version 1.5, 2010/08/07 13:22:14
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version 1.18, 2023/08/07 08:38:51
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*> \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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*> \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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* ===================================================================== |
SUBROUTINE DGGEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, B, LDB, |
SUBROUTINE DGGEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, B, LDB, |
$ ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, ILO, |
$ ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, ILO, |
$ IHI, LSCALE, RSCALE, ABNRM, BBNRM, RCONDE, |
$ IHI, LSCALE, RSCALE, ABNRM, BBNRM, RCONDE, |
$ RCONDV, WORK, LWORK, IWORK, BWORK, INFO ) |
$ RCONDV, WORK, LWORK, IWORK, BWORK, INFO ) |
* |
* |
* -- LAPACK driver routine (version 3.2) -- |
* -- LAPACK driver routine -- |
* -- 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 |
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* |
* |
* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
CHARACTER BALANC, JOBVL, JOBVR, SENSE |
CHARACTER BALANC, JOBVL, JOBVR, SENSE |
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$ VL( LDVL, * ), VR( LDVR, * ), WORK( * ) |
$ VL( LDVL, * ), VR( LDVR, * ), WORK( * ) |
* .. |
* .. |
* |
* |
* Purpose |
|
* ======= |
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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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* Optionally also, it computes a balancing transformation to improve |
|
* the conditioning of the eigenvalues and eigenvectors (ILO, IHI, |
|
* LSCALE, RSCALE, ABNRM, and BBNRM), reciprocal condition numbers for |
|
* the eigenvalues (RCONDE), and reciprocal condition numbers for the |
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* right eigenvectors (RCONDV). |
|
* |
|
* A generalized eigenvalue for a pair of matrices (A,B) is a scalar |
|
* 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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* |
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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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* |
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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 |
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* balancing does. |
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* |
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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) 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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* |
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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) 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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* |
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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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* ALPHAR (output) DOUBLE PRECISION array, dimension (N) |
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* ALPHAI (output) DOUBLE PRECISION array, dimension (N) |
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* BETA (output) DOUBLE PRECISION array, dimension (N) |
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* On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will |
|
* be the generalized eigenvalues. If ALPHAI(j) is zero, then |
|
* the j-th eigenvalue is real; if positive, then the j-th and |
|
* (j+1)-st eigenvalues are a complex conjugate pair, with |
|
* ALPHAI(j+1) negative. |
|
* |
|
* Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j) |
|
* may easily over- or underflow, and BETA(j) may even be zero. |
|
* Thus, the user should avoid naively computing the ratio |
|
* ALPHA/BETA. However, ALPHAR and ALPHAI will be always less |
|
* than and usually comparable with norm(A) in magnitude, and |
|
* BETA always less than and usually comparable with norm(B). |
|
* |
|
* VL (output) DOUBLE PRECISION array, dimension (LDVL,N) |
|
* If JOBVL = 'V', the left eigenvectors u(j) are stored one |
|
* after another in the columns of VL, in the same order as |
|
* their eigenvalues. If the j-th eigenvalue is real, then |
|
* u(j) = VL(:,j), the j-th column of VL. If the j-th and |
|
* (j+1)-th eigenvalues form a complex conjugate pair, then |
|
* u(j) = VL(:,j)+i*VL(:,j+1) and u(j+1) = VL(:,j)-i*VL(:,j+1). |
|
* Each eigenvector will be scaled so the largest component have |
|
* abs(real part) + abs(imag. part) = 1. |
|
* Not referenced if JOBVL = 'N'. |
|
* |
|
* LDVL (input) INTEGER |
|
* The leading dimension of the matrix VL. LDVL >= 1, and |
|
* if JOBVL = 'V', LDVL >= N. |
|
* |
|
* VR (output) DOUBLE PRECISION array, dimension (LDVR,N) |
|
* If JOBVR = 'V', the right eigenvectors v(j) are stored one |
|
* after another in the columns of VR, in the same order as |
|
* their eigenvalues. If the j-th eigenvalue is real, then |
|
* v(j) = VR(:,j), the j-th column of VR. If the j-th and |
|
* (j+1)-th eigenvalues form a complex conjugate pair, then |
|
* v(j) = VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1). |
|
* Each eigenvector will be scaled so the largest component have |
|
* abs(real part) + abs(imag. part) = 1. |
|
* Not referenced if JOBVR = 'N'. |
|
* |
|
* LDVR (input) INTEGER |
|
* The leading dimension of the matrix VR. LDVR >= 1, and |
|
* if JOBVR = 'V', LDVR >= N. |
|
* |
|
* ILO (output) INTEGER |
|
* IHI (output) INTEGER |
|
* ILO and IHI are integer values such that on exit |
|
* A(i,j) = 0 and B(i,j) = 0 if i > j and |
|
* j = 1,...,ILO-1 or i = IHI+1,...,N. |
|
* If BALANC = 'N' or 'S', ILO = 1 and IHI = N. |
|
* |
|
* LSCALE (output) DOUBLE PRECISION array, dimension (N) |
|
* Details of the permutations and scaling factors applied |
|
* to the left side of A and B. If PL(j) is the index of the |
|
* row interchanged with row j, and DL(j) is the scaling |
|
* factor applied to row j, then |
|
* LSCALE(j) = PL(j) for j = 1,...,ILO-1 |
|
* = DL(j) for j = ILO,...,IHI |
|
* = PL(j) for j = IHI+1,...,N. |
|
* 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 |
|
* 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. |
|
* For a complex conjugate pair of eigenvalues two consecutive |
|
* elements of RCONDE are set to the same value. Thus RCONDE(j), |
|
* RCONDV(j), and the j-th columns of VL and VR all correspond |
|
* to the j-th eigenpair. |
|
* If SENSE = 'N or 'V', RCONDE is not referenced. |
|
* |
|
* RCONDV (output) DOUBLE PRECISION array, dimension (N) |
|
* If SENSE = 'V' or 'B', the estimated reciprocal condition |
|
* numbers of the eigenvectors, stored in consecutive elements |
|
* of the array. For a complex eigenvector two consecutive |
|
* elements of RCONDV are set to the same value. 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) DOUBLE PRECISION 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 BALANC = 'S' or 'B', or JOBVL = 'V', or JOBVR = 'V', |
|
* LWORK >= max(1,6*N). |
|
* If SENSE = 'E' or 'B', LWORK >= max(1,10*N). |
|
* If SENSE = 'V' or 'B', LWORK >= 2*N*N+8*N+16. |
|
* |
|
* 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. |
|
* |
|
* IWORK (workspace) INTEGER array, dimension (N+6) |
|
* 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 ALPHAR(j), ALPHAI(j), and BETA(j) |
|
* should be correct for j=INFO+1,...,N. |
|
* > N: =N+1: other than QZ iteration failed in DHGEQZ. |
|
* =N+2: error return from DTGEVC. |
|
* |
|
* 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 .. |
Line 283
|
Line 429
|
* .. External Subroutines .. |
* .. External Subroutines .. |
EXTERNAL DGEQRF, DGGBAK, DGGBAL, DGGHRD, DHGEQZ, DLABAD, |
EXTERNAL DGEQRF, DGGBAK, DGGBAL, DGGHRD, DHGEQZ, DLABAD, |
$ DLACPY, DLASCL, DLASET, DORGQR, DORMQR, DTGEVC, |
$ DLACPY, DLASCL, DLASET, DORGQR, DORMQR, DTGEVC, |
$ DTGSNA, XERBLA |
$ DTGSNA, XERBLA |
* .. |
* .. |
* .. External Functions .. |
* .. External Functions .. |
LOGICAL LSAME |
LOGICAL LSAME |
Line 700
|
Line 846
|
* |
* |
* Undo scaling if necessary |
* Undo scaling if necessary |
* |
* |
|
130 CONTINUE |
|
* |
IF( ILASCL ) THEN |
IF( ILASCL ) THEN |
CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAR, N, IERR ) |
CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAR, N, IERR ) |
CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAI, N, IERR ) |
CALL DLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAI, N, IERR ) |
Line 709
|
Line 857
|
CALL DLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N, IERR ) |
CALL DLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N, IERR ) |
END IF |
END IF |
* |
* |
130 CONTINUE |
|
WORK( 1 ) = MAXWRK |
WORK( 1 ) = MAXWRK |
* |
|
RETURN |
RETURN |
* |
* |
* End of DGGEVX |
* End of DGGEVX |