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Given iterative method: $x_{n+1}=0.7\sin x_n +5 = \phi(x_n)$ for finding solution for $x=0.7\sin x +5$, I want to estimate $|e_6|=|x_6-r|$ as good as possible, with $x_0=5$, where $r$ is exact solution. This method obviously converges, because $\phi$ is contraction, so $r=\phi(r)$ is a fixed point. So, with mean value theorem:

$|e_{n+1}|=|x_{n+1}-r|=|\phi(x_n)-\phi(r)|\le \max_{c\in\mathbb{R}}|\phi'(c)|\cdot |x_n-r|$

and we have:

$|e_n|\le 0.7^n \cdot |e_0|$

But I don't know how can I estimate $|e_0|$ without a computer? I suppose there is some simple way to finish it and with clever observation $|e_0|\le 0.7$. Can anybody help?

xan
  • 2,053

1 Answers1

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You have already shown that the iteration converges to a fixed point $r$ with $r = 0.7 \sin r + 5$, i.e. you know $r - 5 = 0.7 \sin r$. Using $|\sin r| \le 1$ here is the required simple estimate for $e_0$: $$|e_0| = |x_0 - r|= |5-r|= |0.7 \sin r|= 0.7|\sin r| \le 0.7$$

Note: Because $r \approx 4.3463686514876$, the true value is $|e_0| \approx 0.6536313485123$, thus the estimate 0.7 is not so bad.

gammatester
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