南昌大学数值分析英文教材Word文档格式.docx

南昌大学数值分析英文教材Word文档格式.docx

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Wegivesomeimportantbasicfactsfromcalculusasfollows.

Theorem1.1.1（IntermediateValueTheorem）

Letfbeacontinuousfunctionontheinterval[a,b]（thatisf∈C[a.b]）.Thenfrealizeseveryvaluebetweenf（a）andf（b）.Moreprecisely,ifyisanumberbetweenf（a）andf（b）,thenthereexistsanumbercwitha≤c≤bsuchthatf（c）=y.

Theorem1.1.2（ContinuousLimits）

Letfbeacontinuousfunctioninaneighborhoodofx0,andassume

=x0.Then

Inotherwords,limitsmaybebroughtinsidecontinuousfunctions.

Theorem1.1.3（MeanValueTheorem）

Letfbeacontinuouslydifferentiablefunctionontheinterval[a,b]（thatisf∈C1[a,b]）.Thenthereexistsanumberξbetweenaandbsuchthatf’（ξ）=（f（b）–f（a））/（b–a）.

Theorem1.1.4（Rolle’sTheorem）

Letf∈C1[a.b]）andassumethatf（a）=f（b）.Thenthereexistsanumberξbetweenaandbsuchthatf’（ξ）=0.

Theorem1.1.5（Taylor’sTheoremwithRemainder）

Letxandx0berealnumbers,andletfbek+1timescontinuouslydifferentiableontheintervalbetweenxandx0（thatisf∈Ck+1[a,b]）.Thenthereexistsanumberξbetweenxandx0suchthat

1.2TheBisectionMethod

Thisprocessinvolvesfindingaroot,orsolution,ofanequationoftheform（1.1.1）,foragivenfunctionf.Arootofthisequationisalsocalledazeroofthefunctionf.

Thefirsttechnique,basedontheIntermediateValueTheorem,iscalledtheBisection,orBinary-search,method.

Theorem1.2.1

Letf∈C[a.b],satisfyingf（a）f（b）<

0.Thenfhasarootbetweenaandb,thatis,thereexistsanumbercsatisfyinga<

c<

bandf（c）=0.

Proof.

Supposef∈C[a,b],withf（a）andf（b）ofoppositesign.BytheIntermediateValueTheorem（Theorem1.1.1）,thereexistsanumbercin（a,b）withf（c）=0.

Althoughtheprocedurewillworkwhenthereismorethanonerootintheinterval（a,b）,weassumeforsimplicitythattherootinthisintervalisunique.Themethodcallsforarepeatedhalvingofsubintervalsof[a,b]and,ateachstep,locatingthehalfcontainingc.

Tobegin,seta1=aandb1=b,andletc1bethemidpointof[a,b];

thatis,

Iff（c1）=0,thenc=c1,andwearedone.Iff（c1）≠0,thenf（c1）hasthesamesignaseitherf（a1）orf（b1）.Whenf（c1）andf（a1）havethesamesign,c∈（c1,b1）,andweseta2=c1andb2=b1.Whenf（c1）andf（a1）haveoppositesigns,c∈（a1,c1）andweseta2=a1andb2=c1.Wethenreapplytheprocesstotheinterval[a2,b2].ThisproducesthemethoddescribedinAlgorithm1.2.1

Algorithm1.2.1:

Bisection

Tofindasolutiontof（x）=0giventhecontinuousfunctionfontheinterval[a,b],wheref（a）andf（b）haveoppositesigns:

INPUTendpointsa,b;

toleranceTOL;

maximumnumberofiterationsN0,

OUTPUTapproximatesolutionpormessageoffailure.

Step1seti=1;

FA=f（a）;

Step2Whilei<

N0doSteps3-6

Step3Setc=（a+b）/2

FC=f（c）;

Step4IfFC=0or（b-a）/2<

TOLthen

OUTPUT（c）;

//Procedurecompletedsuccessfully.

Stop;

Step5Seti=i+1;

Step6IfFA*FC>

0thenseta=c;

FA=FC;

elsesetb=c;

Step7OUTPUT（‘MethodfailedafterN0iterations,N0=‘,N0）;

STOP;

OtherstoppingprocedurecanbeappliedinStep4ofAlgorithm1.2.1orinanyoftheiterativetechniquesinthischapter.Forexample,wecanselectatoleranceε>

0andgeneratec1,c2,…,cNuntiloneofthefollowingconditionsismet:

|cN-cN-1|<

ε,（1.2.1）

|cN-cN-1|/|cN|<

ε,cN≠0,or（1.2.2）

|f（cN）|<

ε.（1.2.3）

Unfortunately,difficultiescanariseusinganyofthesestoppingcriteria.Forexample,therearesequences{cn}withthepropertythatdifferencescn–cn-1convergestoclosetozerowhilecndifferssignificantlyfromc.Withoutadditionalknowledgeaboutforc.Inequality（1.2.2）isthebeststoppingcriteriontoapplybecauseitcomesclosesttotestingrelativeerror.

Whenusingacomputertogenerateapproximations,itisgoodpracticetosetanupperboundonthenumberofiterations.Thiswilleliminatethepossibilityofenteringaninfiniteloop,asituationthatcanarisewhenthesequencediverges（andalsowhentheprogramisincorrectlycoded）.ThiswasdoneinStep2ofAlgorithm1.2.1wheretheboundN0wassetandtheprocedureterminatedifi>

N0.

NotethattostarttheBisectionAlgorithm,aninterval[a,b]mustbefoundwithf（a）f（b）<

0.Ateachstep