A Simplified Algorithm for a Full-Rank Update Quasi-Newton Method

One of the most common ideas for finding the zero of a nonlinear function is to replace it by a series of suitably chosen linear functions, for which the zeros can easily be determined and the sequence of zeros approximate the zero of the nonlinear function. Widely used classic methods are Newton’s method and a large family of quasi-Newton methods (secant, Broyden’s, discretized Newton, Steffensen’s,...). This strategy can be called “linearization” and such methods may be called “linearization methods”. An efficient linearization method for solving a system of nonlinear equations was developed which showed good stability and convergence properties. It uses an unconventional and simple strategy to improve the performance of classic methods by full-rank update of the Jacobian approximates. It can be considered both as a discretized Newton’s method or as a quasi-Newton method with full-rank update of the Jacobian approximates. A solution to the secant equation presented in [] was based on the Wolfe-Popper procedure [,]. The secant equation was splitted into two equations by introducing an auxiliary variable. A simplified algorithm is given in this paper for the full-rank update procedure described in []. It directly solves the secant equation with the pseudo-inverse of the Jacobian approximate matrix. Numerical examples are shown for demonstration purpose. The convergence and efficiency of the suggested method are discussed and compered with the convergence and efficiency of classic linearization methods.