A First Course in the numerical analysis of differential equations 2ed.pdf

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1、252Classicaliterativemethodsdwherethed×diterationmatrixHandv∈Rareindependentofk.(Ofcourse,bothHandvmustdependonAandb,otherwiseconvergenceisimpossible.)Lemma12.1Givenanarbitrarylinearsystem(12.1),alinearone-stepstationarydscheme(12.3)convergestoauniqueboundedlimitx?∈R,reg

2、ardlessofthechoiceofstartingvaluex[0],ifandonlyifρ(H)<1,whereρ(·)denotesthespectralradius(A.1.5.2).Providedthatρ(H)<1,x?isthecorrectsolutionofthelinearsystem(12.1)ifandonlyifv=(I?H)A?1b.(12.4)ProofLetuscommencebyassumingρ(H)<1.InthiscaseweclaimthatlimHk=O.(12.5)k→∞Toprov

3、ethisstatement,wemakethesimplifyingassumptionthatHhasacompletesetofeigenvectors,hencethatthereexistanonsingulard×dmatrixVandadiagonald×dmatrixDsuchthatH=VDV?1(A.1.5.3andA.1.5.4).HenceH2=VDV?1×VDV?1=VD2V?1,H3=VD3V?1and,ingeneral,itistrivialtoprovebyinductionthatHk=VDk

4、V?1,k=0,1,2,...Therefore,passingtothelimit,limHk=VlimDkV?1.k→∞k→∞TheelementsalongthediagonalofDaretheeigenvaluesofH,henceρ(H)<1impliesDkk?→→∞Oandwededuce(12.5).Ifthesetofeigenvectorsisincomplete,(12.5)canbeprovedjustaseasilybyusingaJordanfactorization(seeA.1.5.6andExer

5、cise12.1).Ournextassertionisthatx[k]=Hkx[0]+(I?H)?1(I?Hk)v,k=0,1,2,...;(12.6)notethatρ(H)<1implies1∈σ(H),whereσ(H)isthesetofalleigenvalues(thespectrum)ofH,thereforetheinverseofI?Hexists.Theproofisbyinduction.Itisobviousthat(12.6)istruefork=0.Hence,letusassumeitfork≥0and

6、attemptitsveri?cationfork+1.Usingthede?nition(12.3)oftheiterativeschemeintandemwiththeinductionassumption(12.6),wereadilyobtainx[k+1]=Hx[k]+v=HHkx[0]+(I?H)?1(I?Hk)v+v=Hk+1x[0]+(I?H)?1(H?Hk+1)+(I?H)?1(I?H)v=Hk+1x[0]+(I?H)?1(I?Hk+1)vandtheproofof(12.6)iscomplete.Letting

7、k→∞in(12.6),(12.5)impliesatoncethattheiterativeprocessconverges,limx[k]=x?:=(I?H)?1v.(12.7)k→∞12.1Linearone-stepstationaryschemes253Wenextconsiderthecaseρ(H)≥1.Providedthat1∈σ(H),thematrixI?Hisinvertibleandx?=(I?H)?1vistheonlypossibleboundedlimitoftheiterativescheme.For

8、,supposetheexistenceofaboundedlimity?.Theny?=limx[k+1]=Hlimx[k]+v=Hy?+v,(12.8)k→∞k→∞thereforey?=x?.Even