The error in interval arithmetic

Abstract

It is widely appreciated that interval arithmetic can yield overly pessimistic results because it ignores correlations among intermediate values. One approach toward understanding this phenomenon begins by disregarding "second-order effects" and rounding errors. An early result of this type is Hansen's proof of Moore's conjecture that the centered form "converges quadratically", i.e., produces exact results under the simplifying assumptions.

Here we give a simple derivation of expressions for the first-order effects of error in interval arithmetic. These results can serve as the basis for proofs of a number of known results.