version 1.3, 2010/08/06 15:28:50
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version 1.19, 2023/08/07 08:39:13
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*> \brief \b DTRSNA |
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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 DTRSNA + dependencies |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtrsna.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/dtrsna.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/dtrsna.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 DTRSNA( JOB, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR, |
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* LDVR, S, SEP, MM, M, WORK, LDWORK, IWORK, |
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* INFO ) |
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* |
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* .. Scalar Arguments .. |
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* CHARACTER HOWMNY, JOB |
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* INTEGER INFO, LDT, LDVL, LDVR, LDWORK, 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 S( * ), SEP( * ), T( LDT, * ), VL( LDVL, * ), |
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* $ VR( LDVR, * ), WORK( LDWORK, * ) |
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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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*> DTRSNA estimates reciprocal condition numbers for specified |
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*> eigenvalues and/or right eigenvectors of a real upper |
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*> quasi-triangular matrix T (or of any matrix Q*T*Q**T with Q |
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*> orthogonal). |
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*> |
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*> T must be in Schur canonical form (as returned by DHSEQR), that is, |
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*> block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each |
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*> 2-by-2 diagonal block has its diagonal elements equal and its |
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*> off-diagonal elements of opposite sign. |
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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 (SEP): |
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*> = 'E': for eigenvalues only (S); |
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*> = 'V': for eigenvectors only (SEP); |
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*> = 'B': for both eigenvalues and eigenvectors (S and SEP). |
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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 matrix T. N >= 0. |
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*> \endverbatim |
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*> |
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*> \param[in] T |
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*> \verbatim |
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*> T is DOUBLE PRECISION array, dimension (LDT,N) |
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*> The upper quasi-triangular matrix T, in Schur canonical form. |
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*> \endverbatim |
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*> |
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*> \param[in] LDT |
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*> \verbatim |
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*> LDT is INTEGER |
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*> The leading dimension of the array T. LDT >= 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 T |
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*> (or of any Q*T*Q**T with Q orthogonal), corresponding to the |
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*> eigenpairs specified by HOWMNY and SELECT. The eigenvectors |
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*> must be stored in consecutive columns of VL, as returned by |
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*> DHSEIN or DTREVC. |
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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. |
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*> LDVL >= 1; and 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 T |
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*> (or of any Q*T*Q**T with Q orthogonal), corresponding to the |
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*> eigenpairs specified by HOWMNY and SELECT. The eigenvectors |
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*> must be stored in consecutive columns of VR, as returned by |
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*> DHSEIN or DTREVC. |
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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. |
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*> LDVR >= 1; and 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), SEP(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] SEP |
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*> \verbatim |
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*> SEP 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 SEP are set to the same value. If |
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*> the eigenvalues cannot be reordered to compute SEP(j), SEP(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', SEP 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 (if JOB = 'E' or 'B') |
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*> and/or SEP (if JOB = 'V' or 'B'). 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/or SEP actually |
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*> used to store the estimated condition numbers. |
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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 (LDWORK,N+6) |
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*> If JOB = 'E', WORK is not referenced. |
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*> \endverbatim |
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*> |
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*> \param[in] LDWORK |
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*> \verbatim |
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*> LDWORK is INTEGER |
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*> The leading dimension of the array WORK. |
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*> LDWORK >= 1; and if JOB = 'V' or 'B', LDWORK >= 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 (2*(N-1)) |
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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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*> \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 an eigenvalue lambda is |
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*> defined as |
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*> |
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*> S(lambda) = |v**T*u| / (norm(u)*norm(v)) |
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*> |
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*> where u and v are the right and left eigenvectors of T corresponding |
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*> to lambda; v**T denotes the transpose of v, and norm(u) |
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*> denotes the Euclidean norm. These reciprocal condition numbers always |
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*> lie between zero (very badly conditioned) and one (very well |
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*> conditioned). If n = 1, S(lambda) is defined to be 1. |
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*> |
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*> An approximate error bound for a computed eigenvalue W(i) is given by |
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*> |
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*> EPS * norm(T) / 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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*> corresponding to lambda is defined as follows. Suppose |
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*> |
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*> T = ( lambda c ) |
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*> ( 0 T22 ) |
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*> |
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*> Then the reciprocal condition number is |
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*> |
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*> SEP( lambda, T22 ) = sigma-min( T22 - lambda*I ) |
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*> |
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*> where sigma-min denotes the smallest singular value. We approximate |
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*> the smallest singular value by the reciprocal of an estimate of the |
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*> one-norm of the inverse of T22 - lambda*I. If n = 1, SEP(1) is |
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*> defined to be abs(T(1,1)). |
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*> |
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*> An approximate error bound for a computed right eigenvector VR(i) |
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*> is given by |
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*> |
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*> EPS * norm(T) / SEP(i) |
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*> \endverbatim |
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*> |
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* ===================================================================== |
SUBROUTINE DTRSNA( JOB, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR, |
SUBROUTINE DTRSNA( JOB, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR, |
$ LDVR, S, SEP, MM, M, WORK, LDWORK, IWORK, |
$ LDVR, S, SEP, MM, M, WORK, LDWORK, IWORK, |
$ INFO ) |
$ INFO ) |
* |
* |
* -- LAPACK routine (version 3.2) -- |
* -- LAPACK computational routine -- |
* -- LAPACK is a software package provided by Univ. of Tennessee, -- |
* -- LAPACK is a software package provided by Univ. of Tennessee, -- |
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
* November 2006 |
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* |
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* Modified to call DLACN2 in place of DLACON, 5 Feb 03, SJH. |
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* |
* |
* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
CHARACTER HOWMNY, JOB |
CHARACTER HOWMNY, JOB |
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$ VR( LDVR, * ), WORK( LDWORK, * ) |
$ VR( LDVR, * ), WORK( LDWORK, * ) |
* .. |
* .. |
* |
* |
* Purpose |
|
* ======= |
|
* |
|
* DTRSNA estimates reciprocal condition numbers for specified |
|
* eigenvalues and/or right eigenvectors of a real upper |
|
* quasi-triangular matrix T (or of any matrix Q*T*Q**T with Q |
|
* orthogonal). |
|
* |
|
* T must be in Schur canonical form (as returned by DHSEQR), that is, |
|
* block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each |
|
* 2-by-2 diagonal block has its diagonal elements equal and its |
|
* off-diagonal elements of opposite sign. |
|
* |
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* 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 (SEP): |
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* = 'E': for eigenvalues only (S); |
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* = 'V': for eigenvectors only (SEP); |
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* = 'B': for both eigenvalues and eigenvectors (S and SEP). |
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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 matrix T. N >= 0. |
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* |
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* T (input) DOUBLE PRECISION array, dimension (LDT,N) |
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* The upper quasi-triangular matrix T, in Schur canonical form. |
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* |
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* LDT (input) INTEGER |
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* The leading dimension of the array T. LDT >= 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 T |
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* (or of any Q*T*Q**T with Q orthogonal), corresponding to the |
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* eigenpairs specified by HOWMNY and SELECT. The eigenvectors |
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* must be stored in consecutive columns of VL, as returned by |
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* DHSEIN or DTREVC. |
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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. |
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* LDVL >= 1; and 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 T |
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* (or of any Q*T*Q**T with Q orthogonal), corresponding to the |
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* eigenpairs specified by HOWMNY and SELECT. The eigenvectors |
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* must be stored in consecutive columns of VR, as returned by |
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* DHSEIN or DTREVC. |
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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. |
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* LDVR >= 1; and 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), SEP(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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* SEP (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 SEP are set to the same value. If |
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* the eigenvalues cannot be reordered to compute SEP(j), SEP(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', SEP 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 (if JOB = 'E' or 'B') |
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* and/or SEP (if JOB = 'V' or 'B'). 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/or SEP actually |
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* used to store the estimated condition numbers. |
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* If HOWMNY = 'A', M is set to N. |
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* |
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* WORK (workspace) DOUBLE PRECISION array, dimension (LDWORK,N+6) |
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* If JOB = 'E', WORK is not referenced. |
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* |
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* LDWORK (input) INTEGER |
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* The leading dimension of the array WORK. |
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* LDWORK >= 1; and if JOB = 'V' or 'B', LDWORK >= N. |
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* |
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* IWORK (workspace) INTEGER array, dimension (2*(N-1)) |
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* If JOB = 'E', IWORK is not referenced. |
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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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* |
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* Further Details |
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* =============== |
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* |
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* The reciprocal of the condition number of an eigenvalue lambda is |
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* defined as |
|
* |
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* S(lambda) = |v'*u| / (norm(u)*norm(v)) |
|
* |
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* where u and v are the right and left eigenvectors of T corresponding |
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* to lambda; v' denotes the conjugate-transpose of v, and norm(u) |
|
* denotes the Euclidean norm. These reciprocal condition numbers always |
|
* lie between zero (very badly conditioned) and one (very well |
|
* conditioned). If n = 1, S(lambda) is defined to be 1. |
|
* |
|
* An approximate error bound for a computed eigenvalue W(i) is given by |
|
* |
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* EPS * norm(T) / S(i) |
|
* |
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* where EPS is the machine precision. |
|
* |
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* The reciprocal of the condition number of the right eigenvector u |
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* corresponding to lambda is defined as follows. Suppose |
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* |
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* T = ( lambda c ) |
|
* ( 0 T22 ) |
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* |
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* Then the reciprocal condition number is |
|
* |
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* SEP( lambda, T22 ) = sigma-min( T22 - lambda*I ) |
|
* |
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* where sigma-min denotes the smallest singular value. We approximate |
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* the smallest singular value by the reciprocal of an estimate of the |
|
* one-norm of the inverse of T22 - lambda*I. If n = 1, SEP(1) is |
|
* defined to be abs(T(1,1)). |
|
* |
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* An approximate error bound for a computed right eigenvector VR(i) |
|
* is given by |
|
* |
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* EPS * norm(T) / SEP(i) |
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* |
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* ===================================================================== |
* ===================================================================== |
* |
* |
* .. Parameters .. |
* .. Parameters .. |
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EXTERNAL LSAME, DDOT, DLAMCH, DLAPY2, DNRM2 |
EXTERNAL LSAME, DDOT, DLAMCH, DLAPY2, DNRM2 |
* .. |
* .. |
* .. External Subroutines .. |
* .. External Subroutines .. |
EXTERNAL DLACN2, DLACPY, DLAQTR, DTREXC, XERBLA |
EXTERNAL DLABAD, DLACN2, DLACPY, DLAQTR, DTREXC, XERBLA |
* .. |
* .. |
* .. Intrinsic Functions .. |
* .. Intrinsic Functions .. |
INTRINSIC ABS, MAX, SQRT |
INTRINSIC ABS, MAX, SQRT |
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* |
* |
* Form |
* Form |
* |
* |
* C' = WORK(2:N,2:N) + i*[rwork(1) ..... rwork(n-1) ] |
* C**T = WORK(2:N,2:N) + i*[rwork(1) ..... rwork(n-1) ] |
* [ mu ] |
* [ mu ] |
* [ .. ] |
* [ .. ] |
* [ .. ] |
* [ .. ] |
* [ mu ] |
* [ mu ] |
* where C' is conjugate transpose of complex matrix C, |
* where C**T is transpose of matrix C, |
* and RWORK is stored starting in the N+1-st column of |
* and RWORK is stored starting in the N+1-st column of |
* WORK. |
* WORK. |
* |
* |
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NN = 2*( N-1 ) |
NN = 2*( N-1 ) |
END IF |
END IF |
* |
* |
* Estimate norm(inv(C')) |
* Estimate norm(inv(C**T)) |
* |
* |
EST = ZERO |
EST = ZERO |
KASE = 0 |
KASE = 0 |
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IF( KASE.EQ.1 ) THEN |
IF( KASE.EQ.1 ) THEN |
IF( N2.EQ.1 ) THEN |
IF( N2.EQ.1 ) THEN |
* |
* |
* Real eigenvalue: solve C'*x = scale*c. |
* Real eigenvalue: solve C**T*x = scale*c. |
* |
* |
CALL DLAQTR( .TRUE., .TRUE., N-1, WORK( 2, 2 ), |
CALL DLAQTR( .TRUE., .TRUE., N-1, WORK( 2, 2 ), |
$ LDWORK, DUMMY, DUMM, SCALE, |
$ LDWORK, DUMMY, DUMM, SCALE, |
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ELSE |
ELSE |
* |
* |
* Complex eigenvalue: solve |
* Complex eigenvalue: solve |
* C'*(p+iq) = scale*(c+id) in real arithmetic. |
* C**T*(p+iq) = scale*(c+id) in real arithmetic. |
* |
* |
CALL DLAQTR( .TRUE., .FALSE., N-1, WORK( 2, 2 ), |
CALL DLAQTR( .TRUE., .FALSE., N-1, WORK( 2, 2 ), |
$ LDWORK, WORK( 1, N+1 ), MU, SCALE, |
$ LDWORK, WORK( 1, N+1 ), MU, SCALE, |