Let's say I have a multivariate function
$$
f:D \to D, D \subset \mathbb R ^n, D \text{ compact},
$$
for which there is no closed form.

That is the only way to evaluate the function is to do it numerically.

For example, the price of some exotic insurance contracts: the rule governing the payout are simple, but have no closed formula.

If I suspect my function $f$ is a contraction, is it possible to prove it numerically?

My first though will be to evaluate numerically $f'$ at 1000 points of $D$.

Is this sufficient?

Probably not, have there been some work on such numerical tests?

My second though will be to assume $f'$ follows a normal distribution, then perform an hypothesis test.

Is this reasonable?

My use case is that, I search for a fixed point of $f$ using dynamic programming.

But the state space becomes too large.

If I can proove $f$ is a contraction, then I avoid the curse of dimensionality.