27

Numerical methods

It happens frequently that the end product of a calculation or piece of analysis

is one or more algebraic or differential equations, or an integral that cannot be

evaluated in closed form or in terms of tabulated or pre-programmed functions.

From the point of view of the physical scientist or engineer, who needs numerical

values for prediction or comparison with experiment, the calculation or analysis

is thus incomplete.

With the ready availability of standard packages on powerful computers for

the numerical solution of equations, both algebraic and differential, and for the

evaluation of integrals, in principle there is no need for the investigator to do

anything other than turn to them. However, it should be a part of every engineer’s

or scientist’s repertoire to have some understanding of the kinds of procedure that

are being put into practice within those packages. The present chapter indicates

(at a simple level) some of the ways in which analytically intractable problems

can be tackled using numerical methods.

In the restricted space available in a book of this nature, it is clearly not

possible to give anything like a full discussion, even of the elementary points that

will be made in this chapter. The limited objective adopted is that of explaining

and illustrating by simple examples some of the basic principles involved. In

many cases, the examples used can be solved in closed form anyway, but this

‘obviousness’ of the answers should not detract from their illustrative usefulness,

and it is hoped that their transparency will help the reader to appreciate some of

the inner workings of the methods described.

The student who proposes to study complicated sets of equations or make

repeated use of the same procedures by, for example, writing computer programs

to carry out the computations, will find it essential to acquire a good under-

standing of topics hardly mentioned here. Amongst these are the sensitivity of

the adopted procedures to errors introduced by the limited accuracy with which

a numerical value can be stored in a computer (rounding errors) and to the