homework-07 - H OMEWORK 7 120202 ESM4A N UMERICAL M ETHODS...

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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 . (b) Derive a closed form for the quadratic B-spline N 2 0 ( u ) over an equidistant knot sequence u i = i . (c) Derive a closed form for the spline s ( u ) = 2 i =0 c i N 2 i ( u ) in B-spline representation with ( c 0 , c 1 , c 2 ) = (1 , 3 , 2) over an equidistant knot sequence u i = i . Problem 20: B-splines. (10 points) (a) Show that n summationdisplay i =0 c r - 1
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