I am trying to devise a Niave Algorithm to find the roots to Quaternion Polynomials.
I thought I had found a solution in the article Newton Method in the Context of Quaternion Analysis, as a Newton Method that works over the Quaternions is exactly what I want. The issue is that, if I understood the method correctly, you need to know the value of the quaternion before you can derive it... at least thats what it looks like in this similar article: Algorithms for quaternion polynomial root-finding. The issue with that is the purpose of this project is to approximate the quaternion roots, which is pointless if I know their exact value.
Is there an iterative algorithm that will do this? i.e. Find at least one root given a Quaternion Polynomial?
Note: I would like a algorithm that could be implemented on Mathematica, so it needs to be regularly reproduceable. No... I'm not asking for the code or anything, I just mean it needs to be an actual algorithm not just like a guess/check approach . Thanks