[ogogmad]

I've just looked up "companion matrix" to double-check what you're saying.

> Computing the matrix exponential is equivalent to finding the eigenvalues of a companion matrix

No it isn't. This is a prank, right?

Just to be clear (assuming you're not talking nonsense on purpose): You're saying that you take a monic polynomial p(x), form its companion matrix C, and then use the exponential function to find the eigenvalues of C. But the eigenvalues of exp(C) are just exp of the eigenvalues of C, which are exp of the roots of p(x). If you're suggesting power iteration (or improvements to that, like the QR algorithm) then what does matrix exponential have to do with that? I think you're confusing the process of raising a matrix to a power (written as M^n) with computing the exponential of a matrix (written as exp(M) or e^M) -- these are not the same operation.