Contents
zggesx - compute for a pair of N-by-N complex nonsymmetric
matrices (A,B), the generalized eigenvalues, the complex
Schur form (S,T),
SUBROUTINE ZGGESX(JOBVSL, JOBVSR, SORT, DELCTG, SENSE, N, A, LDA, B,
LDB, SDIM, ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, RCONDE, RCONDV,
WORK, LWORK, RWORK, IWORK, LIWORK, BWORK, INFO)
CHARACTER * 1 JOBVSL, JOBVSR, SORT, SENSE
DOUBLE COMPLEX A(LDA,*), B(LDB,*), ALPHA(*), BETA(*),
VSL(LDVSL,*), VSR(LDVSR,*), WORK(*)
INTEGER N, LDA, LDB, SDIM, LDVSL, LDVSR, LWORK, LIWORK, INFO
INTEGER IWORK(*)
LOGICAL DELCTG
LOGICAL BWORK(*)
DOUBLE PRECISION RCONDE(*), RCONDV(*), RWORK(*)
SUBROUTINE ZGGESX_64(JOBVSL, JOBVSR, SORT, DELCTG, SENSE, N, A, LDA,
B, LDB, SDIM, ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, RCONDE,
RCONDV, WORK, LWORK, RWORK, IWORK, LIWORK, BWORK, INFO)
CHARACTER * 1 JOBVSL, JOBVSR, SORT, SENSE
DOUBLE COMPLEX A(LDA,*), B(LDB,*), ALPHA(*), BETA(*),
VSL(LDVSL,*), VSR(LDVSR,*), WORK(*)
INTEGER*8 N, LDA, LDB, SDIM, LDVSL, LDVSR, LWORK, LIWORK,
INFO
INTEGER*8 IWORK(*)
LOGICAL*8 DELCTG
LOGICAL*8 BWORK(*)
DOUBLE PRECISION RCONDE(*), RCONDV(*), RWORK(*)
F95 INTERFACE
SUBROUTINE GGESX(JOBVSL, JOBVSR, SORT, [DELCTG], SENSE, [N], A, [LDA],
B, [LDB], SDIM, ALPHA, BETA, VSL, [LDVSL], VSR, [LDVSR], RCONDE,
RCONDV, [WORK], [LWORK], [RWORK], [IWORK], [LIWORK], [BWORK],
[INFO])
CHARACTER(LEN=1) :: JOBVSL, JOBVSR, SORT, SENSE
COMPLEX(8), DIMENSION(:) :: ALPHA, BETA, WORK
COMPLEX(8), DIMENSION(:,:) :: A, B, VSL, VSR
INTEGER :: N, LDA, LDB, SDIM, LDVSL, LDVSR, LWORK, LIWORK,
INFO
INTEGER, DIMENSION(:) :: IWORK
LOGICAL :: DELCTG
LOGICAL, DIMENSION(:) :: BWORK
REAL(8), DIMENSION(:) :: RCONDE, RCONDV, RWORK
SUBROUTINE GGESX_64(JOBVSL, JOBVSR, SORT, [DELCTG], SENSE, [N], A, [LDA],
B, [LDB], SDIM, ALPHA, BETA, VSL, [LDVSL], VSR, [LDVSR], RCONDE,
RCONDV, [WORK], [LWORK], [RWORK], [IWORK], [LIWORK], [BWORK],
[INFO])
CHARACTER(LEN=1) :: JOBVSL, JOBVSR, SORT, SENSE
COMPLEX(8), DIMENSION(:) :: ALPHA, BETA, WORK
COMPLEX(8), DIMENSION(:,:) :: A, B, VSL, VSR
INTEGER(8) :: N, LDA, LDB, SDIM, LDVSL, LDVSR, LWORK,
LIWORK, INFO
INTEGER(8), DIMENSION(:) :: IWORK
LOGICAL(8) :: DELCTG
LOGICAL(8), DIMENSION(:) :: BWORK
REAL(8), DIMENSION(:) :: RCONDE, RCONDV, RWORK
C INTERFACE
#include <sunperf.h>
void zggesx(char jobvsl, char jobvsr, char sort,
int(*delctg)(doublecomplex,doublecomplex), char
sense, int n, doublecomplex *a, int lda, doub-
lecomplex *b, int ldb, int *sdim, doublecomplex
*alpha, doublecomplex *beta, doublecomplex *vsl,
int ldvsl, doublecomplex *vsr, int ldvsr, double
*rconde, double *rcondv, int *info);
void zggesx_64(char jobvsl, char jobvsr, char sort,
long(*delctg)(doublecomplex,doublecomplex), char
sense, long n, doublecomplex *a, long lda, doub-
lecomplex *b, long ldb, long *sdim, doublecomplex
*alpha, doublecomplex *beta, doublecomplex *vsl,
long ldvsl, doublecomplex *vsr, long ldvsr, double
*rconde, double *rcondv, long *info);
zggesx 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 diag-
onal 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 corresponding left
and right eigenspaces (deflating subspaces).
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 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.
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 DELCTG).
DELCTG (input)
LOGICAL FUNCTION of two DOUBLE COMPLEX arguments
DELCTG must be declared EXTERNAL in the calling
subroutine. If SORT = 'N', DELCTG is not refer-
enced. If SORT = 'S', DELCTG 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 DELCTG(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)
Determines which reciprocal condition numbers are
computed. = 'N' : None are computed;
= 'E' : Computed for average of selected eigen-
values only;
= 'V' : Computed for selected deflating subspaces
only;
= 'B' : Computed for both. If SENSE = 'E', 'V',
or 'B', SORT must equal 'S'.
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
DELCTG is true.
ALPHA (output)
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-
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)
If JOBVSL = 'V', VSL will contain the left Schur
vectors. Not referenced if JOBVSL = 'N'.
LDVSL (input)
The leading dimension of the matrix VSL. LDVSL
>=1, and if JOBVSL = 'V', LDVSL >= N.
VSR (input)
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.
RCONDE (output)
If SENSE = 'E' or 'B', RCONDE(1) and RCONDE(2)
contain the reciprocal condition numbers for the
average of the selected eigenvalues. Not refer-
enced if SENSE = 'N' or 'V'.
RCONDV (output)
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)
On exit, if INFO = 0, WORK(1) returns the optimal
LWORK.
LWORK (input)
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)
dimension(8*N) Real workspace.
IWORK (workspace/output)
Not referenced if SENSE = 'N'. On exit, if INFO =
0, IWORK(1) returns the optimal LIWORK.
LIWORK (input)
The dimension of the array WORK. LIWORK >= N+2.
BWORK (workspace)
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 DELCTG=.TRUE. This could also be caused
due to scaling. =N+3: reordering failed in
CTGSEN.