算法设计与分析 - C,C

PolynomialandFFTTopics1.Problem2.Representationofpolynomials3.TheDFTandFFT4.EfficientFFTimplementations5.ConclusionProblemRepresentationofPolynomialsDefinition2Forthepolynomial(1),wehavetwowaysofrepresentingit:CoefficientRepresentation——

(秦九韶算法）Horner’sruleThecoefficientrepresentationisconvenientforcertainoperationsonpolynomials.Forexample,theoperationofevaluatingthepolynomialA(x)atagivenpointx0CoefficientRepresentation——

addingandmultiplicationPoint-valueRepresentationByHorner’srule,ittakesΘ(n2)timetogetapoint-valuerepresentationofpolynomial(1).Ifwechoosexk

cleverly,thecomplexityreducestonlogn.Definition3

Theinverseof

evaluation.Theprocessofdeterminingthecoefficientformofapolynomialfromapointvaluerepresentationiscalledinterpolation.Doestheinterpolationuniquelydetermineapolynomial?Ifnot,theconceptofinterpolationismeaningless.UniquenessofInterpolationLagrangeFormulaWecancomputethecoefficientsofA(x)by(4)intimeΘ(n2).拉格朗日［Lagrange,JosephLouis，1736-1813

●法国数学家。

●涉猎力学，著有分析力学。

●百年以来数学界仍受其理论影响。VirtuesofpointvaluerepresentationFastmultiplicationofpolynomialsincoefficientformCanweusethelinear-timemultiplicationmethodforpolynomialsinpoint-valueformtoexpeditepolynomialmultiplicationincoefficientform?BasicideaofmultiplicationBasicideaofmultiplicationIfwechoose“complexrootsofunity”astheevaluationpointscarefully,wecanproduceapoint-valuerepresentationbytakingtheDiscreteFourierTransformofacoefficientvector.Theinverseoperationinterpolation,canbeperformedbytakingtheinverseDFTofpointvaluepairs.ComplexRootsofUnityAdditiveGroupPropertiesofComplexRootsFourierTransformNowconsidergeneralizationtothecaseofadiscretefunction:DiscreteFourierTransformIdeaofFastFourierTransformRecursiveFFTComplexityofFFTProperty4

Bydivide-and-conquermethod,thetimecostofFFTisT(n)=2T(n/2)+Θ(n)=Θ(nlogn).InterpolationProofDFTn

vsDFT-1nEfficientFFTImplementationButterflyOperationIterative-FFTITERATIVE-FFT(a)1BIT-REVERSE-COPY(a,A)2n←length[a]//nisapowerof2.3fors←1tolgn4dom←2s5ωm←e2πi/m6fork←0ton-1bym7doω←18forj←0tom/2-19dot←ωA[k+j+m/2]10u←A[k+j]11A[k+j]←u+t12A[k+j+m/2]←u-t13ω←ωωmconclusionFourieranalysisisnotlimitedto1-dimensionaldata.Itis