[X,Y,Wx,Wy]=triquad(N,v)

function [X,Y,Wx,Wy]=triquad(N,v)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% triquad.m - Gaussian Quadrature for a triangular domain
%
% This scripts computes the N^2 nodes and weights for a triangle with
% vertices given by the 3x2 vector v. The nodes are produced by collapsing
% the square to a triangle. 
%
% Sample Usage: 
%
% >>[X,Y,Wx,Wy]=triquad(8,[0 0; 0 2; 2 1])
% >>f=@(x,y) exp(x+y);
% >>Q=Wx'*feval(f,X,Y)*Wy;
%
% Reference:  J.N. Lyness, Ronald Cools, A Survey of Numerical Cubature
%             over Triangles (1994)
%             http://citeseer.ist.psu.edu/lyness94survey.html
%
% Written by: Greg von Winckel
% Contact: gregvw(at)math(dot)unm(dot)edu
% http://math.unm.edu/~gregvw
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

n=1:N;  nnk=2*n+1; A=[1/3 repmat(1,1,N)./(nnk.*(nnk+2))];
n=2:N; nnk=nnk(n); B1=2/9; nk=n+1; nnk2=nnk.*nnk;
B=4*(n.*nk).^2./(nnk2.*nnk2-nnk2); ab=[A' [2; B1; B']]; s=sqrt(ab(2:N,2));
[V,X]=eig(diag(ab(1:N,1),0)+diag(s,-1)+diag(s,1));
[X,I]=sort(diag(X)); x=(X+1)/2; wx=ab(1,2)*V(1,I)'.^2/4;

N=N-1; N1=N+1; N2=N+2;  y=cos((2*(N:-1:0)'+1)*pi/(2*N+2));
L=zeros(N1,N2);  y0=2;  iter=0;
while max(abs(y-y0))>eps    
    L(:,1)=1;    L(:,2)=y;   
    for k=2:N1
        L(:,k+1)=( (2*k-1)*y.*L(:,k)-(k-1)*L(:,k-1) )/k;
    end
    Lp=(N2)*( L(:,N1)-y.*L(:,N2) )./(1-y.^2);   
    y0=y;    y=y0-L(:,N2)./Lp;  iter=iter+1;
end

cd=[ 1, 0, 0; -1, 0, 1; 0, 1,-1]*v; 
t=(1+y)/2;  Wx=abs(det(cd(2:3,:)))*wx;  Wy=1./((1-y.^2).*Lp.^2)*(N2/N1)^2;
[tt,xx]=meshgrid(t,x); yy=tt.*xx;
X=cd(1,1)+cd(2,1)*xx+cd(3,1)*yy;    Y=cd(1,2)+cd(2,2)*xx+cd(3,2)*yy;