bv_cvxbook_extra_exercises

When the king saw what she had done he was so

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Unformatted text preview: e form ˙ ˆ F = α1 F1 + · · · + αm Fm , where F1 , . . . , Fm : Rn → Rn are given (basis) functions, and α ∈ Rm is a set of coeﬃcients that we will choose. We will measure the ﬁt using the maximum angle error, J = max i=1,...,N ˆ (q (i) , F (x(i) )) , where (z, w) = cos−1 ((z T w)/ z 2 w 2 ) denotes the angle between nonzero vectors z and w. We are only interested in the case when J is smaller than π/2. (a) Explain how to choose α so as to minimize J using convex optimization. Your method can involve solving multiple convex problems. Be sure to explain how you handle the constraints ˆ F (x(i) ) = 0. 68 (b) Use your method to solve the problem instance with data given in vfield_fit_data.m, with ˆ an aﬃne vector ﬁeld ﬁt, i.e., F (z ) = Az + b. (The matrix A and vector b are the parameters α above.) Give your answer to the nearest degree, as in ‘20◦ < J ⋆ ≤ 21◦ ’. This ﬁle also contains code that plots the vector ﬁeld directions, and also (but commented ˆ ˆ out) the directions of the vector ﬁeld ﬁt, F (x(i) )/ F (x(i) ) 2 . Create this pl...
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