Gauss-Laguerre

Integrates a functio using Gauss-Laguerre cuadrature method

Description

One function produces the Laguerre polynomial and the other integrates

• MATLAB release: MATLAB 6.5 (R13)
• Other Files: polelague.m, ; gausslaguerr.m

[hanna mermelstein]

The idea is quite nice, but the procedure does not work properly. From n >35 the zeros of the Laguerre polynomals start to be complex, what they should not be. Seems, that the roots-routine fails at this point. Up to n=32 the zeros are correctly calculated (see Krylov, Approximate calculation of integrals, Macmillan company, new york : 1962). But the weights (called coefficients in gausslaguerr.m) are completely wrong, thus the integration does not compute the right values at all.

[hanna mermelstein]

The idea is quite nice, but the procedure does not work properly. From n >35 the zeros of the Laguerre polynomals start to be complex, what they should not be. Seems, that the roots-routine fails at this point. Up to n=32 the zeros are correctly calculated (see Krylov, Approximate calculation of integrals, Macmillan company, new york : 1962). But the weights (called coefficients in gausslaguerr.m) are completely wrong, thus the integration does not compute the right values at all.

[zarelly orellana]

hola deseo mucha informacion sobre algoritmos de laguerre enmatlab

[Teddy Qian]

I fixed some mistakes,now it's weights are right,just n<32.
if n>32,the problem is still there.
gausslaguerr.m
function [a,b]=gausslaguerr(n)
p=polelague(n);
x=roots(p(n+1,:));
x=x(x~=0);
for k=1:n
    q(k)=(factorial(n))^2*x(k)*exp(x(k))/(polyval(p(n+2,:),x(k)))^2;
end
a=x;
b=q;

polelague.m
function p=polelague(n)
p(1,1)=1;
p(2,1:2)=[-1 1];
for k=2:n+1
   p(k+1,1:k+1)=((2*(k-2)*[0 p(k,1:k)]+3*[0 p(k,1:k)]-[p(k,1:k) 0]-(k-1).^2*[0 0 p(k-1,1:k-1)]));
end

[Geert V.D.]

The problem with the complex roots is caused by the the 'roots' function of Matlab. This determines the roots as the eigenvalues of the polynomials companion matrix 'CM', wich is such that det(x*I-CM)=P_n(x). To this aim, the general form of the companion matrix is used.
However, when the degree of the polynomial becomes 'too' large, the companion matrix is badly conditioned and yields inaccurate (in this case complex) roots.
The solution to this problem is hence to write your own 'roots' function, with a feasible companion matrix. Since we want the roots to be real, we would like the companion matrix to be symmetrical, as we know that all eigenvalues of a symmetrical matrix are real.
Below is the code for the Gauss-Laguerre, Gauss-Legendre and Gauss-Hermite quadratures. As you can see, the code consists of 2 blocks: first construct a symmetrical companion matrix and then determine the (real) eigenvalues (i.e. the roots of the polynomial). All three functions produce the correct abscissas and weights, for any value n>=2.

function [x, w] = GaussLegendre_2(n)

i = 1:n-1;
a = i./sqrt(4*i.^2-1);
CM = diag(a,1) + diag(a,-1);

[V L] = eig(CM);
[x ind] = sort(diag(L));
V = V(:,ind)';
w = 2 * V(:,1).^2;

function [x, w] = GaussLaguerre_2(n, alpha)

i = 1:n;
a = (2*i-1) + alpha;
b = sqrt( i(1:n-1) .* ((1:n-1) + alpha) );
CM = diag(a) + diag(b,1) + diag(b,-1);

[V L] = eig(CM);
[x ind] = sort(diag(L));
V = V(:,ind)';
w = gamma(alpha+1) .* V(:,1).^2;

function [x, w] = GaussHermite_2(n)

i = 1:n-1;
a = sqrt(i/2);
CM = diag(a,1) + diag(a,-1);

[V L] = eig(CM);
[x ind] = sort(diag(L));
V = V(:,ind)';
w = sqrt(pi) * V(:,1).^2;

[K C]

Geert VD's Code is efficient using the companion matrix.
As a rule of thumb with matlab, you should always avoid using the loops if you can use vectorized methods.

[Johannes Ganze]

Seems that all the Gauss type quadrature methods fail at large n ,for gauss-legendre quadrature the critical point is approximately n=50.