cubic spline函数介绍

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1、CubicSplineInterpolationMAE5093CharlesO’Neill28May2002AbstractAcubicsplineroutinewasdevelopedforunequallyspacedsequentialdatapoints.Cubicsplinetheoryisreviewed.AVisualBasiccomputerprograminExcelwascreatedtofitasplinetoinputdatapoints.Threetestcasesareusedtovalidatetherout

2、ine.Conclusionsregardingthecubicsplineroutinearemade.IntroductionTheobjectiveistofitacubicsplinetodatapoints.Atypicalcurvefitinvolvesformingoneequationthroughallnpoints.Incontrast,asplineallowingeachsegmenttohaveauniqueequationwhilestillconstrainingthecurvefittothedatapro

3、perties.Thispaperwilldevelopthegoverningequationsforacubicspline.AcomputerimplementationusingVisualBasicwillbepresented.Threetestcaseswillvalidatethesplinemethodandthecomputercode.Finally,conclusionswillbediscussed.TheorySplinetheoryissimple.Overnintervals,theroutinefitsn

4、equationssubjecttotheboundaryconditionsofn+1datapoints.ThederivationsofLilley[1]andWheatly[2]areused.Thederivationassumesafunctionalformforthecurvefit.Thisequationformissimplifiedandthensolvedforthecurvefitequation.Theassumedformforthecubicpolynomialcurvefitforeachsegment

5、is,()3()2()y=ax-x+bx-x+cx-x+diiiiiiiwherethespacingbetweensuccessivedatapointsish=x-xii+1istndThecubicsplineconstrainsthefunctionvalue,1derivativeand2derivative.Theroutinemustensurethat)x(y,y¢)x(andy¢¢)x(areequalattheinteriornodepointsforadjacentsegments.Substitutingavari

6、ableSforthepolynomial’ssecondderivativereducesthenumberofequationsfroma,b,c,dforeachsegmenttoonlySforeachsegment.thFortheisegment,theSgoverningequationis,y-yy-y()i+1iii-1hS+2h+2hS+hS=6-i-1i-1i-1iiii+1hhii-1Inmatrixform,thegoverningequationsreducetoatri-diagonalfor

7、m.y3-y2y2-y12(h1+h2)h2S2-hh21h2(h+h)OS223i=6OOhn-2MMy-yy-yh2(h+h)Snn-1-n-1n-2n-2n-2n-1n-1hn-1hn-2S1andSnarezeroforthenaturalsplineboundarycondition.Ifdifferentboundaryconditionsareneeded,theappropriatechangescanbemadetothegoverninge

8、quations.Finally,thecubicsplinepropertiesarefoundbysubstitutingintothefollowingequations.Thesea,