
CGGESX(l) ) CGGESX(l)
NAME
CGGESX  compute for a pair of NbyN 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 NbyN 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 conjugatetranspose of VSR.
Optionally, it also orders the eigenvalues so that a selected cluster of eigenvalues
appears in the leading diagonal blocks of the upper triangular matrix S and the upper tri
angular matrix T; computes a reciprocal 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 col
umns of VSL and VSR then form an orthonormal basis for the corresponding 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 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 nonnegative 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 diagonal of the general
ized 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 (especially 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 com
puted;
= '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 eigenvalues (after sort
ing) 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 eigenvalues. ALPHA(j) and BETA(j),j=1,...,N
are the diagonals of the complex Schur form (S,T). BETA(j) will be nonnegative
real.
Note: the quotients ALPHA(j)/BETA(j) may easily over or underflow, 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 JOB
VSL = 'N'.
LDVSL (input) INTEGER
The leading dimension of the matrix VSL. LDVSL >=1, and if JOBVSL = 'V', LDVSL >=
N.
VSR (output) COMPLEX array, dimension (LDVSR,N)
If JOBVSR = 'V', VSR will contain the right Schur vectors. Not referenced if JOB
VSR = '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 subspaces. 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*(NSDIM)).
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 ith 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) 
