sols_wk3

# sols_wk3 - Homework 3 due Thursday October 28 before 13:30...

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Homework 3 due Thursday October 28 before 13:30 in corre- sponding group folder on door of oﬃce CM110 Consider the sequence { p n } generated by p 0 = 0 , p n +1 = - ( p 3 n + 2) / 6 , n = 0 , 1 , 2 , ... Show that for all n , - 1 3 p n 0 and that the sequence converges to a limit. Give a bound for its rate of convergence. Let p n +1 = g ( p n ) := - ( p 3 n + 2) / 6. We plan to apply the convergence theorem on the interval [ - 1 3 , 0]. Notice that g 0 ( x ) = - x 2 2 0 for x [ - 1 3 , 0] . Hence, we conclude that max x [ - 1 3 , 0] | g 0 ( x ) | ≤ 1 18 < 1 and that as g is monotone decreasing so that g : [ - 1 3 , 0] [ g (0) , g ( - 1 3 )] = [ - 1 3 , - 53 162 ] [ - 1 3 , 0] , i.e. g maps the interval [ - 1 3 , 0] into itself. Hence we can apply the convergence theorem which shows that - 1 3 p n 0 for all n and that the sequence converges to a unique limit, p . As for a bound on its rate of convergence, since the limit p 6 = 0, convergence is linear, since g 0 ( p ) 6 = 0, and so | p n +1 - p | ≤ 1 18 | p n - p | . For the iteration function g ( x ) = 1 + 1 /x + 1 /x 2 , 1. verify that g satisfies the conditions of the contraction mapping theorem on the interval [1 . 75 , 2 . 0]; 2. carry out a few iterations with the starting value p 0 = 1 . 825 and find an upper bound on the magnitude of the error in p
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