Mathematical Methods for Physics and Engineering : A Comprehensive Guide

27.4 NUMERICAL INTEGRATION

has become attached to methods based on randomly generated numbers – in

many ways come into their own when used on multidimensional integrals over

regions with complicated boundaries.

It goes without saying that in order to use random numbers for calculational

purposes a supply of them must be available. There was a time when they were

provided in book form as a two-dimensional array of random digits in the range

0 to 9. The user could generate the successive digits of a random number of

any desired length by selecting their positions in the table in any predetermined

and systematic way. Nowadays all computers and nearly all pocket calculators

offer a function which supplies a sequence of decimal numbers,ξ,that,forall

practical purposes, are randomly and uniformly chosen in the range 0≤ξ<1.

The maximum number of significant figures available in each random number

depends on the precision of the generating device. We will defer the details of

how these numbers are produced until later in this subsection, where it will

also be shown how random numbers distributed in a prescribed way can be

generated.

All integrals of the general form shown in equation (27.34) can, by a suitable

change of variable, be brought to the form

θ=

∫ 1

0

f(x)dx, (27.48)

and we will use this as our standard model.

All approaches to integral evaluation based on random numbers proceed by

estimating a quantity whose expectation value is equal to the sought-for valueθ.

The estimatortmust be unbiased, i.e. we must haveE[t]=θ, and the method

must provide some measure of the likely error in the result. The latter will appear

generally as the variance of the estimate, with its usual statistical interpretation,

and not as a band in which the true answer is known to lie with certainty.

The various approaches really differ from each other only in the degree of

sophistication employed to keep the variance of the estimate ofθsmall. The

overall efficiency of any particular method has to take into account not only the

variance of the estimate but also the computing and book-keeping effort required

to achieve it.

We do not have the space to describe even the most elementary methods in full

detail, but the main thrust of each approach should be apparent to the reader

from the brief descriptions that follow.

Crude Monte Carlo

The most straightforward application is one in which the random numbers are

used to pick sample points at whichf(x) is evaluated. These values are then