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The second test example, Medline SVD, behaves quite differently. The
leading eigenvalues (
) are quite well
separated (the largest one is
and the next one is
), and even if the quotients
are smaller for larger , we will get fast convergence as
indicated in Figure 4.3. The first eigenvalue reaches full
accuracy already at step , and after steps the first 6
eigenvalues are converged. After steps we had 100 eigenvalues.
There is another interesting difference between this well-separated
problem and the L-shaped membrane with its more clustered eigenvalues in that
reorthogonalization is triggered more often. We mark reorthogonalizations
with a dashed vertical line at step , one at , and
every four steps from there on. Since each reorthogonalization involves two vectors,
selective reorthogonalization demands about half as much work as a
full reorthogonalization. In such cases the user is advised
to use full reorthogonalization, since it gives full orthogonality of the
basis vectors at a moderate extra cost in arithmetic work.

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Susan Blackford
2000-11-20