Numerical Analysis is giving me some trouble. In specific, I'm highly confused by our definition of numerical stability. I'm hoping some of you can help me clear my confusion.
Definition. An algorithm $\hat{f}$ for a problem $f$ is called stable if$$ \| \hat{f}(x) - f(\tilde{x}) \|_{\text{rel}} = \mathcal{O}(\varepsilon_{\text{mach}})$$ for some suitably chosen $\tilde{x}$ with $\|x-\tilde{x} \|_{\text{rel}} = \mathcal{O}(\varepsilon_{\text{mach}})$.
So what do I do in practice to prove stability/instability? It may be my unfamiliarity with Landau notation in numerical purposes. Let me try to have a guess.
Proving Stability: Show that there exists a constant $c > 0$ such that: There exists a constant $C > 0$ such that there exists a $\tilde{x}$ with $\|x-\tilde{x} \|_{\text{rel}} < c \varepsilon_{\text{mach}}$ for each sufficiently small $\varepsilon_{\text{mach}} > 0$ such that$$\| \hat{f}(x) - f(\tilde{x}) \|_{\text{rel}} < C \varepsilon_{\text{mach}}$$ Do I read the Landau symbols correctly?
Proving Instability: Show that such $c > 0$ does not exist.
Is that how you would prove stability in practice? For instance, would you start by defining such constants to prove that $x \mapsto \log(1 \oplus \operatorname{fl}(x))$ is instable?
EDIT: Perhaps this seems more convenient: Take $\tilde{x} = (1+c \varepsilon_{\text{mach}})x$ and show that the difference of functions also scales in $\varepsilon_{\text{mach}}$. As for instability, anytime we do that, the result will not decrease as quickly. I guess that's how one should go about that?