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1、2013届数学与应用数学专业毕业设计（论文）英文文献翻译Chapter3InterpolationInterpolationistheprocessofdefiningafunctionthattakesonspecifiedvaluesatspecifiedpoints.Thischapterconcentratesontwocloselyrelatedinterpolants,thepiecewisecubicsplineandtheshape-preservingpiecewisecubicnamed“pchip”.3.1TheInterpolatin
2、gPolynomialWeallknowthattwopointsdetermineastraightline.Moreprecisely,anytwopointsintheplane,and,with,determineauniquefirstdegreepolynomialinwhosegraphpassesthroughthetwopoints.Therearemanydifferentformulasforthepolynomial,buttheyallleadtothesamestraightlinegraph.Thisgeneralizestom
3、orethantwopoints.Givenpointsintheplane,，,withdistinct’s,thereisauniquepolynomialinofdegreelessthanwhosegraphpassesthroughthepoints.Itiseasiesttorememberthat,thenumberofdatapoints,isalsothenumberofcoefficients,althoughsomeoftheleadingcoefficientsmightbezero,sothedegreemightactuallyb
4、elessthan.Again,therearemanydifferentformulasforthepolynomial,buttheyalldefinethesamefunction.Thispolynomialiscalledtheinterpolatingpolynomialbecauseitexactly第17页共25页2013届数学与应用数学专业毕业设计（论文）英文文献翻译re-producesthegivendata.，Later,weexamineotherpolynomials,oflowerdegree,thatonlyapproxima
5、tethedata.Theyarenotinterpolatingpolynomials.ThemostcompactrepresentationoftheinterpolatingpolynomialistheLa-grangeform.Therearetermsinthesumandtermsineachproduct,sothisexpressiondefinesapolynomialofdegreeatmost.Ifisevaluatedat,alltheproductsexceptthetharezero.Furthermore,thethprod
6、uctisequaltoone,sothesumisequaltoandtheinterpolationconditionsaresatisfied.Forexample,considerthefollowingdataset:x=0:3;y=[-5-6-116];Thecommand第17页共25页2013届数学与应用数学专业毕业设计（论文）英文文献翻译disp([x;y])displays0123-5-6-116TheLagrangianformofthepolynomialinterpolatingthisdataisWecanseethateacht
7、ermisofdegreethree,sotheentiresumhasdegreeatmostthree.Becausetheleadingtermdoesnotvanish,thedegreeisactuallythree.Moreover,ifwepluginor3,threeofthetermsvanishandthefourthproducesthecorrespondingvaluefromthedataset.PolynomialsareusuallynotrepresentedintheirLagrangianform.Morefre-que
8、ntly,theyarewrittenassomet
