Iteration Methods For Linear And Nonlinear Complex Systems

Solving many engineering problems can be transformed into getting numerical solutions of partial differential equations, which can be discretized into linear or nonlinear equations. At present, many kinds of numerical methods have been proposed, In this paper we mainly dis-cuss an improved generalized preconditioned modified HSS(GPMHSS) iteration method and its application on solving two classes of linear and nonlinear systems.First, we consider the system of linear equations of the form Ax = b, A= W+iT. By acceler-ating GPMHSS iteration method with successive overrelaxation acceleration (SOR), we give a new iterative method—accelerated GPMHSS (AGPMHSS) iteration method and also the general range of relaxation parameter. Numerical analysis show that the AGPMHSS iteration method effectively improves the convergence speed on the basis of ensuring the accuracy of convergence.Then we consider the nonlinear problem with Jacobi matrix being a large sparse,2×2 com-plex block matrix. SGPMHSS iteration method is proposed to solve the linear systems with the coefficient matrix A being 2×2 complex block matrix and the convergence property is analysed. If SGPMHSS iterative method is used to solve the Newton’s equation, we propose the modified Newton-SGPMHSS (MN-SGPMHSS) iteration method to solve nonlinear systems with Jacobi matrix being 2×2 complex block matrix. The local convergence of this method is proved. Th nu-merical implementations also show that this method has obvious advantages on solving this class of nonlinear equations.