检索结果-内蒙古大学图书馆

A posteriori error bounds for the block-Lanczos method for matrix function approximation

NUMERICAL ALGORITHMS 2025年第2期98卷 903-927页

作者： Xu, Qichen Chen, Tyler Univ Chicago Chicago IL 60637 USA NYU New York NY USA

We extend the error bounds from Chen et al. (SIAM J. matrix Anal. Appl 43(2):787-811, 2022) for the Lanczos method for matrix function approximation to the block algorithm. Numerical experiments suggest that our bounds are fairly robust to changing block size and have the potential for use as a practical stopping criterion. Further experiments work toward a better understanding of how certain hyperparameters should be chosen in order to maximize the quality of the error bounds, even in the previously studied block size one case.

关键词： Error bounds matrix function Block algorithm

来源：评论

学校读者我要写书评

暂无评论

ENHANCED matrix function APPROXIMATION

引用

ELECTRONIC TRANSACTIONS ON NUMERICAL ANALYSIS 2017年 47卷 197-205页

作者： Eshghi, Nasim Reichel, Lothar Spalevic, Miodrag M. Kent State Univ Dept Math Sci Kent OH 44242 USA Univ Belgrade Dept Math Fac Mech Engn Kraljice Marije 16 Belgrade 1112035 35 Serbia

matrix functions of the form f (A) v, where A is a large symmetric matrix, f is a function, and v not equal 0 is a vector, are commonly approximated by first applying a few, say n, steps of the symmetric Lanczos process to A with the initial vector v in order to determine an orthogonal section of A. The latter is represented by a (small) n x n tridiagonal matrix to which f is applied. This approach uses the n first Lanczos vectors provided by the Lanczos process. However, n steps of the Lanczos process yield n + 1 Lanczos vectors. This paper discusses how the (n + 1) st Lanczos vector can be used to improve the quality of the computed approximation of f (A) v. Also the approximation of expressions of the form v(T) f (A) v is considered.

关键词： matrix function symmetric Lanczos process Gauss quadrature

来源：评论

学校读者我要写书评

暂无评论

Computing a matrix function for exponential integrators

引用

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2003年第1期161卷 203-216页

作者： Lu, YY City Univ Hong Kong Dept Math Kowloon Hong Kong Peoples R China

An efficient numerical method is developed for evaluating phi(A), where A is a symmetric matrix and phi is the function defined by phi(x)=(e(x)-1)/x=1+x/2+x(2)/6+.... This matrix function is useful in the so-called exponential integrators for differential equations. In particular, it is related to the exact solution of the ODE system dy/dt=Ay+b, where A and b are t-independent. Our method avoids the eigenvalue decomposition of the matrix A and it requires about 10n(3)/3 operations for a general symmetric n x n matrix. When the matrix is tridiagonal, the required number of operations is only O(n(2)) and it can be further reduced to O(n) if only a column of the matrix function is needed. These efficient schemes for tridiagonal matrices are particularly useful when the Lanczos method is used to calculate the product: of this matrix function (for a large symmetric matrix) with a given vector. (C) 2003 Elsevier B.V. All rights reserved.

关键词： matrix function exponential integrator Chebyshev rational approximation Lanczos method

来源：评论

学校读者我要写书评

暂无评论

Determination of a matrix function using the divided difference method of Newton and the interpolation technique of Hermite

引用

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2009年第1期231卷 67-81页

作者： Dehghan, Mehdi Hajarian, Masoud Amirkabir Univ Technol Dept Appl Math Fac Math & Comp Sci Tehran 15914 Iran

Computing a function f(A) of an n-by-n matrix A is a frequently occurring problem in control theory and other applications. In this paper we introduce an effective approach for the determination of matrix function f (A). We propose a new technique which is based on the extension of Newton divided difference and the interpolation technique of Hermite and using the eigenvalues of the given matrix A. The new algorithm is tested on several problems to show the efficiency of the presented method. Finally, the application of this method in control theory is highlighted. (C) 2009 Elsevier B.V. All rights reserved.

关键词： Cauchy integral Jordan canonical form Newton divided differences matrix function matrix polynomial State-transition matrix Square root of matrix

来源：评论

学校读者我要写书评

暂无评论

Marginal Fisher Analysis With Polynomial matrix function

引用

IEEE ACCESS 2022年 10卷 102451-102461页

作者： Ran, Ruisheng Li, Zheng Wang, Ting Feng, Ji Fang, Bin Chongqing Normal Univ Coll Comp & Informat Sci Coll Intelligent Sci Chongqing Engn Res Ctr Educ Big Data Intelligent Chongqing 401331 Peoples R China Chongqing Univ Coll Comp Sci Chongqing 400044 Peoples R China

Marginal fisher analysis (MFA) is a dimensionality reduction method based on a graph embedding framework. In contrast to traditional linear discriminant analysis (LDA), which requires the data to follow a Gaussian distribution, MFA is suitable for non-Gaussian data, and it has better pattern classification ability. However, MFA has the small-sample-size (SSS) problem. This paper aims to solve the small-sample-size problem while increasing the classification performance of MFA. Based on a matrix function dimensionality reduction framework, the criterion of the MFA method is reconstructed by using the polynomials matrix function transformation, and then a new MFA method is proposed, named PMFA (polynomial marginal fisher analysis). The major contributions of the proposed PMFA method are that it solves the small-sample-size problem of MFA, and it can enlarge the distance between marginal sample points of inter-class, so that it can get better pattern classification performance. Experiments on the public face datasets show that PMFA can get a better classification ability than MFA and its improved methods.

关键词： Principal component analysis Symmetric matrices Face recognition Manifold learning Eigenvalues and eigenfunctions Training data Pattern classification Dimensionality reduction manifold learning marginal fisher analysis matrix function the small-sample-size (SSS) problem

来源：评论

学校读者我要写书评

暂无评论

Solutions to 18 constrained optimization problems on the rank and inertia of the linear matrix function A plus BXB*

引用

MATHEMATICAL AND COMPUTER MODELLING 2012年第3-4期55卷 955-968页

作者： Tian, Yongge Cent Univ Finance & Econ China Econ & Management Acad Beijing 100081 Peoples R China

The inertia of a Hermitian matrix is defined to be a triplet composed by the numbers of the positive, negative and zero eigenvalues of the matrix counted with multiplicities, respectively. If we take the inertia and rank of a Hermitian matrix as objective functions, then they are neither differentiable nor smooth. In this case, maximizing and minimizing the inertia and rank of a Hermitian matrix function could be regarded as a continuous-integer optimization problem. In this paper, we use some pure algebraic operations of matrices and their generalized inverses to derive explicit expansion formulas for calculating the global maximum and minimum ranks and inertias of the linear Hermitian matrix function A + BXB* subject to some rank and definiteness restrictions on the variable matrix X. Various direct consequences of the formulas in characterizing algebraic properties of A + BXB* are also presented. In particular, solutions to a group of constrained optimization problems on the rank and inertia of a partially specified block Hermitian matrix are given. (C) 2011 Elsevier Ltd. All rights reserved.

关键词： matrix function Rank Inertia Maximization Minimization Moore-Penrose inverse Equality Inequality Lowner partial ordering Partially specified matrix

来源：评论

学校读者我要写书评

暂无评论

THE EXTENDED GLOBAL LANCZOS METHOD FOR matrix function APPROXIMATION

引用

ELECTRONIC TRANSACTIONS ON NUMERICAL ANALYSIS 2018年 50卷 144-163页

作者： Bentbib, A. H. El Ghomari, M. Jagels, C. Jbilou, K. Reichel, L. Cadi Ayyad Univ Fac Sci & Technol Lab LAMAI Marrakech Morocco Hanover Coll Dept Math Hanover IN 47243 USA ULCO Calais Lab LMPA 50 Rue F Buisson Calais France Kent State Univ Dept Math Sci Kent OH 44242 USA

The need to compute the trace of a large matrix that is not explicitly known, such as the matrix exp(A), where A is a large symmetric matrix, arises in various applications including in network analysis. The global Lanczos method is a block method that can be applied to compute an approximation of the trace. When the block size is one, this method simplifies to the standard Lanczos method. It is known that for some matrix functions and matrices, the extended Lanczos method, which uses subspaces with both positive and negative powers of A, can give faster convergence than the standard Lanczos method, which uses subspaces with nonnegative powers of A only. This suggests that it may be beneficial to use an extended global Lanczos method instead of the (standard) global Lanczos method. This paper describes an extended global Lanczos method and discusses properties of the associated Gauss-Laurent quadrature rules. Computed examples that illustrate the performance of the extended global Lanczos method are presented.

关键词： extended Krylov subspace extended moment matching Laurent polynomial global Lanczos method matrix function Gauss quadrature rule

来源：评论

学校读者我要写书评

暂无评论

THE LYAPUNOV matrix function AND THE STABILITY OF HYBRID SYSTEMS

引用

SOVIET APPLIED MECHANICS 1985年第4期21卷 386-393页

作者： MARTYNYUK, AA Acad of Sciences of the Ukrainian SSR Mechanics Inst Kiev USSR Acad of Sciences of the Ukrainian SSR Mechanics Inst Kiev USSR

A new class of auxiliary Lyapunov functions (functionals) are proposed for the analysis of stability of hybrid systems. Using the example of a two-component system, the technique of evaluation of the sign-definiteness... 详细信息

关键词： hybrid systems matrix function

来源：评论

学校读者我要写书评

暂无评论

ESTIMATING THE CONDITION NUMBER OF THE FRECHET DERIVATIVE OF A matrix function

引用

SIAM JOURNAL ON SCIENTIFIC COMPUTING 2014年第6期36卷 C617-C634页

作者： Higham, Nicholas J. Relton, Samuel D. Univ Manchester Sch Math Manchester M13 9PL Lancs England

The Frechet derivative L-f of a matrix function f : C-nxn -> C-nxn is used in a variety of applications and several algorithms are available for computing it. We define a condition number for the Frechet derivative and derive upper and lower bounds for it that differ by at most a factor 2. For a wide class of functions we derive an algorithm for estimating the 1-norm condition number that requires O(n(3)) flops given O(n(3)) flops algorithms for evaluating f and L-f;in practice it produces estimates correct to within a factor 6n. Numerical experiments show the new algorithm to be much more reliable than a previous heuristic estimate of conditioning.

关键词： matrix function condition number Frechet derivative Kronecker form matrix exponential matrix logarithm matrix powers matrix pth root MATLAB expm logm sqrtm

来源：评论

学校读者我要写书评

暂无评论

A General matrix function Dimensionality Reduction Framework and Extension for Manifold Learning

引用

IEEE TRANSACTIONS ON CYBERNETICS 2022年第4期52卷 2137-2148页

作者： Ran, Ruisheng Feng, Ji Zhang, Shougui Fang, Bin Chongqing Normal Univ Coll Intelligent Sci Coll Comp & Informat Sci Chongqing 401331 Peoples R China Chongqing Normal Univ Chongqing Engn Res Ctr Educ Big Data Intelligent Chongqing 401331 Peoples R China Chongqing Normal Univ Coll Comp & Informat Sci Chongqing 401331 Peoples R China Chongqing Normal Univ Sch Math Sci Chongqing 401331 Peoples R China Chongqing Univ Coll Comp Sci Chongqing 400044 Peoples R China

Many dimensionality reduction methods in the manifold learning field have the so-called small-sample-size (SSS) problem. Starting from solving the SSS problem, we first summarize the existing dimensionality reduction methods and construct a unified criterion function of these methods. Then, combining the unified criterion with the matrix function, we propose a general matrix function dimensionality reduction framework. This framework is configurable, that is, one can select suitable functions to construct such a matrix transformation framework, and then a series of new dimensionality reduction methods can be derived from this framework. In this article, we discuss how to choose suitable functions from two aspects: 1) solving the SSS problem and 2) improving pattern classification ability. As an extension, with the inverse hyperbolic tangent function and linear function, we propose a new matrix function dimensionality reduction framework. Compared with the existing methods to solve the SSS problem, these new methods can obtain better pattern classification ability and have less computational complexity. The experimental results on handwritten digit, letters databases, and two face databases show the superiority of the new methods.

关键词： Dimensionality reduction Manifolds Learning systems Principal component analysis Laplace equations Cybernetics Databases Dimensionality reduction manifold learning matrix function small-sample-size (SSS) problem

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：