Finite Difference Newton's method for Systems of Non-linear Equations

摘要: Finite difference Newton's method is the one point iteration scheme introduced by Weerakoon (1996) to approximate single roots of nonlinear equations. Proposed scheme replaces the derivative of the function in Newton's method by appropriately chosen forward or backward difference formulae. In this paper the same method is applied to functions of two variables. It is proved that the method is second order convergent. Computational evidence provided here not only supports the theory but goes beyond that, suggesting it is not necessary to have the initial guess within a sufficiently close neighbourhood for the convergence of the proposed method. As problems, such as looping which effect Newton's method, can be overcome with the proposed method by choosing suitable stepsizes, finite difference Newton's method provides convergent results even for functions which do not converge with Newton's iterations.