Tutorial_wk2

# Tutorial_wk2 - p p n< p n-p So p n is a strictly...

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Tutorial week 2 Let g ( x ) = 2 1 + x - 2, p 0 [ a, b ] = [0 , 3] and p n +1 = g ( p n ). Show that { p n } is a strictly decreasing sequence and bounded by 0 deduce that the iteration converges. Use the Convergence of a Monotonic Sequence Theorem. We know g (0) = 0 = p , i.e. p is a ﬁxed point and g 0 (0) = 1. I claim starting with p 0 [0 , 3] the iteration will converge. Notice that on (0,3], 1 2 g 0 ( x ) < 1 . I claim that p n p = 0 for all n , which is obviously true when n = 0 since p 0 p = 0, now p n +1 = g ( p n ) = 2 1 + p n - 2 2 - 2 = 0 . I show that { p n } is a strictly decreasing sequence: p n +1 - p = g ( p n ) - g ( p ) = g 0 ( c )( p n - p ) for c
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Unformatted text preview: ( p, p n ) < p n-p, So { p n } is a strictly decreasing sequence which is bounded below, hence using the monotonic sequence theorem it must converge. • In the homework we showed that the equation x 3 + x + 1 = 0 has one and only one real root, p , and that it lies between-1 and 0. Use the following methods to ﬁnd p correct to three decimal places. – The Regula Falsi method – A ﬁxed point method MATH2051 Tutorial 1 2010-10-14...
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