Simultaneous computation of the row and column rank profiles

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

doi:10.1145/2465506.2465517

Cited by 17 publications

(43 citation statements)

References 10 publications

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“…Although this computation has already been widely studied for numerical computation [8], applications over a finite field have additional specific requirements and constraints that need to be taken into consideration. In particular, rank deficiency is a well defined notion there, and many applications rely on the computation of the echelon form, or the rank profile of the matrix [21,13] only revealed by certain pivoting strategies in the PLUQ factorization algorithm.…”

Section: Parallel Gaussian Eliminationmentioning

confidence: 99%

“…Such an iterative algorithm can be translated into a slab recursive algorithm splitting the row dimension in halves (as implemented in sequential in [12]) or into a slab iterative algorithm. More recently, a more flexible pivoting strategy that results in a tile recursive algorithm, cutting both dimensions simultaneously was presented in [13,14]. As a by product, both row and column rank profiles are also computed simultaneously.…”

Section: Algorithmic Variants For Pluq Factorizationmentioning

confidence: 99%

“…Our focus is on parallel implementations using various pivoting strategies that will reveal the echelon form, or the rank profile of the matrix [21,13]. The latter is a key invariant used in many applications such as Gröbner basis computations [15] and computational number theory [27].…”

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: Parallel Gaussian Eliminationmentioning

confidence: 99%

Section: Algorithmic Variants For Pluq Factorizationmentioning

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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“…In particular, the PLUQ decomposition algorithm of Dumas et al (2013) computes the rank profile matrix of A in O(n 2 r ω−2 ) where r = rank(A). However this estimate may be pessimistic as it does not take into account the left triangular shape of the matrix.…”

Section: From a Pluq Decompositionmentioning

confidence: 99%

“…In order to reach a complexity depending on s and not r, we adapt in Algorithm 2 the tile recursive algorithm of Dumas et al (2013), so that the left triangular structure of the input matrix is preserved and can be used to reduce the amount of computation.…”

Section: A Dedicated Algorithmmentioning

confidence: 99%

Time and space efficient generators for quasiseparable matrices

Pernet

Storjohann

2018

Journal of Symbolic Computation

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The class of quasiseparable matrices is defined by the property that any submatrix entirely below or above the main diagonal has small rank, namely below a bound called the order of quasiseparability. These matrices arise naturally in solving PDE's for particle interaction with the Fast Multi-pole Method (FMM), or computing generalized eigenvalues. From these application fields, structured representations and algorithms have been designed in numerical linear algebra to compute with these matrices in time linear in the matrix dimension and either quadratic or cubic in the quasiseparability order. Motivated by the design of the general purpose exact linear algebra library LinBox, and by algorithmic applications in algebraic computing, we adapt existing techniques introduce novel ones to use quasiseparable matrices in exact linear algebra, where sub-cubic matrix arithmetic is available. In particular, we will show, the connection between the notion of quasiseparability and the rank profile matrix invariant, that we have introduced in 2015. It results in two new structured representations, one being a simpler variation on the hierarchically semiseparable storage, and the second one exploiting the generalized Bruhat decomposition. As a consequence, most basic operations, such as computing the quasiseparability orders, applying a vector, a block vector, multiplying two quasiseparable matrices together, inverting a quasiseparable matrix, can be at least as fast and often faster than previous existing algorithms.

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Parallel Computation of Echelon Forms

Dumas

Gautier

Pernet

et al. 2014

Lecture Notes in Computer Science

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We propose efficient parallel algorithms and implementations on shared memory architectures of LU factorization over a finite field. Compared to the corresponding numerical routines, we have identified three main difficulties specific to linear algebra over finite fields. First, the arithmetic complexity could be dominated by modular reductions. Therefore, it is mandatory to delay as much as possible these reductions while mixing fine-grain parallelizations of tiled iterative and recursive algorithms. Second, fast linear algebra variants, e.g., using Strassen-Winograd algorithm, never suffer from instability and can thus be widely used in cascade with the classical algorithms. There, trade-offs are to be made between size of blocks well suited to those fast variants or to load and communication balancing. Third, many applications over finite fields require the rank profile of the matrix (quite often rank deficient) rather than the solution to a linear system. It is thus important to design parallel algorithms that preserve and compute this rank profile. Moreover, as the rank profile is only discovered during the algorithm, block size has then to be dynamic. We propose and compare several block decomposition: tile iterative with left-looking, right-looking and Crout variants, slab and tile recursive.Experiments demonstrate that the tile recursive variant performs better and matches the performance of reference numerical software when no rank deficiency occur. Furthermore, even in the most heterogeneous case, namely when all pivot blocks are rank deficient, we show that it is possbile to maintain a high efficiency.

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Simultaneous computation of the row and column rank profiles

Cited by 17 publications

References 10 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

Time and space efficient generators for quasiseparable matrices

Parallel Computation of Echelon Forms

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