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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”.3.1TheInterpolatingPolynomialWea
2、llknowthattwopointsdetermineastraightline.Moreprecisely,anytwopointsintheplane,and,with,determineauniquefirstdegreepolynomialinwhosegraphpassesthroughthetwopoints.Therearemanydifferentformulasforthepolynomial,buttheyallleadtothesamestraightlinegraph.Thisgeneralizestomorethantwopoints.Givenpoints
3、intheplane,，,withdistinct’s,thereisauniquepolynomialinofdegreelessthanwhosegraphpassesthroughthepoints.Itiseasiesttorememberthat,thenumberofdatapoints,isalsothenumberofcoefficients,althoughsomeoftheleadingcoefficientsmightbezero,sothedegreemightactuallybelessthan.Again,therearemanydifferentformu
4、lasforthepolynomial,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,thatonlyapproximatethedata.Theyarenotinterpolatingpolynomials.Themostcomp
5、actrepresentationoftheinterpolatingpolynomialistheLa-grangeform.Therearetermsinthesumandtermsineachproduct,sothisexpressiondefinesapolynomialofdegreeatmost.Ifisevaluatedat,alltheproductsexceptthetharezero.Furthermore,thethproductisequaltoone,sothesumisequaltoandtheinterpolationconditionsaresatis
6、fied.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])displays0123-5-6-116TheLagrangianformofthepolynomialinterpolatingthisdataisWecanseethateachtermisofdegreethree,sotheentiresumhasdegreeatmostthree.Becausetheleadingtermdoesnotva
7、nish,thedegreeisactuallythree.Moreover,ifwepluginor3,threeofthetermsvanishandthefourthproducesthecorrespondingvaluefromthedataset.PolynomialsareusuallynotrepresentedintheirLagrangianform.Morefre-quently,theyarewrittenassomet
