Paper 2026/849

On Why and How to Minimize the Arithmetic Complexity of Fast Matrix Multiplication Algorithms

Paul Stankovski Wagner, Lund University

Abstract

Naively multiplying two $2 \times 2$ matri- ces requires eight multiplications and four additions. Strassen showed how to perform the same computation using seven multiplications and 18 additions. By chang- ing basis, Karstadt and Schwartz lowered the number of additions to 12, which they showed to be optimal within this generalized Karstadt-Schwartz (KS) framework. We present improved methods for optimizing the number of additions in Strassen-type matrix multipli- cation schemes for larger matrix sizes, and without any change of basis. Considering fast matrix generation process holistically as consisting of scheme generation and addition reduction, we discuss how to optimize both parts of this pipeline. We indicate that minimizing ad- ditions during the generation process is advantageous. We implement of our methods and use them to optimize the number of additions for schemes with dimensions up to $(n, m, k) = (5, 7, 10)$. Our methods can handle larger dimensions than (what has been published within) the KS framework. We compare our results against solutions within the KS framework on several large sets of schemes. We show that our method performs better relative to the KS framework, the larger the matrix dimensions are. We also apply our algorithms to a large number of schemes where we do not have apples-to-apples results in the KS framework as a comparison. We optimize the arithmetic complexity for two sets of thousands of schemes with the same rank. The number of additions needed after optimization roughly follows a normal distribution. Thus, we need to generate many solutions to minimize arithmetic complexity. Finally, our results on a large set of schemes and our extensive list of future research directions make for a valuable benchmark and facilitate future study of the arithmetic complexity of fast matrix multiplication.

Note: This is a major extension of "The Number of the Beast: Reducing Additions in Fast Matrix Multiplication Algorithms for Dimensions up to 666", presented at ACDA25 and as 2024/2063 on ePrint.