Results for L-Shaped Membrane with Shift-and-Invert.

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 $\sigma=0$. 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 $j=36$ steps. In this run we used full reorthogonalization as advocated above. Even if the number of steps $j$ 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 $A$, so applying their inverses in each step (4.17) needs more work than the application of the original $A$ of (4.8) in the direct Lanczos method.