Convergence Properties.

The convergence is governed by the same orthogonal polynomial theory as in the standard case; see, e.g., [353].

This theory says that we get convergence to those eigenvalues that are represented in the starting vector and faster convergence to those in the ends of the spectrum. The better separated these are from the rest of the eigenvalues, the faster they will converge.

In practical cases, we are often interested just in the lowest eigenvalues, and then it is good that those are among the first to converge in the direct iteration Algorithm 5.4. On the other hand, the relative separation of the lowest eigenvalues is often poor--remember that the separation is relative to the whole spread of the spectrum, not the distance to the origin.

In these cases, and when we want eigenvalues in a specified range, say $\alpha \le \lambda \le \beta$, it is of great advantage to use the shift-and-invert Algorithm 5.5 for an appropriately chosen shift $\sigma$, for instance, in the interval $\alpha \le \sigma \le \beta$.

In the generalized case, shift-and-invert strategies are even better motivated than in the standard case, since we have to solve a system in any case, either in step 4 or in step 9. The shift-and-invert operator $C$, (5.16), most often has very much better separated eigenvalues, needing just $j=50$ instead of several hundreds to get $10$ eigenvalues.