CGGESX - compute for a pair of N-by-N complex nonsymmetric
matrices (A,B), the generalized eigenvalues, the complex
Schur form (S,T),
SYNOPSIS
SUBROUTINE CGGESX( JOBVSL, JOBVSR, SORT, SELCTG, SENSE, N,
A, LDA, B, LDB, SDIM, ALPHA, BETA, VSL,
LDVSL, VSR, LDVSR, RCONDE, RCONDV,
WORK, LWORK, RWORK, IWORK, LIWORK,
BWORK, INFO )
CHARACTER JOBVSL, JOBVSR, SENSE, SORT
INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LIWORK,
LWORK, N, SDIM
LOGICAL BWORK( * )
INTEGER IWORK( * )
REAL RCONDE( 2 ), RCONDV( 2 ), RWORK( * )
COMPLEX A( LDA, * ), ALPHA( * ), B( LDB, * ),
BETA( * ), VSL( LDVSL, * ), VSR( LDVSR,
* ), WORK( * )
LOGICAL SELCTG
EXTERNAL SELCTG
PURPOSE
CGGESX computes for a pair of N-by-N complex nonsymmetric
matrices (A,B), the generalized eigenvalues, the complex
Schur form (S,T), and, optionally, the left and/or right
matrices of 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; computes a reciprocal condition
number for the average of the selected eigenvalues
(RCONDE); and computes a reciprocal condition number for
the right and left deflating subspaces corresponding to
the selected eigenvalues (RCONDV). The leading columns of
VSL and VSR then form an orthonormal basis for the corre
sponding left and right eigenspaces (deflating subspaces).
A generalized eigenvalue for a pair of matrices (A,B) is a
singular. It is usually represented as the pair
(alpha,beta), as there is a reasonable interpretation for
beta=0 or for both being zero.
A pair of matrices (S,T) is in generalized complex Schur
form if T is upper triangular with non-negative diagonal
and S is upper triangular.
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. 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+3 see INFO below).
SENSE (input) CHARACTER
Determines which reciprocal condition numbers are
computed. = 'N' : None are computed;
= 'E' : Computed for average of selected eigenval
ues only;
= 'V' : Computed for selected deflating subspaces
only;
= 'B' : Computed for both. If SENSE = 'E', 'V',
or 'B', SORT must equal 'S'.
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
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) and
BETA(j),j=1,...,N are the diagonals of the com
plex Schur form (S,T). BETA(j) will be non-nega
tive 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.
RCONDE (output) REAL array, dimension ( 2 )
If SENSE = 'E' or 'B', RCONDE(1) and RCONDE(2)
contain the reciprocal condition numbers for the
average of the selected eigenvalues. Not
RCONDV (output) REAL array, dimension ( 2 )
If SENSE = 'V' or 'B', RCONDV(1) and RCONDV(2)
contain the reciprocal condition number for the
selected deflating subspaces. Not referenced if
SENSE = 'N' or 'E'.
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 >= 2*N.
If SENSE = 'E', 'V', or 'B', LWORK >= MAX(2*N,
2*SDIM*(N-SDIM)).
RWORK (workspace) REAL array, dimension ( 8*N )
Real workspace.
IWORK (workspace/output) INTEGER array, dimension
(LIWORK)
Not referenced if SENSE = 'N'. On exit, if INFO =
0, IWORK(1) returns the optimal LIWORK.
LIWORK (input) INTEGER
The dimension of the array WORK. LIWORK >= N+2.
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 failed in CTGSEN.
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