Algorithms for Structured Linear Systems Solving and Their Implementation

Hyun, Seungjoon; Lebreton, Romain; Schost, Éric

doi:10.1145/3087604.3087659

Cited by 4 publications

(2 citation statements)

References 51 publications

(61 reference statements)

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“…In this work, we extend these results by investigating matrices with low displacement rank with respect to any triangular t -band matrices, which we define to be matrices whose non-zero elements all appear in t consecutive diagonals. General and unified algorithms for the most popular classes of matrices with displacement structure have continued to be refined for practicality and precision, such as in [34] and [59]. A second strand of research that inspired our work is the study of orthogonal polynomial transforms, especially that of Driscoll, Healy, and Rockmore [23].…”

Section: A1 Known Resultsmentioning

confidence: 99%

“…We sketch here a simplified randomized algorithm to illustrate the process and give an overview of the proof here in order to analyze the complexity in our case; for more details of this type of algorithm, see [38]. More modern unified algorithms for inverting classic displacement structured matrices exist [34,35,56,57] and can still be adapted to this setting with the same asymptotic bounds.…”

Section: Quasiseparable L Rmentioning

confidence: 99%

See 1 more Smart Citation

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%

Section: Quasiseparable L Rmentioning

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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Fast matrix multiplication and its algebraic neighbourhood

Pan¹

2017

Sb. Math.

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Implementations of Efficient Univariate Polynomial Matrix Algorithms and Application to Bivariate Resultants

Hyun

Neiger

Schost

2019

Proceedings of the 2019 International Symposium on Symbolic and Algebraic Computation

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Complexity bounds for many problems on matrices with univariate polynomial entries have been improved in the last few years. Still, for most related algorithms, efficient implementations are not available, which leaves open the question of the practical impact of these algorithms, e.g. on applications such as decoding some errorcorrecting codes and solving polynomial systems or structured linear systems. In this paper, we discuss implementation aspects for most fundamental operations: multiplication, truncated inversion, approximants, interpolants, kernels, linear system solving, determinant, and basis reduction. We focus on prime fields with a word-size modulus, relying on Shoup's C++ library NTL. Combining these new tools to implement variants of Villard's algorithm for the resultant of generic bivariate polynomials (ISSAC 2018), we get better performance than the state of the art for large parameters.

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Algorithms for Structured Linear Systems Solving and Their Implementation

Cited by 4 publications

References 51 publications

A Two-pronged Progress in Structured Dense Matrix Vector Multiplication

A Two-pronged Progress in Structured Dense Matrix Vector Multiplication

Fast matrix multiplication and its algebraic neighbourhood

Implementations of Efficient Univariate Polynomial Matrix Algorithms and Application to Bivariate Resultants

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