Contents
zgges - 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)
SUBROUTINE ZGGES(JOBVSL, JOBVSR, SORT, DELZTG, N, A, LDA, B, LDB,
SDIM, ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, WORK, LWORK, RWORK,
BWORK, INFO)
CHARACTER * 1 JOBVSL, JOBVSR, SORT
DOUBLE COMPLEX A(LDA,*), B(LDB,*), ALPHA(*), BETA(*),
VSL(LDVSL,*), VSR(LDVSR,*), WORK(*)
INTEGER N, LDA, LDB, SDIM, LDVSL, LDVSR, LWORK, INFO
LOGICAL DELZTG
LOGICAL BWORK(*)
DOUBLE PRECISION RWORK(*)
SUBROUTINE ZGGES_64(JOBVSL, JOBVSR, SORT, DELZTG, N, A, LDA, B, LDB,
SDIM, ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, WORK, LWORK, RWORK,
BWORK, INFO)
CHARACTER * 1 JOBVSL, JOBVSR, SORT
DOUBLE COMPLEX A(LDA,*), B(LDB,*), ALPHA(*), BETA(*),
VSL(LDVSL,*), VSR(LDVSR,*), WORK(*)
INTEGER*8 N, LDA, LDB, SDIM, LDVSL, LDVSR, LWORK, INFO
LOGICAL*8 DELZTG
LOGICAL*8 BWORK(*)
DOUBLE PRECISION RWORK(*)
F95 INTERFACE
SUBROUTINE GGES(JOBVSL, JOBVSR, SORT, [DELZTG], [N], A, [LDA], B, [LDB],
SDIM, ALPHA, BETA, VSL, [LDVSL], VSR, [LDVSR], [WORK], [LWORK],
[RWORK], [BWORK], [INFO])
CHARACTER(LEN=1) :: JOBVSL, JOBVSR, SORT
COMPLEX(8), DIMENSION(:) :: ALPHA, BETA, WORK
COMPLEX(8), DIMENSION(:,:) :: A, B, VSL, VSR
INTEGER :: N, LDA, LDB, SDIM, LDVSL, LDVSR, LWORK, INFO
LOGICAL :: DELZTG
LOGICAL, DIMENSION(:) :: BWORK
REAL(8), DIMENSION(:) :: RWORK
SUBROUTINE GGES_64(JOBVSL, JOBVSR, SORT, [DELZTG], [N], A, [LDA], B,
[LDB], SDIM, ALPHA, BETA, VSL, [LDVSL], VSR, [LDVSR], [WORK],
[LWORK], [RWORK], [BWORK], [INFO])
CHARACTER(LEN=1) :: JOBVSL, JOBVSR, SORT
COMPLEX(8), DIMENSION(:) :: ALPHA, BETA, WORK
COMPLEX(8), DIMENSION(:,:) :: A, B, VSL, VSR
INTEGER(8) :: N, LDA, LDB, SDIM, LDVSL, LDVSR, LWORK, INFO
LOGICAL(8) :: DELZTG
LOGICAL(8), DIMENSION(:) :: BWORK
REAL(8), DIMENSION(:) :: RWORK
C INTERFACE
#include <sunperf.h>
void zgges(char jobvsl, char jobvsr, char sort,
int(*delztg)(doublecomplex,doublecomplex), int n,
doublecomplex *a, int lda, doublecomplex *b, int
ldb, int *sdim, doublecomplex *alpha, doublecom-
plex *beta, doublecomplex *vsl, int ldvsl, doub-
lecomplex *vsr, int ldvsr, int *info);
void zgges_64(char jobvsl, char jobvsr, char sort,
long(*delztg)(doublecomplex,doublecomplex), long
n, doublecomplex *a, long lda, doublecomplex *b,
long ldb, long *sdim, doublecomplex *alpha, doub-
lecomplex *beta, doublecomplex *vsl, long ldvsl,
doublecomplex *vsr, long ldvsr, long *info);
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 diag-
onal 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)
= 'N': do not compute the left Schur vectors;
= 'V': compute the left Schur vectors.
JOBVSR (input)
= 'N': do not compute the right Schur vectors;
= 'V': compute the right Schur vectors.
SORT (input)
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 DELZTG).
DELZTG (input)
LOGICAL FUNCTION of two DOUBLE COMPLEX arguments
DELZTG must be declared EXTERNAL in the calling
subroutine. If SORT = 'N', DELZTG is not refer-
enced. If SORT = 'S', DELZTG is used to select
eigenvalues to sort to the top left of the Schur
form. An eigenvalue ALPHA(j)/BETA(j) is selected
if DELZTG(ALPHA(j),BETA(j)) is true.
Note that a selected complex eigenvalue may no
longer satisfy DELZTG(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) The order of the matrices A, B, VSL, and VSR. N
>= 0.
A (input/output)
DOUBLE 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)
The leading dimension of A. LDA >= max(1,N).
B (input/output)
DOUBLE 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)
The leading dimension of B. LDB >= max(1,N).
SDIM (output)
If SORT = 'N', SDIM = 0. If SORT = 'S', SDIM =
number of eigenvalues (after sorting) for which
DELZTG is true.
ALPHA (output)
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).
BETA (output)
See description of ALPHA.
VSL (input)
DOUBLE COMPLEX array, dimension(LDVSL, N) If
JOBVSL = 'V', VSL will contain the left Schur vec-
tors. Not referenced if JOBVSL = 'N'.
LDVSL (input)
The leading dimension of the matrix VSL. LDVSL >=
1, and if JOBVSL = 'V', LDVSL >= N.
VSR (input)
DOUBLE COMPLEX array, dimension(LDVSR,N) If JOBVSR
= 'V', VSR will contain the right Schur vectors.
Not referenced if JOBVSR = 'N'.
LDVSR (input)
The leading dimension of the matrix VSR. LDVSR >=
1, and if JOBVSR = 'V', LDVSR >= N.
WORK (workspace)
DOUBLE COMPLEX array, dimension(LWORK) On exit, if
INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input)
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.
RWORK (workspace)
DOUBLE PRECISION array, dimension(8*N)
BWORK (workspace)
LOGICAL array, dimension(N) Not referenced if SORT
= 'N'.
INFO (output)
= 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 eigen-
values in the Generalized Schur form no longer
satisfy DELZTG=.TRUE. This could also be caused
due to scaling. =N+3: reordering falied in
CTGSEN.