Chapter 3 Interpolation - MIT OpenCourseWare

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Chapter 3InterpolationInterpolation 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.

[5; 5], as N!1. See the Trefethen textbook on page 44 for an illustration of the Runge phenomenon. (Figure here) If we had done the same numerical experiment for x2[1; 1], the inter-polant would have converged. This shows that the size of the interval matters. Intuitively, there is divergence when the size of the interval is larger than the

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