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1、NumericalAnalysisDamingLiDepartmentofMathematics,ShanghaiJiaoTongUniversity,Shanghai,200240,ChinaEmail:lidaming@sjtu.edu.cnOctober14,2014DamingLiNumericalAnalysisBisectionMethodLetf∈C[a,b]andf(a)f(b)<0,thenfmusthaveazeroin[a,b].Bisectionmethod:Calculatec=(a+b)/2.iff(a)f(c)<0,fhasazeroin[a,c].iff
2、(a)f(c)>0,i.e.,f(b)f(c)<0,fhasazeroin[c,b].iff(a)f(c)=0,thenf(c)=0andazerohasbeenfound.However,itisquiteunlikelythatf(c)illbezerointhecomputerbecauseofroundofferrors.Thus,thestoppingcriterionshouldnotbewhether
3、f(c)
4、=0.Areasonabletolerancemustbeallowed,suchas
5、f(c)
6、<ǫ.Thustherootmustbein[a,c]or[c,
7、b].Theseintervallengtharetheonehalfof[a,b].Repeatingthisprocess,therootlocatesinaninterval,whichlengthisverysmall.Thusthebisectionmethodisalsoknownasthemethodofintervalhalving.DamingLiNumericalAnalysisBisectionMethodToanalyzethebisectionmethod,letusdenotethesuccessiveintervalsthatariseintheproce
8、ssby[a0,b0],[a1,b1],andsoon.Herearesomeobservationsa0≤a1≤a2≤···≤b0b0≥b1≥b2≥···≥a01bn+1−an+1=(bn−an)(n≥0)2Repeatingtheaboveformula,b−a=2−n(b−a)nn00Thuslimbn−liman=0n→∞n→∞DamingLiNumericalAnalysisBisectionMethodLetr=liman=limbnn→∞n→∞Bytakingalimitoftheinequality0≥f(an)f(bn),weobtain0≥
9、f(r)
10、2,whenc
11、ef(r)=0.Supposethatatacertainstageintheprocess,theinterval[an,bn]hasjustbeendefined.Iftheprocessisnowstopped,therootiscertaintolieinthisinterval.Thebestestimateoftherootatthisstageisnotanorbnbutthemidpointoftheinterval:cn=(an+bn)/2.Theerroristhenbound1−(n+1)
12、r−cn
13、≤
14、bn−an
15、≤2(b0−a0)2DamingLiNumeric
16、alAnalysisExercises:P.621,12DamingLiNumericalAnalysisNewton’sMethodThisisalsocalledtheNewton-Raphsoniteration.Letrbeazerooffandletxbeanapproximationtor.Iff′′existsandiscontinuous,thenbyTaylor’sTheorem0=f(r)=f(x+h)=f(x)+hf′(x)+O(h2)whereh=r−x.Ifhissmall(thatis,xisnearr),thenitisreasonabletoignore
17、theO(h2)-termandsolvetheremainingequationforh.Therefore,theresultish=−f(x)/f′(x).Ifxisanapproximationtor,thenx−f(x)/f′(x)shouldbeabetterapproximationtor.Newton’smethodbeginswithanestimatex0ofrandthendefinesinductivelyf(xn)xn+
