ase311_hw6 - ASE 311 Engineering computation Fall 2009...

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Unformatted text preview: ASE 311 Engineering computation, Fall 2009, Homework 6 Due: Friday, October 16, 2:00 PM in the lecture room JGB 2.216 Report all work, including any m-files you have written. You may also find the command diary useful in recording your MATLAB session. Please write clearly and be sure to label for which problem each solution is. Please, staple Part I and Part II separately and make sure that your name is written on both parts. Part I 1. [Matrix inverse] Compute manually the inverse of the matrix ⎡ ⎤ 2 −1 0 = ⎣−1 2 −1 ⎦ 0 −1 2 using the LU factorization of . 2. [Condition number] Compute manually the condition number of from the previous exercise with respect to the column-sum, row-sum and Frobenius norms. Compare with the MATLAB command cond. Part II 3. [Jacobi iteration] Solve the system = , where is the matrix from the first exercise and 0 = 1 1 1 , using MATLAB and the Jacobi method with an initial guess How many iterations do you need in order to have ∣∣ − ∣∣2 < 10−4 ? 4. [Fixed-point iteration] Determine a solution for the nonlinear equations 1( 1, 2( 1, 2) 2) =000 . = = 2 1 2 1 + − 2 2 2 −5=0 −1=0 1 using MATLAB and fixed-point iteration with initial guesses of iterations do you need in order to have ∣∣ ( )∣∣2 < 10−4 ? = 2 = 1. How many 5. [Relaxation] The convergence of iterative methods can sometimes be enhanced by averaging the present and the previous iterates at each step as follows: +1 ← +1 + (1 − ) where is a relaxation parameter to be chosen. Try the relaxation of the fixed-point iteration in the previous exercise using the value = 0.8. Do you observe acceleration of convergence? ...
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