CGGESX(l) ) CGGESX(l)NAME
CGGESX - compute for a pair of N-by-N complex nonsymmetric matrices
(A,B), the generalized eigenvalues, the complex Schur form (S,T),
SYNOPSIS
SUBROUTINE CGGESX( JOBVSL, JOBVSR, SORT, SELCTG, SENSE, N, A, LDA, B,
LDB, SDIM, ALPHA, BETA, VSL, LDVSL, VSR, LDVSR,
RCONDE, RCONDV, WORK, LWORK, RWORK, IWORK, LIWORK,
BWORK, INFO )
CHARACTER JOBVSL, JOBVSR, SENSE, SORT
INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LIWORK, LWORK, N, SDIM
LOGICAL BWORK( * )
INTEGER IWORK( * )
REAL RCONDE( 2 ), RCONDV( 2 ), RWORK( * )
COMPLEX A( LDA, * ), ALPHA( * ), B( LDB, * ), BETA( * ),
VSL( LDVSL, * ), VSR( LDVSR, * ), WORK( * )
LOGICAL SELCTG
EXTERNAL SELCTG
PURPOSE
CGGESX computes for a pair of N-by-N complex nonsymmetric matrices
(A,B), the generalized eigenvalues, the complex 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)**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; computes a recipro‐
cal condition 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 corresponding left and right eigenspaces (deflating sub‐
spaces).
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 for both being zero.
A pair of matrices (S,T) is in generalized complex Schur form if T is
upper triangular with non-negative diagonal and S is upper triangular.
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 SELCTG).
SELCTG (input) LOGICAL FUNCTION of two COMPLEX 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. Note that a selected complex eigenvalue may no longer
satisfy SELCTG(ALPHA(j),BETA(j)) = .TRUE. after ordering, since
ordering may change the value of complex eigenvalues (espe‐
cially if the eigenvalue is ill-conditioned), in this case INFO
is set to N+3 see INFO below).
SENSE (input) CHARACTER
Determines which reciprocal condition numbers are computed. =
'N' : None are computed;
= 'E' : Computed for average of selected eigenvalues only;
= 'V' : Computed for selected deflating subspaces 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) COMPLEX 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 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 SELCTG is true.
ALPHA (output) COMPLEX array, dimension (N)
BETA (output) COMPLEX array, dimension (N) On exit,
ALPHA(j)/BETA(j), j=1,...,N, will be the generalized eigenval‐
ues. ALPHA(j) and BETA(j),j=1,...,N are the diagonals of the
complex Schur form (S,T). 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 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 JOB‐
VSL = 'V', LDVSL >= N.
VSR (output) COMPLEX 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 referenced 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 number for the selected deflating sub‐
spaces. Not referenced if SENSE = 'N' or 'E'.
WORK (workspace/output) COMPLEX array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input) INTEGER
The dimension of the array WORK. LWORK >= 2*N. If SENSE =
'E', 'V', or 'B', LWORK >= MAX(2*N, 2*SDIM*(N-SDIM)).
RWORK (workspace) REAL array, dimension ( 8*N )
Real workspace.
IWORK (workspace/output) INTEGER array, dimension (LIWORK)
Not referenced if SENSE = 'N'. On exit, if INFO = 0, IWORK(1)
returns the optimal LIWORK.
LIWORK (input) INTEGER
The dimension of the array WORK. LIWORK >= N+2.
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
CHGEQZ
=N+2: after reordering, roundoff changed values of some complex
eigenvalues so that leading eigenvalues in the Generalized
Schur form no longer satisfy SELCTG=.TRUE. This could also be
caused due to scaling. =N+3: reordering failed in CTGSEN.
LAPACK version 3.0 15 June 2000 CGGESX(l)
