Correcting matrix products over the ring of integers

Abstract

Let A, B, and C be three $n\times n$ matrices. We investigate the problem of verifying whether $AB=C$ over the ring of integers and finding the correct product AB. Given that C is different from AB by at most k entries, we propose an algorithm that uses $O(\sqrt{k}{n}^2+k^{2}n)$ operations. Let α be the largest absolute value of an entry in A, B, and C. The integers involved in the computation are of $O\left(n^3{\alpha }^2\right)$.