version 1.4, 2010/08/06 15:32:39
|
version 1.10, 2012/07/31 11:06:38
|
Line 1
|
Line 1
|
|
*> \brief <b> ZGGES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices</b> |
|
* |
|
* =========== DOCUMENTATION =========== |
|
* |
|
* Online html documentation available at |
|
* http://www.netlib.org/lapack/explore-html/ |
|
* |
|
*> \htmlonly |
|
*> Download ZGGES + dependencies |
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgges.f"> |
|
*> [TGZ]</a> |
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zgges.f"> |
|
*> [ZIP]</a> |
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgges.f"> |
|
*> [TXT]</a> |
|
*> \endhtmlonly |
|
* |
|
* Definition: |
|
* =========== |
|
* |
|
* SUBROUTINE ZGGES( JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB, |
|
* SDIM, ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, WORK, |
|
* LWORK, RWORK, BWORK, INFO ) |
|
* |
|
* .. Scalar Arguments .. |
|
* CHARACTER JOBVSL, JOBVSR, SORT |
|
* INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N, SDIM |
|
* .. |
|
* .. Array Arguments .. |
|
* LOGICAL BWORK( * ) |
|
* DOUBLE PRECISION RWORK( * ) |
|
* COMPLEX*16 A( LDA, * ), ALPHA( * ), B( LDB, * ), |
|
* $ BETA( * ), VSL( LDVSL, * ), VSR( LDVSR, * ), |
|
* $ WORK( * ) |
|
* .. |
|
* .. Function Arguments .. |
|
* LOGICAL SELCTG |
|
* EXTERNAL SELCTG |
|
* .. |
|
* |
|
* |
|
*> \par Purpose: |
|
* ============= |
|
*> |
|
*> \verbatim |
|
*> |
|
*> ZGGES computes for a pair of N-by-N complex nonsymmetric matrices |
|
*> (A,B), the generalized eigenvalues, the generalized complex Schur |
|
*> form (S, T), and optionally left and/or right Schur vectors (VSL |
|
*> and VSR). This gives the generalized Schur factorization |
|
*> |
|
*> (A,B) = ( (VSL)*S*(VSR)**H, (VSL)*T*(VSR)**H ) |
|
*> |
|
*> where (VSR)**H is the conjugate-transpose of VSR. |
|
*> |
|
*> Optionally, it also orders the eigenvalues so that a selected cluster |
|
*> of eigenvalues appears in the leading diagonal blocks of the upper |
|
*> triangular matrix S and the upper triangular matrix T. The leading |
|
*> columns of VSL and VSR then form an unitary basis for the |
|
*> corresponding left and right eigenspaces (deflating subspaces). |
|
*> |
|
*> (If only the generalized eigenvalues are needed, use the driver |
|
*> ZGGEV instead, which is faster.) |
|
*> |
|
*> A generalized eigenvalue for a pair of matrices (A,B) is a scalar w |
|
*> or a ratio alpha/beta = w, such that A - w*B is singular. It is |
|
*> usually represented as the pair (alpha,beta), as there is a |
|
*> reasonable interpretation for beta=0, and even for both being zero. |
|
*> |
|
*> A pair of matrices (S,T) is in generalized complex Schur form if S |
|
*> and T are upper triangular and, in addition, the diagonal elements |
|
*> of T are non-negative real numbers. |
|
*> \endverbatim |
|
* |
|
* Arguments: |
|
* ========== |
|
* |
|
*> \param[in] JOBVSL |
|
*> \verbatim |
|
*> JOBVSL is CHARACTER*1 |
|
*> = 'N': do not compute the left Schur vectors; |
|
*> = 'V': compute the left Schur vectors. |
|
*> \endverbatim |
|
*> |
|
*> \param[in] JOBVSR |
|
*> \verbatim |
|
*> JOBVSR is CHARACTER*1 |
|
*> = 'N': do not compute the right Schur vectors; |
|
*> = 'V': compute the right Schur vectors. |
|
*> \endverbatim |
|
*> |
|
*> \param[in] SORT |
|
*> \verbatim |
|
*> SORT is CHARACTER*1 |
|
*> Specifies whether or not to order the eigenvalues on the |
|
*> diagonal of the generalized Schur form. |
|
*> = 'N': Eigenvalues are not ordered; |
|
*> = 'S': Eigenvalues are ordered (see SELCTG). |
|
*> \endverbatim |
|
*> |
|
*> \param[in] SELCTG |
|
*> \verbatim |
|
*> SELCTG is a LOGICAL FUNCTION of two COMPLEX*16 arguments |
|
*> SELCTG must be declared EXTERNAL in the calling subroutine. |
|
*> If SORT = 'N', SELCTG is not referenced. |
|
*> If SORT = 'S', SELCTG is used to select eigenvalues to sort |
|
*> to the top left of the Schur form. |
|
*> An eigenvalue ALPHA(j)/BETA(j) is selected if |
|
*> SELCTG(ALPHA(j),BETA(j)) is true. |
|
*> |
|
*> Note that a selected complex eigenvalue may no longer satisfy |
|
*> SELCTG(ALPHA(j),BETA(j)) = .TRUE. after ordering, since |
|
*> ordering may change the value of complex eigenvalues |
|
*> (especially if the eigenvalue is ill-conditioned), in this |
|
*> case INFO is set to N+2 (See INFO below). |
|
*> \endverbatim |
|
*> |
|
*> \param[in] N |
|
*> \verbatim |
|
*> N is INTEGER |
|
*> The order of the matrices A, B, VSL, and VSR. N >= 0. |
|
*> \endverbatim |
|
*> |
|
*> \param[in,out] A |
|
*> \verbatim |
|
*> A is COMPLEX*16 array, dimension (LDA, N) |
|
*> On entry, the first of the pair of matrices. |
|
*> On exit, A has been overwritten by its generalized Schur |
|
*> form S. |
|
*> \endverbatim |
|
*> |
|
*> \param[in] LDA |
|
*> \verbatim |
|
*> LDA is INTEGER |
|
*> The leading dimension of A. LDA >= max(1,N). |
|
*> \endverbatim |
|
*> |
|
*> \param[in,out] B |
|
*> \verbatim |
|
*> B is COMPLEX*16 array, dimension (LDB, N) |
|
*> On entry, the second of the pair of matrices. |
|
*> On exit, B has been overwritten by its generalized Schur |
|
*> form T. |
|
*> \endverbatim |
|
*> |
|
*> \param[in] LDB |
|
*> \verbatim |
|
*> LDB is INTEGER |
|
*> The leading dimension of B. LDB >= max(1,N). |
|
*> \endverbatim |
|
*> |
|
*> \param[out] SDIM |
|
*> \verbatim |
|
*> SDIM is INTEGER |
|
*> If SORT = 'N', SDIM = 0. |
|
*> If SORT = 'S', SDIM = number of eigenvalues (after sorting) |
|
*> for which SELCTG is true. |
|
*> \endverbatim |
|
*> |
|
*> \param[out] ALPHA |
|
*> \verbatim |
|
*> ALPHA is COMPLEX*16 array, dimension (N) |
|
*> \endverbatim |
|
*> |
|
*> \param[out] BETA |
|
*> \verbatim |
|
*> BETA is COMPLEX*16 array, dimension (N) |
|
*> On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the |
|
*> generalized eigenvalues. ALPHA(j), j=1,...,N and BETA(j), |
|
*> j=1,...,N are the diagonals of the complex Schur form (A,B) |
|
*> output by ZGGES. The BETA(j) will be non-negative real. |
|
*> |
|
*> Note: the quotients ALPHA(j)/BETA(j) may easily over- or |
|
*> underflow, and BETA(j) may even be zero. Thus, the user |
|
*> should avoid naively computing the ratio alpha/beta. |
|
*> However, ALPHA will be always less than and usually |
|
*> comparable with norm(A) in magnitude, and BETA always less |
|
*> than and usually comparable with norm(B). |
|
*> \endverbatim |
|
*> |
|
*> \param[out] VSL |
|
*> \verbatim |
|
*> VSL is COMPLEX*16 array, dimension (LDVSL,N) |
|
*> If JOBVSL = 'V', VSL will contain the left Schur vectors. |
|
*> Not referenced if JOBVSL = 'N'. |
|
*> \endverbatim |
|
*> |
|
*> \param[in] LDVSL |
|
*> \verbatim |
|
*> LDVSL is INTEGER |
|
*> The leading dimension of the matrix VSL. LDVSL >= 1, and |
|
*> if JOBVSL = 'V', LDVSL >= N. |
|
*> \endverbatim |
|
*> |
|
*> \param[out] VSR |
|
*> \verbatim |
|
*> VSR is COMPLEX*16 array, dimension (LDVSR,N) |
|
*> If JOBVSR = 'V', VSR will contain the right Schur vectors. |
|
*> Not referenced if JOBVSR = 'N'. |
|
*> \endverbatim |
|
*> |
|
*> \param[in] LDVSR |
|
*> \verbatim |
|
*> LDVSR is INTEGER |
|
*> The leading dimension of the matrix VSR. LDVSR >= 1, and |
|
*> if JOBVSR = 'V', LDVSR >= N. |
|
*> \endverbatim |
|
*> |
|
*> \param[out] WORK |
|
*> \verbatim |
|
*> WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) |
|
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. |
|
*> \endverbatim |
|
*> |
|
*> \param[in] LWORK |
|
*> \verbatim |
|
*> LWORK is INTEGER |
|
*> The dimension of the array WORK. LWORK >= max(1,2*N). |
|
*> For good performance, LWORK must generally be larger. |
|
*> |
|
*> 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. |
|
*> \endverbatim |
|
*> |
|
*> \param[out] RWORK |
|
*> \verbatim |
|
*> RWORK is DOUBLE PRECISION array, dimension (8*N) |
|
*> \endverbatim |
|
*> |
|
*> \param[out] BWORK |
|
*> \verbatim |
|
*> BWORK is LOGICAL array, dimension (N) |
|
*> Not referenced if SORT = 'N'. |
|
*> \endverbatim |
|
*> |
|
*> \param[out] INFO |
|
*> \verbatim |
|
*> INFO is INTEGER |
|
*> = 0: successful exit |
|
*> < 0: if INFO = -i, the i-th argument had an illegal value. |
|
*> =1,...,N: |
|
*> The QZ iteration failed. (A,B) are not in Schur |
|
*> form, but ALPHA(j) and BETA(j) should be correct for |
|
*> j=INFO+1,...,N. |
|
*> > N: =N+1: other than QZ iteration failed in ZHGEQZ |
|
*> =N+2: after reordering, roundoff changed values of |
|
*> some complex eigenvalues so that leading |
|
*> eigenvalues in the Generalized Schur form no |
|
*> longer satisfy SELCTG=.TRUE. This could also |
|
*> be caused due to scaling. |
|
*> =N+3: reordering falied in ZTGSEN. |
|
*> \endverbatim |
|
* |
|
* Authors: |
|
* ======== |
|
* |
|
*> \author Univ. of Tennessee |
|
*> \author Univ. of California Berkeley |
|
*> \author Univ. of Colorado Denver |
|
*> \author NAG Ltd. |
|
* |
|
*> \date November 2011 |
|
* |
|
*> \ingroup complex16GEeigen |
|
* |
|
* ===================================================================== |
SUBROUTINE ZGGES( JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB, |
SUBROUTINE ZGGES( JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB, |
$ SDIM, ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, WORK, |
$ SDIM, ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, WORK, |
$ LWORK, RWORK, BWORK, INFO ) |
$ LWORK, RWORK, BWORK, INFO ) |
* |
* |
* -- LAPACK driver routine (version 3.2) -- |
* -- LAPACK driver routine (version 3.4.0) -- |
* -- LAPACK is a software package provided by Univ. of Tennessee, -- |
* -- LAPACK is a software package provided by Univ. of Tennessee, -- |
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
* November 2006 |
* November 2011 |
* |
* |
* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
CHARACTER JOBVSL, JOBVSR, SORT |
CHARACTER JOBVSL, JOBVSR, SORT |
Line 23
|
Line 291
|
EXTERNAL SELCTG |
EXTERNAL SELCTG |
* .. |
* .. |
* |
* |
* Purpose |
|
* ======= |
|
* |
|
* ZGGES computes for a pair of N-by-N complex nonsymmetric matrices |
|
* (A,B), the generalized eigenvalues, the generalized complex Schur |
|
* form (S, T), and optionally left and/or right Schur vectors (VSL |
|
* and VSR). This gives the generalized Schur factorization |
|
* |
|
* (A,B) = ( (VSL)*S*(VSR)**H, (VSL)*T*(VSR)**H ) |
|
* |
|
* where (VSR)**H is the conjugate-transpose of VSR. |
|
* |
|
* Optionally, it also orders the eigenvalues so that a selected cluster |
|
* of eigenvalues appears in the leading diagonal blocks of the upper |
|
* triangular matrix S and the upper triangular matrix T. The leading |
|
* columns of VSL and VSR then form an unitary basis for the |
|
* corresponding left and right eigenspaces (deflating subspaces). |
|
* |
|
* (If only the generalized eigenvalues are needed, use the driver |
|
* ZGGEV instead, which is faster.) |
|
* |
|
* A generalized eigenvalue for a pair of matrices (A,B) is a scalar w |
|
* or a ratio alpha/beta = w, such that A - w*B is singular. It is |
|
* usually represented as the pair (alpha,beta), as there is a |
|
* reasonable interpretation for beta=0, and even for both being zero. |
|
* |
|
* A pair of matrices (S,T) is in generalized complex Schur form if S |
|
* and T are upper triangular and, in addition, the diagonal elements |
|
* of T are non-negative real numbers. |
|
* |
|
* Arguments |
|
* ========= |
|
* |
|
* JOBVSL (input) CHARACTER*1 |
|
* = 'N': do not compute the left Schur vectors; |
|
* = 'V': compute the left Schur vectors. |
|
* |
|
* JOBVSR (input) CHARACTER*1 |
|
* = 'N': do not compute the right Schur vectors; |
|
* = 'V': compute the right Schur vectors. |
|
* |
|
* SORT (input) CHARACTER*1 |
|
* Specifies whether or not to order the eigenvalues on the |
|
* diagonal of the generalized Schur form. |
|
* = 'N': Eigenvalues are not ordered; |
|
* = 'S': Eigenvalues are ordered (see SELCTG). |
|
* |
|
* SELCTG (external procedure) LOGICAL FUNCTION of two COMPLEX*16 arguments |
|
* SELCTG must be declared EXTERNAL in the calling subroutine. |
|
* If SORT = 'N', SELCTG is not referenced. |
|
* If SORT = 'S', SELCTG is used to select eigenvalues to sort |
|
* to the top left of the Schur form. |
|
* An eigenvalue ALPHA(j)/BETA(j) is selected if |
|
* SELCTG(ALPHA(j),BETA(j)) is true. |
|
* |
|
* Note that a selected complex eigenvalue may no longer satisfy |
|
* SELCTG(ALPHA(j),BETA(j)) = .TRUE. after ordering, since |
|
* ordering may change the value of complex eigenvalues |
|
* (especially if the eigenvalue is ill-conditioned), in this |
|
* case INFO is set to N+2 (See INFO below). |
|
* |
|
* N (input) INTEGER |
|
* The order of the matrices A, B, VSL, and VSR. N >= 0. |
|
* |
|
* A (input/output) COMPLEX*16 array, dimension (LDA, N) |
|
* On entry, the first of the pair of matrices. |
|
* On exit, A has been overwritten by its generalized Schur |
|
* form S. |
|
* |
|
* LDA (input) INTEGER |
|
* The leading dimension of A. LDA >= max(1,N). |
|
* |
|
* B (input/output) COMPLEX*16 array, dimension (LDB, N) |
|
* On entry, the second of the pair of matrices. |
|
* On exit, B has been overwritten by its generalized Schur |
|
* form T. |
|
* |
|
* LDB (input) INTEGER |
|
* The leading dimension of B. LDB >= max(1,N). |
|
* |
|
* SDIM (output) INTEGER |
|
* If SORT = 'N', SDIM = 0. |
|
* If SORT = 'S', SDIM = number of eigenvalues (after sorting) |
|
* for which SELCTG is true. |
|
* |
|
* ALPHA (output) COMPLEX*16 array, dimension (N) |
|
* BETA (output) COMPLEX*16 array, dimension (N) |
|
* On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the |
|
* generalized eigenvalues. ALPHA(j), j=1,...,N and BETA(j), |
|
* j=1,...,N are the diagonals of the complex Schur form (A,B) |
|
* output by ZGGES. The BETA(j) will be non-negative real. |
|
* |
|
* Note: the quotients ALPHA(j)/BETA(j) may easily over- or |
|
* underflow, and BETA(j) may even be zero. Thus, the user |
|
* should avoid naively computing the ratio alpha/beta. |
|
* However, ALPHA will be always less than and usually |
|
* comparable with norm(A) in magnitude, and BETA always less |
|
* than and usually comparable with norm(B). |
|
* |
|
* VSL (output) COMPLEX*16 array, dimension (LDVSL,N) |
|
* If JOBVSL = 'V', VSL will contain the left Schur vectors. |
|
* Not referenced if JOBVSL = 'N'. |
|
* |
|
* LDVSL (input) INTEGER |
|
* The leading dimension of the matrix VSL. LDVSL >= 1, and |
|
* if JOBVSL = 'V', LDVSL >= N. |
|
* |
|
* VSR (output) COMPLEX*16 array, dimension (LDVSR,N) |
|
* If JOBVSR = 'V', VSR will contain the right Schur vectors. |
|
* Not referenced if JOBVSR = 'N'. |
|
* |
|
* LDVSR (input) INTEGER |
|
* The leading dimension of the matrix VSR. LDVSR >= 1, and |
|
* if JOBVSR = 'V', LDVSR >= N. |
|
* |
|
* WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK)) |
|
* On exit, if INFO = 0, WORK(1) returns the optimal LWORK. |
|
* |
|
* LWORK (input) INTEGER |
|
* The dimension of the array WORK. LWORK >= max(1,2*N). |
|
* For good performance, LWORK must generally be larger. |
|
* |
|
* 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. |
|
* |
|
* RWORK (workspace) DOUBLE PRECISION array, dimension (8*N) |
|
* |
|
* BWORK (workspace) LOGICAL array, dimension (N) |
|
* Not referenced if SORT = 'N'. |
|
* |
|
* INFO (output) INTEGER |
|
* = 0: successful exit |
|
* < 0: if INFO = -i, the i-th argument had an illegal value. |
|
* =1,...,N: |
|
* The QZ iteration failed. (A,B) are not in Schur |
|
* form, but ALPHA(j) and BETA(j) should be correct for |
|
* j=INFO+1,...,N. |
|
* > N: =N+1: other than QZ iteration failed in ZHGEQZ |
|
* =N+2: after reordering, roundoff changed values of |
|
* some complex eigenvalues so that leading |
|
* eigenvalues in the Generalized Schur form no |
|
* longer satisfy SELCTG=.TRUE. This could also |
|
* be caused due to scaling. |
|
* =N+3: reordering falied in ZTGSEN. |
|
* |
|
* ===================================================================== |
* ===================================================================== |
* |
* |
* .. Parameters .. |
* .. Parameters .. |