Numerical Integration - Numerical Methods in Engineering with MATLAB - page 227

k ( n )
(2 n )!

f (2 n ) ( c ), E = 0 < c < 1 (6.39)

where k (2) = 0 . 00 285, k (3) = 0 . 000 17, k (4) = 0 . 000 01.

gaussNodes

The function gaussNodes computes the nodal abscissas x i and the corresponding
weights A i usedinGauss-Legendrequadrature. 14

It can be shown that the approxi-

mate values of the abscissas are

cos π

− .

( i 0 25)

x i =

n + 0 . 5

Using these approximations as the starting values, wecompute the nodal ab-
scissas by finding the nonnegativezeros of the Legendre polynomial p n ( x ) with
the Newton-Raphsonmethod (the negativezeros areobtained from symmetry).
Note that gaussNodes calls the subfunction legendre , which returns p n ( t ) and its
derivative.

function [x,A] = gaussNodes(n,tol)
% Computes nodal abscissas x and weights A of
% Gauss-Legendre n-point quadrature.
% USAGE: [x,A] = gaussNodes(n,epsilon,maxIter)
% tol = error tolerance (default is 1.0e4*eps).

if nargin < 2; tol = 1.0e4*eps; end
A=zeros(n,1);x=zeros(n,1);
nRoots = fix(n + 1)/2; % Number of non-neg. roots

fori=1:nRoots
t = cos(pi*(i - 0.25)/(n + 0.5)); % Approx. roots
forj=i:30
[p,dp] = legendre(t,n); % Newton's

dt=-p/dp;t=t+dt; %rootfinding

if abs(dt) < tol % method

x(i) = t; x(n-i+1) = -t;
A(i) = 2/(1-tˆ2)/dpˆ2; % Eq. (6.25)

A(n-i+1) = A(i);
break

14

Thisfunctionis an adaptation of aroutine in Numerical Recipes in Fortran 90 , W. H. Press et al.,
CambridgeUniversity Press (1996).