solhw4 - Solutions for Homework 4 Foundations of...

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Unformatted text preview: Solutions for Homework 4 Foundations of Computational Math 1 Fall 2010 Problem 4.1 Recall that an elementary reflector has the form Q = I + xx T R n n with k x k 2 6 = 0. 4.1.a . Show that Q is orthogonal if and only if =- 2 x T x or = 0 4.1.b . Given v R n , let = k v k and x = v + e 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 =- e 1 Solution: We have Q T Q = ( I + xx T ) T ( I + xx T ) = ( I + xx T )( I + xx T ) = I + 2 xx T + 2 x ( x T x ) x T = I + (2 + 2 x T x ) xx T Since x is arbitrary we must have (2 + 2 x T x ) = 0 =- 2 x T x Now taking x = v + e 1 , we have x T v = v T v + e T 1 v = 2 + 1 x T x = ( v + e 1 ) T ( v + e 1 ) = v T v + 2 1 + 2 = 2( 2 + 1 ) x T x x T v = 2 Finally, Qv = ( I + xx T ) v = v + ( x T v ) x = v + ( x T v ) v + ( x T v ) e 1 ( x T v ) =- 2 x T v x T x =- 1 Qv =- e 1 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 implies careful application of associativity and distribution when combining matrices and vectors. Suppose we are to evaluate z = T 3 T 2 T 1 v = ( I + x 3 y T 3 )( I + x 2 y T 2 )( I + x 1 y T 1 ) v where v R n and z R n . Show that by using the properties of matrix-matrix multiplication and matrix-vector multiplication, the vector z can be evaluated in O ( n ) computations (a good choice of version for an algorithm) or O ( n 3 ) computations (a very bad choice of version for...
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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.

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solhw4 - Solutions for Homework 4 Foundations of...

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