DGGEV(1) LAPACK driver routine (version 3.2) DGGEV(1)NAME
DGGEV - computes for a pair of N-by-N 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 N-by-N real nonsymmetric matrices (A,B)
the generalized eigenvalues, and optionally, the left and/or right gen‐
eralized 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 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.
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 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 gen‐
eralized eigenvalues. If ALPHAI(j) is zero, then the j-th ei‐
genvalue is real; if positive, then the j-th and (j+1)-st ei‐
genvalues are a complex conjugate pair, with ALPHAI(j+1) nega‐
tive. 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 comput‐
ing the ratio alpha/beta. However, ALPHAR and ALPHAI will be
always less than and usually comparable with norm(A) in magni‐
tude, 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 eigen‐
values. 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 eigenvalues
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 is scaled
so the largest component has 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 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 is scaled so the largest component has 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
(MAX(1,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 calcu‐
lates 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 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 itera‐
tion failed in DHGEQZ.
=N+2: error return from DTGEVC.
LAPACK driver routine (version 3November 2008 DGGEV(1)
