## 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

Zbl 0225.68018

### Software:

M4RI; FFLAS-FFPACK; Algorithm 898
Full Text:

### References:

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