# hw7 - φ 2 ( x ), apply the iteration to ±nd the values of...

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Homework 7 Foundations of Computational Math 1 Fall 2011 The solutions will be posted on Monday, 11/21/10 Problem 7.1 7.1.a Let f ( x ) = x 3 - 3 x + 1. This polynomial has three distinct roots. (i) Consider using the iteration function φ 1 ( x ) = 1 3 ( x 3 + 1) Which, if any, of the three roots can you compute with φ 1 ( x ) and how would you choose x (0) for each computable root? (ii) Consider using the iteration function φ 2 ( x ) = 3 2 x - 1 6 ( x 3 + 1) Which, if any, of the three roots can you compute with φ 2 ( x ) and how would you choose x (0) for each computable root? (iii) For each of the roots you identi±ed as computable using either φ 1 ( x ) or

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Unformatted text preview: φ 2 ( x ), apply the iteration to ±nd the values of the roots. (You need not turn in any code, but using a simple program to do this is recommended.) 7.1.b Let φ ( x ) : [ a, b ] → [ a, b ] be a continuous function. Show that if φ ( x ) is a contraction mapping on [ a, b ] then the sequence { x ( k ) } de±ned by x ( k +1) = φ ( x ( k ) ) is a Cauchy sequence. Problem 7.2 Textbook, p. 283, Problem 2 Problem 7.3 Textbook, p. 283, Problem 5 1 Problem 7.4 Textbook, p. 283, Problem 6 Problem 7.5 Textbook, p. 284, Problem 8 2...
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## This note was uploaded on 11/27/2011 for the course MAD 5403 taught by Professor Gallivan during the Spring '09 term at FSU.

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hw7 - φ 2 ( x ), apply the iteration to ±nd the values of...

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