version 1.7, 2010/12/21 13:53:25
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version 1.17, 2023/08/07 08:38:48
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*> \brief <b> DGEEVX 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 DGEGV + dependencies |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgegv.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/dgegv.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/dgegv.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 DGEGV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, ALPHAI, |
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* BETA, VL, LDVL, VR, LDVR, WORK, LWORK, INFO ) |
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* |
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* .. Scalar Arguments .. |
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* CHARACTER JOBVL, JOBVR |
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* INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N |
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* .. |
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* .. Array Arguments .. |
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* DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ), |
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* $ B( LDB, * ), BETA( * ), VL( LDVL, * ), |
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* $ 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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*> This routine is deprecated and has been replaced by routine DGGEV. |
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*> |
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*> DGEGV computes the eigenvalues and, optionally, the left and/or right |
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*> eigenvectors of a real matrix pair (A,B). |
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*> Given two square matrices A and B, |
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*> the generalized nonsymmetric eigenvalue problem (GNEP) is to find the |
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*> eigenvalues lambda and corresponding (non-zero) eigenvectors x such |
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*> that |
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*> |
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*> A*x = lambda*B*x. |
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*> |
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*> An alternate form is to find the eigenvalues mu and corresponding |
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*> eigenvectors y such that |
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*> |
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*> mu*A*y = B*y. |
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*> |
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*> These two forms are equivalent with mu = 1/lambda and x = y if |
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*> neither lambda nor mu is zero. In order to deal with the case that |
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*> lambda or mu is zero or small, two values alpha and beta are returned |
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*> for each eigenvalue, such that lambda = alpha/beta and |
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*> mu = beta/alpha. |
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*> |
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*> The vectors x and y in the above equations are right eigenvectors of |
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*> the matrix pair (A,B). Vectors u and v satisfying |
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*> |
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*> u**H*A = lambda*u**H*B or mu*v**H*A = v**H*B |
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*> |
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*> are left eigenvectors of (A,B). |
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*> |
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*> Note: this routine performs "full balancing" on A and B |
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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] 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 (returned |
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*> in VL). |
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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 (returned |
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*> in VR). |
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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. |
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*> If JOBVL = 'V' or JOBVR = 'V', then on exit A |
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*> contains the real Schur form of A from the generalized Schur |
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*> factorization of the pair (A,B) after balancing. |
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*> If no eigenvectors were computed, then only the diagonal |
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*> blocks from the Schur form will be correct. See DGGHRD and |
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*> DHGEQZ for details. |
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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. |
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*> If JOBVL = 'V' or JOBVR = 'V', then on exit B contains the |
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*> upper triangular matrix obtained from B in the generalized |
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*> Schur factorization of the pair (A,B) after balancing. |
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*> If no eigenvectors were computed, then only those elements of |
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*> B corresponding to the diagonal blocks from the Schur form of |
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*> A will be correct. See DGGHRD and DHGEQZ for details. |
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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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*> The real parts of each scalar alpha defining an eigenvalue of |
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*> GNEP. |
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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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*> The imaginary parts of each scalar alpha defining an |
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*> eigenvalue of GNEP. If ALPHAI(j) is zero, then the j-th |
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*> 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) = -ALPHAI(j). |
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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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*> The scalars beta that define the eigenvalues of GNEP. |
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*> |
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*> Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and |
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*> beta = BETA(j) represent the j-th eigenvalue of the matrix |
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*> pair (A,B), in one of the forms lambda = alpha/beta or |
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*> mu = beta/alpha. Since either lambda or mu may overflow, |
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*> they should not, in general, be computed. |
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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 |
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*> in the columns of VL, in the same order as their eigenvalues. |
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*> If the j-th eigenvalue is real, then u(j) = VL(:,j). |
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*> If the j-th and (j+1)-st eigenvalues form a complex conjugate |
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*> pair, then |
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*> u(j) = VL(:,j) + i*VL(:,j+1) |
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*> and |
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*> u(j+1) = VL(:,j) - i*VL(:,j+1). |
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*> |
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*> Each eigenvector is scaled so that its largest component has |
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*> abs(real part) + abs(imag. part) = 1, except for eigenvectors |
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*> corresponding to an eigenvalue with alpha = beta = 0, which |
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*> are set to zero. |
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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 x(j) are stored |
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*> in the columns of VR, in the same order as their eigenvalues. |
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*> If the j-th eigenvalue is real, then x(j) = VR(:,j). |
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*> If the j-th and (j+1)-st eigenvalues form a complex conjugate |
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*> pair, then |
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*> x(j) = VR(:,j) + i*VR(:,j+1) |
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*> and |
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*> x(j+1) = VR(:,j) - i*VR(:,j+1). |
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*> |
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*> Each eigenvector is scaled so that its largest component has |
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*> abs(real part) + abs(imag. part) = 1, except for eigenvalues |
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*> corresponding to an eigenvalue with alpha = beta = 0, which |
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*> are set to zero. |
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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] 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(1,8*N). |
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*> For good performance, LWORK must generally be larger. |
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*> To compute the optimal value of LWORK, call ILAENV to get |
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*> blocksizes (for DGEQRF, DORMQR, and DORGQR.) Then compute: |
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*> NB -- MAX of the blocksizes for DGEQRF, DORMQR, and DORGQR; |
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*> The optimal LWORK is: |
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*> 2*N + MAX( 6*N, N*(NB+1) ). |
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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] 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: errors that usually indicate LAPACK problems: |
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*> =N+1: error return from DGGBAL |
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*> =N+2: error return from DGEQRF |
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*> =N+3: error return from DORMQR |
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*> =N+4: error return from DORGQR |
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*> =N+5: error return from DGGHRD |
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*> =N+6: error return from DHGEQZ (other than failed |
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*> iteration) |
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*> =N+7: error return from DTGEVC |
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*> =N+8: error return from DGGBAK (computing VL) |
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*> =N+9: error return from DGGBAK (computing VR) |
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*> =N+10: error return from DLASCL (various calls) |
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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 |
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*> --------- |
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*> |
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*> This driver calls DGGBAL to both permute and scale rows and columns |
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*> of A and B. The permutations PL and PR are chosen so that PL*A*PR |
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*> and PL*B*R will be upper triangular except for the diagonal blocks |
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*> A(i:j,i:j) and B(i:j,i:j), with i and j as close together as |
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*> possible. The diagonal scaling matrices DL and DR are chosen so |
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*> that the pair DL*PL*A*PR*DR, DL*PL*B*PR*DR have elements close to |
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*> one (except for the elements that start out zero.) |
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*> |
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*> After the eigenvalues and eigenvectors of the balanced matrices |
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*> have been computed, DGGBAK transforms the eigenvectors back to what |
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*> they would have been (in perfect arithmetic) if they had not been |
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*> balanced. |
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*> |
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*> Contents of A and B on Exit |
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*> -------- -- - --- - -- ---- |
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*> |
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*> If any eigenvectors are computed (either JOBVL='V' or JOBVR='V' or |
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*> both), then on exit the arrays A and B will contain the real Schur |
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*> form[*] of the "balanced" versions of A and B. If no eigenvectors |
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*> are computed, then only the diagonal blocks will be correct. |
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*> |
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*> [*] See DHGEQZ, DGEGS, or read the book "Matrix Computations", |
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*> by Golub & van Loan, pub. by Johns Hopkins U. Press. |
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*> \endverbatim |
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*> |
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* ===================================================================== |
SUBROUTINE DGEGV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, ALPHAI, |
SUBROUTINE DGEGV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, ALPHAI, |
$ BETA, VL, LDVL, VR, LDVR, WORK, LWORK, INFO ) |
$ BETA, VL, LDVL, VR, LDVR, WORK, LWORK, 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 JOBVL, JOBVR |
CHARACTER JOBVL, JOBVR |
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$ VR( LDVR, * ), WORK( * ) |
$ VR( LDVR, * ), WORK( * ) |
* .. |
* .. |
* |
* |
* Purpose |
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* ======= |
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* |
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* This routine is deprecated and has been replaced by routine DGGEV. |
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* |
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* DGEGV computes the eigenvalues and, optionally, the left and/or right |
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* eigenvectors of a real matrix pair (A,B). |
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* Given two square matrices A and B, |
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* the generalized nonsymmetric eigenvalue problem (GNEP) is to find the |
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* eigenvalues lambda and corresponding (non-zero) eigenvectors x such |
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* that |
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* |
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* A*x = lambda*B*x. |
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* |
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* An alternate form is to find the eigenvalues mu and corresponding |
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* eigenvectors y such that |
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* |
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* mu*A*y = B*y. |
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* |
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* These two forms are equivalent with mu = 1/lambda and x = y if |
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* neither lambda nor mu is zero. In order to deal with the case that |
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* lambda or mu is zero or small, two values alpha and beta are returned |
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* for each eigenvalue, such that lambda = alpha/beta and |
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* mu = beta/alpha. |
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* |
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* The vectors x and y in the above equations are right eigenvectors of |
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* the matrix pair (A,B). Vectors u and v satisfying |
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* |
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* u**H*A = lambda*u**H*B or mu*v**H*A = v**H*B |
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* |
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* are left eigenvectors of (A,B). |
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* |
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* Note: this routine performs "full balancing" on A and B -- see |
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* "Further Details", below. |
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* |
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* Arguments |
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* ========= |
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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 (returned |
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* in VL). |
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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 (returned |
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* in VR). |
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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. |
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* If JOBVL = 'V' or JOBVR = 'V', then on exit A |
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* contains the real Schur form of A from the generalized Schur |
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* factorization of the pair (A,B) after balancing. |
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* If no eigenvectors were computed, then only the diagonal |
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* blocks from the Schur form will be correct. See DGGHRD and |
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* DHGEQZ for details. |
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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. |
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* If JOBVL = 'V' or JOBVR = 'V', then on exit B contains the |
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* upper triangular matrix obtained from B in the generalized |
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* Schur factorization of the pair (A,B) after balancing. |
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* If no eigenvectors were computed, then only those elements of |
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* B corresponding to the diagonal blocks from the Schur form of |
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* A will be correct. See DGGHRD and DHGEQZ for details. |
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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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* The real parts of each scalar alpha defining an eigenvalue of |
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* GNEP. |
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* |
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* ALPHAI (output) DOUBLE PRECISION array, dimension (N) |
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* The imaginary parts of each scalar alpha defining an |
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* eigenvalue of GNEP. If ALPHAI(j) is zero, then the j-th |
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* 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) = -ALPHAI(j). |
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* |
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* BETA (output) DOUBLE PRECISION array, dimension (N) |
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* The scalars beta that define the eigenvalues of GNEP. |
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* |
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* Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and |
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* beta = BETA(j) represent the j-th eigenvalue of the matrix |
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* pair (A,B), in one of the forms lambda = alpha/beta or |
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* mu = beta/alpha. Since either lambda or mu may overflow, |
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* they should not, in general, be computed. |
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* |
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* VL (output) DOUBLE PRECISION array, dimension (LDVL,N) |
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* If JOBVL = 'V', the left eigenvectors u(j) are stored |
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* in the columns of VL, in the same order as their eigenvalues. |
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* If the j-th eigenvalue is real, then u(j) = VL(:,j). |
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* If the j-th and (j+1)-st eigenvalues form a complex conjugate |
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* pair, then |
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* u(j) = VL(:,j) + i*VL(:,j+1) |
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* and |
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* u(j+1) = VL(:,j) - i*VL(:,j+1). |
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* |
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* Each eigenvector is scaled so that its largest component has |
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* abs(real part) + abs(imag. part) = 1, except for eigenvectors |
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* corresponding to an eigenvalue with alpha = beta = 0, which |
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* are set to zero. |
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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) DOUBLE PRECISION array, dimension (LDVR,N) |
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* If JOBVR = 'V', the right eigenvectors x(j) are stored |
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* in the columns of VR, in the same order as their eigenvalues. |
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* If the j-th eigenvalue is real, then x(j) = VR(:,j). |
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* If the j-th and (j+1)-st eigenvalues form a complex conjugate |
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* pair, then |
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* x(j) = VR(:,j) + i*VR(:,j+1) |
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* and |
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* x(j+1) = VR(:,j) - i*VR(:,j+1). |
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* |
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* Each eigenvector is scaled so that its largest component has |
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* abs(real part) + abs(imag. part) = 1, except for eigenvalues |
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* corresponding to an eigenvalue with alpha = beta = 0, which |
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* are set to zero. |
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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. |
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* |
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* WORK (workspace/output) 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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* |
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* LWORK (input) INTEGER |
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* The dimension of the array WORK. LWORK >= max(1,8*N). |
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* For good performance, LWORK must generally be larger. |
|
* To compute the optimal value of LWORK, call ILAENV to get |
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* blocksizes (for DGEQRF, DORMQR, and DORGQR.) Then compute: |
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* NB -- MAX of the blocksizes for DGEQRF, DORMQR, and DORGQR; |
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* The optimal LWORK is: |
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* 2*N + MAX( 6*N, N*(NB+1) ). |
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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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* |
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* INFO (output) 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: errors that usually indicate LAPACK problems: |
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* =N+1: error return from DGGBAL |
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* =N+2: error return from DGEQRF |
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* =N+3: error return from DORMQR |
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* =N+4: error return from DORGQR |
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* =N+5: error return from DGGHRD |
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* =N+6: error return from DHGEQZ (other than failed |
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* iteration) |
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* =N+7: error return from DTGEVC |
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* =N+8: error return from DGGBAK (computing VL) |
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* =N+9: error return from DGGBAK (computing VR) |
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* =N+10: error return from DLASCL (various calls) |
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* |
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* Further Details |
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* =============== |
|
* |
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* Balancing |
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* --------- |
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* |
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* This driver calls DGGBAL to both permute and scale rows and columns |
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* of A and B. The permutations PL and PR are chosen so that PL*A*PR |
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* and PL*B*R will be upper triangular except for the diagonal blocks |
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* A(i:j,i:j) and B(i:j,i:j), with i and j as close together as |
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* possible. The diagonal scaling matrices DL and DR are chosen so |
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* that the pair DL*PL*A*PR*DR, DL*PL*B*PR*DR have elements close to |
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* one (except for the elements that start out zero.) |
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* |
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* After the eigenvalues and eigenvectors of the balanced matrices |
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* have been computed, DGGBAK transforms the eigenvectors back to what |
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* they would have been (in perfect arithmetic) if they had not been |
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* balanced. |
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* |
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* Contents of A and B on Exit |
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* -------- -- - --- - -- ---- |
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* |
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* If any eigenvectors are computed (either JOBVL='V' or JOBVR='V' or |
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* both), then on exit the arrays A and B will contain the real Schur |
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* form[*] of the "balanced" versions of A and B. If no eigenvectors |
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* are computed, then only the diagonal blocks will be correct. |
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* |
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* [*] See DHGEQZ, DGEGS, or read the book "Matrix Computations", |
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* by Golub & van Loan, pub. by Johns Hopkins U. Press. |
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* |
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* ===================================================================== |
* ===================================================================== |
* |
* |
* .. Parameters .. |
* .. Parameters .. |