I often read about a function being 'highly non linear'. In my understanding, there is "linear" and "non-linear", so what is this 'highly' about? Is there a formal difference from non linear? How is it defined?
I don't think there's a formal definition. It's my impression that it simply means that not only is it non-linear, but attempting to model it with a linear approximation won't yield reasonable results and may even cause instability in the fitting method. Someone may also use it to simply mean that small input changes can result in counter-intuitively large changes in output.
In a formal sense, I believe one could say that the second derivative differs substantially from zero. If 0 were a "reasonable" approximation to the second derivative over the domain of interest, it is close to linear, but if it's not, the nonlinear effects become very important to capture.
I've rarely heard terms like this apply to relatively simple polynomials, often in practical use it seems to apply to divergent dynamical systems (chaos-theory sort of things), or very non-smooth functions (where much higher-order derivatives are nonzero).
The important aspect missing from the other excellent answers is the domain. E.g., $f(x)=x^2$ is
- highly non-linear on $[-10;10]$ but
- not on either half of the domain (i.e., on both $[-10;0]$ and $[0;10]$ one can possibly use a linear approximation of $f$ without an immediate disaster).
Another example is $f(x)=x^3-x$ which is
- highly non-linear on $[-1;1]$ but
- not on the larger domain $[-10;;10]$
As other mentioned, I don't think there's a formal definition. I would define it as a function which can not be approximated linearly in the typical range of disturbances to the argument. For instance, you have $y=f(x)$, and $\sigma^2=var[x]$. Then if the approximation $f(x+\sigma)\approx f(x)+f'(x)\sigma$ breaks down, then it's highly non-linear. For instance, $f(x)=exp(x^2)$ would be highly non-linear for any $x$ around zero, because its Taylor series are $1+x^2+x^4/2+O(x^5)$.
Informally ... "highly non linear" means "even a blind man can see its not a straight line!" ;) Personally I take it as a danger sign, that it will somehow "blow up in your face" when used with real world examples.
The Tower of Hanoi could be called an example of highly non linear ... the legend being when the monks finish a 64 disk stack, the world will end. If you count total time spent in training, feeding, housing, and motivating everyone to support a thankless boring pointless multi-generational task, I would expect the total cost in man hours to really blow out!
As professional mathematician I can confirm that "highly nonlinear" is not a mathematical precisely defined term. :)
And none of "highly anything" I can think of.
Nonlinear is precise and opposite of linear (obviously).
But linear occurs in two different meanings:
- $f(x) = ax+b$ is called linear function
- only function $f(x) = ax$ (with no constant term $b$) is called linear
To emphasize the difference and presence of the constant term, the first function $(ax + b)$ is also called affine