Interval methods that are guaranteed to underestimate (and the resulting new justification of Kaucher arithmetic)

摘要

One of the main objectives of interval computations is, given the functionf(x1, ...,xn), andnintervals$$bar x_1 ,...,bar x_n$$, to compute the range$$bar y = f(bar x_1 ,...,bar x_n )$$. Traditional methods of interval arithmetic compute anenclosure$$Y supseteq bar y$$for the desired interval$$bar y$$, an enclosure that is often an overestimation. It is desirable to know how close this enclosure is to the desired range interval.