dgges(3P) Sun Performance Library dgges(3P)NAMEdgges - compute for a pair of N-by-N real nonsymmetric matrices (A,B),
SYNOPSIS
SUBROUTINE DGGES(JOBVSL, JOBVSR, SORT, DELCTG, 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 DELCTG
LOGICAL BWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), ALPHAR(*), ALPHAI(*), BETA(*),
VSL(LDVSL,*), VSR(LDVSR,*), WORK(*)
SUBROUTINE DGGES_64(JOBVSL, JOBVSR, SORT, DELCTG, 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 DELCTG
LOGICAL*8 BWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), ALPHAR(*), ALPHAI(*), BETA(*),
VSL(LDVSL,*), VSR(LDVSR,*), WORK(*)
F95 INTERFACE
SUBROUTINE GGES(JOBVSL, JOBVSR, SORT, DELCTG, [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 :: DELCTG
LOGICAL, DIMENSION(:) :: BWORK
REAL(8), DIMENSION(:) :: ALPHAR, ALPHAI, BETA, WORK
REAL(8), DIMENSION(:,:) :: A, B, VSL, VSR
SUBROUTINE GGES_64(JOBVSL, JOBVSR, SORT, DELCTG, [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) :: DELCTG
LOGICAL(8), DIMENSION(:) :: BWORK
REAL(8), DIMENSION(:) :: ALPHAR, ALPHAI, BETA, WORK
REAL(8), DIMENSION(:,:) :: A, B, VSL, VSR
C INTERFACE
#include <sunperf.h>
void dgges(char jobvsl, char jobvsr, char sort, int(*delctg)(dou‐
ble,double,double), 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, int *info);
void dgges_64(char jobvsl, char jobvsr, char sort, long(*delctg)(dou‐
ble,double,double), 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, long
*info);
PURPOSEdgges 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 corre‐
sponding left and right eigenspaces (deflating subspaces).
(If only the generalized eigenvalues are needed, use the driver DGGEV
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 inter‐
pretation 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 trian‐
gular 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 "standard‐
ized" 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 com‐
plex conjugate pair of generalized eigenvalues.
ARGUMENTS
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 DELCTG);
DELCTG (input)
LOGICAL FUNCTION of three DOUBLE PRECISION arguments DELCTG
must be declared EXTERNAL in the calling subroutine. If SORT
= 'N', DELCTG is not referenced. If SORT = 'S', DELCTG is
used to select eigenvalues to sort to the top left of the
Schur form. An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j) is
selected if DELCTG(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 ei‐
genvalue may no longer satisfy DELCTG(ALPHAR(j),ALPHAI(j),
BETA(j)) = .TRUE. after ordering. INFO is to be set to N+2 in
this case.
N (input) The order of the matrices A, B, VSL, and VSR. N >= 0.
A (input/output)
DOUBLE PRECISION array, dimension(LDA,A) 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)
DOUBLE PRECISION array, dimension (LDB, N) On entry, the sec‐
ond 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 ei‐
genvalues (after sorting) for which DELCTG is true. (Complex
conjugate pairs for which DELCTG is true for either eigenval‐
ue count as 2.)
ALPHAR (output)
On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
be the generalized eigenvalues. 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 posi‐
tive, 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 usu‐
ally comparable with norm(A) in magnitude, and BETA always
less than and usually comparable with norm(B).
ALPHAI (output)
See the description for ALPHAR.
BETA (output)
See the description for ALPHAR.
VSL (output)
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 (output)
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.
WORK (workspace)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input)
The dimension of the array WORK. LWORK >= max(8*N,6*N+16).
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.
BWORK (workspace)
dimension(N) Not referenced if SORT = 'N'.
INFO (output)
= 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 DHGEQZ.
=N+2: after reordering, roundoff changed values of some com‐
plex eigenvalues so that leading eigenvalues in the General‐
ized Schur form no longer satisfy DELCTG=.TRUE. This could
also be caused due to scaling. =N+3: reordering failed in
DTGSEN.
6 Mar 2009 dgges(3P)
