Rational Krylov methods for operator functions [Elektronische Ressource] / vorgelegt von Stefan Güttel

Rational Krylov Methods for Operator FunctionsVon der Fakult¨at fu¨r Mathematik und Informatikder Technischen Universita¨t Bergakademie Freiberggenehmigte Dissertation zur Erlangung des akademischen Gradesdoctor rerum naturalium (Dr. rer. nat.),vorgelegt von Dipl.-Math. Stefan Gu¨ttel,geboren am 27.11.1981 in Dresden.Betreuer: Prof. Dr. Michael Eiermann (Freiberg)Gutachter: Prof. Dr. Axel Ruhe (Stockholm)Prof. Dr. Nick Trefethen FRS (Oxford)Verleihung: Freiberg, am 12.03.2010This page is intentionally left almost blank.AbstractWe present a unified and self-contained treatment of rational Krylovmethodsforapproximatingtheproductofafunctionofalinearoperatorwith a vector. With the help of general rational Krylov decompositionswe reveal the connections between seemingly different approximationmethods, such as the Rayleigh–Ritz or shift-and-invert method, and de-rive new methods, for example a restarted rational Krylov method anda related method based on rational interpolation in prescribed nodes.Various theorems known for polynomial Krylov spaces are generalizedtotherationalKrylovcase. Computationalissues,suchasthecomputa-tion of so-called matrix Rayleigh quotients or parallel variants of ratio-nal Arnoldi algorithms, are discussed. We also present novel estimatesfor the error arising from inexact linear system solves and the approx-imation error of the Rayleigh–Ritz method.