# How to find *all* roots of arbitrarily high degree polynomials (in particular, characteristic polynomials)?

In eigenvalue characteristic equation computations of large matrices, it is often necessary to find all the the roots of this polynomial. I understand that there is no general explicit solution for irrational roots of polynomials of degree higher than 5. We might thus use numerical approximations, such as Newton's method, to find a root for the polynomial equation.

My question is, for very high degree polynomials, eg. characteristic equation of degree 100, how can we make systematic/algorithmic computation to find all the 100 roots to this polynomial? Is there a consistent method that will always find all the roots to this eigenvalue characteristic equation?

Thank you!

• This is not an answer to your question about root-finding in general, but it's important to know that for numerical calculation of eigenvalues one should not use the characteristic polynomial. There are other methods that are much better. Commented Nov 7, 2021 at 8:02
• About polynomial roots in general, there are some starting pointers here.
– dxiv
Commented Nov 7, 2021 at 8:06
• Tom Copeland has written an answer here: mathoverflow.net/a/368105/25104 See also square root of 2 in the OEIS: oeis.org/A002193 Commented Nov 7, 2021 at 9:12