One flaw that we had seen in Newton-Cotes methods is as you proceeded to eliminate more terms in order to reduce error it is not uncommon to actually increase error. This is called Runge’s phenomenon and is when the error grows exponentially with degree n. This occurs since with certain high degree sets the higher order derivatives become increasingly large. One way to avoid this is the Gaussian Quadrature method.

Gaussian Quadrature uses unequally spaced points that are unevenly weighted, unlike the equally spaced and unweighted points of Newton-cotes. The domain for this method is [-1,1] and estimates the integral of f(x) over this domain by summing the products of weight of I by the function at the ith point from 1 to n the number of points. (This can be used for domains other than [-1,1] but it must be converted in to this from before the method can be used.)

In order to calculate the Xi and the Wi the Legendre polynomials Pn(x) is used. Xi is the ith root of Pn and Wi = (2 / ((1 – Xi^2) * (Pn’(xi))^2)).

For exaple with low values of n

n = 1, x1 = 0, w1 = 2

n = 2, x1 = (1/3)^(1/2), w1 = 1, x2 = - (1/3)^(1/2), w2 = 1

n=3, x1 = 0, w1 = 8/9, x2,3 = +-(3/5)^(1/2), w2,3 = 5/9

By using such a distribution of points and weights we increase the density of points at the end points, but to accommodate that increase the weight of the more center points. Although calculations of the weights and the distribution of points is more complicated that the evenly spaced equal weighed Newton-Cotes points this extra work results in a more stable and accurate integral.

Sources:

http://pathfinder.scar.utoronto.ca/~dyer/csca57/book_P/node44.html

http://pathfinder.scar.utoronto.ca/~dyer/csca57/book_P/node44.html