ZGGES(l) ) ZGGES(l)NAME
ZGGES - compute for a pair of N-by-N complex nonsymmetric matrices
(A,B), the generalized eigenvalues, the generalized complex Schur form
(S, T), and optionally left and/or right Schur vectors (VSL and VSR)
SYNOPSIS
SUBROUTINE ZGGES( JOBVSL, JOBVSR, SORT, DELCTG, N, A, LDA, B, LDB,
SDIM, ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, WORK,
LWORK, RWORK, BWORK, INFO )
CHARACTER JOBVSL, JOBVSR, SORT
INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N, SDIM
LOGICAL BWORK( * )
DOUBLE PRECISION RWORK( * )
COMPLEX*16 A( LDA, * ), ALPHA( * ), B( LDB, * ), BETA( * ), VSL(
LDVSL, * ), VSR( LDVSR, * ), WORK( * )
LOGICAL DELCTG
EXTERNAL DELCTG
PURPOSE
ZGGES computes for a pair of N-by-N complex nonsymmetric matrices
(A,B), the generalized eigenvalues, the generalized complex Schur form
(S, T), and optionally left and/or right Schur vectors (VSL and VSR).
This gives the generalized Schur factorization
(A,B) = ( (VSL)*S*(VSR)**H, (VSL)*T*(VSR)**H )
where (VSR)**H is the conjugate-transpose of VSR.
Optionally, it also orders the eigenvalues so that a selected cluster
of eigenvalues appears in the leading diagonal blocks of the upper tri‐
angular matrix S and the upper triangular matrix T. The leading columns
of VSL and VSR then form an unitary basis for the corresponding left
and right eigenspaces (deflating subspaces).
(If only the generalized eigenvalues are needed, use the driver ZGGEV
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, and even for both being zero.
A pair of matrices (S,T) is in generalized complex Schur form if S and
T are upper triangular and, in addition, the diagonal elements of T are
non-negative real numbers.
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 diago‐
nal of the generalized Schur form. = 'N': Eigenvalues are not
ordered;
= 'S': Eigenvalues are ordered (see DELZTG).
DELZTG (input) LOGICAL FUNCTION of two COMPLEX*16 arguments
DELZTG must be declared EXTERNAL in the calling subroutine. If
SORT = 'N', DELZTG is not referenced. If SORT = 'S', DELZTG is
used to select eigenvalues to sort to the top left of the Schur
form. An eigenvalue ALPHA(j)/BETA(j) is selected if
DELZTG(ALPHA(j),BETA(j)) is true.
Note that a selected complex eigenvalue may no longer satisfy
DELZTG(ALPHA(j),BETA(j)) = .TRUE. after ordering, since order‐
ing may change the value of complex eigenvalues (especially if
the eigenvalue is ill-conditioned), in this case INFO is set to
N+2 (See INFO below).
N (input) INTEGER
The order of the matrices A, B, VSL, and VSR. N >= 0.
A (input/output) COMPLEX*16 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) COMPLEX*16 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 ei‐
genvalues (after sorting) for which DELZTG is true.
ALPHA (output) COMPLEX*16 array, dimension (N)
BETA (output) COMPLEX*16 array, dimension (N) On exit,
ALPHA(j)/BETA(j), j=1,...,N, will be the generalized eigenval‐
ues. ALPHA(j), j=1,...,N and BETA(j), j=1,...,N are the
diagonals of the complex Schur form (A,B) output by ZGGES. The
BETA(j) will be non-negative real.
Note: the quotients ALPHA(j)/BETA(j) may easily over- or under‐
flow, and BETA(j) may even be zero. Thus, the user should
avoid naively computing the ratio alpha/beta. However, ALPHA
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) COMPLEX*16 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) COMPLEX*16 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.
WORK (workspace/output) COMPLEX*16 array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input) INTEGER
The dimension of the array WORK. LWORK >= max(1,2*N). For
good performance, LWORK must generally be larger.
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.
RWORK (workspace) DOUBLE PRECISION array, dimension (8*N)
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 illegal value.
=1,...,N: The QZ iteration failed. (A,B) are not in Schur
form, but ALPHA(j) and BETA(j) should be correct for
j=INFO+1,...,N. > N: =N+1: other than QZ iteration failed in
ZHGEQZ
=N+2: after reordering, roundoff changed values of some complex
eigenvalues so that leading eigenvalues in the Generalized
Schur form no longer satisfy DELZTG=.TRUE. This could also be
caused due to scaling. =N+3: reordering falied in ZTGSEN.
LAPACK version 3.0 15 June 2000 ZGGES(l)
