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416 lines
13 KiB
416 lines
13 KiB
2 years ago
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*> \brief \b DSYT21
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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 DSYT21( ITYPE, UPLO, N, KBAND, A, LDA, D, E, U, LDU, V,
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* 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, N
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* ..
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* .. Array Arguments ..
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* DOUBLE PRECISION 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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*> DSYT21 generally checks a decomposition of the form
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*>
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*> A = U S U**T
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*>
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*> where **T means transpose, A is symmetric, U is orthogonal, and S is
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*> diagonal (if KBAND=0) or symmetric tridiagonal (if KBAND=1).
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*>
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*> If ITYPE=1, then U is represented as a dense matrix; otherwise U is
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*> expressed as a product of Householder transformations, whose vectors
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*> are stored in the array "V" and whose scaling constants are in "TAU".
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*> We shall use the letter "V" to refer to the product of Householder
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*> transformations (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) = | A - U S U**T | / ( |A| n ulp ) and
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*> RESULT(2) = | I - U U**T | / ( n ulp )
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*>
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*> If ITYPE=2, then:
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*>
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*> RESULT(1) = | A - V S V**T | / ( |A| n ulp )
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*>
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*> If ITYPE=3, then:
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*>
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*> RESULT(1) = | I - V U**T | / ( n ulp )
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*>
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*> For ITYPE > 1, the transformation U is expressed as a product
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*> V = H(1)...H(n-2), where H(j) = I - tau(j) v(j) v(j)**T and each
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*> vector v(j) has its first j elements 0 and the remaining n-j elements
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*> stored in V(j+1:n,j).
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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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*> \param[in] ITYPE
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*> \verbatim
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*> ITYPE is 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**T | / ( |A| n ulp ) and
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*> RESULT(2) = | I - U U**T | / ( n ulp )
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*>
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*> 2: U expressed as a product V of Housholder transformations:
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*> RESULT(1) = | A - V S V**T | / ( |A| n ulp )
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*>
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*> 3: U expressed both as a dense orthogonal matrix and
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*> as a product of Housholder transformations:
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*> RESULT(1) = | I - V U**T | / ( n ulp )
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*> \endverbatim
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*>
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*> \param[in] UPLO
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*> \verbatim
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*> UPLO is CHARACTER
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*> If UPLO='U', the upper triangle of A and V will be used and
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*> the (strictly) lower triangle will not be referenced.
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*> If UPLO='L', the lower triangle of A and V will be used and
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*> the (strictly) upper triangle will not be referenced.
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*> \endverbatim
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*>
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*> \param[in] N
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*> \verbatim
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*> N is INTEGER
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*> The size of the matrix. If it is zero, DSYT21 does nothing.
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*> It must be at least zero.
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*> \endverbatim
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*>
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*> \param[in] KBAND
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*> \verbatim
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*> KBAND is 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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*> \endverbatim
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*>
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*> \param[in] A
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*> \verbatim
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*> A is DOUBLE PRECISION 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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*> \endverbatim
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*>
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*> \param[in] LDA
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*> \verbatim
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*> LDA is 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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*> \endverbatim
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*>
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*> \param[in] D
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*> \verbatim
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*> D is DOUBLE PRECISION array, dimension (N)
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*> The diagonal of the (symmetric tri-) diagonal matrix.
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*> \endverbatim
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*>
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*> \param[in] E
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*> \verbatim
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*> E is DOUBLE PRECISION array, dimension (N-1)
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*> The off-diagonal of the (symmetric tri-) diagonal matrix.
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*> E(1) is the (1,2) and (2,1) element, E(2) is the (2,3) and
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*> (3,2) element, etc.
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*> Not referenced if KBAND=0.
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*> \endverbatim
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*>
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*> \param[in] U
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*> \verbatim
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*> U is DOUBLE PRECISION 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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*> \endverbatim
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*>
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*> \param[in] LDU
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*> \verbatim
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*> LDU is 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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*> \endverbatim
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*>
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*> \param[in] V
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*> \verbatim
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*> V is DOUBLE PRECISION array, dimension (LDV, N)
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*> If ITYPE=2 or 3, the columns of this array contain the
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*> Householder vectors used to describe the orthogonal matrix
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*> in the decomposition. If UPLO='L', then the vectors are in
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*> the lower triangle, if UPLO='U', then in the upper
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*> triangle.
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*> *NOTE* If ITYPE=2 or 3, V is modified and restored. The
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*> subdiagonal (if UPLO='L') or the superdiagonal (if UPLO='U')
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*> is set to one, and later reset to its original value, during
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*> the course of the calculation.
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*> If ITYPE=1, then it is neither referenced nor modified.
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*> \endverbatim
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*>
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*> \param[in] LDV
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*> \verbatim
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*> LDV is 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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*> \endverbatim
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*>
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*> \param[in] TAU
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*> \verbatim
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*> TAU is DOUBLE PRECISION 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)**T 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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*> \endverbatim
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*>
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*> \param[out] WORK
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*> \verbatim
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*> WORK is DOUBLE PRECISION array, dimension (2*N**2)
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*> \endverbatim
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*>
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*> \param[out] RESULT
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*> \verbatim
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*> RESULT is DOUBLE PRECISION 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 ITYPE=1.
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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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*> \ingroup double_eig
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*
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* =====================================================================
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SUBROUTINE DSYT21( ITYPE, UPLO, N, KBAND, A, LDA, D, E, U, LDU, V,
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$ LDV, TAU, WORK, RESULT )
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*
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* -- LAPACK test routine --
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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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*
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* .. Scalar Arguments ..
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CHARACTER UPLO
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INTEGER ITYPE, KBAND, LDA, LDU, LDV, N
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* ..
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* .. Array Arguments ..
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DOUBLE PRECISION 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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DOUBLE PRECISION ZERO, ONE, TEN
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PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, TEN = 10.0D0 )
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* ..
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* .. Local Scalars ..
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LOGICAL LOWER
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CHARACTER CUPLO
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INTEGER IINFO, J, JCOL, JR, JROW
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DOUBLE PRECISION ANORM, ULP, UNFL, VSAVE, WNORM
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* ..
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* .. External Functions ..
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LOGICAL LSAME
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DOUBLE PRECISION DLAMCH, DLANGE, DLANSY
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EXTERNAL LSAME, DLAMCH, DLANGE, DLANSY
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* ..
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* .. External Subroutines ..
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EXTERNAL DGEMM, DLACPY, DLARFY, DLASET, DORM2L, DORM2R,
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$ DSYR, DSYR2
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC DBLE, MAX, MIN
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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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IF( ITYPE.EQ.1 )
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$ RESULT( 2 ) = ZERO
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IF( N.LE.0 )
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$ RETURN
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*
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IF( LSAME( UPLO, 'U' ) ) THEN
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LOWER = .FALSE.
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CUPLO = 'U'
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ELSE
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LOWER = .TRUE.
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CUPLO = 'L'
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END IF
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*
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UNFL = DLAMCH( 'Safe minimum' )
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ULP = DLAMCH( 'Epsilon' )*DLAMCH( 'Base' )
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*
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* Some Error Checks
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*
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IF( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN
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RESULT( 1 ) = TEN / ULP
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RETURN
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END IF
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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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IF( ITYPE.EQ.3 ) THEN
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ANORM = ONE
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ELSE
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ANORM = MAX( DLANSY( '1', CUPLO, N, A, LDA, WORK ), UNFL )
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END IF
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*
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* Compute error matrix:
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*
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IF( ITYPE.EQ.1 ) THEN
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*
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* ITYPE=1: error = A - U S U**T
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*
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CALL DLASET( 'Full', N, N, ZERO, ZERO, WORK, N )
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CALL DLACPY( CUPLO, N, N, A, LDA, WORK, N )
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*
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DO 10 J = 1, N
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CALL DSYR( CUPLO, N, -D( J ), U( 1, J ), 1, WORK, N )
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10 CONTINUE
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*
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IF( N.GT.1 .AND. KBAND.EQ.1 ) THEN
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DO 20 J = 1, N - 1
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CALL DSYR2( CUPLO, N, -E( J ), U( 1, J ), 1, U( 1, J+1 ),
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$ 1, WORK, N )
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20 CONTINUE
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END IF
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WNORM = DLANSY( '1', CUPLO, N, WORK, N, WORK( N**2+1 ) )
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*
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ELSE IF( ITYPE.EQ.2 ) THEN
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*
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* ITYPE=2: error = V S V**T - A
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*
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CALL DLASET( 'Full', N, N, ZERO, ZERO, WORK, N )
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*
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IF( LOWER ) THEN
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WORK( N**2 ) = D( N )
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DO 40 J = N - 1, 1, -1
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IF( KBAND.EQ.1 ) THEN
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WORK( ( N+1 )*( J-1 )+2 ) = ( ONE-TAU( J ) )*E( J )
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DO 30 JR = J + 2, N
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WORK( ( J-1 )*N+JR ) = -TAU( J )*E( J )*V( JR, J )
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30 CONTINUE
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END IF
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*
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VSAVE = V( J+1, J )
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V( J+1, J ) = ONE
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CALL DLARFY( 'L', N-J, V( J+1, J ), 1, TAU( J ),
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$ WORK( ( N+1 )*J+1 ), N, WORK( N**2+1 ) )
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V( J+1, J ) = VSAVE
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WORK( ( N+1 )*( J-1 )+1 ) = D( J )
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40 CONTINUE
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ELSE
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WORK( 1 ) = D( 1 )
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DO 60 J = 1, N - 1
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IF( KBAND.EQ.1 ) THEN
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WORK( ( N+1 )*J ) = ( ONE-TAU( J ) )*E( J )
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DO 50 JR = 1, J - 1
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WORK( J*N+JR ) = -TAU( J )*E( J )*V( JR, J+1 )
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50 CONTINUE
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END IF
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*
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VSAVE = V( J, J+1 )
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V( J, J+1 ) = ONE
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CALL DLARFY( 'U', J, V( 1, J+1 ), 1, TAU( J ), WORK, N,
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$ WORK( N**2+1 ) )
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V( J, J+1 ) = VSAVE
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WORK( ( N+1 )*J+1 ) = D( J+1 )
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60 CONTINUE
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END IF
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*
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DO 90 JCOL = 1, N
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IF( LOWER ) THEN
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DO 70 JROW = JCOL, N
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WORK( JROW+N*( JCOL-1 ) ) = WORK( JROW+N*( JCOL-1 ) )
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$ - A( JROW, JCOL )
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70 CONTINUE
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ELSE
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DO 80 JROW = 1, JCOL
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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
|