Rate of convergence $x_{n+1}=2\sin(x_n)$

Question

$f(x)=\sin(x)-\frac{1}{2}x$ , for $x>0$

We are trying to evaluate the root of function using the following fixed-point iterative method:

$x_{n+1}=2\sin(x_n)$

What is the rate of convergence of the method?

Any direction would be appreciated

Answer 1

You should easily see that $x=0$ is an instable fixed point. Then also that $g([0,\pi])\subset[0,2]\subset[0,\pi]$ so that there is an attracting fixed point inside that interval.

Now you need to find better bounds for that point so that you can then compute a contraction factor as the maximum of the absolute value of the derivative. For instance, it is easy to see that $g([\frac\pi3,\frac{2\pi}3])=[\sqrt3,2]\subset [\frac\pi3,\frac{2\pi}3]$ and $2\cos([\frac\pi3,\frac{2\pi}3])\subset[-1,1]$, which is not sufficient but still closer.

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Numerically you get $$ g([1.85, 1.95])\subset[1.85, 1.95] ~~\text{ with }~~ g'([1.85, 1.95])\subset[-0.75, -0.55] $$ so that $\frac34$ is an upper bound for the convergence rate, $\frac12$ a lower bound.

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One might get analytical results using $$ g(\tfrac\pi2+u)=2\cos(u)\in[2-u^2, 2-u^2+\tfrac1{12}u^4]\subset [2-u^2, 2-\tfrac{47}{48}u^2] ~~\text{ for }~~ u\in[0,\tfrac12]. $$