Generalized Laguerre polynomial

LaguerreGen calculates the generalized Laguerre polynomial L{n, alpha} for real valued alphas

Description

LaguerreGen calculates the generalized Laguerre polynomial L{n, alpha}

This function computes the generalized Laguerre polynomial L{n,alpha}.
If no alpha is supplied, alpha is set to zero and this function calculates the "normal" Laguerre polynomial.

Input:
 - n = nonnegative integer as degree level
 - alpha >= -1 real number (input is optional)

The output is formated as a polynomial vector of degree (n+1) corresponding to MATLAB norms (that is the highest coefficient is the first element).

Possible usage:
 - polyval(LaguerreGen(n, alpha), x) evaluates L{n, alpha}(x)
 - roots(LaguerreGen(n, alpha)) calculates roots of L{n, alpha}

Calculation is done recursively using matrix operations for very fast execution time. The formula is taken from Szegö: Orthogonal Polynomials, 1958, formula (5.1.10)

Acknowledgements

The author wishes to acknowledge the following in the creation of this submission:
AssociatedLaguerrePoly.m

• MATLAB release: MATLAB 7.3 (R2006b)

[John D'Errico]

Not bad. But it could be better.

The help is acceptable, but not what I'd call great. It never carefully defines the inputs, although it does give an example of use of the function, so you can figure out which order they must appear.

Based on the help, I was planning on giving this a rating of 4, until I tripped over a bug. The help states:

% Given nonnegative integer n and alpha >= -1

The zero'th order Laguerre polynomial (for any value of alpha) is well defined.

  L_0(x,alpha) = 1

and since zero is a non-negative integer, the help suggests that LaguerreGen can provide that polynomial. In fact, an error results.

>> LaguerreGen(0,0)

??? Subscripted assignment dimension mismatch.

Error in ==> LaguerreGen at 33
L(2,:)=[zeros(1, n-1), -1, (alpha+1)];

The higher order polynomials are computable, and the code is indeed efficient. I checked a few polynomials up to about order 10, and the coefficients were accurate.

I was a little surprised that no default was supplied for alpha. I.e., when alpha == 0 (or is not supplied) a generalized Laguerre should reduce to a standard Laguerre.

[John D'Errico]

This now properly returns the zero'th order polynomial. The help is better. There is a default for alpha. Preallocation is now used to speed it up. This now looks like a 5 rating to me.

[Geert Van Damme]

I'd like to propose the following elegant alternative: determine the coefficients of the associated Laguerre polynomial of order n, by determining the coefficients of the characteristic polynomial of its companion matrix:

function [c] = Laguerre_coeff(n, alpha)

i = 1:n;
a = (2*i-1) + alpha;
b = sqrt( i(1:n-1) .* ((1:n-1) + alpha) );
CM = diag(a) + diag(b,1) + diag(b,-1);

c = (-1)^n/factorial(n) * poly(CM);

[GHADBANE Ahmed]