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# hw2 - terms of the vector b y t − ˆ y a t = t 2 − a 1...

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ECE 580, Optimization by Vector Space Methods Assignment # 2 Issued: January 30 Due: February 6, 2008 Reading Assignment: Luenberger , Section 10.2, and begin Chapter 3. Problems: 7 Let’s return to Problem # 2 of Assignment 1. Let X = L p [0 , 1], and define three elements of this Banach space, denoted x 1 ,x 2 , and y , with x 1 ( t ) = 1 , x 2 ( t ) = t, y ( t ) = t 2 , t [0 , 1] . For a R 2 we denote ˆ y a = a 1 x 1 + a 2 x 2 X , which is interpreted as an approxima- tion to y . Compute the best approximation: The vector a * R 2 that achieves the minimum, min a bardbl y ˆ y a bardbl . (i) Solve for p = 2 (the Hilbert space case). (ii) Solve for p = . For this it may be easier to consider the representation in
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Unformatted text preview: terms of the vector b , y ( t ) − ˆ y a ( t ) = t 2 − a 1 − a 2 t = ( t − b 2 ) 2 − b 1 What values of b 1 , b 2 will minimize the L ∞ norm? (iii) Consider the general approximation problem with y ( t ) = t n , x i ( t ) = t i-1 for i = 1 , . . . , n and t ∈ [0 , 1]. ±or what values of p could you obtain an expression for a * ? You do not have to compute anything, just explain your reasoning. 8 Luenberger Prob. 10.3 9 Luenberger Prob. 10.4...
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