SGGESX - compute for a pair of N-by-N real nonsymmetric
matrices (A,B), the generalized eigenvalues, the real
Schur form (S,T), and,
SYNOPSIS
SUBROUTINE SGGESX( JOBVSL, JOBVSR, SORT, SELCTG, SENSE, N,
A, LDA, B, LDB, SDIM, ALPHAR, ALPHAI,
BETA, VSL, LDVSL, VSR, LDVSR, RCONDE,
RCONDV, WORK, LWORK, IWORK, LIWORK,
BWORK, INFO )
CHARACTER JOBVSL, JOBVSR, SENSE, SORT
INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LIWORK,
LWORK, N, SDIM
LOGICAL BWORK( * )
INTEGER IWORK( * )
REAL A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
B( LDB, * ), BETA( * ), RCONDE( 2 ),
RCONDV( 2 ), VSL( LDVSL, * ), VSR(
LDVSR, * ), WORK( * )
LOGICAL SELCTG
EXTERNAL SELCTG
PURPOSE
SGGESX computes for a pair of N-by-N real nonsymmetric
matrices (A,B), the generalized eigenvalues, the real
Schur form (S,T), and, 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; computes a reciprocal con
dition 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 corre
sponding 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
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 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 a selected
complex eigenvalue may no longer satisfy
SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) = .TRUE. after
ordering, since ordering may change the value of
complex eigenvalues (especially if the eigenvalue
is ill-conditioned), in this case INFO is set to
N+3.
SENSE (input) CHARACTER
Determines which reciprocal condition numbers are
computed. = 'N' : None are computed;
= 'E' : Computed for average of selected eigenval
ues only;
only;
= 'B' : Computed for both. If SENSE = 'E', 'V',
or 'B', SORT must equal 'S'.
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).
B (input/output) REAL array, dimension (LDB, 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
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.
RCONDE (output) REAL array, dimension ( 2 )
If SENSE = 'E' or 'B', RCONDE(1) and RCONDE(2)
contain the reciprocal condition numbers for the
average of the selected eigenvalues. Not refer
enced if SENSE = 'N' or 'V'.
RCONDV (output) REAL array, dimension ( 2 )
If SENSE = 'V' or 'B', RCONDV(1) and RCONDV(2)
contain the reciprocal condition numbers for the
selected deflating subspaces. Not referenced if
SENSE = 'N' or 'E'.
WORK (workspace/output) REAL array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal
LWORK.
LWORK (input) INTEGER
The dimension of the array WORK. LWORK >=
8*(N+1)+16. If SENSE = 'E', 'V', or 'B', LWORK >=
MAX( 8*(N+1)+16, 2*SDIM*(N-SDIM) ).
IWORK (workspace) INTEGER array, dimension (LIWORK)
Not referenced if SENSE = 'N'.
LIWORK (input) INTEGER
The dimension of the array WORK. LIWORK >= N+6.
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
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.
Further details ===============
An approximate (asymptotic) bound on the average
absolute error of the selected eigenvalues is
EPS * norm((A, B)) / RCONDE( 1 ).
An approximate (asymptotic) bound on the maximum
angular error in the computed deflating subspaces
is
EPS * norm((A, B)) / RCONDV( 2 ).
See LAPACK User's Guide, section 4.11 for more
information.
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