Suppose that we have a bounded polynomial defined on $[0,1]$. I think because it is just polynomial, root finding algorithms would easily and without any instability find all its roots. Am I right?
And what is the fastest and most stable root-finding algorithm for my problem?


Edit: I introduce my polynomial as:
$$ f(x) = B\_n(x) - x$$


$B\_n(x)$ is a bernstein polynomial with positive and strictly increasing coefficients $a\_k$:


$$B\_n(x) = \sum\_{k=1}^n \binom n ka\_kx^k (1-x)^{n-k}$$


$$a\_0=0 < a\_1 < a\_2 < ... < a\_n=1$$



Are root finding algorithms stable for bounded polynomials?
Finding **real** roots is not going to be stable, even if you assume the polynomial to be monic and have bounds on where the interesting stuff happens. As an example, consider $(x^2 + \varepsilon)(x^2 - 2x + 1 - \varepsilon)$. If $\varepsilon < 0$, the only real root is at $0$, if $\varepsilon > 0$ the only real root is at $1$.


If you want a guarantee that the algorithm will converge to a root with bounds on the convergence speed, you'd need to either promise the existence of roots with odd multiplicty, or be willing to accept complex roots as solutions.


