Limited memory technique using trust regions for nonlinear equations

A new trust region subproblem that accelerates convergence for solving symmetric nonlinear equations is defined. To avoid repeatedly computing the trust region subproblem, a line search technique without derivative information is used in the trust region algorithm. Moreover, a limited memory BFGS update is employed to generate an approximated matrix rather than a normal Jacobian matrix or quasi-Newton matrix. Under mild conditions, the global convergence and the superlinear convergence of the given algorithm are established. The numerical results indicate that this method can be beneficial for solving the presented problems.