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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”.
2、3.1TheInterpolatingPolynomialWeallknowthattwopointsdetermineastraightline.Moreprecisely,anytwopointsintheplane,and,with,determineauniquefirstdegreepolynomialinwhosegraphpassesthroughthetwopoints.Therearemanydifferentformulasforthepolynomial,buttheyallleadtothesame
3、straightlinegraph.Thisgeneralizestomorethantwopoints.Givenpointsintheplane,，,withdistinct’s,thereisauniquepolynomialinofdegreelessthanwhosegraphpassesthroughthepoints.Itiseasiesttorememberthat,thenumberofdatapoints,isalsothenumberofcoefficients,althoughsomeofthele
4、adingcoefficientsmightbezero,sothedegreemightactuallybelessthan.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-producesthegi
5、vendata.，Later,weexamineotherpolynomials,oflowerdegree,thatonlyapproximatethedata.Theyarenotinterpolatingpolynomials.ThemostcompactrepresentationoftheinterpolatingpolynomialistheLa-grangeform.Therearetermsinthesumandtermsineachproduct,sothisexpressiondefinesapolyn
6、omialofdegreeatmost.Ifisevaluatedat,alltheproductsexceptthetharezero.Furthermore,thethproductisequaltoone,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
7、文文献翻译disp([x;y])displays0123-5-6-116TheLagrangianformofthepolynomialinterpolatingthisdataisWecanseethateachtermisofdegreethree,sotheentiresumhasdegreeatmostthree.Becausetheleadingtermdoesnotvanish,thedegreeisactuallythree.Moreover,ifwepluginor3,threeofthetermsvani
8、shandthefourthproducesthecorrespondingvaluefromthedataset.PolynomialsareusuallynotrepresentedintheirLagrangianform.Morefre-quently,theyarewrittenassomet
