
CGGES(l) ) CGGES(l)
NAME
CGGES  compute for a pair of NbyN 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 CGGES( JOBVSL, JOBVSR, SORT, SELCTG, 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( * )
REAL RWORK( * )
COMPLEX A( LDA, * ), ALPHA( * ), B( LDB, * ), BETA( * ), VSL( LDVSL, * ), VSR(
LDVSR, * ), WORK( * )
LOGICAL SELCTG
EXTERNAL SELCTG
PURPOSE
CGGES computes for a pair of NbyN 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 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. 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 CGGEV 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, and even for both being zero.
A pair of matrices (S,T) is in generalized complex Schur form if S and T are upper trian
gular and, in addition, the diagonal elements of T are nonnegative 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 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. An eigenvalue ALPHA(j)/BETA(j) is selected if
SELCTG(ALPHA(j),BETA(j)) is true.
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 illconditioned), 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 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), j=1,...,N and
BETA(j), j=1,...,N are the diagonals of the complex Schur form (A,B) output by
CGGES. The 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.
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 >= 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) REAL 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 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 falied
in CTGSEN.
LAPACK version 3.0 15 June 2000 CGGES(l) 
