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(*)
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
void dggesx(char jobvsl, char jobvsr, char sort, logical(*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, logical(*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);
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.
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).