# Preconditioners for symmetrized Toeplitz and multilevel Toeplitz matrices

## TL;DR

This paper introduces novel preconditioners for symmetrized Toeplitz and multilevel Toeplitz matrices, enabling efficient Krylov subspace methods like MINRES with rigorous iteration bounds.

## Contribution

It proposes new ideal preconditioners for symmetrized Toeplitz matrices and analyzes their spectra, addressing a gap in effective preconditioning techniques.

## Key findings

- Preconditioners improve convergence of Krylov methods.
- Spectral analysis supports preconditioner effectiveness.
- Numerical experiments validate theoretical results.

## Abstract

When solving linear systems with nonsymmetric Toeplitz or multilevel Toeplitz matrices using Krylov subspace methods, the coefficient matrix may be symmetrized. The preconditioned MINRES method can then be applied to this symmetrized system, which allows rigorous upper bounds on the number of MINRES iterations to be obtained. However, effective preconditioners for symmetrized (multilevel) Toeplitz matrices are lacking. Here, we propose novel ideal preconditioners, and investigate the spectra of the preconditioned matrices. We show how these preconditioners can be approximated and demonstrate their effectiveness via numerical experiments.