Numerical Methods for Engineers and Scientists 3rd Edition

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, ...