最小二乘法拟合多项式

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1、CurveFittingwithMATLABCurvefittingisausefultoolforrepresentingadatasetinalinearorquadraticfashion.MATLABhastwofunctions,polyfitandpolyval,whichcanquicklyandeasilyfitapolynimialtoasetofdatapoints.Afirstorderpolynomialisthelinearequationthatbestfitsthedata.Apolynomia

2、lcanalsobeusedtofitthedatainaquadraticsense.Asareminder,thegeneralformulaforapolynomialis:Thedegreeofapolynomialisequaltothelargestvalueoftheexponentsoftheindependentvariable,x.Forexample:PolyfitandPolyval"Polyfit"isaMATLABfunctionthatcomputesaleastsquarespolynomia

3、lforagivensetofdata.Polyfitactuallygeneratesthecoefficientsofthepolynomial(whichcanbeusedtosimulateacurvetofitthedata)accordingtothedegreespecified."Polyval"evaluatesapolynomialforagivensetofxvalues.So,polyvalactuallygeneratesacurvetofitthedatabasedonthecoefficient

4、sfoundusingpolyfit.MATLAB'sexplanationofpolyfitandpolyvalare:POLYFIT-PolynomialcurvefittingPOLYFIT(x,y,n)findsthecoefficientsofapolynomialp(x)ofdegreenthatfitsthedata,p(x(i))isapproximatelyequaltoy(i),inaleast-squaressense.POLYVAL-PolynomialevaluationIfVisavectorwh

5、oseelementsarethecoefficientsofapolynomial,thenPOLYVAL(V,s)isthevalueofthepolynomialevaluatedats.Ifsisamatrixorvector,thepolynomialisevaluatedatallpointsins.ThefollowingisanexampleMATLABcurvefitsequenceusingpolyfitandpolyval.Hereistheplotreflectingourfirsttryatfitt

6、ingthedatawithafirstdegreepolynomial(orastraightline).ThecoefficientsgivenbyMATLAByieldthefollowingequationfortheline:Tryingahigherpolynomial:Hereistheplotofthecurvegeneratedfromthe5thorderpolynomial.Thiscurvefitsthedataperfectly.ThecoefficientsgivenbyMATLABforthec

7、urveyieldthefollowingequation: