A Strassen-like matrix multiplication suited for squaring and higher power computation

Abstract

Abstract. Strassen method is not the asymptotically fastest known matrix multiplication algorithm, but it is the most widely used for large matrices on finite fields. Since his manuscript was published, a number of variants have been proposed with various addition complexities. Here we describe a new one. The new variant is as good as those already known for a simple matrix multiplication, but can save operations either when more than two matrices are to be multiplied or for squaring. Moreover it can be proved optimal for this tasks. The biggest gain is shown for n th -power computation, in this scenario the additive complexity can be halved, with respect to original Strassen's.

Sparsifying the Operators of Fast Matrix Multiplication Algorithms

“…However a er some tuning of Algorithms 3 and 4 (see Section 3.4) and the implementation of Algorithm 6 using Z3, all decompositions completed on a PC within a reasonable time. Speci cally, all runs of Algorithms 3 and 4 completed within 40 minutes, while Algorithm 6 took less than one minute, on a PC 4 . It should be remembered that Algorithms 3 and 4 guarantee optimal sparsi cation, while Algorithm 6 has no such guarantee.…”

“…Winograd [43] reduced the leading coe cient of Strassen's algorithm's arithmetic complexity from 7 to 6 by decreasing the number of additions and subtractions in the 2 × 2 base case from 18 to 15 1 . Later, Bodrato [4] introduced the intermediate representation method, that successfully reduces the leading coe cient to 5, for repeated squaring and chain matrix multiplication. Cenk and Hasan [7] presented a non-uniform implementation of Strassen-Winograd's algorithm [43], which also reduces the leading coe cient from 6 to 5, but incurs additional penalties such as a larger memory footprint and higher communication costs.…”

Tensor Rank and Complexity

“…When it comes to algorithms used in practice, there are in fact only few known to beat Strassen's algorithm from 1969. Regarding concrete implementations and papers studying practical issues we refer to [BB15], [Bod10], [DN07], [HJJ + 96], [Smi13] and the references therein.…”

Comparison Study and Operation Collapsing Issues for Serial Implementation of Square Matrix Multiplication Approach Suitable in High Performance Computing Environment

“…Among conventional algorithm, block matrix multiplication algorithm and Strassen's algorithm, the last one has shown better performance for large size matrices, whereas for small matrix, the recursion overhead resulted in inefficiency for Strassen's algorithm. In previous research, that proposed a Strassen-like method where additive complexity has been halved compared to Strassen's method (Bodrato, 2010). The proposed one canperform better for squaring, chain multiplications and general polynomial calculation.…”