Interval arithmetic refinement (for nonlinear systems)

[schillic]

In A brief introduction to interval analysis, Section 7.4, there is an example about the difference of an image of an interval X under function f

$$f(X) = {f(x) | x \in X}$$

and the function extension F to intervals, which is defined as f where all occurrences of x are substituted by X.

Example: f(x) = x * x with X = [-1, 1]
Then f(X) = [0, 1], but F(X) = X * X = [-1, 1].

The article mentions that f(X) \subseteq F(X) holds.

Interval extension (using F) can be refined by partitioning the respective intervals into smaller subintervals and performing the computations for each of them (and taking the interval hull in the end).

By using finer partitioning, this procedure converges to the more precise interval image f(X). See A brief introduction to interval analysis, page 190 f.

Note that interval arithmetic is exact if each interval variable occurs only once in the equations. This means that for linear systems, our interval arithmetic is already most precise.