Mathematical Methods for Physics and Engineering : A Comprehensive Guide

27.4 NUMERICAL INTEGRATION

It will be seen that, by replacing eachnmin the summation byf(x, y, z)nm,

this procedure could be extended to estimate the integral of the functionfover

the volume of the solid. The method has special valueiffis too complicated to

have analytic integrals with respect tox, yandzor if the limits of any of these

integrals are determined by anything other than the simplest combinations of the

other variables. If large values offare known to be concentrated in particular

regions of the integration volume, then some form of stratified sampling should

be used.

It will be apparent that this general method can be extended to integrals

of general functions, bounded but not necessarily continuous, over volumes with

complicated bounding surfaces and, if appropriate, in more than three dimensions.

Random number generation

Earlier in this subsection we showed how to evaluate integrals using sequences of

numbers that we took to be distributed uniformly on the interval 0≤ξ<1. In

reality the sequence of numbers is not truly random, since each is generated in

a mechanistic way from its predecessor and eventually the sequence will repeat

itself. However, the cycle is so long that in practice this is unlikely to be a problem,

and the reproducibility of the sequence can even be turned to advantage when

checking the accuracy of the rest of a calculational program. Much research has

gone into the best ways to produce such ‘pseudo-random’ sequences of numbers.

We do not have space to pursue them here and will limit ourselves to one recipe

that works well in practice.

Given any particular starting (integer) valuex 0 , the following algorithm will

generate a full cycle ofmvalues forξi, uniformly distributed on 0≤ξi<1,

before repeats appear:

xi=axi− 1 +c (modm); ξi=

xi

m

.

Herecis an odd integer andahas the forma=4k+ 1, withkan integer. For

practical reasons, in computers and calculatorsmis taken as a (fairly high) power

of 2, typically the 32nd power.

The uniform distribution can be used to generate random numbersydistributed

according to a more general probability distributionf(y) on the rangea≤y≤b

if the inverse of the indefinite integral offcan be found, either analytically or by

means of a look-up table. In other words, if

F(y)=

∫y

a

f(t)dt,

for whichF(a)=0andF(b)=1,thenF(y) is uniformly distributed on (0,1).

This approach is not limited to finiteaandb;acould be−∞andbcould be∞.

The procedure is thus to select a random numberξfrom a uniform distribution