What's the difference between a well-posed (ill-posed) and well-conditioned (ill-conditioned) problem?
According to Wikipedia:
Even if a problem is well-posed, it may still be ill-conditioned, meaning that a small error in the initial data can result in much larger errors in the answers.
I think this could be explained in a good way by looking at an example.
is well-posed.
However, for some polynomials, 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
$$ w(x) - \alpha x^{19}, \alpha = \pm 2^{-k}, k=23, 24, \dots, 36 $$
The behavior of roots is illustrated in this picture (red is for negative coefficient perturbation, black for positive)
A summary and further info on this topic can be found here.