20095ee103_1_hw3_sols

20095ee103_1_hw3_sols - L. Vandenberghe 10/15/09 EE103...

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Unformatted text preview: L. Vandenberghe 10/15/09 EE103 Homework 3 solutions 1. Exercise 4.5. We can write f ( u 1 , u 2 ) as f ( u 1 , u 2 ) = u 2 1 p 11 + 2 u 1 u 2 p 12 + u 2 2 p 22 + u 1 q 1 + u 2 q 2 + r. For given u 1 and u 2 , this is a linear function of the unknowns p 11 , p 12 , p 22 , q 1 , q 2 , r . For example, f (0 , 1) = 6 means p 22 + q 2 + r = 6 . We therefore obtain the following set of equations: 0 0 1 1 1 1 0 0 1 0 1 1 2 1 1 1 1 1 2 1 1 1 1 1 4 4 1 2 1 4 4 1 2 1 1 p 11 p 12 p 22 q 1 q 2 r = 6 6 3 7 2 6 . >> A = [0 1 1 1; 1 1 1; 1 2 1 1 1 1; 1 2 1 -1 -1 1; 1 4 4 1 2 1; 4 4 1 2 1 1] >> b = [6; 6; 3; 7; 2; 6 ]; >> x = A\b x = 3.0000-2.0000 1.0000-2.0000 0.0000 5.0000 We can check the answer as follows. 1 >> P = [x(1), x(2); x(2), x(3)]; >> q = [x(4); x(5)]; >> r = x(6); >> u = [0;1]; u*P*u + q*u + r ans = 6 >> u = [1;0]; u*P*u + q*u + r ans = 6.0000 >> u = [1;1]; u*P*u + q*u + r ans = 3.0000 >> u = [-1;-1]; u*P*u + q*u + r ans = 7.0000 >> u = [1;2]; u*P*u + q*u + r ans = 2 >> u = [2;1]; u*P*u + q*u + r ans = 6 2. Exercise 4.7. (a) We have integraltext 1 t k dt = 1 / ( k + 1), so the conditions on w are n summationdisplay i =1 w i t k i = 1 k + 1 , k = 0 , 1 , . . . , n 1 . This is a set of linear equations with variables w i : 1 1 1 t 1 t 2 t n t 2 1 t 2 2 t 2 n . . . . . . . . . t n- 1 1 t n- 1 2 t n- 1 n w 1 w 2 . . . w n = 1 1 / 2 1 / 3 . . . 1 /n . (b) This follows from linearity of integration and of the integration rule. If the integration rule is exact for all powers of t up to t n- 1 , then applying it to the function f ( t ) = a + a 1 t + + a n- 1 t n- 1 , gives integraldisplay 1 f ( t ) dt = integraldisplay 1 ( a + a 1 t + + a n- 1 t n- 1 ) dt = a + a 1 integraldisplay 1 t dt + a 2 integraldisplay 1 t 2 dt + + a n- 1 integraldisplay 1 t n- 1 dt 2 = a parenleftBigg summationdisplay i w i parenrightBigg + a 1 parenleftBigg summationdisplay i w i t i parenrightBigg + a 2 parenleftBigg summationdisplay i w i t 2 i parenrightBigg + + a n- 1 parenleftBigg summationdisplay i w i t n- 1 i parenrightBigg = n summationdisplay i =1 w i ( a + a 1 t i + a 2 t 2 i +...
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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_hw3_sols - L. Vandenberghe 10/15/09 EE103...

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