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# homework-05 - H OMEWORK 5 120202 ESM4A N UMERICAL M ETHODS...

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Unformatted text preview: H OMEWORK 5 120202: ESM4A - N UMERICAL M ETHODS Spring 2009 Prof. Dr. Lars Linsen Zymantas Darbenas School of Engineering and Science Jacobs University Due: Friday, March 20, 2009 at noon (in the mailbox labeled "Linsen" in the entrance hall of Research I). Problem 13: Lagrange Interpolation. (10 points) (a) Given knots (ui , vj ) and points pij , i = 0, . . . , 4, j = 0, 1, as below. Determine a bivariate Lagrange polynomial p(u, v) of minimum degree that interpolates the points at the knots. (ui , vj ) pi (0,0) 1 (1,0) 2 (2,0) 3 (4,0) 2 (8,0) 1 (0,1) 2 (1,1) 4 (2,1) 9 (4,1) 4 (8,1) 5 (b) Let p(u, 0) be a curve on the interpolating surface computed above. Use Aitken's algorithm to compute p(6, 0) and compare to the result from evaluating the interpolating surface equation computed above. Problem 14: Newton Interpolation. 1 Given function f (u) = u2 and sample locations 1, 2, 4, and 8. (10 points) (a) Use Newton interpolation to determine a polynomial p(u) of minimum degree that interpolates the function values of f (u) at the given sample locations. Use the devided differences scheme to compute the nodes. (b) Estimate a lowest upper bound for the error |f (u) - p(u)| for u [1, 8]. Problem 15: Polynomial Interpolation. (a) Show that i (10 points) p[u0 . . . ui ] = k=0 pk i j=0,j=k (uk - uj ) , where p[u0 . . . ui ] denotes the divided difference of order i of a polynomial p(u) that interpolates the points pk at the knots uk for k = 0, . . . , i. (b) Show that n |u - ui | i=0 n! 4 b-a n n+1 , where ui = a + i b-a n , i = 0, . . . , n are equidistant knots on the interval [a, b]. ...
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