MATH 4420 Midterm: Math4420Midterm2017
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In class portion: (a) given the linear function f (x) = bt x, where b rn, compute the gradient f , using the directional derivative calculation. Then perform the rst iteration, that is, given a starting point x0, compute x1: consider the least-square approximation problem min x1,x2xj. Following the steps in exercise 3, nd the minimizer x and give a reason for why the steepest descent method converges. Take home portion (due wed, march 22, before class: (a) find (if any), all "critical" points (where f (x) = 0) for the function f (x) = 5x2. Solve analytically, i. e. do not use an iterative method. (b) classify each point as a local min, local max or saddle. (c) apply the newton method with x(0) = (cid:20)1. 0(cid:21), to obtain x(k), for k = 1 . Does it converge: suppose in newton"s method at iteration k, the hessian 2f (x(k)) is not invertible.
