version 1.5, 2010/08/07 13:22:30
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version 1.17, 2023/08/07 08:39:16
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*> \brief <b> ZGEEVX 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 ZGEGS + dependencies |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgegs.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/zgegs.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/zgegs.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 ZGEGS( JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHA, BETA, |
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* VSL, LDVSL, VSR, LDVSR, WORK, LWORK, RWORK, |
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* INFO ) |
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* |
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* .. Scalar Arguments .. |
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* CHARACTER JOBVSL, JOBVSR |
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* INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N |
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* .. |
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* .. Array Arguments .. |
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* DOUBLE PRECISION RWORK( * ) |
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* COMPLEX*16 A( LDA, * ), ALPHA( * ), B( LDB, * ), |
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* $ BETA( * ), VSL( LDVSL, * ), VSR( LDVSR, * ), |
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* $ WORK( * ) |
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* .. |
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* |
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* |
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*> \par Purpose: |
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* ============= |
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*> |
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*> \verbatim |
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*> |
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*> This routine is deprecated and has been replaced by routine ZGGES. |
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*> |
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*> ZGEGS computes the eigenvalues, Schur form, and, optionally, the |
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*> left and or/right Schur vectors of a complex matrix pair (A,B). |
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*> Given two square matrices A and B, the generalized Schur |
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*> factorization has the form |
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*> |
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*> A = Q*S*Z**H, B = Q*T*Z**H |
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*> |
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*> where Q and Z are unitary matrices and S and T are upper triangular. |
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*> The columns of Q are the left Schur vectors |
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*> and the columns of Z are the right Schur vectors. |
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*> |
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*> If only the eigenvalues of (A,B) are needed, the driver routine |
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*> ZGEGV should be used instead. See ZGEGV for a description of the |
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*> eigenvalues of the generalized nonsymmetric eigenvalue problem |
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*> (GNEP). |
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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 (returned in VSL). |
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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 (returned in VSR). |
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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 COMPLEX*16 array, dimension (LDA, N) |
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*> On entry, the matrix A. |
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*> On exit, the upper triangular matrix S from the generalized |
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*> Schur factorization. |
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*> \endverbatim |
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*> |
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*> \param[in] LDA |
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*> \verbatim |
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*> LDA is INTEGER |
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*> The leading dimension of A. LDA >= max(1,N). |
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*> \endverbatim |
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*> |
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*> \param[in,out] B |
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*> \verbatim |
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*> B is COMPLEX*16 array, dimension (LDB, N) |
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*> On entry, the matrix B. |
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*> On exit, the upper triangular matrix T from the generalized |
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*> Schur factorization. |
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*> \endverbatim |
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*> |
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*> \param[in] LDB |
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*> \verbatim |
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*> LDB is INTEGER |
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*> The leading dimension of B. LDB >= max(1,N). |
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*> \endverbatim |
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*> |
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*> \param[out] ALPHA |
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*> \verbatim |
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*> ALPHA is COMPLEX*16 array, dimension (N) |
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*> The complex scalars alpha that define the eigenvalues of |
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*> GNEP. ALPHA(j) = S(j,j), the diagonal element of the Schur |
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*> form of A. |
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*> \endverbatim |
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*> |
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*> \param[out] BETA |
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*> \verbatim |
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*> BETA is COMPLEX*16 array, dimension (N) |
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*> The non-negative real scalars beta that define the |
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*> eigenvalues of GNEP. BETA(j) = T(j,j), the diagonal element |
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*> of the triangular factor T. |
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*> |
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*> Together, the quantities alpha = ALPHA(j) and beta = BETA(j) |
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*> represent the j-th eigenvalue of the matrix pair (A,B), in |
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*> one of the forms lambda = alpha/beta or mu = beta/alpha. |
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*> Since either lambda or mu may overflow, they should not, |
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*> in general, be computed. |
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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 COMPLEX*16 array, dimension (LDVSL,N) |
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*> If JOBVSL = 'V', the matrix of left Schur vectors Q. |
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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 COMPLEX*16 array, dimension (LDVSR,N) |
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*> If JOBVSR = 'V', the matrix of right Schur vectors Z. |
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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 COMPLEX*16 array, dimension (MAX(1,LWORK)) |
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*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. |
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*> \endverbatim |
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*> |
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*> \param[in] LWORK |
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*> \verbatim |
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*> LWORK is INTEGER |
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*> The dimension of the array WORK. LWORK >= max(1,2*N). |
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*> 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 ZGEQRF, ZUNMQR, and CUNGQR.) Then compute: |
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*> NB -- MAX of the blocksizes for ZGEQRF, ZUNMQR, and CUNGQR; |
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*> the optimal LWORK is 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] RWORK |
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*> \verbatim |
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*> RWORK is DOUBLE PRECISION array, dimension (3*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 ALPHA(j) and BETA(j) should be correct for |
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*> j=INFO+1,...,N. |
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*> > N: errors that usually indicate LAPACK problems: |
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*> =N+1: error return from ZGGBAL |
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*> =N+2: error return from ZGEQRF |
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*> =N+3: error return from ZUNMQR |
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*> =N+4: error return from ZUNGQR |
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*> =N+5: error return from ZGGHRD |
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*> =N+6: error return from ZHGEQZ (other than failed |
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*> iteration) |
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*> =N+7: error return from ZGGBAK (computing VSL) |
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*> =N+8: error return from ZGGBAK (computing VSR) |
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*> =N+9: error return from ZLASCL (various places) |
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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 complex16GEeigen |
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* |
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* ===================================================================== |
SUBROUTINE ZGEGS( JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHA, BETA, |
SUBROUTINE ZGEGS( JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHA, BETA, |
$ VSL, LDVSL, VSR, LDVSR, WORK, LWORK, RWORK, |
$ VSL, LDVSL, VSR, LDVSR, WORK, LWORK, RWORK, |
$ INFO ) |
$ 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 |
CHARACTER JOBVSL, JOBVSR |
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$ WORK( * ) |
$ WORK( * ) |
* .. |
* .. |
* |
* |
* Purpose |
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* ======= |
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* |
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* This routine is deprecated and has been replaced by routine ZGGES. |
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* |
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* ZGEGS computes the eigenvalues, Schur form, and, optionally, the |
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* left and or/right Schur vectors of a complex matrix pair (A,B). |
|
* Given two square matrices A and B, the generalized Schur |
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* factorization has the form |
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* |
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* A = Q*S*Z**H, B = Q*T*Z**H |
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* |
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* where Q and Z are unitary matrices and S and T are upper triangular. |
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* The columns of Q are the left Schur vectors |
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* and the columns of Z are the right Schur vectors. |
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* |
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* If only the eigenvalues of (A,B) are needed, the driver routine |
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* ZGEGV should be used instead. See ZGEGV for a description of the |
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* eigenvalues of the generalized nonsymmetric eigenvalue problem |
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* (GNEP). |
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* |
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* Arguments |
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* ========= |
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* |
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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 (returned in VSL). |
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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 (returned in VSR). |
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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) COMPLEX*16 array, dimension (LDA, N) |
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* On entry, the matrix A. |
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* On exit, the upper triangular matrix S from the generalized |
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* Schur factorization. |
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* |
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* LDA (input) INTEGER |
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* The leading dimension of A. LDA >= max(1,N). |
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* |
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* B (input/output) COMPLEX*16 array, dimension (LDB, N) |
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* On entry, the matrix B. |
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* On exit, the upper triangular matrix T from the generalized |
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* Schur factorization. |
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* |
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* LDB (input) INTEGER |
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* The leading dimension of B. LDB >= max(1,N). |
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* |
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* ALPHA (output) COMPLEX*16 array, dimension (N) |
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* The complex scalars alpha that define the eigenvalues of |
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* GNEP. ALPHA(j) = S(j,j), the diagonal element of the Schur |
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* form of A. |
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* |
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* BETA (output) COMPLEX*16 array, dimension (N) |
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* The non-negative real scalars beta that define the |
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* eigenvalues of GNEP. BETA(j) = T(j,j), the diagonal element |
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* of the triangular factor T. |
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* |
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* Together, the quantities alpha = ALPHA(j) and beta = BETA(j) |
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* represent the j-th eigenvalue of the matrix pair (A,B), in |
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* one of the forms lambda = alpha/beta or mu = beta/alpha. |
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* Since either lambda or mu may overflow, they should not, |
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* in general, be computed. |
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* |
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* |
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* VSL (output) COMPLEX*16 array, dimension (LDVSL,N) |
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* If JOBVSL = 'V', the matrix of left Schur vectors Q. |
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* Not referenced if JOBVSL = 'N'. |
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* |
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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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* |
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* VSR (output) COMPLEX*16 array, dimension (LDVSR,N) |
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* If JOBVSR = 'V', the matrix of right Schur vectors Z. |
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* Not referenced if JOBVSR = 'N'. |
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* |
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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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* |
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* WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK)) |
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* On exit, if INFO = 0, WORK(1) returns the optimal LWORK. |
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* |
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* LWORK (input) INTEGER |
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* The dimension of the array WORK. LWORK >= max(1,2*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 ZGEQRF, ZUNMQR, and CUNGQR.) Then compute: |
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* NB -- MAX of the blocksizes for ZGEQRF, ZUNMQR, and CUNGQR; |
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* the optimal LWORK is 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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* RWORK (workspace) DOUBLE PRECISION array, dimension (3*N) |
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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. (A,B) are not in Schur |
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* form, but ALPHA(j) and BETA(j) should be correct for |
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* j=INFO+1,...,N. |
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* > N: errors that usually indicate LAPACK problems: |
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* =N+1: error return from ZGGBAL |
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* =N+2: error return from ZGEQRF |
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* =N+3: error return from ZUNMQR |
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* =N+4: error return from ZUNGQR |
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* =N+5: error return from ZGGHRD |
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* =N+6: error return from ZHGEQZ (other than failed |
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* iteration) |
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* =N+7: error return from ZGGBAK (computing VSL) |
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* =N+8: error return from ZGGBAK (computing VSR) |
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* =N+9: error return from ZLASCL (various places) |
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* |
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* ===================================================================== |
* ===================================================================== |
* |
* |
* .. Parameters .. |
* .. Parameters .. |