In this paper, we propose a nonmonotone adap-tive trust-region method for solving symmetric nonlinear equations problems. The convergent result of the presented method will be estab-lished under favorable conditions. Numerical results are reported.

Yuan, G.L. and Lu, X.W. (2008) A new line search method with trust region for unconstrained optimization. Communications on Applied Nonlinear Analysis, 15(1), 35-49.

Yuan, G.L. and Wei, Z.X. (2008) The superlinear con-vergence analysis of a nonmonotone BFGS algorithm on convex objective functions. Acta Mathematica Sinica, English Series, 24(1), 35-42.

Powell, M.J.D. (1975) Convergence properties of a class of minimization algorithms. Mangasarian, O.L., Meyer, R.R. and Robinson, S.M., Ed., Nonlinear Programming, Academic Press, New York, 2, 1-27.

Schultz, G.A., Schnabel, R.B. and Bryrd, R.H. (1985) A family of trust-region-based algorithms for unconstrained minimization with strong global convergence properties. SIAM Journal on Numerical Analysis, 22(1), 47-67.

Byrd, R.H., Schnabel, R.B. and Schultz G.A. (1987) A trust-region algorithm for nonlinearly constrained opti-mization. SIAM Journal on Numerical Analysis, 24(5), 1152-1170.

Liu, X. and Yuan, Y. (1997) A global convergent, locally superlinearly convergent algorithm for equality con-strained optimization. Research Report, ICM-97-84, Chinese Academy of Sciences, Beijing.

Vardi, A. (1985) A trust-region algorithm for equality constrained minimization: Convergence properties and implementation. SIAM Journal of Numerical Analysis, 22(3), 575-579.

Yuan, Y. (2000) A review of trust region algorithms for optimization. Proceedings of the 4th International Con-gress on Industrial & Applied Mathematics (ICIAM 99), Edinburgh, 271-282.

Yuan, G.L., Meng, S.D. and Wei, Z.X. (2009) A trust- region-based BFGS method with line search technique for symmetric nonlinear equations. Advances in Opera-tions Research, 2009, 1-20.

Zhang, J.L. and Zhang, X.S. (2003) A nonmonotone adaptive trust region method and its convergence. Com-puters and Mathematics with Applications, 45(10-11), 1469-1477.

Griewank, A. (1986) The ‘global’ convergence of Broy-den-like methods with a suitable line search. Journal of the Australian Mathematical Society Series B, 28, 75-92.

Fan, J.Y. (2003) A modified Levenberg-Marquardt algo-rithm for singular system of nonlinear equations. Journal of Computational Mathematics, 21, 625-636.

Yuan, G.L., Wei, Z.X. and Lu, X.W. (2009) A non-monotone trust region method for solving symmetric nonlinear equations. Chinese Quarterly Journal of Mathe- matics, 24, 574-584.

Yuan, G.L. and Lu, X.W. and Wei, Z.X. (2007) A modi-fied trust region method with global convergence for symmetric nonlinear equations. Mathematica Numerica Sinica, 11(3), 225-234.

Li, D. and Fukushima, M. (1999) A global and superlin-ear convergent Gauss-Newton-based BFGS method for symmetric nonlinear equations. SIAM Journal on Nu-merical Analysis, 37(1), 152-172.

Yuan, G.L. and Li, X.R. (2004) An approximate Gauss- Newton-based BFGS method with descent directions for solving symmetric nonlinear equations. OR Transactions, 8(4), 10-26.

Yuan, G.L. and Lu, X.W. and Wei, Z.X. (2009) BFGS trust-region method for symmetric nonlinear equations. Journal of Computational and Applied Mathematics, 230(1), 44-58.

Yuan, G.L., Wang, Z.X. and Wei, Z.X. (2009) A rank-one fitting method with descent direction for solving sym-metric nonlinear equations. International Journal of Communications, Network and System Sciences, 2(6), 555-561.

Yuan, G.L. and Lu, X.W. (2008) A new backtracking inexact BFGS method for symmetric nonlinear equations. Computer and Mathematics with Application, 55(1), 116- 129.