An Introduction to Krylov Subspace Methods

Abstract: Nowadays, many fields of study are have to deal with large and sparse data matrixes, but the most important issue is finding the inverse of these matrixes. Thankfully, Krylov subspace methods can be used in solving these types of problem. However, it is difficult to understand mathematical principles behind these methods. In the first part of the article, Krylov methods are discussed in detail. Thus, readers equipped with a basic knowledge of linear algebra should be able to understand these methods. In this part, the knowledge of Krylov methods are put into some examples for simple implementations of a commonly known Krylov method GMRES. In the second part, the article talks about CG iteration, a wildly known method which is very similar to Krylov methods. By comparison between CG iteration and Krylov methods, readers can get a better comprehension of Krylov methods based on CG iteration. In the third part of the article, aiming to improve the efficiency of Krylov methods, preconditioners are discussed. In addition, the restarting GMRES is briefly introduced to reduce the space consumption of Krylov methods in this part.