Transcription of Chapter 3 Interpolation - MIT OpenCourseWare
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Chapter 3 Interpolation - MIT OpenCourseWare
Chapter 3 InterpolationInterpolation is the problem of fitting a smooth curve through a given set ofpoints, generally as the graph of a function. It is useful at least in data analy-sis ( Interpolation is a form of regression), industrial design, signal processing(digital-to-analog conversion) and in numerical analysis. It is one of thoseimportant recurring concepts in applied mathematics. In this Chapter , wewill immediately put Interpolation to use to formulate high-order quadratureand differentiation Polynomial interpolationGivenN+ 1 pointsxj R, 0 j N, and sample valuesyj=f(xj) ofa function at these points, the polynomial Interpolation problem consists infinding a polynomialpN(x) of degreeNwhich reproduces those values:yj=pN(xj), j= 0,.., other words the graph of the polynomial should pass through the points(xj,yNj). A degree-Npolynomial can be written aspN(x) = n=0annxforsome coefficientsa0,..,aN. For Interpolation , the number of degrees offreedom (N+ 1 coefficients) in the polynomial matches the number of pointswhere the function should be fit.
manner", but this is a problem in approximation rather than interpolation; we will return to it later in the chapter on least-squares. 1. CHAPTER 3. INTERPOLATION Let us rst see how the interpolation problem can be solved numerically in a direct way. Use the expression of p. N.
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