Results for Medline SVD.

The second test example, Medline SVD, behaves quite differently. The leading eigenvalues ( $\lambda_i(A)=\sigma_i(X)^2$) are quite well separated (the largest one is $\lambda_1=3442.5$ and the next one is $\lambda_2=756.6$), and even if the quotients $\lambda_i/\lambda_{i+1}$ are smaller for larger $i$, we will get fast convergence as indicated in Figure 4.3. The first eigenvalue reaches full accuracy already at step $j=14$, and after $j=50$ steps the first 6 eigenvalues are converged. After $j=300$ 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 $j=7$, one at $j=12$, 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.