version 1.6, 2010/08/07 13:22:45
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version 1.15, 2016/08/27 15:27:14
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*> \brief \b ZTGSEN |
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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 ZTGSEN + dependencies |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/ztgsen.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/ztgsen.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/ztgsen.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 ZTGSEN( IJOB, WANTQ, WANTZ, SELECT, N, A, LDA, B, LDB, |
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* ALPHA, BETA, Q, LDQ, Z, LDZ, M, PL, PR, DIF, |
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* WORK, LWORK, IWORK, LIWORK, INFO ) |
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* |
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* .. Scalar Arguments .. |
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* LOGICAL WANTQ, WANTZ |
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* INTEGER IJOB, INFO, LDA, LDB, LDQ, LDZ, LIWORK, LWORK, |
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* $ M, N |
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* DOUBLE PRECISION PL, PR |
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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( * ) |
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* COMPLEX*16 A( LDA, * ), ALPHA( * ), B( LDB, * ), |
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* $ BETA( * ), Q( LDQ, * ), WORK( * ), Z( LDZ, * ) |
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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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*> ZTGSEN reorders the generalized Schur decomposition of a complex |
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*> matrix pair (A, B) (in terms of an unitary equivalence trans- |
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*> formation Q**H * (A, B) * Z), so that a selected cluster of eigenvalues |
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*> appears in the leading diagonal blocks of the pair (A,B). The leading |
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*> columns of Q and Z form unitary bases of the corresponding left and |
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*> right eigenspaces (deflating subspaces). (A, B) must be in |
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*> generalized Schur canonical form, that is, A and B are both upper |
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*> triangular. |
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*> |
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*> ZTGSEN also computes the generalized eigenvalues |
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*> |
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*> w(j)= ALPHA(j) / BETA(j) |
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*> |
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*> of the reordered matrix pair (A, B). |
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*> |
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*> Optionally, the routine computes estimates of reciprocal condition |
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*> numbers for eigenvalues and eigenspaces. These are Difu[(A11,B11), |
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*> (A22,B22)] and Difl[(A11,B11), (A22,B22)], i.e. the separation(s) |
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*> between the matrix pairs (A11, B11) and (A22,B22) that correspond to |
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*> the selected cluster and the eigenvalues outside the cluster, resp., |
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*> and norms of "projections" onto left and right eigenspaces w.r.t. |
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*> the selected cluster in the (1,1)-block. |
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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] IJOB |
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*> \verbatim |
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*> IJOB is integer |
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*> Specifies whether condition numbers are required for the |
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*> cluster of eigenvalues (PL and PR) or the deflating subspaces |
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*> (Difu and Difl): |
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*> =0: Only reorder w.r.t. SELECT. No extras. |
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*> =1: Reciprocal of norms of "projections" onto left and right |
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*> eigenspaces w.r.t. the selected cluster (PL and PR). |
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*> =2: Upper bounds on Difu and Difl. F-norm-based estimate |
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*> (DIF(1:2)). |
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*> =3: Estimate of Difu and Difl. 1-norm-based estimate |
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*> (DIF(1:2)). |
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*> About 5 times as expensive as IJOB = 2. |
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*> =4: Compute PL, PR and DIF (i.e. 0, 1 and 2 above): Economic |
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*> version to get it all. |
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*> =5: Compute PL, PR and DIF (i.e. 0, 1 and 3 above) |
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*> \endverbatim |
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*> |
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*> \param[in] WANTQ |
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*> \verbatim |
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*> WANTQ is LOGICAL |
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*> .TRUE. : update the left transformation matrix Q; |
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*> .FALSE.: do not update Q. |
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*> \endverbatim |
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*> |
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*> \param[in] WANTZ |
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*> \verbatim |
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*> WANTZ is LOGICAL |
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*> .TRUE. : update the right transformation matrix Z; |
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*> .FALSE.: do not update Z. |
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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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*> SELECT specifies the eigenvalues in the selected cluster. To |
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*> select an eigenvalue w(j), SELECT(j) must be set to |
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*> .TRUE.. |
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*> \endverbatim |
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*> |
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*> \param[in] N |
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*> \verbatim |
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*> N is INTEGER |
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*> The order of the matrices A and B. N >= 0. |
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*> \endverbatim |
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*> |
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*> \param[in,out] A |
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*> \verbatim |
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*> A is COMPLEX*16 array, dimension(LDA,N) |
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*> On entry, the upper triangular matrix A, in generalized |
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*> Schur canonical form. |
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*> On exit, A is overwritten by the reordered matrix A. |
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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,out] B |
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*> \verbatim |
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*> B is COMPLEX*16 array, dimension(LDB,N) |
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*> On entry, the upper triangular matrix B, in generalized |
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*> Schur canonical form. |
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*> On exit, B is overwritten by the reordered matrix 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[out] ALPHA |
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*> \verbatim |
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*> ALPHA is COMPLEX*16 array, dimension (N) |
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*> \endverbatim |
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*> |
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*> \param[out] BETA |
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*> \verbatim |
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*> BETA is COMPLEX*16 array, dimension (N) |
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*> |
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*> The diagonal elements of A and B, respectively, |
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*> when the pair (A,B) has been reduced to generalized Schur |
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*> form. ALPHA(i)/BETA(i) i=1,...,N are the generalized |
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*> eigenvalues. |
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*> \endverbatim |
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*> |
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*> \param[in,out] Q |
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*> \verbatim |
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*> Q is COMPLEX*16 array, dimension (LDQ,N) |
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*> On entry, if WANTQ = .TRUE., Q is an N-by-N matrix. |
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*> On exit, Q has been postmultiplied by the left unitary |
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*> transformation matrix which reorder (A, B); The leading M |
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*> columns of Q form orthonormal bases for the specified pair of |
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*> left eigenspaces (deflating subspaces). |
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*> If WANTQ = .FALSE., Q is not referenced. |
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*> \endverbatim |
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*> |
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*> \param[in] LDQ |
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*> \verbatim |
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*> LDQ is INTEGER |
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*> The leading dimension of the array Q. LDQ >= 1. |
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*> If WANTQ = .TRUE., LDQ >= N. |
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*> \endverbatim |
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*> |
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*> \param[in,out] Z |
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*> \verbatim |
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*> Z is COMPLEX*16 array, dimension (LDZ,N) |
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*> On entry, if WANTZ = .TRUE., Z is an N-by-N matrix. |
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*> On exit, Z has been postmultiplied by the left unitary |
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*> transformation matrix which reorder (A, B); The leading M |
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*> columns of Z form orthonormal bases for the specified pair of |
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*> left eigenspaces (deflating subspaces). |
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*> If WANTZ = .FALSE., Z is not referenced. |
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*> \endverbatim |
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*> |
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*> \param[in] LDZ |
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*> \verbatim |
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*> LDZ is INTEGER |
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*> The leading dimension of the array Z. LDZ >= 1. |
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*> If WANTZ = .TRUE., LDZ >= N. |
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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 dimension of the specified pair of left and right |
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*> eigenspaces, (deflating subspaces) 0 <= M <= N. |
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*> \endverbatim |
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*> |
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*> \param[out] PL |
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*> \verbatim |
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*> PL is DOUBLE PRECISION |
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*> \endverbatim |
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*> |
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*> \param[out] PR |
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*> \verbatim |
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*> PR is DOUBLE PRECISION |
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*> |
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*> If IJOB = 1, 4 or 5, PL, PR are lower bounds on the |
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*> reciprocal of the norm of "projections" onto left and right |
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*> eigenspace with respect to the selected cluster. |
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*> 0 < PL, PR <= 1. |
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*> If M = 0 or M = N, PL = PR = 1. |
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*> If IJOB = 0, 2 or 3 PL, PR are 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 (2). |
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*> If IJOB >= 2, DIF(1:2) store the estimates of Difu and Difl. |
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*> If IJOB = 2 or 4, DIF(1:2) are F-norm-based upper bounds on |
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*> Difu and Difl. If IJOB = 3 or 5, DIF(1:2) are 1-norm-based |
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*> estimates of Difu and Difl, computed using reversed |
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*> communication with ZLACN2. |
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*> If M = 0 or N, DIF(1:2) = F-norm([A, B]). |
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*> If IJOB = 0 or 1, DIF is not referenced. |
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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 >= 1 |
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*> If IJOB = 1, 2 or 4, LWORK >= 2*M*(N-M) |
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*> If IJOB = 3 or 5, LWORK >= 4*M*(N-M) |
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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 (MAX(1,LIWORK)) |
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*> On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. |
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*> \endverbatim |
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*> |
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*> \param[in] LIWORK |
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*> \verbatim |
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*> LIWORK is INTEGER |
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*> The dimension of the array IWORK. LIWORK >= 1. |
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*> If IJOB = 1, 2 or 4, LIWORK >= N+2; |
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*> If IJOB = 3 or 5, LIWORK >= MAX(N+2, 2*M*(N-M)); |
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*> |
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*> If LIWORK = -1, then a workspace query is assumed; the |
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*> routine only calculates the optimal size of the IWORK array, |
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*> returns this value as the first entry of the IWORK array, and |
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*> no error message related to LIWORK is issued by XERBLA. |
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*> \endverbatim |
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*> |
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*> \param[out] INFO |
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*> \verbatim |
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*> INFO is INTEGER |
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*> =0: Successful exit. |
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*> <0: If INFO = -i, the i-th argument had an illegal value. |
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*> =1: Reordering of (A, B) failed because the transformed |
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*> matrix pair (A, B) would be too far from generalized |
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*> Schur form; the problem is very ill-conditioned. |
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*> (A, B) may have been partially reordered. |
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*> If requested, 0 is returned in DIF(*), PL and PR. |
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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 June 2016 |
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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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*> ZTGSEN first collects the selected eigenvalues by computing unitary |
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*> U and W that move them to the top left corner of (A, B). In other |
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*> words, the selected eigenvalues are the eigenvalues of (A11, B11) in |
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*> |
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*> U**H*(A, B)*W = (A11 A12) (B11 B12) n1 |
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*> ( 0 A22),( 0 B22) n2 |
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*> n1 n2 n1 n2 |
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*> |
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*> where N = n1+n2 and U**H means the conjugate transpose of U. The first |
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*> n1 columns of U and W span the specified pair of left and right |
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*> eigenspaces (deflating subspaces) of (A, B). |
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*> |
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*> If (A, B) has been obtained from the generalized real Schur |
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*> decomposition of a matrix pair (C, D) = Q*(A, B)*Z**H, then the |
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*> reordered generalized Schur form of (C, D) is given by |
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*> |
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*> (C, D) = (Q*U)*(U**H *(A, B)*W)*(Z*W)**H, |
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*> |
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*> and the first n1 columns of Q*U and Z*W span the corresponding |
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*> deflating subspaces of (C, D) (Q and Z store Q*U and Z*W, resp.). |
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*> |
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*> Note that if the selected eigenvalue is sufficiently ill-conditioned, |
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*> then its value may differ significantly from its value before |
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*> reordering. |
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*> |
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*> The reciprocal condition numbers of the left and right eigenspaces |
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*> spanned by the first n1 columns of U and W (or Q*U and Z*W) may |
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*> be returned in DIF(1:2), corresponding to Difu and Difl, resp. |
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*> |
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*> The Difu and Difl are defined as: |
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*> |
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*> Difu[(A11, B11), (A22, B22)] = sigma-min( Zu ) |
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*> and |
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*> Difl[(A11, B11), (A22, B22)] = Difu[(A22, B22), (A11, B11)], |
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*> |
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*> where sigma-min(Zu) is the smallest singular value of the |
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*> (2*n1*n2)-by-(2*n1*n2) matrix |
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*> |
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*> Zu = [ kron(In2, A11) -kron(A22**H, In1) ] |
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*> [ kron(In2, B11) -kron(B22**H, In1) ]. |
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*> |
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*> Here, Inx is the identity matrix of size nx and A22**H is the |
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*> conjugate transpose of A22. kron(X, Y) is the Kronecker product between |
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*> the matrices X and Y. |
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*> |
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*> When DIF(2) is small, small changes in (A, B) can cause large changes |
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*> in the deflating subspace. An approximate (asymptotic) bound on the |
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*> maximum angular error in the computed deflating subspaces is |
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*> |
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*> EPS * norm((A, B)) / DIF(2), |
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*> |
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*> where EPS is the machine precision. |
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*> |
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*> The reciprocal norm of the projectors on the left and right |
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*> eigenspaces associated with (A11, B11) may be returned in PL and PR. |
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*> They are computed as follows. First we compute L and R so that |
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*> P*(A, B)*Q is block diagonal, where |
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*> |
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*> P = ( I -L ) n1 Q = ( I R ) n1 |
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*> ( 0 I ) n2 and ( 0 I ) n2 |
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*> n1 n2 n1 n2 |
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*> |
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*> and (L, R) is the solution to the generalized Sylvester equation |
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*> |
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*> A11*R - L*A22 = -A12 |
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*> B11*R - L*B22 = -B12 |
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*> |
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*> Then PL = (F-norm(L)**2+1)**(-1/2) and PR = (F-norm(R)**2+1)**(-1/2). |
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*> An approximate (asymptotic) bound on the average absolute error of |
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*> the selected eigenvalues is |
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*> |
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*> EPS * norm((A, B)) / PL. |
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*> |
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*> There are also global error bounds which valid for perturbations up |
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*> to a certain restriction: A lower bound (x) on the smallest |
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*> F-norm(E,F) for which an eigenvalue of (A11, B11) may move and |
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*> coalesce with an eigenvalue of (A22, B22) under perturbation (E,F), |
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*> (i.e. (A + E, B + F), is |
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*> |
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*> x = min(Difu,Difl)/((1/(PL*PL)+1/(PR*PR))**(1/2)+2*max(1/PL,1/PR)). |
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*> |
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*> An approximate bound on x can be computed from DIF(1:2), PL and PR. |
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*> |
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*> If y = ( F-norm(E,F) / x) <= 1, the angles between the perturbed |
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*> (L', R') and unperturbed (L, R) left and right deflating subspaces |
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*> associated with the selected cluster in the (1,1)-blocks can be |
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*> bounded as |
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*> |
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*> max-angle(L, L') <= arctan( y * PL / (1 - y * (1 - PL * PL)**(1/2)) |
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*> max-angle(R, R') <= arctan( y * PR / (1 - y * (1 - PR * PR)**(1/2)) |
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*> |
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*> See LAPACK User's Guide section 4.11 or the following references |
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*> for more information. |
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*> |
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*> Note that if the default method for computing the Frobenius-norm- |
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*> based estimate DIF is not wanted (see ZLATDF), then the parameter |
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*> IDIFJB (see below) should be changed from 3 to 4 (routine ZLATDF |
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*> (IJOB = 2 will be used)). See ZTGSYL for more details. |
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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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*> [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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*> \n |
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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, Report |
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*> 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. |
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*> To appear in Numerical Algorithms, 1996. |
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*> \n |
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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, No 1, |
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*> 1996. |
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*> |
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* ===================================================================== |
SUBROUTINE ZTGSEN( IJOB, WANTQ, WANTZ, SELECT, N, A, LDA, B, LDB, |
SUBROUTINE ZTGSEN( IJOB, WANTQ, WANTZ, SELECT, N, A, LDA, B, LDB, |
$ ALPHA, BETA, Q, LDQ, Z, LDZ, M, PL, PR, DIF, |
$ ALPHA, BETA, Q, LDQ, Z, LDZ, M, PL, PR, DIF, |
$ WORK, LWORK, IWORK, LIWORK, INFO ) |
$ WORK, LWORK, IWORK, LIWORK, INFO ) |
* |
* |
* -- LAPACK routine (version 3.2.2) -- |
* -- LAPACK computational routine (version 3.6.1) -- |
* -- 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..-- |
* January 2007 |
* June 2016 |
* |
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* Modified to call ZLACN2 in place of ZLACON, 10 Feb 03, SJH. |
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* |
* |
* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
LOGICAL WANTQ, WANTZ |
LOGICAL WANTQ, WANTZ |
Line 23
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Line 452
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$ BETA( * ), Q( LDQ, * ), WORK( * ), Z( LDZ, * ) |
$ BETA( * ), Q( LDQ, * ), WORK( * ), Z( LDZ, * ) |
* .. |
* .. |
* |
* |
* Purpose |
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* ======= |
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* |
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* ZTGSEN reorders the generalized Schur decomposition of a complex |
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* matrix pair (A, B) (in terms of an unitary equivalence trans- |
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* formation Q' * (A, B) * Z), so that a selected cluster of eigenvalues |
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* appears in the leading diagonal blocks of the pair (A,B). The leading |
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* columns of Q and Z form unitary bases of the corresponding left and |
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* right eigenspaces (deflating subspaces). (A, B) must be in |
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* generalized Schur canonical form, that is, A and B are both upper |
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* triangular. |
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* |
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* ZTGSEN also computes the generalized eigenvalues |
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* |
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* w(j)= ALPHA(j) / BETA(j) |
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* |
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* of the reordered matrix pair (A, B). |
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* |
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* Optionally, the routine computes estimates of reciprocal condition |
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* numbers for eigenvalues and eigenspaces. These are Difu[(A11,B11), |
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* (A22,B22)] and Difl[(A11,B11), (A22,B22)], i.e. the separation(s) |
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* between the matrix pairs (A11, B11) and (A22,B22) that correspond to |
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* the selected cluster and the eigenvalues outside the cluster, resp., |
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* and norms of "projections" onto left and right eigenspaces w.r.t. |
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* the selected cluster in the (1,1)-block. |
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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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* IJOB (input) integer |
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* Specifies whether condition numbers are required for the |
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* cluster of eigenvalues (PL and PR) or the deflating subspaces |
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* (Difu and Difl): |
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* =0: Only reorder w.r.t. SELECT. No extras. |
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* =1: Reciprocal of norms of "projections" onto left and right |
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* eigenspaces w.r.t. the selected cluster (PL and PR). |
|
* =2: Upper bounds on Difu and Difl. F-norm-based estimate |
|
* (DIF(1:2)). |
|
* =3: Estimate of Difu and Difl. 1-norm-based estimate |
|
* (DIF(1:2)). |
|
* About 5 times as expensive as IJOB = 2. |
|
* =4: Compute PL, PR and DIF (i.e. 0, 1 and 2 above): Economic |
|
* version to get it all. |
|
* =5: Compute PL, PR and DIF (i.e. 0, 1 and 3 above) |
|
* |
|
* WANTQ (input) LOGICAL |
|
* .TRUE. : update the left transformation matrix Q; |
|
* .FALSE.: do not update Q. |
|
* |
|
* WANTZ (input) LOGICAL |
|
* .TRUE. : update the right transformation matrix Z; |
|
* .FALSE.: do not update Z. |
|
* |
|
* SELECT (input) LOGICAL array, dimension (N) |
|
* SELECT specifies the eigenvalues in the selected cluster. To |
|
* select an eigenvalue w(j), SELECT(j) must be set to |
|
* .TRUE.. |
|
* |
|
* N (input) INTEGER |
|
* The order of the matrices A and B. N >= 0. |
|
* |
|
* A (input/output) COMPLEX*16 array, dimension(LDA,N) |
|
* On entry, the upper triangular matrix A, in generalized |
|
* Schur canonical form. |
|
* On exit, A is overwritten by the reordered matrix A. |
|
* |
|
* LDA (input) INTEGER |
|
* The leading dimension of the array A. LDA >= max(1,N). |
|
* |
|
* B (input/output) COMPLEX*16 array, dimension(LDB,N) |
|
* On entry, the upper triangular matrix B, in generalized |
|
* Schur canonical form. |
|
* On exit, B is overwritten by the reordered matrix B. |
|
* |
|
* LDB (input) INTEGER |
|
* The leading dimension of the array B. LDB >= max(1,N). |
|
* |
|
* ALPHA (output) COMPLEX*16 array, dimension (N) |
|
* BETA (output) COMPLEX*16 array, dimension (N) |
|
* The diagonal elements of A and B, respectively, |
|
* when the pair (A,B) has been reduced to generalized Schur |
|
* form. ALPHA(i)/BETA(i) i=1,...,N are the generalized |
|
* eigenvalues. |
|
* |
|
* Q (input/output) COMPLEX*16 array, dimension (LDQ,N) |
|
* On entry, if WANTQ = .TRUE., Q is an N-by-N matrix. |
|
* On exit, Q has been postmultiplied by the left unitary |
|
* transformation matrix which reorder (A, B); The leading M |
|
* columns of Q form orthonormal bases for the specified pair of |
|
* left eigenspaces (deflating subspaces). |
|
* If WANTQ = .FALSE., Q is not referenced. |
|
* |
|
* LDQ (input) INTEGER |
|
* The leading dimension of the array Q. LDQ >= 1. |
|
* If WANTQ = .TRUE., LDQ >= N. |
|
* |
|
* Z (input/output) COMPLEX*16 array, dimension (LDZ,N) |
|
* On entry, if WANTZ = .TRUE., Z is an N-by-N matrix. |
|
* On exit, Z has been postmultiplied by the left unitary |
|
* transformation matrix which reorder (A, B); The leading M |
|
* columns of Z form orthonormal bases for the specified pair of |
|
* left eigenspaces (deflating subspaces). |
|
* If WANTZ = .FALSE., Z is not referenced. |
|
* |
|
* LDZ (input) INTEGER |
|
* The leading dimension of the array Z. LDZ >= 1. |
|
* If WANTZ = .TRUE., LDZ >= N. |
|
* |
|
* M (output) INTEGER |
|
* The dimension of the specified pair of left and right |
|
* eigenspaces, (deflating subspaces) 0 <= M <= N. |
|
* |
|
* PL (output) DOUBLE PRECISION |
|
* PR (output) DOUBLE PRECISION |
|
* If IJOB = 1, 4 or 5, PL, PR are lower bounds on the |
|
* reciprocal of the norm of "projections" onto left and right |
|
* eigenspace with respect to the selected cluster. |
|
* 0 < PL, PR <= 1. |
|
* If M = 0 or M = N, PL = PR = 1. |
|
* If IJOB = 0, 2 or 3 PL, PR are not referenced. |
|
* |
|
* DIF (output) DOUBLE PRECISION array, dimension (2). |
|
* If IJOB >= 2, DIF(1:2) store the estimates of Difu and Difl. |
|
* If IJOB = 2 or 4, DIF(1:2) are F-norm-based upper bounds on |
|
* Difu and Difl. If IJOB = 3 or 5, DIF(1:2) are 1-norm-based |
|
* estimates of Difu and Difl, computed using reversed |
|
* communication with ZLACN2. |
|
* If M = 0 or N, DIF(1:2) = F-norm([A, B]). |
|
* If IJOB = 0 or 1, DIF is not referenced. |
|
* |
|
* 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 >= 1 |
|
* If IJOB = 1, 2 or 4, LWORK >= 2*M*(N-M) |
|
* If IJOB = 3 or 5, LWORK >= 4*M*(N-M) |
|
* |
|
* If LWORK = -1, then a workspace query is assumed; the routine |
|
* only calculates the optimal size of the WORK array, returns |
|
* this value as the first entry of the WORK array, and no error |
|
* message related to LWORK is issued by XERBLA. |
|
* |
|
* IWORK (workspace/output) INTEGER array, dimension (MAX(1,LIWORK)) |
|
* On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. |
|
* |
|
* LIWORK (input) INTEGER |
|
* The dimension of the array IWORK. LIWORK >= 1. |
|
* If IJOB = 1, 2 or 4, LIWORK >= N+2; |
|
* If IJOB = 3 or 5, LIWORK >= MAX(N+2, 2*M*(N-M)); |
|
* |
|
* If LIWORK = -1, then a workspace query is assumed; the |
|
* routine only calculates the optimal size of the IWORK array, |
|
* returns this value as the first entry of the IWORK array, and |
|
* no error message related to LIWORK is issued by XERBLA. |
|
* |
|
* INFO (output) INTEGER |
|
* =0: Successful exit. |
|
* <0: If INFO = -i, the i-th argument had an illegal value. |
|
* =1: Reordering of (A, B) failed because the transformed |
|
* matrix pair (A, B) would be too far from generalized |
|
* Schur form; the problem is very ill-conditioned. |
|
* (A, B) may have been partially reordered. |
|
* If requested, 0 is returned in DIF(*), PL and PR. |
|
* |
|
* |
|
* Further Details |
|
* =============== |
|
* |
|
* ZTGSEN first collects the selected eigenvalues by computing unitary |
|
* U and W that move them to the top left corner of (A, B). In other |
|
* words, the selected eigenvalues are the eigenvalues of (A11, B11) in |
|
* |
|
* U'*(A, B)*W = (A11 A12) (B11 B12) n1 |
|
* ( 0 A22),( 0 B22) n2 |
|
* n1 n2 n1 n2 |
|
* |
|
* where N = n1+n2 and U' means the conjugate transpose of U. The first |
|
* n1 columns of U and W span the specified pair of left and right |
|
* eigenspaces (deflating subspaces) of (A, B). |
|
* |
|
* If (A, B) has been obtained from the generalized real Schur |
|
* decomposition of a matrix pair (C, D) = Q*(A, B)*Z', then the |
|
* reordered generalized Schur form of (C, D) is given by |
|
* |
|
* (C, D) = (Q*U)*(U'*(A, B)*W)*(Z*W)', |
|
* |
|
* and the first n1 columns of Q*U and Z*W span the corresponding |
|
* deflating subspaces of (C, D) (Q and Z store Q*U and Z*W, resp.). |
|
* |
|
* Note that if the selected eigenvalue is sufficiently ill-conditioned, |
|
* then its value may differ significantly from its value before |
|
* reordering. |
|
* |
|
* The reciprocal condition numbers of the left and right eigenspaces |
|
* spanned by the first n1 columns of U and W (or Q*U and Z*W) may |
|
* be returned in DIF(1:2), corresponding to Difu and Difl, resp. |
|
* |
|
* The Difu and Difl are defined as: |
|
* |
|
* Difu[(A11, B11), (A22, B22)] = sigma-min( Zu ) |
|
* and |
|
* Difl[(A11, B11), (A22, B22)] = Difu[(A22, B22), (A11, B11)], |
|
* |
|
* where sigma-min(Zu) is the smallest singular value of the |
|
* (2*n1*n2)-by-(2*n1*n2) matrix |
|
* |
|
* Zu = [ kron(In2, A11) -kron(A22', In1) ] |
|
* [ kron(In2, B11) -kron(B22', In1) ]. |
|
* |
|
* Here, Inx is the identity matrix of size nx and A22' is the |
|
* transpose of A22. kron(X, Y) is the Kronecker product between |
|
* the matrices X and Y. |
|
* |
|
* When DIF(2) is small, small changes in (A, B) can cause large changes |
|
* in the deflating subspace. An approximate (asymptotic) bound on the |
|
* maximum angular error in the computed deflating subspaces is |
|
* |
|
* EPS * norm((A, B)) / DIF(2), |
|
* |
|
* where EPS is the machine precision. |
|
* |
|
* The reciprocal norm of the projectors on the left and right |
|
* eigenspaces associated with (A11, B11) may be returned in PL and PR. |
|
* They are computed as follows. First we compute L and R so that |
|
* P*(A, B)*Q is block diagonal, where |
|
* |
|
* P = ( I -L ) n1 Q = ( I R ) n1 |
|
* ( 0 I ) n2 and ( 0 I ) n2 |
|
* n1 n2 n1 n2 |
|
* |
|
* and (L, R) is the solution to the generalized Sylvester equation |
|
* |
|
* A11*R - L*A22 = -A12 |
|
* B11*R - L*B22 = -B12 |
|
* |
|
* Then PL = (F-norm(L)**2+1)**(-1/2) and PR = (F-norm(R)**2+1)**(-1/2). |
|
* An approximate (asymptotic) bound on the average absolute error of |
|
* the selected eigenvalues is |
|
* |
|
* EPS * norm((A, B)) / PL. |
|
* |
|
* There are also global error bounds which valid for perturbations up |
|
* to a certain restriction: A lower bound (x) on the smallest |
|
* F-norm(E,F) for which an eigenvalue of (A11, B11) may move and |
|
* coalesce with an eigenvalue of (A22, B22) under perturbation (E,F), |
|
* (i.e. (A + E, B + F), is |
|
* |
|
* x = min(Difu,Difl)/((1/(PL*PL)+1/(PR*PR))**(1/2)+2*max(1/PL,1/PR)). |
|
* |
|
* An approximate bound on x can be computed from DIF(1:2), PL and PR. |
|
* |
|
* If y = ( F-norm(E,F) / x) <= 1, the angles between the perturbed |
|
* (L', R') and unperturbed (L, R) left and right deflating subspaces |
|
* associated with the selected cluster in the (1,1)-blocks can be |
|
* bounded as |
|
* |
|
* max-angle(L, L') <= arctan( y * PL / (1 - y * (1 - PL * PL)**(1/2)) |
|
* max-angle(R, R') <= arctan( y * PR / (1 - y * (1 - PR * PR)**(1/2)) |
|
* |
|
* See LAPACK User's Guide section 4.11 or the following references |
|
* for more information. |
|
* |
|
* Note that if the default method for computing the Frobenius-norm- |
|
* based estimate DIF is not wanted (see ZLATDF), then the parameter |
|
* IDIFJB (see below) should be changed from 3 to 4 (routine ZLATDF |
|
* (IJOB = 2 will be used)). See ZTGSYL for more details. |
|
* |
|
* 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 .. |
Line 385
|
Line 518
|
* subspaces. |
* subspaces. |
* |
* |
M = 0 |
M = 0 |
|
IF( .NOT.LQUERY .OR. IJOB.NE.0 ) THEN |
DO 10 K = 1, N |
DO 10 K = 1, N |
ALPHA( K ) = A( K, K ) |
ALPHA( K ) = A( K, K ) |
BETA( K ) = B( K, K ) |
BETA( K ) = B( K, K ) |
Line 396
|
Line 530
|
$ M = M + 1 |
$ M = M + 1 |
END IF |
END IF |
10 CONTINUE |
10 CONTINUE |
|
END IF |
* |
* |
IF( IJOB.EQ.1 .OR. IJOB.EQ.2 .OR. IJOB.EQ.4 ) THEN |
IF( IJOB.EQ.1 .OR. IJOB.EQ.2 .OR. IJOB.EQ.4 ) THEN |
LWMIN = MAX( 1, 2*M*( N-M ) ) |
LWMIN = MAX( 1, 2*M*( N-M ) ) |