Mathematical Methods for Physics and Engineering - Matematica.NET

27Numerical methodsIt happens frequently that the end product of a calculation or piece of analysisis one or more algebraic or differential equations, or an integral that cannot beevaluated 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 numericalvalues for prediction or comparison with experiment, the calculation or analysisis thus incomplete.With the ready availability of standard packages on powerful computers forthe numerical solution of equations, both algebraic and differential, and for theevaluation of integrals, in principle there is no need for the investigator to doanything other than turn to them. However, it should be a part of every engineer’sor scientist’s repertoire to have some understanding of the kinds of procedure thatare being put into practice within those packages. The present chapter indicates(at a simple level) some of the ways in which analytically intractable problemscan be tackled using numerical methods.In the restricted space available in a book of this nature, it is clearly notpossible to give anything like a full discussion, even of the elementary points thatwill be made in this chapter. The limited objective adopted is that of explainingand illustrating by simple examples some of the basic principles involved. Inmany 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 ofthe inner workings of the methods described.The student who proposes to study complicated sets of equations or makerepeated use of the same procedures by, for example, writing computer programsto carry out the computations, will find it essential to acquire a good understandingof topics hardly mentioned here. Amongst these are the sensitivity ofthe adopted procedures to errors introduced by the limited accuracy with whicha numerical value can be stored in a computer (rounding errors) and to the984