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MATH 565X Supplementary Homework Spring 2013 7. If q1 , q2 , . . . qn are a set of vectors in Rn which are orthogonal in the usual sense, and if these are used as search directions to minimize φ(x) = 21 xT Ax − bT x, will the exact solution be obtained after n steps? (Assume A is symmetric and positive definite as usual.) 8. Let f (x) = x21 + x22 + x42 , which has the unique minimizer (0, 0). Carry out two iterates of the BFGS update procedure, with exact line search, using initial guess x0 = (1, 1) and using the identity as the starting approximation to the Hessian.