A non-commutative algorithm for multiplying 4x4 matrices using 48 non-complex multiplications

TL;DR

This work delivers a rational-coefficient variant of a recently discovered $4\times4$ non-commutative matrix multiplication algorithm that achieves $48$ multiplications. By leveraging the tensor-decomposition representation of matrix multiplication and applying a carefully chosen isotropy, the authors map a complex-valued orbit to a rational point, eliminating the need for complex numbers. They provide explicit straight-line programs for the resulting L, R, P representations and also an alternative basis variant, achieving theoretical complexity bounds around $11.65625\,n^{2+\log_4 3}-10.65625\,n^{2}$ and, in the alternative basis, a bound of $o\bigl(n^{2+\log_4 3}\bigr)$. The work extends the Strassen paradigm in a rational-setting and discusses practical implications and future numerical analysis of the proposed schemes. Overall, it advances fast, non-commutative matrix multiplication by delivering a concrete, publicly documented rational decomposition and executable programs enabling broader applicability over rings with invertible $2$.

Abstract

The quest for non-commutative matrix multiplication algorithms in small dimensions has seen a lot of recent improvements recently. In particular, the number of scalar multiplications required to multiply two $4\times4$ matrices was first reduced in \cite{Fawzi:2022aa} from 49 (two recursion levels of Strassen's algorithm) to 47 but only in characteristic 2 or more recently to 48 in \cite{alphaevolve} but over complex numbers. We propose an algorithm in 48 multiplications with only rational coefficients, hence removing the complex number requirement. It was derived from the latter one, under the action of an isotropy which happen to project the algorithm on the field of rational numbers. We also produce a straight line program of this algorithm, reducing the leading constant in the complexity, as well as an alternative basis variant of it, leading to an algorithm running in $7 n^{2+\frac{\log_2 3}{2}} +o\left(n^{2+\frac{log_2 3}{2}}\right)$ operations over any ring containing an inverse of 2.