Associated legendre polynomials fail after certain degree

2017 12 16
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更新时间：2023 1 16
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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

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David Goodmanson
David Goodmanson 2017-12-19

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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.

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ailbeildce
ailbeildce 2018-3-29
(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
David Goodmanson 2018-3-30
You're very welcome. I should probably know, but what is a PM feature?
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?

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Walter Roberson
Walter Roberson 2017-12-16

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If you have the symbolic toolbox you can work with it
https://www.mathworks.com/help/symbolic/legendrep.html

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ailbeildce
ailbeildce 2017-12-17
编辑：ailbeildce 2017-12-17
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
Walter Roberson 2017-12-17
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.

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