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InterpolationandPolynomialApproximation
AcensusofthepopulationoftheUnitedStatesistakenevery10years.Thefollowingtablelistathepopulation,inthousandaofpeople,from1940t01990.
Inreviewingthesedata,wemightaskwhethertheycouldbeusedtoprovideareasonableestimateofthepopulation,say,in19650revenintheyear2010.Predictionsofthistypecanbeobtainedbyusingafunc-tionthatfitsthegivendata.Thisprocessiscalledinterpolationandisthesubjectofthischapter.ThispopulationproblemisconsideredthroughoutthechapterandinExercises240fSection3.1,140fSection3.2,and240fSection3.4.
Oneofthemostusefulandwell-knownclassesoffunctionsmappingthesetofrealnumbersintoitselfistheclassofalgebraicpolynomials,thesetoffunctionsoftheform
公式
wherenisanonnegativeintegerandao,...,anarerealconstants.Onereasonfortheirimportanceisthattheyuniformlyapproximatecontinuousfunctions.Givenanyfunction,definedandcontinuousonaclosedandboundedinterval,thereexistsapolynomialthatisas"close"tothegivenfunctionasdesired.Thisresultisexpressedpreciselyinthefollowingtheorem.(SeeFigure3.1.)
(WeierstrassApproximationTheorem)
Supposethatfisdefinedandcontinuouson[a,b]Foreachε>O,thereexistsapolyno-mialP(x),withthepropertythat
公式
Theproofofthistheoremcanbefoundinmostelementarytextsonrealanalysis(see,forexample,[Bart,pp.165-172]).
Anotherimportantreasonforconsideringtheclassofpolynomialsintheapproximationoffunctionsisthatthederivativeandindefiniteintegralofapolynomialareeasytodetermineandarealsopolynomials.Forthesereasons,polynomialsareoftenusedforapproximatingcontinuousfunctions.
TheTaylorpolynomialswereintroducedinthefirstsectionofthebook.wheretheyweredescribedasoneofthefundamentalbuildingblocksofnumericalanalysis.Giventhisprominence,youmightassumethatpolynomialinterpolationwouldmakeheavyuseofthesefunctions.However,thisisnotthecase.TheTaylorpolynomialsagreeascloselyaspossiblewithagivenfunctionataspecificpoint,buttheyconcentratetheiraccuracynearthatpoint.Agoodinterpolationpolynomialneedstoprovidearelativelyaccurateapproximationoveranentireinterval,andTaylorpolynomialsdonotgenerallydothis.Forexample,supposewecalculatethefirstsixTaylorpolynomialsaboutxo=0forf(x)=e的x次方Sincethederivativesoff(x)arealle^,whichevaluatedatxo=0givesl,theTaylorpolynomialsare
公式
ThegraphsofthepolynomialsareshowninFigure3.2.(Noticethatevenforthehigher-degreepolynomials,theerrorbecomesprogressivelyworseaswemoveawayfromzero.)
Althoughbetterapproximationsareobtainedforf(x)=exifhigher-degreeTaylorpolynomialsareused,thisisnottrueforallfunctions.Consider,asanextremeexample,usingTaylorpolynomialsofvariousdegreesforf(x)=1/xexpandedaboutxo=Itoapproximatef(3)=1/3,Since
and,ingeneral,
theTaylorpolynomialsare
Toapproximatef(3)=1/3BYPn(3)forincreasingvaluesofn,weobtainthevaluesinTable3.1-ratheradramaticfailure!
SincetheTaylorpolynomialshavethepropertythatalltheinformationusedintheapproximationisconcentratedatthesinglepointxo,thetypeofdifficultythatoccurshereisquitecommonandlimitsTaylorpolynomialapproximationtothesituationinwhichapproximationsareneededonlyatpointsclosetoxo.Forordinarycomputationalpurposesitismoreefficienttousemethodsthatincludeinformationatvariouspoints,whichweconsiderintheremainderofthischapter.TheprimaryuseofTaylorpolynomialsinnumericalanalysisisnotforapproximationpurposesbutforthederivationofnumericaltechniquesanderrorestimation.
SincetheTaylorpolynomialsarenotappropriateforinterpolation,altemativemethodsareneeded.Inthissectionwefindapproximatingpolynomialsthataredeterminedsimplybyspecifyingcertainpointsontheplanethroughwhichtheymustpass.
Theproblemofdeterminingapolynomialofdegreeonethatpassesthroughthedistinctpoints(xo,yo)and(x1,y1)isthesameasapproximatingafunctionfforwhichf(x0)=yoandf(x1)=y1bymeansofafirst-degreepolynomialinterpolating,oragreeingwith,thevaluesoffatthegivenpoints.Wefirstdefinethefunctionsandthendefine
Since
wehave
and
SoPistheuniquelinearfunctionpassingthrough(xo,yo)and(x1,y1)(SeeFigure3.3.)
Togeneralizetheconceptoflinearinterpolation,considertheconstructionofapolynomialofdegreeatmostnthatpassesthroughthen+1points(SeeFigure3.4.)
Inthiscaseweneedtoconstruct,foreachk=O,l,...,n,afunctionLn,k(X)withthepropertythatLn.k(Xi)=0wheni≠kandLn,k(Xk)=1.TosatisfyLn,k(Xi)=Oforeachi≠krequiresthatthenumeratorofLn,k(X)containstheterm
TosatisfyLnkXk)=1,thedenominatorofLn,k(X)mustbeequaltothistermevaluatedatx=Xk.Thus,
AsketchofthegraphofatypicalLn,kisshowninFigure3.5.
TheinterpolatingpolynomialiseasilydescribedoncetheformofLn,kisknown.Thispolynomial,calledthenthLagrangeinterpolatingpolynonual.isdefinedinthefollowingtheorem.
Ifxo,x1,...,xnaren+ldistinctnumbersandfisafunctionwhosevaluesaregivenatthesenumbers,thenauniquepolynomialP(x)ofdegreeatmostnexistswithf(xk)=P(xk),foreachk=0,1,...,n.
Thispolynomialisgivenby
where,foreachk=0,1,...,n.’
WewillwriteLn,k(X)simplyasLk(x)whenthereisnoconfusionastoitsdegree.
Usingthenumbers(ornodes)xo=2,x1=2.5,andX2=4tofindthesecondinterpolatingpolynomialforf(x)=1/xrequiresthatwedeterminethecoefficientpolynomialsLo(x),Li(x),andL2(X).Innestedformtheyare
and
have
Sincef(xo)=f
(2)=0.5,f(xi)=f(2.5)=0.4,andf(X2)=f(4)=0.25,
Anapproximationtof(3)=~(seeFigure3.6)is
ComparethistoTable3.1,wherenoTaylorpolynomial,expandedaboutxn=1,couldbeusedtoreasonablyapproximatef(3)=1/3.
WecanuseaCAStoconstructaninterpolatingpolynomial.Forexample,inMapleweuseinterp(X,Y,x);whereXisthelist[xo,...,Xnl,Yisthelist[f(xo),...,f(Xn)],andxisthevariabletobeused.Inthisexamplewecangeneratetheinterpolatingpolynomialp=0.05x2-0.425x+1.15withthecommand
p:
=interp([2,2.5,4].[O,5,0.4,0.25],x);
Toevaluatep(3)asanapproximationtof(3)=~,enter>subs(x=3,p);
whichgives0.325.
Thenextstepistocalculatearemaindertermorboundfortheerrorinvolvedinapproximatingafunctionbyaninterpolatingpolynonual.Thisisdoneinthefollowingtheorem.
Supposexo,x1,...,Xnaredistinctnumbersintheinterval[a,b]andf∈C[a,b].
Then,foreachxin[a,b],anumberξ(x)in(a,b)existswithwhereP(x)istheinterpolatingpolynomialgiveninEq.(3.1).
ProofNotefirstthatifx=Xk,foranyk
choosingξ(Xk)arbitrarilyin(a,b)yieldsEq.
definethefunctiongfortin[a,b]by
wehave
g(xk)=f(Xk)-P(xk)-【f(x)-P(x)]
Moreover,
g(x=f(x)一P(x)一【f(x)一P(x)】i=0i=0(xk一xi)(x-xi)(x-xi)(x-xi)=f(x)-P(x)-[f(x)-P(x)]=0
Thus,g∈Cn+1[a,b],andgiszeroatthen+2distinctnumbersx,XO,Xl,...,Xn.Bythe
GeneralizedRolle'sTheorem,thereexistsanumberξin(a,b)forwhichg(n+1)(n+1)(ξ)=0
SinceP(x)isapolynomialofdegreeatmostn,the(n+l)stderivative,P(n+1)(x),isidenticallyzero.Also,F17=occt-xi)/(x-xl)]isapolynomialofdegree(n+l),so
Equation(3.4)nowbecomes
and,uponsolvingforf(x),wehave
TheerrorformulainTheorem3.3isanimportanttheoreticalresultbecauseLagrangepolynomialsareusedextensivelyforderivingnumericaldifferentiationandintegrationmethods.ErrorboundsforthesetechniquesareobtainedfromtheLagrangeerrorformula.
NotethattheerrorformfortheLagrangepolynomialisquitesimilartothatfortheTaylorpolynomial.ThenthTaylorpolynomialaboutxnconcentratesalltheknowninformationatxnandhasanerrortermoftheform.
TheLagrangepolynomialofdegreenusesinformationatthedistinctnumbersxo,x1...Xnand,inplaceof(x-XO)n,itserrorformulausesaproductofthen+Iterms
(x-xo)(x-XI)...(x-Xn).
Thespecificuseofthiserrorformulaisrestrictedtothosefunctionswhosederivativeshaveknownbounds.
Supposeatableistobepreparedforthefunctionf(x)=ex,forxin[0,1].Assumethenumberofdecimalplacestobegivenperentryisd>8andthatthedifferencebetweenadjacentx-values,thestepsize,ish.Whatshouldhbeforlinearinterpolation(thatis,theLagrangepolynomialofdegreel)togiveanabsoluteerrorofatmost10-6?
Letxo,x1,...bethenumbersatwhichfisevaluated,xbein[0,1],andsupposejsatisfiesxj≤x≤xj+i.Eq.(3.3)impliesthattheerrorinlinearinterpolationis|f(x)-P(x)|=
Sincethestepsizeish,itfollowsthatXj=jh,xj+i=(j+l)hand
Hence
Byconsideringg(x)=(x-jh)(x-(j+l)h),forjh≤x≤(j+l)h,andusingtheExtremeValueTheorem(seeExercise28),wefindthatrj<.x Consequently,theerrorinlinearinterpolationisboundedby anditissufficientforhtobechosensothatwhichimpliesthath<1.72xl0-3. Sincen=(l-O)/hmustbeaninteger,onelogicalchoiceforthestepsizeish=0.001. ThenextexampleillustratesinterpolationforasituationwhentheerrorportionofEq.(3.3)cannotbeused. EXAMPLE3Table3.2listsvaluesofafunctionatvariouspoints.Theapproximationstof(1.5)obtainedbyvariousLagrangepolynomialswillbecompared. Table3.2 Sincel.5isbetweenl.3andl.6,themostappropriatelinearpolyn
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