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# solhw9 - Solutions for Homework 9 Foundations of...

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Solutions for Homework 9 Foundations of Computational Math 1 Fall 2010 Problem 9.1 Let f ( x ) : R R be given by f ( x ) = x 4 5 x 2 + 4 and consider applying Newton’s method for optimization. Here Newton’s method refers to the basic form where the step size is 1 and nothing is done to alter the Hessian to guarantee positive definiteness. Note that f ( x ) is a scalar function of a scalar argument and has the form -5 -4 -3 -2 -1 0 1 2 3 4 5 -8 -6 -4 -2 0 2 4 6 8 (i) What are the values of x that are local minimizers or local maximizers of f ( x ). Justify your answers. (ii) Find the value β > 0 such that f ( x ) has negative curvature for β < x < β , and positive curvature outside the interval, i.e., for x < β or x > β . (iii) What happens to the Newton step at x = β ? (iv) Determine μ ( x ) : R R such that the step of Newton’s method applied to f ( x ) can be written as x k +1 = μ ( x k ) x k . (v) Find the value of α R such that β > α > 0 and Newton’s method cycles and does not converge when x 0 = α or x 0 = α . That is, α = μ ( α ) α and α = μ ( α ) α . (vi) Show that if α < x < α then | μ ( x ) | < 1 1

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(vii) Show that if α < x 0 < α is the initial point for Newton’s method then there is a constant 0 < γ < 1 (possibly dependent on x 0 but independent of k ) such that | x k +1 | < γ | x k | and therefore x k 0.
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solhw9 - Solutions for Homework 9 Foundations of...

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