*> \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