Interactive certificate for the verification of Wiedemann's Krylov sequence: application to the certification of the determinant, the minimal and the characteristic polynomials of sparse matrices

TL;DR

This paper introduces interactive certificates for Wiedemann's Krylov sequence, enabling efficient verification of determinants and polynomials of sparse matrices with nearly linear complexity.

Contribution

It presents algorithms for computing certificates for Krylov sequences of sparse or structured matrices with Monte Carlo verification complexity that is essentially linear.

Findings

Certificates enable efficient verification of matrix properties.

Algorithms work for sparse and structured matrices over abstract fields.

Applications include certification of determinants and characteristic polynomials.

Abstract

Certificates to a linear algebra computation are additional data structures for each output, which can be used by a-possibly randomized- verification algorithm that proves the correctness of each output. Wiede-mann's algorithm projects the Krylov sequence obtained by repeatedly multiplying a vector by a matrix to obtain a linearly recurrent sequence. The minimal polynomial of this sequence divides the minimal polynomial of the matrix. For instance, if the $n\times n$ input matrix is sparse with n 1+o(1) non-zero entries, the computation of the sequence is quadratic in the dimension of the matrix while the computation of the minimal polynomial is n 1+o(1), once that projected Krylov sequence is obtained. In this paper we give algorithms that compute certificates for the Krylov sequence of sparse or structured $n\times n$ matrices over an abstract field, whose Monte Carlo verification complexity can be made essentially linear. As an application this gives certificates for the determinant, the minimal and characteristic polynomials of sparse or structured matrices at the same cost.