function [x,w]=lgwt(N,a,b)
% This script is for computing definite integrals using Legendre-Gauss 
% Quadrature. Computes the Legendre-Gauss nodes and weights  on an interval
% [a,b] with truncation order N
%
% Suppose you have a continuous function f(x) which is defined on [a,b]
% which you can evaluate at any x in [a,b]. Simply evaluate it at all of
% the values contained in the x vector to obtain a vector f. Then compute
% the definite integral using sum(f.*w);
%
% Written by Greg von Winckel - 02/25/2004
  N2=N+1;

  xu=linspace(-1,1,N)';

  % Initial guess
  y=cos((2*(0:N-1)'+1)*pi/(2*N))+(0.27/N)*sin(pi*xu*(N-1)/N2);

  % Legendre-Gauss Vandermonde Matrix
  L=zeros(N,N2);

  % Compute the zeros of the N+1 Legendre Polynomial
  % using the recursion relation and the Newton-Raphson method

  y0=Inf(N,1);

  % Iterate until new points are uniformly within epsilon of old points
  while max(abs(y-y0))>eps
    L(:,1)=1; L(:,2)=y;
    
    for k=2:N
      L(:,k+1)=( (2*k-1)*y.*L(:,k)-(k-1)*L(:,k-1) )/k;
    end
 
    Lp=N2*( L(:,N)-y.*L(:,N2) )./(1-y.^2);   
    
    y0=y;
    y=y0-L(:,N2)./Lp;
  end

  % Linear map from[-1,1] to [a,b]
  x=(a*(1-y)+b*(1+y))/2;      

  % Compute the weights
  w=(b-a)./((1-y.^2).*Lp.^2)*(N2/N)^2;