I am looking for approximation methods for polynomial equations of this kind:
$$x^n -k\cdot x + (k-1) = 0$$
where $n\geq 5$ (typically $n=5, 10, 25, 50, 100$),
$k\in\mathbb{R^+}$,
and $1.05<x<1.15$
Any help is greatly appreciated. You can also just give me some keywords on methods and I will be able to do some further research myself (I hope); that's fine :)
With the conditions as stated, often times there will be no root in the range you give. Since the derivative is $nx^{n-1}-k$, the only critical point occurs at $$b=\left(\frac{k}{n}\right)^{1/(n-1)}.$$ As $1$ is always a root and there is only one other root to the other side of $b$, whenever $k\leq n$ the only other root will be less than $1$.
For instance, to get a root in the range you want the root of the derivative should be between $1$ and $1.15$, imposing the constraint $$ \left(\frac{k}{n}\right)^{1/(n-1)}\leq1.15; $$ that is, you require $$ \bbox[5px, border: 2px solid red]{k\leq n\,1.15^{n-1}.} $$ Note that this is a necessary condition, but not sufficient, for a root to be in your range.
As for finding said root, Newton's method works very fast, with the recursion $$ \bbox[5pt, border: 2pt solid green]{x_{m+1}=x_m-\frac{x_m^n-kx_n+k-1}{nx_m^{n-1}-k}.} $$ Newton's method has the advantage that most often you can start really far from the root and still get there.
Caveat: in this case, with only two roots, the choice of the starting point of the method is easy. In general, though, it could be really complicated, as this example shows.
