SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 3, Page 1414-1428, September 2024. Abstract. We state and give self-contained proofs of semidefinite programming characterizations of the numerical radius and its dual norm for matrices. We show that the computation of the numerical radius and its dual norm within [math] precision are polynomially time computable in the data and [math] using either the ellipsoid method or the short step, primal interior point method. We apply our results to give a simple formula for the spectral and the nuclear norm of a [math] real tensor in terms of the numerical radius and its dual norm.
{"title":"On Semidefinite Programming Characterizations of the Numerical Radius and Its Dual Norm","authors":"Shmuel Friedland, Chi-Kwong Li","doi":"10.1137/23m160356x","DOIUrl":"https://doi.org/10.1137/23m160356x","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 3, Page 1414-1428, September 2024. <br/> Abstract. We state and give self-contained proofs of semidefinite programming characterizations of the numerical radius and its dual norm for matrices. We show that the computation of the numerical radius and its dual norm within [math] precision are polynomially time computable in the data and [math] using either the ellipsoid method or the short step, primal interior point method. We apply our results to give a simple formula for the spectral and the nuclear norm of a [math] real tensor in terms of the numerical radius and its dual norm.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":"30 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141770667","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}
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.
{"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. <br/> 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}
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.
{"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. <br/> 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}
SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 3, Page 1335-1360, September 2024. Abstract. The characterization of the solution set for a class of algebraic Riccati inequalities is studied. This class arises in the passivity analysis of linear time-invariant control systems. Eigenvalue perturbation theory for the Hamiltonian matrix associated with the Riccati inequality is used to analyze the extremal points of the solution set.
{"title":"Eigenstructure Perturbations for a Class of Hamiltonian Matrices and Solutions of Related Riccati Inequalities","authors":"Volker Mehrmann, Hongguo Xu","doi":"10.1137/23m1619563","DOIUrl":"https://doi.org/10.1137/23m1619563","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 3, Page 1335-1360, September 2024. <br/> Abstract. The characterization of the solution set for a class of algebraic Riccati inequalities is studied. This class arises in the passivity analysis of linear time-invariant control systems. Eigenvalue perturbation theory for the Hamiltonian matrix associated with the Riccati inequality is used to analyze the extremal points of the solution set.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":"35 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141737449","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}
Satin K. Gungah, Fawwaz F. Alsubaie, Imad M. Jaimoukha
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.
SIAM 矩阵分析与应用期刊》，第 45 卷，第 3 期，第 1318-1340 页，2024 年 9 月。 摘要多参数矩阵铅笔问题（MPP）是单参数矩阵铅笔问题的一般化：给定一组[math], [math]复数矩阵[math]与[math]，要求找到所有复数标量[math]（不全为零），使得矩阵铅笔[math]失去列秩，以及相应的非零复数向量[math]，使得[math]。我们称[math]元组[math]为特征值，相应的向量[math]为特征向量。这个问题与众所周知的多参数特征值问题有关，只不过只有一支铅笔，而且关键的是，矩阵不一定是正方形的。本文利用我们在 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] 中的初步研究，提出了多参数 MPP 的理论研究及其在[math] optimal model reduction problem 中的应用，给出了双参数 MPP 的完整解决方案。首先，实现了一个膨胀过程，表明双参数 MPP 等价于一组三个 [math] 同步单参数 MPP。这些问题是用克朗内克换元算子（涉及原始矩阵）给出的，它们表现出几种对称性。对这些对称性进行分析后，可将单参数 MPP 的维数缩减为 [math]，从而简化其数值解法。在[数学]的情况下，证明了双参数 MPP 至少有一个解，而且一般都有[数学]解，此外，在秩假设下，克朗内克行列式算子满足换元性质。然后用它来证明双参数 MPP 等价于一组维数为 [math] 的三个同时特征值问题。本文提出了一种通用求解算法，并给出了数值示例，以概述所提算法的程序。
{"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. <br/> 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}
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.
{"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. <br/> 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}
SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 3, Page 1217-1244, September 2024. Abstract. We study the recovery of the underlying graphs or permutations for tensors in the tensor ring or tensor train format. Our proposed algorithms compare the matricization ranks after down-sampling, whose complexity is [math] for [math]th-order tensors. We prove that our algorithms can almost surely recover the correct graph or permutation when tensor entries can be observed without noise. We further establish the robustness of our algorithms against observational noise. The theoretical results are validated by numerical experiments.
{"title":"One-Dimensional Tensor Network Recovery","authors":"Ziang Chen, Jianfeng Lu, Anru Zhang","doi":"10.1137/23m159888x","DOIUrl":"https://doi.org/10.1137/23m159888x","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 3, Page 1217-1244, September 2024. <br/> Abstract. We study the recovery of the underlying graphs or permutations for tensors in the tensor ring or tensor train format. Our proposed algorithms compare the matricization ranks after down-sampling, whose complexity is [math] for [math]th-order tensors. We prove that our algorithms can almost surely recover the correct graph or permutation when tensor entries can be observed without noise. We further establish the robustness of our algorithms against observational noise. The theoretical results are validated by numerical experiments.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":"7 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141527080","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}
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.
{"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. <br/> 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}
SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 2, Page 1215-1215, June 2024. Abstract. A typo in the paper [M. Hladík, SIAM J. Matrix Anal. Appl., 44 (2023), pp. 175–195] is corrected.
{"title":"Erratum: Properties of the Solution Set of Absolute Value Equations and the Related Matrix Classes","authors":"Milan Hladík","doi":"10.1137/24m1635715","DOIUrl":"https://doi.org/10.1137/24m1635715","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 2, Page 1215-1215, June 2024. <br/> Abstract. A typo in the paper [M. Hladík, SIAM J. Matrix Anal. Appl., 44 (2023), pp. 175–195] is corrected.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":"28 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2024-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141509741","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}
SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 2, Page 1183-1214, June 2024. Abstract. This paper develops a class of algorithms for general linear systems and eigenvalue problems. These algorithms apply fast randomized dimension reduction (“sketching”) to accelerate standard subspace projection methods, such as GMRES and Rayleigh–Ritz. This modification makes it possible to incorporate nontraditional bases for the approximation subspace that are easier to construct. When the basis is numerically full rank, the new algorithms have accuracy similar to classic methods but run faster and may use less storage. For model problems, numerical experiments show large advantages over the optimized MATLAB routines, including a [math] speedup over [math] and a [math] speedup over [math].
{"title":"Fast and Accurate Randomized Algorithms for Linear Systems and Eigenvalue Problems","authors":"Yuji Nakatsukasa, Joel A. Tropp","doi":"10.1137/23m1565413","DOIUrl":"https://doi.org/10.1137/23m1565413","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 2, Page 1183-1214, June 2024. <br/> Abstract. This paper develops a class of algorithms for general linear systems and eigenvalue problems. These algorithms apply fast randomized dimension reduction (“sketching”) to accelerate standard subspace projection methods, such as GMRES and Rayleigh–Ritz. This modification makes it possible to incorporate nontraditional bases for the approximation subspace that are easier to construct. When the basis is numerically full rank, the new algorithms have accuracy similar to classic methods but run faster and may use less storage. For model problems, numerical experiments show large advantages over the optimized MATLAB routines, including a [math] speedup over [math] and a [math] speedup over [math].","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":"134 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2024-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141509740","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}
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