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MATH 565X Supplementary Homework Spring 2013 9. Carry out two iterations of the Nelder-Mead simplex method for the problem of minimizing f (x) = cos2 x in R1 , with initial simplex vertices ∆0 = {1, 2}. 10. Carry out two iterations of the golden section search method for the problem of minimizing f (x) = cos2 x in R1 , with initial bracketing triplet {1, 1.5, 2}. 11. An important problem in signals processing amounts to finding parameters c1 , . . . cn , λ1 , . . . λn such that n X ci e−λi t ≈ f (t) i=1 for a given ’signal’ function f (t). Here the ≈ symbol means that the sense in which the finite exponential sum fits the given f (t) has yet to be specified – we can’t expect equality to hold for every t, no matter what choice of parameters is made, but we might be able to get a good approximation over some suitable range of t values. One approach is to set this up as a specific nonlinear least squares problem as follows. Let x = (x1 , . . . x2n ) = (c1 , . . . cn , λ1 , . . . λn ) be the vector of parameters to be determined, let sj = f (tj ) be given ’samples’ of f for j = 1, . . . 2n, and set rj (x) = n X −λi tj ci e − sj = i=1 n X xi e−xi+n tj − sj i=1 We then obtain the parameters as the solution of the nonlinear least square problem min ||r(x)||2 x∈R2n It is known that there is at most one solution of r(x) = 0. a) Find a general expression for J(x). b) Let n = 2, t = (0, .3, .6, .9), s = (2.7, 1.48, .819, .458) and use starting guess x0 = (1, 1, 1, 2) to do one of the following. 1. Carry out two iterations of the Gauss-Newton method by paper/calculator and report the results, including value of the residual at each step, or 2. Program the Gauss-Newton method to carry out these iterations, and report the result and number of iterations required to reduce the residual to below 10−4 . Many other techniques have been developed for this problem.