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1、gammheaderwillbeprovidedbythepublisherStructuredEigenvalueProblemsHeikeFassbender∗1andDanielKressner∗∗21InstitutComputationalMathematics,TUBraunschweig,D-38023Braunschweig2DepartmentofMathematics,UniversityofZagreb,Croatia;DepartmentofComputingSci-ence,U
2、me˚aUniversity,SwedenReceived15April2005,revised26May2005,accepted1June2005KeywordsStructuredmatrix,eigenvalue,invariantsubspace,numericalmethods,software.MSC(2000)65-F15Mosteigenvalueproblemsarisinginpracticeareknowntobestructured.Structureisoftenintrod
3、ucedbydiscretizationandlinearizationtechniquesbutmayalsobeaconsequenceofpropertiesinducedbytheoriginalproblem.Preservingthisstructurecanhelppreservephys-icallyrelevantsymmetriesintheeigenvaluesofthematrixandmayimprovetheaccuracyandefficiencyofaneigenvalue
4、computation.Thepurposeofthisbriefsurveyistohighlightthesefactsforsomecommonmatrixstructures.Thisincludesatreatmentofrathergeneralconceptssuchasstructuredconditionnumbersandbackwarderrorsaswellasanoverviewofalgorithmsandapplicationsforseveralmatrixclasses
5、includingsymmetric,skew-symmetric,persymmetric,blockcyclic,Hamiltonian,symplecticandorthogonalmatrices.Copyrightlinewillbeprovidedbythepublisher1IntroductionThissurveyisconcernedwithcomputingeigenvalues,eigenvectorsandinvariantsubspacesofastructuredsquar
6、ematrixA.Inthescopeofthispaper,ann×nmatrixAisconsideredtobestructuredifitsn2entriesdependonlessthann2parameters.Itusuallytakesalongprocessofsimplifications,linearizationsanddiscretizationsbeforeonecomesupwiththeproblemofcomputingtheeigenvaluesorinvariants
7、ubspacesofamatrix.Thesetechniquestypicallyleadtohighlystructuredmatrixrepresentations,which,forexample,maycontainredundancyorinheritsomephysicalpropertiesfromtheoriginalproblem.Asasimpleexample,letusconsideraquadraticeigenvalueproblemoftheform(λ2I+λC+K)x
8、=0,(1)nwhereC∈Rn×nisskew-symmetric(C=−CT),K∈Rn×nissymmetric(K=KT),andIndenotesthen×nidentitymatrix.Eigenvalueproblemsofthistypearise,e.g.,fromgyroscopicsystems[96,117]orMaxwellequations[108];theyhavethephysicallyrelevantpr
