Relative condition number — ill-conditioned versus well-conditioned

Question

Currently, I am learning about the relative condition number ($\kappa$), and how one considers a function well-conditioned or ill-conditioned. From my understanding, a large $\kappa$ indicates that a function is ill-conditioned, while a small $\kappa$ indicates that a function is well-conditioned. However, is there a range where $\kappa$ can be labeled as large (ill-conditioned) or small (well-conditioned)?

For example, the function $x \mapsto x^3$ would give a value of $\kappa = 3$ and $x \mapsto x^\frac13$ would give a value of $\kappa = \frac13$. However, these do not give me the idea if $\kappa$ is large or small and, hence, I am unable to tell if they are well-conditioned or ill-conditioned.

Answer 1

Well the post is 4 years old but I think it deserves an answer. Probably the author has found the solution.

If we mean this condition number, then a large $k$ value means that the function is ill-conditioned. This means that if a small change is made in its elements (as it happens when we put data in the computer to make calculations) the solution of the system might differ a lot from the real solution.

It seems that if the change in $x$ is $\bar{x}=x(1+\varepsilon)$ then $k(x)=\frac{|f(x)-f(x+\varepsilon x)|}{|\varepsilon f(x)}$ as we can see here. We then expect a relative error $\frac{|f(x)-f(x+\varepsilon x)|}{|f(x)|}\approx k(x)|\varepsilon|$.

For matrices I was taught that a condition number generated by a subordinate norm and of order larger than $10^5$ is bad (and the matrix is ill-conditioned). A condition number of order (approximately) $1-10^2$ is good (so the matrix is well-conditioned). The range $10^3-10^4$ is something between the two cases (well and ill).

I think that the things are similar for functions. When $|f(x)-f(x+\varepsilon x)|$ is of the same order as $\varepsilon$ is then the condition number is good. I am not familiar with function's condition number but I think that the examples indicate two functions of good condition.