Associated legendre polynomials fail after certain degree

Hi,

I am using legendre polynomials for an application on spherical harmonics. However the code

legendre(170,0.5)

where 170 is the degree/order fail, giving me Inf or NaN. Is this considered a bug or is there way to aid the issue using higher precision somehow?

Best

Answer 1

[David Goodmanson]

Hi ailbeildce,

Try legendre(n,x,'norm') or legendre(n,x,'sch'). Each of these normalizes the associated legendre function slightly differently, and both leave out a factor in front that gets out of hand in a big way as m gets large [where m is the upper parameter in Pmn, 0<=m<=n, and m=0 corresponds to the usual Pn].

With either of those options, n can go up to at least 2400.

You can see what the factors are in 'doc legendre'. You will have to check, but I think the 'norm' option for Pmn gives you

 Int{-1,1} Pmn(x)^2 dx = 1,

appropriate for spherical harmonics.

[ailbeildce]

(As Mathworks doesn't have a PM feature) I'd like to thank you for this answer. I noticed you've been answering a few questions I was struggling and although it's been a while from the time of this question, thank you so much for helping me!

[David Goodmanson]

You're very welcome. I should probably know, but what is a PM feature?

[Elvis Alexander Agüero Vera]

I guess he refers to a private mesage.

Somewhat related question: I also need to calculate with efficiency the derivatives of the legendre Polynomials. I would appreciate a fast way of computing that.

Also, why is it that

f = matlabFunction(diff(legendreP(50, x), x))

is so unstable for degrees greater than, say, 50?

Answer 2

[Walter Roberson]

If you have the symbolic toolbox you can work with it

https://www.mathworks.com/help/symbolic/legendrep.html

[ailbeildce]

legendre() gives out more information than legendreP. Although I don't know if there's a way to generate Y_l^m where m!=0 with legendreP.

[Walter Roberson]

For integer m you can see https://en.wikipedia.org/wiki/Associated_Legendre_polynomials#Definition_for_non-negative_integer_parameters_%E2%84%93_and_m which the formula given in terms of derivatives. As the different orders correspond to different numbers of derivatives of the Legendre polynomial, you can find the different orders in a loop.