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sols_wk4

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

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Homework 4 due Thursday November 4 before 13:30 in corre- sponding group folder on door of oﬃce 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 0 . Use this formula, starting with p 0 = 0 . 5, and Aitken’s acceleration method to find 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 fixed point, p = g ( p ) satisfies 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 0 ( -∞ , ), sin p 0 [ - 1 , 1] = p 1 = g ( p 0 ) = 1 - sin p 0 [0 , 2] so we need only restrict our attention to a fixed 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 0 ( 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.

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

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