Convergence of the modified SOR–Newton method for non-smooth equations

Abstract

Motivated by Chen [On the convergence of SOR methods for nonsmooth equations. Numer. Linear Algebra Appl. 9 (2002), pp. 81–92], in this paper, we further investigate a modified SOR–Newton (MSOR–Newton) method for solving a system of nonlinear equations F(x)=0, where F is strongly monotone and locally Lipschitz continuous but not necessarily differentiable. The convergence interval of the parameter in the MSOR–Newton method is given. Compared with that of the SOR–Newton method, this interval can be enlarged. Furthermore, when the B-differential of F(x) is difficult to compute, a simple replacement can be used, which can reduce the computational load. Numerical examples show that at the same cost of computational complexity, this MSOR–Newton method can converge faster than the corresponding SOR–Newton method by choosing a suitable parameter.

Keywords:

• nonlinear equations
• non-smooth
• semismooth
• MSOR–Newton method
• convergence

1999 AMS Subject Classification:

• 65F10

Acknowledgements

The authors thank the referees for their many valuable comments and suggestions, which have helped in the improvement of the paper. This work was supported by the National Natural Science Foundation of China under grant 10971102, the Natural Science Foundation of Jiangsu Province of China under grant BK2009398 and the National Excellent Doctoral Thesis Award of China (200720).