Rational Krylov Methods for Operator Functions

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 Rayleigh–Ritz or shift-and-invert method, and de-rive 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 computa-tion of so-called matrix Rayleigh quotients or parallel variants of ratio-nal Arnoldi algorithms, are discussed. We also present novel estimates for the error arising from inexact linear system solves and the approx-imation error of the Rayleigh–Ritz method. Rational Krylov method