Mathematical Methods for Physics and Engineering : A Comprehensive Guide

27.4 NUMERICAL INTEGRATION

factor is treated accurately in Gauss–Chebyshev integration. Thus

∫ 1

− 1

f(x)

√

1 −x^2

dx≈

∑n

i=1

wif(xi), (27.44)

where the integration pointsxiare the zeros of the Chebyshev polynomials of

the first kindTn(x)andwiare the corresponding weights. Fortunately, both sets

are analytic and can be written compactly for allnas

xi=cos

(i−^12 )π

n

,wi=

π

n

fori=1,... ,n. (27.45)

Note that, for any givenn, all points are weighted equally and that no special

action is required to deal with the integrable singularities atx=±1; they are

dealt with automatically through the weight function.

For integrals involving factors of the form (1−x^2 )^1 /^2 , the corresponding formula,

based on Chebyshev polynomials of the second kindUn(x), is

∫ 1

− 1

f(x)

√

1 −x^2 dx≈

∑n

i=1

wif(xi), (27.46)

with integration points and weights given, fori=1,... ,n,by

xi=cos

iπ

n+1

,wi=

π

n+1

sin^2

iπ

n+1

. (27.47)

For discussions of the many other schemes available, as well as their relative

merits, the reader is referred to books devoted specifically to the theory of

numerical analysis. There, details of integration points and weights, as well as

quantitative estimates of the error involved in replacing an integral by a finite

sum, will be found. Table 27.9 gives the points and weights for a selection of

Gauss–Laguerre and Gauss–Hermite schemes.§

27.4.4 Monte Carlo methods

Surprising as it may at first seem, random numbers may be used to carry out

numerical integration. The random element comes in principally when selecting

the points at which the integrand is evaluated, and naturally does not extend to

the actual values of the integrand!

For the most part we will continue to use as our model one-dimensional

integrals between finite limits, as typified by equation (27.34). Extensions to cover

infinite or multidimensional integrals will be indicated briefly at the end of the

section. It should be noted here, however, that Monte Carlo methods – the name

§They, and those presented in table 27.8 for Gauss–Legendre integration, are taken from the much

more comprehensive sets to be found in M. Abramowitz and I. A. Stegun (eds),Handbook of

Mathematical Functions(New York: Dover, 1965).