  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.
