SGEEV(l) ) SGEEV(l)NAME
SGEEV - compute for an N-by-N real nonsymmetric matrix A, the eigenval‐
ues and, optionally, the left and/or right eigenvectors
SYNOPSIS
SUBROUTINE SGEEV( JOBVL, JOBVR, N, A, LDA, WR, WI, VL, LDVL, VR, LDVR,
WORK, LWORK, INFO )
CHARACTER JOBVL, JOBVR
INTEGER INFO, LDA, LDVL, LDVR, LWORK, N
REAL A( LDA, * ), VL( LDVL, * ), VR( LDVR, * ), WI( * ),
WORK( * ), WR( * )
PURPOSE
SGEEV computes for an N-by-N real nonsymmetric matrix A, the eigenval‐
ues and, optionally, the left and/or right eigenvectors. The right
eigenvector v(j) of A satisfies
A * v(j) = lambda(j) * v(j)
where lambda(j) is its eigenvalue.
The left eigenvector u(j) of A satisfies
u(j)**H * A = lambda(j) * u(j)**H
where u(j)**H denotes the conjugate transpose of u(j).
The computed eigenvectors are normalized to have Euclidean norm equal
to 1 and largest component real.
ARGUMENTS
JOBVL (input) CHARACTER*1
= 'N': left eigenvectors of A are not computed;
= 'V': left eigenvectors of A are computed.
JOBVR (input) CHARACTER*1
= 'N': right eigenvectors of A are not computed;
= 'V': right eigenvectors of A are computed.
N (input) INTEGER
The order of the matrix A. N >= 0.
A (input/output) REAL array, dimension (LDA,N)
On entry, the N-by-N matrix A. On exit, A has been overwrit‐
ten.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,N).
WR (output) REAL array, dimension (N)
WI (output) REAL array, dimension (N) WR and WI contain
the real and imaginary parts, respectively, of the computed ei‐
genvalues. Complex conjugate pairs of eigenvalues appear con‐
secutively with the eigenvalue having the positive imaginary
part first.
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 JOBVL = 'N', VL is not referenced. If the j-th ei‐
genvalue is real, then u(j) = VL(:,j), the j-th column of VL.
If the j-th and (j+1)-st 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).
LDVL (input) INTEGER
The leading dimension of the array VL. LDVL >= 1; 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 JOBVR = 'N', VR is not referenced. 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)-st eigenvalues form a complex conju‐
gate pair, then v(j) = VR(:,j) + i*VR(:,j+1) and
v(j+1) = VR(:,j) - i*VR(:,j+1).
LDVR (input) INTEGER
The leading dimension of the array VR. LDVR >= 1; 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(1,3*N), and if
JOBVL = 'V' or JOBVR = 'V', LWORK >= 4*N. For good perfor‐
mance, 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.
> 0: if INFO = i, the QR algorithm failed to compute all the
eigenvalues, and no eigenvectors have been computed; elements
i+1:N of WR and WI contain eigenvalues which have converged.
LAPACK version 3.0 15 June 2000 SGEEV(l)
