## Abstract

In interval arithmetics, special care has been brought to the definition of interval extension functions that compute narrow interval images. In particular, when a function f is monotonic w.r.t. a variable in a given domain, it is well-known that the monotonicity-based interval extension of f computes a sharper image than the natural interval extension does. This paper presents a so-called "occurrence grouping" interval extension [f] _{og} of a function f. When f is not monotonic w.r.t. a variable x in a given domain, we try to transform f into a new function f ^{og} that is monotonic w.r.t. two subsets x _{a} and x _{b} of the occurrences of x: f ^{og} is increasing w.r.t. x _{a} and decreasing w.r.t. x _{b} . [f] _{og} is the interval extension by monotonicity of f ^{og} and produces a sharper interval image than the natural extension does. For finding a good occurrence grouping, we propose a linear program and an algorithm that minimize a Taylor-based over-estimate of the image diameter of [f] _{og} . Experiments show the benefits of this new interval extension for solving systems of nonlinear equations.

Original language | English |
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Pages (from-to) | 173-188 |

Number of pages | 16 |

Journal | Computing |

Volume | 94 |

Issue number | 2-4 |

DOIs | |

State | Published - Mar 2012 |

Externally published | Yes |

## Keywords

- Interval extension
- Intervals
- Monotonicity
- Occurrence grouping