I'm working on an algorithm in which I need to approximate the behavior of a polynomial by computing its roots to some $\epsilon$ precision. The problem can be defined as follows:
Let $f(x) = x^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$ be a univariate monic polynomial whose roots we cannot compute exactly. Let $A = \{a_{n-1},...,a_0\}$.
Let $g(x) = (x-\hat{x_1})(x-\hat{x_2})...(x-\hat{x_n})$ be an approximation polynomial for $f$. Assume $\{\hat{x_1}, \hat{x_2}, ..., \hat{x_n}\}$ values are ordered from smallest to largest. Let $\{x_1,x_2,...,x_n\}$ be the set of zeroes of $f$ similarly ordered.
We are guaranteed that each $x_i$ value is approximated by $\hat{x_i}$ to within $\epsilon$. Therefore, for all $i$, $|x_i - \hat{x_i}| \leq \epsilon$.
For a closed interval $[s,t]$, we want to find an error bound $h(A,s,t,\epsilon)$ such that $$\forall \hat{x}\in [s,t].\exists x. |\hat{x}-x| \leq h(A,s,t,\epsilon) \land (\operatorname{sign}(g(\hat{x})) = \operatorname{sign}(f(x)))$$ where $\operatorname{sign}(y) = (-1,0,1)$ for $(y < 0, y = 0, y > 0)$ respectively.
I don't need an exact bound for $h(A,s,t,\epsilon)$, but any sort of bound in which $\lim_{\epsilon\rightarrow 0} h(A,s,t,\epsilon) = 0$ and $h(A,s,t,\epsilon)$ monotonically decreases with $\epsilon$ (for small enough $\epsilon$ similar to the gap theorem) should be good enough for my purposes.
I realize that the formulation looks similar to the Stone-Weierstrass theorem, but in my problem, I'm not trying to bound the distance between $f(x)$ and $g(x)$, but rather the distance between two points $x$ and $\hat{x}$ such that $f(x)$ and $g(\hat{x})$ have the same sign. Sorry if my notation or terminology is off as my background is not in mathematics. Thanks