On Minimizing the Number of Multiplications Necessary for Matrix Multiplication

Hopcroft, John E.; Kerr, Lígia Regina Franco Sansigolo

doi:10.1137/0120004

Cited by 144 publications

(89 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…For all sizes (m, n, p), if one of m, n, p is equal to 2, we take into account the cost of Hopcroft and Kerr's algorithm [14]. For square matrices, we also determine the cost provided by trilinear aggregating techniques.…”

Section: Filling the Tablesmentioning

confidence: 99%

Optimization techniques for small matrix multiplication

Drevet

Islam

Schost

2011

ACM Commun. Comput. Algebra

View full text Add to dashboard Cite

The complexity of matrix multiplication has attracted a lot of attention in the last forty years. In this paper, instead of considering asymptotic aspects of this problem, we are interested in reducing the cost of multiplication for matrices of small size, say up to 30. Following previous work in a similar vein by Probert & Fischer, Smith, and Mezzarobba, we base our approach on previous algorithms for small matrices, due to Strassen, Winograd, Pan, Laderman, . . . and show how to exploit these standard algorithms in an improved way. We illustrate the use of our results by generating multiplication code over various rings, such as integers, polynomials, differential operators or linear recurrence operators.

show abstract

Section: Filling the Tablesmentioning

confidence: 99%

Optimization techniques for small matrix multiplication

Drevet

Islam

Schost

2011

ACM Commun. Comput. Algebra

View full text Add to dashboard Cite

show abstract

“…In 1969, Strassen [20] presented an explicit algorithm for multiplying 2 × 2 matrices using seven multiplications. In the opposite direction, Hopcroft and Kerr [12] and Winograd [22] proved independently that there is no algorithm for multiplying 2×2 matrices using only six multiplications.…”

Section: Introductionmentioning

confidence: 99%

The border rank of the multiplication of $2\times 2$ matrices is seven

Landsberg

2005

J. Amer. Math. Soc.

View full text Add to dashboard Cite

“…The arithmetic cost of such algorithms has been extensively studied (see [19], [9], [14], [24], [20], [21], [15] and further details in [10]). Additionally, their communication cost is asymptotically lower than that of the classical algorithm [4].…”

Section: Introductionmentioning

confidence: 99%

Communication-Optimal Parallel Recursive Rectangular Matrix Multiplication

Demmel

Eliahu

Fox

et al. 2013

2013 IEEE 27th International Symposium on Parallel and Distributed Processing

View full text Add to dashboard Cite

Abstract-Communication-optimal algorithms are known for square matrix multiplication. Here, we obtain the first communication-optimal algorithm for all dimensions of rectangular matrices. Combining the dimension-splitting technique of Frigo, Leiserson, Prokop and Ramachandran (1999) with the recursive BFS/DFS approach of Ballard, Demmel, Holtz, Lipshitz and Schwartz (2012) allows for a communication-optimal as well as cache-and network-oblivious algorithm. Moreover, the implementation is simple: approximately 50 lines of code for the shared-memory version. Since the new algorithm minimizes communication across the network, between NUMA domains, and between levels of cache, it performs well in practice on both shared-and distributed-memory machines. We show significant speedups over existing parallel linear algebra libraries both on a 32-core shared-memory machine and on a distributed-memory supercomputer.

show abstract

On Minimizing the Number of Multiplications Necessary for Matrix Multiplication

Cited by 144 publications

References 4 publications

Optimization techniques for small matrix multiplication

Optimization techniques for small matrix multiplication

The border rank of the multiplication of $2\times 2$ matrices is seven

Communication-Optimal Parallel Recursive Rectangular Matrix Multiplication

Contact Info

Product

Resources

About