Interval and twin arithmetics (Q1375597)

Let \(f\) be a continuous real valued function defined on an \(n\)-dimensional compact interval \(X_1\times X_2\times \cdots \times X_n\) as a term consisting of constants, variables, unary, binary operations and elementary functions. It is well known that the range \(R\) of \(f\) is enclosed by the so-called interval arithmetic evaluation \(f(X_1, X_2, \dots,X_n)\) of \(f\) which consists in replacing the variables \(x_i\) in the term \(f(x_1, x_2, \dots, x_n)\) by the corresponding intervals \(X_i\) and the operations and elementary functions by their corresponding interval counterparts. Using standard interval arithmetic yields an enclosure of \(R\) which may overestimate \(R\) due to so-called dependencies of the input data. There are interval arithmetics which take into account such dependencies. The author shortly reviews them. In addition, he generalizes the concept of standard intervals by introducing twins \(T\) which are defined as pairs \(T= (X_\ell,X)\). Here, \(X_\ell\) is a compact interval or the empty set, and \(X\) is a compact interval. Twins and a corresponding arithmetic aim to enclose an unknown interval \(I\) by \(X_\ell\), \(X\) such that \(X_\ell \subseteq I\subseteq X\) which is abbreviated by \(I \sqsubseteq T\). Usually \(X_\ell\) is called an inner enclosure of \(I\) while \(X\) is named an outer one. According to the purpose of twins the definition of its arithmetic is guided by the properties \[ \begin{aligned} I\sqsubseteq T & \Rightarrow \diamond I\sqsubseteq\diamond T,\\ I_1 \sqsubseteq T_1\;\&\;I_2 \sqsubseteq T_2 & \Rightarrow I_1 \circ I_2 \sqsubseteq T_1 \circ T_2, \end{aligned} \] where \(I\), \(I_1\), \(I_2\) are intervals, \(T\), \(T_1\), \(T_2\) are twins, and \(\diamond\) and \(\circ\) are any unary and any binary operation, respectively. In order to exploit some additional informations on the type of monotonicity of \(f\) directed twins \((T, \alpha) \) are introduced where \(\alpha \in \{+,-\}\). The component \(\alpha\) is related to the monotone behaviour of \(f\), and the corresponding arithmetic is adapted to it. Details can be found in the paper. Examples illustrate the application of twin arithmetic and its directed extension.