Abstract
Given a nonempty set of functions
where a = x 0 < ... < x n = b are known nodes and w i , i = 0, ..., n, d i , i = 1, ..., n, known compact intervals, the main aim of the present paper is to show that the functions
exist, are in F, and are easily computable. This is achieved essentially by giving simple formulas for computing two vectors \(\tilde l, \tilde u \in {\mathbb{R}^{n + 1}}\) with the properties
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\(\tilde l \leqslant \tilde u\) implies
$$\begin{gathered} \tilde l, \tilde u \in T: = \{ \xi = {({\xi _0},...,{\xi _n})^T} \in {\mathbb{R}^{n + 1}}: \hfill \\ {\xi _i} \in {w_i}, i = 0,...,n, and \hfill \\ {\xi _i} - {\xi _{i - 1}} \in {d_i}({x_i} - {x_{i - 1}}),i = 1,...,n\} \hfill \\ \end{gathered} $$and that \([\tilde l,\tilde u]\) is the interval hull of (the tolerance polyhedron) T;
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\(\tilde l \leqslant \tilde u\) iff T ≠ ø iff F ≠ ø.
\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{f} ,\bar f\) can serve for solving the following problem:
Assume that μ is a monotonically increasing functional on the set of Lipschitz-continuous functions f : [a, b] → ℝ (e.g. μ(f) = ſ b a f (x) dx or μ(f)= min f ([a, b]) or μ (f) = max f ([a, b])), and that the available information about a function g : [a, b] →ℝ is “g ∈ F,” then the problem is to find the best possible interval inclusion of μ (g). Obviously, this inclusion is given by the interval \([\mu (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{f} ),\mu (\bar f)].\) Complete formulas for computing this interval are given for the case μ (f) = ſ b a f(x) dx.
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References
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© 1999 Springer Science+Business Media Dordrecht
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Heindl, G. (1999). A Representation of the Interval Hull of a Tolerance Polyhedron Describing Inclusions of Function Values and Slopes. In: Csendes, T. (eds) Developments in Reliable Computing. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-1247-7_21
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DOI: https://doi.org/10.1007/978-94-017-1247-7_21
Publisher Name: Springer, Dordrecht
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