sgges - compute for a pair of N-by-N real nonsymmetric matrices (A,B),
SUBROUTINE SGGES( JOBVSL, JOBVSR, SORT, SELCTG, 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 SELCTG LOGICAL BWORK(*) REAL A(LDA,*), B(LDB,*), ALPHAR(*), ALPHAI(*), BETA(*), VSL(LDVSL,*), VSR(LDVSR,*), WORK(*)
SUBROUTINE SGGES_64( JOBVSL, JOBVSR, SORT, SELCTG, 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 SELCTG LOGICAL*8 BWORK(*) REAL A(LDA,*), B(LDB,*), ALPHAR(*), ALPHAI(*), BETA(*), VSL(LDVSL,*), VSR(LDVSR,*), WORK(*)
SUBROUTINE GGES( JOBVSL, JOBVSR, SORT, SELCTG, [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 :: SELCTG LOGICAL, DIMENSION(:) :: BWORK REAL, DIMENSION(:) :: ALPHAR, ALPHAI, BETA, WORK REAL, DIMENSION(:,:) :: A, B, VSL, VSR
SUBROUTINE GGES_64( JOBVSL, JOBVSR, SORT, SELCTG, [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) :: SELCTG LOGICAL(8), DIMENSION(:) :: BWORK REAL, DIMENSION(:) :: ALPHAR, ALPHAI, BETA, WORK REAL, DIMENSION(:,:) :: A, B, VSL, VSR
#include <sunperf.h>
void sgges(char jobvsl, char jobvsr, char sort, logical(*selctg)(float,float,float), 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, int *info);
void sgges_64(char jobvsl, char jobvsr, char sort, logical(*selctg)(float,float,float), 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, long *info);
sgges 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 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.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.
= '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 in the ill-conditioned case, a selected complex
eigenvalue may no longer satisfy SELCTG(ALPHAR(j),ALPHAI(j),
BETA(j)) = .TRUE. after ordering. INFO is to be set to N+2
in this case.
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).
WORK(1) returns the optimal LWORK.
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.
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.