Some Root Finding With Extensions to Higher Dimensions

Abstract

Root finding is an issue in scientific computing. Because most nonlinear problems in science and engineering can be considered as the root finding problems, directly or indirectly. The research in numerical modeling for root finding is still going on. In this study, fixed point iterative methods for solving simple real roots of nonlinear equations, which improve convergence of some existing methods, are thorough. Derivative estimations up to the third order (in root finding, some recent ideas) are applied in Taylor’s approximation of a nonlinear equation by a cubic model to achieve efficient iterative methods. We may also discuss possible extensions to two dimensions and consider Newton’s method and Halley’s method in 1D and 2D problem solving. Several examples for test of efficiency and convergence analyses using C++ are offered. And some engineering applications of root finding are conferred. Graphical demonstrations are supported with matlab basic tools.

Keywords: engineering applications, derivative estimations, iterative methods, simple roots, Taylor’s approximation.