A Proposal on Development and Analysis of New Iterative Schemes for Solving Nonlinear Equations
CHAPTER ONE
Aim and objectives
The following research objectives will help in analyzing the work
- To review iterative schemes between 1998 and 2012 which have been developed from Adomian decomposition method, Homotopy perturbation method and variants of Newton-Raphson’s method for solving nonlinear equations.
- To develop new schemes that could compete with previous schemes and probably have further advantages.
- To compare the new schemes with the existing known iterative schemes.
CHAPTER TWO
LITERATURE REVIEW
Introduction
A numerical method for determining zeroes of a functional equation, f ( x) = 0 is generally an Iterative method that will converge to zero of the function, f(x). It is simply a method which produces an approximate rather than exact solution. As one would presume, each algorithm has its advantages and disadvantages and therefore selecting the right algorithm for a given problem is never easy. Various methods for solving one variable nonlinear equations are presented in the literature. Probably the easiest numerical method for solving a nonlinear equation is the Newton (Newton-Raphson) method as already stated in chapter 1. This method have local convergent and will converge to complex zeros only if the initial guess is complex. However, it can be suitably modified to compute zeros of complex polynomials and transcendental equations. When Newton’s method does converge, the convergence is quadratic, ie the order of convergence is two.
The Newton-Raphson algorithm is derived from Taylor series expansion of nonlinear equation. The higher order terms (second order and higher order derivatives of the series) are neglected assuming that the initial guess for the iterative process is closer to the solution. Hence the equation for Newton- Raphson method is one of the reduced forms of Taylor series expansion. The Newton-Raphson method is an iterative process for solving other non-linear equations. In the iterative process, the first order derivative of the non-linear equation is calculated at an initial guess of the variables for the first iteration. The change in variables is then calculated by solving linear equations that contain the first order derivative and input vector. The change in variable is used to update the variables in each iteration.
Generalizations of Newton’s method
Several researchers generalized this method and offered methods which suggest convergence with higher orders in comparison with Newton’s method. The following works are variants of Newton-Raphson method.
Jisheng et al. (2006) presented a new modification of Newton’s method for solving non-linear equations. Analysis of convergence showed that the new method is cubically convergent. Per iteration, the new method requires two evaluations of the function and one evaluation of its first derivative. Thus, the new method is preferable if the computational costs of the first derivative are equal or more than those of the function itself. Its practical utility was demonstrated by numerical examples.
CHAPTER THREE
RESEARCH DESIGN AND METHODOLOGY
The researcher will used experimental design, analysis of variance; transformations, model validation and residual analysis. Factorial design with fixed, random and mixed effects. The design was suitable for the study as the study sought to development and analysis of new iterative schemes for solving nonlinear equation
METHOD OF DATA ANALYSIS
The researcher will employ nonlinear series of polynomials. It helps to assess the decomposing the nonlinear term into a series of polynomials
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