The aggregation and cancellation techniques as a practical tool for faster matrix multiplication

Abstract

The main purpose of this paper is to present a fast matrix multiplication algorithm taken from the paper of Laderman et al. (Linear Algebra Appl. 162–164 (1992) 557) in a refined compact “analytical” form and to demonstrate that it can be implemented as quite efficient computer code. Our improved presentation enables us to simplify substantially the analysis of the computational complexity and numerical stability of the algorithm as well as its computer implementation. The algorithm multiplies two N × N matrices using O( N 2.7760) arithmetic operations. In the case where N = 18·48 k , for a positive integer k, the total number of flops required by the algorithm is 4.894 N 2.7760−16.165 N 2, which may be compared to a similar estimate for the Winograd algorithm, 3.732 N 2.8074−5 N 2 flops, N = 8·2 k , the latter being current record bound among all known practical algorithms. Moreover, we present a pseudo-code of the algorithm which demonstrates its very moderate working memory requirements, much smaller than that of the best available implementations of Strassen and Winograd algorithms. For matrices of medium-large size (say, 2000⩽ N<10,000) we consider one-level algorithms and compare them with the (multilevel) Strassen and Winograd algorithms. The results of numerical tests clearly indicate that our accelerated matrix multiplication routines implementing two or three disjoint product-based algorithm are comparable in computational time with an implementation of Winograd algorithm and clearly outperform it with respect to working space and (especially) numerical stability. The tests were performed for the matrices of the order of up to 7000, both in double and single precision.