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Matrix Anal. Appl."],"published-print":{"date-parts":[[2019,1]]},"abstract":"<jats:p>We study the structured condition number of differentiable maps between smooth matrix manifolds, extending previous results to maps that are only $\\mathbb{R}$-differentiable for complex manifolds. We present algorithms to compute the structured condition number. As special cases of smooth manifolds, we analyze automorphism groups, and Lie and Jordan algebras associated with a scalar product. For such manifolds, we derive a lower bound on the structured condition number that is cheaper to compute than the structured condition number. We provide numerical comparisons between the structured and unstructured condition numbers for the principal matrix logarithm and principal matrix square root of matrices in automorphism groups as well as for the map between matrices in automorphism groups and their polar decomposition. We show that our lower bound can be used as a good estimate for the structured condition number when the matrix argument is well conditioned. We show that the structured and unstructured condition numbers can differ by many orders of magnitude, thus motivating the development of algorithms preserving structure.<\/jats:p>","DOI":"10.1137\/17m1148943","type":"journal-article","created":{"date-parts":[[2019,6,25]],"date-time":"2019-06-25T11:41:31Z","timestamp":1561462891000},"page":"774-799","source":"Crossref","is-referenced-by-count":10,"title":["The Structured Condition Number of a Differentiable Map between Matrix Manifolds, with Applications"],"prefix":"10.1137","volume":"40","author":[{"given":"Bahar","family":"Arslan","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1775-041X","authenticated-orcid":true,"given":"Vanni","family":"Noferini","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1011-2570","authenticated-orcid":true,"given":"Fran\u00e7oise","family":"Tisseur","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2019,6,25]]},"reference":[{"key":"atypb1","doi-asserted-by":"crossref","unstructured":"P. B\u00fcrgisser and F. Cucker,\n                      Condition. The Geometry of Numerical Algorithms\n                      , Springer-Verlag, Berlin, 2013.","DOI":"10.1007\/978-3-642-38896-5"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.13001\/1081-3810.1128"},{"key":"atypb3","first-page":"263","author":"Dedieu J.-P.","year":"1996","journal-title":"RI"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2004.12.012"},{"key":"atypb5","unstructured":"G. H. Golub and C. F. Van Loan,\n                      Matrix Computations\n                      , 4th ed., Johns Hopkins University Press, Baltimore, MD, 2013."},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1080\/03081087308817015"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1080\/03081088108817379"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1137\/0907079"},{"key":"atypb10","doi-asserted-by":"crossref","unstructured":"N. J. Higham,\n                      Functions of Matrices: Theory and Computation\n                      , SIAM, Philadelphia, 2008,https:\/\/doi.org\/10.1137\/1.9780898717778.","DOI":"10.1137\/1.9780898717778"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479804442218"},{"key":"atypb12","doi-asserted-by":"crossref","unstructured":"R. A. Horn and C. R. Johnson,\n                      Topics in Matrix Analysis\n                      , Cambridge University Press, New York, 1991.","DOI":"10.1017\/CBO9780511840371"},{"key":"atypb13","unstructured":"D. P. Jagger,\n                      MATLAB Toolbox for Classical Matrix Groups\n                      , M.Sc. Thesis, University of Manchester, Manchester, England, 2003."},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(94)90322-0"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1137\/0610014"},{"key":"atypb16","unstructured":"P. Lancaster and M. Tismenetsky,\n                      The Theory of Matrices\n                      , 2nd ed., Academic Press, London, 1985."},{"key":"atypb17","doi-asserted-by":"crossref","unstructured":"J. M. Lee,\n                      Introduction to Smooth Manifolds\n                      , Springer-Verlag, New York, 2012.","DOI":"10.1007\/978-1-4419-9982-5"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1137\/040619363"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1137\/16M1072851"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1137\/0703023"},{"key":"atypb21","doi-asserted-by":"crossref","unstructured":"K. Tapp,\n                      Matrix Groups for Undergraduates\n                      , American Mathematical Society, Providence, RI, 2005.","DOI":"10.1090\/stml\/029"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1016\/S0024-3795(03)00370-7"}],"container-title":["SIAM Journal on Matrix Analysis and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/17M1148943","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:15:25Z","timestamp":1787328925000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/17M1148943"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,1]]},"references-count":21,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2019,1]]}},"alternative-id":["10.1137\/17M1148943"],"URL":"https:\/\/doi.org\/10.1137\/17m1148943","relation":{},"ISSN":["0895-4798","1095-7162"],"issn-type":[{"value":"0895-4798","type":"print"},{"value":"1095-7162","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,1]]}}}