULT( 2 ),
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$ TAU( * ), U( LDU, * ), V( LDV, * ), 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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REAL ZERO, ONE
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PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0 )
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* ..
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* .. Local Scalars ..
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INTEGER J, JJ, JJ1, JJ2, NN, NNP1
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REAL ANORM, ULP, UNFL, WNORM
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* ..
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* .. External Functions ..
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REAL SLAMCH, SLANSY
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EXTERNAL SLAMCH, SLANSY
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* ..
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* .. External Subroutines ..
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EXTERNAL SGEMM, SSYMM
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC MAX, MIN, REAL
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* ..
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* .. Executable Statements ..
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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 .OR. M.LE.0 )
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$ RETURN
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*
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UNFL = SLAMCH( 'Safe minimum' )
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ULP = SLAMCH( 'Precision' )
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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( SLANSY( '1', UPLO, N, A, LDA, WORK ), UNFL )
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*
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* Compute error matrix:
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*
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* ITYPE=1: error = U' A U - S
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*
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CALL SSYMM( 'L', UPLO, N, M, ONE, A, LDA, U, LDU, ZERO, WORK, N )
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NN = N*N
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NNP1 = NN + 1
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CALL SGEMM( 'T', 'N', M, M, N, ONE, U, LDU, WORK, N, ZERO,
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$ WORK( NNP1 ), N )
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DO 10 J = 1, M
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JJ = NN + ( J-1 )*N + J
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WORK( JJ ) = WORK( JJ ) - D( J )
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10 CONTINUE
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IF( KBAND.EQ.1 .AND. N.GT.1 ) THEN
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DO 20 J = 2, M
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JJ1 = NN + ( J-1 )*N + J - 1
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JJ2 = NN + ( J-2 )*N + J
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WORK( JJ1 ) = WORK( JJ1 ) - E( J-1 )
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WORK( JJ2 ) = WORK( JJ2 ) - E( J-1 )
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20 CONTINUE
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END IF
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WNORM = SLANSY( '1', UPLO, M, WORK( NNP1 ), N, WORK( 1 ) )
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*
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IF( ANORM.GT.WNORM ) THEN
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RESULT( 1 ) = ( WNORM / ANORM ) / ( M*ULP )
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ELSE
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IF( ANORM.LT.ONE ) THEN
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RESULT( 1 ) = ( MIN( WNORM, M*ANORM ) / ANORM ) / ( M*ULP )
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ELSE
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RESULT( 1 ) = MIN( WNORM / ANORM, REAL( M ) ) / ( M*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 U'U - I
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*
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IF( ITYPE.EQ.1 )
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$ CALL SORT01( 'Columns', N, M, U, LDU, WORK, 2*N*N,
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$ RESULT( 2 ) )
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*
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RETURN
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*
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* End of SSYT22
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*
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END
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