基于Toeplitz和Hankel矩阵的快速多项式变换
Fast polynomial transforms based on Toeplitz and Hankel matrices
AI总结：
本文提出基于Toeplitz和Hankel矩阵的快速多项式变换方法，利用对角缩放Hadamard积实现高效的多项式系数转换，提出O(N(logN)^2)算法，适用于高精度计算。
摘要：
许多标准转换矩阵可以在对角缩放Hadamard积中分解，涉及Toeplitz和Hankel矩阵。这使得我们能够基于快速傅里叶变换，推导出将一次多项式系数从一个基转换到另一个基的O(N(logN)^2)算法。数值结果表明，该方法与最先进的技术相竞争，无需预计算成本，可以用少量代码实现，并且易于适应扩展精度算术。
英文摘要：
Many standard conversion matrices between coefficients in classical orthogonal polynomial expansions can be decomposed using diagonally-scaled Hadamard products involving Toeplitz and Hankel matrices. This allows us to derive $\smash{\mathcal{O}(N(\log N)^2)}$ algorithms, based on the fast Fourier transform, for converting coefficients of a degree $N$ polynomial in one polynomial basis to coefficients in another. Numerical results show that this approach is competitive with state-of-the-art techniques, requires no precomputational cost, can be implemented in a handful of lines of code, and is easily adapted to extended precision arithmetic.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
