Nearest Ω-stable matrix via Riemannian optimization
We study the problem of finding the nearest Ω -stable matrix to a certain matrix A , i.e., the nearest matrix with all its eigenvalues in a prescribed closed set Ω . Distances are measured in the Frobenius norm. An important special case is finding the nearest Hurwitz or Schur stable matrix, which h...
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| Veröffentlicht in: | Numerische Mathematik 2021-08, Vol.148 (4), p.817-851 |
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| creator | Noferini, Vanni Poloni, Federico |
| description | We study the problem of finding the nearest
Ω
-stable matrix to a certain matrix
A
, i.e., the nearest matrix with all its eigenvalues in a prescribed closed set
Ω
. Distances are measured in the Frobenius norm. An important special case is finding the nearest Hurwitz or Schur stable matrix, which has applications in systems theory. We describe a reformulation of the task as an optimization problem on the Riemannian manifold of orthogonal (or unitary) matrices. The problem can then be solved using standard methods from the theory of Riemannian optimization. The resulting algorithm is remarkably fast on small-scale and medium-scale matrices, and returns directly a Schur factorization of the minimizer, sidestepping the numerical difficulties associated with eigenvalues with high multiplicity. |
| doi_str_mv | 10.1007/s00211-021-01217-4 |
| format | Article |
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Ω
-stable matrix to a certain matrix
A
, i.e., the nearest matrix with all its eigenvalues in a prescribed closed set
Ω
. Distances are measured in the Frobenius norm. An important special case is finding the nearest Hurwitz or Schur stable matrix, which has applications in systems theory. We describe a reformulation of the task as an optimization problem on the Riemannian manifold of orthogonal (or unitary) matrices. The problem can then be solved using standard methods from the theory of Riemannian optimization. The resulting algorithm is remarkably fast on small-scale and medium-scale matrices, and returns directly a Schur factorization of the minimizer, sidestepping the numerical difficulties associated with eigenvalues with high multiplicity.</description><identifier>ISSN: 0029-599X</identifier><identifier>EISSN: 0945-3245</identifier><identifier>DOI: 10.1007/s00211-021-01217-4</identifier><language>eng</language><publisher>Berlin/Heidelberg: Springer Berlin Heidelberg</publisher><subject>Algorithms ; Eigenvalues ; Mathematical and Computational Engineering ; Mathematical and Computational Physics ; Mathematical Methods in Physics ; Mathematics ; Mathematics and Statistics ; Numerical Analysis ; Numerical and Computational Physics ; Optimization ; Riemann manifold ; Simulation ; System theory ; Systems theory ; Theoretical</subject><ispartof>Numerische Mathematik, 2021-08, Vol.148 (4), p.817-851</ispartof><rights>The Author(s) 2021</rights><rights>The Author(s) 2021. This work is published under http://creativecommons.org/licenses/by/4.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s00211-021-01217-4$$EPDF$$P50$$Gspringer$$Hfree_for_read</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s00211-021-01217-4$$EHTML$$P50$$Gspringer$$Hfree_for_read</linktohtml><link.rule.ids>314,780,784,27924,27925,41488,42557,51319</link.rule.ids></links><search><creatorcontrib>Noferini, Vanni</creatorcontrib><creatorcontrib>Poloni, Federico</creatorcontrib><title>Nearest Ω-stable matrix via Riemannian optimization</title><title>Numerische Mathematik</title><addtitle>Numer. Math</addtitle><description>We study the problem of finding the nearest
Ω
-stable matrix to a certain matrix
A
, i.e., the nearest matrix with all its eigenvalues in a prescribed closed set
Ω
. Distances are measured in the Frobenius norm. An important special case is finding the nearest Hurwitz or Schur stable matrix, which has applications in systems theory. We describe a reformulation of the task as an optimization problem on the Riemannian manifold of orthogonal (or unitary) matrices. The problem can then be solved using standard methods from the theory of Riemannian optimization. The resulting algorithm is remarkably fast on small-scale and medium-scale matrices, and returns directly a Schur factorization of the minimizer, sidestepping the numerical difficulties associated with eigenvalues with high multiplicity.</description><subject>Algorithms</subject><subject>Eigenvalues</subject><subject>Mathematical and Computational Engineering</subject><subject>Mathematical and Computational Physics</subject><subject>Mathematical Methods in Physics</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Numerical Analysis</subject><subject>Numerical and Computational Physics</subject><subject>Optimization</subject><subject>Riemann manifold</subject><subject>Simulation</subject><subject>System theory</subject><subject>Systems theory</subject><subject>Theoretical</subject><issn>0029-599X</issn><issn>0945-3245</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2021</creationdate><recordtype>article</recordtype><sourceid>C6C</sourceid><recordid>eNpFkM9KxDAQh4MouK6-gKeC5-hMMmmSoyzqCouCKHgLaTeVLNs_Nl0R38iX8ZmsVvDymzl8_Gb4GDtFOEcAfZEABCIfgwMK1Jz22AwsKS4Fqf1xB2G5svb5kB2ltAFAnRPOGN0F34c0ZF-fPA2-2Ias9kMf37O36LOHGGrfNNE3WdsNsY4ffohtc8wOKr9N4eRvztnT9dXjYslX9ze3i8sV74TQAycqpAg6l9XaFqZQNgRjgMCQoIKE9LhWhTfSVoiecmMV5EpVAowsVVmVcs7Opt6ub19345du0-76ZjzphMolGq2MHik5UanrY_MS-n8Kwf3ocZMeN4b71eNIfgOrLFdJ</recordid><startdate>20210801</startdate><enddate>20210801</enddate><creator>Noferini, Vanni</creator><creator>Poloni, Federico</creator><general>Springer Berlin Heidelberg</general><general>Springer Nature B.V</general><scope>C6C</scope></search><sort><creationdate>20210801</creationdate><title>Nearest Ω-stable matrix via Riemannian optimization</title><author>Noferini, Vanni ; Poloni, Federico</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-p227t-44b32e763fd9b8b59ee880408424b423a1d5ba839f11a468950655f2083c5cfc3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2021</creationdate><topic>Algorithms</topic><topic>Eigenvalues</topic><topic>Mathematical and Computational Engineering</topic><topic>Mathematical and Computational Physics</topic><topic>Mathematical Methods in Physics</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Numerical Analysis</topic><topic>Numerical and Computational Physics</topic><topic>Optimization</topic><topic>Riemann manifold</topic><topic>Simulation</topic><topic>System theory</topic><topic>Systems theory</topic><topic>Theoretical</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Noferini, Vanni</creatorcontrib><creatorcontrib>Poloni, Federico</creatorcontrib><collection>Springer Nature OA Free Journals</collection><jtitle>Numerische Mathematik</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Noferini, Vanni</au><au>Poloni, Federico</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Nearest Ω-stable matrix via Riemannian optimization</atitle><jtitle>Numerische Mathematik</jtitle><stitle>Numer. Math</stitle><date>2021-08-01</date><risdate>2021</risdate><volume>148</volume><issue>4</issue><spage>817</spage><epage>851</epage><pages>817-851</pages><issn>0029-599X</issn><eissn>0945-3245</eissn><abstract>We study the problem of finding the nearest
Ω
-stable matrix to a certain matrix
A
, i.e., the nearest matrix with all its eigenvalues in a prescribed closed set
Ω
. Distances are measured in the Frobenius norm. An important special case is finding the nearest Hurwitz or Schur stable matrix, which has applications in systems theory. We describe a reformulation of the task as an optimization problem on the Riemannian manifold of orthogonal (or unitary) matrices. The problem can then be solved using standard methods from the theory of Riemannian optimization. The resulting algorithm is remarkably fast on small-scale and medium-scale matrices, and returns directly a Schur factorization of the minimizer, sidestepping the numerical difficulties associated with eigenvalues with high multiplicity.</abstract><cop>Berlin/Heidelberg</cop><pub>Springer Berlin Heidelberg</pub><doi>10.1007/s00211-021-01217-4</doi><tpages>35</tpages><oa>free_for_read</oa></addata></record> |
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| subjects | Algorithms Eigenvalues Mathematical and Computational Engineering Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Numerical Analysis Numerical and Computational Physics Optimization Riemann manifold Simulation System theory Systems theory Theoretical |
| title | Nearest Ω-stable matrix via Riemannian optimization |
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