Modified Newton-GSOR method for solving complex nonlinear systems with symmetric Jacobian matrices

Abstract

In this paper, a new method for solving complex nonlinear systems with symmetric Jacobian matrices is proposed-modified Newton generalized successive overrelaxation (MN-GSOR) iteration method. The new MN-GSOR method helps us to solve nonlinear systems using the GSOR method as an inner iterative approximation for solving the Newton equations. Next, we use the Hölder continuous condition instead of the Lipschitz assumption to analyze and prove the convergence properties of the MN-GSOR method. Numerical experiments are conducted to show the superiority and effectiveness of MN-GSOR method, and comparisons are made to see the advantages over some other recently proposed methods. Furthermore, when the imaginary part of the Jacobian matrix is symmetric but non-definite, some recently proposed methods are not applicable, but the MN-GSOR method still works well. Keywords Complex systems of nonlinear equations • Convergence analysis • Modified Newton-PMHSS • Modified Newton-GSOR • Modified Newton-HSS Mathematics Subject Classification 60F10 • 60F50 with F : D ⊂ C n → C n being nonlinear and continuously differentiable. Communicated by Zhong-Zhi Bai.