HW4suggestions - , 1) ± xy 2 + x 2 y + x 4 = 3 x 3 y 5-2 x...

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Jacobs University, Bremen School of Engineering and Science Prof. Dr. Lars Linsen, Orif Ibrogimov Spring Term 2010 Homework 4 120202: ESM4A - Numerical Methods Homework Problems 4.1. a) Using the Bisection method find a positive solution of the equation 4 - e x - 2 x 2 = 0 with a precision of ε = 0 . 01. (answer: 0.8828125) b) Perform four iterations of Newton’s method for the polynomial p ( x ) = 4 x 3 - 2 x 2 + 3 starting with x 0 = - 1. c) Using the Secant method find a positive solution of the following equation x 3 - 0 . 2 x 2 - 0 . 2 x - 1 . 2 = 0 with a precision of 0.002. ( 7+4+9=20 points ) 4.2. Perform two iterations of Newton’s method on the system starting with (1
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Unformatted text preview: , 1) ± xy 2 + x 2 y + x 4 = 3 x 3 y 5-2 x 5 y-x 2 =-2 ( 5 points ) 4.3. (a) Prove that Newton’s iteration will diverge for the function f ( x ) = 7 x 4 + 3 x 2 + π no matter what (real) starting point is selected. (b) (Bonus) Let f ∈ C 2 ( R ) be a convex, increasing function. Show that if f has a zero, then zero is unique and the Newton iteration will converge to it from any starting point. ( 5 points + 5 bonus points ) Due: 05.03.10, at 3 pm (in the mailbox labeled Linsen in the entrance hall of Res.I )...
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