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Unformatted text preview: H OMEWORK 1
120202: ESM4A - N UMERICAL M ETHODS Spring 2009
Prof. Dr. Lars Linsen Zymantas Darbenas School of Engineering and Science Jacobs University Due: Friday, February 20, 2009 at noon (in the mailbox labeled "Linsen" in the entrance hall of Research I). Problem 1: Taylor series. Let f (x) = ln( x ). 2 (a) Derive the Taylor series of f at 2. (b) Show that the derived Taylor series represents the function f for x [2, 3]. (c) Compute ln(1.5) using the truncated Taylor series with terms up to order 3. Problem 2: Number representation. (10 points) The solutions x1 and x2 of a quadratic equation ax2 + bx + c = 0 can be found by the equation -b b2 - 4ac . x1,2 = 2a (a) Compute the solutions to the quadratic equation with a = 1, b = 200, and c = 0.000015 using floating point representations with 10 bit precision (for the mantissa) and base 10. (b) Compute the roundoff error. (c) Compute an upper and a lower bound for the number of lost significant bits when executing the problematic subtraction. Problem 3: Gaussian elimination with scaled partial pivoting. Given matrix 1 0 4 1 2 1 10 1 A= -1 5 4 3 and vector b = 5 2 2 2 (10 points) 1 2 . 3 4 (10 points) (a) Check whether Gaussian elimination with scaled partial pivoting can be applied to solve Ax = b. (b) Solve Ax = b for x using Gaussian elimination with scaled partial pivoting. ...
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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.
- Spring '10