hw5_solS09

hw5_solS09 - EE 103, Spring '09, Prof SEJ: HW 5 Sol....

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EE 103, Spring ’09, Prof SEJ: HW 5 Sol. Page 1 of 9 Applied Numerical Computing Instructor: Prof. S. E. Jacobsen HW 5 Solution Students: Distributed HW solutions are a component of the course and should be fully understood. Prob 1: (a) 12 () ( 2 ,4 ) f xx x  . Therefore, at 00 (0.5,0.5) , ( 1, 2) TT xd   (there is no need to normalize, since that merely changes the value of , the stepsize. The line-search problem is 22 min ( ) min(0.5 ) 2(0.5 2 ) g   We then have 1 5 18 1 '( ) 2(0.5 ) 8(0.5 2 ) 0 5/18 0.5 1 0.22222 0.5 2 0.05555 ( ) 0.05555 g x fx       The First Two Directions, Using Steepest Descent
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EE 103, Spring ’09, Prof SEJ: HW 5 Sol. Page 2 of 9 (b) The Newton Direction for this Specific Example There are two ways to understand why the Newton direction, for this example, points directly to the optimal solution. The first way is to recognize that f is quadratic; the Newton direction points directly to the optimal solution of the quadratic approximation which, in this case, is the function f (there is a sufficient condition, called “convexity” , which guarantees that setting the gradient of the quadratic approximation equal to zero produces the minimum of the quadratic approximation). The second way is to directly compute the Newton direction. Since 12 () ( 2 ,4 ) 20 () 04 f f xx x Hx    we have that the Newton direction is 01 0 0 1/2 0 1 0 1/4 2 T Nf dH xf x 
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This note was uploaded on 06/01/2010 for the course EE EE 103 taught by Professor Jacobsen during the Spring '09 term at UCLA.

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hw5_solS09 - EE 103, Spring '09, Prof SEJ: HW 5 Sol....

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