Page 170 - Excel for Scientists and Engineers: Numerical Methods

Chapter 8

Roots of Equations

Many problems in science and engineering can be expressed in the form of
an equation in a single unknown, i.e., y = F(x). A value of x that makes y = 0 is
called a root of the function; often the solution to a scientific problem is a root of
a function. If the function to be solved is a quadratic equation, there is a familiar
formula to find the two roots of the expression. But for almost all other
functions, similar formulas aren't available; the roots must be obtained by
successive approximations, beginning with an initial estimate and then refining it.
This chapter presents a number of methods for obtaining the roots or zeroes of a
function.

A Graphical Method
As a preliminary step in finding the roots of a complicated or unfamiliar
function, it is helpful to make a chart of the function, in order to get preliminary
estimates of the roots, and indeed to find out how many roots there are. A cubic
equation such as the one shown in equation 8- 1 and Figure 8- 1,

y = x3 + 0. 13x2 - 0.0005~ - 0.0009 (8-1)
always has three roots, either three real roots as in Figure 8-1, or one real and two
imaginary roots. Figure 8-27 later in this chapter shows an example of the latter
case.
0.0004 -

0.0002 -

>
-0.0002 -
-0.0004 L I
I
-0.0006 ' 4

-0.20 -0.10 0.00 0.10
X

Figure 8-1. A regular polynomial with three real roots.

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