In numerical analysis, I was asked whether Bairstow's algorithm convergence rate is quadratic.
My initial feeling was that it does, since it is essentially Newton's method for a system of non-linear equations, and Newton's method converges quadratically in one dimension (When f is from R to R).
A quick search on the internet seemed to indicate that indeed, the rate is quadratic as long as the roots are of multiplicity 1.
However, I have real trouble proving this. I was trying to imitate the proof of Newton's method in one dimesnsion found in Kincaid and Cheney's book "Numerical Analysis", but ran into some trouble since there isn't a simple "second derivative" for a vector valued function (unless I want to use tensors, and that's way beyond our course material).
So I'm stuck. How do I show that the rate of convergence is quadratic (or that it's not if the roots aren't simple)?