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1、Chapter3,InterpolationandExtrapolationInterpolation&Extrapolation(xi,yi)Findananalyticfunctionf(x)thatpassesthroughgivenNpointsexactly.低级多项式高级多项式低级多项式高级多项式PolynomialInterpolationUsepolynomialofdegreeN-1tofitexactlywithNdatapoints(xi,yi),i=1,2,…,N.Thecoefficientciisdeterminedbya
2、systemoflinearequationsVandermondeMatrixEquationButitisnotadvisabletosolvethissystemnumericallybecauseofill-conditioning.ConditionNumbercond(A)=
3、
4、A
5、
6、·
7、
8、A-1
9、
10、Forsingularmatrix,cond(A)=∞Alinearsystemisill-conditionedifcond(A)isverylarge.NormsVectorp-normMatrixnormsup:supremumComm
11、onlyUsedNormsVectornormMatrixnormWhereμisthemaximumeigenvalueofmatrixATA.Lagrange’sFormulaItcanbeverifiedthatthesolutiontotheVandermondeequationisgivenbytheformulabelow:li(x)hasthepropertyli(xi)=1;li(xk)=0,k≠i.Joseph-LouisLagrange(1736-1813)Italian-Frenchmathematicianassociated
12、withmanyclassicmathematicsandphysics–Lagrangemultipliersinminimizationofafunction,Lagrange’sinterpolationformula,Lagrange’stheoremingrouptheoryandnumbertheory,andtheLagrangian(L=T-V)inmechanicsandLagrangeequations.Neville’sAlgorithmEvaluateLagrange’sinterpolationformulaf(x)atx,
13、giventhedatapoints(xi,yi).InterpolationtableauP(x)x1:y1=P1P12x2:y2=P2P123P23P1234x3:y3=P3P234P34x4:y4=P4Pi,i+1,i+2,…,i+nisapolynomialofdegreeninxthatpassesthroughthepoints(xi,yi),(xi+1,yi+1),…,(xi+n,yi+n)exactly.DetermineP12fromP1&P2GiventhevalueP1andP2atx=x1andx2,wefindlineari
14、nterpolationP12(x)=λ(x)P1+[1-λ(x)]P2SinceP12(x1)=P1andP12(x2)=P2,wemusthaveλ(x1)=1,λ(x2)=0soλ(x)=λ12=(x-x2)/(x1-x2)DetermineP123fromP12&P23WewriteP123(x)=λ(x)P12(x)+[1-λ(x)]P23(x)P123(x2)=P2alreadyforanychoiceofλ(x).WerequirethatP123(x1)=P12(x1)=P1andP123(x3)=P23(x3)=P3,thusλ(x
15、1)=1,λ(x3)=0OrRecursionRelationforPGiventwom-pointinterpolatedvaluePconstructedfrompointi,i+1,i+2,…,i+m-1,andi+1,i+2,…,i+m,thenextlevelm+1pointinterpolationfromitoi+misaconvexcombination:UseSmallDifferenceC&DP1P2P3P4P12P23P34P123P234P1234C1,1=P12-P1D1,1C1,2C1,3D1,2D1,3C2,1=P123
16、-P12C2,2D2,1C3,1=P1234-P123D3,1=P1234-P234D2,2=P234-P3
