GNU/Linux man pages
Livre :
Expressions régulières,
Syntaxe et mise en oeuvre :
GNU/Linux |
CentOS 4.8 |
i386 |
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cgges(l) |
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CGGES - compute 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)
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SUBROUTINE CGGES( |
JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB, SDIM, ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, WORK, LWORK, RWORK, BWORK, INFO ) | ||
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CHARACTER |
JOBVSL, JOBVSR, SORT | ||
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INTEGER |
INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N, SDIM | ||
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LOGICAL |
BWORK( * ) | ||
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REAL |
RWORK( * ) | ||
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COMPLEX |
A( LDA, * ), ALPHA( * ), B( LDB, * ), BETA( * ), VSL( LDVSL, * ), VSR( LDVSR, * ), WORK( * ) | ||
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LOGICAL |
SELCTG | ||
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EXTERNAL |
SELCTG |
CGGES 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 CGGEV 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.
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 (input) LOGICAL FUNCTION of two COMPLEX 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 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 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 array, dimension (N)
BETA (output) COMPLEX 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 CGGES. 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 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 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 array, dimension (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) REAL 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 CHGEQZ
=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 CTGSEN.
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cgges(l) | |