Inclusion isotony

The inclusion isotonicity represents one of the fundamental properties of the interval calculation is. In this case, in general, the range of values of a function to be restricted can be achieved whereby a more accurate result. However, it should be noted that different representations of a function that are equivalent to one another can lead to different value range enclosures. The range of values ​​is always included, but never underestimated. The aim is to get as close as possible to the desired result or to limit the range of values ​​as far as possible.

Inclusion isotony is an important property of intervals when the interval is widened. It is mainly used in the field of interval analysis and numerical mathematics .

definition

Specifically, it says that any function is contained in its interval extension for all , that is, it includes all values ​​of . In mathematical terms, this means: ${\displaystyle f(x)}$ ${\displaystyle F(X)}$ ${\displaystyle x\in X}$ ${\displaystyle F(X)}$ ${\displaystyle f(x)}$

${\displaystyle x\in X\Rightarrow f(x)\in F(X)\wedge X\subseteq Y\Rightarrow F(X)\subseteq F(Y)}$

Any extension of the interval that has this property is called an isotonic of inclusion. The operations of interval arithmetic involved here then satisfy:

${\displaystyle X_{1}\subseteq Y_{1},X_{2}\subseteq Y_{2}\Rightarrow X_{1}\odot X_{2}\subseteq Y_{1}\odot Y_{2}}$

Individual evidence

1. ↑ Dobner H.-J., Nonnenmacher A., ​​Mlynski DA Automatic differentiation and interval arithmetic for liquid crystal simulation . Electrical Engineering 80 (1997). Springer-Verlag 1997, p. 179
2. ^ Moore RE, Kearfott RB, Cloud MJ Introduction to Interval Analysis . SIAM, USA, 2009.
3. ↑ ( Memento of the original from January 29, 2016 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice.