# Chapter 3

# Solving Nonlinear Equations

**Core Topics**

Estimation of errors in numerical solutions (3.2).

Bisection method (3.3).

Regula falsi method (3.4).

Newton's method (3.5).

Secant method (3.6).

Fixed-point iteration method (3.7).

Use of MATLAB built-in Functions for solving nonlinear equations (3.8).

**Complementary Topics**

Equations with multiple roots (3.9).

System of nonlinear equations (3.10).

## 3.1 BACKGROUND

Equations need to be solved in all areas of science and engineering. An equation of one variable can be written in the form:

A solution to the equation (also called a *root* of the equation) is a numerical value of *x* that satisfies the equation. Graphically, as shown in Fig. 3-1, the solution is the point where the function *f*(*x*) crosses or touches the *x*-axis. An equation might have no solution or can have one or several (possibly many) roots.

When the equation is simple, the value of *x* can be determined analytically. This is the case when *x* can be written explicitly by applying mathematical operations, or when a known formula (such as the formula for solving a quadratic equation) can be used to determine the exact value of *x*. In many situations, however, it is impossible to determine the root of an equation analytically. For example, the area of a segment *A*_{S} of a circle with radius *r* (shaded area in Fig. 3-2) is given by:

Figure 3-1: Illustration of equations with no, one, ...