检索结果-内蒙古大学图书馆

Decay bounds for the numerical quasiseparable preservation in matrix functions

LINEAR ALGEBRA AND ITS APPLICATIONS 2017年 516卷 212-242页

作者： Massei, Stefano Robol, Leonardo Scuola Normale Super Pisa Pisa Italy Katholieke Univ Leuven Dept Comp Sci Leuven Belgium

Given matrices A and B such that B = f (A), where f (z) is a holomorphic function, we analyze the relation between the singular values of the off-diagonal submatrices of A and B. We provide a family of bounds which depend on the interplay between the spectrum of the argument A and the singularities of the function. In particular, these bounds guarantee the numerical preservation of quasiseparable structures under mild hypotheses. We extend the Dunford-Cauchy integral formula to the case in which some Poles are contained inside the contour of integration. We use this tool together with the technology of hierarchical matrices (H-matrices) for the effective computation of matrix functions with quasiseparable arguments. (C) 2016 Elsevier Inc. All rights reserved.

关键词： matrix functions Quasiseparable matrices Off-diagonal singular values Decay bounds Exponential decay H-matrices

来源：评论

学校读者我要写书评

暂无评论

The scaling and modified squaring method for matrix functions related to the exponential

引用

APPLIED NUMERICAL MATHEMATICS 2009年第3-4期59卷 783-799页

作者： Skaflestad, B. Wright, W. M. La Trobe Univ Dept Math Bundoora Vic 3086 Australia NTNU Dept Math N-7491 Trondheim Norway

In recent years, there has been a resurgence in the construction and implementation of exponential integrators. which are numerical methods specifically designed for he numerical solution of spatially discretized semi-linear partial differential equations. Exponential integrators use the matrix exponential and related matrix functions within the formulation of the numerical method. The scaling and squaring method is the most widely used method for computing the matrix exponential. The aim of this paper is to discuss the efficient and accurate evaluation of the matrix exponential and related matrix functions Using a scaling and modified squaring method. (c) 2008 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： matrix exponential matrix functions phi-functions Pade approximation Scaling and modified squaring method Exponential integrators

来源：评论

学校读者我要写书评

暂无评论

Computing performability measures in Markov chains by means of matrix functions

引用

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2020年 368卷 112534-000页

作者： Masetti, G. Robol, L. Dept Comp Sci Largo B Pontecorvo 3 I-56127 Pisa Italy Dept Math Largo B Pontecorvo 5 I-56127 Pisa Italy Inst Sci & Technol A Faedo Via G Moruzzi 1 I-56124 Pisa Italy

We discuss the efficient computation of performance, reliability, and availability measures for Markov chains;these metrics - and the ones obtained by combining them, are often called performability measures. We show that this computational problem can be recasted as the evaluation of a bilinear form induced by appropriate matrix functions, and thus solved by leveraging the fast methods available for this task. We provide a comprehensive analysis of the theory required to translate the problem from the language of Markov chains to the one of matrix functions. The advantages of this new formulation are discussed, and it is shown that this setting allows to easily study the sensitivities of the measures with respect to the model parameters. Numerical experiments confirm the effectiveness of our approach;the tests we have run show that we can outperform the solvers available in state of the art commercial packages on a representative set of large scale examples. (C) 2019 Elsevier B.V. All rights reserved.

关键词： Markov chains Performance measures Availability Reliability matrix functions

来源：评论

学校读者我要写书评

暂无评论

Inversion of matrices and matrix functions as a nonlinear discrete system: Stability and sensitivity analysis

引用

IEE PROCEEDINGS-CONTROL THEORY AND APPLICATIONS 2001年第1期148卷 43-48页

作者： Balestrino, A Cannata, G Univ Pisa Dept Elect Syst & Automat Engn DSEA I-56126 Pisa Italy Univ Genoa Dept Commun Comp & Syst Sci DIST I-16145 Genoa Italy

A technique for the recursive inversion of matrices and matrix functions is presented. The proposed method can be modelled as a discrete nonlinear dynamic system. Convergence, stability and robustness properties are d... 详细信息

关键词： Algebra convergence Nonlinear control systems nonlinear dynamical systems robustness Discrete control systems inverse problems discrete time systems discrete time system matrix functions Stability in control theory inverse problem sensitivity analysis stability matrix algebra Control system analysis and synthesis methods nonlinear dynamic system

来源：评论

学校读者我要写书评

暂无评论

ERROR ESTIMATES FOR POLYNOMIAL KRYLOV APPROXIMATIONS TO matrix functions

引用

SIAM JOURNAL ON matrix ANALYSIS AND APPLICATIONS 2008年第4期30卷 1546-1565页

作者： Diele, Fasma Moret, Igor Ragni, Stefania Univ Bari Fac Econ I-70124 Bari Italy Univ Trieste Dipartimento Matemat & Informat I-34127 Trieste Italy CNR Ist Applicaz Calcolo M Picone I-70126 Bari Italy

In this paper we are interested in the polynomial Krylov approximations for the computation of phi(A)upsilon, where A is a square matrix, v represents a given vector, and. is a suitable function which can be employed in modern integrators for differential problems. Our aim consists of proposing and analyzing innovative a posteriori error estimates which allow a good control of the approximation procedure. The effectiveness of the results we provide is tested on some numerical examples of interest.

关键词： Krylov methods matrix functions exponential-like functions error estimates

来源：评论

学校读者我要写书评

暂无评论

MONOTONE CONVERGENCE OF THE LANCZOS APPROXIMATIONS TO matrix functions OF HERMITIAN MATRICES

引用

ELECTRONIC TRANSACTIONS ON NUMERICAL ANALYSIS 2009年 35卷 118-128页

作者： Frommer, Andreas Univ Wuppertal Fachbereich Math & Nat Wissensch D-42097 Wuppertal Germany

When A is a Hermitian matrix, the action f(A)b of a matrix function f(A) on a vector b can efficiently be approximated via the Lanczos method. In this note we use M-matrix theory to establish that the 2-norm of the error of the sequence of approximations is monotonically decreasing if f is a Stieltjes transform and A is positive definite. We discuss the relation of our approach to a recent, more general monotonicity result of Druskin for Laplace transforms. We also extend the class of functions to certain product type functions. This yields, for example, monotonicity when approximating sign(A)b with A indefinite if the Lanczos method is performed for A(2) rather than A.

关键词： matrix functions Lanczos method Galerkin approximation monotone convergence error estimates

来源：评论

学校读者我要写书评

暂无评论

Nonlocal constitutive laws generated by matrix functions: Lattice dynamics models and their continuum limits

引用

INTERNATIONAL JOURNAL OF ENGINEERING SCIENCE 2014年 80卷 106-123页

作者： Michelitsch, Thomas M. Collet, Bernard Wang, Xingjun Univ Paris 06 Sorbonne Univ CNRS UMR 7190 Inst Jean le Rond dAlembert F-75252 Paris 05 France Quiannan Normal Coll Nationalities Dept Phys Duyun 558000 Guizhou Peoples R China

We analyze one-dimensional discrete and quasi-continuous linear chains of N >> 1 equidistant and identical mass points with periodic boundary conditions and generalized nonlocal interparticle interactions in the harmonic approximation. We introduce elastic potentials which define by Hamilton's principle discrete "Laplacian operators" ("Laplacian matrices") which are operator functions (N x N-matrix functions) of the Laplacian of the Born-von-Karman linear chain with next neighbor interactions. The non-locality of the constitutive law of the present model is a natural consequence of the non-diagonality of these Laplacian matrix functions in the N dimensional vector space of particle displacement fields where the periodic boundary conditions (cyclic boundary conditions) and as a consequence the (Bloch-)eigenvectors of the linear chain are maintained. In the quasi-continuum limit (long-wave limit) the Laplacian matrices yield "Laplacian convolution kernels" (and the related elastic modulus kernels) of the non-local constitutive law. The elastic stability is guaranteed by the positiveness of the elastic potentials. We establish criteria for "weak" and "strong" nonlocality of the constitutive behavior which can be controlled by scaling behavior of material constants in the continuum limit when the interparticle spacing h -> 0. The approach provides a general method to generate physically admissible (elastically stable) non-local constitutive laws by means of "simple" Laplacian matrix functions. The model can be generalized to model non-locality in n = 2, 3, ... dimensions of the physical space. (C) 2014 Elsevier Ltd. All rights reserved.

关键词： Linear chain Nonlocal elasticity Nonlocal constitutive law Laplacian operator Long-wave continuum limit matrix functions Convolution kernels

来源：评论

学校读者我要写书评

暂无评论

ON THE STABILITY OF NETWORK INDICES DEFINED BY MEANS OF matrix functions

引用

SIAM JOURNAL ON matrix ANALYSIS AND APPLICATIONS 2018年第4期39卷 1521-1546页

作者： Pozza, Stefano Tudisco, Francesco Charles Univ Prague Dept Numer Math Fac Math & Phys Prague 18675 Czech Republic Univ Strathclyde Dept Math & Stat Glasgow G11 X14 Lanark Scotland

Identifying important components in a network is one of the major goals of network analysis. Popular and effective measures of importance of a node or a set of nodes are defined in terms of suitable entries of functions of matrices f(A). These kinds of measures are particularly relevant as they are able to capture the global structure of connections involving a node. However, computing the entries of f(A) requires a significant computational effort. In this work we address the problem of estimating the changes in the entries of f(A) with respect to changes in the edge structure. Intuition suggests that, if the topology, or the overall weight, of the connections in the new graph (G) over tilde is not significantly distorted, relevant components in G maintain their leading role in (G) over tilde. We propose several bounds giving mathematical reasoning to such intuition and showing, in particular, that the magnitude of the variation of the entry f(A)(kl) decays exponentially with the shortest-path distance in G that separates either k or l from the set of nodes touched by the edges that are perturbed. Moreover, we propose a simple method that exploits the computation of f(A) to simultaneously compute the all-pairs shortest-path distances of G, with essentially no additional cost. The proposed hounds are particularly relevant when the nodes whose edge connection tends to change more often or tends to be more often affected by noise have a marginal role in the graph and are distant from the most central nodes.

关键词： centrality indices stability decay bounds matrix functions Faber polynomials geodesic distance

来源：评论

学校读者我要写书评

暂无评论

ANALYSIS OF PROBING TECHNIQUES FOR SPARSE APPROXIMATION AND TRACE ESTIMATION OF DECAYING matrix functions

引用

SIAM JOURNAL ON matrix ANALYSIS AND APPLICATIONS 2021年第3期42卷 1290-1318页

作者： Frommer, Andreas Schimmel, Claudia Schweitzer, Marcel Berg Univ Wuppertal Dept Math D-42097 Wuppertal Germany Heinrich Heine Univ Dusseldorf Math Nat Wissensch Fak Univ Str 1 D-40225 Dusseldorf Germany

The computation of matrix functions f(A), or related quantities like their trace, is an important but challenging task, in particular, for large and sparse matrices A. In recent years, probing methods have become an often considered tool in this context, as they allow one to replace the computation of f(A) or tr(f(A)) by the evaluation of (a small number of) quantities of the form f(A)v or v(T) f(A)(v), respectively. These quantities can then be efficiently computed by standard techniques like, e.g., Krylov subspace methods. It is well known that probing methods are particularly efficient when f(A) is approximately sparse, e.g., when the entries of f(A) show a strong off-diagonal decay, but a rigorous error analysis is lacking so far. In this paper we develop new theoretical results on the existence of sparse approximations for f(A) and error bounds for probing methods based on graph colorings. As a by-product, by carefully inspecting the proofs of these error bounds, we also gain new insights into when to stop the Krylov iteration used for approximating f(A)v or v(T) f(A)(v), thus allowing for a practically efficient implementation of the probing methods.

关键词： matrix functions sparse approximations trace decay bounds graph coloring probing method Krylov subspace method

来源：评论

学校读者我要写书评

暂无评论

Stopping criteria for rational matrix functions of Hermitian and symmetric matrices

引用

SIAM JOURNAL ON SCIENTIFIC COMPUTING 2008年第3期30卷 1387-1412页

作者： Frommer, Andreas Simoncini, Valeria Berg Univ Wuppertal Fachbereich Mat & Nat Wissensch D-42097 Wuppertal Germany Univ Bologna Dipartmento Matemat I-40127 Bologna Italy

Building upon earlier work by Golub, Meurant, Strakos, and Tichy, we derive new a posteriori error bounds for Krylov subspace approximations to f(A)b, the action of a function f of a real symmetric or complex Hermitian matrix A on a vector b. To this purpose we assume that a rational function in partial fraction expansion form is used to approximate f, and the Krylov subspace approximations are obtained as linear combinations of Galerkin approximations to the individual terms in the partial fraction expansion. Our error estimates come at very low computational cost. In certain important special situations they can be shown to actually be lower bounds of the error. Our numerical results include experiments with the matrix exponential, as used in exponential integrators, and with the matrix sign function, as used in lattice quantum chromodynamics simulations, and demonstrate the accuracy of the estimates. The use of our error estimates within acceleration procedures is also discussed.

关键词： matrix functions partial fraction expansions error estimates error bounds Lanczos method Arnoldi method CG iteration

来源：评论

学校读者我要写书评

暂无评论

建议与咨询留下您的常用邮箱和电话号码，以便我们向您反馈解决方案和替代方法

时间限定

文献类型

馆藏选择

核心期刊

语言

文献类型

帮助

文字说明：

检索规则说明：

检索范例：

分类表

所选分类

限定检索结果

文献类型

馆藏范围

日期分布

学科分类号

主题

机构

作者

语言

请选择保存的检索档案：

请选择收藏分类：

通借通还

建议与咨询 留下您的常用邮箱和电话号码，以便我们向您反馈解决方案和替代方法

时间限定

文献类型

馆藏选择

核心期刊

语言

文献类型

帮助

文字说明：

检索规则说明：

检索范例：

分类表

所选分类

限定检索结果

文献类型

馆藏范围

日期分布

学科分类号

主题

机构

作者

语言

请选择保存的检索档案： 新增检索档案 确定 取消

请选择收藏分类： 新增自定义分类 确定 取消

通借通还

建议与咨询留下您的常用邮箱和电话号码，以便我们向您反馈解决方案和替代方法

请选择保存的检索档案：

请选择收藏分类：