solhw2

solhw2 - Solutions for Homework 2 Foundations of...

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Unformatted text preview: Solutions for Homework 2 Foundations of Computational Math 1 Fall 2011 Problem 2.1 Let n = 4 and consider the lower triangular system Lx = f of the form 1 λ 21 1 λ 31 λ 32 1 λ 41 λ 42 λ 43 1 ξ 1 ξ 2 ξ 3 ξ 4 = φ 1 φ 2 φ 3 φ 4 Recall, that it was shown in class that the column-oriented algorithm could be derived from a factorization L = L 1 L 2 L 3 where L i was an elementary unit lower triangular matrix associated with the i-th column of L . Show that the row-oriented algorithm can be derived from a factorization of L of the form L = R 2 R 3 R 4 where R i is associated with the i-th row of L . Solution: We can define R i in a manner similar to the column forms used for L i . Specifically, define for the case n = 4 R 2 = 1 0 0 0 λ 21 1 0 0 0 1 0 0 0 1 R 3 = 1 0 0 1 0 0 λ 31 λ 32 1 0 0 1 R 4 = 1 1 1 λ 41 λ 42 λ 43 1 It is straighforward to verify that this satisfies L = R 2 R 3 R 4 . To see that the pattern holds for any n note that R j = I + e j r T j r T j e k = 0 , j ≤ k ≤ n R i R j = R i ( I + e j r T j ) = R i + R i e j r T j = R i + e j r T j , since i < j 1 and therefore we simply put the nonzeros of j-th row from R j into the zero positions in the j-th row of R i to form the product. This easily generalizes when R i is replaced with a matrix with a set of rows whose indices are all less than...
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solhw2 - Solutions for Homework 2 Foundations of...

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