Can a numerically unstable algorithm be useful?
What a numerically unstable algorithm can be useful for? (except of finding solutions on short intervals)
Suppose I have an unstable numerical algorithm (and no other known algorithms) for solution of an important problem. What utility can I have from knowledge about such an algorithm?
Sorry for an imprecise question, I am only in beginning of understanding this, but it is very important for me. So I dare to ask here an imprecise soft-question.
I do not try to define numerical stability precisely in this question, because the question is anyway philosophical.
If you know something about the classes of problems for which the instability manifests itself or you have some measure of the magnitude of the instability, then you can use that information to decide. For example, Gaussian elimination with partial pivoting is the default algorithm for solving linear systems even though it is known to be numerically unstable on certain problems. This is not really an issue, however, because these types of problems rarely show up in applications for whatever reason.