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Unformatted text preview: H OMEWORK 8
120202: ESM4A - N UMERICAL M ETHODS Spring 2009
Prof. Dr. Lars Linsen Zymantas Darbenas School of Engineering and Science Jacobs University Due: Friday, April 24, 2009 at noon (in the mailbox labeled "Linsen" in the entrance hall of Research I). Problem 22: Spline interpolation. (10 points) (a) Derive the collocation matrix for periodic quadratic spline interpolation using B-spline represen3 5 7 9 tations over the knot sequence 2 , 2 , 2 , 2 , 11 . 2 (b) Given knots ui = i + 2 for i = 1, . . . , 4, points (p1 , . . . , p4 ) = (1, 4, 1, 3), and endpoint derivatives d1 = d4 = 1. Use clamped cubic spline interpolation to compute an interpolating spline in B-spline representation. Sketch the graph of the spline curve and the control polygon. Problem 23: Uniform subdivision. (10 points) (a) Given uniform B-splines Nin (u) over knot sequence Z and uniform B-splines Min (u) over knot sequence 1 Z with 2 ci Nin (u) = bn Min (u). i
i i Show that bn i = 1 n-1 2 (bi + n-1 bi+1 ) 4 with b0 2i = b0 2i+1 = ci for n > 0. (b) Given uniform spline s(u) = i=0 ci Ni2 (u) in B-spline representation and control points (c0 , . . . , c4 ) = (1, 2, 2, 1, 1). Apply three iterations of uniform subdivision and compute the errors sup min |s(u) - ck | i
i=1 u 3 for k = 0, . . . , 3, where ck denotes the ith control point after k subdivision iterations. i 3 Hint: minu |s(u) - c0 | is obtained for u = i + 2 . i Problem 24: Least squares method. Let f (x) = a(x - 1)2 + b cos(x) + c sin( x) with f (0) = 0, f (1) = 1, and f (2) = 1. 2 (10 points) (a) Derive the normal equations for the best approximative solution to a, b, and c in the least-squares sense. (b) Find the least-squares solution for a, b, and c. (c) Sketch the graph of function f (x) over interval [0, 2]. ...
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This note was uploaded on 05/18/2010 for the course MATHEMATIC 120102 taught by Professor Xxxxxxxxxyyyyyy during the Spring '10 term at Jacobs University Bremen.
- Spring '10