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