In this section, we combine the ideas of the inexact Cayley transformation, the (Jacobi) Davidson method, and the rational Krylov method. Roughly speaking, we still have the same method as the inexact Cayley transform Arnoldi method or the preconditioned Lanczos method, with the only difference that the zero and the pole may be updated on each iteration. As with the preconditioned Lanczos method, the Ritz vectors are computed from the Hessenberg matrices. In addition, the Ritz values are also computed from the recurrence relation.
We now discuss the method in detail, using the algorithm of the rational Krylov method, designed for the inexact Cayley transforms, given below.
Let us analyze this algorithm step by step. The solution of the linear system leads to the relations (11.2) where is a Ritz vector and is the associated Ritz value from the previous iteration, such that the right-hand side is the residual . Since , we can also write , where is called the continuation vector. After the Gram-Schmidt orthogonalization, we have , so we can rewrite (11.2) as
Before we continue, we must give some properties of the exact rational
The following lemma explains how to compute Ritz values in the rational
As with the (Jacobi) Davidson method, the RKS method applies an inexact Cayley transform to a vector. The difference lies in the way the Ritz pairs are computed. In the (Jacobi) Davidson method, the Ritz pairs result from a Galerkin projection of on the subspace. In the RKS method, the Ritz pairs are computed from the recurrence relation using Lemma 11.1, assuming that .
With the inexact Cayley transform, the transformation can be a large perturbation of the exact Cayley transform, but can still be used to compute one eigenpair, when is well chosen. The same is true here. The inexact rational Krylov method delivers an (or ) that is not random but small in the direction of the desired eigenpair, as long as the various parameters are properly set.
A Ritz pair computed by Algorithm 11.4 produces residual