Modified Iterative Method for Computing the Approximate Solutions of Nonlinear Equations

Abstract

This thesis concentrates on developing a Modi ed Iterative Method for computing the approximate solutions of nonlinear equations. We discus the concept of Error Analysis, Errors in Numerical Methods, Approximation and Convergence. Newton's method is discussed and proved. This study is set out to construct or develop a Modi ed Iterative Method for computing the approximate solutions of nonlinear equations by using Taylor Series expansion and Adomian Decomposition Method (ADM). The Taylor series is used in this study due to its higher possibility of convergence since it is a power series. In the same vain, the Adomian Decomposition method is a semi analytical method which decomposes the nonlinear equations into a series of functions thereby making the convergence of these functions much easier. The convergence of this method is proved to be of order 2. The Modi eld Iterative Method is a modi cation based on Newton - Raphson's method. Matlab R2020a is used to compute the solutions of some numerical examples with the proposed modi ed method. The computation of the approximated solutions of the method are compared with some existing iterative methods in literature such as Newton's method, Karthikeyan's method and External Touch Algorithm method. Then we discussed the accuracy of the proposed modi ed iterative method when applied to single variable nonlinear equations. The study pointed out that, the modi ed method is comparable with the existing methods. Finally we concluded that the modi ed iterative method is more accurate than the Newton's method, the External Touch Algorithm method and even to some extent, the Karthikeyen's method.