chap07 - Interpolation Polynomial Interpolation Piecewise...

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InterpolationPolynomial InterpolationPiecewise Polynomial InterpolationScientific Computing: An Introductory SurveyChapter 7 – InterpolationProf. Michael T. HeathDepartment of Computer ScienceUniversity of Illinois at Urbana-ChampaignCopyright c2002. Reproduction permittedfor noncommercial, educational use only.Michael T. HeathScientific Computing1 / 56
InterpolationPolynomial InterpolationPiecewise Polynomial InterpolationOutline1Interpolation2Polynomial Interpolation3Piecewise Polynomial InterpolationMichael T. HeathScientific Computing2 / 56
InterpolationPolynomial InterpolationPiecewise Polynomial InterpolationMotivationChoosing InterpolantExistence and UniquenessInterpolationBasic interpolation problem: for given data(t1, y1),(t2, y2), . . .(tm, ym)witht1< t2<· · ·< tmdetermine functionf:RRsuch thatf(ti) =yi,i= 1, . . . , mfisinterpolating function, orinterpolant, for given dataAdditional data might be prescribed, such as slope ofinterpolant at given pointsAdditional constraints might be imposed, such assmoothness, monotonicity, or convexity of interpolantfcould be function of more than one variable, but we willconsider only one-dimensional caseMichael T. HeathScientific Computing3 / 56
InterpolationPolynomial InterpolationPiecewise Polynomial InterpolationMotivationChoosing InterpolantExistence and UniquenessPurposes for InterpolationPlotting smooth curve through discrete data pointsReading between lines of tableDifferentiating or integrating tabular dataQuick and easy evaluation of mathematical functionReplacing complicated function by simple oneMichael T. HeathScientific Computing4 / 56
InterpolationPolynomial InterpolationPiecewise Polynomial InterpolationMotivationChoosing InterpolantExistence and UniquenessInterpolation vs ApproximationBy definition, interpolating function fits given data pointsexactlyInterpolation is inappropriate if data points subject tosignificant errorsIt is usually preferable to smooth noisy data, for exampleby least squares approximationApproximation is also more appropriate for special functionlibrariesMichael T. HeathScientific Computing5 / 56
InterpolationPolynomial InterpolationPiecewise Polynomial InterpolationMotivationChoosing InterpolantExistence and UniquenessIssues in InterpolationArbitrarily many functions interpolate given set of data pointsWhat form should interpolating function have?How should interpolant behave between data points?Should interpolant inherit properties of data, such asmonotonicity, convexity, or periodicity?Are parameters that define interpolating functionmeaningful?If function and data are plotted, should results be visuallypleasing?Michael T. HeathScientific Computing6 / 56
InterpolationPolynomial InterpolationPiecewise Polynomial InterpolationMotivationChoosing InterpolantExistence and UniquenessChoosing InterpolantChoice of function for interpolation based onHow easy interpolating function is to work withdetermining its parameters

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