$\mathcal{O}(1)$ Computation of Legendre Polynomials and Gauss--Legendre Nodes and Weights for Parallel Computing

Abstract

A self-contained set of algorithms is proposed for the fast evaluation of Legendre polynomials of arbitrary degree and argument $\in [-1,1]$. More specifically the time required to evaluate any Legendre polynomial, regardless of argument and degree, is bounded by a constant; i.e., the complexity is $\mathcal{O}(1)$. The proposed algorithm also immediately yields an $\mathcal{O}(1)$ algorithm for computing an arbitrary Gauss--Legendre quadrature node. Such a capability is crucial for efficiently performing certain parallel computations with high order Legendre polynomials, such as computing an integral in parallel by means of Gauss--Legendre quadrature and the parallel evaluation of Legendre series. In order to achieve the $\mathcal{O}(1)$ complexity, novel efficient asymptotic expansions are derived and used alongside known results. A C++ implementation is available from the authors that includes the evaluation routines of the Legendre polynomials and Gauss--Legendre quadrature rules.