翻译：插值与拟合

《翻译：插值与拟合》由会员上传分享，免费在线阅读，更多相关内容在行业资料-天天文库。

1、PADEAPPROXIMATIONBYRATIONALFUNCTION129Wecanapplythisformulatogetthepolynomialapproximationdirectlyforagivenfunctionf(x),withouthavingtoresorttotheLagrangeorNewtonpolynomial.Givenafunction,thedegreeoftheapproximatepolynomial,andtheleft/rightboundarypointsoftheinterva

2、l,theaboveMATLABroutine“cheby()”usesthisformulatomaketheChebyshevpolynomialapproximation.ThefollowingexampleillustratesthatthisformulagivesthesameapproximatepolynomialfunctionascouldbeobtainedbyapplyingtheNewtonpolynomialwiththeChebyshevnodes.Example3.1.Approximatio

3、nbyChebyshevPolynomial.Considertheproblemoffindingthesecond-degree(N=2)polynomialtoapproximatethefunction.Wemakethefollowingprogram“do_cheby.m”,whichusestheMATLABroutine“cheby()”forthisjobandusesLagrange/NewtonpolynomialwiththeChebyshevnodestodothesamejob.Readerscan

4、runthisprogramtocheckiftheresultsarethesame.3.4PADEAPPROXIMATIONBYRATIONALFUNCTIONPadeapproximationtriestoapproximateafunctionf(x)aroundapointxobyarationalfunction(3.4.1)whereareknown.Howdowefindsucharationalfunction?WewritetheTaylorseriesexpansionoff(x)uptodegreeM+

5、Natx=xoas130INTERPOLATIONANDCURVEFITTINGAssuming＝0forsimplicity,wegetthecoefficientsofsuchthat(3.4.3)bysolvingthefollowingequations:（3.4.4a）（3.4.4b）Here,wemustfirstsolveEq.(3.4.4b)forandthensubstitutedi’sintoEq.(3.4.4a)toobtainTheMATLABroutine“padeap()”implementsthi

6、sschemetofindthecoefficientvectorsofthenumerator/denominatorpolynomialofthePadeapproximationforagivenfunctionf(x).Notethefollowingthings:žThederivativesuptoorder(M+N)arecomputednumericallybyusingtheroutine“difapx()”,thatwillbeintroducedinSection5.3.žInordertocompute

7、thevaluesofthePadeapproximatefunction,wesubstituteforxinwhichhasbeenobtainedwiththeassumptionthat＝0.PADEAPPROXIMATIONBYRATIONALFUNCTION131Example3.2.PadeApproximationfor.Let’sfindthePadeapproximationforaround＝0.WemaketheMATLABprogram“do_pade.m”,whichusestheroutine“p

8、adeap()”forthisjobandusesitagainwithnooutputargumenttoseethegraphicresultsasdepictedinFig.3.6.Toconfirmandsupportthisresultfromtheanalytic