GEMMW: A Portable Level 3 BLAS Winograd Variant of Strassen's Matrix-Matrix Multiply Algorithm

Douglas, Craig C.; Heroux, Michael A.; Slishman, Gordon; Smith, Roger M.

doi:10.1006/jcph.1994.1001

Cited by 63 publications

(56 citation statements)

References 3 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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“…For example, the schedule proposed in [DHSS94], with two temporaries only, can be revisited for the sequence seen in §2.2 as follows:…”

Section: S Schedulingmentioning

confidence: 99%

“…We compare the internal cubic GP-Pari implementation (Version 2.3.2, with GMP 4.2.1) and a GP-Pari script implementing Strassen-Winograd's ("SW", one level only) scheduled as in [DHSS94] with another scripts using the new proposed sequence described in §2.2 (one recursion level), then one implementing ψ(A) → ψ(A 2 ) which is the building brick for exponentiation, and a last one using also fig.1 for 2×2 (or 3×3) matrices. The last column gives percentage difference between ψ and non-ψ implementation.…”

Section: T Timingsmentioning

confidence: 99%

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

Bodrato

2010

Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation

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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.

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“…These MAs make the algorithm faster, however they make it weakly numerically stable and not unstable [Higham 2002]. As the starting point for our hybrid adaptive algorithm we use Winograd's algorithm (e.g., [Douglas et al 1994]), which requires only 15 MAs. Thus, Winograd's algorithm has, like the original by Strassen, asymptotic operation count O(n 2.81 ), but it has a smaller constant factor and thus fewer operations than Strassen's algorithm.…”

Section: Introductionmentioning

confidence: 99%

“…(3) At every recursive step, we use only 3 temporary matrices, which is the minimum number possible [Douglas et al 1994]. Furthermore, we differ from Douglas et al work in that we do not perform redundant computations for odd-size matrices.…”

Section: Introductionmentioning

confidence: 99%

Adaptive Winograd's matrix multiplications

D'Alberto

Nicolau

2009

ACM Trans. Math. Softw.

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Modern architectures have complex memory hierarchies and increasing parallelism (e.g., multicores). These features make achieving and maintaining good performance across rapidly changing architectures increasingly difficult. Performance has become a complex trade-off, not just a simple matter of counting cost of simple CPU operations.We present a novel, hybrid, and adaptive recursive Strassen-Winograd's matrix multiplication (MM) that uses automatically tuned linear algebra software (ATLAS) or GotoBLAS. Our algorithm applies to any size and shape matrices stored in either row or column major layout (in double-precision in this work) and thus is efficiently applicable to both C and FORTRAN implementations. In addition, our algorithm divides the computation into equivalent in-complexity sub-MMs and does not require any extra computation to combine the intermediary sub-MM results.We achieve up to 22% execution-time reduction versus GotoBLAS/ATLAS alone for a single core system and up to 19% for a 2 dual-core processor system. Most importantly, even for small matrices such as 1500×1500, our approach attains already 10% execution-time reduction and, for MM of matrices larger than 3000×3000, it delivers performance that would correspond, for a classic O(n 3 ) algorithm, to faster-than-processor peak performance (i.e., our algorithm delivers the equivalent of 5 GFLOPS performance on a system with 4.4 GFLOPS peak performance and where GotoBLAS achieves only 4 GFLOPS). This is a result of the savings in operations (and thus FLOPS). Therefore, our algorithm is faster than any classic MM algorithms could ever be for matrices of this size. Furthermore, we present experimental evidence based on established methodologies found in the literature that our algorithm is, for a family of matrices, as accurate as the classic algorithms.

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GEMMW: A Portable Level 3 BLAS Winograd Variant of Strassen's Matrix-Matrix Multiply Algorithm

Cited by 63 publications

References 3 publications

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

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

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

Adaptive Winograd's matrix multiplications

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