Computing Function of Large Matrices by a Preconditioned Rational Krylov Method

Rational Krylov methods are a powerful alternative for computing the product of a function of a large matrix times a given vector. However, the creation of the underlying rational subspaces requires solving sequences of large linear systems, a delicate task that can require intensive computational resources and should be monitored to avoid the creation of subspace different to those required whenever, e.g., the underlying matrices are ill-conditioned. We propose the use of robust preconditioned iterative techniques to speedup the underlying process. We also discuss briefly how the inexact solution of these linear systems can affect the computed subspace. A preliminary test approximating a fractional power of the Laplacian matrix is included.