A norm descent derivative-free algorithm for solving large-scale nonlinear symmetric equations

In this paper, we propose a norm descent derivative-free algorithm for solving large-scale nonlinear symmetric equations without involving any information of the gradient or Jacobian matrix by using some approximate substitutions. The proposed algorithm is extended from an efficient three-term conjugate gradient method for solving unconstrained optimization problems, and inherits some nice properties such as simple structure, low storage requirements and symmetric property. Under some appropriate conditions, the global convergence is proved. Finally, the numerical experiments and comparisons show that the proposed algorithm is very effective for large-scale problems.