Implementation of Strassen's algorithm for matrix multiplication

Huss–Lederman, Steven; Jacobson, Elaine; Tsao, Anna; Turnbull, T.; Johnson, Jeremy

doi:10.1145/369028.369096

Cited by 84 publications

(77 citation statements)

References 15 publications

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“…The best asymptotic complexity for this computation has been successively improved since then, down to O`n 2.376´i n [5] (see [3,4] for a review), but Strassen-Winograd's still remains one of the most practicable. Former studies on how to turn this algorithm into practice can be found in [2,9,10,6] and references therein for numerical computation and in [15,7] for computations over a finite field.…”

Section: Introductionmentioning

confidence: 99%

Memory efficient scheduling of Strassen-Winograd's matrix multiplication algorithm

Boyer

Dumas

Pernet

et al. 2009

Proceedings of the 2009 International Symposium on Symbolic and Algebraic Computation

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We propose several new schedules for Strassen-Winograd's matrix multiplication algorithm, they reduce the extra memory allocation requirements by three different means: by introducing a few pre-additions, by overwriting the input matrices, or by using a first recursive level of classical multiplication. In particular, we show two fully in-place schedules: one having the same number of operations, if the input matrices can be overwritten; the other one, slightly increasing the constant of the leading term of the complexity, if the input matrices are read-only. Many of these schedules have been found by an implementation of an exhaustive search algorithm based on a pebble game.

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Section: Introductionmentioning

confidence: 99%

Memory efficient scheduling of Strassen-Winograd's matrix multiplication algorithm

Boyer

Dumas

Pernet

et al. 2009

Proceedings of the 2009 International Symposium on Symbolic and Algebraic Computation

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“…This problem of cut-off criteria is essential in scientific computing. For instance, in coding matrix multiplication, one is faced with choosing among Strassen multiplication, classical multiplication and others; see [12] for details.…”

Section: Optimizing Balanced Bivariate Multiplicationmentioning

confidence: 99%

FFT-Based Dense Polynomial Arithmetic on Multi-cores

Maza

Xie

2010

High Performance Computing Systems and Applications

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Abstract. We report efficient implementation techniques for FFT-based dense multivariate polynomial arithmetic over finite fields, targeting multi-cores. We have extended a preliminary study dedicated to polynomial multiplication and obtained a complete set of efficient parallel routines in Cilk++ for polynomial arithmetic such as normal form computation. Since bivariate multiplication applied to balanced data is a good kernel for these routines, we provide an in-depth study on the performance and the cut-off criteria of our different implementations for this operation. We also show that, not only optimized parallel multiplication can improve the performance of higher-level algorithms such as normal form computation but also this composition is necessary for parallel normal form computation to reach peak performance on a variety of problems that we have tested.

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“…Then once the product C − = A − B − is computed, one resolves the remaining rows and columns of C from the remaining rows and columns of A and B that are not in A − and B − (cf. [Huss-Lederman et al 1996]). For those remaining pieces Strassen-Winograd is not used but an implementation which does not cut the matrices into submatrices.…”

Section: Increasing the Number Of Precomputation Tablesmentioning

confidence: 99%

Algorithm 898

Albrecht

Bard

Hart

2010

ACM Trans. Math. Softw.

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We describe an efficient implementation of a hierarchy of algorithms for multiplication of dense matrices over the field with two elements (F 2 ). In particular we present our implementation -in the M4RI library -of Strassen-Winograd matrix multiplication and the "Method of the Four Russians for Multiplication" (M4RM) and compare it against other available implementations. Good performance is demonstrated on AMD's Opteron processor and particulary good performance on Intel's Core 2 Duo processor. The open-source M4RI library is available as a stand-alone package as well as part of the Sage mathematics system.In machine terms, addition in F 2 is logical-XOR, and multiplication is logical-AND, thus a machine word of 64 bits allows one to operate on 64 elements of F 2 in parallel: at most one CPU cycle for 64 parallel additions or multiplications. As such, element-wise operations over F 2 are relatively cheap. In fact, in this paper, we conclude that the actual bottlenecks are memory reads and writes and issues of data locality. We present our empirical findings in relation to minimizing these and give an analysis thereof.

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Implementation of Strassen's algorithm for matrix multiplication

Cited by 84 publications

References 15 publications

Memory efficient scheduling of Strassen-Winograd's matrix multiplication algorithm

Memory efficient scheduling of Strassen-Winograd's matrix multiplication algorithm

FFT-Based Dense Polynomial Arithmetic on Multi-cores

Algorithm 898

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