Approximate Methods of Inverse Preconditioners for Solving the Linear Algebraic Systems

ABSTRACT: The efficiency and robustness of iterative methods can be improved using a preconditioner that causes a change in the original matrix implicitly or explicitly. Usually preconditioners are constructed using the structure of the coefficient matrix. Therefore a preconditioner which works well for one class of matrices may fail to give good results for an other class. The focus of this study is to analyze, the efficiency of approximate inverse preconditioners for solving linear systems that arises from the discretization of the Poisson equation on a rectangle with Dirichlete boundary conditions. To realize this first, geometric construction of second order and a class of third order iterative methods for approximating a simple root of the nonlinear equation ( ) are investigated. Then by the generalization of these methods to Banach spaces, and applying them to the equation ( ) , Newton and Chebyshev iterative methods for matrix inversion are studied. These methods are applied to solve linear system of equations obtained from difference analog of Dirichlet problem of Laplace’s equation on a rectangle. The research is proceeded with the numerical results achieved and some discussions are made based on these results. Keywords: Chebyshev’s method, approximate inverse preconditioner, finite difference scheme, Laplace equation. …………………………………………………………………………………………………………………………