Numerical Example.

We report results on the generalized Hermitian shift-and-invert Lanczos algorithm for one of the examples in §4.4.6, the L-shaped membrane, but now with a 9-point finite difference approximation.

We use the same grid spacing $h=1/64$ and get two sparse symmetric positive definite matrices $A$ and $B$ of order $n=2945$. They have a very regular sparse band structure, now with at most 9 nonzero elements filled in each row. We used the shift $\sigma=0$ to seek the 8 smallest eigenvalues. The residual estimates (5.21) at each step $j$ are plotted in Figure 5.1.

When we compare the results on this generalized problem to those for the standard Hermitian eigenproblem of the same order in Figure 4.4 (p. ) in §4.4.6, we have to bear in mind that this 9-point approximation is a more accurate one to the partial differential equation than the 5-point approximation used there.