SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 3, Page 1689-1719, September 2024. Abstract. We consider the inverse eigenvalue problem of constructing a substochastic matrix from the given spectrum parameters with the corresponding eigenvector constraints. This substochastic inverse eigenvalue problem (SstIEP) with the specific eigenvector constraints is formulated into a nonconvex optimization problem (NcOP). The solvability for SstIEP with the specific eigenvector constraints is equivalent to identifying the attainability of a zero optimal value for the formulated NcOP. When the optimal objective value is zero, the corresponding optimal solution to the formulated NcOP is just the substochastic matrix that we wish to construct. We develop the alternating minimization algorithm to solve the formulated NcOP, and its convergence is established by developing a novel method to obtain the boundedness of the optimal solution. Some numerical experiments are conducted to demonstrate the efficiency of the proposed method.
{"title":"On Substochastic Inverse Eigenvalue Problems with the Corresponding Eigenvector Constraints","authors":"Yujie Liu, Dacheng Yao, Hanqin Zhang","doi":"10.1137/23m1547305","DOIUrl":"https://doi.org/10.1137/23m1547305","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 3, Page 1689-1719, September 2024. <br/> Abstract. We consider the inverse eigenvalue problem of constructing a substochastic matrix from the given spectrum parameters with the corresponding eigenvector constraints. This substochastic inverse eigenvalue problem (SstIEP) with the specific eigenvector constraints is formulated into a nonconvex optimization problem (NcOP). The solvability for SstIEP with the specific eigenvector constraints is equivalent to identifying the attainability of a zero optimal value for the formulated NcOP. When the optimal objective value is zero, the corresponding optimal solution to the formulated NcOP is just the substochastic matrix that we wish to construct. We develop the alternating minimization algorithm to solve the formulated NcOP, and its convergence is established by developing a novel method to obtain the boundedness of the optimal solution. Some numerical experiments are conducted to demonstrate the efficiency of the proposed method.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142218077","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 1669-1688, September 2024. Abstract. We consider the problem of computing tractable approximations of time-dependent [math] large positive semidefinite (PSD) matrices defined as solutions of a matrix differential equation. We propose to use “low-rank plus diagonal” PSD matrices as approximations that can be stored with a memory cost being linear in the high dimension [math]. To constrain the solution of the differential equation to remain in that subset, we project the derivative at all times onto the tangent space to the subset, following the methodology of dynamical low-rank approximation. We derive a closed-form formula for the projection and show that after some manipulations, it can be computed with a numerical cost being linear in [math], allowing for tractable implementation. Contrary to previous approaches based on pure low-rank approximations, the addition of the diagonal term allows for our approximations to be invertible matrices that can moreover be inverted with linear cost in [math]. We apply the technique to Riccati-like equations, then to two particular problems: first, a low-rank approximation to our recent Wasserstein gradient flow for Gaussian approximation of posterior distributions in approximate Bayesian inference and, second, a novel low-rank approximation of the Kalman filter for high-dimensional systems. Numerical simulations illustrate the results.
{"title":"Low-Rank Plus Diagonal Approximations for Riccati-Like Matrix Differential Equations","authors":"Silvère Bonnabel, Marc Lambert, Francis Bach","doi":"10.1137/23m1587610","DOIUrl":"https://doi.org/10.1137/23m1587610","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 3, Page 1669-1688, September 2024. <br/> Abstract. We consider the problem of computing tractable approximations of time-dependent [math] large positive semidefinite (PSD) matrices defined as solutions of a matrix differential equation. We propose to use “low-rank plus diagonal” PSD matrices as approximations that can be stored with a memory cost being linear in the high dimension [math]. To constrain the solution of the differential equation to remain in that subset, we project the derivative at all times onto the tangent space to the subset, following the methodology of dynamical low-rank approximation. We derive a closed-form formula for the projection and show that after some manipulations, it can be computed with a numerical cost being linear in [math], allowing for tractable implementation. Contrary to previous approaches based on pure low-rank approximations, the addition of the diagonal term allows for our approximations to be invertible matrices that can moreover be inverted with linear cost in [math]. We apply the technique to Riccati-like equations, then to two particular problems: first, a low-rank approximation to our recent Wasserstein gradient flow for Gaussian approximation of posterior distributions in approximate Bayesian inference and, second, a novel low-rank approximation of the Kalman filter for high-dimensional systems. Numerical simulations illustrate the results.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142227048","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 1643-1668, September 2024. Abstract. We study the problem of estimating the frequencies of several complex sinusoids with constant amplitude (CA) (also called constant modulus) from multichannel signals of their superposition. To exploit the CA property for frequency estimation in the framework of atomic norm minimization (ANM), we introduce multiple positive-semidenite block matrices composed of Hankel and Toeplitz submatrices and formulate the ANM problem as a convex structured low-rank approximation (SLRA) problem. The proposed SLRA is a semidenite programming and has substantial differences from existing such formulations without using the CA property. The proposed approach is termed as SLRA-based ANM for CA frequency estimation (SACA). We provide theoretical guarantees and extensive simulations that validate the advantages of SACA.
{"title":"Multichannel Frequency Estimation with Constant Amplitude via Convex Structured Low-Rank Approximation","authors":"Xunmeng Wu, Zai Yang, Zongben Xu","doi":"10.1137/23m1587737","DOIUrl":"https://doi.org/10.1137/23m1587737","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 3, Page 1643-1668, September 2024. <br/> Abstract. We study the problem of estimating the frequencies of several complex sinusoids with constant amplitude (CA) (also called constant modulus) from multichannel signals of their superposition. To exploit the CA property for frequency estimation in the framework of atomic norm minimization (ANM), we introduce multiple positive-semidenite block matrices composed of Hankel and Toeplitz submatrices and formulate the ANM problem as a convex structured low-rank approximation (SLRA) problem. The proposed SLRA is a semidenite programming and has substantial differences from existing such formulations without using the CA property. The proposed approach is termed as SLRA-based ANM for CA frequency estimation (SACA). We provide theoretical guarantees and extensive simulations that validate the advantages of SACA.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142218078","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}
Joshua Pickard, Can Chen, Cooper Stansbury, Amit Surana, Anthony M. Bloch, Indika Rajapakse
SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 3, Page 1621-1642, September 2024. Abstract. Hypergraphs and tensors extend classic graph and matrix theories to account for multiway relationships, which are ubiquitous in engineering, biological, and social systems. While the Kronecker product is a potent tool for analyzing the coupling of systems in a graph or matrix context, its utility in studying multiway interactions, such as those represented by tensors and hypergraphs, remains elusive. In this article, we present a comprehensive exploration of algebraic, structural, and spectral properties of the tensor Kronecker product. We express Tucker and tensor train decompositions and various tensor eigenvalues in terms of the tensor Kronecker product. Additionally, we utilize the tensor Kronecker product to form Kronecker hypergraphs, which are tensor-based hypergraph products, and investigate the structure and stability of polynomial dynamics on Kronecker hypergraphs. Finally, we provide numerical examples to demonstrate the utility of the tensor Kronecker product in computing Z-eigenvalues, performing various tensor decompositions, and determining the stability of polynomial systems.
{"title":"Kronecker Product of Tensors and Hypergraphs: Structure and Dynamics","authors":"Joshua Pickard, Can Chen, Cooper Stansbury, Amit Surana, Anthony M. Bloch, Indika Rajapakse","doi":"10.1137/23m1592547","DOIUrl":"https://doi.org/10.1137/23m1592547","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 3, Page 1621-1642, September 2024. <br/> Abstract. Hypergraphs and tensors extend classic graph and matrix theories to account for multiway relationships, which are ubiquitous in engineering, biological, and social systems. While the Kronecker product is a potent tool for analyzing the coupling of systems in a graph or matrix context, its utility in studying multiway interactions, such as those represented by tensors and hypergraphs, remains elusive. In this article, we present a comprehensive exploration of algebraic, structural, and spectral properties of the tensor Kronecker product. We express Tucker and tensor train decompositions and various tensor eigenvalues in terms of the tensor Kronecker product. Additionally, we utilize the tensor Kronecker product to form Kronecker hypergraphs, which are tensor-based hypergraph products, and investigate the structure and stability of polynomial dynamics on Kronecker hypergraphs. Finally, we provide numerical examples to demonstrate the utility of the tensor Kronecker product in computing Z-eigenvalues, performing various tensor decompositions, and determining the stability of polynomial systems.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142227050","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 1599-1620, September 2024. Abstract. Gaussian elimination (GE) is the most used dense linear solver. Error analysis of GE with selected pivoting strategies on well-conditioned systems can focus on studying the behavior of growth factors. Although exponential growth is possible with GE with partial pivoting (GEPP), growth tends to stay much smaller in practice. Support for this behavior was provided recently by Huang and Tikhomirov’s average-case analysis of GEPP, which showed GEPP growth factors for Gaussian matrices stay at most polynomial with very high probability. GE with complete pivoting (GECP) has also seen a lot of recent interest, with improvements to both lower and upper bounds on worst-case GECP growth provided by Bisain, Edelman, and Urschel in 2023. We are interested in studying how GEPP and GECP behave on the same linear systems as well as studying large growth on particular subclasses of matrices, including orthogonal matrices. Moreover, as a means to better address the question of why large growth is rarely encountered, we further study matrices with a large difference in growth between using GEPP and GECP, and we explore how the smaller growth strategy dominates behavior in a small neighborhood of the initial matrix.
{"title":"Growth Factors of Orthogonal Matrices and Local Behavior of Gaussian Elimination with Partial and Complete Pivoting","authors":"John Peca-Medlin","doi":"10.1137/23m1597733","DOIUrl":"https://doi.org/10.1137/23m1597733","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 3, Page 1599-1620, September 2024. <br/> Abstract. Gaussian elimination (GE) is the most used dense linear solver. Error analysis of GE with selected pivoting strategies on well-conditioned systems can focus on studying the behavior of growth factors. Although exponential growth is possible with GE with partial pivoting (GEPP), growth tends to stay much smaller in practice. Support for this behavior was provided recently by Huang and Tikhomirov’s average-case analysis of GEPP, which showed GEPP growth factors for Gaussian matrices stay at most polynomial with very high probability. GE with complete pivoting (GECP) has also seen a lot of recent interest, with improvements to both lower and upper bounds on worst-case GECP growth provided by Bisain, Edelman, and Urschel in 2023. We are interested in studying how GEPP and GECP behave on the same linear systems as well as studying large growth on particular subclasses of matrices, including orthogonal matrices. Moreover, as a means to better address the question of why large growth is rarely encountered, we further study matrices with a large difference in growth between using GEPP and GECP, and we explore how the smaller growth strategy dominates behavior in a small neighborhood of the initial matrix.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142218079","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 1573-1598, September 2024. Abstract. This article is about finding the limit set for banded Toeplitz matrices. Our main result is a new approach to approximate the limit set [math], where [math] is the symbol of the banded Toeplitz matrix. The new approach is geometrical and based on the formula [math], where [math] is a scaling factor, i.e., [math], and [math] denotes the spectrum. We show that the full intersection can be approximated by the intersection for a finite number of [math]’s and that the intersection of polygon approximations for [math] yields an approximating polygon for [math] that converges to [math] in the Hausdorff metric. Further, we show that one can slightly expand the polygon approximations for [math] to ensure that they contain [math]. Then, taking the intersection yields an approximating superset of [math] which converges to [math] in the Hausdorff metric and is guaranteed to contain [math]. Combining the established algebraic (root-finding) method with our approximating superset, we are able to give an explicit bound on the Hausdorff distance to the true limit set. We implement the algorithm in Python and test it. It performs on par to and better in some cases than existing algorithms. We argue, but do not prove, that the average time complexity of the algorithm is [math], where [math] is the number of [math]’s and [math] is the number of vertices for the polygons approximating [math]. Further, we argue that the distance from [math] to both the approximating polygon and the approximating superset decreases as [math] for most of [math], where [math] is the number of elementary operations required by the algorithm.
{"title":"A Geometric Approach to Approximating the Limit Set of Eigenvalues for Banded Toeplitz Matrices","authors":"Teodor Bucht, Jacob S. Christiansen","doi":"10.1137/23m1587804","DOIUrl":"https://doi.org/10.1137/23m1587804","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 3, Page 1573-1598, September 2024. <br/> Abstract. This article is about finding the limit set for banded Toeplitz matrices. Our main result is a new approach to approximate the limit set [math], where [math] is the symbol of the banded Toeplitz matrix. The new approach is geometrical and based on the formula [math], where [math] is a scaling factor, i.e., [math], and [math] denotes the spectrum. We show that the full intersection can be approximated by the intersection for a finite number of [math]’s and that the intersection of polygon approximations for [math] yields an approximating polygon for [math] that converges to [math] in the Hausdorff metric. Further, we show that one can slightly expand the polygon approximations for [math] to ensure that they contain [math]. Then, taking the intersection yields an approximating superset of [math] which converges to [math] in the Hausdorff metric and is guaranteed to contain [math]. Combining the established algebraic (root-finding) method with our approximating superset, we are able to give an explicit bound on the Hausdorff distance to the true limit set. We implement the algorithm in Python and test it. It performs on par to and better in some cases than existing algorithms. We argue, but do not prove, that the average time complexity of the algorithm is [math], where [math] is the number of [math]’s and [math] is the number of vertices for the polygons approximating [math]. Further, we argue that the distance from [math] to both the approximating polygon and the approximating superset decreases as [math] for most of [math], where [math] is the number of elementary operations required by the algorithm.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142218091","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}
Alexander Cloninger, Gal Mishne, Andreas Oslandsbotn, Sawyer J. Robertson, Zhengchao Wan, Yusu Wang
SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 3, Page 1541-1572, September 2024. Abstract. We investigate the concept of effective resistance in connection graphs, expanding its traditional application from undirected graphs. We propose a robust definition of effective resistance in connection graphs by focusing on the duality of Dirichlet-type and Poisson-type problems on connection graphs. Additionally, we delve into random walks, taking into account both node transitions and vector rotations. This approach introduces novel concepts of effective conductance and resistance matrices for connection graphs, capturing mean rotation matrices corresponding to random walk transitions. Thereby, it provides new theoretical insights for network analysis and optimization.
{"title":"Random Walks, Conductance, and Resistance for the Connection Graph Laplacian","authors":"Alexander Cloninger, Gal Mishne, Andreas Oslandsbotn, Sawyer J. Robertson, Zhengchao Wan, Yusu Wang","doi":"10.1137/23m1595400","DOIUrl":"https://doi.org/10.1137/23m1595400","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 3, Page 1541-1572, September 2024. <br/> Abstract. We investigate the concept of effective resistance in connection graphs, expanding its traditional application from undirected graphs. We propose a robust definition of effective resistance in connection graphs by focusing on the duality of Dirichlet-type and Poisson-type problems on connection graphs. Additionally, we delve into random walks, taking into account both node transitions and vector rotations. This approach introduces novel concepts of effective conductance and resistance matrices for connection graphs, capturing mean rotation matrices corresponding to random walk transitions. Thereby, it provides new theoretical insights for network analysis and optimization.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142218090","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}
Christos Boutsikas, Petros Drineas, Ilse C. F. Ipsen
SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 3, Page 1518-1540, September 2024. Abstract. We perturb a real matrix [math] of full column rank, and derive lower bounds for the smallest singular values of the perturbed matrix, in terms of normwise absolute perturbations. Our bounds, which extend existing lower-order expressions, demonstrate the potential increase in the smallest singular values and represent a qualitative model for the increase in the small singular values after a matrix has been downcast to a lower arithmetic precision. Numerical experiments confirm the qualitative validity of this model and its ability to predict singular values changes in the presence of decreased arithmetic precision.
{"title":"Small Singular Values Can Increase in Lower Precision","authors":"Christos Boutsikas, Petros Drineas, Ilse C. F. Ipsen","doi":"10.1137/23m1557209","DOIUrl":"https://doi.org/10.1137/23m1557209","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 3, Page 1518-1540, September 2024. <br/> Abstract. We perturb a real matrix [math] of full column rank, and derive lower bounds for the smallest singular values of the perturbed matrix, in terms of normwise absolute perturbations. Our bounds, which extend existing lower-order expressions, demonstrate the potential increase in the smallest singular values and represent a qualitative model for the increase in the small singular values after a matrix has been downcast to a lower arithmetic precision. Numerical experiments confirm the qualitative validity of this model and its ability to predict singular values changes in the presence of decreased arithmetic precision.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142218092","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":"Reorthogonalized Block Classical Gram–Schmidt Using Two Cholesky-Based TSQR Algorithms","authors":"Jesse L. Barlow","doi":"10.1137/23m1605387","DOIUrl":"https://doi.org/10.1137/23m1605387","url":null,"abstract":"","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141924462","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 1455-1486, September 2024. Abstract. A two dimensional eigenvalue problem (2DEVP) of a Hermitian matrix pair [math] is introduced in this paper. The 2DEVP can be regarded as a linear algebra formulation of the well-known eigenvalue optimization problem of the parameter matrix [math]. We first present fundamental properties of the 2DEVP, such as the existence and variational characterizations of 2D-eigenvalues, and then devise a Rayleigh quotient iteration (RQI)-like algorithm, 2DRQI in short, for computing a 2D-eigentriplet of the 2DEVP. The efficacy of the 2DRQI is demonstrated by large scale eigenvalue optimization problems arising from the minmax of Rayleigh quotients and the distance to instability of a stable matrix.
{"title":"Variational Characterization and Rayleigh Quotient Iteration of 2D Eigenvalue Problem with Applications","authors":"Tianyi Lu, Yangfeng Su, Zhaojun Bai","doi":"10.1137/22m1472589","DOIUrl":"https://doi.org/10.1137/22m1472589","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 3, Page 1455-1486, September 2024. <br/> Abstract. A two dimensional eigenvalue problem (2DEVP) of a Hermitian matrix pair [math] is introduced in this paper. The 2DEVP can be regarded as a linear algebra formulation of the well-known eigenvalue optimization problem of the parameter matrix [math]. We first present fundamental properties of the 2DEVP, such as the existence and variational characterizations of 2D-eigenvalues, and then devise a Rayleigh quotient iteration (RQI)-like algorithm, 2DRQI in short, for computing a 2D-eigentriplet of the 2DEVP. The efficacy of the 2DRQI is demonstrated by large scale eigenvalue optimization problems arising from the minmax of Rayleigh quotients and the distance to instability of a stable matrix.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141943696","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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