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1、OnADIMethodforSylvesterEquationsNinoslavTruharRen-CangLiTechnicalReport2008-02http://www.uta.edu/math/preprint/OnADIMethodforSylvesterEquationsNinoslavTruhar∗Ren-CangLi†February8,2007AbstractThispaperisconcernedwithnumericalsolutionsoflargescaleSylvesterequationsAX−XB=C,Lyapu
2、novequationsasaspecialcaseinparticularincluded,withChavingverysmallrank.ForstableLyapunovequations,Penzl(2000)andLiandWhite(2002)demonstratedthatthesocalledCholeskyfactoredADImethodwithdecentshiftparameterscanbeveryeffective.InthispaperwepresentageneralizationofCholeskyfactore
3、dADIforSylvesterequations.WealsodemonstratethatoftenmuchmoreaccuratesolutionsthanADIsolutionscanbegottenbyperformingGalerkinprojectionviathecolumnspaceandrowspaceofthecomputedapproximatesolutions.1IntroductionAnm×nSylvesterequationtakestheformAX−XB=C,(1.1)whereA,B,andCarem×m,
4、n×n,andm×n,respectively,andunknownmatrixXism×n.ALyapunovequationisaspecialcasewithm=n,B=−A∗,andC=C∗,wherethestarsuperscripttakescomplexconjugateandtranspose.Equation(1.1)hasauniquesolutionifAandBhavenocommoneigenvalues,whichwillbeassumedthroughoutthispaper.Sylvesterequationsa
5、ppearfrequentlyinmanyareasofappliedmathematics,boththeoreticallyandpractically.WereferthereadertotheelegantsurveybyBhatiaandRosenthal[6]andreferencesthereinforahistoryoftheequationandmanyinterestingandimportanttheoreticalresults.Sylvesterequationsplayvitalrolesinanumberofappl
6、icationssuchasmatrixeigen-decompositions[13],controltheoryandmodelreduction[2,32],numericalsolutionstomatrixdifferentialRiccatiequations[9,10],andimageprocessing[7].∗DepartmentofMathematics,J.J.StrossmayerUniversityofOsijek,TrgLjudevitaGaja6,31000Osijek,Croatia.Email:ntruhar@m
7、athos.hr.SupportedinpartbytheNationalScienceFoundationunderGrantNo.235-2352818-1042.PartofthisworkwasdonewhilethisauthorwasavisitingprofessorattheDepartmentofMathematics,UniversityofTexasatArlington,Arlington,TX,USA.†DepartmentofMathematics,UniversityofTexasatArlington,P.O.Bo
8、x19408,Arlington,TX76019-0408,USA.Email:rcli@uta.edu.Supportedinpart
