{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:30:37Z","timestamp":1787329837740,"version":"build-2736575974"},"reference-count":47,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","funder":[{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["EP\/R009821\/1"],"award-info":[{"award-number":["EP\/R009821\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Matrix Anal. Appl."],"published-print":{"date-parts":[[2019,1]]},"abstract":"<jats:p>When solving linear systems with nonsymmetric Toeplitz or multilevel Toeplitz matrices using Krylov subspace methods, the coefficient matrix may be symmetrized. The preconditioned MINRES method can then be applied to this symmetrized system, which allows rigorous upper bounds on the number of MINRES iterations to be obtained. However, effective preconditioners for symmetrized (multilevel) Toeplitz matrices are lacking. Here, we propose novel ideal preconditioners and investigate the spectra of the preconditioned matrices. We show how these preconditioners can be approximated and demonstrate their effectiveness via numerical experiments.<\/jats:p>","DOI":"10.1137\/18m1205406","type":"journal-article","created":{"date-parts":[[2019,7,16]],"date-time":"2019-07-16T12:56:48Z","timestamp":1563281808000},"page":"870-887","source":"Crossref","is-referenced-by-count":21,"title":["Preconditioners for Symmetrized Toeplitz and Multilevel Toeplitz Matrices"],"prefix":"10.1137","volume":"40","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1527-3178","authenticated-orcid":true,"given":"J.","family":"Pestana","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2019,7,16]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1007\/BF02510405"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1007\/BF00319101"},{"key":"atypb3","first-page":"107","volume":"45","author":"Breiten T.","year":"2016","journal-title":"Electron. Trans. Numer. Anal."},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(93)90226-E"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1099-1506(199601\/02)3:1<45::AID-NLA70>3.0.CO;2-T"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1137\/0729066"},{"key":"atypb7","doi-asserted-by":"crossref","unstructured":"R. H.F. Chan and X.Q. Jin,\n                      An Introduction to Iterative Toeplitz Solvers\n                      , Fund. Alg. 5, SIAM, 2007,https:\/\/doi.org\/10.1137\/1.9780898718850.","DOI":"10.1137\/1.9780898718850"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1137\/0909051"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1002\/nla.2007"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2015.11.061"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1016\/S0024-3795(99)00251-7"},{"key":"atypb12","doi-asserted-by":"crossref","unstructured":"P. Ferrari, I. Furci, S. Hon, M. Ayman Mursaleen, and S. Serra-Capizzano,\n                      The Eigenvalue Distribution of Special 2-by-2 Block Matrix Sequences, with Applications to the Case of Symmetrized Toeplitz Structures\n                      , preprint,https:\/\/arxiv.org\/abs\/1810.03326, 2018.","DOI":"10.1137\/18M1207399"},{"key":"atypb13","doi-asserted-by":"crossref","unstructured":"C. Garoni and S. Serra-Capizzano,\n                      Generalized Locally Toeplitz Sequences: Theory and Applications. Vol.\n                      I, Springer, 2017.","DOI":"10.1007\/978-3-319-53679-8"},{"key":"atypb14","doi-asserted-by":"crossref","unstructured":"I. Gohberg, P. Lancaster, and L. Rodman,\n                      Indefinite Linear Algebra and Applications\n                      , Birkh\u00e4user Verlag, Basel, 2005.","DOI":"10.1007\/s00020-005-1356-6"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479894275030"},{"key":"atypb16","unstructured":"U. Grenander and G. Szeg\u00f6,\n                      Toeplitz Forms and Their Applications\n                      , 2nd ed., AMS Chelsea Publishing, 2001."},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1007\/BF02140682"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.6028\/jres.049.044"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1137\/0611009"},{"key":"atypb20","unstructured":"R. A. Horn and C. R. Johnson,\n                      Matrix Analysis\n                      , Cambridge University Press, 1990."},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1002\/nla.396"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2006.08.032"},{"key":"atypb23","doi-asserted-by":"crossref","unstructured":"M. Mazza and J. Pestana,\n                      Spectral properties of flipped Toeplitz matrices and related preconditioning\n                      , BIT, to appear,https:\/\/doi.org\/10.1007\/s10543-018-0740-y.","DOI":"10.1007\/s10543-018-0740-y"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2004.01.033"},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1016\/j.apnum.2005.02.008"},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2017.08.064"},{"key":"atypb27","unstructured":"M. K. 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Pestana,\n                      Nonstandard Inner Products and Preconditioned Iterative Methods\n                      , Ph.D. thesis, University of Oxford, 2011."},{"key":"atypb35","doi-asserted-by":"publisher","DOI":"10.1137\/140974213"},{"key":"atypb36","doi-asserted-by":"publisher","DOI":"10.1016\/S0024-3795(98)10144-1"},{"key":"atypb37","doi-asserted-by":"publisher","DOI":"10.1137\/0907058"},{"key":"atypb38","doi-asserted-by":"publisher","DOI":"10.1007\/BF01934269"},{"key":"atypb39","doi-asserted-by":"publisher","DOI":"10.1023\/A:1022329526925"},{"key":"atypb40","doi-asserted-by":"publisher","DOI":"10.1016\/S0024-3795(01)00361-5"},{"key":"atypb41","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479897331941"},{"key":"atypb42","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-03-01506-0"},{"key":"atypb43","doi-asserted-by":"publisher","DOI":"10.1002\/sapm1986742171"},{"key":"atypb44","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2003.07.013"},{"key":"atypb45","doi-asserted-by":"publisher","DOI":"10.1137\/0613030"},{"key":"atypb46","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2010.07.011"},{"key":"atypb47","doi-asserted-by":"publisher","DOI":"10.4171\/ZAA\/350"}],"container-title":["SIAM Journal on Matrix Analysis and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/18M1205406","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:35:56Z","timestamp":1787326556000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/18M1205406"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,1]]},"references-count":47,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2019,1]]}},"alternative-id":["10.1137\/18M1205406"],"URL":"https:\/\/doi.org\/10.1137\/18m1205406","relation":{},"ISSN":["0895-4798","1095-7162"],"issn-type":[{"value":"0895-4798","type":"print"},{"value":"1095-7162","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,1]]}}}