### A Midpoint PhenomenonA Midpoint Phenomenon

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 Author Goldstein, A. J. ♦ Richman, P. L. Source ACM Digital Library Content type Text Publisher Association for Computing Machinery (ACM) File Format PDF Copyright Year ©1973 Language English
 Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science Abstract Finite-precision interval arithmetic evaluation of a function ƒ of $\textit{n}$ variables at an $\textit{n}-dimensional$ rectangle $\textit{T}$ which is the Cartesian product of intervals yields an interval which is denoted by $\textit{F}(\textit{T}).$ Correspondingly, finite-precision real arithmetic evaluation of ƒ at the midpoint $\textit{m}(\textit{T})$ of $\textit{T}$ yields a number which is denoted by $\textit{f}(\textit{m}(\textit{T}))$ &Egr; $\textit{F}(\textit{T}).$ Often, $\textit{f}(\textit{m}(\textit{T}))$ is surprisingly close to $\textit{m}(\textit{F}(\textit{T})).$ 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 $\textit{t}$ &Egr; $\textit{T},$ then as the maximum edge length $\textit{w}(\textit{T})$ of $\textit{T}$ approaches zero, $[\textit{m}(\textit{F}(\textit{T}))$ - $ƒ(\textit{m}(\textit{T}))]/\textit{w}(\textit{F}(\textit{T}))$ = $\textit{O}(\textit{w}(\textit{T})),$ where $\textit{F}(\textit{T})$ and $ƒ(\textit{m}(\textit{T}))$ denote the infinite-precision results corresponding to $\textit{F}(\textit{T})$ and $\textit{f}(\textit{m}(\textit{T})),$ respectively. More precise results are derived when ƒ is one of +, -, ×, or /. ISSN 00045411 Age Range 18 to 22 years ♦ above 22 year Educational Use Research Education Level UG and PG Learning Resource Type Article Publisher Date 1973-04-01 Publisher Place New York e-ISSN 1557735X Journal Journal of the ACM (JACM) Volume Number 20 Issue Number 2 Page Count 4 Starting Page 301 Ending Page 304

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Source: ACM Digital Library