hw5_solutions - Homework 5 Math 471 Fall 2007 Assigned...

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Homework 5 Math 471, Fall 2007 Assigned: Friday, October 5, 2007 Due: Friday, October 12, 2007 Include a cover page Clearly label all plots using title , xlabel , ylabel , legend Use the subplot command to compare multiple plots Include printouts of all Matlab code, labeled with your name, date, section, etc. (1) (Pivoting) (a) Prove that the matrix 0 1 1 1 does not have an LU decomposition. Use direct decomposition. Suppose that 0 1 1 1 = 1 0 21 1 u 11 u 12 0 u 22 = u 11 u 12 21 u 11 21 u 12 + u 22 . Then u 11 = 0. But this implies 1 = 21 u 11 = 0. (b) Does the system 0 1 1 1 x y = a b have a unique solution for all a, b R ? (Why?) Yes. det 0 1 1 1 = - 1 . (c) How can you modify the system in part (b) so that LU decomposition applies? Switch the two rows. 1
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2 (2) (Gaussian Elimination with Partial Pivoting) Do page 169, # 7(a) by hand. r = 1 2 3 2 3 1 - 4 4 1 4 9 3 4 6 0 r = 2 1 3 0 5 / 2 - 1 - 17 / 2 4 1 4 9 0 13 / 4 3 - 27 / 4 r = 2 3 1 0 0 - 43 / 13 - 43 / 13 4 1 4 9 0 13 / 4 3 - 27 / 4 (3) (Implementing PLU
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