Evaluation of Gauss-Legendre curves
This work provides faster evaluation methods for Gauss-Legendre curves, which are used in geometric modeling and numerical analysis, but the improvement is incremental over existing techniques.
The paper presents efficient O(n^2+dn) methods for evaluating Gauss-Legendre curves of degree n in E^d and O(Mdn+dn^2) algorithms for multipoint evaluation, using new polynomial representations and recurrence relations.
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.