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hw4 - Homework 4 Foundations of Computational Math 1 Fall...

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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 x 2 = 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 = ± v and x = v + γe 1 . Assuming that x = 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 reflector-based orthogonal factorization, we have used elementary transformations, T i , that can be characterized as rank-1 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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