version 1.6, 2010/08/13 21:04:15
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version 1.9, 2011/11/21 20:43:22
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*> \brief \b ZTGSNA |
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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 ZTGSNA + dependencies |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/ztgsna.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/ztgsna.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/ztgsna.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 ZTGSNA( JOB, HOWMNY, SELECT, N, A, LDA, B, LDB, VL, |
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* LDVL, VR, LDVR, S, DIF, MM, M, WORK, LWORK, |
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* IWORK, INFO ) |
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* |
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* .. Scalar Arguments .. |
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* CHARACTER HOWMNY, JOB |
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* INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, M, MM, N |
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* .. |
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* .. Array Arguments .. |
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* LOGICAL SELECT( * ) |
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* INTEGER IWORK( * ) |
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* DOUBLE PRECISION DIF( * ), S( * ) |
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* COMPLEX*16 A( LDA, * ), B( LDB, * ), VL( LDVL, * ), |
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* $ VR( LDVR, * ), WORK( * ) |
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* .. |
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* |
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* |
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*> \par Purpose: |
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* ============= |
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*> |
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*> \verbatim |
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*> |
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*> ZTGSNA estimates reciprocal condition numbers for specified |
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*> eigenvalues and/or eigenvectors of a matrix pair (A, B). |
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*> |
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*> (A, B) must be in generalized Schur canonical form, that is, A and |
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*> B are both upper triangular. |
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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] JOB |
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*> \verbatim |
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*> JOB is CHARACTER*1 |
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*> Specifies whether condition numbers are required for |
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*> eigenvalues (S) or eigenvectors (DIF): |
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*> = 'E': for eigenvalues only (S); |
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*> = 'V': for eigenvectors only (DIF); |
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*> = 'B': for both eigenvalues and eigenvectors (S and DIF). |
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*> \endverbatim |
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*> |
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*> \param[in] HOWMNY |
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*> \verbatim |
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*> HOWMNY is CHARACTER*1 |
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*> = 'A': compute condition numbers for all eigenpairs; |
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*> = 'S': compute condition numbers for selected eigenpairs |
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*> specified by the array SELECT. |
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*> \endverbatim |
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*> |
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*> \param[in] SELECT |
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*> \verbatim |
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*> SELECT is LOGICAL array, dimension (N) |
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*> If HOWMNY = 'S', SELECT specifies the eigenpairs for which |
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*> condition numbers are required. To select condition numbers |
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*> for the corresponding j-th eigenvalue and/or eigenvector, |
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*> SELECT(j) must be set to .TRUE.. |
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*> If HOWMNY = 'A', SELECT is not referenced. |
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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 square matrix pair (A, B). N >= 0. |
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*> \endverbatim |
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*> |
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*> \param[in] A |
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*> \verbatim |
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*> A is COMPLEX*16 array, dimension (LDA,N) |
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*> The upper triangular matrix A in the pair (A,B). |
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*> \endverbatim |
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*> |
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*> \param[in] LDA |
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*> \verbatim |
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*> LDA is INTEGER |
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*> The leading dimension of the array A. LDA >= max(1,N). |
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*> \endverbatim |
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*> |
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*> \param[in] B |
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*> \verbatim |
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*> B is COMPLEX*16 array, dimension (LDB,N) |
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*> The upper triangular matrix B in the pair (A, B). |
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*> \endverbatim |
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*> |
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*> \param[in] LDB |
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*> \verbatim |
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*> LDB is INTEGER |
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*> The leading dimension of the array B. LDB >= max(1,N). |
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*> \endverbatim |
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*> |
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*> \param[in] VL |
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*> \verbatim |
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*> VL is COMPLEX*16 array, dimension (LDVL,M) |
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*> IF JOB = 'E' or 'B', VL must contain left eigenvectors of |
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*> (A, B), corresponding to the eigenpairs specified by HOWMNY |
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*> and SELECT. The eigenvectors must be stored in consecutive |
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*> columns of VL, as returned by ZTGEVC. |
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*> If JOB = 'V', VL is not referenced. |
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*> \endverbatim |
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*> |
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*> \param[in] LDVL |
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*> \verbatim |
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*> LDVL is INTEGER |
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*> The leading dimension of the array VL. LDVL >= 1; and |
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*> If JOB = 'E' or 'B', LDVL >= N. |
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*> \endverbatim |
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*> |
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*> \param[in] VR |
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*> \verbatim |
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*> VR is COMPLEX*16 array, dimension (LDVR,M) |
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*> IF JOB = 'E' or 'B', VR must contain right eigenvectors of |
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*> (A, B), corresponding to the eigenpairs specified by HOWMNY |
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*> and SELECT. The eigenvectors must be stored in consecutive |
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*> columns of VR, as returned by ZTGEVC. |
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*> If JOB = 'V', VR is not referenced. |
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*> \endverbatim |
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*> |
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*> \param[in] LDVR |
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*> \verbatim |
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*> LDVR is INTEGER |
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*> The leading dimension of the array VR. LDVR >= 1; |
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*> If JOB = 'E' or 'B', LDVR >= N. |
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*> \endverbatim |
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*> |
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*> \param[out] S |
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*> \verbatim |
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*> S is DOUBLE PRECISION array, dimension (MM) |
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*> If JOB = 'E' or 'B', the reciprocal condition numbers of the |
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*> selected eigenvalues, stored in consecutive elements of the |
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*> array. |
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*> If JOB = 'V', S is not referenced. |
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*> \endverbatim |
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*> |
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*> \param[out] DIF |
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*> \verbatim |
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*> DIF is DOUBLE PRECISION array, dimension (MM) |
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*> If JOB = 'V' or 'B', the estimated reciprocal condition |
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*> numbers of the selected eigenvectors, stored in consecutive |
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*> elements of the array. |
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*> If the eigenvalues cannot be reordered to compute DIF(j), |
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*> DIF(j) is set to 0; this can only occur when the true value |
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*> would be very small anyway. |
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*> For each eigenvalue/vector specified by SELECT, DIF stores |
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*> a Frobenius norm-based estimate of Difl. |
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*> If JOB = 'E', DIF is not referenced. |
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*> \endverbatim |
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*> |
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*> \param[in] MM |
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*> \verbatim |
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*> MM is INTEGER |
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*> The number of elements in the arrays S and DIF. MM >= M. |
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*> \endverbatim |
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*> |
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*> \param[out] M |
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*> \verbatim |
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*> M is INTEGER |
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*> The number of elements of the arrays S and DIF used to store |
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*> the specified condition numbers; for each selected eigenvalue |
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*> one element is used. If HOWMNY = 'A', M is set to 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,N). |
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*> If JOB = 'V' or 'B', LWORK >= max(1,2*N*N). |
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*> \endverbatim |
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*> |
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*> \param[out] IWORK |
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*> \verbatim |
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*> IWORK is INTEGER array, dimension (N+2) |
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*> If JOB = 'E', IWORK is not referenced. |
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*> \endverbatim |
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*> |
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*> \param[out] INFO |
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*> \verbatim |
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*> INFO is INTEGER |
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*> = 0: Successful exit |
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*> < 0: If INFO = -i, the i-th argument had an illegal value |
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*> \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 November 2011 |
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* |
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*> \ingroup complex16OTHERcomputational |
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* |
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*> \par Further Details: |
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* ===================== |
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*> |
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*> \verbatim |
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*> |
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*> The reciprocal of the condition number of the i-th generalized |
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*> eigenvalue w = (a, b) is defined as |
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*> |
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*> S(I) = (|v**HAu|**2 + |v**HBu|**2)**(1/2) / (norm(u)*norm(v)) |
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*> |
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*> where u and v are the right and left eigenvectors of (A, B) |
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*> corresponding to w; |z| denotes the absolute value of the complex |
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*> number, and norm(u) denotes the 2-norm of the vector u. The pair |
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*> (a, b) corresponds to an eigenvalue w = a/b (= v**HAu/v**HBu) of the |
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*> matrix pair (A, B). If both a and b equal zero, then (A,B) is |
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*> singular and S(I) = -1 is returned. |
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*> |
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*> An approximate error bound on the chordal distance between the i-th |
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*> computed generalized eigenvalue w and the corresponding exact |
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*> eigenvalue lambda is |
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*> |
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*> chord(w, lambda) <= EPS * norm(A, B) / S(I), |
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*> |
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*> where EPS is the machine precision. |
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*> |
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*> The reciprocal of the condition number of the right eigenvector u |
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*> and left eigenvector v corresponding to the generalized eigenvalue w |
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*> is defined as follows. Suppose |
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*> |
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*> (A, B) = ( a * ) ( b * ) 1 |
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*> ( 0 A22 ),( 0 B22 ) n-1 |
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*> 1 n-1 1 n-1 |
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*> |
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*> Then the reciprocal condition number DIF(I) is |
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*> |
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*> Difl[(a, b), (A22, B22)] = sigma-min( Zl ) |
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*> |
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*> where sigma-min(Zl) denotes the smallest singular value of |
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*> |
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*> Zl = [ kron(a, In-1) -kron(1, A22) ] |
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*> [ kron(b, In-1) -kron(1, B22) ]. |
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*> |
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*> Here In-1 is the identity matrix of size n-1 and X**H is the conjugate |
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*> transpose of X. kron(X, Y) is the Kronecker product between the |
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*> matrices X and Y. |
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*> |
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*> We approximate the smallest singular value of Zl with an upper |
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*> bound. This is done by ZLATDF. |
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*> |
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*> An approximate error bound for a computed eigenvector VL(i) or |
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*> VR(i) is given by |
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*> |
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*> EPS * norm(A, B) / DIF(i). |
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*> |
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*> See ref. [2-3] for more details and further references. |
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*> \endverbatim |
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* |
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*> \par Contributors: |
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* ================== |
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*> |
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*> Bo Kagstrom and Peter Poromaa, Department of Computing Science, |
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*> Umea University, S-901 87 Umea, Sweden. |
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* |
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*> \par References: |
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* ================ |
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*> |
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*> \verbatim |
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*> |
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*> [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the |
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*> Generalized Real Schur Form of a Regular Matrix Pair (A, B), in |
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*> M.S. Moonen et al (eds), Linear Algebra for Large Scale and |
|
*> Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218. |
|
*> |
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*> [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified |
|
*> Eigenvalues of a Regular Matrix Pair (A, B) and Condition |
|
*> Estimation: Theory, Algorithms and Software, Report |
|
*> UMINF - 94.04, Department of Computing Science, Umea University, |
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*> S-901 87 Umea, Sweden, 1994. Also as LAPACK Working Note 87. |
|
*> To appear in Numerical Algorithms, 1996. |
|
*> |
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*> [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software |
|
*> for Solving the Generalized Sylvester Equation and Estimating the |
|
*> Separation between Regular Matrix Pairs, Report UMINF - 93.23, |
|
*> Department of Computing Science, Umea University, S-901 87 Umea, |
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*> Sweden, December 1993, Revised April 1994, Also as LAPACK Working |
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*> Note 75. |
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*> To appear in ACM Trans. on Math. Software, Vol 22, No 1, 1996. |
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*> \endverbatim |
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*> |
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* ===================================================================== |
SUBROUTINE ZTGSNA( JOB, HOWMNY, SELECT, N, A, LDA, B, LDB, VL, |
SUBROUTINE ZTGSNA( JOB, HOWMNY, SELECT, N, A, LDA, B, LDB, VL, |
$ LDVL, VR, LDVR, S, DIF, MM, M, WORK, LWORK, |
$ LDVL, VR, LDVR, S, DIF, MM, M, WORK, LWORK, |
$ IWORK, INFO ) |
$ IWORK, INFO ) |
* |
* |
* -- LAPACK routine (version 3.2) -- |
* -- LAPACK computational routine (version 3.4.0) -- |
* -- 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 |
* November 2011 |
* |
* |
* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
CHARACTER HOWMNY, JOB |
CHARACTER HOWMNY, JOB |
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$ VR( LDVR, * ), WORK( * ) |
$ VR( LDVR, * ), WORK( * ) |
* .. |
* .. |
* |
* |
* Purpose |
|
* ======= |
|
* |
|
* ZTGSNA estimates reciprocal condition numbers for specified |
|
* eigenvalues and/or eigenvectors of a matrix pair (A, B). |
|
* |
|
* (A, B) must be in generalized Schur canonical form, that is, A and |
|
* B are both upper triangular. |
|
* |
|
* Arguments |
|
* ========= |
|
* |
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* JOB (input) CHARACTER*1 |
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* Specifies whether condition numbers are required for |
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* eigenvalues (S) or eigenvectors (DIF): |
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* = 'E': for eigenvalues only (S); |
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* = 'V': for eigenvectors only (DIF); |
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* = 'B': for both eigenvalues and eigenvectors (S and DIF). |
|
* |
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* HOWMNY (input) CHARACTER*1 |
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* = 'A': compute condition numbers for all eigenpairs; |
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* = 'S': compute condition numbers for selected eigenpairs |
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* specified by the array SELECT. |
|
* |
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* SELECT (input) LOGICAL array, dimension (N) |
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* If HOWMNY = 'S', SELECT specifies the eigenpairs for which |
|
* condition numbers are required. To select condition numbers |
|
* for the corresponding j-th eigenvalue and/or eigenvector, |
|
* SELECT(j) must be set to .TRUE.. |
|
* If HOWMNY = 'A', SELECT is not referenced. |
|
* |
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* N (input) INTEGER |
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* The order of the square matrix pair (A, B). N >= 0. |
|
* |
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* A (input) COMPLEX*16 array, dimension (LDA,N) |
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* The upper triangular matrix A in the pair (A,B). |
|
* |
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* LDA (input) INTEGER |
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* The leading dimension of the array A. LDA >= max(1,N). |
|
* |
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* B (input) COMPLEX*16 array, dimension (LDB,N) |
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* The upper triangular matrix B in the pair (A, B). |
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* |
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* LDB (input) INTEGER |
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* The leading dimension of the array B. LDB >= max(1,N). |
|
* |
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* VL (input) COMPLEX*16 array, dimension (LDVL,M) |
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* IF JOB = 'E' or 'B', VL must contain left eigenvectors of |
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* (A, B), corresponding to the eigenpairs specified by HOWMNY |
|
* and SELECT. The eigenvectors must be stored in consecutive |
|
* columns of VL, as returned by ZTGEVC. |
|
* If JOB = 'V', VL is not referenced. |
|
* |
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* LDVL (input) INTEGER |
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* The leading dimension of the array VL. LDVL >= 1; and |
|
* If JOB = 'E' or 'B', LDVL >= N. |
|
* |
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* VR (input) COMPLEX*16 array, dimension (LDVR,M) |
|
* IF JOB = 'E' or 'B', VR must contain right eigenvectors of |
|
* (A, B), corresponding to the eigenpairs specified by HOWMNY |
|
* and SELECT. The eigenvectors must be stored in consecutive |
|
* columns of VR, as returned by ZTGEVC. |
|
* If JOB = 'V', VR is not referenced. |
|
* |
|
* LDVR (input) INTEGER |
|
* The leading dimension of the array VR. LDVR >= 1; |
|
* If JOB = 'E' or 'B', LDVR >= N. |
|
* |
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* S (output) DOUBLE PRECISION array, dimension (MM) |
|
* If JOB = 'E' or 'B', the reciprocal condition numbers of the |
|
* selected eigenvalues, stored in consecutive elements of the |
|
* array. |
|
* If JOB = 'V', S is not referenced. |
|
* |
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* DIF (output) DOUBLE PRECISION array, dimension (MM) |
|
* If JOB = 'V' or 'B', the estimated reciprocal condition |
|
* numbers of the selected eigenvectors, stored in consecutive |
|
* elements of the array. |
|
* If the eigenvalues cannot be reordered to compute DIF(j), |
|
* DIF(j) is set to 0; this can only occur when the true value |
|
* would be very small anyway. |
|
* For each eigenvalue/vector specified by SELECT, DIF stores |
|
* a Frobenius norm-based estimate of Difl. |
|
* If JOB = 'E', DIF is not referenced. |
|
* |
|
* MM (input) INTEGER |
|
* The number of elements in the arrays S and DIF. MM >= M. |
|
* |
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* M (output) INTEGER |
|
* The number of elements of the arrays S and DIF used to store |
|
* the specified condition numbers; for each selected eigenvalue |
|
* one element is used. If HOWMNY = 'A', M is set to N. |
|
* |
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* WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK)) |
|
* On exit, if INFO = 0, WORK(1) returns the optimal LWORK. |
|
* |
|
* LWORK (input) INTEGER |
|
* The dimension of the array WORK. LWORK >= max(1,N). |
|
* If JOB = 'V' or 'B', LWORK >= max(1,2*N*N). |
|
* |
|
* IWORK (workspace) INTEGER array, dimension (N+2) |
|
* If JOB = 'E', IWORK is not referenced. |
|
* |
|
* INFO (output) INTEGER |
|
* = 0: Successful exit |
|
* < 0: If INFO = -i, the i-th argument had an illegal value |
|
* |
|
* Further Details |
|
* =============== |
|
* |
|
* The reciprocal of the condition number of the i-th generalized |
|
* eigenvalue w = (a, b) is defined as |
|
* |
|
* S(I) = (|v'Au|**2 + |v'Bu|**2)**(1/2) / (norm(u)*norm(v)) |
|
* |
|
* where u and v are the right and left eigenvectors of (A, B) |
|
* corresponding to w; |z| denotes the absolute value of the complex |
|
* number, and norm(u) denotes the 2-norm of the vector u. The pair |
|
* (a, b) corresponds to an eigenvalue w = a/b (= v'Au/v'Bu) of the |
|
* matrix pair (A, B). If both a and b equal zero, then (A,B) is |
|
* singular and S(I) = -1 is returned. |
|
* |
|
* An approximate error bound on the chordal distance between the i-th |
|
* computed generalized eigenvalue w and the corresponding exact |
|
* eigenvalue lambda is |
|
* |
|
* chord(w, lambda) <= EPS * norm(A, B) / S(I), |
|
* |
|
* where EPS is the machine precision. |
|
* |
|
* The reciprocal of the condition number of the right eigenvector u |
|
* and left eigenvector v corresponding to the generalized eigenvalue w |
|
* is defined as follows. Suppose |
|
* |
|
* (A, B) = ( a * ) ( b * ) 1 |
|
* ( 0 A22 ),( 0 B22 ) n-1 |
|
* 1 n-1 1 n-1 |
|
* |
|
* Then the reciprocal condition number DIF(I) is |
|
* |
|
* Difl[(a, b), (A22, B22)] = sigma-min( Zl ) |
|
* |
|
* where sigma-min(Zl) denotes the smallest singular value of |
|
* |
|
* Zl = [ kron(a, In-1) -kron(1, A22) ] |
|
* [ kron(b, In-1) -kron(1, B22) ]. |
|
* |
|
* Here In-1 is the identity matrix of size n-1 and X' is the conjugate |
|
* transpose of X. kron(X, Y) is the Kronecker product between the |
|
* matrices X and Y. |
|
* |
|
* We approximate the smallest singular value of Zl with an upper |
|
* bound. This is done by ZLATDF. |
|
* |
|
* An approximate error bound for a computed eigenvector VL(i) or |
|
* VR(i) is given by |
|
* |
|
* EPS * norm(A, B) / DIF(i). |
|
* |
|
* See ref. [2-3] for more details and further references. |
|
* |
|
* Based on contributions by |
|
* Bo Kagstrom and Peter Poromaa, Department of Computing Science, |
|
* Umea University, S-901 87 Umea, Sweden. |
|
* |
|
* References |
|
* ========== |
|
* |
|
* [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the |
|
* Generalized Real Schur Form of a Regular Matrix Pair (A, B), in |
|
* M.S. Moonen et al (eds), Linear Algebra for Large Scale and |
|
* Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218. |
|
* |
|
* [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified |
|
* Eigenvalues of a Regular Matrix Pair (A, B) and Condition |
|
* Estimation: Theory, Algorithms and Software, Report |
|
* UMINF - 94.04, Department of Computing Science, Umea University, |
|
* S-901 87 Umea, Sweden, 1994. Also as LAPACK Working Note 87. |
|
* To appear in Numerical Algorithms, 1996. |
|
* |
|
* [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software |
|
* for Solving the Generalized Sylvester Equation and Estimating the |
|
* Separation between Regular Matrix Pairs, Report UMINF - 93.23, |
|
* Department of Computing Science, Umea University, S-901 87 Umea, |
|
* Sweden, December 1993, Revised April 1994, Also as LAPACK Working |
|
* Note 75. |
|
* To appear in ACM Trans. on Math. Software, Vol 22, No 1, 1996. |
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* .. Parameters .. |
* .. Parameters .. |