一种适用于通用3x3矩阵乘法的58次加法、秩23算法方案

摘要

本文提出了一种针对一般非交换环上精确3×3矩阵乘法的新型最优算法，通过仅需58次标量加法的秩23方案，将此前无需基变换的最佳加法复杂度从60次进一步降低。该成果是通过将三元受限翻转图探索与贪心交集消减的公共子表达式消除技术相结合的自动化搜索发现的。所得方案仅使用{-1, 0, 1}范围内的系数，确保了算法在任意域上的高效性与可移植性。标量运算总次数从83次减少至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.

一种适用于通用3x3矩阵乘法的58次加法、秩23算法方案

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

摘要

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