# Rational Krylov Methods for Operator Functions

Güttel, Stefan (2010) Rational Krylov Methods for Operator Functions. Doctoral thesis, Technische UniversitÃ¤t Bergakademie Freiberg.

## Abstract

We present a unified and self-contained treatment of rational Krylov methods for approximating the product of a function of a linear operator with a vector. With the help of general rational Krylov decompositions we reveal the connections between seemingly different approximation methods, such as the Rayleighâ��Ritz or shift-and-invert method, and derive new methods, for example a restarted rational Krylov method and a related method based on rational interpolation in prescribed nodes. Various theorems known for polynomial Krylov spaces are generalized to the rational Krylov case. Computational issues, such as the computation of so-called matrix Rayleigh quotients or parallel variants of rational Arnoldi algorithms, are discussed. We also present novel estimates for the error arising from inexact linear system solves and the approximation error of the Rayleighâ��Ritz method. Rational Krylov methods involve several parameters and we discuss their optimal choice by considering the underlying rational approximation problems. In particular, we present different classes of optimal parameters and collect formulas for the associated convergence rates. Often the parameters leading to best convergence rates are not optimal in terms of computation time required by the resulting rational Krylov method. We explain this observation and present new approaches for computing parameters that are preferable for computations. We give a heuristic explanation of superlinear convergence effects observed with the Rayleighâ��Ritz method, utilizing a new theory of the convergence of rational Ritz values. All theoretical results are tested and illustrated by numerical examples. Numerous links to the historical and recent literature are included.

Item Type: Thesis (Doctoral) rational Krylov, operator function, Rayleigh-Ritz, Ritz values, error estimation, parameter optimization, parallel computing MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis Stefan Güttel 24 Oct 2017 30 Oct 2017 20:26 http://eprints.maths.manchester.ac.uk/id/eprint/2586