
DGGEV(l) ) DGGEV(l)
NAME
DGGEV  compute for a pair of NbyN real nonsymmetric matrices (A,B)
SYNOPSIS
SUBROUTINE DGGEV( 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
DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ), B( LDB, * ), BETA( * ),
VL( LDVL, * ), VR( LDVR, * ), WORK( * )
PURPOSE
DGGEV computes for a pair of NbyN real nonsymmetric matrices (A,B) the generalized ei
genvalues, 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  lambda*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.
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 conjugatetranspose of u(j).
ARGUMENTS
JOBVL (input) CHARACTER*1
= 'N': do not compute the left generalized eigenvectors;
= 'V': compute the left generalized eigenvectors.
JOBVR (input) CHARACTER*1
= 'N': do not compute the right generalized eigenvectors;
= 'V': compute the right generalized eigenvectors.
N (input) INTEGER
The order of the matrices A, B, VL, and VR. N >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA, N)
On entry, the matrix A in the pair (A,B). On exit, A has been overwritten.
LDA (input) INTEGER
The leading dimension of A. LDA >= max(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB, N)
On entry, the matrix B in the pair (A,B). On exit, B has been overwritten.
LDB (input) INTEGER
The leading dimension of B. LDB >= max(1,N).
ALPHAR (output) DOUBLE PRECISION array, dimension (N)
ALPHAI (output) DOUBLE PRECISION array, dimension (N) BETA (output) DOUBLE
PRECISION array, dimension (N) On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j),
j=1,...,N, will be the generalized eigenvalues. If ALPHAI(j) is zero, then the j
th eigenvalue is real; if positive, then the jth and (j+1)st eigenvalues are a
complex conjugate pair, with ALPHAI(j+1) negative.
Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j) may easily over or
underflow, and BETA(j) may even be zero. Thus, the user should avoid naively com
puting 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) DOUBLE PRECISION 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 jth eigenvalue is
real, then u(j) = VL(:,j), the jth column of VL. If the jth 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) DOUBLE PRECISION 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 jth eigenvalue is
real, then v(j) = VR(:,j), the jth column of VR. If the jth 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) DOUBLE PRECISION 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 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.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = i, the ith argument had an illegal value.
= 1,...,N: The QZ iteration failed. No eigenvectors 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 DHGEQZ.
=N+2: error return from DTGEVC.
LAPACK version 3.0 15 June 2000 DGGEV(l) 
