A quasi-Newton algorithm for large-scale nonlinear equations

In this paper, the algorithm for large-scale nonlinear equations is designed by the following steps: (i) a conjugate gradient (CG) algorithm is designed as a sub-algorithm to obtain the initial points of the main algorithm, where the sub-algorithm's initial point does not have any restrictions; (ii) a quasi-Newton algorithm with the initial points given by sub-algorithm is defined as main algorithm, where a new nonmonotone line search technique is presented to get the step length ak. The given nonmonotone line search technique can avoid computing the Jacobian matrix. The global convergence and the 1 + q-order convergent rate of the main algorithm are established under suitable conditions. Numerical results show that the proposed method is competitive with a similar method for large-scale problems.