SGGEV(3) LAPACK driver routine (version 3.1) SGGEV(3)
NAME
SGGEV - for a pair of N-by-N real nonsymmetric matrices (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 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) REAL 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) REAL 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) REAL array, dimension (N)
ALPHAI (output) REAL array, dimension (N) BETA (output)
REAL array, dimension (N) On exit, (ALPHAR(j) +
ALPHAI(j)*i)/BETA(j), j=1,...,N, will be the generalized eigen-
values. 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 ALPHAI(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 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) 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 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) REAL 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 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 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 SHGEQZ.
=N+2: error return from STGEVC.
LAPACK driver routine (version 3.N1o)vember 2006 SGGEV(3)