Background : The other day I managed to find a numerical approach to estimate the position of poles and zeroes of a Meromorphic function. The issue is that sometimes it does not converge perfectly it would be useful to find a way to refine the solution found.
So given a set of found poles $\{\hat p_k\}$ and zeroes $\{\hat h_k\}$ of an estimated meromorphic function $$\hat g(z) = \frac{\displaystyle \prod_k(z-\hat h_k)}{\displaystyle \prod_k (z-\hat p_k)}$$ approximating the actual underlying meromorphic function $$g(z) = \frac{\displaystyle \prod_k(z-h_k)}{\displaystyle \prod_k (z- p_k)}$$
Let us assume that we therefore would like to update $\hat h_k, \hat p_k$ to closer match $h_k, p_k$
Own work :
An Idea I have been having is to construct the quotient function:
$$Q(z) = \frac{g(z)}{\hat g(z)} = \frac{\displaystyle \prod_k(z- h_k)(z- \hat p_k)}{\displaystyle \prod_k (z- p_k)(z- \hat h_k)}$$
If perfect match, then this quotient function shall be identically $= 1$ everywhere in the complex plane.
Maybe I can measure in local neighborhoods how much it "differs" from $1$ in some sense?
Example of meromorphic function with 3 poles, 3 holes and the quotient function after making an approximation.