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1、Universiteit-UtrechtDepartmentofMathematicsJacobi-Davidsonalgorithmsforvariouseigenproblems-Aworkingdocument-byGerardL.G.Sleijpen,HenkA.VanderVorst,andZhaojunBaiPreprintnr.1114August17,1999JACOBI-DAVIDSONALGORITHMSFORVARIOUSEIGENPROBLEMS-AWORKINGDOCUMENT-zGERARDL.G.SLEIJPEN,HENKA.VANDERVOR
2、ST,ANDZHAOJUNBAIThesectionsinthisreportwillappearassectionsinthebook:Z.Bai,J.Demmel,J.Dongarra,A.Ruhe,andH.VanderVorst(editors),TemplatesfortheSolutionofAlgebraicEigenvalueProblems:APracticalGuide,SIAM,toappearin2000.ThenumberingoftheChaptersandSectionsinthisreportreferstothenumbersthatthese
3、sectionswillhaveinthatbook.Thepage-numbersandtheequation-numbers,however,aredierent.DepartementofMathematics,UtrechtUniversity,P.O.Box80.010,NL-3508TAUtrecht,TheNetherlands.E-mail:sleijpen@math.uu.nl,vorst@math.uu.nl.zDepartmentofMathematics,UniversityofKentucky,Lexington,KY,E-mail:bai@ms.u
4、ky.edu.CHAPTER4HermitianEigenproblem4.7Jacobi-DavidsonmethodsG.SleijpenandH.vanderVorstTheLanczosmethod(see[2,Section4.4])isquiteeectivetocomputeeigenvaluesintheendsofthespectrumofAiftheseeigenvaluearewellseparatedfromtheremainingspectrum,orwhenit?1canbeappliedto(A?I),forsomereasonableguess
5、foraneigenvalue.?1Ifnoneoftheseconditionsisfullled,forinstanceifthecomputationofavector(A?I)yforgivenyisnotfeasiblewithadirectsolver,thenvariantsoftheJacobi-Davidsonmethod[32]oeranattractivealternative.4.7.1BasicTheoryTheJacobi-Davidsonmethodisbasedonacombinationoftwobasicprinciples.Therst
6、oneistoapplyaRitz-GalerkinapproachfortheeigenproblemAx=x,withrespecttosomegivensubspacespannedbyanorthonormalbasisv;:::;v.TheusageofotherthanKrylovsubspaces1mwassuggestedbyDavidson[7]whoalsosuggestedspecicchoices,dierentfromtheonesthatwewilltake,fortheconstructionoforthonormalbasisvectorsv.
7、TheRitz-GalerkinconditionisjAVs?Vs?fv;:::;vg;mm1mwhichleadstothereducedsystemVAVs?s=0;(4.1)mm(m)(m)whereVdenotesthematrixwithcolumnsvtov.Thisequationhasmsolutions(;s).m1mjj(m)(m)(m)Thempairs(;uVs)arecalledtheRitzvaluesandRitzvectorsofAwithres
