homework-03 - H OMEWORK 3 120202 ESM4A N UMERICAL M ETHODS...

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Unformatted text preview: H OMEWORK 3 120202: ESM4A - N UMERICAL M ETHODS Spring 2009 Prof. Dr. Lars Linsen Zymantas Darbenas School of Engineering and Science Jacobs University Due: Friday, March 6, 2009 at noon (in the mailbox labeled "Linsen" in the entrance hall of Research I). Problem 7: Consistency and Stability. (10 points) Consider the following iterative method for locating roots of a non-linear function f within an interval [a, b]: b-a xn+1 := xn - f (xn ), n > 0, f (b) - f (a) with f C 1 [a, b] and f (a) = f (b). (a) Show that the method is consistent. b-a (b) Show that the method is stable, if f (r) f (b)-f (a) (0, 2) with r being the root. Problem 8: Newton's Method. Let f (x) = |x|. (10 points) (a) Apply three steps of Newton's method for finding roots of non-linear equations starting with x0 = 1. (b) For what starting points x0 does the Newton iteration converge? Why does this not contradict the convergence theorem presented in class? Problem 9: Bisection and Secant Method. Given function f (x) = cos(x) - x3 and starting points x0 = 0 and x1 = 1. (10 points) (a) Check whether the bisection and the secant method can be applied to locate the root of f (without validating proximity postulate). (b) Apply two steps of the schemes (if applicable). ...
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This note was uploaded on 05/18/2010 for the course MATHEMATIC 120102 taught by Professor Xxxxxxxxxyyyyyy during the Spring '10 term at Jacobs University Bremen.

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