Evaluation of Gauss-Legendre curves

TL;DR

This paper introduces efficient algorithms for evaluating Gauss-Legendre curves and their derivatives using new polynomial representations and recurrence relations, optimizing computational complexity.

Contribution

The paper develops novel polynomial representations and recurrence-based algorithms for fast evaluation of Gauss-Legendre curves in multiple dimensions.

Findings

Achieves $O(n^2+dn)$ complexity for evaluating Gauss-Legendre curves.

Proposes multipoint evaluation algorithms with $O(Mdn+dn^2)$ complexity.

Provides new bases for representing Gauss-Legendre polynomials and derivatives.

Abstract

We present new representations of Gauss--Legendre polynomials and their derivatives in the shifted power basis and in bases related to symmetric orthogonal Jacobi polynomials. Using these representations and certain recurrence relations, we propose efficient $O(n^2+dn)$ methods for evaluating a Gauss--Legendre curve of degree $n$ in $\mathbb E^d$. We also propose algorithms for multipoint evaluation with computational complexity $O(Mdn+dn^2)$, where $M$ is the number of evaluation points.