On the arithmetic complexity of Strassen-like matrix multiplications

摘要

The Strassen algorithm for multiplying 2 x 2 matrices requires seven multiplications and 18 additions. The recursive use of this algorithm for matrices of dimension n yields a total arithmetic complexity of (7n(2.81) - 6n(2)) for n = 2(k). Winograd showed that using seven multiplications for this kind of matrix multiplication is optimal. Therefore, any algorithm for multiplying 2 x 2 matrices with seven multiplications is called a Strassen-like algorithm. Winograd also discovered an additively optimal Strassen-like algorithm with 15 additions. This algorithm is called the Winograd's variant, whose arithmetic complexity is (6n(2.81) - 5n(2)) for n = 2(k) and (3.73n(2.81) - 5n(2)) for n = 8 . 2(k), which is the best-known bound for Strassen-like multiplications. This paper proposes a method that reduces the complexity of Winograd's variant to (5n(2.81) + 0.5n(2.59) + 2n(2.32) - 6.5n(2)) for n = 2(k). It is also shown that the total arithmetic complexity can be improved to (3.55n(2.81) + 0.148n(2.59) + 1.02n(2.32) - 6.5n(2)) for n = 8 . 2(k), which, to the best of our knowledge, improves the best-known bound for a Strassen-like matrix multiplication algorithm. (C) 2016 Elsevier Ltd. All rights reserved.