We discuss the results for a small example that can be easily repeated.
The unsymmetric matrix is of dimension
and it is
tridiagonal. The diagonal entries are
, the codiagonal
entries are
,
.
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We use Algorithms 7.18 and 7.19 for the
computation of the eigenvalues closest to the target value
. Since these eigenvalues are located in the interior of
the spectrum, we expect some advantage from Algorithm 7.19,
which is designed for interior eigenvalues. Indeed, as we will see, the
usage of harmonic Ritz values leads to an advantage here.
We have carried out the experiments in MATLAB. The input parameters have
been chosen as follows. The starting vector has been chosen with
random entries (with
in MATLAB). The tolerance is
. The subspace dimension parameters are
,
. The correction equations
in (27) of Algorithm 7.18 and
(34) of Algorithm 7.19
have been solved approximately
with five steps of GMRES.
We show graphically in Figure 7.5 the norm of the residual
vector as a function of the iteration number. Each time when the norm is
less than , then we have determined an eigenvalue in
acceptable approximation, and the iteration is continued with deflation
for the next eigenvalue. The left picture represents the results
obtained with Algorithm 7.18, and we see that there is no
convergence detected within
Jacobi-Davidson steps. In the right
picture we see the results for Algorithm 7.19, and now
eigenvalues have been discovered within
iterations.