Is there always a numerically stable algorithm for each problem?

Question

Is there always a numerically stable algorithm for each problem?

Maybe there is a problem(s) for which there is no numerically stable algorithm?

Anyone can say something about it?

Answer 1

The answer is no, there is not. Any uncomputable number will do ($\pi$ is an example of a computable number - we can calculate it with arbitrary precision). Take Chaitin's constant, for example, which is a halting probability. It turns out we can't compute it with arbitrary precision like we can with $e$, $\pi$, $\sqrt{2}$ and so on (and this is actually proven).