Contents
dgges - compute for a pair of N-by-N real nonsymmetric
matrices (A,B),
SUBROUTINE DGGES(JOBVSL, JOBVSR, SORT, DELCTG, N, A, LDA, B, LDB,
SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, WORK, LWORK,
BWORK, INFO)
CHARACTER * 1 JOBVSL, JOBVSR, SORT
INTEGER N, LDA, LDB, SDIM, LDVSL, LDVSR, LWORK, INFO
LOGICAL DELCTG
LOGICAL BWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), ALPHAR(*), ALPHAI(*),
BETA(*), VSL(LDVSL,*), VSR(LDVSR,*), WORK(*)
SUBROUTINE DGGES_64(JOBVSL, JOBVSR, SORT, DELCTG, N, A, LDA, B, LDB,
SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, WORK, LWORK,
BWORK, INFO)
CHARACTER * 1 JOBVSL, JOBVSR, SORT
INTEGER*8 N, LDA, LDB, SDIM, LDVSL, LDVSR, LWORK, INFO
LOGICAL*8 DELCTG
LOGICAL*8 BWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), ALPHAR(*), ALPHAI(*),
BETA(*), VSL(LDVSL,*), VSR(LDVSR,*), WORK(*)
F95 INTERFACE
SUBROUTINE GGES(JOBVSL, JOBVSR, SORT, [DELCTG], [N], A, [LDA], B, [LDB],
SDIM, ALPHAR, ALPHAI, BETA, VSL, [LDVSL], VSR, [LDVSR], [WORK],
[LWORK], [BWORK], [INFO])
CHARACTER(LEN=1) :: JOBVSL, JOBVSR, SORT
INTEGER :: N, LDA, LDB, SDIM, LDVSL, LDVSR, LWORK, INFO
LOGICAL :: DELCTG
LOGICAL, DIMENSION(:) :: BWORK
REAL(8), DIMENSION(:) :: ALPHAR, ALPHAI, BETA, WORK
REAL(8), DIMENSION(:,:) :: A, B, VSL, VSR
SUBROUTINE GGES_64(JOBVSL, JOBVSR, SORT, [DELCTG], [N], A, [LDA], B,
[LDB], SDIM, ALPHAR, ALPHAI, BETA, VSL, [LDVSL], VSR, [LDVSR],
[WORK], [LWORK], [BWORK], [INFO])
CHARACTER(LEN=1) :: JOBVSL, JOBVSR, SORT
INTEGER(8) :: N, LDA, LDB, SDIM, LDVSL, LDVSR, LWORK, INFO
LOGICAL(8) :: DELCTG
LOGICAL(8), DIMENSION(:) :: BWORK
REAL(8), DIMENSION(:) :: ALPHAR, ALPHAI, BETA, WORK
REAL(8), DIMENSION(:,:) :: A, B, VSL, VSR
C INTERFACE
#include <sunperf.h>
void dgges(char jobvsl, char jobvsr, char sort,
int(*delctg)(double,double,double), int n, double
*a, int lda, double *b, int ldb, int *sdim, double
*alphar, double *alphai, double *beta, double
*vsl, int ldvsl, double *vsr, int ldvsr, int
*info);
void dgges_64(char jobvsl, char jobvsr, char sort,
long(*delctg)(double,double,double), long n, dou-
ble *a, long lda, double *b, long ldb, long *sdim,
double *alphar, double *alphai, double *beta, dou-
ble *vsl, long ldvsl, double *vsr, long ldvsr,
long *info);
dgges computes for a pair of N-by-N real nonsymmetric
matrices (A,B), the generalized eigenvalues, the generalized
real Schur form (S,T), optionally, the left and/or right
matrices of Schur vectors (VSL and VSR). This gives the gen-
eralized Schur factorization
(A,B) = ( (VSL)*S*(VSR)**T, (VSL)*T*(VSR)**T )
Optionally, it also orders the eigenvalues so that a
selected cluster of eigenvalues appears in the leading diag-
onal blocks of the upper quasi-triangular matrix S and the
upper triangular matrix T.The leading columns of VSL and VSR
then form an orthonormal basis for the corresponding left
and right eigenspaces (deflating subspaces).
(If only the generalized eigenvalues are needed, use the
driver SGGEV 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 or both being zero.
A pair of matrices (S,T) is in generalized real Schur form
if T is upper triangular with non-negative diagonal and S is
block upper triangular with 1-by-1 and 2-by-2 blocks. 1-
by-1 blocks correspond to real generalized eigenvalues,
while 2-by-2 blocks of S will be "standardized" by making
the corresponding elements of T have the form:
[ a 0 ]
[ 0 b ]
and the pair of corresponding 2-by-2 blocks in S and T will
have a complex conjugate pair of generalized eigenvalues.
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 three DOUBLE PRECISION argu-
ments DELCTG must be declared EXTERNAL in the cal-
ling 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
(ALPHAR(j)+ALPHAI(j))/BETA(j) is selected if
DELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e.
if either one of a complex conjugate pair of
eigenvalues is selected, then both complex eigen-
values are selected.
Note that in the ill-conditioned case, a selected
complex eigenvalue may no longer satisfy
DELCTG(ALPHAR(j),ALPHAI(j), BETA(j)) = .TRUE.
after ordering. INFO is to be set to N+2 in this
case.
N (input) The order of the matrices A, B, VSL, and VSR. N
>= 0.
A (input/output)
DOUBLE PRECISION array, dimension(LDA,A) 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 PRECISION 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. (Complex conjugate pairs for
which DELCTG is true for either eigenvalue count
as 2.)
ALPHAR (output)
On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j),
j=1,...,N, will be the generalized eigenvalues.
ALPHAR(j) + ALPHAI(j)*i, and BETA(j),j=1,...,N
are the diagonals of the complex Schur form (S,T)
that would result if the 2-by-2 diagonal blocks of
the real Schur form of (A,B) were further reduced
to triangular form using 2-by-2 complex unitary
transformations. If ALPHAI(j) is zero, then the
j-th eigenvalue is real; if positive, then the j-
th and (j+1)-st eigenvalues are a complex conju-
gate pair, with ALPHAI(j+1) negative.
Note: the quotients ALPHAR(j)/BETA(j) and
ALPHAI(j)/BETA(j) may easily over- or underflow,
and BETA(j) may even be zero. Thus, the user
should avoid naively computing the ratio. How-
ever, ALPHAR and ALPHAI will be always less than
and usually comparable with norm(A) in magnitude,
and BETA always less than and usually comparable
with norm(B).
ALPHAI (output)
See the description for ALPHAR.
BETA (output)
See the description for ALPHAR.
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.
WORK (workspace)
On exit, if INFO = 0, WORK(1) returns the optimal
LWORK.
LWORK (input)
The dimension of the array WORK. LWORK >= 8*N+16.
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.
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 ALPHAR(j), ALPHAI(j), and
BETA(j) should be correct for j=INFO+1,...,N. >
N: =N+1: other than QZ iteration failed in
SHGEQZ.
=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
STGSEN.