20105ee103_1_hw3_sols

20105ee103_1_hw3_sols - L. Vandenberghe 10/14/10 EE103...

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Unformatted text preview: L. Vandenberghe 10/14/10 EE103 Homework 3 solutions 1. Exercise 4.4. We get five equations c + c 1 t + c 2 t 2 1 + d 1 t + d 2 t 2 = y for five different values of t and y . These equations are nonlinear in the unknowns c , c 1 , c 2 , d 1 , d 2 . We can convert them to linear equations by multiplying by the denominator: c + c 1 t + c 2 t 2 = y (1 + d 1 t + d 2 t 2 ) . This gives five linear equations c + c 1 t 1 + c 2 t 2 1 y 1 d 1 t 1 + y 1 d 2 t 2 1 = y 1 c + c 1 t 2 + c 2 t 2 2 y 2 d 1 t 2 + y 2 d 2 t 2 2 = y 2 c + c 1 t 3 + c 2 t 2 3 y 3 d 1 t 3 + y 3 d 2 t 2 3 = y 3 c + c 1 t 4 + c 2 t 2 4 y 4 d 1 t 4 + y 4 d 2 t 2 4 = y 4 c + c 1 t 5 + c 2 t 2 5 y 5 d 1 t 5 + y 5 d 2 t 2 5 = y 5 where t 1 = 1 , t 2 = 2 , t 3 = 3 , t 4 = 4 , t 5 = 5 and y 1 = 2 . 3 , y 2 = 4 . 8 , y 3 = 8 . 9 , y 4 = 16 . 9 , y 5 = 41 . . In matrix-vector notation 1 t 1 t 2 1 y 1 t 1 y 1 t 2 1 1 t 2 t 2 2 y 2 t 2 y 2 t 2 2 1 t 3 t 2 3 y 3 t 3 y 3 t 2 3 1 t 4 t 2 4 y 4 t 4 y 4 t 2 4 1 t 5 t 2 5 y 5 t 5 y 5 t 2 5 c c 1 c 2 d 1 d 2 = y 1 y 2 y 3 y 4 y 5 . It can be solved by the following Matlab code: >> t = [1; 2; 3; 4; 5]; >> y = [2.3; 4.8; 8.9; 16.9; 41.0]; >> x = [ ones(5,1), t, t.^2,-y.*t,-y.*(t.^2) ] \ y x = 1 0.6150 1.1972-0.2411-0.3468 0.0299 2. Exercise 5.2. (a) An example is shown below, but there are many other correct answers. In general, A will be lower triangular if the nodes are numbered so that each node has a lower number than its parent (the next node on the path leading to the root node), an arc connecting two non-root nodes is given the lowest of the labels of the two nodes, an arc connecting a non-root node to the root node is given the label of the non-root node. R 5 2 1 4 3 6 1 3 5 6 2 4 The node-arc incidence matrix for the numbering in the example is A = 1 0 0 1 0 0 1 0 0 1 1 0 1 1 1 1 0 0 1 ....
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This note was uploaded on 02/03/2011 for the course EE 103 taught by Professor Vandenberghe,lieven during the Spring '08 term at UCLA.

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20105ee103_1_hw3_sols - L. Vandenberghe 10/14/10 EE103...

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