Strassen's Matrix Multiplication Algorithm for Matrices of Arbitrary Order

Abstract

The well known algorithm of Volker Strassen for matrix multiplication can only be used for $(m2^k \times m2^k)$ matrices. For arbitrary $(n \times n)$ matrices one has to add zero rows and columns to the given matrices to use Strassen's algorithm. Strassen gave a strategy of how to set $m$ and $k$ for arbitrary $n$ to ensure $n\leq m2^k$. In this paper we study the number $d$ of additional zero rows and columns and the influence on the number of flops used by the algorithm in the worst case ($d=n/16$), best case ($d=1$) and in the average case ($d\approx n/48$). The aim of this work is to give a detailed analysis of the number of additional zero rows and columns and the additional work caused by Strassen's bad parameters. Strassen used the parameters $m$ and $k$ to show that his matrix multiplication algorithm needs less than $4.7n^{\log_2 7}$ flops. We can show in this paper, that these parameters cause an additional work of approx. 20 % in the worst case in comparison to the optimal strategy for the worst case. This is the main reason for the search for better parameters.