SGGES - 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 JOBVSL, JOBVSR, SORT
INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N,
SDIM
LOGICAL BWORK( * )
REAL A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
B( LDB, * ), BETA( * ), VSL( LDVSL, * ),
VSR( LDVSR, * ), WORK( * )
LOGICAL SELCTG
EXTERNAL SELCTG
PURPOSE
SGGES computes for a pair of N-by-N real nonsymmetric
matrices (A,B), the generalized eigenvalues, the general
ized 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 correspond
ing 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,
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 eigen
values.
ARGUMENTS
JOBVSL (input) CHARACTER*1
= 'N': do not compute the left Schur vectors;
= 'V': compute the left Schur vectors.
JOBVSR (input) CHARACTER*1
= 'N': do not compute the right Schur vectors;
= 'V': compute the right Schur vectors.
SORT (input) CHARACTER*1
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 refer
enced. 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 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.
N (input) INTEGER
The order of the matrices A, B, VSL, and VSR. N
>= 0.
A (input/output) REAL array, dimension (LDA, N)
On entry, the first of the pair of matrices. On
exit, A has been overwritten by its generalized
Schur form S.
LDA (input) INTEGER
The leading dimension of A. LDA >= max(1,N).
On entry, the second of the pair of matrices. On
exit, B has been overwritten by its generalized
Schur form T.
LDB (input) INTEGER
The leading dimension of B. LDB >= max(1,N).
SDIM (output) INTEGER
If SORT = 'N', SDIM = 0. If SORT = 'S', SDIM =
number of eigenvalues (after sorting) for which
SELCTG is true. (Complex conjugate pairs for
which SELCTG is true for either eigenvalue count
as 2.)
ALPHAR (output) REAL array, dimension (N)
ALPHAI (output) REAL array, dimension (N) BETA
(output) REAL array, dimension (N) 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 diago
nals 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 trian
gular form using 2-by-2 complex unitary transfor
mations. 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. How
ever, 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).
VSL (output) REAL array, dimension (LDVSL,N)
If JOBVSL = 'V', VSL will contain the left Schur
vectors. Not referenced if JOBVSL = 'N'.
LDVSL (input) INTEGER
The leading dimension of the matrix VSL. LDVSL
>=1, and if JOBVSL = 'V', LDVSL >= N.
VSR (output) REAL array, dimension (LDVSR,N)
If JOBVSR = 'V', VSR will contain the right Schur
vectors. Not referenced if JOBVSR = 'N'.
LDVSR (input) INTEGER
The leading dimension of the matrix VSR. LDVSR >=
1, and if JOBVSR = 'V', LDVSR >= N.
On exit, if INFO = 0, WORK(1) returns the optimal
LWORK.
LWORK (input) INTEGER
The dimension of the array WORK. LWORK >= 8*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) LOGICAL array, dimension (N)
Not referenced if SORT = 'N'.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an ille
gal 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 eigenval
ues in the Generalized Schur form no longer sat
isfy SELCTG=.TRUE. This could also be caused due
to scaling. =N+3: reordering failed in STGSEN.
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