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On the arithmetic complexity of Strassen-like matrix multiplications. (English) Zbl 1353.68309
Summary: The Strassen algorithm for multiplying \(2 \times 2\) matrices requires seven multiplications and 18 additions. The recursive use of this algorithm for matrices of dimension \(n\) yields a total arithmetic complexity of \((7 n^{2.81} - 6 n^2)\) for \(n = 2^k\). S. Winograd [Linear Algebra Appl. 4, 381–388 (1971; Zbl 0225.68018)] showed that using seven multiplications for this kind of matrix multiplication is optimal. Therefore, any algorithm for multiplying \(2\times 2\) 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 \((6 n^{2.81} - 5 n^2)\) for \(n = 2^k\) and \((3.73 n^{2.81} - 5 n^2)\) for \(n = 8 \cdot 2^k\), which is the best-known bound for Strassen-like multiplications. This paper proposes a method that reduces the complexity of Winograd’s variant to \((5 n^{2.81} + 0.5 n^{2.59} + 2 n^{2.32} - 6.5 n^2)\) for \(n = 2^k\). It is also shown that the total arithmetic complexity can be improved to \((3.55 n^{2.81} + 0.148 n^{2.59} + 1.02 n^{2.32} - 6.5n^2)\) for \(n = 8 \cdot 2^k\), which, to the best of our knowledge, improves the best-known bound for a Strassen-like matrix multiplication algorithm.

MSC:
68W30 Symbolic computation and algebraic computation
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