869 lines
29 KiB
Fortran
869 lines
29 KiB
Fortran
*> \brief <b> DGGEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices</b>
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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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*> \htmlonly
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*> Download DGGEVX + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dggevx.f">
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*> [TGZ]</a>
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dggevx.f">
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*> [ZIP]</a>
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dggevx.f">
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*> [TXT]</a>
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*> \endhtmlonly
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*
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* Definition:
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* ===========
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*
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* SUBROUTINE DGGEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, B, LDB,
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* ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, ILO,
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* IHI, LSCALE, RSCALE, ABNRM, BBNRM, RCONDE,
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* RCONDV, WORK, LWORK, IWORK, BWORK, INFO )
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*
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* .. Scalar Arguments ..
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* CHARACTER BALANC, JOBVL, JOBVR, SENSE
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* INTEGER IHI, ILO, INFO, LDA, LDB, LDVL, LDVR, LWORK, N
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* DOUBLE PRECISION ABNRM, BBNRM
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* ..
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* .. Array Arguments ..
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* LOGICAL BWORK( * )
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* INTEGER IWORK( * )
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* DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
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* $ B( LDB, * ), BETA( * ), LSCALE( * ),
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* $ RCONDE( * ), RCONDV( * ), RSCALE( * ),
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* $ VL( LDVL, * ), VR( LDVR, * ), 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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*> DGGEVX computes for a pair of N-by-N real nonsymmetric matrices (A,B)
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*> the generalized eigenvalues, and optionally, the left and/or right
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*> generalized eigenvectors.
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*>
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*> Optionally also, it computes a balancing transformation to improve
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*> the conditioning of the eigenvalues and eigenvectors (ILO, IHI,
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*> LSCALE, RSCALE, ABNRM, and BBNRM), reciprocal condition numbers for
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*> the eigenvalues (RCONDE), and reciprocal condition numbers for the
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*> right eigenvectors (RCONDV).
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*>
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*> A generalized eigenvalue for a pair of matrices (A,B) is a scalar
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*> lambda or a ratio alpha/beta = lambda, such that A - lambda*B is
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*> singular. It is usually represented as the pair (alpha,beta), as
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*> there is a reasonable interpretation for beta=0, and even for both
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*> being zero.
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*>
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*> The right eigenvector v(j) corresponding to the eigenvalue lambda(j)
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*> of (A,B) satisfies
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*>
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*> A * v(j) = lambda(j) * B * v(j) .
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*>
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*> The left eigenvector u(j) corresponding to the eigenvalue lambda(j)
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*> of (A,B) satisfies
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*>
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*> u(j)**H * A = lambda(j) * u(j)**H * B.
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*>
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*> where u(j)**H is the conjugate-transpose of u(j).
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*>
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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] BALANC
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*> \verbatim
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*> BALANC is CHARACTER*1
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*> Specifies the balance option to be performed.
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*> = 'N': do not diagonally scale or permute;
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*> = 'P': permute only;
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*> = 'S': scale only;
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*> = 'B': both permute and scale.
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*> Computed reciprocal condition numbers will be for the
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*> matrices after permuting and/or balancing. Permuting does
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*> not change condition numbers (in exact arithmetic), but
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*> balancing does.
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*> \endverbatim
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*>
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*> \param[in] JOBVL
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*> \verbatim
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*> JOBVL is CHARACTER*1
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*> = 'N': do not compute the left generalized eigenvectors;
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*> = 'V': compute the left generalized eigenvectors.
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*> \endverbatim
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*>
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*> \param[in] JOBVR
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*> \verbatim
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*> JOBVR is CHARACTER*1
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*> = 'N': do not compute the right generalized eigenvectors;
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*> = 'V': compute the right generalized eigenvectors.
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*> \endverbatim
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*>
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*> \param[in] SENSE
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*> \verbatim
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*> SENSE is CHARACTER*1
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*> Determines which reciprocal condition numbers are computed.
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*> = 'N': none are computed;
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*> = 'E': computed for eigenvalues only;
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*> = 'V': computed for eigenvectors only;
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*> = 'B': computed for eigenvalues and eigenvectors.
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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 matrices A, B, VL, and VR. N >= 0.
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*> \endverbatim
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*>
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*> \param[in,out] A
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*> \verbatim
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*> A is DOUBLE PRECISION array, dimension (LDA, N)
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*> On entry, the matrix A in the pair (A,B).
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*> On exit, A has been overwritten. If JOBVL='V' or JOBVR='V'
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*> or both, then A contains the first part of the real Schur
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*> form of the "balanced" versions of the input A and B.
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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. LDA >= max(1,N).
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*> \endverbatim
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*>
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*> \param[in,out] B
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*> \verbatim
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*> B is DOUBLE PRECISION array, dimension (LDB, N)
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*> On entry, the matrix B in the pair (A,B).
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*> On exit, B has been overwritten. If JOBVL='V' or JOBVR='V'
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*> or both, then B contains the second part of the real Schur
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*> form of the "balanced" versions of the input A and B.
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*> \endverbatim
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*>
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*> \param[in] LDB
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*> \verbatim
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*> LDB is INTEGER
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*> The leading dimension of B. LDB >= max(1,N).
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*> \endverbatim
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*>
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*> \param[out] ALPHAR
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*> \verbatim
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*> ALPHAR is DOUBLE PRECISION array, dimension (N)
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*> \endverbatim
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*>
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*> \param[out] ALPHAI
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*> \verbatim
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*> ALPHAI is DOUBLE PRECISION array, dimension (N)
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*> \endverbatim
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*>
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*> \param[out] BETA
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*> \verbatim
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*> BETA is DOUBLE PRECISION array, dimension (N)
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*> On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
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*> be the generalized eigenvalues. If ALPHAI(j) is zero, then
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*> the j-th eigenvalue is real; if positive, then the j-th and
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*> (j+1)-st eigenvalues are a complex conjugate pair, with
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*> ALPHAI(j+1) negative.
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*>
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*> Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j)
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*> may easily over- or underflow, and BETA(j) may even be zero.
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*> Thus, the user should avoid naively computing the ratio
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*> ALPHA/BETA. However, ALPHAR and ALPHAI will be always less
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*> than and usually comparable with norm(A) in magnitude, and
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*> BETA always less than and usually comparable with norm(B).
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*> \endverbatim
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*>
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*> \param[out] VL
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*> \verbatim
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*> VL is DOUBLE PRECISION array, dimension (LDVL,N)
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*> If JOBVL = 'V', the left eigenvectors u(j) are stored one
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*> after another in the columns of VL, in the same order as
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*> their eigenvalues. If the j-th eigenvalue is real, then
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*> u(j) = VL(:,j), the j-th column of VL. If the j-th and
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*> (j+1)-th eigenvalues form a complex conjugate pair, then
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*> u(j) = VL(:,j)+i*VL(:,j+1) and u(j+1) = VL(:,j)-i*VL(:,j+1).
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*> Each eigenvector will be scaled so the largest component have
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*> abs(real part) + abs(imag. part) = 1.
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*> Not referenced if JOBVL = 'N'.
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*> \endverbatim
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*>
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*> \param[in] LDVL
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*> \verbatim
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*> LDVL is INTEGER
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*> The leading dimension of the matrix VL. LDVL >= 1, and
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*> if JOBVL = 'V', LDVL >= N.
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*> \endverbatim
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*>
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*> \param[out] VR
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*> \verbatim
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*> VR is DOUBLE PRECISION array, dimension (LDVR,N)
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*> If JOBVR = 'V', the right eigenvectors v(j) are stored one
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*> after another in the columns of VR, in the same order as
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*> their eigenvalues. If the j-th eigenvalue is real, then
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*> v(j) = VR(:,j), the j-th column of VR. If the j-th and
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*> (j+1)-th eigenvalues form a complex conjugate pair, then
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*> v(j) = VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1).
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*> Each eigenvector will be scaled so the largest component have
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*> abs(real part) + abs(imag. part) = 1.
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*> Not referenced if JOBVR = 'N'.
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*> \endverbatim
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*>
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*> \param[in] LDVR
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*> \verbatim
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*> LDVR is INTEGER
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*> The leading dimension of the matrix VR. LDVR >= 1, and
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*> if JOBVR = 'V', LDVR >= N.
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*> \endverbatim
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*>
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*> \param[out] ILO
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*> \verbatim
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*> ILO is INTEGER
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*> \endverbatim
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*>
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*> \param[out] IHI
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*> \verbatim
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*> IHI is INTEGER
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*> ILO and IHI are integer values such that on exit
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*> A(i,j) = 0 and B(i,j) = 0 if i > j and
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*> j = 1,...,ILO-1 or i = IHI+1,...,N.
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*> If BALANC = 'N' or 'S', ILO = 1 and IHI = N.
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*> \endverbatim
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*>
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*> \param[out] LSCALE
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*> \verbatim
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*> LSCALE is DOUBLE PRECISION array, dimension (N)
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*> Details of the permutations and scaling factors applied
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*> to the left side of A and B. If PL(j) is the index of the
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*> row interchanged with row j, and DL(j) is the scaling
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*> factor applied to row j, then
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*> LSCALE(j) = PL(j) for j = 1,...,ILO-1
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*> = DL(j) for j = ILO,...,IHI
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*> = PL(j) for j = IHI+1,...,N.
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*> The order in which the interchanges are made is N to IHI+1,
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*> then 1 to ILO-1.
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*> \endverbatim
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*>
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*> \param[out] RSCALE
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*> \verbatim
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*> RSCALE is DOUBLE PRECISION array, dimension (N)
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*> Details of the permutations and scaling factors applied
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*> to the right side of A and B. If PR(j) is the index of the
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*> column interchanged with column j, and DR(j) is the scaling
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*> factor applied to column j, then
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*> RSCALE(j) = PR(j) for j = 1,...,ILO-1
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*> = DR(j) for j = ILO,...,IHI
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*> = PR(j) for j = IHI+1,...,N
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*> The order in which the interchanges are made is N to IHI+1,
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*> then 1 to ILO-1.
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*> \endverbatim
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*>
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*> \param[out] ABNRM
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*> \verbatim
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*> ABNRM is DOUBLE PRECISION
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*> The one-norm of the balanced matrix A.
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*> \endverbatim
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*>
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*> \param[out] BBNRM
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*> \verbatim
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*> BBNRM is DOUBLE PRECISION
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*> The one-norm of the balanced matrix B.
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*> \endverbatim
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*>
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*> \param[out] RCONDE
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*> \verbatim
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*> RCONDE is DOUBLE PRECISION array, dimension (N)
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*> If SENSE = 'E' or 'B', the reciprocal condition numbers of
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*> the eigenvalues, stored in consecutive elements of the array.
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*> For a complex conjugate pair of eigenvalues two consecutive
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*> elements of RCONDE are set to the same value. Thus RCONDE(j),
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*> RCONDV(j), and the j-th columns of VL and VR all correspond
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*> to the j-th eigenpair.
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*> If SENSE = 'N or 'V', RCONDE is not referenced.
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*> \endverbatim
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*>
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*> \param[out] RCONDV
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*> \verbatim
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*> RCONDV is DOUBLE PRECISION array, dimension (N)
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*> If SENSE = 'V' or 'B', the estimated reciprocal condition
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*> numbers of the eigenvectors, stored in consecutive elements
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*> of the array. For a complex eigenvector two consecutive
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*> elements of RCONDV are set to the same value. If the
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*> eigenvalues cannot be reordered to compute RCONDV(j),
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*> RCONDV(j) is set to 0; this can only occur when the true
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*> value would be very small anyway.
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*> If SENSE = 'N' or 'E', RCONDV 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 (MAX(1,LWORK))
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*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
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*> \endverbatim
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*>
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*> \param[in] LWORK
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*> \verbatim
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*> LWORK is INTEGER
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*> The dimension of the array WORK. LWORK >= max(1,2*N).
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*> If BALANC = 'S' or 'B', or JOBVL = 'V', or JOBVR = 'V',
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*> LWORK >= max(1,6*N).
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*> If SENSE = 'E' or 'B', LWORK >= max(1,10*N).
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*> If SENSE = 'V' or 'B', LWORK >= 2*N*N+8*N+16.
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*>
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*> If LWORK = -1, then a workspace query is assumed; the routine
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*> only calculates the optimal size of the WORK array, returns
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*> this value as the first entry of the WORK array, and no error
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*> message related to LWORK is issued by XERBLA.
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*> \endverbatim
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*>
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*> \param[out] IWORK
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*> \verbatim
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*> IWORK is INTEGER array, dimension (N+6)
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*> If SENSE = 'E', IWORK is not referenced.
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*> \endverbatim
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*>
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*> \param[out] BWORK
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*> \verbatim
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*> BWORK is LOGICAL array, dimension (N)
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*> If SENSE = 'N', BWORK is not referenced.
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*> \endverbatim
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*>
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*> \param[out] INFO
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*> \verbatim
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*> INFO is INTEGER
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*> = 0: successful exit
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*> < 0: if INFO = -i, the i-th argument had an illegal value.
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*> = 1,...,N:
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*> The QZ iteration failed. No eigenvectors have been
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*> calculated, but ALPHAR(j), ALPHAI(j), and BETA(j)
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*> should be correct for j=INFO+1,...,N.
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*> > N: =N+1: other than QZ iteration failed in DHGEQZ.
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*> =N+2: error return from DTGEVC.
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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 April 2012
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*
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*> \ingroup doubleGEeigen
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*
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*> \par Further Details:
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* =====================
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*>
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*> \verbatim
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*>
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*> Balancing a matrix pair (A,B) includes, first, permuting rows and
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*> columns to isolate eigenvalues, second, applying diagonal similarity
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*> transformation to the rows and columns to make the rows and columns
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*> as close in norm as possible. The computed reciprocal condition
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*> numbers correspond to the balanced matrix. Permuting rows and columns
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*> will not change the condition numbers (in exact arithmetic) but
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*> diagonal scaling will. For further explanation of balancing, see
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*> section 4.11.1.2 of LAPACK Users' Guide.
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*>
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*> An approximate error bound on the chordal distance between the i-th
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*> computed generalized eigenvalue w and the corresponding exact
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*> eigenvalue lambda is
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*>
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*> chord(w, lambda) <= EPS * norm(ABNRM, BBNRM) / RCONDE(I)
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*>
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*> An approximate error bound for the angle between the i-th computed
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*> eigenvector VL(i) or VR(i) is given by
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*>
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*> EPS * norm(ABNRM, BBNRM) / DIF(i).
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*>
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*> For further explanation of the reciprocal condition numbers RCONDE
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*> and RCONDV, see section 4.11 of LAPACK User's Guide.
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*> \endverbatim
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*>
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* =====================================================================
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SUBROUTINE DGGEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, B, LDB,
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$ ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, ILO,
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$ IHI, LSCALE, RSCALE, ABNRM, BBNRM, RCONDE,
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$ RCONDV, WORK, LWORK, IWORK, BWORK, INFO )
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*
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* -- LAPACK driver routine (version 3.4.1) --
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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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* April 2012
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*
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* .. Scalar Arguments ..
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CHARACTER BALANC, JOBVL, JOBVR, SENSE
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INTEGER IHI, ILO, INFO, LDA, LDB, LDVL, LDVR, LWORK, N
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DOUBLE PRECISION ABNRM, BBNRM
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* ..
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* .. Array Arguments ..
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LOGICAL BWORK( * )
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INTEGER IWORK( * )
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DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
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$ B( LDB, * ), BETA( * ), LSCALE( * ),
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$ RCONDE( * ), RCONDV( * ), RSCALE( * ),
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$ VL( LDVL, * ), VR( LDVR, * ), 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
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PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
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* ..
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* .. Local Scalars ..
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LOGICAL ILASCL, ILBSCL, ILV, ILVL, ILVR, LQUERY, NOSCL,
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$ PAIR, WANTSB, WANTSE, WANTSN, WANTSV
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CHARACTER CHTEMP
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INTEGER I, ICOLS, IERR, IJOBVL, IJOBVR, IN, IROWS,
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$ ITAU, IWRK, IWRK1, J, JC, JR, M, MAXWRK,
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$ MINWRK, MM
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DOUBLE PRECISION ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS,
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$ SMLNUM, TEMP
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* ..
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* .. Local Arrays ..
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LOGICAL LDUMMA( 1 )
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* ..
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* .. External Subroutines ..
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EXTERNAL DGEQRF, DGGBAK, DGGBAL, DGGHRD, DHGEQZ, DLABAD,
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$ DLACPY, DLASCL, DLASET, DORGQR, DORMQR, DTGEVC,
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$ DTGSNA, XERBLA
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* ..
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* .. External Functions ..
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LOGICAL LSAME
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INTEGER ILAENV
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DOUBLE PRECISION DLAMCH, DLANGE
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EXTERNAL LSAME, ILAENV, DLAMCH, DLANGE
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC ABS, MAX, SQRT
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* ..
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* .. Executable Statements ..
|
|
*
|
|
* Decode the input arguments
|
|
*
|
|
IF( LSAME( JOBVL, 'N' ) ) THEN
|
|
IJOBVL = 1
|
|
ILVL = .FALSE.
|
|
ELSE IF( LSAME( JOBVL, 'V' ) ) THEN
|
|
IJOBVL = 2
|
|
ILVL = .TRUE.
|
|
ELSE
|
|
IJOBVL = -1
|
|
ILVL = .FALSE.
|
|
END IF
|
|
*
|
|
IF( LSAME( JOBVR, 'N' ) ) THEN
|
|
IJOBVR = 1
|
|
ILVR = .FALSE.
|
|
ELSE IF( LSAME( JOBVR, 'V' ) ) THEN
|
|
IJOBVR = 2
|
|
ILVR = .TRUE.
|
|
ELSE
|
|
IJOBVR = -1
|
|
ILVR = .FALSE.
|
|
END IF
|
|
ILV = ILVL .OR. ILVR
|
|
*
|
|
NOSCL = LSAME( BALANC, 'N' ) .OR. LSAME( BALANC, 'P' )
|
|
WANTSN = LSAME( SENSE, 'N' )
|
|
WANTSE = LSAME( SENSE, 'E' )
|
|
WANTSV = LSAME( SENSE, 'V' )
|
|
WANTSB = LSAME( SENSE, 'B' )
|
|
*
|
|
* Test the input arguments
|
|
*
|
|
INFO = 0
|
|
LQUERY = ( LWORK.EQ.-1 )
|
|
IF( .NOT.( LSAME( BALANC, 'N' ) .OR. LSAME( BALANC,
|
|
$ 'S' ) .OR. LSAME( BALANC, 'P' ) .OR. LSAME( BALANC, 'B' ) ) )
|
|
$ THEN
|
|
INFO = -1
|
|
ELSE IF( IJOBVL.LE.0 ) THEN
|
|
INFO = -2
|
|
ELSE IF( IJOBVR.LE.0 ) THEN
|
|
INFO = -3
|
|
ELSE IF( .NOT.( WANTSN .OR. WANTSE .OR. WANTSB .OR. WANTSV ) )
|
|
$ THEN
|
|
INFO = -4
|
|
ELSE IF( N.LT.0 ) THEN
|
|
INFO = -5
|
|
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
|
|
INFO = -7
|
|
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
|
|
INFO = -9
|
|
ELSE IF( LDVL.LT.1 .OR. ( ILVL .AND. LDVL.LT.N ) ) THEN
|
|
INFO = -14
|
|
ELSE IF( LDVR.LT.1 .OR. ( ILVR .AND. LDVR.LT.N ) ) THEN
|
|
INFO = -16
|
|
END IF
|
|
*
|
|
* Compute workspace
|
|
* (Note: Comments in the code beginning "Workspace:" describe the
|
|
* minimal amount of workspace needed at that point in the code,
|
|
* as well as the preferred amount for good performance.
|
|
* NB refers to the optimal block size for the immediately
|
|
* following subroutine, as returned by ILAENV. The workspace is
|
|
* computed assuming ILO = 1 and IHI = N, the worst case.)
|
|
*
|
|
IF( INFO.EQ.0 ) THEN
|
|
IF( N.EQ.0 ) THEN
|
|
MINWRK = 1
|
|
MAXWRK = 1
|
|
ELSE
|
|
IF( NOSCL .AND. .NOT.ILV ) THEN
|
|
MINWRK = 2*N
|
|
ELSE
|
|
MINWRK = 6*N
|
|
END IF
|
|
IF( WANTSE .OR. WANTSB ) THEN
|
|
MINWRK = 10*N
|
|
END IF
|
|
IF( WANTSV .OR. WANTSB ) THEN
|
|
MINWRK = MAX( MINWRK, 2*N*( N + 4 ) + 16 )
|
|
END IF
|
|
MAXWRK = MINWRK
|
|
MAXWRK = MAX( MAXWRK,
|
|
$ N + N*ILAENV( 1, 'DGEQRF', ' ', N, 1, N, 0 ) )
|
|
MAXWRK = MAX( MAXWRK,
|
|
$ N + N*ILAENV( 1, 'DORMQR', ' ', N, 1, N, 0 ) )
|
|
IF( ILVL ) THEN
|
|
MAXWRK = MAX( MAXWRK, N +
|
|
$ N*ILAENV( 1, 'DORGQR', ' ', N, 1, N, 0 ) )
|
|
END IF
|
|
END IF
|
|
WORK( 1 ) = MAXWRK
|
|
*
|
|
IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
|
|
INFO = -26
|
|
END IF
|
|
END IF
|
|
*
|
|
IF( INFO.NE.0 ) THEN
|
|
CALL XERBLA( 'DGGEVX', -INFO )
|
|
RETURN
|
|
ELSE IF( LQUERY ) THEN
|
|
RETURN
|
|
END IF
|
|
*
|
|
* Quick return if possible
|
|
*
|
|
IF( N.EQ.0 )
|
|
$ RETURN
|
|
*
|
|
*
|
|
* Get machine constants
|
|
*
|
|
EPS = DLAMCH( 'P' )
|
|
SMLNUM = DLAMCH( 'S' )
|
|
BIGNUM = ONE / SMLNUM
|
|
CALL DLABAD( SMLNUM, BIGNUM )
|
|
SMLNUM = SQRT( SMLNUM ) / EPS
|
|
BIGNUM = ONE / SMLNUM
|
|
*
|
|
* Scale A if max element outside range [SMLNUM,BIGNUM]
|
|
*
|
|
ANRM = DLANGE( 'M', N, N, A, LDA, WORK )
|
|
ILASCL = .FALSE.
|
|
IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
|
|
ANRMTO = SMLNUM
|
|
ILASCL = .TRUE.
|
|
ELSE IF( ANRM.GT.BIGNUM ) THEN
|
|
ANRMTO = BIGNUM
|
|
ILASCL = .TRUE.
|
|
END IF
|
|
IF( ILASCL )
|
|
$ CALL DLASCL( 'G', 0, 0, ANRM, ANRMTO, N, N, A, LDA, IERR )
|
|
*
|
|
* Scale B if max element outside range [SMLNUM,BIGNUM]
|
|
*
|
|
BNRM = DLANGE( 'M', N, N, B, LDB, WORK )
|
|
ILBSCL = .FALSE.
|
|
IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
|
|
BNRMTO = SMLNUM
|
|
ILBSCL = .TRUE.
|
|
ELSE IF( BNRM.GT.BIGNUM ) THEN
|
|
BNRMTO = BIGNUM
|
|
ILBSCL = .TRUE.
|
|
END IF
|
|
IF( ILBSCL )
|
|
$ CALL DLASCL( 'G', 0, 0, BNRM, BNRMTO, N, N, B, LDB, IERR )
|
|
*
|
|
* Permute and/or balance the matrix pair (A,B)
|
|
* (Workspace: need 6*N if BALANC = 'S' or 'B', 1 otherwise)
|
|
*
|
|
CALL DGGBAL( BALANC, N, A, LDA, B, LDB, ILO, IHI, LSCALE, RSCALE,
|
|
$ WORK, IERR )
|
|
*
|
|
* Compute ABNRM and BBNRM
|
|
*
|
|
ABNRM = DLANGE( '1', N, N, A, LDA, WORK( 1 ) )
|
|
IF( ILASCL ) THEN
|
|
WORK( 1 ) = ABNRM
|
|
CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, 1, 1, WORK( 1 ), 1,
|
|
$ IERR )
|
|
ABNRM = WORK( 1 )
|
|
END IF
|
|
*
|
|
BBNRM = DLANGE( '1', N, N, B, LDB, WORK( 1 ) )
|
|
IF( ILBSCL ) THEN
|
|
WORK( 1 ) = BBNRM
|
|
CALL DLASCL( 'G', 0, 0, BNRMTO, BNRM, 1, 1, WORK( 1 ), 1,
|
|
$ IERR )
|
|
BBNRM = WORK( 1 )
|
|
END IF
|
|
*
|
|
* Reduce B to triangular form (QR decomposition of B)
|
|
* (Workspace: need N, prefer N*NB )
|
|
*
|
|
IROWS = IHI + 1 - ILO
|
|
IF( ILV .OR. .NOT.WANTSN ) THEN
|
|
ICOLS = N + 1 - ILO
|
|
ELSE
|
|
ICOLS = IROWS
|
|
END IF
|
|
ITAU = 1
|
|
IWRK = ITAU + IROWS
|
|
CALL DGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ),
|
|
$ WORK( IWRK ), LWORK+1-IWRK, IERR )
|
|
*
|
|
* Apply the orthogonal transformation to A
|
|
* (Workspace: need N, prefer N*NB)
|
|
*
|
|
CALL DORMQR( 'L', 'T', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB,
|
|
$ WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWRK ),
|
|
$ LWORK+1-IWRK, IERR )
|
|
*
|
|
* Initialize VL and/or VR
|
|
* (Workspace: need N, prefer N*NB)
|
|
*
|
|
IF( ILVL ) THEN
|
|
CALL DLASET( 'Full', N, N, ZERO, ONE, VL, LDVL )
|
|
IF( IROWS.GT.1 ) THEN
|
|
CALL DLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB,
|
|
$ VL( ILO+1, ILO ), LDVL )
|
|
END IF
|
|
CALL DORGQR( IROWS, IROWS, IROWS, VL( ILO, ILO ), LDVL,
|
|
$ WORK( ITAU ), WORK( IWRK ), LWORK+1-IWRK, IERR )
|
|
END IF
|
|
*
|
|
IF( ILVR )
|
|
$ CALL DLASET( 'Full', N, N, ZERO, ONE, VR, LDVR )
|
|
*
|
|
* Reduce to generalized Hessenberg form
|
|
* (Workspace: none needed)
|
|
*
|
|
IF( ILV .OR. .NOT.WANTSN ) THEN
|
|
*
|
|
* Eigenvectors requested -- work on whole matrix.
|
|
*
|
|
CALL DGGHRD( JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB, VL,
|
|
$ LDVL, VR, LDVR, IERR )
|
|
ELSE
|
|
CALL DGGHRD( 'N', 'N', IROWS, 1, IROWS, A( ILO, ILO ), LDA,
|
|
$ B( ILO, ILO ), LDB, VL, LDVL, VR, LDVR, IERR )
|
|
END IF
|
|
*
|
|
* Perform QZ algorithm (Compute eigenvalues, and optionally, the
|
|
* Schur forms and Schur vectors)
|
|
* (Workspace: need N)
|
|
*
|
|
IF( ILV .OR. .NOT.WANTSN ) THEN
|
|
CHTEMP = 'S'
|
|
ELSE
|
|
CHTEMP = 'E'
|
|
END IF
|
|
*
|
|
CALL DHGEQZ( CHTEMP, JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB,
|
|
$ ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, WORK,
|
|
$ LWORK, IERR )
|
|
IF( IERR.NE.0 ) THEN
|
|
IF( IERR.GT.0 .AND. IERR.LE.N ) THEN
|
|
INFO = IERR
|
|
ELSE IF( IERR.GT.N .AND. IERR.LE.2*N ) THEN
|
|
INFO = IERR - N
|
|
ELSE
|
|
INFO = N + 1
|
|
END IF
|
|
GO TO 130
|
|
END IF
|
|
*
|
|
* Compute Eigenvectors and estimate condition numbers if desired
|
|
* (Workspace: DTGEVC: need 6*N
|
|
* DTGSNA: need 2*N*(N+2)+16 if SENSE = 'V' or 'B',
|
|
* need N otherwise )
|
|
*
|
|
IF( ILV .OR. .NOT.WANTSN ) THEN
|
|
IF( ILV ) THEN
|
|
IF( ILVL ) THEN
|
|
IF( ILVR ) THEN
|
|
CHTEMP = 'B'
|
|
ELSE
|
|
CHTEMP = 'L'
|
|
END IF
|
|
ELSE
|
|
CHTEMP = 'R'
|
|
END IF
|
|
*
|
|
CALL DTGEVC( CHTEMP, 'B', LDUMMA, N, A, LDA, B, LDB, VL,
|
|
$ LDVL, VR, LDVR, N, IN, WORK, IERR )
|
|
IF( IERR.NE.0 ) THEN
|
|
INFO = N + 2
|
|
GO TO 130
|
|
END IF
|
|
END IF
|
|
*
|
|
IF( .NOT.WANTSN ) THEN
|
|
*
|
|
* compute eigenvectors (DTGEVC) and estimate condition
|
|
* numbers (DTGSNA). Note that the definition of the condition
|
|
* number is not invariant under transformation (u,v) to
|
|
* (Q*u, Z*v), where (u,v) are eigenvectors of the generalized
|
|
* Schur form (S,T), Q and Z are orthogonal matrices. In order
|
|
* to avoid using extra 2*N*N workspace, we have to recalculate
|
|
* eigenvectors and estimate one condition numbers at a time.
|
|
*
|
|
PAIR = .FALSE.
|
|
DO 20 I = 1, N
|
|
*
|
|
IF( PAIR ) THEN
|
|
PAIR = .FALSE.
|
|
GO TO 20
|
|
END IF
|
|
MM = 1
|
|
IF( I.LT.N ) THEN
|
|
IF( A( I+1, I ).NE.ZERO ) THEN
|
|
PAIR = .TRUE.
|
|
MM = 2
|
|
END IF
|
|
END IF
|
|
*
|
|
DO 10 J = 1, N
|
|
BWORK( J ) = .FALSE.
|
|
10 CONTINUE
|
|
IF( MM.EQ.1 ) THEN
|
|
BWORK( I ) = .TRUE.
|
|
ELSE IF( MM.EQ.2 ) THEN
|
|
BWORK( I ) = .TRUE.
|
|
BWORK( I+1 ) = .TRUE.
|
|
END IF
|
|
*
|
|
IWRK = MM*N + 1
|
|
IWRK1 = IWRK + MM*N
|
|
*
|
|
* Compute a pair of left and right eigenvectors.
|
|
* (compute workspace: need up to 4*N + 6*N)
|
|
*
|
|
IF( WANTSE .OR. WANTSB ) THEN
|
|
CALL DTGEVC( 'B', 'S', BWORK, N, A, LDA, B, LDB,
|
|
$ WORK( 1 ), N, WORK( IWRK ), N, MM, M,
|
|
$ WORK( IWRK1 ), IERR )
|
|
IF( IERR.NE.0 ) THEN
|
|
INFO = N + 2
|
|
GO TO 130
|
|
END IF
|
|
END IF
|
|
*
|
|
CALL DTGSNA( SENSE, 'S', BWORK, N, A, LDA, B, LDB,
|
|
$ WORK( 1 ), N, WORK( IWRK ), N, RCONDE( I ),
|
|
$ RCONDV( I ), MM, M, WORK( IWRK1 ),
|
|
$ LWORK-IWRK1+1, IWORK, IERR )
|
|
*
|
|
20 CONTINUE
|
|
END IF
|
|
END IF
|
|
*
|
|
* Undo balancing on VL and VR and normalization
|
|
* (Workspace: none needed)
|
|
*
|
|
IF( ILVL ) THEN
|
|
CALL DGGBAK( BALANC, 'L', N, ILO, IHI, LSCALE, RSCALE, N, VL,
|
|
$ LDVL, IERR )
|
|
*
|
|
DO 70 JC = 1, N
|
|
IF( ALPHAI( JC ).LT.ZERO )
|
|
$ GO TO 70
|
|
TEMP = ZERO
|
|
IF( ALPHAI( JC ).EQ.ZERO ) THEN
|
|
DO 30 JR = 1, N
|
|
TEMP = MAX( TEMP, ABS( VL( JR, JC ) ) )
|
|
30 CONTINUE
|
|
ELSE
|
|
DO 40 JR = 1, N
|
|
TEMP = MAX( TEMP, ABS( VL( JR, JC ) )+
|
|
$ ABS( VL( JR, JC+1 ) ) )
|
|
40 CONTINUE
|
|
END IF
|
|
IF( TEMP.LT.SMLNUM )
|
|
$ GO TO 70
|
|
TEMP = ONE / TEMP
|
|
IF( ALPHAI( JC ).EQ.ZERO ) THEN
|
|
DO 50 JR = 1, N
|
|
VL( JR, JC ) = VL( JR, JC )*TEMP
|
|
50 CONTINUE
|
|
ELSE
|
|
DO 60 JR = 1, N
|
|
VL( JR, JC ) = VL( JR, JC )*TEMP
|
|
VL( JR, JC+1 ) = VL( JR, JC+1 )*TEMP
|
|
60 CONTINUE
|
|
END IF
|
|
70 CONTINUE
|
|
END IF
|
|
IF( ILVR ) THEN
|
|
CALL DGGBAK( BALANC, 'R', N, ILO, IHI, LSCALE, RSCALE, N, VR,
|
|
$ LDVR, IERR )
|
|
DO 120 JC = 1, N
|
|
IF( ALPHAI( JC ).LT.ZERO )
|
|
$ GO TO 120
|
|
TEMP = ZERO
|
|
IF( ALPHAI( JC ).EQ.ZERO ) THEN
|
|
DO 80 JR = 1, N
|
|
TEMP = MAX( TEMP, ABS( VR( JR, JC ) ) )
|
|
80 CONTINUE
|
|
ELSE
|
|
DO 90 JR = 1, N
|
|
TEMP = MAX( TEMP, ABS( VR( JR, JC ) )+
|
|
$ ABS( VR( JR, JC+1 ) ) )
|
|
90 CONTINUE
|
|
END IF
|
|
IF( TEMP.LT.SMLNUM )
|
|
$ GO TO 120
|
|
TEMP = ONE / TEMP
|
|
IF( ALPHAI( JC ).EQ.ZERO ) THEN
|
|
DO 100 JR = 1, N
|
|
VR( JR, JC ) = VR( JR, JC )*TEMP
|
|
100 CONTINUE
|
|
ELSE
|
|
DO 110 JR = 1, N
|
|
VR( JR, JC ) = VR( JR, JC )*TEMP
|
|
VR( JR, JC+1 ) = VR( JR, JC+1 )*TEMP
|
|
110 CONTINUE
|
|
END IF
|
|
120 CONTINUE
|
|
END IF
|
|
*
|
|
* Undo scaling if necessary
|
|
*
|
|
130 CONTINUE
|
|
*
|
|
IF( ILASCL ) THEN
|
|
CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAR, N, IERR )
|
|
CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAI, N, IERR )
|
|
END IF
|
|
*
|
|
IF( ILBSCL ) THEN
|
|
CALL DLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N, IERR )
|
|
END IF
|
|
*
|
|
WORK( 1 ) = MAXWRK
|
|
RETURN
|
|
*
|
|
* End of DGGEVX
|
|
*
|
|
END
|