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Unformatted text preview: Homework 4 Foundations of Computational Math 1 Fall 2010 The solutions will be posted on Wednesday, 9/27/10 Problem 4.1 Recall that an elementary reflector has the form Q = I + αxx T ∈ R n × n with k x k 2 6 = 0. 4.1.a . Show that Q is orthogonal if and only if α = 2 x T x or α = 0 4.1.b . Given v ∈ R n , let γ = ±k v k and x = v + γe 1 . Assuming that x 6 = v show that x T x x T v = 2 4.1.c . Using the definitions and results above show that Qv = γe 1 Problem 4.2 4.2.a This part of the problem concerns the computational complexity question of operation count. For both LU factorization and Householder reflectorbased orthogonal factorization, we have used elementary transformations, T i , that can be characterized as rank1 updates to the identity matrix, i.e., T i = I + x i y T i , x i ∈ R n and y i ∈ R n Gauss transforms and Householder reflectors differ in the definitions of the vectors x i and y i . Maintaining computational efficiency in terms of a reasonable operation count usually....
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This note was uploaded on 07/25/2011 for the course MAD 5403 taught by Professor Gallivan during the Spring '11 term at University of Florida.
 Spring '11
 Gallivan

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