fixed point iteration of a single variable with unknown constants

Okay so given this simple looking fixed point iteration, i.e., $$x_{n+1}=g(x_n)$$

$$x_{n+1} = g(x) = -b - \frac{c}{x_n}$$

The idea is to find a region in the space of $$(b,c) \in \mathbb{R^2}$$ that will converge for all good starting points $$x_0$$ (ones that are relatively close to $$x$$) with the error being reduced by more than $$1/2$$ on each iteration, in other words the iteration is $$\mathcal{O}(1/2^n)$$.

I understand the brute force way of computing all the possible combinations, which my computer is working on, checking the error and then plotting the points that fit the criteria on a 2-D plot. It seems like there should be a way to figure this out more concisely, I envisage there is a way to take partials of $$g(x)$$ or optimize a potential of some sort or other and figure out the region analytically.

Step 1: If the iteration does converge, it would converge to some equilibrium point of $$x_{n+1} = -b - \dfrac{c}{x_{n}}$$, which is a root of $$x = -b - \dfrac{c}{x}$$.
That is, possible convergence points would be $$\dfrac{-b \pm \sqrt{b^2 - 4c}}{2}$$.
Step2: fix one of them, say $$X = \dfrac{-b + \sqrt{b^2 - 4c}}{2}$$, and consider $$\epsilon_{n} = x_{n} - X$$. Compute $$\epsilon_{n+1}$$. Obviously it depends on $$x_{n}$$, and you want $$|\epsilon_{n+1}| < |\dfrac{\epsilon_{n}}{{2}}|$$. This is a condition for $$x$$s you are looking for.
• I very well might misread your question. Are you interested in $b,c$ pairs for which an iteration converges fast for certain $x$, or, given $b,c$ you are interested in an initial $x$? – user58697 Jul 7 at 0:30