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- W5087298 abstract "AHISTORYOFINVERSEITERATIONILSEC.F.IPSENy1.TheMetho d.Inverseiterationisametho dforcomputingeigenfunctionsoflinearop erators.HereweconsiderinverseiterationforthesolutionofalgebraiceigenvalueproblemAx=x;whereAisarealorcomplexsquarematrix,thescalaraneigenvalue,andnon-zerovectorxisaneigenector.Inerseiterationthemetho dofchoicewhenonehasatone'sdisp osalapproximationstoasp eci edsubsetofeigenvaluesA,forwhichonewantstocomputeasso ciatedeigenvectors.Themetho disfrequentlyusedinstructuralmechanics,forinstance,whentheextremeeigenvaluesandcorresp ondingectorsofHermitianp ositive-(semi)de niteeigenproblemsaresought[2,312131830].GivenanapproximationtoeigenvalueofA,inerseiterationgeneratesasequenceofvectorsxkfromagivenstartingector0bysolvingthelinearsystems1(A I)xk=s1;k1:Thescalarskisresp onsiblefornormalisingx.Usuallyonechosessothatk=1insomenorm.Ifeverythinggo eswell,thesequenceofiteratesxkconvergestoaneigenvectorasso ciatedwithalueclosestto.Inparticularwhenthereisonlyasingleeigenvalueclosestto,thenthexkconergeanectorasso ciatedwith,providedthestartingvectorx0containsatributionofthiseigenvector.Sincexkisamultipleoftheector(AI)ky0andsince1=isthelargesteigenvalueof(A I)1,contributioninxkeigendirectionasso ciatedwithincreasesfasterask!1thanthecontributionsothereigenvalues.Inerseiterationismathematicallyidentical2tothep owermetho dappliedto(A I)1.In1979Wilkinsonremarked:`Inverseiterationisnowthemostwidelyusedmetho dforcomputingeigenvectorscorresp ondingtoselectedalueswhichhaealreadyb eencomputedmoreorlessaccurately'[26,p339].Alo okatsoftwareinthepublicdomainshowsthatthisisstilltrueto day[1,25,28].Yet,aftersomanyears,theb ehaviourofmetho disstillnotcompletelyundersto o d{neitherinexactnoroatingp ointarithmetic[4].Asaconsequence,evenhighlysophisticated,stateoftheartsoftwarelikeLAPACKstillcomesacrossmatricesforwhichitcannotcomputeeigenvectorstohighaccuracywithoutadrasticsacri ceineciency.Thisarticledescrib esthee ortsundertakenonb ehalfofinverseiteration3fromitsToapp earin:HelmutWielandt,MathematischeWerke,MathematicalWorks,VolumeII:Ma-trixTheoryandAnalysis,Hupp ert,B.Schneider,H.,eds.,WalterdeGruyter,Berlin.yDepartmentofMathematics,NorthCarolinaStateUniversity,P.O.Box8205,Raleigh,NC27695-8205,USA(ipsen@math.ncsu.edu) .Thisworkassupp ortedinpartbyNSFgrantCCR-9102853.1HereIdenotestheidentitymatrix.2Mathematicalidentitymeansthatequalitholdsinexactarithmeticbutnotnecessarily niteprecisionarithmeticb ecausethetwometho dsdeterminetheiriteratesbymeansofdi erenop erations.3WeconcentrateonthesolutionofordinaryeigenvalueproblemratherthangeneralisedeigenvalueproblemAx=B x;andwedonotdiscusstechniquesforconergenceaccelerationsuchasRayleighquotientiterationwheretheshiftchangeswitheachiteration.1" @default.
- W5087298 created "2016-06-24" @default.
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- W5087298 date "1994-01-01" @default.
- W5087298 modified "2023-09-23" @default.
- W5087298 title "A history of inverse iteration" @default.
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