数学与应用数学专业论文英文文献翻译.docx

数学与应用数学专业论文英文文献翻译.docx

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数学与应用数学专业论文英文文献翻译

数学与应用数学专业论文英文文献翻译

Chapter3

Interpolation

Interpolationistheprocessofdefiningafunctionthattakesonspecifiedvaluesatspecifiedpoints.Thischapterconcentratesontwocloselyrelatedinterpolants,thepiecewisecubicsplineandtheshape-preservingpiecewisecubicnamed“pchip”.

3.1TheInterpolatingPolynomial

Weallknowthattwopointsdetermineastraightline.Moreprecisely,anytwopointsintheplane,and,with,determineauniquefirstdegreepolynomialinwhosegraphpassesthroughthetwopoints.Therearemanydifferentformulasforthepolynomial,buttheyallleadtothesamestraightlinegraph.

Thisgeneralizestomorethantwopoints.Givenpointsintheplane,，,withdistinct’s,thereisauniquepolynomialinofdegreelessthanwhosegraphpassesthroughthepoints.Itiseasiesttorememberthat,thenumberofdatapoints,isalsothenumberofcoefficients,althoughsomeoftheleadingcoefficientsmightbezero,sothedegreemightactuallybelessthan.Again,therearemanydifferentformulasforthepolynomial,buttheyalldefinethesamefunction.

Thispolynomialiscalledtheinterpolatingpolynomialbecauseitexactlyre-producesthegivendata.

，

Later,weexamineotherpolynomials,oflowerdegree,thatonlyapproximatethedata.Theyarenotinterpolatingpolynomials.

ThemostcompactrepresentationoftheinterpolatingpolynomialistheLa-grangeform.

Therearetermsinthesumandtermsineachproduct,sothisexpressiondefinesapolynomialofdegreeatmost.Ifisevaluatedat,alltheproductsexceptthetharezero.Furthermore,thethproductisequaltoone,sothesumisequaltoandtheinterpolationconditionsaresatisfied.

Forexample,considerthefollowingdataset:

x=0:

3;

y=[-5-6-116];

Thecommand

disp（[x;y]）

displays

0123

-5-6-116

TheLagrangianformofthepolynomialinterpolatingthisdatais

Wecanseethateachtermisofdegreethree,sotheentiresumhasdegreeatmostthree.Becausetheleadingtermdoesnotvanish,thedegreeisactuallythree.Moreover,ifwepluginor3,threeofthetermsvanishandthefourthproducesthecorrespondingvaluefromthedataset.

PolynomialsareusuallynotrepresentedintheirLagrangianform.Morefre-quently,theyarewrittenassomethinglike

Thesimplepowersofxarecalledmonomialsandthisformofapolynomialissaidtobeusingthepowerform.

Thecoefficientsofaninterpolatingpolynomialusingitspowerform,

can,inprinciple,becomputedbysolvingasystemofsimultaneouslinearequations

ThematrixofthislinearsystemisknownasaVandermondematrix.Itselementsare

ThecolumnsofaVandermondematrixaresometimeswrittenintheoppositeorder,butpolynomialcoefficientvectorsinMatlabalwayshavethehighestpowerfirst.

TheMatlabfunctionvandergeneratesVandermondematrices.Forourex-ampledataset,

V=vander（x）

generates

V=

0001

1111

8421

27931

Then

c=V\y’

computesthecoefficients

c=

1.0000

0.0000

-2.0000

-5.0000

Infact,theexampledatawasgeneratedfromthepolynomial.

OneoftheexercisesasksyoutoshowthatVandermondematricesarenonsin-gularifthepointsaredistinct.ButanotheroneoftheexercisesasksyoutoshowthataVandermondematrixcanbeverybadlyconditioned.Consequently,usingthepowerformandtheVandermondematrixisasatisfactorytechniqueforproblemsinvolvingafewwell-spacedandwell-scaleddatapoints.Butasageneral-purposeapproach,itisdangerous.

Inthischapter,wedescribeseveralMatlabfunctionsthatimplementvariousinterpolationalgorithms.Allofthemhavethecallingsequence

v=interp（x,y,u）

Thefirsttwoinputarguments,and,arevectorsofthesamelengththatdefinetheinterpolatingpoints.Thethirdinputargument,,isavectorofpointswherethefunctionistobeevaluated.Theoutput,v,isthesamelengthasuandhaselements

Ourfirstsuchinterpolationfunction,polyinterp,isbasedontheLagrangeform.ThecodeusesMatlabarrayoperationstoevaluatethepolynomialatallthecomponentsofusimultaneously.

functionv=polyinterp（x,y,u）

n=length（x）;

v=zeros（size（u））;

fork=1:

n

w=ones（size（u））;

forj=[1:

k-1k+1:

n]

w=（u-x（j））./（x（k）-x（j））.*w;

end

end

v=v+w*y（k）;

Toillustratepolyinterp,createavectorofdenselyspacedevaluationpoints.

u=-.25:

.01:

3.25;

Then

v=polyinterp（x,y,u）;

plot（x,y,’o’,u,v,’-’）

createsfigure3.1.

Figure3.1.polyinterp

Thepolyinterpfunctionalsoworkscorrectlywithsymbolicvariables.Forexample,create

symx=sym（’x’）

Thenevaluateanddisplaythesymbolicformoftheinterpolatingpolynomialwith

P=polyinterp（x,y,symx）

pretty（P）

produces

-5（-1/3x+1）（-1/2x+1）（-x+1）-6（-1/2x+3/2）（-x+2）x

-1/2（-x+3）（x-1）x+16/3（x-2）（1/2x-1/2）x

Thisexpressionisarearrange