Parallel Computation of Echelon Forms

Dumas, J.; Gautier, Thierry; Pernet, Clément; Sultan, Ziad

doi:10.1007/978-3-319-09873-9_42

Cited by 13 publications

(20 citation statements)

References 12 publications

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“…In section 3 we study different iterative and recursive variants and cutting strategies for the parallel matrix multiplication pfgemm and compare them with our best iterative standard parallel matrix multiplication [11]. In section 4 we show three parallel algorithms for the pftrsm routine: an iterative variant, a recursive variant and a hybrid combination.…”

Section: Methodology Of Experimentsmentioning

confidence: 99%

“…This is one argument in favor of a coarse granularity in our algorithms. Table 1 taken from [11] shows the impact of the block size for iterative and recur-…”

Section: Ingredients For the Design Of Parallel Kernelsmentioning

confidence: 99%

“…In [11], parallel iterative and recursive implementations of exact PLUQ decomposition revealing the rank profiles of the matrix are presented. It is there shown that the parallel recursive implementation behaves best in terms of performance.…”

Section: Parallel Gaussian Eliminationmentioning

confidence: 99%

“…In a previous work [11], we presented our first investigations on the parallelization of exact dense Gaussian elimination for multicore computers. The focus there was on exploring the variants of parallel exact Gaussian elimination algorithms (block iterative or recursive, using slabs or tiles, etc) and showed how and why the tile recursive variant outperforms the others.…”

Section: Introductionmentioning

confidence: 99%

“…Our approach is to also apply recursive algorithms with a task based parallelization, for these building blocks. We explore six variants for the parallelization of the matrix multiplication (five recursive variants compared to the block iterative algorithm of [11]) and introduce a new hybrid parallel algorithm for the triangular system solve with matrix right hand side. We then recall the slab and tile recursive Gaussian elimination algorithms and present their performance using these new building blocks.…”

Section: Introductionmentioning

confidence: 99%

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Recursion based parallelization of exact dense linear algebra routines for Gaussian elimination

et al. 2016

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We present block algorithms and their implementation for the parallelization of sub-cubic Gaussian elimination on shared memory architectures. Contrarily to the classical cubic algorithms in parallel numerical linear algebra, we focus here on recursive algorithms and coarse grain parallelization. Indeed, sub-cubic matrix arithmetic can only be achieved through recursive algorithms making coarse grain block algorithms perform more efficiently than fine grain ones. This work is motivated by the design and implementation of dense linear algebra over a finite field, where fast matrix multiplication is used extensively and where costly modular reductions also advocate for coarse grain block decomposition. We incrementally build efficient kernels, for matrix multiplication first, then triangular system solving, on top of which a recursive PLUQ decomposition algorithm is built. We study the parallelization of these kernels using several algorithmic variants: either iterative or recursive and using different splitting strategies. Experiments show that recursive adaptive methods for matrix multiplication, hybrid recursive-iterative methods for triangular system solve and tile recursive versions of the PLUQ decomposition, together with various data mapping policies, provide the best performance on a 32 cores NUMA architecture. Overall, we show that the overhead of modular reductions is more than compensated by the fast linear algebra algorithms and that exact dense linear algebra matches the performance of full rank reference numerical software even in the presence of rank deficiencies.

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Section: Methodology Of Experimentsmentioning

confidence: 99%

“…This is one argument in favor of a coarse granularity in our algorithms. Table 1 taken from [11] shows the impact of the block size for iterative and recur-…”

Section: Ingredients For the Design Of Parallel Kernelsmentioning

confidence: 99%

Section: Parallel Gaussian Eliminationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Recursion based parallelization of exact dense linear algebra routines for Gaussian elimination

et al. 2016

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Solving Boolean polynomial systems by parallelizing characteristic set method for cyber‐physical systems

Zhao

Zhu

et al. 2020

Softw Pract Exp

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Summary Many cyber‐attach schemes and coding models established by algebra tools are build to address the problem of security of cyber‐pysical systems (CPS). As an important field of algebra computing, Boolean Polynomial System Solving (PoSSo) problem plays a very important role in many algebra applications. In this article, we propose an efficient Parallel Boolean Characteristic Set method (PBCS) under the high‐performance computing environment to improve the efficiency of solving Boolean polynomial systems. The PBCS is implemented based on the state‐of‐the‐art Boolean Characteristic Set method (BCS). It adopts a master‐slave parallel pattern, and distributes tasks based on the polynomial sets after initial zero decomposition. We design a strategy of dynamically reallocating tasks to ameliorate load imbalance, which is caused by dynamical zero decomposition of polynomials. Furthermore, we improve its performance by optimizing the parameter settings of PBCS, including the maximum number of polynomial branches that trigger the dynamic allocation policy and the scheduling time. Experimental results with solving several Boolean polynomial systems confirm that PBCS is efficient and scalable, especially for the equations generating from stream ciphers that have block triangular structure. Moreover, the method also has good scalability. It shows a stable speedup as well even extending to the size of thousands of CPU cores.

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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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Parallel Computation of Echelon Forms

Cited by 13 publications

References 12 publications

Recursion based parallelization of exact dense linear algebra routines for Gaussian elimination

Recursion based parallelization of exact dense linear algebra routines for Gaussian elimination

Solving Boolean polynomial systems by parallelizing characteristic set method for cyber‐physical systems

Elements of Design for Containers and Solutions in the LinBox Library

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