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# homework-08 - H OMEWORK 8 120202 ESM4A N UMERICAL M ETHODS...

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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 represen- tations over the knot sequence 3 2 , 5 2 , 7 2 , 9 2 , 11 2 . (b) Given knots u i = i + 2 for i = 1 , . . . , 4 , points ( p 1 , . . . , p 4 ) = (1 , 4 , 1 , 3) , and endpoint derivatives d 1 = d 4 = 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 N n i ( u ) over knot sequence Z and uniform B-splines M n i ( u ) over knot sequence 1 2 Z with summationdisplay i c i N n i ( u ) = summationdisplay i b n i M n i ( u ) . Show that b n i = 1 2 ( b n - 1 i + b n - 1 i +1 ) with b 0 2 i = b 0 2 i +1 = c i for
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