version 1.4, 2010/08/06 15:32:23
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version 1.14, 2017/06/17 10:53:47
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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 DGEGS + dependencies |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgegs.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/dgegs.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/dgegs.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 DGEGS( JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHAR, |
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* ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, WORK, |
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* LWORK, 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 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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* |
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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 DGGES. |
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*> |
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*> DGEGS computes the eigenvalues, real Schur form, and, optionally, |
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*> left and or/right Schur vectors of a real matrix pair (A,B). |
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*> Given two square matrices A and B, the generalized real Schur |
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*> factorization has the form |
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*> |
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*> A = Q*S*Z**T, B = Q*T*Z**T |
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*> |
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*> where Q and Z are orthogonal matrices, T is upper triangular, and S |
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*> is an upper quasi-triangular matrix with 1-by-1 and 2-by-2 diagonal |
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*> blocks, the 2-by-2 blocks corresponding to complex conjugate pairs |
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*> of eigenvalues of (A,B). 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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*> DGEGV should be used instead. See DGEGV 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 DOUBLE PRECISION array, dimension (LDA, N) |
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*> On entry, the matrix A. |
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*> On exit, the upper quasi-triangular matrix S from the |
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*> generalized real 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 DOUBLE PRECISION 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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*> real 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] 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 |
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*> of 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 (j+1)-st |
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*> 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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*> 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] 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', 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 DOUBLE PRECISION 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 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(1,4*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 2*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. (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: 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 DGGBAK (computing VSL) |
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*> =N+8: error return from DGGBAK (computing VSR) |
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*> =N+9: error return from DLASCL (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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*> \date December 2016 |
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* |
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*> \ingroup doubleGEeigen |
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* |
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* ===================================================================== |
SUBROUTINE DGEGS( JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHAR, |
SUBROUTINE DGEGS( JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHAR, |
$ ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, WORK, |
$ ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, WORK, |
$ LWORK, INFO ) |
$ LWORK, INFO ) |
* |
* |
* -- LAPACK driver routine (version 3.2) -- |
* -- LAPACK driver routine (version 3.7.0) -- |
* -- 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 |
* December 2016 |
* |
* |
* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
CHARACTER JOBVSL, JOBVSR |
CHARACTER JOBVSL, JOBVSR |
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$ VSR( LDVSR, * ), WORK( * ) |
$ VSR( LDVSR, * ), WORK( * ) |
* .. |
* .. |
* |
* |
* Purpose |
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* ======= |
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* |
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* This routine is deprecated and has been replaced by routine DGGES. |
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* |
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* DGEGS computes the eigenvalues, real Schur form, and, optionally, |
|
* left and or/right Schur vectors of a real matrix pair (A,B). |
|
* Given two square matrices A and B, the generalized real Schur |
|
* factorization has the form |
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* |
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* A = Q*S*Z**T, B = Q*T*Z**T |
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* |
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* where Q and Z are orthogonal matrices, T is upper triangular, and S |
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* is an upper quasi-triangular matrix with 1-by-1 and 2-by-2 diagonal |
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* blocks, the 2-by-2 blocks corresponding to complex conjugate pairs |
|
* of eigenvalues of (A,B). 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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* DGEGV should be used instead. See DGEGV 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) DOUBLE PRECISION array, dimension (LDA, N) |
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* On entry, the matrix A. |
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* On exit, the upper quasi-triangular matrix S from the |
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* generalized real 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) DOUBLE PRECISION 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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* real 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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* ALPHAR (output) DOUBLE PRECISION array, dimension (N) |
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* The real parts of each scalar alpha defining an eigenvalue |
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* of 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 (j+1)-st |
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* 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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* 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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* VSL (output) DOUBLE PRECISION 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) DOUBLE PRECISION 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) 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(1,4*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 2*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. (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: 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 DGGBAK (computing VSL) |
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* =N+8: error return from DGGBAK (computing VSR) |
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* =N+9: error return from DLASCL (various places) |
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* |
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* ===================================================================== |
* ===================================================================== |
* |
* |
* .. Parameters .. |
* .. Parameters .. |