The Fundamental Theorems of Interval Analysis
M. H. van Emden, B. Moa
TL;DR
This work addresses the lack of a rigorous foundation in interval analysis by precisely defining the function computed by an expression and proving fundamental convergence theorems for interval evaluations. It develops a set-theoretic semantics for expressions, introducing a distribution function $\delta$ to handle variable sharing and showing that the interval extension of an expression can be expressed as a composition involving the operation symbol and the canonical set extensions. The authors prove that canonical interval extensions of Cauchy-continuous operations yield continuous set extensions, ensuring that interval results converge to the true function value as input intervals shrink. The approach provides a rigorous alternative to topology-based treatments and grounds interval arithmetic in established set-theoretic concepts, with broad applicability to interval methods.
Abstract
Expressions are not functions. Confusing the two concepts or failing to define the function that is computed by an expression weakens the rigour of interval arithmetic. We give such a definition and continue with the required re-statements and proofs of the fundamental theorems of interval arithmetic and interval analysis. Revision Feb. 10, 2009: added reference to and acknowledgement of P. Taylor.