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":null,"pages":null},"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":null,"pages":null},"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}
{"title":"Preconditioner Design via Bregman Divergences","authors":"Andreas A. Bock, Martin S. Andersen","doi":"10.1137/23m1566637","DOIUrl":"https://doi.org/10.1137/23m1566637","url":null,"abstract":"","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141372531","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 1114-1147, June 2024. Abstract. The spectral decomposition of a real skew-symmetric matrix is shown to be equivalent to a specific structured singular value decomposition (SVD) of the matrix. Based on such equivalence, we propose a skew-symmetric Lanczos bidiagonalization (SSLBD) method to compute extremal singular values and the corresponding singular vectors of the matrix, from which its extremal conjugate eigenpairs are recovered pairwise in real arithmetic. A number of convergence results on the method are established, and accuracy estimates for approximate singular triplets are given. In finite precision arithmetic, it is proven that the semi-orthogonality of each set of the computed left and right Lanczos basis vectors and the semi-biorthogonality of two sets of basis vectors are needed to compute the singular values accurately and to make the method work as if it does in exact arithmetic. A commonly used efficient partial reorthogonalization strategy is adapted to maintain the desired semi-orthogonality and semi-biorthogonality. For practical purpose, an implicitly restarted SSLBD algorithm is developed with partial reorthogonalization. Numerical experiments illustrate the effectiveness and overall efficiency of the algorithm.
{"title":"A Skew-Symmetric Lanczos Bidiagonalization Method for Computing Several Extremal Eigenpairs of a Large Skew-Symmetric Matrix","authors":"Jinzhi Huang, Zhongxiao Jia","doi":"10.1137/23m1553029","DOIUrl":"https://doi.org/10.1137/23m1553029","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 2, Page 1114-1147, June 2024. <br/> Abstract. The spectral decomposition of a real skew-symmetric matrix is shown to be equivalent to a specific structured singular value decomposition (SVD) of the matrix. Based on such equivalence, we propose a skew-symmetric Lanczos bidiagonalization (SSLBD) method to compute extremal singular values and the corresponding singular vectors of the matrix, from which its extremal conjugate eigenpairs are recovered pairwise in real arithmetic. A number of convergence results on the method are established, and accuracy estimates for approximate singular triplets are given. In finite precision arithmetic, it is proven that the semi-orthogonality of each set of the computed left and right Lanczos basis vectors and the semi-biorthogonality of two sets of basis vectors are needed to compute the singular values accurately and to make the method work as if it does in exact arithmetic. A commonly used efficient partial reorthogonalization strategy is adapted to maintain the desired semi-orthogonality and semi-biorthogonality. For practical purpose, an implicitly restarted SSLBD algorithm is developed with partial reorthogonalization. Numerical experiments illustrate the effectiveness and overall efficiency of the algorithm.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141257265","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}
Cyrus Mostajeran, Nathaël Da Costa, Graham Van Goffrier, Rodolphe Sepulchre
SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 2, Page 1089-1113, June 2024. Abstract. Differential geometric approaches to the analysis and processing of data in the form of symmetric positive definite (SPD) matrices have had notable successful applications to numerous fields, including computer vision, medical imaging, and machine learning. The dominant geometric paradigm for such applications has consisted of a few Riemannian geometries associated with spectral computations that are costly at high scale and in high dimensions. We present a route to a scalable geometric framework for the analysis and processing of SPD-valued data based on the efficient computation of extreme generalized eigenvalues through the Hilbert and Thompson geometries of the semidefinite cone. We explore a particular geodesic space structure based on Thompson geometry in detail and establish several properties associated with this structure. Furthermore, we define a novel inductive mean of SPD matrices based on this geometry and prove its existence and uniqueness for a given finite collection of points. Finally, we state and prove a number of desirable properties that are satisfied by this mean.
SIAM 矩阵分析与应用期刊》，第 45 卷，第 2 期，第 1089-1113 页，2024 年 6 月。 摘要。以对称正定（SPD）矩阵形式分析和处理数据的微分几何方法在计算机视觉、医学成像和机器学习等众多领域都有显著的成功应用。此类应用的主流几何范式包括一些与光谱计算相关的黎曼几何图形，这些几何图形在高尺度和高维度下耗费巨大。我们通过半定锥的希尔伯特和汤普森几何图形，在高效计算极端广义特征值的基础上，为分析和处理 SPD 值数据提出了一个可扩展的几何框架。我们详细探讨了基于汤普森几何的特定大地空间结构，并建立了与该结构相关的若干属性。此外，我们还基于这种几何结构定义了一种新颖的 SPD 矩阵归纳平均值，并证明了它对于给定有限点集合的存在性和唯一性。最后，我们陈述并证明了该均值所满足的一系列理想属性。
{"title":"Differential Geometry with Extreme Eigenvalues in the Positive Semidefinite Cone","authors":"Cyrus Mostajeran, Nathaël Da Costa, Graham Van Goffrier, Rodolphe Sepulchre","doi":"10.1137/23m1563906","DOIUrl":"https://doi.org/10.1137/23m1563906","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 2, Page 1089-1113, June 2024. <br/> Abstract. Differential geometric approaches to the analysis and processing of data in the form of symmetric positive definite (SPD) matrices have had notable successful applications to numerous fields, including computer vision, medical imaging, and machine learning. The dominant geometric paradigm for such applications has consisted of a few Riemannian geometries associated with spectral computations that are costly at high scale and in high dimensions. We present a route to a scalable geometric framework for the analysis and processing of SPD-valued data based on the efficient computation of extreme generalized eigenvalues through the Hilbert and Thompson geometries of the semidefinite cone. We explore a particular geodesic space structure based on Thompson geometry in detail and establish several properties associated with this structure. Furthermore, we define a novel inductive mean of SPD matrices based on this geometry and prove its existence and uniqueness for a given finite collection of points. Finally, we state and prove a number of desirable properties that are satisfied by this mean.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141551311","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 1054-1075, June 2024. Abstract. This paper presents a memory-efficient, first-order method for low multilinear rank approximation of high-order, high-dimensional tensors. In our method, we exploit the second-order information of the cost function and the constraints to suggest a new Riemannian metric on the Grassmann manifold. We use a Riemmanian coordinate descent method for solving the problem and also provide a global convergence analysis matching that of the coordinate descent method in the Euclidean setting. We also show that each step of our method with the unit step size is actually a step of the orthogonal iteration algorithm. Experimental results show the computational advantage of our method for high-dimensional tensors.
{"title":"Riemannian Preconditioned Coordinate Descent for Low Multilinear Rank Approximation","authors":"Mohammad Hamed, Reshad Hosseini","doi":"10.1137/21m1463896","DOIUrl":"https://doi.org/10.1137/21m1463896","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 2, Page 1054-1075, June 2024. <br/> Abstract. This paper presents a memory-efficient, first-order method for low multilinear rank approximation of high-order, high-dimensional tensors. In our method, we exploit the second-order information of the cost function and the constraints to suggest a new Riemannian metric on the Grassmann manifold. We use a Riemmanian coordinate descent method for solving the problem and also provide a global convergence analysis matching that of the coordinate descent method in the Euclidean setting. We also show that each step of our method with the unit step size is actually a step of the orthogonal iteration algorithm. Experimental results show the computational advantage of our method for high-dimensional tensors.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141152353","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}
{"title":"Probabilistic Rounding Error Analysis of Modified Gram–Schmidt","authors":"Qinmeng Zou","doi":"10.1137/23m1585817","DOIUrl":"https://doi.org/10.1137/23m1585817","url":null,"abstract":"","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141114215","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 1035-1053, June 2024. Abstract. We study the tangential interpolation problem for a passive transfer function in standard state-space form. We derive new interpolation conditions based on the computation of a deflating subspace associated with a selection of spectral zeros of a parameterized para-Hermitian transfer function. We show that this technique improves the robustness of the low order model and that it can also be applied to nonpassive systems, provided they have sufficiently many spectral zeros in the open right half-plane. We analyze the accuracy needed for the computation of the deflating subspace, in order to still have a passive lower order model and we derive a novel selection procedure of spectral zeros in order to obtain low order models with a small approximation error.
{"title":"Parameterized Interpolation of Passive Systems","authors":"Peter Benner, Pawan Goyal, Paul Van Dooren","doi":"10.1137/23m1580528","DOIUrl":"https://doi.org/10.1137/23m1580528","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 2, Page 1035-1053, June 2024. <br/> Abstract. We study the tangential interpolation problem for a passive transfer function in standard state-space form. We derive new interpolation conditions based on the computation of a deflating subspace associated with a selection of spectral zeros of a parameterized para-Hermitian transfer function. We show that this technique improves the robustness of the low order model and that it can also be applied to nonpassive systems, provided they have sufficiently many spectral zeros in the open right half-plane. We analyze the accuracy needed for the computation of the deflating subspace, in order to still have a passive lower order model and we derive a novel selection procedure of spectral zeros in order to obtain low order models with a small approximation error.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141152360","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}
Ivo Dravins, Stefano Serra-Capizzano, Maya Neytcheva
SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 2, Page 1007-1034, June 2024. Abstract. The use of high order fully implicit Runge–Kutta methods is of significant importance in the context of the numerical solution of transient partial differential equations, in particular when solving large scale problems due to fine space resolution with many millions of spatial degrees of freedom and long time intervals. In this study we consider strongly [math]-stable implicit Runge–Kutta methods of arbitrary order of accuracy, based on Radau quadratures, for which efficient preconditioners have been introduced. A refined spectral analysis of the corresponding matrices and matrix sequences is presented, both in terms of localization and asymptotic global distribution of the eigenvalues. Specific expressions of the eigenvectors are also obtained. The given study fully agrees with the numerically observed spectral behavior and substantially improves the theoretical studies done in this direction so far. Concluding remarks and open problems end the current work, with specific attention to the potential generalizations of the hereby suggested general approach.
{"title":"Spectral Analysis of Preconditioned Matrices Arising from Stage-Parallel Implicit Runge–Kutta Methods of Arbitrarily High Order","authors":"Ivo Dravins, Stefano Serra-Capizzano, Maya Neytcheva","doi":"10.1137/23m1552498","DOIUrl":"https://doi.org/10.1137/23m1552498","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 2, Page 1007-1034, June 2024. <br/> Abstract. The use of high order fully implicit Runge–Kutta methods is of significant importance in the context of the numerical solution of transient partial differential equations, in particular when solving large scale problems due to fine space resolution with many millions of spatial degrees of freedom and long time intervals. In this study we consider strongly [math]-stable implicit Runge–Kutta methods of arbitrary order of accuracy, based on Radau quadratures, for which efficient preconditioners have been introduced. A refined spectral analysis of the corresponding matrices and matrix sequences is presented, both in terms of localization and asymptotic global distribution of the eigenvalues. Specific expressions of the eigenvectors are also obtained. The given study fully agrees with the numerically observed spectral behavior and substantially improves the theoretical studies done in this direction so far. Concluding remarks and open problems end the current work, with specific attention to the potential generalizations of the hereby suggested general approach.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140931836","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}
El Houcine Bergou, Soumia Boucherouite, Aritra Dutta, Xin Li, Anna Ma
SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 2, Page 992-1006, June 2024. Abstract. Large-scale linear systems, [math], frequently arise in practice and demand effective iterative solvers. Often, these systems are noisy due to operational errors or faulty data-collection processes. In the past decade, the randomized Kaczmarz algorithm (RK) has been studied extensively as an efficient iterative solver for such systems. However, the convergence study of RK in the noisy regime is limited and considers measurement noise in the right-hand side vector, [math]. Unfortunately, in practice, that is not always the case; the coefficient matrix [math] can also be noisy. In this paper, we analyze the convergence of RK for doubly noisy linear systems, i.e., when the coefficient matrix, [math], has additive or multiplicative noise, and [math] is also noisy. In our analyses, the quantity [math] influences the convergence of RK, where [math] represents a noisy version of [math]. We claim that our analysis is robust and realistically applicable, as we do not require information about the noiseless coefficient matrix, [math], and by considering different conditions on noise, we can control the convergence of RK. We perform numerical experiments to substantiate our theoretical findings.
{"title":"A Note on the Randomized Kaczmarz Algorithm for Solving Doubly Noisy Linear Systems","authors":"El Houcine Bergou, Soumia Boucherouite, Aritra Dutta, Xin Li, Anna Ma","doi":"10.1137/23m155712x","DOIUrl":"https://doi.org/10.1137/23m155712x","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 2, Page 992-1006, June 2024. <br/> Abstract. Large-scale linear systems, [math], frequently arise in practice and demand effective iterative solvers. Often, these systems are noisy due to operational errors or faulty data-collection processes. In the past decade, the randomized Kaczmarz algorithm (RK) has been studied extensively as an efficient iterative solver for such systems. However, the convergence study of RK in the noisy regime is limited and considers measurement noise in the right-hand side vector, [math]. Unfortunately, in practice, that is not always the case; the coefficient matrix [math] can also be noisy. In this paper, we analyze the convergence of RK for doubly noisy linear systems, i.e., when the coefficient matrix, [math], has additive or multiplicative noise, and [math] is also noisy. In our analyses, the quantity [math] influences the convergence of RK, where [math] represents a noisy version of [math]. We claim that our analysis is robust and realistically applicable, as we do not require information about the noiseless coefficient matrix, [math], and by considering different conditions on noise, we can control the convergence of RK. We perform numerical experiments to substantiate our theoretical findings.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140886693","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：