A weakly stable algorithm for general Toeplitz systems

TL;DR

This paper presents a fast algorithm for QR factorization of Toeplitz and Hankel matrices that is weakly stable, providing a practical method for solving linear systems and least squares problems involving these structured matrices.

Contribution

It introduces a new fast QR factorization algorithm that is weakly stable for Toeplitz and Hankel matrices, enabling efficient solutions to related linear systems.

Findings

The algorithm is weakly stable with R^T.R close to A^T.A.

It provides a practical approach for solving Toeplitz and Hankel linear systems.

Applicable to least squares problems with these structured matrices.

Abstract

We show that a fast algorithm for the QR factorization of a Toeplitz or Hankel matrix A is weakly stable in the sense that R^T.R is close to A^T.A. Thus, when the algorithm is used to solve the semi-normal equations R^T.Rx = A^Tb, we obtain a weakly stable method for the solution of a nonsingular Toeplitz or Hankel linear system Ax = b. The algorithm also applies to the solution of the full-rank Toeplitz or Hankel least squares problem.