final05

# final05 - L Vandenberghe EE103 Final Exam Solutions Problem...

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Unformatted text preview: L. Vandenberghe 6/16/05 EE103 Final Exam Solutions Problem 1 (20 points). The figure shows the function f ( x ) = (1 + x ) − 1 1 + ( x − 1) evaluated in IEEE double precision arithmetic in the interval [10 − 16 , 10 − 15 ], using the Matlab command ((1+x)-1)/(1+(x-1)) to evaluate f ( x ). x 1 2 10 − 16 10 − 15 We notice that the computed function is piecewise-constant, instead of a constant 1. 1. What are the endpoints of the intervals on which the computed values are constant? 2. What are the computed values on each interval? Carefully explain your answers. Solution. The first figure shows the rounded values of the numerator and denominator. The second plot shows the result of the division. x (1 + x ) − 1 1 + ( x − 1) ǫ M 2 ǫ M 3 ǫ M 4 ǫ M 5 ǫ M 6 ǫ M 7 ǫ M 8 ǫ M 9 ǫ M 2 ǫ M 4 ǫ M 6 ǫ M 8 ǫ M 10 ǫ M ǫ M 2 ǫ M 3 ǫ M 4 ǫ M 5 ǫ M 6 ǫ M 7 ǫ M 8 ǫ M 9 ǫ M 2 1 2 / 3 4 / 3 1 4 / 5 6 / 5 1 6 / 7 8 / 7 1 8 / 9 2 Problem 2 (20 points). The figure shows m = 20 points ( t i , y i ) as circles. These points are well approximated by a function of the form f ( t ) = αt β e γt . (An example is shown in dashed line.) 1 2 3 4 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 t f ( t ) Explain how you would compute values of the parameters α , β , γ such that αt β i e γt i ≈ y i , i = 1 , . . . , m, (1) using the following two methods. 1. The Gauss-Newton method applied to the nonlinear least-squares problem minimize m summationdisplay i =1 parenleftBig αt β i e γt i − y i parenrightBig 2 with variables α , β , γ . Your description should include a clear statement of the linear least-squares problems you solve at each iteration. You do not have to include a line search. 2. Solving a single linear least-squares problem, obtained by selecting a suitable error function for (1) and/or making a change of variables. Clearly state the least-squares problem, the relation between its variables and the parameters α , β , γ , and the error function you choose to measure the quality of fit in (1). Solution. 1. This problem is a nonlinear least-squares problem minimize g ( x ) = ∑ m i =1 r i ( x ) 2 with variables x = ( α, β, γ ) and r i ( α, β, γ ) = αt β i e γt i − y i . Starting at some initial estimate x (for example, computed with the method of part 2), we repeatedly solve linear least-squares problems minimize bardbl A ( k ) x − b ( k ) bardbl 2 3 with b ( k ) = Ax ( k ) − r ( k ) and A ( k ) = t β 1 e γt 1 α (log t 1 ) t...
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final05 - L Vandenberghe EE103 Final Exam Solutions Problem...

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