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(*)
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
#include <sunperf.h>
void zggesx(char jobvsl, char jobvsr, char sort, logical(*delctg)(COMPLEX*16,COMPLEX*16), char sense, int n, doublecomplex *a, int lda, doublecomplex *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, logical(*delctg)(COMPLEX*16,COMPLEX*16), char sense, long n, doublecomplex *a, long lda, doublecomplex *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 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 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.
= 'N': do not compute the left Schur vectors;
= 'V': compute the left Schur vectors.
= 'N': do not compute the right Schur vectors;
= 'V': compute the right Schur vectors.
= 'S': Eigenvalues are ordered (see DELCTG).
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).
= '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'.
ALPHA(j)
and BETA(j),j =1,...,N are
the diagonals 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.
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).
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(1)
and RCONDV(2)
contain the
reciprocal condition number for the selected deflating
subspaces.
Not referenced if SENSE = 'N' or 'E'.
WORK(1)
returns the optimal LWORK.
dimension(8*N)
Real workspace.
IWORK(1)
returns the optimal LIWORK.
dimension(N)
Not referenced if SORT = 'N'.
= 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 complex eigenvalues so that leading eigenvalues in the Generalized Schur form no longer satisfy DELCTG =.TRUE. This could also be caused due to scaling. =N+3: reordering failed in CTGSEN.