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 is an element of [-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 O(1). The proposed algorithm also immediately yields an 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 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