sggev(3)

NAME

SGGEV - compute for a pair of N-by-N real nonsymmetric ma

trices (A,B)

SYNOPSIS

SUBROUTINE SGGEV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR,
ALPHAI, BETA, VL, LDVL, VR, LDVR, WORK, LWORK, INFO )
    CHARACTER     JOBVL, JOBVR
    INTEGER       INFO, LDA, LDB, LDVL, LDVR, LWORK, N
    REAL          A( LDA, * ), ALPHAI( * ), ALPHAR(  *  ),
B( LDB, * ), BETA( * ), VL( LDVL, * ), VR( LDVR, * ), WORK( * )

PURPOSE

SGGEV computes for a pair of N-by-N real nonsymmetric ma

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

left and/or right generalized eigenvectors.

A generalized eigenvalue for a pair of matrices (A,B) is a

scalar lambda or a ratio alpha/beta = lambda, such that A - lamb

da*B is singular. It is usually represented as the pair (al

pha,beta), as there is a reasonable interpretation for beta=0,

and even for both being zero.

The right eigenvector v(j) corresponding to the eigenvalue

lambda(j) of (A,B) satisfies

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

The left eigenvector u(j) corresponding to the eigenvalue

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) REAL 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) REAL 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).

ALPHAR (output) REAL array, dimension (N)

ALPHAI (output) REAL array, dimension (N) BETA

(output) REAL array, dimension (N) On exit, (ALPHAR(j) + AL

PHAI(j)*i)/BETA(j), j=1,...,N, will be the generalized eigenval

ues. If ALPHAI(j) is zero, then the j-th eigenvalue is real; if

positive, then the j-th and (j+1)-st eigenvalues are a complex

conjugate pair, with ALPHAI(j+1) negative.

Note: the quotients ALPHAR(j)/BETA(j) and AL

PHAI(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, ALPHAR and ALPHAI will be always less

than and usually comparable with norm(A) in magnitude, and BETA

always less than and usually comparable with norm(B).

VL (output) REAL array, dimension (LDVL,N)

If JOBVL = 'V', the left eigenvectors u(j) are

stored one after another in the columns of VL, in the same order

as their eigenvalues. If the j-th eigenvalue is real, then u(j) =

VL(:,j), the j-th column of VL. If the j-th and (j+1)-th eigen

values form a complex conjugate pair, then u(j) =

VL(:,j)+i*VL(:,j+1) and u(j+1) = VL(:,j)-i*VL(:,j+1). Each

eigenvector will be scaled so the largest component 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) REAL array, dimension (LDVR,N)

If JOBVR = 'V', the right eigenvectors v(j) are

stored one after another in the columns of VR, in the same order

as their eigenvalues. If the j-th eigenvalue is real, then v(j) =

VR(:,j), the j-th column of VR. If the j-th and (j+1)-th eigen

values form a complex conjugate pair, then v(j) =

VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1). Each

eigenvector will be scaled so the largest component 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) REAL 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(1,8*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.

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 ALPHAR(j), ALPHAI(j), and BETA(j)

should be correct for j=INFO+1,...,N. > N: =N+1: other than QZ

iteration failed in SHGEQZ.
=N+2: error return from STGEVC.

LAPACK version 3.0 15 June 2000

SGGEV(3)