Laguerre polynomials of large degree.

Dear MATLAB community members,

Has anyone experience issues with MATLABs laguerreL function with very large values of n,

I noticed that the results for large values of x and m in laguerreL(m,x) the answer of MATHEMATICA and MATLAB also do not match. Is it a known issue due to (possible use) of hypergeometric representation in laguerreL? I would like to use it to approximate an integral involving Laguerre polynomial products, where the integrand is highly oscillatory. Using quadgk, I can get some outputs; but is there a way to verify these solutions specially at higher values m and n; given discrepencies in MATHEMATICA, using recursive definitions, or built-in laguerreL at large m and x?

Thank you in advance!

Answer 1

[John D'Errico]

If you are using double precision, then OF COURSE high order polynomials of any form in MATLAB should not be trusted!!!! At least, not if you are using double precision. High order polynomials and double precision don't mix well.

But that just means you need to be careful. I don't really understand the problem.

help laguerreL

--- help for sym/laguerreL ---

  LAGUERREL   Laguerre's L function and Laguerre polynomials. 
 
     Y = LAGUERREL(N,A,X) is the generalized Laguerre 
     function with parameter A. It is an N-th degree
     polynomial in X if N is a nonnegative integer.

    Documentation for sym/laguerreL
       doc sym/laguerreL

    Other uses of laguerreL

       double/laguerreL    single/laguerreL

And here we can find a form for those polynomials:

https://en.wikipedia.org/wiki/Laguerre_polynomials

syms x
laguerreL(50,0,x)

ans = 

That first term coefficient should be 1/factorial(50).

factorial(sym(50))

ans = 

30414093201713378043612608166064768844377641568960512000000000000

The second term coefficient should be -50^2/factorial(50).

50^2/factorial(sym(50))

ans = 

It looks to be working to me, to full precision, as long as I use symbolic tools. If you are trying to use double precision arithmetic, AND high order polynomials, well, you get what you should expect.

If you have some specific issue, then you need to show EXACTLY what you are doing where you see a discrepancy.