zgges(3P) Sun Performance Library zgges(3P)NAMEzgges - 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)
SYNOPSIS
SUBROUTINE ZGGES(JOBVSL, JOBVSR, SORT, DELCTG, 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 DELCTG
LOGICAL BWORK(*)
DOUBLE PRECISION RWORK(*)
SUBROUTINE ZGGES_64(JOBVSL, JOBVSR, SORT, DELCTG, 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 DELCTG
LOGICAL*8 BWORK(*)
DOUBLE PRECISION RWORK(*)
F95 INTERFACE
SUBROUTINE GGES(JOBVSL, JOBVSR, SORT, DELCTG, [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 :: DELCTG
LOGICAL, DIMENSION(:) :: BWORK
REAL(8), DIMENSION(:) :: RWORK
SUBROUTINE GGES_64(JOBVSL, JOBVSR, SORT, DELCTG, [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) :: DELCTG
LOGICAL(8), DIMENSION(:) :: BWORK
REAL(8), DIMENSION(:) :: RWORK
C INTERFACE
#include <sunperf.h>
void zgges(char jobvsl, char jobvsr, char sort, int(*delztg)(doublecom‐
plex,doublecomplex), int n, doublecomplex *a, int lda, dou‐
blecomplex *b, int ldb, int *sdim, doublecomplex *alpha, dou‐
blecomplex *beta, doublecomplex *vsl, int ldvsl, doublecom‐
plex *vsr, int ldvsr, int *info);
void zgges_64(char jobvsl, char jobvsr, char sort, long(*delztg)(dou‐
blecomplex,doublecomplex), long n, doublecomplex *a, long
lda, doublecomplex *b, long ldb, long *sdim, doublecomplex
*alpha, doublecomplex *beta, doublecomplex *vsl, long ldvsl,
doublecomplex *vsr, long ldvsr, long *info);
PURPOSEzgges 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 tri‐
angular 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 ZGGEV
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 inter‐
pretation 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.
ARGUMENTS
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 referenced. If SORT = 'S', DELCTG is used
to select eigenvalues to sort to the top left of the Schur
form. An eigenvalue ALPHA(j)/BETA(j) is selected if
DELCTG(ALPHA(j),BETA(j)) is true.
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 (espe‐
cially 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 ei‐
genvalues (after sorting) for which DELCTG is true.
ALPHA (output)
On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the general‐
ized eigenvalues. ALPHA(j), j=1,...,N and BETA(j),
j=1,...,N are the diagonals of the complex Schur form (A,B)
output by ZGGES. 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. How‐
ever, ALPHA will be always less than and usually comparable
with norm(A) in magnitude, and BETA always less than and usu‐
ally comparable with norm(B).
BETA (output)
See description of ALPHA.
VSL (output)
DOUBLE COMPLEX array, dimension(LDVSL, N) 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 (output)
DOUBLE COMPLEX array, dimension(LDVSR,N) If JOBVSR = 'V', VSR
will contain the right Schur vectors. Not referenced if JOB‐
VSR = '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 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)
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 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 ZHGEQZ
=N+2: after reordering, roundoff changed values of some com‐
plex eigenvalues so that leading eigenvalues in the General‐
ized Schur form no longer satisfy DELCTG=.TRUE. This could
also be caused due to scaling. =N+3: reordering falied in
ZTGSEN.
6 Mar 2009 zgges(3P)
