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3.1 Consider the Newton's method with constant stepsize applied to minimize the function f(x) = ||x||3, where || . || is the Euclidean norm and x ER". Answer the following questions: (a) Identify the range of stepsize for which convergence is obtained, and show it includes the unit stepsize. (3) The following formula may be useful: (A+CBCT)-1 = -A-C(B-H+CTA-'C)-'CTA =A 1 1 (b) Show that for any stepsize within the range in (a), the method converges linearly to x* = 0. [2] 3.2 Starting with the initial solution x = [1, 2]" and H I, use the Davidon-Fletcher-Powell (DFP) method with exact line search to locate a minimizer of the quadratic function 5 3 f(x) = *** +01:32 +33 +381 +212. 3.1 Consider the Newton's method with constant stepsize applied to minimize the function f(x) = ||x||3, where || . || is the Euclidean norm and x ER". Answer the following questions: (a) Identify the range of stepsize for which convergence is obtained, and show it includes the unit stepsize. (3) The following formula may be useful: (A+CBCT)-1 = -A-C(B-H+CTA-'C)-'CTA =A 1 1 (b) Show that for any stepsize within the range in (a), the method converges linearly to x* = 0. [2] 3.2 Starting with the initial solution x = [1, 2]" and H I, use the Davidon-Fletcher-Powell (DFP) method with exact line search to locate a minimizer of the quadratic function 5 3 f(x) = *** +01:32 +33 +381 +212

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