4.1 Introduction

A common problem encountered in engineering analysis is this: given a function ( x), determine the values of x for which ( x)=0. The solutions (values of x) are known as the roots of the equation ( x)=0, or the zeroes of the function ( x).

Before proceeding further, it might be helpful to review the concept of a function. The equation

contains three elements: an input value x, an output value y and the rule for computing y. The function is said to be given if the rule is specified. In numerical computing the rule is invariably a computer algorithm. It may be a function statement, such as

or a complex procedure containing hundreds or thousands of lines of code. As long as the algorithm produces an output y for each input x, it qualifies as a function.

The roots of equations may be real or complex. The complex roots are seldom computed, since they rarely have physical significance. An exception is the polynomial equation

where the complex roots maybe meaningful (as in the analysis of damped vibrations, for example). For the time being, we will concentrate on finding the real roots of equations. Complex zeroes of polynomials are treated near the end of this chapter.

In general, an equation...