Conference of the International Linear Algebra Society (ILAS2013) edition:18 location:Providence, RI date:3-7 June 2013
In a previous contribution, we introduced the Newton rational Krylov method for solving the nonlinear eigenvalue problem (NLEP):
A(λ)x = 0.
Now, we present a practical two-level rational Krylov algorithm for solving large-scale NLEPs. At the first level, the large-scale NLEP is projected yielding a small NLEP at the second level. Therefore, the method consists of two nested iterations. In the outer iteration, we construct an orthogonal basis V and project A(λ) onto it. In the inner iteration, we solve the small NLEP A(λ)x = 0 with A(λ) = V∗ A(λ)V . We use polynomial interpolation of A(λ) resulting, after linearization, in a generalized linear eigenvalue problem with a companion-type structure. The interpolating matrix polynomial is connected with the invariant pairs in such a way it allows locking of invariant pairs and implicit restarting of the algorithm. We also illustrate the method with numerical examples and give a number of scenarios where the method performs very well.