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

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 $\omega$.