An effective trust-region-based approach for symmetric nonlinear systems

Abstract

This paper presents a new trust-region procedure for solving symmetric nonlinear systems of equations having several variables. The proposed approach takes advantage of the combination of both an effective adaptive trust-region radius and a non-monotone strategy. It is believed that the selection of an appropriate adaptive radius and the application of a suitable non-monotone strategy can improve the efficiency and robustness of the trust-region framework as well as decrease the computational costs of the algorithm by decreasing the required number of subproblems to be solved. The global convergence and the quadratic convergence of the proposed approach are proved without the non-degeneracy assumption of the exact Jacobian. The preliminary numerical results of the proposed algorithm indicating the promising behaviour of the new procedure for solving nonlinear systems are also reported.

Keywords:

• nonlinear equations
• trust-region framework
• adaptive radius
• non-monotone technique
• convergence theory

2010 AMS Subject Classifications:

• 90C30
• 93E24
• 34A34

Acknowledgements

The authors are grateful to two anonymous referees for their valuable comments and suggestions. They also thank the research council of the University of Vienna, Bu-Ali Sina University and Rzai University for supporting this research.