Efficient computation of Laguerre polynomials

TL;DR

This paper introduces an efficient algorithm and a Fortran 90 module for computing Laguerre polynomials with high accuracy across a broad parameter range, utilizing recurrence relations and asymptotic expansions.

Contribution

It presents a novel, accurate, and efficient computational method and software implementation for Laguerre polynomials covering various parameter regimes.

Findings

Achieves relative accuracy close to 10^{-12} for Laguerre polynomial computations.

Uses recurrence relations and asymptotic expansions depending on parameter regions.

Provides a Fortran 90 module (LaguerrePol) for practical implementation.

Abstract

An efficient algorithm and a Fortran 90 module (LaguerrePol) for computing Laguerre polynomials $L^{(\alpha)}_n(z)$ are presented. The standard three-term recurrence relation satisfied by the polynomials and different types of asymptotic expansions valid for $n$ large and $\alpha$ small, are used depending on the parameter region. Based on tests of contiguous relations in the parameter $\alpha$ and the degree $n$ satisfied by the polynomials, we claim that a relative accuracy close or better than $10^{-12}$ can be obtained using the module LaguerrePol for computing the functions $L^{(\alpha)}_n(z)$ in the parameter range $z \ge 0$, $-1 < \alpha \le 5$, $n \ge 0$.