Derivative Free Conjugate Gradient Method via Broyden’s Update for solving symmetric systems of nonlinear equations

Abstract

The applications of mathematics in many areas of computing, scientific and engineering research mostly give rise to a systems of nonlinear equations. Various iterative methods have been developed to solve such equations, this includes Newton method, Quasi-Newton’s etc. Over the years, there has been significant theoretical study on quasi-Newton methods for solving such systems, but unfortunately the methods suffers setback. To overcome such problems, a Derivative free Method for Solving Symmetric Systems of Nonlinear Equations Using Broyden’s Update is presented. The modification is achieved by simply approximating the inverse Hessian matrix to with (δk and I represents acceleration parameter and an identity matrix respectively) without computing any derivative. The method uses the symmetric structure of the system sufficiently and the generalized classical Broyden’s update method for unconstrained optimization problems. The squared norm merit function is used, both the direction and the line search technique are derivative-free, this attractive feature of the proposed method makes it to have a very low storage requirement thereby solving large scale problems successfully. In an effort to solve nonlinear problems of the form F(x) = 0, 0, x ∈ Rn different initial starting points were used on a set of benchmark test problems, the output is based on number of iterations and CPU time. A comparison between the proposed method and the classical methods were made and found that the proposed method is efficient, robust and outperformed the existing method.