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HW6suggestions - Jacobs University Bremen School of...

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Jacobs University, Bremen School of Engineering and Science Prof. Dr. Lars Linsen, Orif Ibrogimov Spring Term 2010 Homework 6 120202: ESM4A - Numerical Methods Homework Problems 6.1. Suppose that the function 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]. Show that the error is less then 30 . 8 × 10 - 7 on this interval. ( 7 points ) 6.2. Given knots ( u i , v j ) and points p ij , i = 0 , . . . , 4, j = 0 , 1, as below. Determine a bivariate Lagrange polynomial p ( u, v ) of minimum degree that interpolates the points at the knots. ( u i , v j ) (0, 0) (1, 0) (2, 0) (4, 0) (8, 0) (0, 1) (1, 1) (2, 1) (4, 1) (8, 1) P i 1 2 -3 2 1 2 4 9 4 5 ( 6 points ) 6.3. Given knots u 0 = 0 , u 1 = 1 2 and u 2 = 1 and points p 0 = 1 , p 1 = 2 . 25 and p 2 = 10. (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 ) = 2 and p ( u 2 ) = 20 at the endpoints of the interval. (b) Compute respective generalized Hermite polynomials and the interpolating curve p ( u ). ( 7 points ) 6.4. (a) Show that for each Bernstein polynomial
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