Given the fixed point iteration function $g(x)=e^{-x}$, I want to find the order of convergence of $g$. (I know yet that the iteration method converges to the fixed point for every starting point).
Now, the definition of order of convergence is the following:
An iterative method is said to converge at the fixed point with order $a\geq 1$ if $\displaystyle{\lim\limits_{i \rightarrow \infty}\frac{|x_{i+1}-\bar{x}|}{|x_i -\bar{x}|^a}}=\alpha \in \mathbb{R}_+$ (some textbooks require that if $a=1$ then $\alpha \in (0,1]$.
My guess is that the iterative method is linearly convergent, meaning $a=1$, my guess is because the derivative of $g(x)$ is never zero. Also, if I try to picture graphic of the method, then it seems to converge quite rapidly, more than I would expect from a linear order convergence (which, I think, is something pretty slow, or at least slower than quadratic convergence or more, $a\geq 2$).
Best Answer
The asymptotic convergence rate is based on the derivative of $g$ at the fixed point. You don't know the fixed point exactly, but you can give a simple interval bound for it using the intermediate value theorem. This bound will tell you that the derivative is nonzero at the fixed point, which implies linear convergence. Specifically $\alpha$ is the absolute value of the derivative at the fixed point.
(By the way, I'd advise you to take a look at weaker versions of the definition of the order of convergence. That one, although it is intuitive, is almost never actually applicable.)