c数值算法程序大全c3-2

c数值计算 程序大全

3.2RationalFunctionInterpolationandExtrapolation111

3.2RationalFunctionInterpolationand

Extrapolation

Somefunctionsarenotwellapproximatedbypolynomials,butarewellapproximatedbyrationalfunctions,thatisquotientsofpolynomials.Wede-notebyRi(i+1)...(i+m)arationalfunctionpassingthroughthem+1points(xi,yi)...(xi+m,yi+m).Moreexplicitly,suppose

RPi(i+1)...(i+m)=

µ(x)p0+p1x+···+pµxµQ=

ν(x)q0+q1x+···+qνxν

(3.2.1)

Sincethereareµ+ν+1unknownp’sandq’s(q0beingarbitrary),wemusthave

m+1=µ+ν+1

(3.2.2)

Inspecifyingarationalfunctioninterpolatingfunction,youmustgivethedesiredorderofboththenumeratorandthedenominator.

Rationalfunctionsaresometimessuperiortopolynomials,roughlyspeaking,becauseoftheirabilitytomodelfunctionswithpoles,thatis,zerosofthedenominatorofequation(3.2.1).Thesepolesmightoccurforrealvaluesofx,ifthefunctiontobeinterpolateditselfhaspoles.Moreoften,thefunctionf(x)is niteforall niterealx,buthasananalyticcontinuationwithpolesinthecomplexx-plane.Suchpolescanthemselvesruinapolynomialapproximation,evenonerestrictedtorealvaluesofx,justastheycanruintheconvergenceofanin nitepowerseriesinx.Ifyoudrawacircleinthecomplexplanearoundyourmtabulatedpoints,thenyoushouldnotexpectpolynomialinterpolationtobegoodunlessthenearestpoleisratherfaroutsidethecircle.Arationalfunctionapproximation,bycontrast,willstay“good”aslongasithasenoughpowersofxinitsdenominatortoaccountfor(cancel)anynearbypoles.

Fortheinterpolationproblem,arationalfunctionisconstructedsoastogothroughachosensetoftabulatedfunctionalvalues.However,weshouldalsomentioninpassingthatrationalfunctionapproximationscanbeusedinanalyticwork.Onesometimesconstructsarationalfunctionapproximationbythecriterionthattherationalfunctionofequation(3.2.1)itselfhaveapowerseriesexpansionthatagreeswiththe rstm+1termsofthepowerseriesexpansionofthedesiredfunctionf(x).ThisiscalledPade´approximation,andisdiscussedin§5.12.BulirschandStoerfoundanalgorithmoftheNevilletypewhichperformsrationalfunctionextrapolationontabulateddata.Atableaulikethatofequation(3.1.2)isconstructedcolumnbycolumn,leadingtoaresultandanerrorestimate.TheBulirsch-Stoeralgorithmproducestheso-calleddiagonalrationalfunction,withthedegreesofnumeratoranddenominatorequal(ifmiseven)orwiththedegreeofthedenominatorlargerbyone(ifmisodd,cf.equation3.2.2above).Forthederivationofthealgorithm,referto[1].Thealgorithmissummarizedbyarecurrence

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