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To illustrate the effect of applying a Lanczos algorithm to a shift-and-invert
operator (4.14), we follow the convergence for the L-shaped
membrane matrix with a shift applied at the origin . The
picture in Figure 4.4 is now very similar to the Medline
SVD example, and we get full accuracy in the six smallest eigenvalues
after steps. In this run we used full reorthogonalization as
advocated above. Even if the number of steps is reduced by more than a
factor of 10, we have to take into consideration that the
factorization (4.16) takes some time and that the
factors are denser than the original , so applying
their inverses in each step (4.17) needs more work than the
application of the original
of (4.8) in the direct Lanczos method.

Susan Blackford
2000-11-20