sgges(3P) Sun Performance Library sgges(3P)NAMEsgges - compute for a pair of N-by-N real nonsymmetric matrices (A,B),
SYNOPSIS
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(*)
F95 INTERFACE
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
C INTERFACE
#include <sunperf.h>
void sgges(char jobvsl, char jobvsr, char sort,
int(*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,
long(*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);
PURPOSEsgges 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 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 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 SELCTG);
SELCTG (input)
LOGICAL FUNCTION of three REAL arguments SELCTG must be
declared EXTERNAL in the calling subroutine. If SORT = 'N',
SELCTG is not referenced. If SORT = 'S', SELCTG 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
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 ei‐
genvalue 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.
N (input) The order of the matrices A, B, VSL, and VSR. N >= 0.
A (input/output)
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)
On entry, the second 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 SELCTG is true. (Complex
conjugate pairs for which SELCTG 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 SHGEQZ.
=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 SELCTG=.TRUE. This could
also be caused due to scaling. =N+3: reordering failed in
STGSEN.
6 Mar 2009 sgges(3P)
