A unified theory for Krylov algorithms

Large systems of linear equations arise in many different scientific applications. For example, partial differential equations discretized with the finite difference or finite element method yield a system of equations. Large systems can be solved with either sparse factorization techniques or iterative methods. These two approaches can be combined into a method that uses approximate factorization preconditioning for an iterative method. Krylov algorithms are iterative numerical methods for large unsymmetric systems of linear equations. In this paper, we set up a general theoretical framework for Krylov algorithms and so highlight their common features. We first introduce the conception of orthogonality between linear subspaces. We then formulate a unified definition for Krylov algorithms. On this basis, we study some of their common properties. This work may give useful hints on formulating new better iterative methods for unsymmetric problems.