*> \brief \b DSYT21 * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * * Definition: * =========== * * SUBROUTINE DSYT21( ITYPE, UPLO, N, KBAND, A, LDA, D, E, U, LDU, V, * LDV, TAU, WORK, RESULT ) * * .. Scalar Arguments .. * CHARACTER UPLO * INTEGER ITYPE, KBAND, LDA, LDU, LDV, N * .. * .. Array Arguments .. * DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), RESULT( 2 ), * $ TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> DSYT21 generally checks a decomposition of the form *> *> A = U S U**T *> *> where **T means transpose, A is symmetric, U is orthogonal, 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 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) = | A - U S U**T | / ( |A| n ulp ) and *> RESULT(2) = | I - U U**T | / ( n ulp ) *> *> If ITYPE=2, then: *> *> RESULT(1) = | A - V S V**T | / ( |A| n ulp ) *> *> If ITYPE=3, then: *> *> RESULT(1) = | I - V U**T | / ( n ulp ) *> *> For ITYPE > 1, the transformation U is expressed as a product *> V = H(1)...H(n-2), where H(j) = I - tau(j) v(j) v(j)**T and each *> vector v(j) has its first j elements 0 and the remaining n-j elements *> stored in V(j+1:n,j). *> \endverbatim * * Arguments: * ========== * *> \param[in] ITYPE *> \verbatim *> ITYPE is INTEGER *> Specifies the type of tests to be performed. *> 1: U expressed as a dense orthogonal matrix: *> RESULT(1) = | A - U S U**T | / ( |A| n ulp ) and *> RESULT(2) = | I - U U**T | / ( n ulp ) *> *> 2: U expressed as a product V of Housholder transformations: *> RESULT(1) = | A - V S V**T | / ( |A| n ulp ) *> *> 3: U expressed both as a dense orthogonal matrix and *> as a product of Housholder transformations: *> RESULT(1) = | I - V U**T | / ( n ulp ) *> \endverbatim *> *> \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, DSYT21 does nothing. *> It must be at least zero. *> \endverbatim *> *> \param[in] KBAND *> \verbatim *> KBAND is 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. *> \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 N. *> \endverbatim *> *> \param[in] D *> \verbatim *> D is DOUBLE PRECISION array, dimension (N) *> The diagonal of the (symmetric tri-) diagonal matrix. *> \endverbatim *> *> \param[in] E *> \verbatim *> E is DOUBLE PRECISION array, dimension (N-1) *> The off-diagonal of the (symmetric tri-) diagonal matrix. *> E(1) is the (1,2) and (2,1) element, E(2) is the (2,3) and *> (3,2) element, etc. *> Not referenced if KBAND=0. *> \endverbatim *> *> \param[in] U *> \verbatim *> U is 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. *> \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[in] V *> \verbatim *> V is DOUBLE PRECISION array, dimension (LDV, N) *> If ITYPE=2 or 3, the columns of this array contain the *> Householder vectors used to describe the orthogonal matrix *> in the decomposition. If UPLO='L', then the vectors are in *> the lower triangle, if UPLO='U', then in the upper *> triangle. *> *NOTE* If ITYPE=2 or 3, V is modified and restored. The *> subdiagonal (if UPLO='L') or the superdiagonal (if UPLO='U') *> is set to one, and later reset to its original value, during *> the course of the calculation. *> If ITYPE=1, then it is neither referenced nor modified. *> \endverbatim *> *> \param[in] LDV *> \verbatim *> LDV is INTEGER *> The leading dimension of V. LDV must be at least N and *> at least 1. *> \endverbatim *> *> \param[in] TAU *> \verbatim *> TAU is DOUBLE PRECISION array, dimension (N) *> If ITYPE >= 2, then TAU(j) is the scalar factor of *> v(j) v(j)**T in the Householder transformation H(j) of *> the product U = H(1)...H(n-2) *> If ITYPE < 2, then TAU is not referenced. *> \endverbatim *> *> \param[out] WORK *> \verbatim *> WORK is DOUBLE PRECISION array, dimension (2*N**2) *> \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. *> RESULT(1) is always modified. RESULT(2) is modified only *> if ITYPE=1. *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \ingroup double_eig * * ===================================================================== SUBROUTINE DSYT21( ITYPE, UPLO, N, KBAND, A, LDA, D, E, U, LDU, V, $ LDV, TAU, WORK, RESULT ) * * -- LAPACK test routine -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * * .. Scalar Arguments .. CHARACTER UPLO INTEGER ITYPE, KBAND, LDA, LDU, LDV, N * .. * .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), RESULT( 2 ), $ TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * ) * .. * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ZERO, ONE, TEN PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, TEN = 10.0D0 ) * .. * .. Local Scalars .. LOGICAL LOWER CHARACTER CUPLO INTEGER IINFO, J, JCOL, JR, JROW DOUBLE PRECISION ANORM, ULP, UNFL, VSAVE, WNORM * .. * .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DLAMCH, DLANGE, DLANSY EXTERNAL LSAME, DLAMCH, DLANGE, DLANSY * .. * .. External Subroutines .. EXTERNAL DGEMM, DLACPY, DLARFY, DLASET, DORM2L, DORM2R, $ DSYR, DSYR2 * .. * .. Intrinsic Functions .. INTRINSIC DBLE, MAX, MIN * .. * .. Executable Statements .. * RESULT( 1 ) = ZERO IF( ITYPE.EQ.1 ) $ RESULT( 2 ) = ZERO IF( N.LE.0 ) $ RETURN * 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 * IF( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN RESULT( 1 ) = TEN / ULP RETURN END IF * * Do Test 1 * * Norm of A: * IF( ITYPE.EQ.3 ) THEN ANORM = ONE ELSE ANORM = MAX( DLANSY( '1', CUPLO, N, A, LDA, WORK ), UNFL ) END IF * * Compute error matrix: * IF( ITYPE.EQ.1 ) THEN * * ITYPE=1: error = A - U S U**T * CALL DLASET( 'Full', N, N, ZERO, ZERO, WORK, N ) CALL DLACPY( CUPLO, N, N, A, LDA, WORK, N ) * DO 10 J = 1, N CALL DSYR( CUPLO, N, -D( J ), U( 1, J ), 1, WORK, N ) 10 CONTINUE * IF( N.GT.1 .AND. KBAND.EQ.1 ) THEN DO 20 J = 1, N - 1 CALL DSYR2( CUPLO, N, -E( J ), U( 1, J ), 1, U( 1, J+1 ), $ 1, WORK, N ) 20 CONTINUE END IF WNORM = DLANSY( '1', CUPLO, N, WORK, N, WORK( N**2+1 ) ) * ELSE IF( ITYPE.EQ.2 ) THEN * * ITYPE=2: error = V S V**T - A * CALL DLASET( 'Full', N, N, ZERO, ZERO, WORK, N ) * IF( LOWER ) THEN WORK( N**2 ) = D( N ) DO 40 J = N - 1, 1, -1 IF( KBAND.EQ.1 ) THEN WORK( ( N+1 )*( J-1 )+2 ) = ( ONE-TAU( J ) )*E( J ) DO 30 JR = J + 2, N WORK( ( J-1 )*N+JR ) = -TAU( J )*E( J )*V( JR, J ) 30 CONTINUE END IF * VSAVE = V( J+1, J ) V( J+1, J ) = ONE CALL DLARFY( 'L', N-J, V( J+1, J ), 1, TAU( J ), $ WORK( ( N+1 )*J+1 ), N, WORK( N**2+1 ) ) V( J+1, J ) = VSAVE WORK( ( N+1 )*( J-1 )+1 ) = D( J ) 40 CONTINUE ELSE WORK( 1 ) = D( 1 ) DO 60 J = 1, N - 1 IF( KBAND.EQ.1 ) THEN WORK( ( N+1 )*J ) = ( ONE-TAU( J ) )*E( J ) DO 50 JR = 1, J - 1 WORK( J*N+JR ) = -TAU( J )*E( J )*V( JR, J+1 ) 50 CONTINUE END IF * VSAVE = V( J, J+1 ) V( J, J+1 ) = ONE CALL DLARFY( 'U', J, V( 1, J+1 ), 1, TAU( J ), WORK, N, $ WORK( N**2+1 ) ) V( J, J+1 ) = VSAVE WORK( ( N+1 )*J+1 ) = D( J+1 ) 60 CONTINUE END IF * DO 90 JCOL = 1, N IF( LOWER ) THEN DO 70 JROW = JCOL, N WORK( JROW+N*( JCOL-1 ) ) = WORK( JROW+N*( JCOL-1 ) ) $ - A( JROW, JCOL ) 70 CONTINUE ELSE DO 80 JROW = 1, JCOL WORK( JROW+N*( JCOL-1 ) ) = WORK( JROW+N*( JCOL-1 ) ) $ - A( JROW, JCOL ) 80 CONTINUE END IF 90 CONTINUE WNORM = DLANSY( '1', CUPLO, N, WORK, N, WORK( N**2+1 ) ) * ELSE IF( ITYPE.EQ.3 ) THEN * * ITYPE=3: error = U V**T - I * IF( N.LT.2 ) $ RETURN CALL DLACPY( ' ', N, N, U, LDU, WORK, N ) IF( LOWER ) THEN CALL DORM2R( 'R', 'T', N, N-1, N-1, V( 2, 1 ), LDV, TAU, $ WORK( N+1 ), N, WORK( N**2+1 ), IINFO ) ELSE CALL DORM2L( 'R', 'T', N, N-1, N-1, V( 1, 2 ), LDV, TAU, $ WORK, N, WORK( N**2+1 ), IINFO ) END IF IF( IINFO.NE.0 ) THEN RESULT( 1 ) = TEN / ULP RETURN END IF * DO 100 J = 1, N WORK( ( N+1 )*( J-1 )+1 ) = WORK( ( N+1 )*( J-1 )+1 ) - ONE 100 CONTINUE * WNORM = DLANGE( '1', N, N, WORK, N, WORK( N**2+1 ) ) END IF * 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 U U**T - I * IF( ITYPE.EQ.1 ) THEN CALL DGEMM( 'N', 'C', N, N, N, ONE, U, LDU, U, LDU, ZERO, WORK, $ N ) * DO 110 J = 1, N WORK( ( N+1 )*( J-1 )+1 ) = WORK( ( N+1 )*( J-1 )+1 ) - ONE 110 CONTINUE * RESULT( 2 ) = MIN( DLANGE( '1', N, N, WORK, N, $ WORK( N**2+1 ) ), DBLE( N ) ) / ( N*ULP ) END IF * RETURN * * End of DSYT21 * END