Computing the Characteristic Polynomial of Generic Toeplitz-like and Hankel-like Matrices

TL;DR

This paper introduces new randomized algorithms for efficiently computing the characteristic and minimal polynomials of structured matrices like Toeplitz and Hankel, reducing computational complexity significantly over previous methods.

Contribution

It presents novel algorithms based on structured projections and Coppersmith's block Wiedemann method, achieving faster computation of polynomials for Toeplitz-like and Hankel-like matrices.

Findings

Randomized Monte Carlo algorithm for minimal polynomial with complexity $ ilde O(n^{ ext{exponent}})$

Efficient characteristic polynomial computation for generic Toeplitz+Hankel matrices

Reduced complexity from $O(n^2)$ to less than $1.86$ and $1.58$ exponents for specific cases

Abstract

New algorithms are presented for computing annihilating polynomials of Toeplitz, Hankel, and more generally Toeplitz+ Hankel-like matrices over a field. Our approach follows works on Coppersmith's block Wiedemann method with structured projections, which have been recently successfully applied for computing the bivariate resultant. A first baby-step/giant step approach -- directly derived using known techniques on structured matrices -- gives a randomized Monte Carlo algorithm for the minimal polynomial of an $n\times n$ Toeplitz or Hankel-like matrix of displacement rank $\alpha$ using $\tilde O(n^{\omega - c(\omega)} \alpha^{c(\omega)})$ arithmetic operations, where $\omega$ is the exponent of matrix multiplication and $c(2.373)\approx 0.523$ for the best known value of $\omega$. For generic Toeplitz+Hankel-like matrices a second algorithm computes the characteristic polynomial in $\tilde O(n^{2-1/\omega})$ operations when the displacement rank is considered constant. Previous algorithms required $O(n^2)$ operations while the exponents presented here are respectively less than $1.86$ and $1.58$ with the best known estimate for $\omega$.