{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:29:12Z","timestamp":1787336952313,"version":"build-2736575974"},"reference-count":33,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Matrix Anal. Appl."],"published-print":{"date-parts":[[2003,1]]},"abstract":"<jats:p>We present a Krylov subspace-type projection method for a quadratic matrix polynomial $\\lambda^2 I -\\lambda A - B$ that works directly with A and B without going through any linearization. We discuss a special case when one matrix is a low rank perturbation of the other matrix. We also apply the method to solve quadratically constrained linear least squares problem through a reformulation of Gander, Golub, and von Matt as a quadratic eigenvalue problem, and we demonstrate the effectiveness of this approach. Numerical examples are given to illustrate the efficiency of the algorithms.<\/jats:p>","DOI":"10.1137\/s0895479802409390","type":"journal-article","created":{"date-parts":[[2003,11,7]],"date-time":"2003-11-07T11:19:22Z","timestamp":1068203962000},"page":"405-428","source":"Crossref","is-referenced-by-count":25,"title":["A Krylov Subspace Method for Quadratic Matrix Polynomials with Application to Constrained Least Squares Problems"],"prefix":"10.1137","volume":"25","author":[{"given":"Ren-Cang","family":"Li","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Qiang","family":"Ye","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,31]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1090\/qam\/42792"},{"key":"R2","unstructured":"Z. Bai,\n                      personal communication\n                      , 2000."},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898719581"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1023\/A:1022383005969"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1007\/s002110100339"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1007\/s002119900077"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611971446"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1980-0583502-2"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1137\/0113073"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1007\/BF01396656"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(89)90494-1"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1007\/BF01385796"},{"key":"R13","volume-title":"Matrix polynomials","author":"Gohberg I.","year":"1982"},{"key":"R14","volume-title":"Matrix computations","author":"Golub Gene","year":"1996"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1007\/BF02149761"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898719697"},{"key":"R17","unstructured":"L. Hoffnung, R.\u2010C. Li, and Q. Ye,\n                      Krylov Type Subspace Methods for Matrix Polynomials\n                      , Research report 2002\u201008, Department of Mathematics, University of Kentucky, 2002, available online from http:\/\/www.ms.uky.edu\/\u223cmath\/MAreport; Linear Algebra Appl., to appear."},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.6028\/jres.045.026"},{"key":"R19","doi-asserted-by":"crossref","unstructured":"R. Lehoucq, D. C. Sorensen, and C. Yang,\n                      ARPACK User\u2019s Guide: Solution of Large\u2010Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods\n                      , SIAM, Philadelphia, 1998.","DOI":"10.1137\/1.9780898719628"},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1137\/S106482759935174X"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1137\/0904038"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611971163"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827500378167"},{"key":"R24","doi-asserted-by":"publisher","DOI":"10.1137\/S105262349928887X"},{"key":"R25","doi-asserted-by":"publisher","DOI":"10.1007\/BF01731936"},{"key":"R26","doi-asserted-by":"publisher","DOI":"10.1137\/0719026"},{"key":"R27","doi-asserted-by":"publisher","DOI":"10.1137\/S1052623494274374"},{"key":"R28","volume-title":"Numerical methods for large eigenvalue problems","author":"Saad Youcef","year":"1992"},{"key":"R29","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898718003"},{"key":"R30","doi-asserted-by":"publisher","DOI":"10.1137\/S0036144500381988"},{"key":"R31","volume-title":"The algebraic eigenvalue problem","author":"Wilkinson J.","year":"1965"},{"key":"R32","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1991-1068826-4"},{"key":"R33","doi-asserted-by":"publisher","DOI":"10.1016\/S0168-9274(98)00002-6"}],"container-title":["SIAM Journal on Matrix Analysis and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/S0895479802409390","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:38:00Z","timestamp":1787333880000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/S0895479802409390"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2003,1]]},"references-count":33,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2003,1]]}},"alternative-id":["10.1137\/S0895479802409390"],"URL":"https:\/\/doi.org\/10.1137\/s0895479802409390","relation":{},"ISSN":["0895-4798","1095-7162"],"issn-type":[{"value":"0895-4798","type":"print"},{"value":"1095-7162","type":"electronic"}],"subject":[],"published":{"date-parts":[[2003,1]]}}}