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Unformatted text preview: H OMEWORK 7
120202: ESM4A - N UMERICAL M ETHODS Spring 2009
Prof. Dr. Lars Linsen Zymantas Darbenas School of Engineering and Science Jacobs University Due: Friday, April 17, 2009 at noon (in the mailbox labeled "Linsen" in the entrance hall of Research I). Problem 19: Splines. (10 points) (a) Derive the natural quadratic spline s(u), u [-1, 1], with knots -1, 0, and 1 and interpolation constraints s(-1) = 0 and s(0) = 2.
2 (b) Derive a closed form for the quadratic B-spline N0 (u) over an equidistant knot sequence ui = i. (c) Derive a closed form for the spline s(u) = i=0 ci Ni2 (u) in B-spline representation with (c0 , c1 , c2 ) = (1, 3, 2) over an equidistant knot sequence ui = i. Problem 20: B-splines. (a) Show that
n r-1 ci Nin-r+1 (u) = i=0 i=1 n 2 (10 points) cr Nin-r (u) i with
n-r+1 r-1 n-r+1 r-1 ci , )ci-1 + i cr = (1 - i i n-r+1 i = u - ui , ui+n-r+1 - ui and u [un , un+1 ]. (b) Show that n n d n N n-1 (u) - N n-1 (u). N (u) = du i ui+n - ui i ui+n+1 - ui+1 i+1 Problem 21: Evaluation of cubic B-spline. Given equidistant knots ui = i + 2. (10 points) (a) Use the de Boor algorithm to estimate values Ni3 (uj ) for all cubic B-splines Ni3 (u) and all knots uj . (b) Use the de Boor algorithm to estimate values Ni3 (3.5) for all cubic B-splines Ni3 (u). ...
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