A 58-Addition, Rank-23 Scheme for General 3x3 Matrix Multiplication

TL;DR

This paper introduces a new efficient algorithm for exact 3x3 matrix multiplication over general rings, reducing scalar additions to 58 and total operations to 81, through automated search techniques.

Contribution

It presents the first rank-23 scheme with only 58 scalar additions for 3x3 matrix multiplication, improving previous additive complexity without basis change.

Findings

Achieves a rank-23 scheme with 58 additions

Reduces total scalar operations from 83 to 81

Uses coefficients from {-1, 0, 1} for efficiency

Abstract

This paper presents a new state-of-the-art algorithm for exact $3\times3$ matrix multiplication over general non-commutative rings, achieving a rank-23 scheme with only 58 scalar additions. This improves the previous best additive complexity of 60 additions without a change of basis. The result was discovered through an automated search combining ternary-restricted flip-graph exploration with greedy intersection reduction for common subexpression elimination. The resulting scheme uses only coefficients from $\{-1, 0, 1\}$, ensuring both efficiency and portability across arbitrary fields. The total scalar operation count is reduced from 83 to 81.