A Midpoint Phenomenon

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Abstract
Finite-precision interval arithmetic evaluation of a function ƒ of n variables at an n -dimensional rectangle T which is the Cartesian product of intervals yields an interval which is denoted by F ( T ). Correspondingly, finite-precision real arithmetic evaluation of ƒ at the midpoint m ( T ) of T yields a number which is denoted by f ( m ( T )) Ε F ( T ). Often, f ( m ( T )) is surprisingly close to m ( F ( T )). The purpose of this note is to provide some insight into this phenomenon by examining the case of infinite precision and rational functions. It is shown that if the gradient of ƒ is nonzero at a fixed point t Ε T , then as the maximum edge length w ( T ) of T approaches zero, [ m ( F ( T )) - ƒ( m ( T ))]/ w ( F ( T )) = O ( w ( T )), where F ( T ) and ƒ( m ( T )) denote the infinite-precision results corresponding to F ( T ) and f ( m ( T )), respectively. More precise results are derived when ƒ is one of +, -, ×, or /.

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