Mathematical Methods for Physics and Engineering - Matematica.NET

27.4 NUMERICAL INTEGRATION27.4.3 Gaussian integrationIn the cases considered in the previous two subsections, the function f wasmimicked by linear and quadratic functions. These yield exact answers if fitself is a linear or quadratic function (respectively) of x. This process couldbe continued by increasing the order of the polynomial mimicking-function soas to increase the accuracy with which more complicated functions f could benumerically integrated. However, the same effect can be achieved with less effortby not insisting upon equally spaced points x i .The detailed analysis of such methods of numerical integration, in which theintegration points are not equally spaced and the weightings given to the values ateach 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 functionwith a weighted sum of mutually orthogonal polynomials. The polynomials, F n (x),are chosen to be orthogonal with respect to a particular weight function w(x), i.e.∫ baF n (x)F m (x)w(x) dx = k n δ nm ,where k n is some constant that may depend upon n. Often the weight function isunity and the polynomials are mutually orthogonal in the most straightforwardsense; this is the case for Gauss–Legendre integration for which the appropriatepolynomials are the Legendre polynomials, P n (x). This particular scheme isdiscussed in more detail below.Other schemes cover cases in which one or both of the integral limits a and bare not finite. For example, if the limits are 0 and ∞ and the integrand containsa negative exponential function e −αx , a simple change of variable can cast itinto a form for which Gauss–Laguerre integration would be particularly wellsuited. This form of quadrature is based upon the Laguerre polynomials, forwhich the appropriate weight function is w(x) =e −x . Advantage is taken of this,and the handling of the exponential factor in the integrand is effectively carriedout analytically. If the other factors in the integrand can be well mimicked bylow-order polynomials, then a Gauss–Laguerre integration using only a modestnumber of points gives accurate results.If we also add that the integral over the range −∞ to ∞ of an integrandcontaining an explicit factor exp(−βx 2 ) may be conveniently calculated using ascheme based on the Hermite polynomials, the reader will appreciate the closeconnection between the various Gaussian quadrature schemes and the sets ofeigenfunctions discussed in chapter 18. As noted above, the Gauss–Legendrescheme, 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 rangethough the Legendre polynomials P l (x) on which it is based are only defined1005