On finding multiplicities of characteristic polynomial factors of black-box matrices

TL;DR

This paper introduces algorithms for computing the characteristic polynomial of black-box matrices using minimal polynomial information, demonstrating practical efficiency and speedups in graph theory applications.

Contribution

It presents new algorithms and heuristics for characteristic polynomial computation from minimal polynomial data for black-box matrices, applicable over integers and finite fields.

Findings

Algorithms perform efficiently in practice

Significant speedups achieved in graph theory applications

Effective for matrices over integers and finite fields

Abstract

We present algorithms and heuristics to compute the characteristic polynomial of a matrix given its minimal polynomial. The matrix is represented as a black-box, i.e., by a function to compute its matrix-vector product. The methods apply to matrices either over the integers or over a large enough finite field. Experiments show that these methods perform efficiently in practice. Combined in an adaptive strategy, these algorithms reach significant speedups in practice for some integer matrices arising in an application from graph theory.