🚀 go-pugleaf

Suppose I have an arbitrary function H(s), which  can be assumed as 
being of the form F(s)/G(s), where F and G are polynomials that very 
likely have complex roots. However, the function is not available in 
explicit form, but it can be numerically evaluated at any point with
calculator's accuracy (12 digits).

I would like to approximate the roots of F(s) and G(s) so as to express
H(s) explicitly in terms of s.
I suppose Newton's method would work satisfactorily,
unfortunately I don't have a clue on how to approximate the derivative
of  a complex-valued function (note that s is complex as well).

Are there any commonly known algorithms for that?

Thanks in advance for any help. (post or e-mail either is fine)

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