Legendre Laguerre and Hermite - Gauss Quadrature

Nodes and weights for Legendre Laguerre and Hermite - Gauss Quadrature

Description

This .zip file contains 3 mfiles for computing the nodes and weights for Legendre Laguerre and Hermite - Gauss Quadrature of any order n.

Contrary to most of the files in use, the included files are all based on a symmetrical companion matrix, the eigenvalues of which are always real and correspond to the roots of the respective polynomials.

Hence, there is a 100% certainty of avoiding complex roots at high orders. It is known that the latter issue often causes huge numerical troubles.

• MATLAB release: MATLAB 7.0.4 (R14SP2)

[Marios Karaoulis]

Very useful program.

How can I calculate the GaussLegendre in the [0 1] space?

[Marios Karaoulis]

One addition I did is this

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

a and b are user provided, and I think now that you can shift from [1 1] to [a b]

[David Holdaway]

Very nice submission, simple and fast

[David Holdaway]

Oh and I forgot to add this required you insist x = 0 is the middle value for odd via
if mod(n,2)==1
    x(ceil(n/2))=0;
end

[David Holdaway]

I added this on the end to refine the eigenvalues via newtons method

if mod(n,2)==1
    x(ceil(n/2))=0;
end

    z=x(x>=0);
   success = false;
    for its=1:50 %maxit=50, usually will take 2
        p1 = pi^(-1/4);
        p2=0;
        for j=1:n %make hermite we need
            p3=p2;
            p2=p1;
            p1=z.*sqrt(2/j).*p2-sqrt((j-1)/j).*p3;
        end
    pp = sqrt(2*n).*p2;
    z1=z;
    z=z1-p1./pp;
    if all(abs(z-z1)< 20*eps)
        success = true;
        break
    end
    end
    if ~success
        warning('failed to converge to desired accuracy')
    end
    w(x>=0) = 2./pp.^2;
    w(x<=0) = flipud(w(x>=0));
    x(x>=0) = z;
    x(x<=0) = -flipud(z);