dgegs - routine is deprecated and has been replaced by routine SGGES
SUBROUTINE DGEGS( JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHAR, ALPHAI, * BETA, VSL, LDVSL, VSR, LDVSR, WORK, LDWORK, INFO) CHARACTER * 1 JOBVSL, JOBVSR INTEGER N, LDA, LDB, LDVSL, LDVSR, LDWORK, INFO DOUBLE PRECISION A(LDA,*), B(LDB,*), ALPHAR(*), ALPHAI(*), BETA(*), VSL(LDVSL,*), VSR(LDVSR,*), WORK(*)
SUBROUTINE DGEGS_64( JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHAR, * ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, WORK, LDWORK, INFO) CHARACTER * 1 JOBVSL, JOBVSR INTEGER*8 N, LDA, LDB, LDVSL, LDVSR, LDWORK, INFO DOUBLE PRECISION A(LDA,*), B(LDB,*), ALPHAR(*), ALPHAI(*), BETA(*), VSL(LDVSL,*), VSR(LDVSR,*), WORK(*)
SUBROUTINE GEGS( JOBVSL, JOBVSR, [N], A, [LDA], B, [LDB], ALPHAR, * ALPHAI, BETA, VSL, [LDVSL], VSR, [LDVSR], [WORK], [LDWORK], [INFO]) CHARACTER(LEN=1) :: JOBVSL, JOBVSR INTEGER :: N, LDA, LDB, LDVSL, LDVSR, LDWORK, INFO REAL(8), DIMENSION(:) :: ALPHAR, ALPHAI, BETA, WORK REAL(8), DIMENSION(:,:) :: A, B, VSL, VSR
SUBROUTINE GEGS_64( JOBVSL, JOBVSR, [N], A, [LDA], B, [LDB], ALPHAR, * ALPHAI, BETA, VSL, [LDVSL], VSR, [LDVSR], [WORK], [LDWORK], [INFO]) CHARACTER(LEN=1) :: JOBVSL, JOBVSR INTEGER(8) :: N, LDA, LDB, LDVSL, LDVSR, LDWORK, INFO REAL(8), DIMENSION(:) :: ALPHAR, ALPHAI, BETA, WORK REAL(8), DIMENSION(:,:) :: A, B, VSL, VSR
#include <sunperf.h>
void dgegs(char jobvsl, char jobvsr, int n, double *a, int lda, double *b, int ldb, double *alphar, double *alphai, double *beta, double *vsl, int ldvsl, double *vsr, int ldvsr, int *info);
void dgegs_64(char jobvsl, char jobvsr, long n, double *a, long lda, double *b, long ldb, double *alphar, double *alphai, double *beta, double *vsl, long ldvsl, double *vsr, long ldvsr, long *info);
dgegs routine is deprecated and has been replaced by routine SGGES.
SGEGS computes for a pair of N-by-N real nonsymmetric matrices A, B: the generalized eigenvalues (alphar +/- alphai*i, beta), the real Schur form (A, B), and optionally left and/or right Schur vectors (VSL and VSR).
(If only the generalized eigenvalues are needed, use the driver SGEGV instead.)
A generalized eigenvalue for a pair of matrices (A,B) is, roughly speaking, 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, and even for both being zero. A good beginning reference is the book, ``Matrix Computations'', by G. Golub & C. van Loan (Johns Hopkins U. Press)
The (generalized) Schur form of a pair of matrices is the result of multiplying both matrices on the left by one orthogonal matrix and both on the right by another orthogonal matrix, these two orthogonal matrices being chosen so as to bring the pair of matrices into (real) Schur form.
A pair of matrices A, B is in generalized real Schur form if B is upper triangular with non-negative diagonal and A 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 A will be ``standardized'' by making the corresponding elements of B have the form:
[ a 0 ]
[ 0 b ]
and the pair of corresponding 2-by-2 blocks in A and B will have a complex conjugate pair of generalized eigenvalues.
The left and right Schur vectors are the columns of VSL and VSR, respectively, where VSL and VSR are the orthogonal matrices which reduce A and B to Schur form:
Schur form of (A,B) = ( (VSL)**T A (VSR), (VSL)**T B (VSR) )
= '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.
ALPHAR(j) + ALPHAI(j)*i,
j =1,...,N and BETA(j),j =1,...,N are the diagonals of the
complex Schur form (A,B) 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
alpha/beta. 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 LDWORK.
If LDWORK = -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 LDWORK is issued by XERBLA.
= 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: errors that usually indicate LAPACK problems:
=N+1: error return from SGGBAL
=N+2: error return from SGEQRF
=N+3: error return from SORMQR
=N+4: error return from SORGQR
=N+5: error return from SGGHRD
=N+6: error return from SHGEQZ (other than failed iteration) =N+7: error return from SGGBAK (computing VSL)
=N+8: error return from SGGBAK (computing VSR)
=N+9: error return from SLASCL (various places)