MHB Zeros of generalised Laguerre polynomial

[ognik]

Hi - does anyone know of a program library/subroutine/some other source, to find the zeros of a generalised Laguerre polynomial? ie. $ L^{\alpha}_N (x_i) = 0 $

 

[Ackbach]

In Mathematica (or Wolfram Development Platform), the command

LaguerreL[n,a,x]

yields the generalized Laguerre polynomials $L_{n}^{a}(x)$. You could then use the power of Mathematica to find zeros as much as you like. Is this for a particular value of $a$? And to what order of polynomial are you intending to go? Mathematica can find symbolic roots all the up to 4th-order. Naturally, it might be difficult to find a fifth-order, since a general formula does not exist. If you have a particular value of $a$ in mind, then you can get Mathematica to find the roots numerically fairly easily, as well.

 

[ognik]

Thanks - I should have said I wanted something I could include in a program (Fortran); and I will need N up to potentially 100 ...

For general interest, the subs I found are:

LAGUERRE_POLYNOMIAL - Laguerre Polynomials

https://github.com/exciting/exciting/blob/master/src/src_gw/gaulag.f90

Back to Mathematica, I tried Roots[LaguerreL[4, 2, x] == 0, x] - but that gives me a string of calculations for each root, what do I need to do please, to simplify/just get the number for each root?

 

[topsquark]

You aren't getting a string of calculations, it's telling you the exact values of the 4 roots. If all you need are approximations, try N[Solve[LaguerreL[4, 2, x] == 0, x]]. That will give you a list of the approximate values of the 4 roots. If you need more precision in the answer you can use N[Solve[LaguerreL[4, 2, x] == 0, x],6] to get the answer to six decimal places. Be warned, though, depending on what algorithms Mathematica has to use to get the solutions you may not be able to get six decimal precision.

-Dan

 

[ognik]

That makes rather good sense, and is all useful info, thanks.
My next thought is - is there a way to get Mathematica to output to a file, like a csv?

BTW, it does return values for N > 4, I tried up to 100 ... but only 7 decimal places.