A Neglected Theorem - Numerical Analysis via Variable Interval Computing

Abstract

A neglected theorem in numerical computing is proved. In science labs, the computing of a scientific formula $y_0 = E(x_0)$ is often approximated by computing an ‘adequately close’ finite decimal f, namely, E(f). In essence, numerical computing is approximate computing: A finite decimal, say $f = 0.333$, can be rounded from any reals in [0.3325, 0.333), called rounding interval; so a computation of f is an approximate computing for any reals in this rounding interval. A real number, based on Cantor’s infinite decimal representation, can be regarded as a convergent sequence of rounding intervals. The computing of such sequences is variable interval computing – hope to be a useful approach to numerical analysis.