version 1.1.1.1, 2010/01/26 15:22:46
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version 1.18, 2023/08/07 08:38:50
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*> \brief <b> DGGES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors 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 DGGES + dependencies |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgges.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/dgges.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/dgges.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 DGGES( JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB, |
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* SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, |
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* LDVSR, WORK, LWORK, BWORK, INFO ) |
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* |
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* .. Scalar Arguments .. |
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* CHARACTER JOBVSL, JOBVSR, SORT |
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* INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N, SDIM |
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* .. |
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* .. Array Arguments .. |
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* LOGICAL BWORK( * ) |
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* DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ), |
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* $ B( LDB, * ), BETA( * ), VSL( LDVSL, * ), |
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* $ VSR( LDVSR, * ), WORK( * ) |
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* .. |
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* .. Function Arguments .. |
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* LOGICAL SELCTG |
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* EXTERNAL SELCTG |
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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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*> DGGES computes for a pair of N-by-N real nonsymmetric matrices (A,B), |
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*> the generalized eigenvalues, the generalized real Schur form (S,T), |
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*> optionally, the left and/or right matrices of Schur vectors (VSL and |
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*> VSR). This gives the generalized Schur factorization |
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*> |
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*> (A,B) = ( (VSL)*S*(VSR)**T, (VSL)*T*(VSR)**T ) |
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*> |
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*> Optionally, it also orders the eigenvalues so that a selected cluster |
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*> of eigenvalues appears in the leading diagonal blocks of the upper |
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*> quasi-triangular matrix S and the upper triangular matrix T.The |
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*> leading columns of VSL and VSR then form an orthonormal basis for the |
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*> corresponding left and right eigenspaces (deflating subspaces). |
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*> |
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*> (If only the generalized eigenvalues are needed, use the driver |
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*> DGGEV instead, which is faster.) |
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*> |
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*> A generalized eigenvalue for a pair of matrices (A,B) is a scalar w |
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*> or a ratio alpha/beta = w, such that A - w*B is singular. It is |
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*> usually represented as the pair (alpha,beta), as there is a |
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*> reasonable interpretation for beta=0 or both being zero. |
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*> |
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*> A pair of matrices (S,T) is in generalized real Schur form if T is |
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*> upper triangular with non-negative diagonal and S is block upper |
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*> triangular with 1-by-1 and 2-by-2 blocks. 1-by-1 blocks correspond |
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*> to real generalized eigenvalues, while 2-by-2 blocks of S will be |
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*> "standardized" by making the corresponding elements of T have the |
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*> form: |
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*> [ a 0 ] |
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*> [ 0 b ] |
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*> |
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*> and the pair of corresponding 2-by-2 blocks in S and T will have a |
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*> complex conjugate pair of generalized eigenvalues. |
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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] JOBVSL |
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*> \verbatim |
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*> JOBVSL is CHARACTER*1 |
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*> = 'N': do not compute the left Schur vectors; |
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*> = 'V': compute the left Schur vectors. |
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*> \endverbatim |
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*> |
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*> \param[in] JOBVSR |
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*> \verbatim |
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*> JOBVSR is CHARACTER*1 |
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*> = 'N': do not compute the right Schur vectors; |
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*> = 'V': compute the right Schur vectors. |
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*> \endverbatim |
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*> |
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*> \param[in] SORT |
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*> \verbatim |
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*> SORT is CHARACTER*1 |
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*> Specifies whether or not to order the eigenvalues on the |
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*> diagonal of the generalized Schur form. |
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*> = 'N': Eigenvalues are not ordered; |
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*> = 'S': Eigenvalues are ordered (see SELCTG); |
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*> \endverbatim |
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*> |
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*> \param[in] SELCTG |
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*> \verbatim |
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*> SELCTG is a LOGICAL FUNCTION of three DOUBLE PRECISION arguments |
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*> SELCTG must be declared EXTERNAL in the calling subroutine. |
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*> If SORT = 'N', SELCTG is not referenced. |
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*> If SORT = 'S', SELCTG is used to select eigenvalues to sort |
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*> to the top left of the Schur form. |
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*> An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j) is selected if |
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*> SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e. if either |
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*> one of a complex conjugate pair of eigenvalues is selected, |
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*> then both complex eigenvalues are selected. |
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*> |
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*> Note that in the ill-conditioned case, a selected complex |
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*> eigenvalue may no longer satisfy SELCTG(ALPHAR(j),ALPHAI(j), |
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*> BETA(j)) = .TRUE. after ordering. INFO is to be set to N+2 |
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*> in this case. |
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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, VSL, and VSR. 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 first of the pair of matrices. |
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*> On exit, A has been overwritten by its generalized Schur |
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*> form S. |
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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 second of the pair of matrices. |
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*> On exit, B has been overwritten by its generalized Schur |
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*> form T. |
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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] SDIM |
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*> \verbatim |
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*> SDIM is INTEGER |
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*> If SORT = 'N', SDIM = 0. |
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*> If SORT = 'S', SDIM = number of eigenvalues (after sorting) |
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*> for which SELCTG is true. (Complex conjugate pairs for which |
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*> SELCTG is true for either eigenvalue count as 2.) |
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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. ALPHAR(j) + ALPHAI(j)*i, |
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*> and BETA(j),j=1,...,N are the diagonals of the complex Schur |
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*> form (S,T) that would result if the 2-by-2 diagonal blocks of |
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*> the real Schur form of (A,B) were further reduced to |
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*> triangular form using 2-by-2 complex unitary transformations. |
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*> If ALPHAI(j) is zero, then the j-th eigenvalue is real; if |
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*> positive, then the j-th and (j+1)-st eigenvalues are a |
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*> complex conjugate pair, with 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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*> However, ALPHAR and ALPHAI will be always less than and |
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*> usually comparable with norm(A) in magnitude, and BETA always |
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*> less than and usually comparable with norm(B). |
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*> \endverbatim |
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*> |
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*> \param[out] VSL |
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*> \verbatim |
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*> VSL is DOUBLE PRECISION array, dimension (LDVSL,N) |
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*> If JOBVSL = 'V', VSL will contain the left Schur vectors. |
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*> Not referenced if JOBVSL = 'N'. |
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*> \endverbatim |
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*> |
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*> \param[in] LDVSL |
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*> \verbatim |
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*> LDVSL is INTEGER |
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*> The leading dimension of the matrix VSL. LDVSL >=1, and |
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*> if JOBVSL = 'V', LDVSL >= N. |
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*> \endverbatim |
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*> |
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*> \param[out] VSR |
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*> \verbatim |
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*> VSR is DOUBLE PRECISION array, dimension (LDVSR,N) |
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*> If JOBVSR = 'V', VSR will contain the right Schur vectors. |
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*> Not referenced if JOBVSR = 'N'. |
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*> \endverbatim |
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*> |
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*> \param[in] LDVSR |
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*> \verbatim |
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*> LDVSR is INTEGER |
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*> The leading dimension of the matrix VSR. LDVSR >= 1, and |
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*> if JOBVSR = 'V', LDVSR >= 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. |
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*> If N = 0, LWORK >= 1, else LWORK >= 8*N+16. |
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*> For good performance , LWORK must generally be larger. |
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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] BWORK |
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*> \verbatim |
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*> BWORK is LOGICAL array, dimension (N) |
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*> Not referenced if SORT = 'N'. |
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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. (A,B) are not in Schur |
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*> form, but ALPHAR(j), ALPHAI(j), and BETA(j) should |
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*> 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: after reordering, roundoff changed values of |
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*> some complex eigenvalues so that leading |
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*> eigenvalues in the Generalized Schur form no |
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*> longer satisfy SELCTG=.TRUE. This could also |
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*> be caused due to scaling. |
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*> =N+3: reordering failed in DTGSEN. |
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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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* ===================================================================== |
SUBROUTINE DGGES( JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB, |
SUBROUTINE DGGES( JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB, |
$ SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, |
$ SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, |
$ LDVSR, WORK, LWORK, BWORK, INFO ) |
$ LDVSR, WORK, LWORK, 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 JOBVSL, JOBVSR, SORT |
CHARACTER JOBVSL, JOBVSR, SORT |
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Line 301
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EXTERNAL SELCTG |
EXTERNAL SELCTG |
* .. |
* .. |
* |
* |
* Purpose |
|
* ======= |
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* |
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* DGGES computes for a pair of N-by-N real nonsymmetric matrices (A,B), |
|
* the generalized eigenvalues, the generalized real Schur form (S,T), |
|
* optionally, the left and/or right matrices of Schur vectors (VSL and |
|
* VSR). This gives the generalized Schur factorization |
|
* |
|
* (A,B) = ( (VSL)*S*(VSR)**T, (VSL)*T*(VSR)**T ) |
|
* |
|
* Optionally, it also orders the eigenvalues so that a selected cluster |
|
* of eigenvalues appears in the leading diagonal blocks of the upper |
|
* quasi-triangular matrix S and the upper triangular matrix T.The |
|
* leading columns of VSL and VSR then form an orthonormal basis for the |
|
* corresponding left and right eigenspaces (deflating subspaces). |
|
* |
|
* (If only the generalized eigenvalues are needed, use the driver |
|
* DGGEV instead, which is faster.) |
|
* |
|
* A generalized eigenvalue for a pair of matrices (A,B) is a scalar w |
|
* or a ratio alpha/beta = w, such that A - w*B is singular. It is |
|
* usually represented as the pair (alpha,beta), as there is a |
|
* reasonable interpretation for beta=0 or both being zero. |
|
* |
|
* A pair of matrices (S,T) is in generalized real Schur form if T is |
|
* upper triangular with non-negative diagonal and S is block upper |
|
* triangular with 1-by-1 and 2-by-2 blocks. 1-by-1 blocks correspond |
|
* to real generalized eigenvalues, while 2-by-2 blocks of S will be |
|
* "standardized" by making the corresponding elements of T have the |
|
* form: |
|
* [ a 0 ] |
|
* [ 0 b ] |
|
* |
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* and the pair of corresponding 2-by-2 blocks in S and T will have a |
|
* complex conjugate pair of generalized eigenvalues. |
|
* |
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* |
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* Arguments |
|
* ========= |
|
* |
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* JOBVSL (input) CHARACTER*1 |
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* = 'N': do not compute the left Schur vectors; |
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* = 'V': compute the left Schur vectors. |
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* |
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* JOBVSR (input) CHARACTER*1 |
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* = 'N': do not compute the right Schur vectors; |
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* = 'V': compute the right Schur vectors. |
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* |
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* SORT (input) CHARACTER*1 |
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* Specifies whether or not to order the eigenvalues on the |
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* diagonal of the generalized Schur form. |
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* = 'N': Eigenvalues are not ordered; |
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* = 'S': Eigenvalues are ordered (see SELCTG); |
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* |
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* SELCTG (external procedure) LOGICAL FUNCTION of three DOUBLE PRECISION arguments |
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* SELCTG must be declared EXTERNAL in the calling subroutine. |
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* If SORT = 'N', SELCTG is not referenced. |
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* If SORT = 'S', SELCTG is used to select eigenvalues to sort |
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* to the top left of the Schur form. |
|
* An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j) is selected if |
|
* SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e. if either |
|
* one of a complex conjugate pair of eigenvalues is selected, |
|
* then both complex eigenvalues are selected. |
|
* |
|
* Note that in the ill-conditioned case, a selected complex |
|
* eigenvalue may no longer satisfy SELCTG(ALPHAR(j),ALPHAI(j), |
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* BETA(j)) = .TRUE. after ordering. INFO is to be set to N+2 |
|
* in this case. |
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* |
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* N (input) INTEGER |
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* The order of the matrices A, B, VSL, and VSR. 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 first of the pair of matrices. |
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* On exit, A has been overwritten by its generalized Schur |
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* form S. |
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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 second of the pair of matrices. |
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* On exit, B has been overwritten by its generalized Schur |
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* form T. |
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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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* SDIM (output) INTEGER |
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* If SORT = 'N', SDIM = 0. |
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* If SORT = 'S', SDIM = number of eigenvalues (after sorting) |
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* for which SELCTG is true. (Complex conjugate pairs for which |
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* SELCTG is true for either eigenvalue count as 2.) |
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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. ALPHAR(j) + ALPHAI(j)*i, |
|
* and BETA(j),j=1,...,N are the diagonals of the complex Schur |
|
* form (S,T) that would result if the 2-by-2 diagonal blocks of |
|
* the real Schur form of (A,B) were further reduced to |
|
* triangular form using 2-by-2 complex unitary transformations. |
|
* 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. |
|
* 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). |
|
* |
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* VSL (output) DOUBLE PRECISION array, dimension (LDVSL,N) |
|
* If JOBVSL = 'V', VSL will contain the left Schur vectors. |
|
* Not referenced if JOBVSL = 'N'. |
|
* |
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* LDVSL (input) INTEGER |
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* The leading dimension of the matrix VSL. LDVSL >=1, and |
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* if JOBVSL = 'V', LDVSL >= N. |
|
* |
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* VSR (output) DOUBLE PRECISION array, dimension (LDVSR,N) |
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* If JOBVSR = 'V', VSR will contain the right Schur vectors. |
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* Not referenced if JOBVSR = 'N'. |
|
* |
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* LDVSR (input) INTEGER |
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* The leading dimension of the matrix VSR. LDVSR >= 1, and |
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* if JOBVSR = 'V', LDVSR >= N. |
|
* |
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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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* LWORK (input) INTEGER |
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* The dimension of the array WORK. |
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* If N = 0, LWORK >= 1, else LWORK >= 8*N+16. |
|
* For good performance , LWORK must generally be larger. |
|
* |
|
* 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. |
|
* |
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* BWORK (workspace) LOGICAL array, dimension (N) |
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* Not referenced if SORT = 'N'. |
|
* |
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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. |
|
* = 1,...,N: |
|
* The QZ iteration failed. (A,B) are not in Schur |
|
* form, 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: after reordering, roundoff changed values of |
|
* some complex eigenvalues so that leading |
|
* eigenvalues in the Generalized Schur form no |
|
* longer satisfy SELCTG=.TRUE. This could also |
|
* be caused due to scaling. |
|
* =N+3: reordering failed in DTGSEN. |
|
* |
|
* ===================================================================== |
* ===================================================================== |
* |
* |
* .. Parameters .. |
* .. Parameters .. |