Contents
dggesx - compute for a pair of N-by-N real nonsymmetric
matrices (A,B), the generalized eigenvalues, the real Schur
form (S,T), and,
SUBROUTINE DGGESX(JOBVSL, JOBVSR, SORT, DELCTG, SENSE, N, A, LDA, B,
LDB, SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, RCONDE,
RCONDV, WORK, LWORK, IWORK, LIWORK, BWORK, INFO)
CHARACTER * 1 JOBVSL, JOBVSR, SORT, SENSE
INTEGER N, LDA, LDB, SDIM, LDVSL, LDVSR, LWORK, LIWORK, INFO
INTEGER IWORK(*)
LOGICAL DELCTG
LOGICAL BWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), ALPHAR(*), ALPHAI(*),
BETA(*), VSL(LDVSL,*), VSR(LDVSR,*), RCONDE(*), RCONDV(*),
WORK(*)
SUBROUTINE DGGESX_64(JOBVSL, JOBVSR, SORT, DELCTG, SENSE, N, A, LDA,
B, LDB, SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR,
RCONDE, RCONDV, WORK, LWORK, IWORK, LIWORK, BWORK, INFO)
CHARACTER * 1 JOBVSL, JOBVSR, SORT, SENSE
INTEGER*8 N, LDA, LDB, SDIM, LDVSL, LDVSR, LWORK, LIWORK,
INFO
INTEGER*8 IWORK(*)
LOGICAL*8 DELCTG
LOGICAL*8 BWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), ALPHAR(*), ALPHAI(*),
BETA(*), VSL(LDVSL,*), VSR(LDVSR,*), RCONDE(*), RCONDV(*),
WORK(*)
F95 INTERFACE
SUBROUTINE GGESX(JOBVSL, JOBVSR, SORT, [DELCTG], SENSE, [N], A, [LDA],
B, [LDB], SDIM, ALPHAR, ALPHAI, BETA, VSL, [LDVSL], VSR, [LDVSR],
RCONDE, RCONDV, [WORK], [LWORK], [IWORK], [LIWORK], [BWORK],
[INFO])
CHARACTER(LEN=1) :: JOBVSL, JOBVSR, SORT, SENSE
INTEGER :: N, LDA, LDB, SDIM, LDVSL, LDVSR, LWORK, LIWORK,
INFO
INTEGER, DIMENSION(:) :: IWORK
LOGICAL :: DELCTG
LOGICAL, DIMENSION(:) :: BWORK
REAL(8), DIMENSION(:) :: ALPHAR, ALPHAI, BETA, RCONDE,
RCONDV, WORK
REAL(8), DIMENSION(:,:) :: A, B, VSL, VSR
SUBROUTINE GGESX_64(JOBVSL, JOBVSR, SORT, [DELCTG], SENSE, [N], A, [LDA],
B, [LDB], SDIM, ALPHAR, ALPHAI, BETA, VSL, [LDVSL], VSR, [LDVSR],
RCONDE, RCONDV, [WORK], [LWORK], [IWORK], [LIWORK], [BWORK],
[INFO])
CHARACTER(LEN=1) :: JOBVSL, JOBVSR, SORT, SENSE
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(:) :: ALPHAR, ALPHAI, BETA, RCONDE,
RCONDV, WORK
REAL(8), DIMENSION(:,:) :: A, B, VSL, VSR
C INTERFACE
#include <sunperf.h>
void dggesx(char jobvsl, char jobvsr, char sort,
int(*delctg)(double,double,double), char sense,
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, double *rconde, double *rcondv, int *info);
void dggesx_64(char jobvsl, char jobvsr, char sort,
long(*delctg)(double,double,double), char sense,
long n, double *a, long lda, double *b, long ldb,
long *sdim, double *alphar, double *alphai, double
*beta, double *vsl, long ldvsl, double *vsr, long
ldvsr, double *rconde, double *rcondv, long
*info);
dggesx computes for a pair of N-by-N real nonsymmetric
matrices (A,B), the generalized eigenvalues, the real 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)**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; 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 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 a selected complex
eigenvalue may no longer satisfy
DELCTG(ALPHAR(j),ALPHAI(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.
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 PRECISION 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 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)
DOUBLE PRECISION array, dimension(N) 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 diago-
nals 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 tri-
angular 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)
DOUBLE PRECISION array, dimension(N) See the
description for ALPHAR.
BETA (output)
DOUBLE PRECISION arary, dimension(N) See the
description for ALPHAR.
VSL (input)
DOUBLE PRECISION array, dimension(LDVSL,N) If
JOBVSL = 'V', VSL will contain the left Schur vec-
tors. Not referenced if JOBVSL = 'N'.
LDVSL (input)
The leading dimension of the matrix VSL. LDVSL
>=1, and if JOBVSL = 'V', LDVSL >= N.
VSR (input)
DOUBLE PRECISION array, dimension(LDVSR,N) 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 numbers for the
selected deflating subspaces. Not referenced if
SENSE = 'N' or 'E'.
WORK (workspace)
DOUBLE PRECISION array, dimension(LWORK) On exit,
if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input)
The dimension of the array WORK. LWORK >=
8*(N+1)+16. If SENSE = 'E', 'V', or 'B', LWORK >=
MAX( 8*(N+1)+16, 2*SDIM*(N-SDIM) ).
IWORK (workspace)
INTEGER array, dimension(LIWORK) Not referenced if
SENSE = 'N'.
LIWORK (input)
The dimension of the array WORK. LIWORK >= N+6.
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 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.
Further details ===============
An approximate (asymptotic) bound on the average
absolute error of the selected eigenvalues is
EPS * norm((A, B)) / RCONDE( 1 ).
An approximate (asymptotic) bound on the maximum
angular error in the computed deflating subspaces
is
EPS * norm((A, B)) / RCONDV( 2 ).
See LAPACK User's Guide, section 4.11 for more
information.