Interval arithmetic: From principles to implementation
TL;DR: It is shown that the IEEE standard's specification of operations involving the signed infinities, signed zeros, and the exact/inexact flag are such as to make a correct and optimal implementation more efficient.
read more
Abstract: We start with a mathematical definition of a real interval as a closed, connected set of reals. Interval arithmetic operations (addition, subtraction, multiplication, and division) are likewise defined mathematically and we provide algorithms for computing these operations assuming exact real arithmetic. Next, we define interval arithmetic operations on intervals with IEEE 754 floating point endpoints to be sound and optimal approximations of the real interval operations and we show that the IEEE standard's specification of operations involving the signed infinities, signed zeros, and the exact/inexact flag are such as to make a correct and optimal implementation more efficient. From the resulting theorems, we derive data that are sufficiently detailed to convert directly to a program for efficiently implementing the interval operations. Finally, we extend these results to the case of general intervals, which are defined as connected sets of reals that are not necessarily closed.
read more
Chat with Paper
AI Agents for this Paper
Find similar papers on Google Scholar, PubMed and Arxiv
Write a critical review of this paper
Analyze citations of this paper to find unaddressed research gaps
Citations
•Book
Handbook of Constraint Programming
Francesca Rossi,Peter van Beek,Toby Walsh +2 more
- 01 Jan 2006
TL;DR: Researchers from other fields should find in this handbook an effective way to learn about constraint programming and to possibly use some of the constraint programming concepts and techniques in their work, thus providing a means for a fruitful cross-fertilization among different research areas.
Efficient Subwindow Search: A Branch and Bound Framework for Object Localization
TL;DR: A simple yet powerful branch and bound scheme that allows efficient maximization of a large class of quality functions over all possible subimages and converges to a globally optimal solution typically in linear or even sublinear time, in contrast to the quadratic scaling of exhaustive or sliding window search.
Safety verification of hybrid systems by constraint propagation-based abstraction refinement
Stefan Ratschan,Zhikun She +1 more
TL;DR: This paper starts from a classical method that uses interval arithmetic to check whether trajectories can move over the boundaries in a rectangular grid and improves it by developing an additional refinement step that employs interval-constraint propagation to add information to the abstraction without introducing new grid elements.
243
A Review of Trimming in Isogeometric Analysis: Challenges, Data Exchange and Simulation Aspects
TL;DR: The treatment of trimmed geometries in the context of design, data exchange, and computational simulation is reviewed.
Safety Verification of Hybrid Systems by Constraint Propagation Based Abstraction Refinement
Stefan Ratschan,Zhikun She,Manfred Morari,Lothar Thiele +3 more
- 01 Jan 2005
TL;DR: In this article, an interval-constraint propagation is used to add information to the abstraction without introducing new grid elements and the resulting method allows switching conditions, initial states, and unsafe states to be described by complex constraints, instead of sets that correspond to grid elements.
170
References
•Book
Introduction to Interval Computation
Götz Alefeld,Jürgen Herzberger,J. Rokne +2 more
- 02 Jan 2016
TL;DR: In this paper, the authors present an ALGOLGOL-based approach for the inclusion of complex Zeros of polynomials of a function of one real variable in a system of linear systems of equations.
2.1K
Global optimization using interval analysis
TL;DR: This Second Edition of Global Optimization Using Interval Analysis expands and improves various aspects of its forerunner and features significant new discussions, such as those on the use of consistency methods to enhance algorithm performance.
1.8K
Large-Signal Behavior of Junction Transistors
J. J. Ebers,John L. Moll +1 more
- 01 Dec 1954
TL;DR: In this paper, a generalized two-terminal-pair theory of junction transistors is presented which is applicable, on a dc basis, in all regions of operation, and the transition of the transistor switch from open to closed, or vice versa, is discussed, including the effects of minority carrier storage.
603
Related Papers (5)
Ramon E. Moore
- 01 Jan 1987
Arnold Neumaier
- 01 Jan 1990
Luc Jaulin,Michel Kieffer,Olivier Didrit,Eric Walter +3 more
- 30 Aug 2001
[...]
Lotfi A. Zadeh
- 01 Aug 1996
Götz Alefeld,Jürgen Herzberger,J. Rokne +2 more
- 02 Jan 2016