{"title":"Analyzing the Influence of Agents in Trust Networks: Applying Nonsmooth Eigensensitivity Theory to a Graph Centrality Problem","authors":"Jon Donnelly, Peter G. Stechlinski","doi":"10.1137/21m146884x","DOIUrl":"https://doi.org/10.1137/21m146884x","url":null,"abstract":"","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":"44 1","pages":"1271-1298"},"PeriodicalIF":1.5,"publicationDate":"2023-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"64315182","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}
Vanni Noferini, Lauri Nyman, Javier Pérez, María C. Quintana
Zeros of rational transfer function matrices are the eigenvalues of associated polynomial system matrices under minimality conditions. In this paper, we define a structured condition number for a simple eigenvalue of a (locally) minimal polynomial system matrix , which in turn is a simple zero of its transfer function matrix . Since any rational matrix can be written as the transfer function of a polynomial system matrix, our analysis yields a structured perturbation theory for simple zeros of rational matrices . To capture all the zeros of , regardless of whether they are poles, we consider the notion of root vectors. As corollaries of the main results, we pay particular attention to the special case of being not a pole of since in this case the results get simpler and can be useful in practice. We also compare our structured condition number with Tisseur’s unstructured condition number for eigenvalues of matrix polynomials and show that the latter can be unboundedly larger. Finally, we corroborate our analysis by numerical experiments.
{"title":"Perturbation Theory of Transfer Function Matrices","authors":"Vanni Noferini, Lauri Nyman, Javier Pérez, María C. Quintana","doi":"10.1137/22m1509825","DOIUrl":"https://doi.org/10.1137/22m1509825","url":null,"abstract":"Zeros of rational transfer function matrices are the eigenvalues of associated polynomial system matrices under minimality conditions. In this paper, we define a structured condition number for a simple eigenvalue of a (locally) minimal polynomial system matrix , which in turn is a simple zero of its transfer function matrix . Since any rational matrix can be written as the transfer function of a polynomial system matrix, our analysis yields a structured perturbation theory for simple zeros of rational matrices . To capture all the zeros of , regardless of whether they are poles, we consider the notion of root vectors. As corollaries of the main results, we pay particular attention to the special case of being not a pole of since in this case the results get simpler and can be useful in practice. We also compare our structured condition number with Tisseur’s unstructured condition number for eigenvalues of matrix polynomials and show that the latter can be unboundedly larger. Finally, we corroborate our analysis by numerical experiments.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":"64 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136081554","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":"Eigenvalue Embedding of Damped Vibroacoustic System with No-Spillover","authors":"K. Zhao, Zhong Y. Liu","doi":"10.1137/22m1527416","DOIUrl":"https://doi.org/10.1137/22m1527416","url":null,"abstract":"","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":"289 1","pages":"1189-1217"},"PeriodicalIF":1.5,"publicationDate":"2023-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76183506","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}
In this paper, we revisit the greatest common right divisor (GCRD) extraction from a set of polynomial matrices , , with coefficients in a generic field and with common column dimension . We give necessary and sufficient conditions for a matrix to be a GCRD using the Smith normal form of the compound matrix obtained by concatenating vertically, where . We also describe the complete set of degrees of freedom for the solution , and we link it to the Smith form and Hermite form of . We then give an algorithm for constructing a particular minimum size solution for this problem when or , using state-space techniques. This new method works directly on the coefficient matrices of , using orthogonal transformations only. The method is based on the staircase algorithm, applied to a particular pencil derived from a generalized state-space model of .
{"title":"Revisiting the Matrix Polynomial Greatest Common Divisor","authors":"Vanni Noferini, Paul Van Dooren","doi":"10.1137/22m1531993","DOIUrl":"https://doi.org/10.1137/22m1531993","url":null,"abstract":"In this paper, we revisit the greatest common right divisor (GCRD) extraction from a set of polynomial matrices , , with coefficients in a generic field and with common column dimension . We give necessary and sufficient conditions for a matrix to be a GCRD using the Smith normal form of the compound matrix obtained by concatenating vertically, where . We also describe the complete set of degrees of freedom for the solution , and we link it to the Smith form and Hermite form of . We then give an algorithm for constructing a particular minimum size solution for this problem when or , using state-space techniques. This new method works directly on the coefficient matrices of , using orthogonal transformations only. The method is based on the staircase algorithm, applied to a particular pencil derived from a generalized state-space model of .","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135795066","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 Householder QR Factorization","authors":"Michael P. Connolly, N. Higham","doi":"10.1137/22m1514817","DOIUrl":"https://doi.org/10.1137/22m1514817","url":null,"abstract":"","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":"19 1","pages":"1146-1163"},"PeriodicalIF":1.5,"publicationDate":"2023-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88001085","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 44, Issue 3, Page 1096-1121, September 2023. Abstract. We analyze self-dual polyhedral cones and prove several properties about their slack matrices. In particular, we show that self-duality is equivalent to the existence of a positive semidefinite (PSD) slack. Beyond that, we show that if the underlying cone is irreducible, then the corresponding PSD slacks are not only doubly nonnegative matrices (DNN) but are extreme rays of the cone of DNN matrices, which correspond to a family of extreme rays not previously described. More surprisingly, we show that, unless the cone is simplicial, PSD slacks not only fail to be completely positive matrices but they also lie outside the cone of completely positive semidefinite matrices. Finally, we show how one can use semidefinite programming to probe the existence of self-dual cones with given combinatorics. Our results are given for polyhedral cones but we also discuss some consequences for negatively self-polar polytopes.
{"title":"Self-Dual Polyhedral Cones and Their Slack Matrices","authors":"João Gouveia, Bruno F. Lourenço","doi":"10.1137/22m1519869","DOIUrl":"https://doi.org/10.1137/22m1519869","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 44, Issue 3, Page 1096-1121, September 2023. <br/> Abstract. We analyze self-dual polyhedral cones and prove several properties about their slack matrices. In particular, we show that self-duality is equivalent to the existence of a positive semidefinite (PSD) slack. Beyond that, we show that if the underlying cone is irreducible, then the corresponding PSD slacks are not only doubly nonnegative matrices (DNN) but are extreme rays of the cone of DNN matrices, which correspond to a family of extreme rays not previously described. More surprisingly, we show that, unless the cone is simplicial, PSD slacks not only fail to be completely positive matrices but they also lie outside the cone of completely positive semidefinite matrices. Finally, we show how one can use semidefinite programming to probe the existence of self-dual cones with given combinatorics. Our results are given for polyhedral cones but we also discuss some consequences for negatively self-polar polytopes.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":"12 3","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138513879","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 44, Issue 3, Page 1073-1095, September 2023. Abstract. The computation of [math], the action of a matrix function on a vector, is a task arising in many areas of scientific computing. In many applications, the matrix [math] is sparse but so large that only a rather small number of Krylov basis vectors can be stored. Here we discuss a new approach to overcome this limitation by randomized sketching combined with an integral representation of [math]. Two different approximation methods are introduced, one based on sketched FOM and another based on sketched GMRES. The convergence of the latter method is analyzed for Stieltjes functions of positive real matrices. We also derive a closed-form expression for the sketched FOM approximant and bound its distance to the full FOM approximant. Numerical experiments demonstrate the potential of the presented sketching approaches.
{"title":"Randomized Sketching for Krylov Approximations of Large-Scale Matrix Functions","authors":"Stefan Güttel, Marcel Schweitzer","doi":"10.1137/22m1518062","DOIUrl":"https://doi.org/10.1137/22m1518062","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 44, Issue 3, Page 1073-1095, September 2023. <br/> Abstract. The computation of [math], the action of a matrix function on a vector, is a task arising in many areas of scientific computing. In many applications, the matrix [math] is sparse but so large that only a rather small number of Krylov basis vectors can be stored. Here we discuss a new approach to overcome this limitation by randomized sketching combined with an integral representation of [math]. Two different approximation methods are introduced, one based on sketched FOM and another based on sketched GMRES. The convergence of the latter method is analyzed for Stieltjes functions of positive real matrices. We also derive a closed-form expression for the sketched FOM approximant and bound its distance to the full FOM approximant. Numerical experiments demonstrate the potential of the presented sketching approaches.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":"232 6","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138513876","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}
Christopher Brissette, Andy Huang, George M. Slota
{"title":"Spectrum Consistent Coarsening Approximates Edge Weights","authors":"Christopher Brissette, Andy Huang, George M. Slota","doi":"10.1137/21m1458119","DOIUrl":"https://doi.org/10.1137/21m1458119","url":null,"abstract":"","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":"17 1","pages":"1032-1046"},"PeriodicalIF":1.5,"publicationDate":"2023-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86361376","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}
Tensor Kronecker products, the natural generalization of the matrix Kronecker product, are independently emerging in multiple research communities. Like their matrix counterpart, the tensor generalization gives structure for implicit multiplication and factorization theorems. We present a theorem that decouples the dominant eigenvectors of tensor Kronecker products, which is a rare generalization from matrix theory to tensor eigenvectors. This theorem implies low-rank structure ought to be present in the iterates of tensor power methods on Kronecker products. We investigate low-rank structure in the network alignment algorithm TAME, a power method heuristic. Using the low-rank structure directly or via a new heuristic embedding approach, we produce new algorithms which are faster while improving or maintaining accuracy, and which scale to problems that cannot be realistically handled with existing techniques.
{"title":"Dominant Z-Eigenpairs of Tensor Kronecker Products Decouple","authors":"Charles Colley, Huda Nassar, D. Gleich","doi":"10.1137/22m1502008","DOIUrl":"https://doi.org/10.1137/22m1502008","url":null,"abstract":"Tensor Kronecker products, the natural generalization of the matrix Kronecker product, are independently emerging in multiple research communities. Like their matrix counterpart, the tensor generalization gives structure for implicit multiplication and factorization theorems. We present a theorem that decouples the dominant eigenvectors of tensor Kronecker products, which is a rare generalization from matrix theory to tensor eigenvectors. This theorem implies low-rank structure ought to be present in the iterates of tensor power methods on Kronecker products. We investigate low-rank structure in the network alignment algorithm TAME, a power method heuristic. Using the low-rank structure directly or via a new heuristic embedding approach, we produce new algorithms which are faster while improving or maintaining accuracy, and which scale to problems that cannot be realistically handled with existing techniques.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":"2 1","pages":"1006-1031"},"PeriodicalIF":1.5,"publicationDate":"2023-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77295857","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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