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382 lines
10 KiB
382 lines
10 KiB
*> \brief \b DGET22
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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 DGET22( TRANSA, TRANSE, TRANSW, N, A, LDA, E, LDE, WR,
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* WI, WORK, RESULT )
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*
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* .. Scalar Arguments ..
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* CHARACTER TRANSA, TRANSE, TRANSW
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* INTEGER LDA, LDE, N
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* ..
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* .. Array Arguments ..
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* DOUBLE PRECISION A( LDA, * ), E( LDE, * ), RESULT( 2 ), WI( * ),
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* $ WORK( * ), WR( * )
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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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*> DGET22 does an eigenvector check.
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*>
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*> The basic test is:
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*>
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*> RESULT(1) = | A E - E W | / ( |A| |E| ulp )
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*>
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*> using the 1-norm. It also tests the normalization of E:
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*>
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*> RESULT(2) = max | m-norm(E(j)) - 1 | / ( n ulp )
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*> j
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*>
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*> where E(j) is the j-th eigenvector, and m-norm is the max-norm of a
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*> vector. If an eigenvector is complex, as determined from WI(j)
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*> nonzero, then the max-norm of the vector ( er + i*ei ) is the maximum
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*> of
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*> |er(1)| + |ei(1)|, ... , |er(n)| + |ei(n)|
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*>
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*> W is a block diagonal matrix, with a 1 by 1 block for each real
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*> eigenvalue and a 2 by 2 block for each complex conjugate pair.
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*> If eigenvalues j and j+1 are a complex conjugate pair, so that
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*> WR(j) = WR(j+1) = wr and WI(j) = - WI(j+1) = wi, then the 2 by 2
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*> block corresponding to the pair will be:
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*>
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*> ( wr wi )
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*> ( -wi wr )
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*>
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*> Such a block multiplying an n by 2 matrix ( ur ui ) on the right
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*> will be the same as multiplying ur + i*ui by wr + i*wi.
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*>
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*> To handle various schemes for storage of left eigenvectors, there are
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*> options to use A-transpose instead of A, E-transpose instead of E,
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*> and/or W-transpose instead of W.
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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] TRANSA
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*> \verbatim
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*> TRANSA is CHARACTER*1
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*> Specifies whether or not A is transposed.
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*> = 'N': No transpose
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*> = 'T': Transpose
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*> = 'C': Conjugate transpose (= Transpose)
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*> \endverbatim
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*>
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*> \param[in] TRANSE
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*> \verbatim
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*> TRANSE is CHARACTER*1
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*> Specifies whether or not E is transposed.
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*> = 'N': No transpose, eigenvectors are in columns of E
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*> = 'T': Transpose, eigenvectors are in rows of E
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*> = 'C': Conjugate transpose (= Transpose)
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*> \endverbatim
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*>
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*> \param[in] TRANSW
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*> \verbatim
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*> TRANSW is CHARACTER*1
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*> Specifies whether or not W is transposed.
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*> = 'N': No transpose
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*> = 'T': Transpose, use -WI(j) instead of WI(j)
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*> = 'C': Conjugate transpose, use -WI(j) instead of WI(j)
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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 order of the matrix A. N >= 0.
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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 matrix whose eigenvectors are in E.
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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 the array A. LDA >= max(1,N).
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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 (LDE,N)
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*> The matrix of eigenvectors. If TRANSE = 'N', the eigenvectors
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*> are stored in the columns of E, if TRANSE = 'T' or 'C', the
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*> eigenvectors are stored in the rows of E.
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*> \endverbatim
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*>
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*> \param[in] LDE
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*> \verbatim
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*> LDE is INTEGER
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*> The leading dimension of the array E. LDE >= max(1,N).
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*> \endverbatim
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*>
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*> \param[in] WR
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*> \verbatim
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*> WR is DOUBLE PRECISION array, dimension (N)
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*> \endverbatim
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*>
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*> \param[in] WI
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*> \verbatim
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*> WI is DOUBLE PRECISION array, dimension (N)
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*>
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*> The real and imaginary parts of the eigenvalues of A.
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*> Purely real eigenvalues are indicated by WI(j) = 0.
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*> Complex conjugate pairs are indicated by WR(j)=WR(j+1) and
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*> WI(j) = - WI(j+1) non-zero; the real part is assumed to be
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*> stored in the j-th row/column and the imaginary part in
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*> the (j+1)-th row/column.
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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*(N+1))
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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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*> RESULT(1) = | A E - E W | / ( |A| |E| ulp )
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*> RESULT(2) = max | m-norm(E(j)) - 1 | / ( n ulp )
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*> j
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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 DGET22( TRANSA, TRANSE, TRANSW, N, A, LDA, E, LDE, WR,
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$ WI, 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 TRANSA, TRANSE, TRANSW
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INTEGER LDA, LDE, N
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* ..
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* .. Array Arguments ..
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DOUBLE PRECISION A( LDA, * ), E( LDE, * ), RESULT( 2 ), WI( * ),
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$ WORK( * ), WR( * )
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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
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PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
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* ..
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* .. Local Scalars ..
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CHARACTER NORMA, NORME
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INTEGER IECOL, IEROW, INCE, IPAIR, ITRNSE, J, JCOL,
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$ JVEC
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DOUBLE PRECISION ANORM, ENORM, ENRMAX, ENRMIN, ERRNRM, TEMP1,
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$ ULP, UNFL
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* ..
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* .. Local Arrays ..
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DOUBLE PRECISION WMAT( 2, 2 )
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* ..
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* .. External Functions ..
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LOGICAL LSAME
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DOUBLE PRECISION DLAMCH, DLANGE
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EXTERNAL LSAME, DLAMCH, DLANGE
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* ..
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* .. External Subroutines ..
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EXTERNAL DAXPY, DGEMM, DLASET
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC ABS, DBLE, MAX, MIN
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* ..
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* .. Executable Statements ..
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*
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* Initialize RESULT (in case N=0)
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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 )
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$ RETURN
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*
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UNFL = DLAMCH( 'Safe minimum' )
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ULP = DLAMCH( 'Precision' )
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*
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ITRNSE = 0
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INCE = 1
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NORMA = 'O'
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NORME = 'O'
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*
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IF( LSAME( TRANSA, 'T' ) .OR. LSAME( TRANSA, 'C' ) ) THEN
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NORMA = 'I'
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END IF
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IF( LSAME( TRANSE, 'T' ) .OR. LSAME( TRANSE, 'C' ) ) THEN
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NORME = 'I'
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ITRNSE = 1
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INCE = LDE
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END IF
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*
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* Check normalization of E
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*
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ENRMIN = ONE / ULP
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ENRMAX = ZERO
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IF( ITRNSE.EQ.0 ) THEN
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*
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* Eigenvectors are column vectors.
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*
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IPAIR = 0
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DO 30 JVEC = 1, N
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TEMP1 = ZERO
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IF( IPAIR.EQ.0 .AND. JVEC.LT.N .AND. WI( JVEC ).NE.ZERO )
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$ IPAIR = 1
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IF( IPAIR.EQ.1 ) THEN
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*
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* Complex eigenvector
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*
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DO 10 J = 1, N
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TEMP1 = MAX( TEMP1, ABS( E( J, JVEC ) )+
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$ ABS( E( J, JVEC+1 ) ) )
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10 CONTINUE
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ENRMIN = MIN( ENRMIN, TEMP1 )
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ENRMAX = MAX( ENRMAX, TEMP1 )
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IPAIR = 2
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ELSE IF( IPAIR.EQ.2 ) THEN
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IPAIR = 0
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ELSE
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*
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* Real eigenvector
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*
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DO 20 J = 1, N
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TEMP1 = MAX( TEMP1, ABS( E( J, JVEC ) ) )
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20 CONTINUE
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ENRMIN = MIN( ENRMIN, TEMP1 )
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ENRMAX = MAX( ENRMAX, TEMP1 )
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IPAIR = 0
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END IF
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30 CONTINUE
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*
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ELSE
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*
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* Eigenvectors are row vectors.
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*
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DO 40 JVEC = 1, N
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WORK( JVEC ) = ZERO
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40 CONTINUE
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*
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DO 60 J = 1, N
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IPAIR = 0
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DO 50 JVEC = 1, N
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IF( IPAIR.EQ.0 .AND. JVEC.LT.N .AND. WI( JVEC ).NE.ZERO )
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$ IPAIR = 1
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IF( IPAIR.EQ.1 ) THEN
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WORK( JVEC ) = MAX( WORK( JVEC ),
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$ ABS( E( J, JVEC ) )+ABS( E( J,
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$ JVEC+1 ) ) )
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WORK( JVEC+1 ) = WORK( JVEC )
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ELSE IF( IPAIR.EQ.2 ) THEN
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IPAIR = 0
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ELSE
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WORK( JVEC ) = MAX( WORK( JVEC ),
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$ ABS( E( J, JVEC ) ) )
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IPAIR = 0
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END IF
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50 CONTINUE
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60 CONTINUE
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*
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DO 70 JVEC = 1, N
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ENRMIN = MIN( ENRMIN, WORK( JVEC ) )
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ENRMAX = MAX( ENRMAX, WORK( JVEC ) )
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70 CONTINUE
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END IF
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*
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* Norm of A:
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*
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ANORM = MAX( DLANGE( NORMA, N, N, A, LDA, WORK ), UNFL )
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*
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* Norm of E:
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*
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ENORM = MAX( DLANGE( NORME, N, N, E, LDE, WORK ), ULP )
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*
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* Norm of error:
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*
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* Error = AE - EW
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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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IPAIR = 0
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IEROW = 1
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IECOL = 1
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*
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DO 80 JCOL = 1, N
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IF( ITRNSE.EQ.1 ) THEN
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IEROW = JCOL
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ELSE
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IECOL = JCOL
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END IF
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*
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IF( IPAIR.EQ.0 .AND. WI( JCOL ).NE.ZERO )
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$ IPAIR = 1
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*
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IF( IPAIR.EQ.1 ) THEN
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WMAT( 1, 1 ) = WR( JCOL )
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WMAT( 2, 1 ) = -WI( JCOL )
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WMAT( 1, 2 ) = WI( JCOL )
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WMAT( 2, 2 ) = WR( JCOL )
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CALL DGEMM( TRANSE, TRANSW, N, 2, 2, ONE, E( IEROW, IECOL ),
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$ LDE, WMAT, 2, ZERO, WORK( N*( JCOL-1 )+1 ), N )
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IPAIR = 2
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ELSE IF( IPAIR.EQ.2 ) THEN
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IPAIR = 0
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*
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ELSE
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*
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CALL DAXPY( N, WR( JCOL ), E( IEROW, IECOL ), INCE,
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$ WORK( N*( JCOL-1 )+1 ), 1 )
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IPAIR = 0
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END IF
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*
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80 CONTINUE
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*
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CALL DGEMM( TRANSA, TRANSE, N, N, N, ONE, A, LDA, E, LDE, -ONE,
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$ WORK, N )
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*
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ERRNRM = DLANGE( 'One', N, N, WORK, N, WORK( N*N+1 ) ) / ENORM
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*
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* Compute RESULT(1) (avoiding under/overflow)
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*
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IF( ANORM.GT.ERRNRM ) THEN
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RESULT( 1 ) = ( ERRNRM / ANORM ) / ULP
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ELSE
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IF( ANORM.LT.ONE ) THEN
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RESULT( 1 ) = ONE / ULP
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ELSE
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RESULT( 1 ) = MIN( ERRNRM / ANORM, ONE ) / ULP
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END IF
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END IF
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*
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* Compute RESULT(2) : the normalization error in E.
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*
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RESULT( 2 ) = MAX( ABS( ENRMAX-ONE ), ABS( ENRMIN-ONE ) ) /
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$ ( DBLE( N )*ULP )
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*
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RETURN
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*
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* End of DGET22
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*
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END
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