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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”.3.1TheInterpola
2、tingPolynomialWeallknowthattwopointsdetermineastraightline.Moreprecisely,anytwopointsintheplane,and,with,determineauniquefirstdegreepolynomialinwhosegraphpassesthroughthetwopoints.Therearemanydifferentformulasforthepolynomial,buttheyallleadtothesamestraightlinegraph.Thisgenerali
3、zestomorethantwopoints.Givenpointsintheplane,，,withdistinct’s,thereisauniquepolynomialinofdegreelessthanwhosegraphpassesthroughthepoints.Itiseasiesttorememberthat,thenumberofdatapoints,isalsothenumberofcoefficients,althoughsomeoftheleadingcoefficientsmightbezero,sothedegreemight
4、actuallybelessthan.Again,therearemanydifferentformulasforthepolynomial,buttheyalldefinethesamefunction.Thispolynomialiscalledtheinterpolatingpolynomialbecauseitexactly第19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,thato
5、nlyapproximatethedata.Theyarenotinterpolatingpolynomials.ThemostcompactrepresentationoftheinterpolatingpolynomialistheLa-grangeform.Therearetermsinthesumandtermsineachproduct,sothisexpressiondefinesapolynomialofdegreeatmost.Ifisevaluatedat,alltheproductsexceptthetharezero.Furthe
6、rmore,thethproductisequaltoone,sothesumisequaltoandtheinterpolationconditionsaresatisfied.Forexample,considerthefollowingdataset:x=0:3;y=[-5-6-116];Thecommand第19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])displays0123-5-6-116TheLagrangianformofthepolynomialinterpolatingthisdatai
7、sWecanseethateachtermisofdegreethree,sotheentiresumhasdegreeatmostthree.Becausetheleadingtermdoesnotvanish,thedegreeisactuallythree.Moreover,ifwepluginor3,threeofthetermsvanishandthefourthproducesthecorrespondingvaluefromthedataset.PolynomialsareusuallynotrepresentedintheirLagra
8、ngianform.Morefre-quently,theyarewrittenassomet
