I am writing a piece of software and I'm trying to avoid root finding polynomials for efficiency purposes. I have two polynomials with complex coefficients, where the roots of both polynomials are known. In fact, the internal representation I use just stores the complex roots and a complex scaling factor, not the polynomial coefficients themselves, although it's trivial to compute these. Essentially, my goal is to sum the two polynomials and express the result in the same internal representation which I use for the addends. The mathematical challenge here is then to find the roots of the sum (and the scaling factor, which is trivial). As I've mentioned, I would very much like to find a way to do this more efficiently than just finding the coefficients of the sum and computing the roots from there - although I will resort to this method as a last resort if what I'm asking for isn't possible.
EDIT: I am putting a bounty on this question, so I want to clarify my standard for a good answer. A good answer will either:
- Explain or overview an algorithm which accomplishes my task, preferably with references to relevant papers with more information.
- Give an explanation for why the task is infeasible, possibly with references or suggestions for another angle of attack, but this is not necessary.