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Unformatted text preview: 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 Newtons method for optimization. Here Newtons 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 1 2 3 4 5-8-6-4-2 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 > 0 such that f ( x ) has negative curvature for < x < , and positive curvature outside the interval, i.e., for x < or x > . (iii) What happens to the Newton step at x = ? (iv) Determine ( x ) : R R such that the step of Newtons method applied to f ( x ) can be written as x k +1 = ( x k ) x k . (v) Find the value of R such that > > 0 and Newtons method cycles and does not converge when x = or x = . That is, = ( ) and = ( ) . (vi) Show that if < x < then | ( x ) | < 1 1 (vii) Show that if < x < is the initial point for Newtons method then there is a constant 0 < < 1 (possibly dependent on x but independent of...
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This note was uploaded on 07/25/2011 for the course MAD 5403 taught by Professor Gallivan during the Spring '11 term at University of Florida.
- Spring '11