version 1.8, 2011/07/22 07:38:12
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version 1.9, 2011/11/21 20:43:06
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*> \brief \b DTGSNA |
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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 DTGSNA + dependencies |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtgsna.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/dtgsna.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/dtgsna.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 DTGSNA( 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 A( LDA, * ), B( LDB, * ), DIF( * ), S( * ), |
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* $ VL( LDVL, * ), 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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*> DTGSNA estimates reciprocal condition numbers for specified |
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*> eigenvalues and/or eigenvectors of a matrix pair (A, B) in |
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*> generalized real Schur canonical form (or of any matrix pair |
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*> (Q*A*Z**T, Q*B*Z**T) with orthogonal matrices Q and Z, where |
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*> Z**T denotes the transpose of Z. |
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*> |
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*> (A, B) must be in generalized real Schur form (as returned by DGGES), |
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*> i.e. A is block upper triangular with 1-by-1 and 2-by-2 diagonal |
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*> blocks. B is upper triangular. |
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*> |
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*> \endverbatim |
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* |
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* Arguments: |
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* ========== |
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* |
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*> \param[in] 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 eigenpair corresponding to a real eigenvalue w(j), |
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*> SELECT(j) must be set to .TRUE.. To select condition numbers |
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*> corresponding to a complex conjugate pair of eigenvalues w(j) |
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*> and w(j+1), either SELECT(j) or SELECT(j+1) or both, must be |
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*> 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 DOUBLE PRECISION array, dimension (LDA,N) |
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*> The upper quasi-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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DTGEVC. |
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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. |
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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 DOUBLE PRECISION 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 ov VR, as returned by DTGEVC. |
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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. For a complex conjugate pair of eigenvalues two |
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*> consecutive elements of S are set to the same value. Thus |
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*> S(j), DIF(j), and the j-th columns of VL and VR all |
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*> correspond to the same eigenpair (but not in general the |
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*> j-th eigenpair, unless all eigenpairs are selected). |
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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. For a complex eigenvector two |
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*> consecutive elements of DIF are set to the same value. If |
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*> the eigenvalues cannot be reordered to compute DIF(j), DIF(j) |
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*> is set to 0; this can only occur when the true value would be |
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*> very small anyway. |
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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 real |
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*> eigenvalue one element is used, and for each selected complex |
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*> conjugate pair of eigenvalues, two elements are used. |
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*> 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 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,N). |
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*> If JOB = 'V' or 'B' LWORK >= 2*N*(N+2)+16. |
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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] IWORK |
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*> \verbatim |
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*> IWORK is INTEGER array, dimension (N + 6) |
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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 doubleOTHERcomputational |
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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 a generalized eigenvalue |
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*> w = (a, b) is defined as |
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*> |
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*> S(w) = (|u**TAv|**2 + |u**TBv|**2)**(1/2) / (norm(u)*norm(v)) |
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*> |
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*> where u and v are the left and right 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. |
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*> The pair (a, b) corresponds to an eigenvalue w = a/b (= u**TAv/u**TBv) |
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*> of the 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 DIF(i) of 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: |
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*> |
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*> a) If the i-th eigenvalue w = (a,b) is real |
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*> |
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*> Suppose U and V are orthogonal transformations such that |
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*> |
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*> U**T*(A, B)*V = (S, T) = ( a * ) ( b * ) 1 |
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*> ( 0 S22 ),( 0 T22 ) 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), (S22, T22)) = sigma-min( Zl ), |
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*> |
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*> where sigma-min(Zl) denotes the smallest singular value of the |
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*> 2(n-1)-by-2(n-1) matrix |
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*> |
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*> Zl = [ kron(a, In-1) -kron(1, S22) ] |
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*> [ kron(b, In-1) -kron(1, T22) ] . |
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*> |
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*> Here In-1 is the identity matrix of size n-1. kron(X, Y) is the |
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*> Kronecker product between the matrices X and Y. |
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*> |
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*> Note that if the default method for computing DIF(i) is wanted |
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*> (see DLATDF), then the parameter DIFDRI (see below) should be |
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*> changed from 3 to 4 (routine DLATDF(IJOB = 2 will be used)). |
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*> See DTGSYL for more details. |
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*> |
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*> b) If the i-th and (i+1)-th eigenvalues are complex conjugate pair, |
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*> |
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*> Suppose U and V are orthogonal transformations such that |
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*> |
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*> U**T*(A, B)*V = (S, T) = ( S11 * ) ( T11 * ) 2 |
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*> ( 0 S22 ),( 0 T22) n-2 |
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*> 2 n-2 2 n-2 |
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*> |
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*> and (S11, T11) corresponds to the complex conjugate eigenvalue |
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*> pair (w, conjg(w)). There exist unitary matrices U1 and V1 such |
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*> that |
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*> |
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*> U1**T*S11*V1 = ( s11 s12 ) and U1**T*T11*V1 = ( t11 t12 ) |
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*> ( 0 s22 ) ( 0 t22 ) |
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*> |
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*> where the generalized eigenvalues w = s11/t11 and |
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*> conjg(w) = s22/t22. |
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*> |
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*> Then the reciprocal condition number DIF(i) is bounded by |
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*> |
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*> min( d1, max( 1, |real(s11)/real(s22)| )*d2 ) |
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*> |
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*> where, d1 = Difl((s11, t11), (s22, t22)) = sigma-min(Z1), where |
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*> Z1 is the complex 2-by-2 matrix |
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*> |
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*> Z1 = [ s11 -s22 ] |
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*> [ t11 -t22 ], |
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*> |
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*> This is done by computing (using real arithmetic) the |
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*> roots of the characteristical polynomial det(Z1**T * Z1 - lambda I), |
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*> where Z1**T denotes the transpose of Z1 and det(X) denotes |
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*> the determinant of X. |
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*> |
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*> and d2 is an upper bound on Difl((S11, T11), (S22, T22)), i.e. an |
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*> upper bound on sigma-min(Z2), where Z2 is (2n-2)-by-(2n-2) |
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*> |
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*> Z2 = [ kron(S11**T, In-2) -kron(I2, S22) ] |
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*> [ kron(T11**T, In-2) -kron(I2, T22) ] |
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*> |
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*> Note that if the default method for computing DIF is wanted (see |
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*> DLATDF), then the parameter DIFDRI (see below) should be changed |
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*> from 3 to 4 (routine DLATDF(IJOB = 2 will be used)). See DTGSYL |
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*> for more details. |
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*> |
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*> For each eigenvalue/vector specified by SELECT, DIF stores a |
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*> Frobenius norm-based estimate of Difl. |
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*> |
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*> An approximate error bound for the i-th 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 |
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*> Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218. |
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*> |
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*> [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified |
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*> Eigenvalues of a Regular Matrix Pair (A, B) and Condition |
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*> Estimation: Theory, Algorithms and Software, |
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*> Report UMINF - 94.04, Department of Computing Science, Umea |
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*> University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working |
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*> Note 87. To appear in Numerical Algorithms, 1996. |
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*> |
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*> [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software |
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*> for Solving the Generalized Sylvester Equation and Estimating the |
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*> Separation between Regular Matrix Pairs, Report UMINF - 93.23, |
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*> 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. To appear in ACM Trans. on Math. Software, Vol 22, |
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*> No 1, 1996. |
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*> \endverbatim |
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*> |
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* ===================================================================== |
SUBROUTINE DTGSNA( JOB, HOWMNY, SELECT, N, A, LDA, B, LDB, VL, |
SUBROUTINE DTGSNA( 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 3.3.1) -- |
* -- 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..-- |
* -- April 2011 -- |
* November 2011 |
* |
* |
* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
CHARACTER HOWMNY, JOB |
CHARACTER HOWMNY, JOB |
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$ VL( LDVL, * ), VR( LDVR, * ), WORK( * ) |
$ VL( LDVL, * ), VR( LDVR, * ), WORK( * ) |
* .. |
* .. |
* |
* |
* Purpose |
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* ======= |
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* |
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* DTGSNA estimates reciprocal condition numbers for specified |
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* eigenvalues and/or eigenvectors of a matrix pair (A, B) in |
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* generalized real Schur canonical form (or of any matrix pair |
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* (Q*A*Z**T, Q*B*Z**T) with orthogonal matrices Q and Z, where |
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* Z**T denotes the transpose of Z. |
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* |
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* (A, B) must be in generalized real Schur form (as returned by DGGES), |
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* i.e. A is block upper triangular with 1-by-1 and 2-by-2 diagonal |
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* blocks. B is upper triangular. |
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* |
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* |
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* Arguments |
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* ========= |
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* |
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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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* |
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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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* |
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* SELECT (input) 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 eigenpair corresponding to a real eigenvalue w(j), |
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* SELECT(j) must be set to .TRUE.. To select condition numbers |
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* corresponding to a complex conjugate pair of eigenvalues w(j) |
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* and w(j+1), either SELECT(j) or SELECT(j+1) or both, must be |
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* set to .TRUE.. |
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* If HOWMNY = 'A', SELECT is not referenced. |
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* |
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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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* |
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* A (input) DOUBLE PRECISION array, dimension (LDA,N) |
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* The upper quasi-triangular matrix A in the pair (A,B). |
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* |
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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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* |
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* B (input) DOUBLE PRECISION 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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* |
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* VL (input) DOUBLE PRECISION 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 DTGEVC. |
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* If JOB = 'V', VL is not referenced. |
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* |
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* LDVL (input) INTEGER |
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* The leading dimension of the array VL. LDVL >= 1. |
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* If JOB = 'E' or 'B', LDVL >= N. |
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* |
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* VR (input) DOUBLE PRECISION 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 ov VR, as returned by DTGEVC. |
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* If JOB = 'V', VR is not referenced. |
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* |
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* LDVR (input) 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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* |
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* S (output) 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. For a complex conjugate pair of eigenvalues two |
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* consecutive elements of S are set to the same value. Thus |
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* S(j), DIF(j), and the j-th columns of VL and VR all |
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* correspond to the same eigenpair (but not in general the |
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* j-th eigenpair, unless all eigenpairs are selected). |
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* If JOB = 'V', S is not referenced. |
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* |
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* DIF (output) 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. For a complex eigenvector two |
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* consecutive elements of DIF are set to the same value. If |
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* the eigenvalues cannot be reordered to compute DIF(j), DIF(j) |
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* is set to 0; this can only occur when the true value would be |
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* very small anyway. |
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* If JOB = 'E', DIF is not referenced. |
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* |
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* MM (input) INTEGER |
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* The number of elements in the arrays S and DIF. MM >= M. |
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* |
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* M (output) 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 real |
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* eigenvalue one element is used, and for each selected complex |
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* conjugate pair of eigenvalues, two elements are used. |
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* If HOWMNY = 'A', M is set to 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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* |
|
* LWORK (input) INTEGER |
|
* The dimension of the array WORK. LWORK >= max(1,N). |
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* If JOB = 'V' or 'B' LWORK >= 2*N*(N+2)+16. |
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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 |
|
* message related to LWORK is issued by XERBLA. |
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* |
|
* IWORK (workspace) INTEGER array, dimension (N + 6) |
|
* If JOB = 'E', IWORK is not referenced. |
|
* |
|
* INFO (output) INTEGER |
|
* =0: Successful exit |
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* <0: If INFO = -i, the i-th argument had an illegal value |
|
* |
|
* |
|
* Further Details |
|
* =============== |
|
* |
|
* The reciprocal of the condition number of a generalized eigenvalue |
|
* w = (a, b) is defined as |
|
* |
|
* S(w) = (|u**TAv|**2 + |u**TBv|**2)**(1/2) / (norm(u)*norm(v)) |
|
* |
|
* where u and v are the left and right 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 (= u**TAv/u**TBv) |
|
* 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 DIF(i) of right eigenvector u |
|
* and left eigenvector v corresponding to the generalized eigenvalue w |
|
* is defined as follows: |
|
* |
|
* a) If the i-th eigenvalue w = (a,b) is real |
|
* |
|
* Suppose U and V are orthogonal transformations such that |
|
* |
|
* U**T*(A, B)*V = (S, T) = ( a * ) ( b * ) 1 |
|
* ( 0 S22 ),( 0 T22 ) n-1 |
|
* 1 n-1 1 n-1 |
|
* |
|
* Then the reciprocal condition number DIF(i) is |
|
* |
|
* Difl((a, b), (S22, T22)) = sigma-min( Zl ), |
|
* |
|
* where sigma-min(Zl) denotes the smallest singular value of the |
|
* 2(n-1)-by-2(n-1) matrix |
|
* |
|
* Zl = [ kron(a, In-1) -kron(1, S22) ] |
|
* [ kron(b, In-1) -kron(1, T22) ] . |
|
* |
|
* Here In-1 is the identity matrix of size n-1. kron(X, Y) is the |
|
* Kronecker product between the matrices X and Y. |
|
* |
|
* Note that if the default method for computing DIF(i) is wanted |
|
* (see DLATDF), then the parameter DIFDRI (see below) should be |
|
* changed from 3 to 4 (routine DLATDF(IJOB = 2 will be used)). |
|
* See DTGSYL for more details. |
|
* |
|
* b) If the i-th and (i+1)-th eigenvalues are complex conjugate pair, |
|
* |
|
* Suppose U and V are orthogonal transformations such that |
|
* |
|
* U**T*(A, B)*V = (S, T) = ( S11 * ) ( T11 * ) 2 |
|
* ( 0 S22 ),( 0 T22) n-2 |
|
* 2 n-2 2 n-2 |
|
* |
|
* and (S11, T11) corresponds to the complex conjugate eigenvalue |
|
* pair (w, conjg(w)). There exist unitary matrices U1 and V1 such |
|
* that |
|
* |
|
* U1**T*S11*V1 = ( s11 s12 ) and U1**T*T11*V1 = ( t11 t12 ) |
|
* ( 0 s22 ) ( 0 t22 ) |
|
* |
|
* where the generalized eigenvalues w = s11/t11 and |
|
* conjg(w) = s22/t22. |
|
* |
|
* Then the reciprocal condition number DIF(i) is bounded by |
|
* |
|
* min( d1, max( 1, |real(s11)/real(s22)| )*d2 ) |
|
* |
|
* where, d1 = Difl((s11, t11), (s22, t22)) = sigma-min(Z1), where |
|
* Z1 is the complex 2-by-2 matrix |
|
* |
|
* Z1 = [ s11 -s22 ] |
|
* [ t11 -t22 ], |
|
* |
|
* This is done by computing (using real arithmetic) the |
|
* roots of the characteristical polynomial det(Z1**T * Z1 - lambda I), |
|
* where Z1**T denotes the transpose of Z1 and det(X) denotes |
|
* the determinant of X. |
|
* |
|
* and d2 is an upper bound on Difl((S11, T11), (S22, T22)), i.e. an |
|
* upper bound on sigma-min(Z2), where Z2 is (2n-2)-by-(2n-2) |
|
* |
|
* Z2 = [ kron(S11**T, In-2) -kron(I2, S22) ] |
|
* [ kron(T11**T, In-2) -kron(I2, T22) ] |
|
* |
|
* Note that if the default method for computing DIF is wanted (see |
|
* DLATDF), then the parameter DIFDRI (see below) should be changed |
|
* from 3 to 4 (routine DLATDF(IJOB = 2 will be used)). See DTGSYL |
|
* for more details. |
|
* |
|
* For each eigenvalue/vector specified by SELECT, DIF stores a |
|
* Frobenius norm-based estimate of Difl. |
|
* |
|
* An approximate error bound for the i-th 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. |
|
* |
|
* ===================================================================== |
* ===================================================================== |
* |
* |
* .. Parameters .. |
* .. Parameters .. |