The equivalence of algebraic stability and AN-stability

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1、BIT27(1987),510-533THEEQUIVALENCEOFALGEBRAICSTABILITYANDAN-STABILITYJ.C.BUTCHERDepartmentofComputerScience,UniversityofAuckland,Auckland,NewZealandAbstract.Itisshownthat,undercertainrestrictions,AN-stabilityisequivalenttoalgebraicstabilityforgenerallinearmethods.Therestrictionshavethe.p

2、urposeofexcludingfromconsiderationmethodswhichcanbereplacedbysimplermethodsinvariousspecificwaysandAN-stabilityistobeinter-pretedinthestrongsense.ThisresultgeneralizesknownresultsforRunge-Kuttaandforone-legmethods.CRclassification:5.17.AMSclassification.65L20.1.Introduction.Inapreviousp

3、aper[4],variouslevelsofAN-stabilityforgenerallinearmethodswereintroducedandsomerelationshipsbetweenthemwerediscussed.TheseriousgapremainingfromthatpaperisthatthequestionwasnotresolvedastowhetherornotAN-stability,inthestrongsense,impliesalgebraicstability.Subjecttocertainrestrictionsonth

4、emethod,tantamounttoexcludingsometypesofreducibility,wewillshowinthispaperthatAN-stabilityisequivalenttoalgebraicstability.ThisisaverysatisfactoryresultinthatitgeneralizeswhatisalreadyknownabouttheimportantspecialcasesofRunge-Kuttamethods[1],[7]andone-legmethods[8].Theproofitselfhassome

5、thingincommonwiththeproofsofeachoftheimportantspecialcases.Inparticular,itmakesuseofageneralizationoftheAfunctionsusedin[8]butwithnon-negativityoftherealpartofarationalfunctionofacomplexvariablereplacedbythepositivesemidefinitenessoftheHermitianpartofamatrixvaluedfunctionofacomplexvaria

6、ble.Insection2,inadditiontostatingtheconditionstobeimposedonthemethodsunderstudy,anumberofdefinitionswillbegiven,includingdefinitionsoftwoclassesofpartitionedmatricesAandB.TheclassAisthegeneralizationtomultistagemethodsofAfunctionsandBiscloselyrelatedtothesetofReceivedFebruary1987.Revis

7、edAugust1987.THEEQUIVALENCEOFALGEBRAICSTABILITY...511algebraicallystablemethods.Alsoinsection2wewillstatetwotheorems,thefirstbeingthatA=Bandthesecondbeingthemainresultofthepaper.Somepreliminarylemmaswillbeintroducedinsection3andthelemmasrelatingAandBmatricesrequiredfortheproofo