efficiently determining if a polynomial has a root in the interval [0,T]

Question

I have polynomials of nontrivial degree (4+) and need to robustly and efficiently determine whether or not they have a root in the interval [0,T]. The precise location or number

Answer 1

It's not that efficient, but is quite reliable. You can construct the polynomial's Companion Matrix (A sparse matrix whose eigenvalues are the polynomial's roots).

There are efficient eigenvalue algorithms that can find eigenvalues in a given interval. One of them is the inverse iteration (Can find eigenvalues closest to some input value. Just give the middle point of the interval as the above value).