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In the previous experiment there was no sign of the influence of
subspace iteration. In this experiment however we will clearly see
the effect of the subspace iteration. For this example again two
clusters of eigenvalues were chosen and
, one
expects a clear view of the convergence of the subspace iteration in
this case. Because the LanczosRitz values approximate the extreme
eigenvalues, it will at least take steps before the dominant
eigenvalues are approximated. After these steps one can expect to see
the convergence of the subspace iteration. The first figure (left of
Figure 6.3) shows for each step
in
the algorithm the norms of the blocks
for ,
the lines correspond to one particular submatrix, i.e., the norm of
this submatrix is shown after every step in the algorithm.
The second figure (right of Figure 6.3) is constructed
in an analogous way as in Experiment 6.4.2. In Figure 6.4 it can be seen in which step the Ritz values
approximate the most extreme eigenvalues well enough, this is also the
point from which the convergence behavior starts in Figure 6.3.
Figure 6.3:
Equally spaced eigenvalues in two clusters and

Figure 6.4:
Lanczos behavior of equally spaced eigenvalues in two clusters and

It is clearly seen in Figure 6.3 that the subspace iteration starts with a small delay (as soon
as the LanczosRitz values approximate the dominant eigenvalues well enough the convergence behavior starts).
Next: Experiment 4
Up: Numerical experiments
Previous: Experiment 2
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Raf Vandebril
20040503