← 研踪 PaperTrailarXiv:2604.17331 · v2
Evaluation of Gauss-Legendre curves
首次提交 2026-04-19 16:53
版本更新 2026-10-29 03:53
雷达观测 2026-10-30 10:22
摘要
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 $O(n^2+dn)$ 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. Numerical tests show that the approach using the shifted power basis is very fast, but it works well only for small $n$ (approximately up to 20), whereas the method employing Jacobi polynomials gives good numerical results even for large $n$ (around 100).
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v2 · 2026-10-29 03:53 · 最新观测版本
Evaluation of Gauss-Legendre curves
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 $O(n^2+dn)$ 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. Numerical tests show that the approach using the shifted power basis is very fast, but it works well only for small $n$ (approximately up to 20), whereas the method employing Jacobi polynomials gives good numerical results even for large $n$ (around 100).
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