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homework-06 - H OMEWORK 6 120202 ESM4A N UMERICAL M ETHODS...

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H OMEWORK 6 120202: ESM4A - N UMERICAL M ETHODS Spring 2009 Prof. Dr. Lars Linsen Zymantas Darbenas School of Engineering and Science Jacobs University Due: Friday, March 27, 2009 at noon (in the mailbox labeled “Linsen” in the entrance hall of Research I). Problem 16: Generalized Hermite interpolation. (10 points) Given knots u 0 = 0 , u 1 = 3 , and u 2 = 9 and points p 0 = 1 , p 1 = 2 , and p 2 = 1 . (a) Use the generalized Hermite interpolation scheme to develop the interpolation conditions for a curve p ( u ) of minimal degree that interpolates the given points at the respective knots and the derivatives p ( u 0 ) = 1 and p ( u 2 ) = 0 at the endpoints of the interval. (b) Compute the respective generalized Hermite polynomials and the interpolating curve p ( u ) . Problem 17: Bernstein polynomials. (10 points) (a) Show that the Bernstein polynomials B n i ( u ) , i = 0 ,...,n , are linearly independent ( u [0 , 1] ). (b) Show that for each Bernstein polynomial B n +1 i ( u ) with i ∈ { 0 ,...,n + 1 } the following equation
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