ASE311 HW4 pdf

# ASE311 HW4 pdf - ASE 311 Engineering computation Fall 2009...

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Unformatted text preview: ASE 311 Engineering computation, Fall 2009, Homework 4 Due: Friday, September 25, 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. Three matrices are deﬁned as [ 1 3 [] = , 0 −1 [ 2 2 [] = , 1 0 [ 4 15 [] = 5 9 Compute manually [][] − [][], [][] − [][] and [][] − [][]. Check with MATLAB. 2. [Transpose of a product] Let [] be a × - and [] a × -matrix, respectively. Prove that ([][]) = [] [] Part II 3. Solve the system of equations 2 + 6 + 10 = 0 + 3 + 3 = 2 3 + 14 + 28 = −8 manually using Gauss elimination. 4. Solve the system of equations from the previous exercise using MATLAB and a) Cramer’s rule b) Built-in function inv c) Left division with the backslash operator ∖ 5. Solve the tridiagonal system ⎡ 2 −1 ⎢−1 2 −1 ⎢ ⎢ .. ⎢ . ⎢ ⎣ −1 ⎤⎡ ⎡ ⎤ 1 ⎥⎢ ⎥ ⎢1⎥ ⎥⎢ ⎥ ⎢ ⎥ ⎥⎢ ⎥ ⎢.⎥ ⎥⎢ ⎥ = ⎢.⎥ ⎥⎢ ⎥ ⎢.⎥ ⎦ ⎣−1 ⎦ ⎣1⎦ 2 −1 −1 2 1 1 2 . . . ⎤ using MATLAB and naive Gauss elimination for = 100, 200, 300, 400, 500. Report the value of 1 for each case and measure the CPU time used by your solver routine with the command cputime. Plot the time as a function of . Hint: You can create the coeﬃcient matrix using the command A = -diag(ones(n-1,1),-1)+2*diag(ones(n,1))-diag(ones(n-1,1),1); ...
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