2.1 Introduction

In this chapter we study one of the fundamental problems of numerical analysis, namely the numerical solutions of nonlinear equations. Most equations arising in practice are nonlinear and are rarely of a form that allows the roots to be determined exactly. Consequently, numerical methods are used to solve nonlinear algebraic equations when the equations prove intractable to ordinary mathematical techniques. These numerical methods are all iterative, and they may be used for equations that contain one or several variables. These techniques can be divided into two categories; one-point (needing one initial approximation) and two-point (needing two initial approximations) methods. A nonlinear equation may be considered any one of the following types:

1. An equation may be an algebraic equation (a polynomial equation of degree n) expressible in the form:

   
   

   where a _n , a _{n ?1}, , a ₁, and a ₀ are constants. For example, the following equations are nonlinear:

   
   

2. The power of the unknown variable involved in the equation must be difficult to manipulate. For example, the following equations are nonlinear:

   
   

3. An equation may be a transcendental equation which involves trigonometric functions, exponential functions, and logarithmic functions. For example, the following equations are nonlinear:

   
   

All the numerical methods described in this chapter are applicable to general nonlinear functions. Special techniques do, however, exist for certain restricted classes of functions, such as polynomials. Some of these are described in the books by [58] and [93]. The iterative methods we will discuss in...