通用3x3矩阵乘法的58次加法、秩23算法

摘要

本文提出一种针对一般非交换环上精确3×3矩阵乘法的全新最优算法，通过秩23方案仅需58次标量加法运算。该结果在不改变基的前提下，将先前最佳的60次加法复杂度进一步提升。此发现源于结合三元限制翻转图探索与贪婪交集约简的自动搜索方法，实现了公共子表达式消除。最终方案仅使用{-1, 0, 1}范围内的系数，确保算法在任意域上兼具高效性与可移植性。标量运算总量从83次降至81次。

English

This paper presents a new state-of-the-art algorithm for exact 3times3 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.

通用3x3矩阵乘法的58次加法、秩23算法

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

摘要

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