CGGES - compute for a pair of N-by-N complex nonsymmetric
matrices (A,B), the generalized eigenvalues, the general
ized complex Schur form (S, T), and optionally left and/or
right Schur vectors (VSL and VSR)
SYNOPSIS
SUBROUTINE CGGES( JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA,
B, LDB, SDIM, ALPHA, BETA, VSL, LDVSL,
VSR, LDVSR, WORK, LWORK, RWORK, BWORK,
INFO )
CHARACTER JOBVSL, JOBVSR, SORT
INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N,
SDIM
LOGICAL BWORK( * )
REAL RWORK( * )
COMPLEX A( LDA, * ), ALPHA( * ), B( LDB, * ),
BETA( * ), VSL( LDVSL, * ), VSR( LDVSR,
* ), WORK( * )
LOGICAL SELCTG
EXTERNAL SELCTG
PURPOSE
CGGES computes for a pair of N-by-N complex nonsymmetric
matrices (A,B), the generalized eigenvalues, the general
ized complex Schur form (S, T), and optionally left and/or
right Schur vectors (VSL and VSR). This gives the general
ized 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.
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 (input) LOGICAL FUNCTION of two COMPLEX arguments
SELCTG must be declared EXTERNAL in the calling
subroutine. If SORT = 'N', SELCTG is not refer
enced. 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).
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 com
plex 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 gen
erally 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.
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 ille
gal 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 eigenval
ues in the Generalized Schur form no longer sat
isfy SELCTG=.TRUE. This could also be caused due
to scaling. =N+3: reordering falied in CTGSEN.
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