Fall2011-HW5-soln - Homework 5 Solution Sketches Math 371,...

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Homework 5 Solution Sketches Math 371, Fall 2011 Assigned: Sunday, October 9, 201 1 Due: Thursday, October 20, 2011 (1) (Pivoting) (a) Prove that the matrix ± 0 1 1 1 ² does not have an LU decomposition. Use direct decomposition. Suppose that ± 0 1 1 1 ² = ± 1 0 ü 21 1 ²± u 11 u 12 0 u 22 ² = ± u 11 u 12 ü 21 u 11 ü 21 u 12 + u 22 ² . Then u 11 = 0. But this implies 1 = ü 21 u 11 = 0. (b) Does the system ± 0 1 1 1 ² ± x y ² = ± a b ² have a unique solution for all a,b R ? (Why?) Yes. The matrix is non-singular, since det ± 0 1 1 1 ² = - 1 . (c) How can you modify the system in part (b) so that LU decomposition applies? Switch the two rows. (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 1
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2 (3) (Implementing Gaussian Elimination) Below is a program that computes the
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Fall2011-HW5-soln - Homework 5 Solution Sketches Math 371,...

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