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Legendre Laguerre and Hermite - Gauss Quadrature

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19 Feb 2010 (Updated )

Nodes and weights for Legendre Laguerre and Hermite - Gauss Quadrature

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File Information
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.

Acknowledgements

This file inspired Nodal Dg and Correction Procedure Via Reconstruction (Cpr).

MATLAB release MATLAB 7.0.4 (R14SP2)
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Comments and Ratings (9)
04 Oct 2013 Manuel Diaz

Is it me or these routines are not working correctly in Matlab 2013b?
PS. I have been using them since 2010.

04 Oct 2013 Manuel Diaz  
29 Jan 2013 Giuseppe Cascarino

Simple and fast.

06 Jan 2013 afu2007 ??

nice!

01 May 2012 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);

01 May 2012 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

30 Apr 2012 David Holdaway

Very nice submission, simple and fast

20 Oct 2010 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]

20 Oct 2010 Marios Karaoulis

Very useful program.

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

Updates
19 Feb 2010

No updates were made. Only the description was changed: it should read "the eigenvalues of the companion matrix" instead of "the roots of the companion matrix". Obviously, a matrix has no roots.

21 Feb 2010

Some minor changes to the code's comments.

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