Is there a faster way to symbolically integrate Legendre polynomials?

[member 428835]

Hi PF!

I'm doing several operations involving integrating Legendre polynomials. Since I am trying to loop through, my approach is something like

N = 5;
a = 0.5;
x =linspace(-1,1,100);
for k=1:N

    v(:,k) = legendreP(k+1,x)-legendreP(k+1,a)/legendreP(1,a)*legendreP(1,x);

end% for j

where I then perform Gram-Schmidt over each $v$ vector. Desired accuracy implies I need to have $x$ have about 1000 components (actually more). I would like to do this symbolically, but I'm unsure how to build $v$ as a function handle (the looping is throwing me off). Any help?

I have it working well in Mathematica, but it's a little slower than I'd prefer.

 

[DrClaude]

Desired accuracy implies I need to have $x$ have about 1000 components (actually more).

That's because you are not using the right approach. You should not be sampling the function at regular intervals. Check out these links:
https://en.wikipedia.org/wiki/Gaussian_quadrature
http://mathworld.wolfram.com/Legendre-GaussQuadrature.html
http://homepage.divms.uiowa.edu/~atkinson/ftp/ENA_Materials/Overheads/sec_5-3.pdf

Edit: Found this also: https://www.mathworks.com/matlabcen...0-legendre-gauss-quadrature-weights-and-nodes

 

[mfig]

You can define V as a function handle this way.

V = @(x,k,a) legendreP(k+1,x)-legendreP(k+1,a)/legendreP(1,a)*legendreP(1,x);

Then to do the integration, you can do this:

integral(@(x) V(x,1,0.05),-1,1)

 

[member 428835]

You can define V as a function handle this way.

V = @(x,k,a) legendreP(k+1,x)-legendreP(k+1,a)/legendreP(1,a)*legendreP(1,x);

Then to do the integration, you can do this:

integral(@(x) V(x,1,0.05),-1,1)

But how can I loop through this so V contains multiple values of a, something like V being a vector, where each entry corresponds to a Legendre polynomial with a different a value?

 

[mfig]

If you post what you are trying to do with V, meaning the code that uses V, perhaps I can help. :-)

 

[member 428835]

If you post what you are trying to do with V, meaning the code that uses V, perhaps I can help. :-)

The code is very long. I can send you it if you'd like but perhaps this way would be easier: let's say I'm trying to build a matrix $$A_{ij} = \int_0^1\sin(i x)\sin(j x)\,dx$$ One way to compute this integral is to make a matrix of sines such that the rows represent a node $x_i$ and the columns represent a harmonic. Example
$$
S = \begin{bmatrix}
\sin(x) & \sin(2x)&\sin (3x)
\end{bmatrix}
$$

Then to compute $\int_0^1 \sin(ix)\sin(jx)\,dx$ I take $trapz(S(:,i).*S(:,j))dx$. However, isn't there a way to define $S$ (matrix of sines) as a vector of function handles rather than a matrix, so I wouldn't have to explicitly specify the $x$ nodes?

 

[mfig]

O.k., I think I see. What is wrong with the previous suggestion? Option 2 doesn't depend on specified x nodes, and is very accurate by default.

 

[member 428835]

Thanks!