IMA Conference on Numerical Linear Algebra and Optimisation edition:4th location:Birmingham date:3-5 September 2014
We present a new framework of Compact Rational Krylov (CORK) methods for solving the nonlinear eigenvalue problem (NLEP): A(λ)x = 0. For many years linearizations are used for solving polynomial eigenvalue problems. On the other hand, for the general nonlinear case, A(λ) can first be approximated by a (rational) matrix polynomial and then a convenient linearization is used. The major disadvantage of methods based on linearizations is the growing memory cost with the iteration count, i.e., in general the memory cost is proportional to the degree of the polynomial. However, the CORK family of rational Krylov methods exploits the structure of the linearization pencils and uses a generalization of the compact Arnoldi decomposition. In this way, the extra memory cost due to the linearization of the original eigenvalue problems is negligible.