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# HW5suggestions - f x = x sin x is approximated by a...

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Jacobs University, Bremen School of Engineering and Science Prof. Dr. Lars Linsen, Orif Ibrogimov Spring Term 2010 Homework 5 120202: ESM4A - Numerical Methods Homework Problems 5.1. (a) Find the Lagrange and Newton forms of the interpolating polynomial for these data: x -2 0 1 f ( x ) 0 1 -1 (b) For the function y = sin πx determine interpolation polynomial of Lagrange, taking points x i = 1 i +2 , i = 0 , 1 , 2. ( 4.5+3.5=8 points ) 5.2. (a) Suppose that n N ∪ { 0 } and we are given a knot sequence u 0 < u 1 < . . . < u n and points p 0 , . . . , p n . Prove that there is a unique polynomial P ( u ) of degree n such that P ( u i ) = p i for i = 0 , . . . , n . (b) Show that the Lagrange polynomials have the property of partition of unity, i.e., prove that the following identity n X i =0 L n i ( u ) 1 holds for all u R . ( 6.5+7.5=14 points ) 5.3. Suppose that the function
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Unformatted text preview: f ( x ) = x sin x is approximated by a polynomial of degree nine that interpolates our function f at ten points in the interval [0 , 1], then show that the error is less then 30 . 8 × 10-7 on this interval. ( 8 points ) 5.4. (Bonus) Let x , x 1 , x 2 , . . . , x n be arbitrary integers, x < x 1 < x 2 . . . < x n . Show that every polynomial of n th degree of the form x n + a 1 x n-1 + a 2 x n-2 + . . . + a n assumes at the points x , x 1 , x 2 , . . . , x n values, at least one of which is of absolute value ≥ n ! 2 n . ( 10 bonus points ) Due: 12.03.10, at 3 pm (in the mailbox labeled Linsen in the entrance hall of Res.I )...
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