20095ee103_1_hw7_sols

20095ee103_1_hw7_sols - L Vandenberghe EE103 Homework 7...

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Unformatted text preview: L. Vandenberghe 11/19/09 EE103 Homework 7 solutions 1. Exercise 12.1(b). The figure shows g ( x ) and ( E − x ) /R for E = 15, R = 3. 1 2 3 4 1 2 3 4 5 6 7 8 There are three solutions that we can find by iterating x := x − f ( x ) f ′ ( x ) = x − x 3 − 6 x 2 + 10 x − ( E − x ) /R 3 x 2 − 12 x + 10 + 1 /R . Starting at 0 . 5, we obtain the first solution 0.8239. Starting at 2, we obtain another solution 1.7948. Starting at 3 . 5, we obtain the third solution 3.3813. 2. Exercise 12.5. We first plot the two ellipses. The easiest way to do this is to transform them to unit circles centered at the origin. For the first ellipse we use a change of variables bracketleftBigg y 1 y 2 bracketrightBigg = bracketleftBigg x 1 − 1 √ 2( x 2 − 1) bracketrightBigg . This transforms the ellipse in a unit circle described by y 2 1 + y 2 2 = 1. For the second ellipse we use the change of variables bracketleftBigg y 1 y 2 bracketrightBigg = radicalBig 3 / 2 1 / √ 2 parenleftBiggbracketleftBigg 1 1 1 − 1 bracketrightBiggbracketleftBigg x 1 x 2 bracketrightBigg − bracketleftBigg 2 1 bracketrightBiggparenrightBigg . The Matlab code and figure are as follows. 1 t = linspace(0, 2*pi, 1000); c1 = [1; 1]; ellipse1= [c1(1) + cos(t); c1(2) + sqrt(0.5)*sin(t)]; plot(ellipse1(1,:), ellipse1(2,:)); hold on c2 = [2; 1]; ellipse2= [1 1; 1 -1]\ ... [c2(1) + sqrt(2/3)*cos(t); c2(2) + sqrt(2)*sin(t)]; plot(ellipse2(1,:), ellipse2(2,:),’--’); 0.5 1 1.5 2 2.5-0.5 0.5 1 1.5 2 x 1 x 2 The Newton iteration is x := x − Df ( x ) − 1 f ( x ) with f ( x ) = bracketleftBigg ( x 1 − 1) 2 + 2( x 2 − 1) 2 − 1 3( x 1 + x 2 − 2) 2 + ( x 1 − x 2 − 1) 2 − 2 bracketrightBigg Df ( x ) = bracketleftBigg 2( x 1 − 1) 4( x 2 − 1) 6( x 1 + x 2 − 2) + 2( x 1 − x 2 − 1) 6( x 1 + x 2 − 2) − 2( x...
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This note was uploaded on 05/10/2010 for the course EE EE103 taught by Professor Jacobson during the Spring '09 term at UCLA.

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20095ee103_1_hw7_sols - L Vandenberghe EE103 Homework 7...

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