442 lines
14 KiB
Fortran
442 lines
14 KiB
Fortran
*> \brief \b DSPT21
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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 DSPT21( ITYPE, UPLO, N, KBAND, AP, D, E, U, LDU, VP,
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* 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, LDU, N
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* ..
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* .. Array Arguments ..
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* DOUBLE PRECISION AP( * ), D( * ), E( * ), RESULT( 2 ), TAU( * ),
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* $ U( LDU, * ), VP( * ), 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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*> DSPT21 generally checks a decomposition of the form
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*>
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*> A = U S U'
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*>
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*> where ' means transpose, A is symmetric (stored in packed format), U
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*> is orthogonal, and S is diagonal (if KBAND=0) or symmetric
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*> tridiagonal (if KBAND=1). If ITYPE=1, then U is represented as a
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*> dense matrix, otherwise the U is expressed as a product of
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*> Householder transformations, whose vectors are stored in the array
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*> "V" and whose scaling constants are in "TAU"; we shall use the
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*> letter "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) = | A - U S U' | / ( |A| n ulp ) *andC> RESULT(2) = | I - UU' | / ( 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' | / ( |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 - VU' | / ( n ulp )
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*>
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*> Packed storage means that, for example, if UPLO='U', then the columns
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*> of the upper triangle of A are stored one after another, so that
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*> A(1,j+1) immediately follows A(j,j) in the array AP. Similarly, if
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*> UPLO='L', then the columns of the lower triangle of A are stored one
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*> after another in AP, so that A(j+1,j+1) immediately follows A(n,j)
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*> in the array AP. This means that A(i,j) is stored in:
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*>
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*> AP( i + j*(j-1)/2 ) if UPLO='U'
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*>
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*> AP( i + (2*n-j)*(j-1)/2 ) if UPLO='L'
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*>
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*> The array VP bears the same relation to the matrix V that A does to
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*> AP.
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*>
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*> For ITYPE > 1, the transformation U is expressed as a product
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*> of Householder transformations:
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*>
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*> If UPLO='U', then V = H(n-1)...H(1), where
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*>
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*> H(j) = I - tau(j) v(j) v(j)'
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*>
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*> and the first j-1 elements of v(j) are stored in V(1:j-1,j+1),
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*> (i.e., VP( j*(j+1)/2 + 1 : j*(j+1)/2 + j-1 ) ),
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*> the j-th element is 1, and the last n-j elements are 0.
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*>
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*> If UPLO='L', then V = H(1)...H(n-1), where
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*>
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*> H(j) = I - tau(j) v(j) v(j)'
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*>
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*> and the first j elements of v(j) are 0, the (j+1)-st is 1, and the
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*> (j+2)-nd through n-th elements are stored in V(j+2:n,j) (i.e.,
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*> in VP( (2*n-j)*(j-1)/2 + j+2 : (2*n-j)*(j-1)/2 + n ) .)
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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' | / ( |A| n ulp ) *andC> RESULT(2) = | I - UU' | / ( 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' | / ( |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 - VU' | / ( 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', AP and VP are considered to contain the upper
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*> triangle of A and V.
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*> If UPLO='L', AP and VP are considered to contain the lower
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*> triangle of A and V.
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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, DSPT21 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] AP
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*> \verbatim
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*> AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
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*> The original (unfactored) matrix. It is assumed to be
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*> symmetric, and contains the columns of just the upper
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*> triangle (UPLO='U') or only the lower triangle (UPLO='L'),
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*> packed one after another.
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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] VP
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*> \verbatim
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*> VP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
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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, as described in purpose.
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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] 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)' 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 (N**2+N)
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*> Workspace.
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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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*> \date November 2011
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*
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*> \ingroup double_eig
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*
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* =====================================================================
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SUBROUTINE DSPT21( ITYPE, UPLO, N, KBAND, AP, D, E, U, LDU, VP,
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$ 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, LDU, N
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* ..
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* .. Array Arguments ..
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DOUBLE PRECISION AP( * ), D( * ), E( * ), RESULT( 2 ), TAU( * ),
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$ U( LDU, * ), VP( * ), 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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DOUBLE PRECISION HALF
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PARAMETER ( HALF = 1.0D+0 / 2.0D+0 )
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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, JP, JP1, JR, LAP
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DOUBLE PRECISION ANORM, TEMP, 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 DDOT, DLAMCH, DLANGE, DLANSP
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EXTERNAL LSAME, DDOT, DLAMCH, DLANGE, DLANSP
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* ..
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* .. External Subroutines ..
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EXTERNAL DAXPY, DCOPY, DGEMM, DLACPY, DLASET, DOPMTR,
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$ DSPMV, DSPR, DSPR2
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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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* 1) Constants
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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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LAP = ( N*( N+1 ) ) / 2
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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( DLANSP( '1', CUPLO, N, AP, 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'
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*
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CALL DLASET( 'Full', N, N, ZERO, ZERO, WORK, N )
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CALL DCOPY( LAP, AP, 1, WORK, 1 )
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*
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DO 10 J = 1, N
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CALL DSPR( CUPLO, N, -D( J ), U( 1, J ), 1, WORK )
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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 DSPR2( CUPLO, N, -E( J ), U( 1, J ), 1, U( 1, J+1 ),
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$ 1, WORK )
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20 CONTINUE
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END IF
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WNORM = DLANSP( '1', CUPLO, N, WORK, 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' - 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( LAP ) = D( N )
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DO 40 J = N - 1, 1, -1
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JP = ( ( 2*N-J )*( J-1 ) ) / 2
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JP1 = JP + N - J
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IF( KBAND.EQ.1 ) THEN
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WORK( JP+J+1 ) = ( ONE-TAU( J ) )*E( J )
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DO 30 JR = J + 2, N
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WORK( JP+JR ) = -TAU( J )*E( J )*VP( JP+JR )
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30 CONTINUE
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END IF
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*
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IF( TAU( J ).NE.ZERO ) THEN
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VSAVE = VP( JP+J+1 )
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VP( JP+J+1 ) = ONE
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CALL DSPMV( 'L', N-J, ONE, WORK( JP1+J+1 ),
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$ VP( JP+J+1 ), 1, ZERO, WORK( LAP+1 ), 1 )
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TEMP = -HALF*TAU( J )*DDOT( N-J, WORK( LAP+1 ), 1,
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$ VP( JP+J+1 ), 1 )
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CALL DAXPY( N-J, TEMP, VP( JP+J+1 ), 1, WORK( LAP+1 ),
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$ 1 )
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CALL DSPR2( 'L', N-J, -TAU( J ), VP( JP+J+1 ), 1,
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$ WORK( LAP+1 ), 1, WORK( JP1+J+1 ) )
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VP( JP+J+1 ) = VSAVE
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END IF
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WORK( JP+J ) = 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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JP = ( J*( J-1 ) ) / 2
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JP1 = JP + J
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IF( KBAND.EQ.1 ) THEN
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WORK( JP1+J ) = ( ONE-TAU( J ) )*E( J )
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DO 50 JR = 1, J - 1
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WORK( JP1+JR ) = -TAU( J )*E( J )*VP( JP1+JR )
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50 CONTINUE
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END IF
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*
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IF( TAU( J ).NE.ZERO ) THEN
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VSAVE = VP( JP1+J )
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VP( JP1+J ) = ONE
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CALL DSPMV( 'U', J, ONE, WORK, VP( JP1+1 ), 1, ZERO,
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$ WORK( LAP+1 ), 1 )
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TEMP = -HALF*TAU( J )*DDOT( J, WORK( LAP+1 ), 1,
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$ VP( JP1+1 ), 1 )
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CALL DAXPY( J, TEMP, VP( JP1+1 ), 1, WORK( LAP+1 ),
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$ 1 )
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CALL DSPR2( 'U', J, -TAU( J ), VP( JP1+1 ), 1,
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$ WORK( LAP+1 ), 1, WORK )
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VP( JP1+J ) = VSAVE
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END IF
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WORK( JP1+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 70 J = 1, LAP
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WORK( J ) = WORK( J ) - AP( J )
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70 CONTINUE
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WNORM = DLANSP( '1', CUPLO, N, WORK, WORK( LAP+1 ) )
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*
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ELSE IF( ITYPE.EQ.3 ) THEN
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*
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* ITYPE=3: error = U V' - I
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*
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IF( N.LT.2 )
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$ RETURN
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CALL DLACPY( ' ', N, N, U, LDU, WORK, N )
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CALL DOPMTR( 'R', CUPLO, 'T', N, N, VP, TAU, WORK, N,
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$ WORK( N**2+1 ), IINFO )
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IF( IINFO.NE.0 ) 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 80 J = 1, N
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WORK( ( N+1 )*( J-1 )+1 ) = WORK( ( N+1 )*( J-1 )+1 ) - ONE
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80 CONTINUE
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*
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WNORM = DLANGE( '1', N, N, WORK, N, WORK( N**2+1 ) )
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END IF
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*
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IF( ANORM.GT.WNORM ) THEN
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RESULT( 1 ) = ( WNORM / ANORM ) / ( N*ULP )
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ELSE
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IF( ANORM.LT.ONE ) THEN
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RESULT( 1 ) = ( MIN( WNORM, N*ANORM ) / ANORM ) / ( N*ULP )
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ELSE
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RESULT( 1 ) = MIN( WNORM / ANORM, DBLE( N ) ) / ( N*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 UU' - I
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*
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IF( ITYPE.EQ.1 ) THEN
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CALL DGEMM( 'N', 'C', N, N, N, ONE, U, LDU, U, LDU, ZERO, WORK,
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$ N )
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*
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DO 90 J = 1, N
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WORK( ( N+1 )*( J-1 )+1 ) = WORK( ( N+1 )*( J-1 )+1 ) - ONE
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90 CONTINUE
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*
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RESULT( 2 ) = MIN( DLANGE( '1', N, N, WORK, N,
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$ WORK( N**2+1 ) ), DBLE( N ) ) / ( N*ULP )
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END IF
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*
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RETURN
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*
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* End of DSPT21
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*
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END
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