HW10suggestions - h = π 4 . Hint : First using the...

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Jacobs University, Bremen School of Engineering and Science Prof. Dr. Lars Linsen, Orif Ibrogimov Spring Term 2010 Homework 10 120202: ESM4A - Numerical Methods Homework Problems 10.1. We seek to compute the integral Z 1 0 cos ± πx 2 2 ² dx with an error of magnitude 0 . 001. (a) Apply the recursive Simpson’s rule to obtain it. (b)Apply the composite trapezoid rule to obtain it. ( 5+5=10 points ) 10.2. (a) Show that if f ( t ) can be approximated by a trigonometric polynomial of degree n so that magnitude of error is less than ε ( t (0 , 2 π )), then the error with the use of the trapezoidal rule (with h = 2 π n +1 ) on the integral 1 2 π R 2 π 0 f ( t ) dt is less than 2 ε . (b) Use above to explain the sensationally good result in problem 9.3(b), when
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Unformatted text preview: h = π 4 . Hint : First using the Interpolation error theorem(pages 11-14 in Lecture 11)show that the function g ( x ) = exp(2-1 / 2 x ) can be approximated by a seventh-degree algebraic polynomial for x ∈ [-1 , 1] with error < 2 . 5 · 10-8 . ( 8+2=10 points ) 10.3. Show that the formula Z 1-1 f ( x )(1-x 2 )-1 / 2 dx = π n n X k =1 f ± cos 2 k-1 2 n π ² is exact for all polynomials of degree 2 n-1. Hint : Use problem 9.3(c) from Homework 9. ( 10 points ) Due: 30.04.10, at 3 pm (in the mailbox labeled Linsen in the entrance hall of Res.I )...
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