Numerically find Zeros in Complex functions

[Raghnar]

I have this non-trivial complex function based on.

$$f(E)=\sum_{2,\omega}\frac{h(1,2,\omega)}{E-E_{2}-\hbar\omega+i\delta}$$

So is a sum of this denominator that rises many poles and zeros.
I want to find all the zeros (computationally, analitically, I don't mind) a in a fairly efficient way (that must be done like thousands times, so I can't make a night for one iteration)

If you have any ideas or suggestions I'm all ears

 

[Raghnar]

No-one?
You can give me also some references or generic advices, you don't know or don't have time to give the answer...

 

[HallsofIvy]

I have this non-trivial complex function based on.

$$f(E)=\sum_{2,\omega}\frac{h(1,2,\omega)}{E-E_{2}-\hbar\omega+i\delta}$$

So is a sum of this denominator that rises many poles and zeros.
I want to find all the zeros (computationally, analitically, I don't mind) a in a fairly efficient way (that must be done like thousands times, so I can't make a night for one iteration)

If you have any ideas or suggestions I'm all ears

Obviously the zeros of this function depend strongly on the zeros of $h(1,2,\omega)$ and you have given no information about that function.

 

[Raghnar]

Obviously the zeros of this function depend strongly on the zeros of $h(1,2,\omega)$ and you have given no information about that function.

$h(1,2,\omega)$ is not a function but are matrix elements of the discreet parameters 1,2 (particles) and omega (phonons).
Really I think is not the issue here, there is always ten (usually many more) of nonzero $h(1,2,\omega)$ in which the problem remains open. I cannot hope that h is trivially zero almost everywhere and comes to save the day! ;)

I'm sorry for haven't been clear