Fast polynomial transforms based on Toeplitz and Hankel matrices

TL;DR

This paper introduces fast algorithms for polynomial basis conversion using Toeplitz and Hankel matrices, achieving near-linear complexity with FFT-based methods, and demonstrates their efficiency and simplicity in implementation.

Contribution

The paper presents a novel approach to polynomial coefficient conversion leveraging Toeplitz and Hankel matrix decompositions, enabling efficient $ ext{O}(N( ext{log} N)^2)$ algorithms.

Findings

Algorithms are competitive with state-of-the-art methods.

No precomputational cost is required.

Easy to implement and adaptable to extended precision.

Abstract

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.