Effect on existing roots of polynomial when adding small higher-order term

Question

How do existing roots of a polynomial change when adding higher-order term with a small coefficient? Given a sufficiently small coefficient of the new higher-order term, the existing roots shouldn't change much but is there any theorem around this?

Answer 1

No, a very small coefficient perturbation can affect the roots dramatically, meaning lets say a root changes tens orders of magnitude more than coefficient change. This is actually (for many people) one of the most surprising results in mathematics. Most of the time, this will not be the case, however this still poses a difficult problem in numerical analysis.

Following example is from a Matlab blog post called "Wilkinson’s Polynomials"

Observe this polynomial

$$ w(x) = \prod_{i=1}^{20} (x - i) = (x-1)(x-2) \ldots (x-20)$$

It has roots $1, 2, \dots, 20$.

Now observe this family of polynomials

$$ w(x) - \alpha x^{19}, \alpha = +- 2^{-k}, k=23, 24, \dots, 36 $$

The behavior of roots is illustrated in this picture (red is for negative coefficient perturbation, black for positive)

The root movement is huge, given that coefficient change is less than $2^{-23}$.

A summary and further info on this topic can be found here.