Contents
sggesx - compute for a pair of N-by-N real nonsymmetric
matrices (A,B), the generalized eigenvalues, the real Schur
form (S,T), and,
SUBROUTINE SGGESX(JOBVSL, JOBVSR, SORT, SELCTG, 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 SELCTG
LOGICAL BWORK(*)
REAL A(LDA,*), B(LDB,*), ALPHAR(*), ALPHAI(*), BETA(*),
VSL(LDVSL,*), VSR(LDVSR,*), RCONDE(*), RCONDV(*), WORK(*)
SUBROUTINE SGGESX_64(JOBVSL, JOBVSR, SORT, SELCTG, 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 SELCTG
LOGICAL*8 BWORK(*)
REAL A(LDA,*), B(LDB,*), ALPHAR(*), ALPHAI(*), BETA(*),
VSL(LDVSL,*), VSR(LDVSR,*), RCONDE(*), RCONDV(*), WORK(*)
F95 INTERFACE
SUBROUTINE GGESX(JOBVSL, JOBVSR, SORT, [SELCTG], 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 :: SELCTG
LOGICAL, DIMENSION(:) :: BWORK
REAL, DIMENSION(:) :: ALPHAR, ALPHAI, BETA, RCONDE, RCONDV,
WORK
REAL, DIMENSION(:,:) :: A, B, VSL, VSR
SUBROUTINE GGESX_64(JOBVSL, JOBVSR, SORT, [SELCTG], 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) :: SELCTG
LOGICAL(8), DIMENSION(:) :: BWORK
REAL, DIMENSION(:) :: ALPHAR, ALPHAI, BETA, RCONDE, RCONDV,
WORK
REAL, DIMENSION(:,:) :: A, B, VSL, VSR
C INTERFACE
#include <sunperf.h>
void sggesx(char jobvsl, char jobvsr, char sort,
int(*selctg)(float,float,float), char sense, int
n, float *a, int lda, float *b, int ldb, int
*sdim, float *alphar, float *alphai, float *beta,
float *vsl, int ldvsl, float *vsr, int ldvsr,
float *rconde, float *rcondv, int *info);
void sggesx_64(char jobvsl, char jobvsr, char sort,
long(*selctg)(float,float,float), char sense, long
n, float *a, long lda, float *b, long ldb, long
*sdim, float *alphar, float *alphai, float *beta,
float *vsl, long ldvsl, float *vsr, long ldvsr,
float *rconde, float *rcondv, long *info);
sggesx 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 SELCTG).
SELCTG (input)
LOGICAL FUNCTION of three REAL arguments SELCTG
must be declared EXTERNAL in the calling subrou-
tine. If SORT = 'N', SELCTG is not referenced.
If SORT = 'S', SELCTG is used to select eigen-
values to sort to the top left of the Schur form.
An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j) is
selected if SELCTG(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
eigenvalues are selected. Note that a selected
complex eigenvalue may no longer satisfy
SELCTG(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)
REAL 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)
REAL 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
SELCTG is true. (Complex conjugate pairs for
which SELCTG is true for either eigenvalue count
as 2.)
ALPHAR (output)
REAL array, dimension(N) On exit, (ALPHAR(j) +
ALPHAI(j)*i)/BETA(j), j=1,...,N, will be the gen-
eralized eigenvalues. ALPHAR(j) + ALPHAI(j)*i and
BETA(j),j=1,...,N are the diagonals of the com-
plex 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 conjugate 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)
REAL array, dimension(N) See the description for
ALPHAR.
BETA (output)
REAL array, dimension(N) See the description for
ALPHAR.
VSL (input)
REAL 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 (input)
REAL 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)
REAL 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
eigenvalues in the Generalized Schur form no
longer satisfy SELCTG=.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.