HW8suggestions - + 1 ∞ X j = i N k +1 j ( u ). If k = 1,...

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Jacobs University, Bremen School of Engineering and Science Prof. Dr. Lars Linsen, Orif Ibrogimov Spring Term 2010 Homework 8 120202: ESM4A - Numerical Methods Homework Problems 8.1. a) Derive the natural quadratic spline s ( u ) , u [0 , 3], with knots 0, 2, and 3 and interpolation constraints s (0) = 0 and s (2) = 2. b) Derive a closed form for the quadratic B-spline N 2 0 ( u ) over an equidistant knot sequence u i = i . c) Determine the value of ( a, b, c ) that makes the function f ( x ) = ± x 3 x [0 , 1] 1 2 ( x - 1) 3 + a ( x - 1) 2 + b ( x - 1) + c x [1 , 3] a cubic spline. Is it a natural cubic spline? ( ? points ) 8.2. Prove that (a) For all k N , sup -∞ <u< | X i = -∞ c i N k i ( u ) | ≤ sup -∞ <i< | c i | . (b) For k 2, d du N k i ( u ) = n u i + k - u i N n - 1 i ( u ) - n u i + n +1 - u i +1 N n - 1 i +1 ( u ). (c) For k 2, Z u -∞ N k i ( s ) ds = t i + k +1 - t i k
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Unformatted text preview: + 1 ∞ X j = i N k +1 j ( u ). If k = 1, the relations in (b) and (c) holds for all u except for the knots. ( ? points ) 8.3. Consider the class of functions on [ a, b ], which are summable with square of its second derivative, W 2 2 [ a, b ]. Consider interpolate function u ( x ) ∈ W 2 2 [ a, b ] , u ( x i ) = f ( x i ) i = 0 , 1 , . . . , n, which minimizes the functional J ( u ) = Z b a ² d 2 u dx 2 ³ 2 dx. Show that such function is natural cubic spline. ( 10 points ) Due: 16.04.10, at 3 pm (in the mailbox labeled Linsen in the entrance hall of Res.I )...
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