Mathematical Methods for Physics and Engineering : A Comprehensive Guide

27.4 NUMERICAL INTEGRATION

27.4.3 Gaussian integration

In the cases considered in the previous two subsections, the functionf was

mimicked by linear and quadratic functions. These yield exact answers iff

itself is a linear or quadratic function (respectively) ofx. This process could

be continued by increasing the order of the polynomial mimicking-function so

as to increase the accuracy with which more complicated functionsfcould be

numerically integrated. However, the same effect can be achieved with less effort

by not insisting upon equally spaced pointsxi.

The detailed analysis of such methods of numerical integration, in which the

integration points are not equally spaced and the weightings given to the values at

each point do not fall into a few simple groups, is too long to be given in full here.

Suffice it to say that the methods are based upon mimicking the given function

with a weighted sum of mutually orthogonal polynomials. The polynomials,Fn(x),

are chosen to be orthogonal with respect to a particular weight functionw(x), i.e.

∫b

a

Fn(x)Fm(x)w(x)dx=knδnm,

whereknis some constant that may depend uponn. Often the weight function is

unity and the polynomials are mutually orthogonal in the most straightforward

sense; this is the case for Gauss–Legendre integration for which the appropriate

polynomials are the Legendre polynomials,Pn(x). This particular scheme is

discussed in more detail below.

Other schemes cover cases in which one or both of the integral limitsaandb

are not finite. For example, if the limits are 0 and∞and the integrand contains

a negative exponential functione−αx, a simple change of variable can cast it

into a form for which Gauss–Laguerre integration would be particularly well

suited. This form of quadrature is based upon the Laguerre polynomials, for

which the appropriate weight function isw(x)=e−x. Advantage is taken of this,

and the handling of the exponential factor in the integrand is effectively carried

out analytically. If the other factors in the integrand can be well mimicked by

low-order polynomials, then a Gauss–Laguerre integration using only a modest

number of points gives accurate results.

If we also add that the integral over the range−∞to∞of an integrand

containing an explicit factor exp(−βx^2 ) may be conveniently calculated using a

scheme based on the Hermite polynomials, the reader will appreciate the close

connection between the various Gaussian quadrature schemes and the sets of

eigenfunctions discussed in chapter 18. As noted above, the Gauss–Legendre

scheme, which we discuss next, is just such a scheme, though its weight function,

being unity throughout the range, is not explicitly displayed in the integrand.

Gauss–Legendre quadrature can be applied to integrals over any finite range

though the Legendre polynomialsP(x) on which it is based are only defined