homework-02 - H OMEWORK 2 120202: ESM4A - N UMERICAL M...

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Unformatted text preview: H OMEWORK 2 120202: ESM4A - N UMERICAL M ETHODS Spring 2009 Prof. Dr. Lars Linsen Zymantas Darbenas School of Engineering and Science Jacobs University Due: Friday, February 27, 2009 at noon (in the mailbox labeled "Linsen" in the entrance hall of Research I). Problem 4: LU decomposition with scaled partial pivoting. Given matrix 1 1 0 4 A = 2 1 10 and vector b = 2 . 3 -1 5 4 (10 points) (a) Compute the LU decomposition of matrix A using Gaussian elimination with scaled partial pivoting. (b) Solve Ax = b for x using the LU decomposition. Problem 5: Convergence of iterative solutions. Let A be a strictly diagonal dominant matrix, i.e., |aii | > (a) the Jacobi iteration and (b) the Gauss-Seidel iteration for solving the linear equation system Ax = b converge for any starting point. Problem 6: Gauss-Seidel iteration. Given the system of linear equations 5x1 + 2x2 - 2x3 x1 + 6x2 - 2x3 x2 + 4x3 = 10 (10 points) (10 points) i=j |aij | for all rows i. Show that = -6 = 12 (a) For what starting points can we apply the Gauss-Seidel iteration? Explain your answer. (b) Given starting point x0 = 0, execute the first three steps of a Gauss-Seidel iteration, i. e., compute x1 , x2 , and x3 . ...
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