Time and space efficient generators for quasiseparable matrices

Pernet, Clément; Storjohann, Arne

doi:10.1016/j.jsc.2017.07.010

Cited by 8 publications

(13 citation statements)

References 32 publications

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“…These types of matrices turn out to be highly useful for devising fast practical solutions of certain structured systems of equations. Quasiseparable matrices in particular, which appear in our Theorem 1.2, are actively being researched with recent fast representations and algorithms in [61,62]. Just as how the four main displacement structures are closely tied to polynomial operations [57], the work of Bella, Eidelman, Gohberg, and Olshevsky show deep connections between computations with rank-structured matrices and with polynomials [8].…”

Section: A1 Known Resultsmentioning

confidence: 99%

A Two-pronged Progress in Structured Dense Matrix Vector Multiplication

De¹,

Gu²,

Puttagunta³

et al. 2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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Matrix-vector multiplication is one of the most fundamental computing primitives. Given a matrix A ∈ F N ×N and a vector b ∈ F N , it is known that in the worst case Θ(N 2 ) operations over F are needed to compute Ab. Many types of structured matrices do admit faster multiplication. However, even given a matrix A that is known to have this property, it is hard in general to recover a representation of A exposing the actual fast multiplication algorithm. Additionally, it is not known in general whether the inverses of such structured matrices can be computed or multiplied quickly. A broad question is thus to identify classes of structured dense matrices that can be represented with O(N ) parameters, and for which matrix-vector multiplication (and ideally other operations such as solvers) can be performed in a sub-quadratic number of operations.One such class of structured matrices that admit nearlinear matrix-vector multiplication are the orthogonal polynomial transforms whose rows correspond to a family of orthogonal polynomials. Other well known classes include the Toeplitz, Hankel, Vandermonde, Cauchy matrices and their extensions (e.g. confluent Cauchy-like matrices) that are all special cases of a low displacement rank property.In this paper, we make progress on two fronts: 1. We introduce the notion of recurrence width of matrices.For matrices A with constant recurrence width, we design algorithms to compute both Ab and A T b with a near-linear number of operations. This notion of width is finer than all the above classes of structured matrices and thus we can compute near-linear matrixvector multiplication for all of them using the same core algorithm. Furthermore, we show that it is possible to solve the harder problems of recovering the structured parameterization of a matrix with low recurrence width, and computing matrix-vector product with its inverse in near-linear time. 2. We additionally adapt our algorithm to a matrix-vector multiplication algorithm for a much more general class of matrices with displacement structure: those with low displacement rank with respect to quasiseparable matrices. This result is a novel connection between matrices with displacement structure and those with rank structure, two large but previously separate classes of structured matrices. This class includes Toeplitzplus-Hankel-like matrices, the Discrete Trigonometric Transforms, and more, and captures all previously known matrices with displacement structure under a unified parameterization and algorithm.Our work unifies, generalizes, and simplifies existing stateof-the-art results in structured matrix-vector multiplication. Finally, we show how applications in areas such as multipoint evaluations of multivariate polynomials can be reduced to problems involving low recurrence width matrices.

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Section: A1 Known Resultsmentioning

confidence: 99%

A Two-pronged Progress in Structured Dense Matrix Vector Multiplication

De¹,

Gu²,

Puttagunta³

et al. 2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…Like other LU algorithms, it gives a gain of r/n times when applied to matrices of small rank r. But it is also efficient for full rank sparse matrices. An example of this type of matrices that is important in applications is considered in the work [4].…”

Section: Discussionmentioning

confidence: 99%

LDU factorization

Malaschonok¹

2020

Preprint

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LU-factorization of matrices is one of the fundamental algorithms of linear algebra. The widespread use of supercomputers with distributed memory requires a review of traditional algorithms, which were based on the common memory of a computer. Matrix block recursive algorithms are a class of algorithms that provide coarse-grained parallelization. The block recursive LU factorization algorithm was obtained in 2010. This algorithm is called LEU-factorization. It, like the traditional LU-algorithm, is designed for matrices over number fields. However, it does not solve the problem of numerical instability. We propose a generalization of the LEU algorithm to the case of a commutative domain and its field of quotients. This LDU factorization algorithm decomposes the matrix over the commutative domain into a product of three matrices, in which the matrices L and U belong to the commutative domain, and the elements of the weighted truncated permutation matrix D are the elements inverse to the product of some pair of minors. All elements are calculated without errors, so the problem of instability does not arise.The work was partially supported by the Academy of Sciences of Ukraine, project (2020) "Creation of an open data e-platform for the collective use centers".

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“…These quasiseparable matrices arise naturally in solving PDEs for particle interaction with the Fast Multi-pole Method (FMM). The efficiency of application of the block-recursive algorithm of the Bruhat decomposition to the quasiseparable matrices is studied in [21].…”

Section: Solving Ode's and Pdesmentioning

confidence: 99%

Recursive Matrix Algorithms in Commutative Domain for Cluster with Distributed Memory

Malaschonok

Ilchenko

2018

2018 Ivannikov Memorial Workshop (IVMEM)

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We give an overview of the theoretical results for matrix block-recursive algorithms in commutative domains and present the results of experiments that we conducted with new parallel programs based on these algorithms on a supercomputer MVS-10P at the Joint Supercomputer Center of the Russian Academy of Science. To demonstrate a scalability of these programs we measure the running time of the program for a different number of processors and plot the graphs of efficiency factor. Also we present the main application areas in which such parallel algorithms are used. It is concluded that this class of algorithms allows to obtain efficient parallel programs on clusters with distributed memory.Index Terms-block-recursive matrix algorithms, commutative domain, factorization of matrices, matrix inversion, distributed memory

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Time and space efficient generators for quasiseparable matrices

Cited by 8 publications

References 32 publications

A Two-pronged Progress in Structured Dense Matrix Vector Multiplication

A Two-pronged Progress in Structured Dense Matrix Vector Multiplication

LDU factorization

Recursive Matrix Algorithms in Commutative Domain for Cluster with Distributed Memory

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