TITLE: Finding all roots of a polynomial
QUESTION [19 upvotes]: Is it possible, for an arbitrary polynomial in one variable with integer coefficients, to determine the roots of the polynomial in the Complex Field to arbitrary accuracy? When I was looking into this, I found some papers on homotopy continuation that seem to solve this problem (for the Real solutions at least), is that correct? Or are there restrictions on whether homotopy continuation will work? Does the solution region need to be bounded?

REPLY [5 votes]: I give a course on matrix analysis, and I like to mention the following thing. It is equivalent to finding the roots of polynomials, or to finding the eigenvalues of matrices, because to a polynomial $P$, you can associate its companion matrix $C_P$, and to a square matrix $M$, you can associate its characteristic polynomial $\chi_M$.
Now, because the calculation of $\chi_M$ is costly, and direct solving methods are ill-conditionned, it is usually a bad idea to use polynomial solving in order to compute eigenvalues.
Instead, the efficient idea is to apply the QR method to the companion matrix $C_P$ when you want to calculate the roots of $P$. This way has several advantages:

the method does converge. Proven.
it is fast. Actually, each iteration produces a Hessenberg matrix (only one non-zero sub-diagonal) and this ensures that the cost of an iteration is only $O(n^2)$, instead of $O(n^3)$.
it is stable. Because an iteration acts as a unitary conjugation. Therefore the round-off errors only add up, but are not amplified.

I agree that because of the round-off errors, this produces on a computer approximations of limited (but outstanding) precision. What you can do is to stop after a few hundred iterations, and then apply Newton's method or something similar, starting from one of the approximate eigenevalues.