The aim of this work is to discuss the Krylov subspace methods used to solve algebraic equation systems in a comparative way. The reason why the Krylov methods are important is as they can be properly used for a large system encountered in different problems. Krylov subspace methods represent a class of iterative methods that play a significant part the iteration of large-scale sparse matrix equations. The most significant issue here is to consider the inverse of the large and sparse data matrices used in various fields, because it may not always be possible in terms of computation costs to find the inverse of those matrices. Various methods need to be improved and accelerated to achieve high efficiency to find inverses of large and prone to be ill-conditioned matrices. The Krylov subspace iteration methods, which had the greatest impact on the development and practice of science fields have led to a major change. Krylov methods are counted to be one of the most popular classes of numerical methods for solving those types algebraic equation systems. The results produced by the Krylov methods are compared with the results of rival methods. Advantages and disadvantages of the Krylov subspace methods are illustratively discussed.
Anahtar Kelimeler: Krylov subspace method, Linear equations systems, Iterative method