Let $N(s)$ and $D(s)$ be two polynomials in $s \in \mathbb C$ of degrees $m$ and $n$, respectively, with $m
For a given $k$, let $R_k$ be the set of roots of $P(s)$: $R_k = \{ r_i \in \mathbb C | P(r_i)=0, i = 1,\ldots,n \}$
Assume for some $k^\prime$, I have already computed $R_{k^\prime}$.
Is there a numerically efficient method to compute the set of roots $R_{k^{\prime\prime}}$ for an adjacent value $k^{\prime\prime}$ of $k^\prime$?
Or, less specifically, given a perturbation $\delta = |k^{\prime\prime} - k^{\prime}|$, is there a way to compute the maximum change in the location of the roots if it is possible to pair the roots in $R_{k^{\prime\prime}}$ with those in $R_{k^{\prime}}$?