TITLE: Fixed Point Iteration is Not Converging to the Desired Root
QUESTION [7 upvotes]: What is the fixed point for the following function?
$f(x) = 2sin(\pi x) + x = 0$ between $[1,2]$
I expressed $f(x)$ as $x=g(x)$ such that $g(x)$ can be $- 2sin(\pi x)$ or $\frac{1}{\pi}sin^{-1}(\frac{-x}{2})$. However, in both cases, the method is not coverging to a root lying between in the interval $[1, 2]$. What to do? Please help me.

REPLY [3 votes]: This is the visual interpretation of @Robert Israel's answer.
Captured from my old exercise book ($\epsilon$ used is $0.0005$).

Basically, the problem lies within how we truncate the sine-shaped graph when we draw arcsine graph $y = \frac{1}{\pi} sin^{-1}(-\pi/2)$, while in fact we are solving an equation involving sine, that is, $2sin(\pi x)+x=0 $. The $y_1$ and $y_2$ in the picture above is the "extended version" of the arcsine graph.