On the Reuse of Additions in Matrix Multiplication

Abstract

We consider the problem of multiplying pairs of matrices by means of quadratic algorithms in terms of the reuse of additions. We show that if such an algorithm is to be significantly faster than the naïve matrix multiplication method then it must reuse additions to a great extent. (For example, any quadratic or bilinear algorithm for $n \times n$ matrix multiplication that does not reuse additions, except when reusing nonscalar steps, requires at least $n^{3}/8 - n^{2}/4 + n/8$ arithmetic operations.)