The analysis is given for the Cayley transform only, though shift-and-invert could be used as well. The reasons are threefold. First, the shift-and-invert Arnoldi method and the Cayley transform Arnoldi method produce the same Ritz vectors. Second, it makes a link with the (Jacobi) Davidson methods easier to establish. Third, the Cayley transform leads to a more natural approach to the problem.
In the Arnoldi method applied to the Cayley transform,
, we must
sequence of linear systems
By putting all the for
in and in , we obtain
Note that for a fixed and , depends on and the way each of the linear systems (11.2) are solved. When (11.2) is solved by a preconditioner or a stationary linear system solver, i.e., for , then is independent of and and so is .
Eigenpairs are computed from . In Arnoldi's method, this happens by the Hessenberg matrix that arises from the orthogonalization of ; in the Davidson method, one uses the projection with and , e.g., .
contains the eigenvectors associated with the well-separated extreme eigenvalues of . This is a perturbed , so the relation with is partially lost, and accurately computing eigenpairs of may be difficult. To see which parameters play a role in this perturbation, from Theorems 4.3 and 4.4 in , it can be shown that if has distinct eigenvalues, then for each eigenpair of there is an eigenpair of such that
This section can be concluded as follows.
The remaining question is now how to compute the eigenpairs of or how to exploit them for computing eigenpairs of . In §11.2.3 and §11.2.4, we discuss the Rayleigh-Ritz technique; i.e., eigenpairs are computed from the orthogonal projection of on the . In §11.2.7, the eigenpairs are computed from the eigenpairs of directly by use of the rational Krylov recurrence relation. §11.2.6 presents a Lanczos algorithm that uses the recurrence relation for the eigenvectors and the Rayleigh-Ritz projection for the eigenvalues.
Note that and cannot be chosen too far away from each other. Suppose the eigenvalue is wanted. From the conclusion given above, the convergence is faster when is chosen close to , and the computed eigenvalue can only be accurate when is close to . In theory, one could select , which is usually used in the Davidson method. Since in that case, there is a high risk of stagnation in the latter method. A robust selection is used in the Jacobi-Davidson method, which we discuss in §11.2.5.