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

Boyer, Brice; Dumas, J.; Pernet, Clément; Zhou, Wei

doi:10.1145/1576702.1576713

Cited by 31 publications

(27 citation statements)

References 14 publications

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“…In this section, we provide schedules for Bini's approximate multiplication, in a similar fashion to [BDPZ09]. These schedules are then implemented.…”

Section: Memory Usage and Schedulingmentioning

confidence: 99%

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Matrix Multiplication Over Word-Size Modular Rings Using Approximate Formulas

Boyer

Dumas

2016

ACM Trans. Math. Softw.

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Matrix multiplication over word-size modular rings using approximate formulae. AbstractBini's approximate formula (or border rank) for matrix multiplication achieves a better complexity than Strassen's matrix multiplication formula. In this paper, we show a novel way to use the approximate formula in the special case where the ring is Z/pZ. Besides, we show an implementation à la FFLAS-FFPACK, where p is a word-size prime number, that improves on state-of-the-art Z/pZ matrix multiplication implementations.

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“…In this section, we provide schedules for Bini's approximate multiplication, in a similar fashion to [BDPZ09]. These schedules are then implemented.…”

Section: Memory Usage and Schedulingmentioning

confidence: 99%

“…This requirement is smaller than Strassen-Winograd's (cf. [BDPZ09]) where X is of size m /2 × max ( k /2, n /2) and Y has size k /2 × n /2. For instance, for m = n = k, we have 5 12 m 2 for Bini's and 1 2 m 2 for Strassen-Winograd's.…”

Section: Lemma 1 the Extra Memory Used For One Level Of Recursion Inmentioning

confidence: 99%

Matrix Multiplication Over Word-Size Modular Rings Using Approximate Formulas

Boyer

Dumas

2016

ACM Trans. Math. Softw.

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“…Even if the asymptotically fastest algorithms [6,22] are not practicable, matrix multiplication happens to be also the most efficient building block in practice [7,17], thus making the reduction trees (or DAGs) from complexity theory still highly relevant in practice. The development of efficient exact linear algebra software then consists in making these reductions effective: fine tuning of the building block, taking the best advantage of the available computer arithmetic [10,2,1,9,19] and minimizing the memory footprint [3]; improving existing reductions in the leading constant of their time and space complexities [20].…”

Section: Reductions To Building Blocksmentioning

confidence: 99%

Exact Linear Algebra Algorithmic

Pernet

2015

Proceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation

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Exact linear algebra is a core component of many symbolic and algebraic computations, as it often delivers competitive theoretical complexities and also better harnesses the efficiency of modern computing infrastructures. In this tutorial we will present an overview on the recent advances in exact linear algebra algorithmic and implementation techniques, and highlight the few key ideas that have proven successful in their design. As an illustration, we will study in more details the computation of some matrix normal forms over a finite field or the ring of polynomials, specific to computer algebra. In particular, we will give a special care to the design and implementation of parallel exact linear algebra routines, trying to emphasize the similarities and distinctness with parallel numerical linear algebra. We aim to provide the working computer algebraist with a set of best practices for the use or the design of exact linear algebra software, together with an overview on a few still unresolved algorithmic problems in the field.

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“…The routine fgemm in Fflas uses by default the classic schedules for the multiplication and the product with accumulation (cf. [5]), but we also implement the low memory routines therein. The new algorithms are competitive and can reach sizes that were limiting.…”

Section: New Algorithms For the Mul Solutionmentioning

confidence: 99%

Elements of Design for Containers and Solutions in the LinBox Library

Boyer

Dumas

Giorgi

et al. 2014

Mathematical Software – ICMS 2014

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We describe in this paper new design techniques used in the C++ exact linear algebra library LinBox, intended to make the library safer and easier to use, while keeping it generic and efficient. First, we review the new simplified structure for containers, based on our founding scope allocation model. We explain design choices and their impact on coding: unification of our matrix classes, clearer model for matrices and submatrices, etc. Then we present a variation of the strategy design pattern that is comprised of a controller-plugin system: the controller (solution) chooses among plug-ins (algorithms) that always call back the controllers for subtasks. We give examples using the solution mul. Finally we present a benchmark architecture that serves two purposes: Providing the user with easier ways to produce graphs; Creating a framework for automatically tuning the library and supporting regression testing.

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

Cited by 31 publications

References 14 publications

Matrix Multiplication Over Word-Size Modular Rings Using Approximate Formulas

Matrix Multiplication Over Word-Size Modular Rings Using Approximate Formulas

Exact Linear Algebra Algorithmic

Elements of Design for Containers and Solutions in the LinBox Library

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