zgges(3)

NAME

ZGGES - compute for a pair of N-by-N complex nonsymmetric

matrices (A,B), the generalized eigenvalues, the generalized com

plex 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 com

plex Schur form (S, T), and optionally left and/or right Schur

vectors (VSL and VSR). This gives the generalized Schur factor

ization

(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 se

lected cluster of eigenvalues appears in the leading diagonal

blocks of the upper triangular 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 singu

lar. 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 triangular and, in addition, the diago

nal 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 diagonal of the generalized Schur form. = 'N': Eigenval

ues are not ordered;
= 'S': Eigenvalues are ordered (see DELZTG).

DELZTG (input) LOGICAL FUNCTION of two COMPLEX*16 argu

ments

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 ordering may change the value of complex eigenvalues (espe

cially 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 eigenvalues (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

eigenvalues. 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 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 com

parable 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 larg

er.

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 ille

gal 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 Gen

eralized Schur form no longer satisfy DELZTG=.TRUE. This could

also be caused due to scaling. =N+3: reordering falied in ZT

GSEN.

LAPACK version 3.0 15 June 2000

ZGGES(3)