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(*)
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
#include <sunperf.h>
void sggesx(char jobvsl, char jobvsr, char sort, logical(*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, logical(*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 diagonal 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.
= '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 SELCTG).
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.
= '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'.
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 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.
However, 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).
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 numbers for the selected deflating
subspaces.
Not referenced if SENSE = 'N' or 'E'.
WORK(1) returns the optimal LWORK.
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 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.