A new approximate descent derivative-free algorithm for large-scale nonlinear symmetric equations

In this paper, an approximate descent three-term derivative-free algorithm is developed for a large-scale system of nonlinear symmetric equations where the gradients and the difference of the gradients are computed approximately in order to avoid computing and storing the corresponding Jacobian matrices or their approximate matrices. The new method enjoys the sufficient descent property independent of the accuracy of line search strategies and the error bounds of these approximations are established. Under some mild conditions and a nonmonotone line search technique, the global and local convergence properties are established respectively. Numerical results indicate that the proposed algorithm outperforms the other similar ones available in the literature.