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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-preservingpiecewisecubicn
2、amed“pchip”.3.1TheInterpolatingPolynomialWeallknowthattwopointsdetermineastraightline.Moreprecisely,anytwopointsintheplane,and,with,determineauniquefirstdegreepolynomialinwhosegraphpassesthroughthetwopoints.Therearemanydifferentformulasforthepolynomial
3、,buttheyallleadtothesamestraightlinegraph.Thisgeneralizestomorethantwopoints.Givenpointsintheplane,，,withdistinct’s,thereisauniquepolynomialinofdegreelessthanwhosegraphpassesthroughthepoints.Itiseasiesttorememberthat,thenumberofdatapoints,isalsothenumb
4、erofcoefficients,althoughsomeoftheleadingcoefficientsmightbezero,sothedegreemightactuallybelessthan.Again,therearemanydifferentformulasforthepolynomial,buttheyalldefinethesamefunction.Thispolynomialiscalledtheinterpolatingpolynomialbecauseitexactly第17页
5、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,thatonlyapproximatethedata.Theyarenotinterpolatingpolynomials.ThemostcompactrepresentationoftheinterpolatingpolynomialistheLa-grangeform.Thereareterm
6、sinthesumandtermsineachproduct,sothisexpressiondefinesapolynomialofdegreeatmost.Ifisevaluatedat,alltheproductsexceptthetharezero.Furthermore,thethproductisequaltoone,sothesumisequaltoandtheinterpolationconditionsaresatisfied.Forexample,considerthefollo
7、wingdataset: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])displays0123-5-6-116TheLagrangianformofthepolynomialinterpolatingthisdataisWecanseethateachtermisofdegreethree,sotheentiresumhasdegreeatmostthree.Becausetheleadingt
8、ermdoesnotvanish,thedegreeisactuallythree.Moreover,ifwepluginor3,threeofthetermsvanishandthefourthproducesthecorrespondingvaluefromthedataset.PolynomialsareusuallynotrepresentedintheirLagrangianform.Morefre-quently,theyarewrittenassomet
