SGGEV - compute 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 option
ally, 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 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.
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 eigenvalues. If ALPHAI(j) is
zero, then the j-th eigenvalue is real; if posi
tive, 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 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 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 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.
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 eigenvalues
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 gen
erally 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 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 itera
tion failed in SHGEQZ.
=N+2: error return from STGEVC.
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