Superfast solution of Toeplitz systems based on syzygy reduction

TL;DR

This paper introduces a novel superfast algorithm for solving Toeplitz systems by leveraging syzygy modules of polynomials, achieving computational complexity of O(n log^2 n).

Contribution

It establishes a new connection between Toeplitz matrices and syzygy modules, enabling a more efficient solution method.

Findings

Achieves solution of Toeplitz systems in O(n log^2 n) time.

Shows that the module of syzygies is generated by two elements.

Reinterprets the solution as a polynomial remainder problem.

Abstract

We present a new superfast algorithm for solving Toeplitz systems. This algorithm is based on a relation between the solution of such problems and syzygies of polynomials or moving lines. We show an explicit connection between the generators of a Toeplitz matrix and the generators of the corresponding module of syzygies. We show that this module is generated by two elements and the solution of a Toeplitz system T u=g can be reinterpreted as the remainder of a vector depending on g, by these two generators. We obtain these generators and this remainder with computational complexity O(n log^2 n) for a Toeplitz matrix of size nxn.