Arnoldi Method with Inexact Cayley Transform

In this paragraph, we discuss the case of the inexact Cayley transform within a Galerkin projection framework. On each iteration, a Ritz value $\theta$ is chosen as zero. We restrict ourselves to the case where the linear system (11.2) is solved by a preconditioner $M$ or a stationary linear system solver such that $w_j = M^{-1} (A-\theta B) y_j$ for $j=1, \ldots,k-1$. Thus we can use the Arnoldi method to build the Krylov space of $T_{\mathrm{IC}} \equiv M^{-1}(A-\theta B)$.

The disadvantage of this method is that an update of $\theta$ requires a complete restart of the Arnoldi process.

The asymptotic convergence towards, e.g., $\lambda_1$, is governed by the separation of the eigenvalues of $T_{\mathrm{IC}} = M^{-1} (A - \lambda_1 B) \approx (A - \mu B)^{-1} (A-\lambda_1 B)$. As for the SI, the separation improves when the pole $\mu$ is closer to $\lambda_1$.

The separation of the eigenvalues of the exact transformation improves when $\mu$ is very close to the Ritz value $\theta$, but the corresponding linear systems are more difficult to solve than for a $\mu$ a bit away from the spectrum. In [323], the linear systems are solved with preconditioners such as incomplete factorizations and multigrid. The numerical examples show faster convergence of Algorithm 11.1 when $\mu$ lies a bit away from the spectrum.