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

by   Clement Pernet, et al.

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× n Toeplitz or Hankel-like matrix of displacement rank α using Õ(n^ω - c(ω)α^c(ω)) arithmetic operations, where ω is the exponent of matrix multiplication and c(2.373)≈ 0.523 for the best known value of ω. For generic Toeplitz+Hankel-like matrices a second algorithm computes the characteristic polynomial in Õ(n^2-1/ω) 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 ω.


page 1

page 2

page 3

page 4


Deterministic computation of the characteristic polynomial in the time of matrix multiplication

This paper describes an algorithm which computes the characteristic poly...

Linear Time Interactive Certificates for the Minimal Polynomial and the Determinant of a Sparse Matrix

Certificates to a linear algebra computation are additional data structu...

On a fast and nearly division-free algorithm for the characteristic polynomial

We review the Preparata-Sarwate algorithm, a simple O(n^3.5) method for ...

On matrices with displacement structure: generalized operators and faster algorithms

For matrices with displacement structure, basic operations like multipli...

Computing the Characteristic Polynomial of a Finite Rank Two Drinfeld Module

Motivated by finding analogues of elliptic curve point counting techniqu...

A Fast Randomized Geometric Algorithm for Computing Riemann-Roch Spaces

We propose a probabilistic Las Vegas variant of Brill-Noether's algorith...

Computing Permanents on a Trellis

The problem of computing the permanent of a matrix has attracted interes...