cggesx(3P) Sun Performance Library cggesx(3P)NAMEcggesx - 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 * 1 JOBVSL, JOBVSR, SORT, SENSE
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 SELCTG
LOGICAL BWORK(*)
REAL RCONDE(*), RCONDV(*), RWORK(*)
SUBROUTINE CGGESX_64(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 * 1 JOBVSL, JOBVSR, SORT, SENSE
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 SELCTG
LOGICAL*8 BWORK(*)
REAL RCONDE(*), RCONDV(*), RWORK(*)
F95 INTERFACE
SUBROUTINE GGESX(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(LEN=1) :: JOBVSL, JOBVSR, SORT, SENSE
COMPLEX, DIMENSION(:) :: ALPHA, BETA, WORK
COMPLEX, DIMENSION(:,:) :: A, B, VSL, VSR
INTEGER :: N, LDA, LDB, SDIM, LDVSL, LDVSR, LWORK, LIWORK, INFO
INTEGER, DIMENSION(:) :: IWORK
LOGICAL :: SELCTG
LOGICAL, DIMENSION(:) :: BWORK
REAL, DIMENSION(:) :: RCONDE, RCONDV, RWORK
SUBROUTINE GGESX_64(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(LEN=1) :: JOBVSL, JOBVSR, SORT, SENSE
COMPLEX, DIMENSION(:) :: ALPHA, BETA, WORK
COMPLEX, DIMENSION(:,:) :: A, B, VSL, VSR
INTEGER(8) :: N, LDA, LDB, SDIM, LDVSL, LDVSR, LWORK, LIWORK, INFO
INTEGER(8), DIMENSION(:) :: IWORK
LOGICAL(8) :: SELCTG
LOGICAL(8), DIMENSION(:) :: BWORK
REAL, DIMENSION(:) :: RCONDE, RCONDV, RWORK
C INTERFACE
#include <sunperf.h>
void cggesx(char jobvsl, char jobvsr, char sort, int(*selctg)(com‐
plex,complex), char sense, int n, complex *a, int lda, com‐
plex *b, int ldb, int *sdim, complex *alpha, complex *beta,
complex *vsl, int ldvsl, complex *vsr, int ldvsr, float
*rconde, float *rcondv, int *info);
void cggesx_64(char jobvsl, char jobvsr, char sort, long(*selctg)(com‐
plex,complex), char sense, long n, complex *a, long lda, com‐
plex *b, long ldb, long *sdim, complex *alpha, complex *beta,
complex *vsl, long ldvsl, complex *vsr, long ldvsr, float
*rconde, float *rcondv, long *info);
PURPOSEcggesx 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 tri‐
angular matrix S and the upper triangular matrix T; computes a recipro‐
cal 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 sub‐
spaces).
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 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)
= '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 SELCTG).
SELCTG (input)
LOGICAL FUNCTION of two COMPLEX arguments SELCTG must be
declared EXTERNAL in the calling subroutine. If SORT = 'N',
SELCTG is not referenced. 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 (espe‐
cially 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 eigenvalues 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)
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)
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 SELCTG 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 diago‐
nals of the complex 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. 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)
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)
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 referenced 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 sub‐
spaces. 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 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 CHGEQZ
=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 SELCTG=.TRUE. This could
also be caused due to scaling. =N+3: reordering failed in
CTGSEN.
6 Mar 2009 cggesx(3P)
