Computing for Numerical Methods Using Visual C++

Chapter 6. Nonlinear Equations

• 6.1 Introduction

• 6.2 Existence of Solutions

• 6.3 Bisection Method

• 6.4 False Position Method

• 6.5 Newton—Raphson Method

• 6.6 Secant Method

• 6.7 Fixed-Point Iteration Method

• 6.8 Visual Solution: Code6

• 6.9 Summary

  • Numerical Exercises

  • Programming Challenges

INTRODUCTION

As described in the previous chapter, a nonlinear equation is an equation that has one or more nonlinear terms in its expression, or an equation that is not in the form of Equation (5.1). In its general form, a nonlinear equation involving n variables, x₁, x₂, x₃,...,x_n, can be expressed as

Equation 6.1. 

Some examples of nonlinear equations are given below:

3 — 3x2 + x3 = −2,	nonlinear because of the presence of nonlinear terms x2 and x 3.
x + sin y = −1,	nonlinear because of the presence of nonlinear term sin y.
xy = −1	nonlinear because the sum of indices in the variable term is 2.
	nonlinear because variable x is in the denominator.

Finding the root(s) of a nonlinear equation is one of the most fundamental problems involving nonlinear equations. The problem is stated as follows:

Given a function f(x) where f(x)= 0 exists, find the value of x.

Alternatively, the problem is also called finding the zeros of a function. Graphically, this problem can be described as finding one or more points along the x -axis where the ...