sols_wk4 - Homework 4 due Thursday November 4 before 13:30...

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Unformatted text preview: Homework 4 due Thursday November 4 before 13:30 in corre-sponding group folder on door of oce CM110 Prove that the sequence generated by the iterative formula p n +1 = 1-sin p n converges to the real root of the equation sin x + x-1 = 0 for every real starting value p . Use this formula, starting with p = 0 . 5, and Aitkens acceleration method to nd that root correct to three decimal places. The iterative method is p n +1 = g ( p n ) where g ( x ) = 1-sin x . Hence a xed point, p = g ( p ) satises p = 1-sin p f ( p ) := sin p + p-1 = 0 (0.1) i.e. p is a root of the equation f ( p ) = 0. For any starting value p (- , ), sin p [-1 , 1] = p 1 = g ( p ) = 1-sin p [0 , 2] so we need only restrict our attention to a xed point in the interval [0 , 2]. We show directly that equation (0.1) has a unique solution p = 1-sin p [0 , 2]. Let f ( x ) = sin x + x-1, then f ( x ) = cos x + 1 > 0, for all x [0 , 2]. Moreover we note that f (0) =-1 < 0 and f (2) = 1 + sin 2 > 0. We deduce that f ( ) is a strictly increasing continuous function which changes sign, hence from the Intermediate Value theorem there exists a unique p [0 , 2] such that f ( p ) = 0. In order to prove that the sequence { p n } converges to p , our rst thought is to check whether the convergence theorem can be applied on [0 , 2]. Since g 00 ( x ) =...
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sols_wk4 - Homework 4 due Thursday November 4 before 13:30...

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