zggev(3)

NAME

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

matrices (A,B), the generalized eigenvalues, and optionally, the

left and/or right generalized eigenvectors

SYNOPSIS

SUBROUTINE ZGGEV( JOBVL, JOBVR, N, A, LDA, B, LDB,  ALPHA,
BETA, VL, LDVL, VR, LDVR, WORK, LWORK, RWORK, INFO )
    CHARACTER     JOBVL, JOBVR
    INTEGER       INFO, LDA, LDB, LDVL, LDVR, LWORK, N
    DOUBLE        PRECISION RWORK( * )
    COMPLEX*16     A(  LDA,  * ), ALPHA( * ), B( LDB, * ),
BETA( * ), VL( LDVL, * ), VR( LDVR, * ), WORK( * )

PURPOSE

ZGGEV computes for a pair of N-by-N complex nonsymmetric

matrices (A,B), the generalized eigenvalues, and optionally, the

left and/or right generalized eigenvectors. A generalized eigen

value for a pair of matrices (A,B) is a scalar lambda or a ratio

alpha/beta = lambda, such that A - lambda*B is singular. It is

usually represented as the pair (alpha,beta), as there is a rea

sonable interpretation for beta=0, and even for both being zero.

The right generalized eigenvector v(j) corresponding to

the generalized eigenvalue lambda(j) of (A,B) satisfies

A * v(j) = lambda(j) * B * v(j).

The left generalized eigenvector u(j) corresponding to the

generalized eigenvalues lambda(j) of (A,B) satisfies

u(j)**H * A = lambda(j) * u(j)**H * B

where u(j)**H is the conjugate-transpose of u(j).

ARGUMENTS

JOBVL (input) CHARACTER*1

= 'N': do not compute the left generalized eigen

vectors;
= 'V': compute the left generalized eigenvectors.

JOBVR (input) CHARACTER*1

= 'N': do not compute the right generalized

eigenvectors;
= 'V': compute the right generalized eigenvec

tors.

N (input) INTEGER

The order of the matrices A, B, VL, and VR. N >=

0.

A (input/output) COMPLEX*16 array, dimension (LDA,

N)

On entry, the matrix A in the pair (A,B). On ex

it, A has been overwritten.

LDA (input) INTEGER

The leading dimension of A. LDA >= max(1,N).

B (input/output) COMPLEX*16 array, dimension (LDB,

N)

On entry, the matrix B in the pair (A,B). On ex

it, B has been overwritten.

LDB (input) INTEGER

The leading dimension of B. LDB >= max(1,N).

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.

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).

VL (output) COMPLEX*16 array, dimension (LDVL,N)

If JOBVL = 'V', the left generalized eigenvectors

u(j) are stored one after another in the columns of VL, in the

same order as their eigenvalues. Each eigenvector will be scaled

so the largest component will have abs(real part) + abs(imag.

part) = 1. Not referenced if JOBVL = 'N'.

LDVL (input) INTEGER

The leading dimension of the matrix VL. LDVL >= 1,

and if JOBVL = 'V', LDVL >= N.

VR (output) COMPLEX*16 array, dimension (LDVR,N)

If JOBVR = 'V', the right generalized eigenvectors

v(j) are stored one after another in the columns of VR, in the

same order as their eigenvalues. Each eigenvector will be scaled

so the largest component will have abs(real part) + abs(imag.

part) = 1. Not referenced if JOBVR = 'N'.

LDVR (input) INTEGER

The leading dimension of the matrix VR. LDVR >= 1,

and if JOBVR = 'V', LDVR >= 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/output) DOUBLE PRECISION array, dimen

sion (8*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. No eigenvec

tors have been calculated, but ALPHA(j) and BETA(j) should be

correct for j=INFO+1,...,N. > N: =N+1: other then QZ iteration

failed in DHGEQZ,
=N+2: error return from DTGEVC.

LAPACK version 3.0 15 June 2000

ZGGEV(3)