Algorithm for min‐range multiplication of affine forms

Affine arithmetic produces guaranteed enclosures for computed quantities, taking into account any uncertainties in the input data as well as round‐off errors. Elementary operations on affine forms are redefined so they result in affine forms. Affine‐linear operations result straightforwardly in affine forms. Non‐linear operators, such as multiplication, must be approximated by affine forms. Choosing the appropriate approximation is a big challenge. The reason is that different approximations may be more accurate for specific purposes. This paper presents an efficient method for computing the minimum range (min‐range) affine approximation of the product of arbitrary affine forms that do not contain zero properly. Numerical experiments are carried out to demonstrate the essential features of the proposed approach, especially its usefulness for bounding ranges of functions for global optimisation and for finding roots of functions.