solhw8 - Solutions for Homework 8 Foundations of...

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Math 1 Fall 2010 Problem 8.1 Textbook, page 330, Problem 6 Solution: We are given G : R 3 R 3 to define the iteration used to solve a system of nonlinear equations: x ( k +1) = G ( x ( k ) x ( k +1) 1 x ( k +1) 2 x ( k +1) 3 = - 1 81 cos x ( k ) 1 + 1 9 ( x ( k ) 2 ) 2 + 1 3 sin x ( k ) 3 1 3 sin x ( k ) 1 + 1 3 cos x ( k ) 3 - 1 9 cos x ( k ) 1 + 1 3 x ( k ) 2 + 1 6 sin x ( k ) 3 We know there is a fixed point at x * = 0 1 3 0 To show that there is a nontrivial region around x * in which convergence occurs for any x (0) we check the sufficient condition that the spectral radius of the Jacobian of G is strictly less than one at the fixed point. We have J G ( x ) = 1 81 sin x 1 2 9 x 2 1 3 cos x 3 1 3 cos x 1 0 - 1 3 sin x 3 1 9 sin x 1 1 3 1 6 cos x 3 J G ( x * ) = 0 2 27 1 3
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solhw8 - Solutions for Homework 8 Foundations of...

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