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Unformatted text preview: AMATH 341 / CM 271 / CS 371 Assignment 5 Due : Wednesday March 10, 2010 Instructor: K. D. Papoulia 1. In this question you will prove the convergence order for Newtons method. Let x be the root of f ( x ) = 0 and x k be the k th approximation of this root by Newtons method. Assume that f satisfies the conditions of the convergence theorem for Newtons method. (a) Show that the error, e k = x k x , satisfies e k +1 = e k f ( x + e k ) f ( x + e k ) . (1) (b) Since f C 2 , Taylors Theorem shows that (1) can be written e k +1 = e k f ( x ) e k + 1 2 f ( k ) e 2 k f ( x ) + f ( k ) e k , (2) for some k , k between x and x + e k . Show that (2) can be written | e k +1 | = c k | e k | 2 , where c k = f ( k ) 1 2 f ( k ) f ( x ) + f ( k ) e k (c) Show that lim k c k = f ( x ) 2 f ( x ) . Hint: use the fact that lim k x k = x . 2. Consider the degree-4 polynomial f ( x ) = ( x 2 . 2)( x 2 . 3)( x 2 . 4)( x 2 . 5) con- sidered in an earlier problem set. For each of the four roots of f , find a fixed point iteration in which g ( x ) has the form x cf ( x ) for some scalar c (may be different for each root) that converges to that root. Experimentally determine an interval of starting values for each of the four choices of roots that ensures convergence. 3. (a) Suppose g is a continuously differentiable function and g ( x ) = x . Suppose that | g ( x ) | > 1. Show that fixed point iteration based on g will never converge to x , assuming that no iterate x k happens to exactly be equal to x . (b) Consider the function given by f ( x ) = exp(3 x ) 9 x 2 2 + 1 9 . (3) 1 Plot the functions f ( x ) and g ( x ) = x f ( x ) using ezplot and provide the resulting plot, which has a root at x = 0. Using fixed point iteration on g ( x ) with a su- ciently small tolerance, create a table of the results you get with the initial condition x = 1. Why are we unable to achieve convergence? Suggest a different g ( x ) and x that would give us convergence with fixed point iteration. Limit consideration to g ( x ) of the form x cf ( x ) for a scalar c . 4. Consider the problem of finding a three-dimensional 5-sided box (i.e., one side is open) with volume 1 and with minimal surface area, where we count as surface area only the 5 sides that are present. Write a Matlab program to solve this optimization problem using Newtons method for multivariate optimization. Terminate when the norm of the gradient is less than 10 15 . Terminate also if the method seems to be diverging, e.g., one variable gets larger than 10 or smaller than 1 / 10. Try some different starting points to see if the same solution is always obtained. The variables of the problem are x = ( l ; w ), the length and width of the box. The third linear dimension is 1 / ( lw ) due to the volume constraint. Try posing the objective function) due to the volume constraint....
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This note was uploaded on 06/21/2010 for the course CS 5212 taught by Professor Papoulia,katerina during the Spring '10 term at Waterloo.
- Spring '10