Convergent Homotopy Analysis Method for Solving Linear Systems

By using homotopy analysis method (HAM), we introduce an iterative method for solving linear systems. This method (HAM) can be used to accelerate the convergence of the basic iterative methods. We also show that by applying HAM to a divergent iterative scheme, it is possible to construct a convergent homotopy-series solution when the iteration matrix G of the iterative scheme has particular properties such as being symmetric, having real eigenvalues. Numerical experiments are given to show the efficiency of the new method. 1. Introduction Computational simulation of scientific and engineering problems often depend on solving linear system of equations. Such systems frequently arise from discrete approximation to partial differential equations. Systems of linear equations can be solved either by direct or by iterative methods. Iterative methods are ideally suited for solving large and sparse systems. For the numerical solution of a large nonsingular linear system, where is given, is known, and is unknown, one class of iterative methods is based on a splitting of the matrix , that is, where is taken to be invertible and cheap to invert, which mean that a linear system with matrix coefficient is much more economical to solve than (1). Based on (2), (1) can be written in the fixed-point form which yields the following iterative scheme for the solution of (1): A sufficient and necessary condition for (4) to converge to the solution of (1) is , where denotes spectral radius. Some effective splitting iterative methods and preconditioning methods were presented for solving the linear system of (1), see [1–9]. Recently, Keramati [10], Yusufo？lu [11], and Liu [12] applied the homotopy perturbation method to obtain the solution of linear systems and deduced the conditions for checking the convergence of homotopy series. In this work, we show how the homotopy analysis method may be regarded as an acceleration procedure based on the iterative method (4). We observe that it is not necessary that the basic method (4) be convergent. When , it is sufficient that the eigenvalues , of iteration matrix satisfy the relation , , (or , ). When , by applying the homotopy analysis method to the basic iterative method (4), we can improve the rate of convergence of the iterative method (4). This paper is organized as follows. In Section 2, we introduce the basic concept of HAM, derive the conditions for convergence of the homotopy series, and apply the homotopy analysis method to the Jacobi, Richardson, SSOR, and SAOR methods. In Section 3, some numerical examples are presented