In this paragraph, we discuss the case of the inexact Cayley transform within a Galerkin projection framework. On each iteration, a Ritz value is chosen as zero. We restrict ourselves to the case where the linear system (11.2) is solved by a preconditioner or a stationary linear system solver such that for . Thus we can use the Arnoldi method to build the Krylov space of .
The disadvantage of this method is that an update of requires a complete restart of the Arnoldi process.
The asymptotic convergence towards, e.g., , is governed by the separation of the eigenvalues of . As for the SI, the separation improves when the pole is closer to .
The separation of the eigenvalues of the exact transformation improves when is very close to the Ritz value , but the corresponding linear systems are more difficult to solve than for a a bit away from the spectrum. In , the linear systems are solved with preconditioners such as incomplete factorizations and multigrid. The numerical examples show faster convergence of Algorithm 11.1 when lies a bit away from the spectrum.