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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.

In this chapter, we will immediately put interpolation to use to formulate high-order quadrature and di erentiation rules. 3.1 Polynomial interpolation Given N+ 1 points x j 2R, 0 j N, and sample values y j = f(x j) of a function at these points, the polynomial interpolation problem consists in nding a polynomial p

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