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Methods Numerical II Problem set 2 - due February 13, 2006 1. LaGrange Interpolation Given n + 1 data points (x0 , y0 )(x1 , y1 )...(xn , yn ), the nth degree polynomial which interpolates these points can be written as: n Pn (x) = k=0 yk Ln,k (x) (1) where (x xj ) j=k (xk xj ) Ln,k (x) = Show that n (2) Ln,k (x) = 1 k=0 (3) for all x. 2. Write a program to perform polynomial interpolation of a given function y = f (x) at n + 1 data points. a) For n = 2, 4, 8 test this on any fourth degree polynomial for f (x). Use evenly spaced points. (Of course, you will evaluate your interpolating polynomial at many more points on the interval besides your data Note points.) that for n = 4, 8 there should be no interpolation error. b) Let 1 f (x) = and 1 1 + x2 (4) xj = 5 + jh, f or j = 0, 1, 2, ..., n, h = 10/n (5) For n = 2, 4, 8, 10, 11, 12, 13, ..., 19, 20 plot the graph (in the interval [-5,5]) of the interpolating polynomial Pn (x). Plot the function f (x) on the same axis. Does the interpolation error get smaller when more points are included? c) Same as part (b), but use the Chebyshev points as nodes: 1 2j + 1 xj = [b + a (b a) cos( )], 2 2(n + 1) j = 0, 1, ...n, a = 5, b=5 (6) d) How do your results in (b) and (c) compare? Give a theoretical explanation for these results. 2

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