{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T02:07:25Z","timestamp":1740103645619,"version":"3.37.3"},"reference-count":19,"publisher":"Wiley","issue":"3","license":[{"start":{"date-parts":[[2014,11,13]],"date-time":"2014-11-13T00:00:00Z","timestamp":1415836800000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"funder":[{"name":"NSF","award":["1115834"],"award-info":[{"award-number":["1115834"]}]},{"DOI":"10.13039\/501100001809","name":"NSFC","doi-asserted-by":"crossref","award":["10261012"],"award-info":[{"award-number":["10261012"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100001809","name":"NSFC","doi-asserted-by":"crossref","award":["11428104"],"award-info":[{"award-number":["11428104"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Numerical Linear Algebra App"],"published-print":{"date-parts":[[2015,5]]},"abstract":"Summary<\/jats:title>Recently, Guo and Lin [SIAM J. Matrix Anal. Appl.<\/jats:italic>, 31 (2010), 2784\u20132801] proposed an efficient numerical method to solve the palindromic quadratic eigenvalue problem (PQEP) (\u03bb<\/jats:italic>2<\/jats:sup>A<\/jats:italic>T<\/jats:sup>+\u03bb<\/jats:italic>Q<\/jats:italic> + A<\/jats:italic>)z<\/jats:italic> = 0 arising from the vibration analysis of high speed trains, where have special structures: both Q<\/jats:italic> and A<\/jats:italic> are, among others, m<\/jats:italic> \u00d7 m<\/jats:italic> block matrices with each block being k<\/jats:italic> \u00d7 k<\/jats:italic> (thus, n<\/jats:italic> = m<\/jats:italic>k<\/jats:italic>), and moreover, Q<\/jats:italic> is block tridiagonal, and A<\/jats:italic> has only one nonzero block in the (1,m<\/jats:italic>)th block position. The key intermediate step of the method is the computation of the so\u2010called stabilizing solution to the n<\/jats:italic> \u00d7 n<\/jats:italic> nonlinear matrix equation X<\/jats:italic> + A<\/jats:italic>T<\/jats:sup>X<\/jats:italic>\u22121<\/jats:sup>A<\/jats:italic> = Q<\/jats:italic> via the doubling algorithm. The aim of this article is to propose an improvement to this key step through solving a new nonlinear matrix equation having the same form but of only k<\/jats:italic> \u00d7 k<\/jats:italic> in size. This new and much smaller matrix equation can also be solved by the doubling algorithm. For the same accuracy, it takes the same number of doubling iterations to solve both the larger and the new smaller matrix equations, but each doubling iterative step on the larger equation takes about 4.8 as many flops than the step on the smaller equation. Replacing Guo's and Lin's key intermediate step by our modified one leads to an alternative method for the PQEP. This alternative method is faster, but the improvement in speed is not as dramatic as just for solving the respective nonlinear matrix equations and levels off as m<\/jats:italic> increases. Numerical examples are presented to show the effectiveness of the new method. Copyright \u00a9 2014 John Wiley & Sons, Ltd.<\/jats:p>","DOI":"10.1002\/nla.1962","type":"journal-article","created":{"date-parts":[[2014,11,13]],"date-time":"2014-11-13T08:06:18Z","timestamp":1415865978000},"page":"393-409","source":"Crossref","is-referenced-by-count":7,"title":["A new look at the doubling algorithm for a structured palindromic quadratic eigenvalue problem"],"prefix":"10.1002","volume":"22","author":[{"given":"Linzhang","family":"Lu","sequence":"first","affiliation":[{"name":"School of Mathematical Science Xiamen University Xiamen 361005 China"},{"name":"School of Mathematics and Computer Science Guizhou Normal University Guiyang 550001 China"}]},{"given":"Fei","family":"Yuan","sequence":"additional","affiliation":[{"name":"School of Mathematical Science Xiamen University Xiamen 361005 China"}]},{"given":"Ren\u2010Cang","family":"Li","sequence":"additional","affiliation":[{"name":"Department of Mathematics University of Texas at Arlington Arlington TX 76019 USA"}]}],"member":"311","published-online":{"date-parts":[[2014,11,13]]},"reference":[{"volume-title":"Numerische L\u00f6sung von quadratischen Eigenwertproblemen mit Anwendung in der Schienendynamik","year":"2004","author":"Hilliges A","key":"e_1_2_10_2_1"},{"key":"e_1_2_10_3_1","unstructured":"HilligesA MehlC MehrmannV.On the solution of palindromic eigenvalue problems. 4th European Congress on Computational Methods in Applied Sciences and Engineerings (ECCOMAS) Jyv\u00e4skyl\u00e4 Finland 2004."},{"key":"e_1_2_10_4_1","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-010-0297-4"},{"key":"e_1_2_10_5_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2007.07.016"},{"key":"e_1_2_10_6_1","doi-asserted-by":"publisher","DOI":"10.1137\/090763196"},{"issue":"9","key":"e_1_2_10_7_1","article-title":"Accurate eigenvalues for fast trains","volume":"37","author":"Ipsen ICF","year":"2004","journal-title":"SIAM News"},{"key":"e_1_2_10_8_1","doi-asserted-by":"publisher","DOI":"10.1137\/050628362"},{"key":"e_1_2_10_9_1","doi-asserted-by":"publisher","DOI":"10.1137\/080713550"},{"key":"e_1_2_10_10_1","doi-asserted-by":"publisher","DOI":"10.1137\/0902014"},{"key":"e_1_2_10_11_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2003.12.010"},{"key":"e_1_2_10_12_1","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-05-01748-5"},{"key":"e_1_2_10_13_1","doi-asserted-by":"publisher","DOI":"10.1093\/imanum\/20.4.499"},{"key":"e_1_2_10_14_1","doi-asserted-by":"publisher","DOI":"10.1137\/S0036144500381988"},{"key":"e_1_2_10_15_1","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898719604"},{"key":"e_1_2_10_16_1","doi-asserted-by":"publisher","DOI":"10.1137\/080717304"},{"volume-title":"Computational Methods of Linear Algebra","year":"1959","author":"Faddeeva VN","key":"e_1_2_10_17_1"},{"volume-title":"Robust and Optimal Control","year":"1995","author":"Zhou K","key":"e_1_2_10_18_1"},{"key":"e_1_2_10_19_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0024-3795(01)00341-X"},{"key":"e_1_2_10_20_1","unstructured":"LuL YuanF LiRC.An improved structure\u2010preserving doubling algorithm for a structured palindromic quadratic eigenvalue problem 2014\u201002 Department of Mathematics University of Texas at Arlington January2014."}],"container-title":["Numerical Linear Algebra with Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fnla.1962","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fnla.1962","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/nla.1962","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,9,16]],"date-time":"2023-09-16T09:59:52Z","timestamp":1694858392000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/nla.1962"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,11,13]]},"references-count":19,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2015,5]]}},"alternative-id":["10.1002\/nla.1962"],"URL":"https:\/\/doi.org\/10.1002\/nla.1962","archive":["Portico"],"relation":{},"ISSN":["1070-5325","1099-1506"],"issn-type":[{"type":"print","value":"1070-5325"},{"type":"electronic","value":"1099-1506"}],"subject":[],"published":{"date-parts":[[2014,11,13]]}}}