version 1.3, 2010/08/06 15:28:53
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version 1.10, 2012/07/31 11:06:38
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*> \brief <b> ZGGEVX 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 ZGGEVX + dependencies |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zggevx.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/zggevx.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/zggevx.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 ZGGEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, B, LDB, |
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* ALPHA, BETA, VL, LDVL, VR, LDVR, ILO, IHI, |
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* LSCALE, RSCALE, ABNRM, BBNRM, RCONDE, RCONDV, |
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* WORK, LWORK, RWORK, 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 LSCALE( * ), RCONDE( * ), RCONDV( * ), |
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* $ RSCALE( * ), RWORK( * ) |
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* COMPLEX*16 A( LDA, * ), ALPHA( * ), B( LDB, * ), |
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* $ BETA( * ), VL( LDVL, * ), VR( LDVR, * ), |
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* $ 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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*> ZGGEVX computes for a pair of N-by-N complex nonsymmetric matrices |
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*> (A,B) the generalized eigenvalues, and optionally, the left and/or |
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*> right generalized eigenvectors. |
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*> |
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*> Optionally, it also 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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*> A * v(j) = lambda(j) * B * v(j) . |
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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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*> u(j)**H * A = lambda(j) * u(j)**H * B. |
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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 COMPLEX*16 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 complex 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 COMPLEX*16 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 complex |
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*> Schur 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] ALPHA |
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*> \verbatim |
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*> ALPHA is COMPLEX*16 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 COMPLEX*16 array, dimension (N) |
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*> On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the generalized |
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*> eigenvalues. |
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*> |
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*> Note: the quotient ALPHA(j)/BETA(j) ) may easily over- or |
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*> underflow, and BETA(j) may even be zero. Thus, the user |
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*> should avoid naively computing the ratio ALPHA/BETA. |
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*> However, ALPHA will be always less than and usually |
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*> comparable with norm(A) in magnitude, and BETA always less |
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*> 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 COMPLEX*16 array, dimension (LDVL,N) |
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*> If JOBVL = 'V', the left generalized eigenvectors u(j) are |
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*> stored one after another in the columns of VL, in the same |
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*> order as their eigenvalues. |
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*> Each eigenvector will be scaled so the largest component |
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*> will have 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 COMPLEX*16 array, dimension (LDVR,N) |
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*> If JOBVR = 'V', the right generalized eigenvectors v(j) are |
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*> stored one after another in the columns of VR, in the same |
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*> order as their eigenvalues. |
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*> Each eigenvector will be scaled so the largest component |
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*> will have 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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*> 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 JOB = '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. If the eigenvalues cannot be reordered to |
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*> compute RCONDV(j), RCONDV(j) is set to 0; this can only occur |
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*> when the true 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 COMPLEX*16 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 SENSE = 'E', LWORK >= max(1,4*N). |
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*> If SENSE = 'V' or 'B', LWORK >= max(1,2*N*N+2*N). |
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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] RWORK |
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*> \verbatim |
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*> RWORK is DOUBLE PRECISION array, dimension (lrwork) |
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*> lrwork must be at least max(1,6*N) if BALANC = 'S' or 'B', |
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*> and at least max(1,2*N) otherwise. |
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*> Real workspace. |
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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+2) |
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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 ALPHA(j) and BETA(j) should be correct |
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*> for j=INFO+1,...,N. |
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*> > N: =N+1: other than QZ iteration failed in ZHGEQZ. |
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*> =N+2: error return from ZTGEVC. |
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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 complex16GEeigen |
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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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* ===================================================================== |
SUBROUTINE ZGGEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, B, LDB, |
SUBROUTINE ZGGEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, B, LDB, |
$ ALPHA, BETA, VL, LDVL, VR, LDVR, ILO, IHI, |
$ ALPHA, BETA, VL, LDVL, VR, LDVR, ILO, IHI, |
$ LSCALE, RSCALE, ABNRM, BBNRM, RCONDE, RCONDV, |
$ LSCALE, RSCALE, ABNRM, BBNRM, RCONDE, RCONDV, |
$ WORK, LWORK, RWORK, IWORK, BWORK, INFO ) |
$ WORK, LWORK, RWORK, IWORK, BWORK, INFO ) |
* |
* |
* -- LAPACK driver routine (version 3.2) -- |
* -- LAPACK driver routine (version 3.4.1) -- |
* -- LAPACK is a software package provided by Univ. of Tennessee, -- |
* -- LAPACK is a software package provided by Univ. of Tennessee, -- |
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
* November 2006 |
* April 2012 |
* |
* |
* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
CHARACTER BALANC, JOBVL, JOBVR, SENSE |
CHARACTER BALANC, JOBVL, JOBVR, SENSE |
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$ WORK( * ) |
$ WORK( * ) |
* .. |
* .. |
* |
* |
* Purpose |
* ===================================================================== |
* ======= |
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* |
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* ZGGEVX computes for a pair of N-by-N complex nonsymmetric matrices |
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* (A,B) the generalized eigenvalues, and optionally, the left and/or |
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* right generalized eigenvectors. |
|
* |
|
* Optionally, it also computes a balancing transformation to improve |
|
* the conditioning of the eigenvalues and eigenvectors (ILO, IHI, |
|
* 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 |
|
* singular. It is usually represented as the pair (alpha,beta), as |
|
* there is a reasonable interpretation for beta=0, and even for both |
|
* being zero. |
|
* |
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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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* A * v(j) = lambda(j) * B * v(j) . |
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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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* u(j)**H * A = lambda(j) * u(j)**H * B. |
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* where u(j)**H is the conjugate-transpose of u(j). |
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* |
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* |
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* Arguments |
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* ========= |
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* |
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* BALANC (input) 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 |
|
* balancing does. |
|
* |
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* JOBVL (input) 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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* |
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* JOBVR (input) 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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* |
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* SENSE (input) 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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* |
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* N (input) INTEGER |
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* The order of the matrices A, B, VL, and VR. N >= 0. |
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* |
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* A (input/output) COMPLEX*16 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 complex Schur |
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* form of the "balanced" versions of the input A and B. |
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* |
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* LDA (input) INTEGER |
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* The leading dimension of A. LDA >= max(1,N). |
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* |
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* B (input/output) COMPLEX*16 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 complex |
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* Schur form of the "balanced" versions of the input A and B. |
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* |
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* LDB (input) INTEGER |
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* The leading dimension of B. LDB >= max(1,N). |
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* |
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* ALPHA (output) COMPLEX*16 array, dimension (N) |
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* BETA (output) COMPLEX*16 array, dimension (N) |
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* On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the generalized |
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* eigenvalues. |
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* |
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* Note: the quotient ALPHA(j)/BETA(j) ) may easily over- or |
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* underflow, and BETA(j) may even be zero. Thus, the user |
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* should avoid naively computing the ratio ALPHA/BETA. |
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* However, ALPHA will be always less than and usually |
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* comparable with norm(A) in magnitude, and BETA always less |
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* than and usually comparable with norm(B). |
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* |
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* VL (output) COMPLEX*16 array, dimension (LDVL,N) |
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* If JOBVL = 'V', the left generalized eigenvectors u(j) are |
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* stored one after another in the columns of VL, in the same |
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* order as their eigenvalues. |
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* Each eigenvector will be scaled so the largest component |
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* will have abs(real part) + abs(imag. part) = 1. |
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* Not referenced if JOBVL = 'N'. |
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* |
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* LDVL (input) 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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* |
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* VR (output) COMPLEX*16 array, dimension (LDVR,N) |
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* If JOBVR = 'V', the right generalized eigenvectors v(j) are |
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* stored one after another in the columns of VR, in the same |
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* order as their eigenvalues. |
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* Each eigenvector will be scaled so the largest component |
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* will have abs(real part) + abs(imag. part) = 1. |
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* Not referenced if JOBVR = 'N'. |
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* |
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* LDVR (input) INTEGER |
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* The leading dimension of the matrix VR. LDVR >= 1, and |
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* if JOBVR = 'V', LDVR >= N. |
|
* |
|
* ILO (output) INTEGER |
|
* IHI (output) 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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* |
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* LSCALE (output) DOUBLE PRECISION array, dimension (N) |
|
* 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 |
|
* 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 |
|
* = PL(j) for j = IHI+1,...,N. |
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* The order in which the interchanges are made is N to IHI+1, |
|
* then 1 to ILO-1. |
|
* |
|
* RSCALE (output) DOUBLE PRECISION array, dimension (N) |
|
* Details of the permutations and scaling factors applied |
|
* to the right side of A and B. If PR(j) is the index of the |
|
* column interchanged with column j, and DR(j) is the scaling |
|
* factor applied to column j, then |
|
* RSCALE(j) = PR(j) for j = 1,...,ILO-1 |
|
* = DR(j) for j = ILO,...,IHI |
|
* = PR(j) for j = IHI+1,...,N |
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* The order in which the interchanges are made is N to IHI+1, |
|
* then 1 to ILO-1. |
|
* |
|
* ABNRM (output) DOUBLE PRECISION |
|
* The one-norm of the balanced matrix A. |
|
* |
|
* BBNRM (output) DOUBLE PRECISION |
|
* The one-norm of the balanced matrix B. |
|
* |
|
* RCONDE (output) DOUBLE PRECISION array, dimension (N) |
|
* If SENSE = 'E' or 'B', the reciprocal condition numbers of |
|
* the eigenvalues, stored in consecutive elements of the array. |
|
* If SENSE = 'N' or 'V', RCONDE is not referenced. |
|
* |
|
* RCONDV (output) DOUBLE PRECISION array, dimension (N) |
|
* If JOB = 'V' or 'B', the estimated reciprocal condition |
|
* numbers of the eigenvectors, stored in consecutive elements |
|
* of the array. If the eigenvalues cannot be reordered to |
|
* compute RCONDV(j), RCONDV(j) is set to 0; this can only occur |
|
* when the true value would be very small anyway. |
|
* If SENSE = 'N' or 'E', RCONDV is not referenced. |
|
* |
|
* WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK)) |
|
* On exit, if INFO = 0, WORK(1) returns the optimal LWORK. |
|
* |
|
* LWORK (input) INTEGER |
|
* The dimension of the array WORK. LWORK >= max(1,2*N). |
|
* If SENSE = 'E', LWORK >= max(1,4*N). |
|
* If SENSE = 'V' or 'B', LWORK >= max(1,2*N*N+2*N). |
|
* |
|
* If LWORK = -1, then a workspace query is assumed; the routine |
|
* only calculates the optimal size of the WORK array, returns |
|
* this value as the first entry of the WORK array, and no error |
|
* message related to LWORK is issued by XERBLA. |
|
* |
|
* RWORK (workspace) REAL array, dimension (lrwork) |
|
* lrwork must be at least max(1,6*N) if BALANC = 'S' or 'B', |
|
* and at least max(1,2*N) otherwise. |
|
* Real workspace. |
|
* |
|
* IWORK (workspace) INTEGER array, dimension (N+2) |
|
* If SENSE = 'E', IWORK is not referenced. |
|
* |
|
* BWORK (workspace) LOGICAL array, dimension (N) |
|
* If SENSE = 'N', BWORK is not referenced. |
|
* |
|
* INFO (output) INTEGER |
|
* = 0: successful exit |
|
* < 0: if INFO = -i, the i-th argument had an illegal value. |
|
* = 1,...,N: |
|
* The QZ iteration failed. No eigenvectors have been |
|
* calculated, but ALPHA(j) and BETA(j) should be correct |
|
* for j=INFO+1,...,N. |
|
* > N: =N+1: other than QZ iteration failed in ZHGEQZ. |
|
* =N+2: error return from ZTGEVC. |
|
* |
|
* Further Details |
|
* =============== |
|
* |
|
* Balancing a matrix pair (A,B) includes, first, permuting rows and |
|
* columns to isolate eigenvalues, second, applying diagonal similarity |
|
* transformation to the rows and columns to make the rows and columns |
|
* as close in norm as possible. The computed reciprocal condition |
|
* numbers correspond to the balanced matrix. Permuting rows and columns |
|
* will not change the condition numbers (in exact arithmetic) but |
|
* diagonal scaling will. For further explanation of balancing, see |
|
* section 4.11.1.2 of LAPACK Users' Guide. |
|
* |
|
* An approximate error bound on the chordal distance between the i-th |
|
* computed generalized eigenvalue w and the corresponding exact |
|
* eigenvalue lambda is |
|
* |
|
* chord(w, lambda) <= EPS * norm(ABNRM, BBNRM) / RCONDE(I) |
|
* |
|
* An approximate error bound for the angle between the i-th computed |
|
* eigenvector VL(i) or VR(i) is given by |
|
* |
|
* EPS * norm(ABNRM, BBNRM) / DIF(i). |
|
* |
|
* For further explanation of the reciprocal condition numbers RCONDE |
|
* and RCONDV, see section 4.11 of LAPACK User's Guide. |
|
* |
* |
* .. Parameters .. |
* .. Parameters .. |
DOUBLE PRECISION ZERO, ONE |
DOUBLE PRECISION ZERO, ONE |
Line 637
|
Line 788
|
* |
* |
* Undo scaling if necessary |
* Undo scaling if necessary |
* |
* |
|
90 CONTINUE |
|
* |
IF( ILASCL ) |
IF( ILASCL ) |
$ CALL ZLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHA, N, IERR ) |
$ CALL ZLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHA, N, IERR ) |
* |
* |
IF( ILBSCL ) |
IF( ILBSCL ) |
$ CALL ZLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N, IERR ) |
$ CALL ZLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N, IERR ) |
* |
* |
90 CONTINUE |
|
WORK( 1 ) = MAXWRK |
WORK( 1 ) = MAXWRK |
* |
|
RETURN |
RETURN |
* |
* |
* End of ZGGEVX |
* End of ZGGEVX |