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1、Algebraiceigenvalueproblem:SomefundamentaltheoryandsomefundamentalalgorithmsYunkaiZhouDepartmentofMathematicsSouthernMethodistUniversityDallas,Texas75075yzhou@smu.eduJuly23,2013Acknowledgement:NSFgrantDMS-1228271.Thanks:AihuiZhou,XiaoyingDai,JunFang,XingyuGa
2、o,SunJiaChang1Overview2Somefundamentaltheory3Fundamentalbutelementaryeigen-algorithmsY.ZhouSummerschoolforeigenvalueproblems2/52OverviewofeigenproblemsPDE:Au=f−→Au=λuStandardeigenvalueproblem:Axi=λixi,i=1,2,···,nGeneralizedeigenvalueproblem:Axi=λiBxi,i=1,2,·
3、··,nY.ZhouSummerschoolforeigenvalueproblems3/52OverviewofeigenproblemsPDE:Au=f−→Au=λuStandardeigenvalueproblem:Axi=λixi,i=1,2,···,nGeneralizedeigenvalueproblem:Axi=λiBxi,i=1,2,···,nHxiAxiλi=H,dependsnonlinearlyonxixiBxiAmayalsodependonxiQuiteoften,needmoreth
4、anjustoneeigenvalue(e.g.,i=1,2,···,kw)Y.ZhouSummerschoolforeigenvalueproblems3/52Applicationsofeigenvalueproblems1MaterialsScience;Quantumchemistry/physics(Schrodingereqn.(1926),Kohn-Shameqn.(1965))¨2Photoniccrystals,waveguide(Helmholtzeqn.,(187?),Maxwelleqn
5、.(1873)),3Structuremechanics;Dynamicalsystems(stabilityissue)4Economicalsystem;network(stability,evolution)5Biologicalsystem;Medical-bio-informatics6Discreteproblems(Graphicalspectraltheory)7Random/Stochasticproblems(Markovchain)8Informationretrieval;Datamin
6、ing;Image/signalprocessing9...Threecurses(orblessings?)facingmodernapplications:Dimensionality;Nonlinearity;StochAsticY.ZhouSummerschoolforeigenvalueproblems4/52Generalized−→StandardAx=λBx.IfBisnonsingular,thenB-1Ax=λx.HHIfA=A,B=B≻0,thenAx=λBxenjoysfarnicerp
7、roperties.HIfB=LL,thenL-1AL-Hy=λy,y=LHx.ButfactoringBmaybeexpensive.Y.ZhouSummerschoolforeigenvalueproblems5/521Overview2Somefundamentaltheory3Fundamentalbutelementaryeigen-algorithmsY.ZhouSummerschoolforeigenvalueproblems6/52Summaryof6majormatrixdecompositi
8、ons(1)LUdecompositionA=LUwhereLisunitlowertriangular,UisuppertriangularCholeskydecomposition(2ndoneforhermitianPSDmatrices):HHA=LDL,A=RRwhereRisuppertriangular,andLisunituppertriangularQRdecompo
