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displacement structure of the core inverse

LINEAR & MULTILINEAR ALGEBRA 2022年第2期70卷 203-214页

作者： Ma, Haifeng Harbin Normal Univ Sch Math Sci Harbin 150025 Peoples R China

A matrix A is said to possess an UV-displacement structure if rank is small compared with the rank of A. Estimates for the rank are presented, where is the core inverse of A. The computation of the displacement is als... 详细信息

关键词： Core inverse displacement structure displacement rank Schur decomposition

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displacement structure of Core-EP Inverses

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Wuhan University Journal of Natural Sciences 2020年第6期25卷 483-488页

作者： MENG Xianchun WANG Hongxing LIU Ailing School of Mathematics and Statistics Hechi UniversityYizhou 546300GuangxiChina School of Mathematics and Physics Guangxi University for NationalitiesNanning 530006GuangxiChina

In this paper,we study the displacement rank of the Core-EP *** Sylvester displacement and generalized displacement are *** present upper bounds for the ranks of the displacements of the Core-EP *** experiments are pr... 详细信息

关键词： core-EP invers displacement structure Hermitian matrix

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Bounds on the Singular Values of Matrices with displacement structure

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SIAM REVIEW 2019年第2期61卷 319-344页

作者： Beckermann, Bernhard Townsend, Alex UST Lille UFR Math UMR 8524 CNRS Lab Paul PainleveEquipe ANO EDP F-59655 Villeneuve Dascq France Cornell Univ Dept Math White Hall Ithaca NY 14853 USA

Matrices with displacement structure, such as Pick, Vandermonde, and Hankel matrices, appear in a diverse range of applications. In this paper, we use an extremal problem involving rational functions to derive explicit bounds on the singular values of such matrices. For example, we show that the kth singular value of a real n x n positive definite Hankel matrix, H n , is bounded by C rho(-k/log n)parallel to H-n parallel to(2) with explicitly given constants C > 0 and rho > 1, where parallel to H-n parallel to(2) is the spectral norm. This means that a real n x n positive definite Hankel matrix can be approximated, up to an accuracy of epsilon parallel to H-n parallel to(2) with 0 < epsilon < 1, by a rank O(log n log(1/epsilon)) matrix. Analogous results are obtained for Pick, Cauchy, real Vandermonde, Lowner, and certain Krylov matrices.

关键词： singular values displacement structure Zolotarev rational

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ON THE SINGULAR VALUES OF MATRICES WITH displacement structure

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SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS 2017年第4期38卷 1227-1248页

作者： Beckermann, Bernhard Townsend, Alex UMR 8524 CNRS Lab Paul Painleve Equipe ANO EDP UFR MathUST Lille F-59655 Villeneuve Dascq France Cornell Univ Dept Math Ithaca NY 14853 USA

Matrices with displacement structure such as Pick, Vandermonde, and Hankel matrices appear in a diverse range of applications. In this paper, we use an extremal problem involving rational functions to derive explicit bounds on the singular values of such matrices. For example, we show that the k th singular value of a real n x n positive de finite Hankel matrix, H-n, is bounded by C rho(-k/logn) || H-n || 2 with explicitly given constants C > 0 and rho > 1, where || H-n ||2 is the spectral norm. This means that a real n x n positive de fi nite Hankel matrix can be approximated, up to an accuracy of is an element of|| H-n || 2 with 0 < is an element of < 1, by a rank O (log n log(1/is an element of)) matrix. Analogous results are obtained for Pick, Cauchy, real Vandermonde, Lowner, and certain Krylov matrices.

关键词： singular values displacement structure Zolotarev rational

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SUPERFAST DIRECT INVERSION OF THE NONUNIFORM DISCRETE FOURIER TRANSFORM VIA HIERARCHICALLY SEMISEPARABLE LEAST SQUARES

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2025年第3期47卷 A1702-A1732页

作者： Wilber, Heather Epperly, Ethan N. Barnett, Alex H. Univ Washington Dept Appl Math Seattle WA 98105 USA CALTECH Div Comp & Math Sci Pasadena CA 91125 USA Flatiron Inst Ctr Computat Math New York NY 10010 USA

A direct solver is introduced for solving overdetermined linear systems involving nonuniform discrete Fourier transform matrices. Such matrices can be transformed into a Cauchy-like form that has hierarchical low rank structure. The rank structure of this matrix is explained, and it is shown that the ranks of the relevant submatrices grow only logarithmically with the number of columns of the matrix. A fast rank-structured hierarchical approximation method based on this analysis is developed, along with a hierarchical least-squares solver for these and related systems. This result is a direct method for inverting nonuniform discrete transforms with a complexity that is usually nearly linear with respect to the degrees of freedom in the problem. This solver is benchmarked against various iterative and direct solvers in the setting of inverting the one-dimensional type-II (or forward) transform, for a range of condition numbers and problem sizes (up to 4 \times 106 by 2 \times 106). These experiments demonstrate that this method is especially useful for large problems with multiple right-hand sides.

关键词： nonuniform discrete Fourier transform Vandermonde rank structured least squares displacement structure hierarchical matrices rectangular linear systems

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On the singular values of matrices with high displacement rank

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LINEAR ALGEBRA AND ITS APPLICATIONS 2018年 548卷 19-41页

作者： Townsend, Alex Wilber, Heather Cornell Univ Dept Math White Hall Ithaca NY 14853 USA Cornell Univ Ctr Appl Math Ithaca NY 14853 USA

We introduce a new ADI-based low rank solver for AX - XB = F, where F has rapidly decaying singular values. Our approach results in both theoretical and practical gains, including (1) the derivation of new bounds on singular values for classes of matrices with high displacement rank, (2) a practical algorithm for solving certain Lyapunov and Sylvester matrix equations with high rank right-hand sides, and (3) a collection of low rank Poisson solvers that achieve spectral accuracy and optimal computational complexity. (C) 2018 Elsevier Inc. All rights reserved.

关键词： Low rank approximation Sylvester matrix equation Fast Poisson solver Alternating direction implicit method Singular values displacement structure

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Fast Algorithms for displacement and Low-Rank structured Matrices 18

Fast Algorithms for Displacement and Low-Rank Structured Mat...

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43rd ACM International Symposium on Symbolic and Algebraic Computation (ISSAC)

作者： Chandrasekaran, Shivkumar Govindarajan, Nithin Rajagopal, Abhejit Univ Calif Santa Barbara Santa Barbara CA 93106 USA

Fast matrix algorithms, displacement structure, low-rank, sequentially semi-separable, hierarchically semi-separable, Toeplitz, Hankel, Vandermonde, Cauchy, Hilbert, Fast multipole method

ISBN: (纸本)9781450355506

Fast matrix algorithms, displacement structure, low-rank, sequentially semi-separable, hierarchically semi-separable, Toeplitz, Hankel, Vandermonde, Cauchy, Hilbert, Fast multipole method

关键词： Fast matrix algorithms displacement structure low-rank sequentially semi-separable hierarchically semi-separable Toeplitz Hankel Vandermonde Cauchy Hilbert Fast multipole method

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FAST COMPUTATION OF THE MATRIX EXPONENTIAL FOR A TOEPLITZ MATRIX

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SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS 2018年第1期39卷 23-47页

作者： Kressner, Daniel Luce, Robert Ecole Polytech Fed Lausanne Stn 8 CH-1015 Lausanne Switzerland

The computation of the matrix exponential is a ubiquitous operation in numerical mathematics, and for a general, unstructured n x n matrix it can be computed in O(n(3)) operations. An interesting problem arises if the input matrix is a Toeplitz matrix, for example as the result of discretizing integral equations with a time invariant kernel. In this case it is not obvious how to take advantage of the Toeplitz structure, as the exponential of a Toeplitz matrix is, in general, not a Toeplitz matrix itself. The main contribution of this work are fast algorithms for the computation of the Toeplitz matrix exponential. The algorithms have provable quadratic complexity if the spectrum is real, or sectorial, or, more generally, if the imaginary parts of the rightmost eigenvalues do not vary too much. They may be efficient even outside these spectral constraints. They are based on the scaling and squaring framework, and their analysis connects classical results from rational approximation theory to matrices of low displacement rank. As an example, the developed methods are applied to Merton's jump-diffusion model for option pricing.

关键词： Toeplitz matrix matrix exponential displacement structure rational approximation option pricing

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A Fast Schur-Euclid-Type Algorithm for Quasiseparable Polynomials 21st

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Conference on Applications of Computer Algebra (ACA)

作者： Perera, Sirani M. Olshevsky, Vadim Embry Riddle Aeronaut Univ Daytona Beach FL 32114 USA Univ Connecticut Storrs CT USA

ISBN: (纸本)9783319569321;9783319569307

In this paper, a fast O(n(2)) algorithm is presented for computing recursive triangular factorization of a Bezoutian matrix associated with quasiseparable polynomials via a displacement equation. The new algorithm applies to a fairly general class of quasiseparable polynomials that includes real orthogonal, Szegd polynomials, and several other important classes of polynomials, e.g., those defined by banded tiessenberg matrices. While the algorithm can be seen as a Schur-type for the Bezoutian matrix it can also be seen as a Euclid-type for quasiseparable polynomials via factorization of a displacement equation. The process, i.e., fast Euclid-type algorithm for quasiseparable polynomials or Schur-type algorithm for Bezoutian associated with quasiseparable polynomials, is carried out with the help of a displacement equation satisfied by Bezoutian and Confederate matrices leading to O (n(2)) complexity.

关键词： Quasiseparable matrices Bezoutians Euclid algorithm Schur algorithm Fast algorithms displacement structure

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displacement structure approach to q-adic polynomial-Vandermonde and related matrices

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LINEAR ALGEBRA AND ITS APPLICATIONS 2010年第2期433卷 319-332页

作者： Zhang, Zhengzhi Yang, Zhenghong Li, Cheng China Agr Univ Coll Sci Dept Math Beijing 100083 Peoples R China Beijing Global Informat Ctr Exploitat & Applicat Beijing 100094 Peoples R China Xian Univ Finance & Econ Inst Stat Xian 710100 Peoples R China

In the present paper a new class of the so-called q-adic polynomial-Vandermonde-like matrices over an arbitrary non-algebraically closed field is introduced. This class generalizes both the simple and the confluent polynomial-Vandermonde-like matrices over the complex field, and the q-adic Vandermonde and the q-adic Chebyshev-Vandermonde-like matrices studied earlier by different authors. Three kinds of displacement structures and two kinds of fast inversion formulas are obtained for this class of matrices by using displacement structure matrix method, which generalize the corresponding results of the polynomial-Vandermonde-like and the q-adic Vandermonde-like matrices. (C) 2010 Elsevier Inc. All rights reserved.

关键词： Vandermonde matrix q-Adic polynomial-Vandermonde matrix displacement structure Fast inversion formula

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