A Note on the Comparison of a Class of Preconditioned Iterative Methods

The preconditioned iterative methods are mainly categorized into implicit preconditioned methods and explicit preconditioned methods. In this manuscript we review implicit preconditioned methods, based on incomplete and approximate factorization, and explicit preconditioned methods, based on sparse approximate inverses and explicit approximate inverses. Additionally we present the modified Moore-Penrose conditions and theoretical estimates on the iteration matrix of the explicit preconditioned method, based on explicit approximate inverses. Finally, the performance of the preconditioned iterative methods is illustrated by solving characteristic 2D elliptic problem and numerical results are given. The theoretical estimates were in qualitative agreement with the numerical results.