Convergence analysis of Krylov subspace methods

Abstract

AbstractOne of the most powerful tools for solving large and sparse systems of linear algebraic equations is a class of iterative methods called Krylov subspace methods. Their significant advantages like low memory requirements and good approximation properties make them very popular, and they are widely used in applications throughout science and engineering. The use of the Krylov subspaces in iterative methods for linear systems is even counted among the “Top 10” algorithmic ideas of the 20th century. Convergence analysis of these methods is not only of a great theoretical importance but it can also help to answer practically relevant questions about improving the performance of these methods. As we show, the question about the convergence behavior leads to complicated nonlinear problems. Despite intense research efforts, these problems are not well understood in some cases. The goal of this survey is to summarize known convergence results for three well‐known Krylov subspace methods (CG, MINRES and GMRES) and to formulate open questions in this area. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)