SIAM Journal on Matrix Analysis and Applications最新文献_第2页

On Semidefinite Programming Characterizations of the Numerical Radius and Its Dual Norm 论数值半径及其双规范的半有限编程特征

IF 1.5 2区数学

SIAM Journal on Matrix Analysis and Applications Pub Date : 2024-07-23 DOI: 10.1137/23m160356x

Shmuel Friedland, Chi-Kwong Li

引用次数: 0

Spectral Transformation for the Dense Symmetric Semidefinite Generalized Eigenvalue Problem 密集对称半无限广义特征值问题的谱变换

IF 1.5 2区数学

SIAM Journal on Matrix Analysis and Applications Pub Date : 2024-07-22 DOI: 10.1137/24m162916x

Michael Stewart

{"title":"Spectral Transformation for the Dense Symmetric Semidefinite Generalized Eigenvalue Problem","authors":"Michael Stewart","doi":"10.1137/24m162916x","DOIUrl":"https://doi.org/10.1137/24m162916x","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 3, Page 1392-1413, September 2024. Abstract. The spectral transformation Lanczos method for the sparse symmetric definite generalized eigenvalue problem for matrices [math] and [math] is an iterative method that addresses the case of semidefinite or ill-conditioned [math] using a shifted and inverted formulation of the problem. This paper proposes the same approach for dense problems and shows that with a shift chosen in accordance with certain constraints, the algorithm can conditionally ensure that every computed shifted and inverted eigenvalue is close to the exact shifted and inverted eigenvalue of a pair of matrices close to [math] and [math]. Under the same assumptions on the shift, the analysis of the algorithm for the shifted and inverted problem leads to useful error bounds for the original problem, including a bound that shows how a single shift that is of moderate size in a scaled sense can be chosen so that every computed generalized eigenvalue corresponds to a generalized eigenvalue of a pair of matrices close to [math] and [math]. The computed generalized eigenvectors give a relative residual that depends on the distance between the corresponding generalized eigenvalue and the shift. If the shift is of moderate size, then relative residuals are small for generalized eigenvalues that are not much larger than the shift. Larger shifts give small relative residuals for generalized eigenvalues that are not much larger or smaller than the shift.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":"22 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141737447","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Single-Pass Nyström Approximation in Mixed Precision 混合精度下的单程尼斯特伦近似法

IF 1.5 2区数学

SIAM Journal on Matrix Analysis and Applications Pub Date : 2024-07-19 DOI: 10.1137/22m154079x

Erin Carson, Ieva Daužickaitė

{"title":"Single-Pass Nyström Approximation in Mixed Precision","authors":"Erin Carson, Ieva Daužickaitė","doi":"10.1137/22m154079x","DOIUrl":"https://doi.org/10.1137/22m154079x","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 3, Page 1361-1391, September 2024. Abstract. Low-rank matrix approximations appear in a number of scientific computing applications. We consider the Nyström method for approximating a positive semidefinite matrix [math]. In the case that [math] is very large or its entries can only be accessed once, a single-pass version may be necessary. In this work, we perform a complete rounding error analysis of the single-pass Nyström method in two precisions, where the computation of the expensive matrix product with [math] is assumed to be performed in the lower of the two precisions. Our analysis gives insight into how the sketching matrix and shift should be chosen to ensure stability, implementation aspects which have been commented on in the literature but not yet rigorously justified. We further develop a heuristic to determine how to pick the lower precision, which confirms the general intuition that the lower the desired rank of the approximation, the lower the precision we can use without detriment. We also demonstrate that our mixed precision Nyström method can be used to inexpensively construct limited memory preconditioners for the conjugate gradient method and derive a bound on the condition number of the resulting preconditioned coefficient matrix. We present numerical experiments on a set of matrices with various spectral decays and demonstrate the utility of our mixed precision approach.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":"15 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2024-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141770662","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Eigenstructure Perturbations for a Class of Hamiltonian Matrices and Solutions of Related Riccati Inequalities 一类哈密尔顿矩阵的特征结构扰动及相关里卡提不等式的解

IF 1.5 2区数学

SIAM Journal on Matrix Analysis and Applications Pub Date : 2024-07-18 DOI: 10.1137/23m1619563

Volker Mehrmann, Hongguo Xu

引用次数: 0

On the Two-Parameter Matrix Pencil Problem 关于双参数矩阵铅笔问题

IF 1.5 2区数学

SIAM Journal on Matrix Analysis and Applications Pub Date : 2024-07-11 DOI: 10.1137/23m1545963

Satin K. Gungah, Fawwaz F. Alsubaie, Imad M. Jaimoukha

{"title":"On the Two-Parameter Matrix Pencil Problem","authors":"Satin K. Gungah, Fawwaz F. Alsubaie, Imad M. Jaimoukha","doi":"10.1137/23m1545963","DOIUrl":"https://doi.org/10.1137/23m1545963","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 3, Page 1318-1340, September 2024. Abstract. The multiparameter matrix pencil problem (MPP) is a generalization of the one-parameter MPP: Given a set of [math], [math] complex matrices [math] with [math], it is required to find all complex scalars [math], not all zero, such that the matrix pencil [math] loses column rank and the corresponding nonzero complex vector [math] such that [math]. We call the [math]-tuple [math] an eigenvalue and the corresponding vector [math] an eigenvector. This problem is related to the well-known multiparameter eigenvalue problem, except that there is only one pencil and, crucially, the matrices are not necessarily square. This paper uses our preliminary investigation in F. F. Alsubaie [[math] Optimal Model Reduction for Linear Dynamic Systems and the Solution of Multiparameter Matrix Pencil Problems, PhD thesis, Imperial College London, 2019], which presents a theoretical study of the multiparameter MPP and its applications in the [math] optimal model reduction problem, to give a full solution to the two-parameter MPP. First, an inflation process is implemented to show that the two-parameter MPP is equivalent to a set of three [math] simultaneous one-parameter MPPs. These problems are given in terms of Kronecker commutator operators (involving the original matrices) that exhibit several symmetries. These symmetries are analyzed and are then used to deflate the dimensions of the one-parameter MPPs to [math], thus simplifying their numerical solution. In the case in which [math], it is shown that the two-parameter MPP has at least one solution and generically [math] solutions, and furthermore that, under a rank assumption, the Kronecker determinant operators satisfy a commutativity property. This is then used to show that the two-parameter MPP is equivalent to a set of three simultaneous eigenvalue problems of dimension [math]. A general solution algorithm is presented and numerical examples are given to outline the procedure of the proposed algorithm.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":"18 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141614677","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

On Adaptive Stochastic Heavy Ball Momentum for Solving Linear Systems 论求解线性系统的自适应随机重球动量

IF 1.5 2区数学

SIAM Journal on Matrix Analysis and Applications Pub Date : 2024-07-09 DOI: 10.1137/23m1575883

Yun Zeng, Deren Han, Yansheng Su, Jiaxin Xie

{"title":"On Adaptive Stochastic Heavy Ball Momentum for Solving Linear Systems","authors":"Yun Zeng, Deren Han, Yansheng Su, Jiaxin Xie","doi":"10.1137/23m1575883","DOIUrl":"https://doi.org/10.1137/23m1575883","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 3, Page 1259-1286, September 2024. Abstract. The stochastic heavy ball momentum (SHBM) method has gained considerable popularity as a scalable approach for solving large-scale optimization problems. However, one limitation of this method is its reliance on prior knowledge of certain problem parameters, such as singular values of a matrix. In this paper, we propose an adaptive variant of the SHBM method for solving stochastic problems that are reformulated from linear systems using user-defined distributions. Our adaptive SHBM (ASHBM) method utilizes iterative information to update the parameters, addressing an open problem in the literature regarding the adaptive learning of momentum parameters. We prove that our method converges linearly in expectation, with a better convergence bound compared to the basic method. Notably, we demonstrate that the deterministic version of our ASHBM algorithm can be reformulated as a variant of the conjugate gradient (CG) method, inheriting many of its appealing properties, such as finite-time convergence. Consequently, the ASHBM method can be further generalized to develop a brand-new framework of the stochastic CG method for solving linear systems. Our theoretical results are supported by numerical experiments.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":"14 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141570800","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

One-Dimensional Tensor Network Recovery 一维张量网络恢复

IF 1.5 2区数学

SIAM Journal on Matrix Analysis and Applications Pub Date : 2024-07-01 DOI: 10.1137/23m159888x

Ziang Chen, Jianfeng Lu, Anru Zhang

引用次数: 0

On Compatible Transfer Operators in Nonsymmetric Algebraic Multigrid 论非对称代数多网格中的兼容转移算子

IF 1.5 2区数学

SIAM Journal on Matrix Analysis and Applications Pub Date : 2024-07-01 DOI: 10.1137/23m1586069

Ben S. Southworth, Thomas A. Manteuffel

{"title":"On Compatible Transfer Operators in Nonsymmetric Algebraic Multigrid","authors":"Ben S. Southworth, Thomas A. Manteuffel","doi":"10.1137/23m1586069","DOIUrl":"https://doi.org/10.1137/23m1586069","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 3, Page 1245-1258, September 2024. Abstract. The standard goal for an effective algebraic multigrid (AMG) algorithm is to develop relaxation and coarse-grid correction schemes that attenuate complementary error modes. In the nonsymmetric setting, coarse-grid correction [math] will almost certainly be nonorthogonal (and divergent) in any known standard product, meaning [math]. This introduces a new consideration, that one wants coarse-grid correction to be as close to orthogonal as possible, in an appropriate norm. In addition, due to nonorthogonality, [math] may actually amplify certain error modes that are in the range of interpolation. Relaxation must then not only be complementary to interpolation, but also rapidly eliminate any error amplified by the nonorthogonal correction, or the algorithm may diverge. This paper develops analytic formulae on how to construct “compatible” transfer operators in nonsymmetric AMG such that [math] in some standard matrix-induced norm. Discussion is provided on different options for the norm in the nonsymmetric setting, the relation between “ideal” transfer operators in different norms, and insight into the convergence of nonsymmetric reduction-based AMG.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":"21 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141509739","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Erratum: Properties of the Solution Set of Absolute Value Equations and the Related Matrix Classes 勘误：绝对值方程解集及相关矩阵类的性质

IF 1.5 2区数学

SIAM Journal on Matrix Analysis and Applications Pub Date : 2024-06-26 DOI: 10.1137/24m1635715

Milan Hladík

引用次数: 0

Fast and Accurate Randomized Algorithms for Linear Systems and Eigenvalue Problems 线性系统和特征值问题的快速准确随机算法

IF 1.5 2区数学

SIAM Journal on Matrix Analysis and Applications Pub Date : 2024-06-20 DOI: 10.1137/23m1565413

Yuji Nakatsukasa, Joel A. Tropp

引用次数: 0