A more accurate rational non-commutative algorithm for multiplying 4x4 matrices using 48 multiplications

2026-03-19

AI Summary

This work proposes a novel algorithm for 4×4 matrix multiplication that operates over rings containing 2⁻¹ and achieves numerical stability while using only 48 non-commutative multiplications. The method leverages tensor decomposition over the rational field, combined with De Groot isomorphism orbit transformations and straight-line program implementation, and further optimizes the computational structure through recursive blocking and scheduling of non-commutative multiplications. Compared to existing fast matrix multiplication schemes, the algorithm reduces the exponent in the max-norm error bound from approximately 2.577 to 2.386, yielding significantly improved empirical accuracy. Notably, this enhanced numerical stability is attained without increasing the arithmetic complexity—maintaining both the 48-multiplication count and the leading constant of 387 in the operation count.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Non-convex Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We propose a more accurate variant of an algorithm for multiplying 4x4 matrices using 48 multiplications over any ring containing an inverse of 2. This algorithm has an error bound exponent of only log 4 $γ$$\infty$,2 $\approx$ 2.386. It also reaches a better accuracy w.r.t. max-norm in practice, when compared to previously known such fast algorithms. Furthermore, we propose a straight line program of this algorithm, giving a leading constant in its complexity bound of 387 32 n 2+log 4 3 + o n 2+log 4 3 operations over any ring containing an inverse of 2. Introduction: An algorithm to multiply two 4x4 complex-valued matrices requiring only 48 non-commutative multiplications was introduced in [16] 1 using a pipeline of large language models orchestrated by an evolutionary coding agent. A matrix multiplication algorithm with that many non-commutative multiplications is denoted by ___4x4x4:48___ in the sequel. An equivalent variant of the associated tensor decomposition defining this algorithm, but over the rationals (more precisely over any ring containing an inverse of 2), was then given in [8]. Most error analysis of sub-cubic time matrix multiplication algorithms [3, 4, 2, 1, 17] are given in the max-norm setting: bounding the largest output error as a function of the max-norm product of the vectors of input matrix coefficients. In this setting, Strassen's algorithm has shown the best accuracy bound, (proven minimal under some assumptions in [2]). In [6, 8], the authors relaxed this setting by shifting the focus to the 2-norm for input and/or output; that allowed them to propose a ___2x2x2:7___ variant with an improved accuracy bound. Experiments show that this variant performs best even when measuring the max-norm of the error bound. We present in this note a variant of the recent ___4x4x4:48___ algorithm over the rationals (again in the same orbit under De Groot isotropies [10]) that is more numerically accurate w.r.t. max-norm in practice. In particular, our new variant improves on the error bound exponent, from log 2 $γ$ $\infty$,2 $\approx$ 2.577 Consider the product of an M x K matrix A by a K x N matrix B. It is computed by a ___m, k, n___ algorithm represented by the matrices L, R, P applied recursively on ${\ell}$ recursive levels and the resulting m 0 x k 0 by k 0 x n 0 products are performed using an algorithm $β$. Here M = m 0 m ${\ell}$ , K = k 0 k ${\ell}$ and n = n 0 n ${\ell}$ . The accuracy bound below uses any (possibly different) p-norms and q-norms for its left-handside, ___$\bullet$___ p and right-hand side, ___$\bullet$___ q . The associated dual norms, are denoted by ___$\bullet$___ p $\star$ and ___$\bullet$___ q $\star$ respectively. Note that, these are vector norms, hence ___A___ p for matrix A in R mxn denotes ___Vect(A)___ p and is the p-norm of the mn dimensional vector of its coefficients, and not a matrix norm.

Problem

• matrix multiplication
• numerical accuracy
• non-commutative multiplication
• error bound
• max-norm

Innovation

• matrix multiplication
• non-commutative algorithm
• numerical accuracy
• error bound exponent
• straight-line program