One of the main problems of interval computations is to compute the range of a given function f over given intervals. For a linear function, we can feasibly estimate its range, but for quadratic (and for more complex) functions, the problem of computing the exact range is NP-hard. So, if we limit ourselves to feasible algorithms, we have to compute enclosures instead of the actual ranges. It is known that asymptotically the smallest possible excess width of these enclosures is O(Δ2), where Δ is the largest half-width of the input intervals. This asymptotics is attained for the Mean Value method, one of the most widely used methods for estimating the range.
The excess width is caused by quadratic (and higher order) terms in the function f. It is therefore desirable to come up with an estimation method for which the excess width decreases when the maximum of this quadratic term decreases. In the Mean Value method, while the excess width is bounded by O(Δ2), we cannot guarantee that the excess width decreases with the size of the quadratic term. In this paper, we show that, by using Grothendieck inequality, we can create a modification of the Mean Value method in which the quadratic term is estimated accurately modulo a small multiplicative constant -- i.e., in which the excess width is guaranteed to be bounded by 3.6 times the size of the quadratic term.