## Proving Conjectures by Use of Interval Arithmetic (2001)

Venue: | Facius Axel: Perspective on Enclosure Methods |

Citations: | 7 - 0 self |

### BibTeX

@INPROCEEDINGS{Universitit01provingconjectures,

author = {Bergische Universitit and Andreas Frommer and Andreas Frommer},

title = {Proving Conjectures by Use of Interval Arithmetic},

booktitle = {Facius Axel: Perspective on Enclosure Methods},

year = {2001},

pages = {pp.},

publisher = {Springer}

}

### OpenURL

### Abstract

Machine interval arithmetic has become an important tool in computer assisted proofs in analysis. Usually, an interval arithmetic computation is just one of many ingredients in such a proof. The purpose of this contribution is to highlight and to summarize the role of interval arithmetic in some outstanding results obtained in computer assisted analysis. 'Outstanding' is defined through the observation that the importance of a mathematical result is at least to some extent indicated by the fact that it has been formulated as a 'conjecture' prior to its proof.

### Citations

1085 |
The Algebraic Eigenvalue Problem
- Wilkinson
- 1965
(Show Context)
Citation Context ...orks with a different arithmetic. During the past 50 years, error analysis of (floating point) machine algorithms has thus been a topic research subject in numerical analysis. Backward error analysis =-=[14,23,24]-=- has proven very useful in this context. The idea is to interpret the output of the machine algorithm as the result of the theoretical algorithm with a different input and to establish bounds on the d... |

847 |
Accuracy and Stability of Numerical Algorithms
- Higham
- 2002
(Show Context)
Citation Context ...orks with a different arithmetic. During the past 50 years, error analysis of (floating point) machine algorithms has thus been a topic research subject in numerical analysis. Backward error analysis =-=[14,23,24]-=- has proven very useful in this context. The idea is to interpret the output of the machine algorithm as the result of the theoretical algorithm with a different input and to establish bounds on the d... |

450 |
An introduction to interval computations
- Alefeld, Herzberger
- 1983
(Show Context)
Citation Context ... the original input. This is the place where interval arithmetic comes into play. 'Correct' interval arithmetic is the set theoretic extension of the arithmetic on the reals to compact intervals, see =-=[3]-=-. As such it also suffers from the fact that it cannot be exactly matched onto a floating point counter part. But through the judicious use of directed roundings on floating point operations with the ... |

133 |
Rounding Errors in Algebraic Processes
- Wilkinson
- 1963
(Show Context)
Citation Context ...orks with a different arithmetic. During the past 50 years, error analysis of (floating point) machine algorithms has thus been a topic research subject in numerical analysis. Backward error analysis =-=[14,23,24]-=- has proven very useful in this context. The idea is to interpret the output of the machine algorithm as the result of the theoretical algorithm with a different input and to establish bounds on the d... |

107 | A proof of the Kepler conjecture
- Hales
(Show Context)
Citation Context ...mplete, accepted proof has not yet been published as a reviewed Proving Conjectures by Using Interval Arithmetic 5 journal article. But the work of Hales and Ferguson, as explained on Hale's web site =-=[11]-=-, was presented at several conferences and is now about to be accepted as a proof of Kepler's conjecture. Our summary here is based on [1,10,11]. 2.1 The Paper Work The ratio of space filled with ball... |

74 |
L.: Computer Arithmetic in Theory and Practice
- Kulisch, Miranker
- 1981
(Show Context)
Citation Context ...rithmetic counter partswill give a result asb which satisfies ab_Daob. With the rounding modes available in the IEEE Standard 754 , the result asb can in addition be made closest possible to aob, see =-=[16]-=- or [3, Ch. 4]. We again distinguish between the theoretical algorithm and the machine algorithm. Due to the containment property, it is possible to identify crucial relations which, when observed for... |

45 |
Computer–Assisted Proof of the Feigenbaum Conjectures
- Lanford
- 1982
(Show Context)
Citation Context ...re regard them as conjectures as well. We will describe three pieces of work where interval arithmetic was used to turn such conjectures into theorems. The first dates back to 1982, where Lanford III =-=[17]-=- proved what he called the 'Feigenbaum conjectures' on some universal features displayed by infinite sequences of period doubling bifurcations. Among other things, Lanford proves the first seven digit... |

19 | Double bubbles minimize
- Hass, Schlafly
- 2000
(Show Context)
Citation Context ...al volumes. If the area is a and v either volume, then a a -- 243rv 2. The proof of this conjecture was announced by Hass, Hutchings and Schlafiy [12] in 1995. The complete paper recently appeared as =-=[13]-=- and our presentation is based on that paper. Fig. 2. double bubble and torus bubble 3.1 The Paper Work One first has to find categories of surfaces which are known to contain a minimizing one. Classe... |

18 |
Thomas-Fermi model: The second correction
- Schwinger
- 1981
(Show Context)
Citation Context ...chwinger conjecture resides in the fact that it improves by 'one order' the 'Scott conjecture' E(Z) -- -coZ ?/ + Z 2 + O(ZV),-/ The Dirac-Schwinger conjecture appeared in a paper by Schwinger in 1981 =-=[21]-=-; it was proved by Fefferman and Seco in a series of eight papers published in 1994/95. Their computer assisted proof even gives a more quantitative 1 result for the o(Z/S)-term by replacing it by O(Z... |

16 | The double bubble conjecture
- Hass, Hutchings, et al.
- 1995
(Show Context)
Citation Context ...bble is the surface of smallest area enclosing two equal volumes. If the area is a and v either volume, then a a -- 243rv 2. The proof of this conjecture was announced by Hass, Hutchings and Schlafiy =-=[12]-=- in 1995. The complete paper recently appeared as [13] and our presentation is based on that paper. Fig. 2. double bubble and torus bubble 3.1 The Paper Work One first has to find categories of surfac... |

9 |
Efficient numerical validation of solutions of nonlinear systems
- Alefeld, Gienger, et al.
- 1994
(Show Context)
Citation Context ...ition columns (the 'methanol-8-problem' ) does have a zero, but was not able to prove it using analytical tools. A computer assisted proof for this conjecture was given by Alefeld , Gienger and Potra =-=[2]-=- in 1994 using the method just described. Relation b) above is crucial to verifying branch and bound methods in global optimization [15]. Here, the bounding step checks strict inequalities between the... |

9 |
Rigorous Global Search
- Kearfott
(Show Context)
Citation Context ...oof for this conjecture was given by Alefeld , Gienger and Potra [2] in 1994 using the method just described. Relation b) above is crucial to verifying branch and bound methods in global optimization =-=[15]-=-. Here, the bounding step checks strict inequalities between the range of the objective function over a given interval vector and a candidate optimum. 4 Andreas Frommer Machine interval arithmetic has... |

5 |
AWA: software for the computation of guaranteed bounds for solutions of ordinary initial value problems. Institut für Angewandte Mathematik, Universität Karlsruhe
- Lohner
- 1994
(Show Context)
Citation Context ...l operator representing the ode is contracting. To this purpose, machine interval arithmetic computations are used. As opposed to what may now be considered the standard approach (see the program AWA =-=[18]-=-, e.g.), [9] explictly computes (and uses) a bound on the Lipschitz constant of the integral operator. Note also that due to the singularities at 0 and c, one has to take special care of 'end' interva... |

4 | Interval arithmetic in quantum mechanics
- Fefferman, Seco
- 1996
(Show Context)
Citation Context ...s published in 1994/95. Their computer assisted proof even gives a more quantitative 1 result for the o(Z/S)-term by replacing it by O(Z /-) with Zo - 2ss' Our presentation here is based on the paper =-=[9]-=- by Fefferman and Seco. 4.1 The Paper Work The asymptotic extension stated in the conjecture is not at all proved directly. It follows - after quite a bit of mathematics - from a strong inequality to ... |

3 |
Every planar graph is four colorable, part I: discharging
- Appel, Haken
- 1977
(Show Context)
Citation Context ...ter assisted proofs in discrete mathematics have a long and successful history. As the most prominent example in this area, let us just mention the proof of the Four Colour Theorem by Appel and Haken =-=[5,6]-=- When a computer is used to prove a result in analysis, the picture becomes quite a bit more involved. This is due to the fact that the basic quantities in analysis are the real numbers, an infinite c... |

3 |
Every planar graph is four colorable, part II: reducibility
- Appel, Haken
- 1977
(Show Context)
Citation Context ...ter assisted proofs in discrete mathematics have a long and successful history. As the most prominent example in this area, let us just mention the proof of the Four Colour Theorem by Appel and Haken =-=[5,6]-=- When a computer is used to prove a result in analysis, the picture becomes quite a bit more involved. This is due to the fact that the basic quantities in analysis are the real numbers, an infinite c... |

2 |
Soap Bubbles. Dover Publ
- Boys
- 1959
(Show Context)
Citation Context ...from the University of Illinois at Urbana-Champaign. The double bubble conjecture can be traced back to at least the year 1911, where this conjecture was formulated based on physical experiments, see =-=[7]-=-. Conjecture 2 (Double Bubble Conjecture). The double bubble is the surface of smallest area enclosing two equal volumes. If the area is a and v either volume, then a a -- 243rv 2. The proof of this c... |

1 | The dynamics of the Jouanolou foliation on the complex projectire 2-space. Ergodic Theory Dyn. Sys
- Camacho, Figueiredo
(Show Context)
Citation Context ...) mod 1, p-sfor two different values of k and compute the homoclinic point with a guaranteed accuracy of up to 10 decimal digits. We end by briefly describing the results of Camacho and de Figueiredo =-=[8]-=- on the dynamics of the Jouanolou foliation . Without even explaining terminology, suffice it to say that in this article the authors prove that four interesting foliations do not have non-trivial min... |

1 |
A collection of nonlinear model problems
- Mor
- 1990
(Show Context)
Citation Context ...we have just described a general procedure which one can use to prove lots of conjectures, each conjecture saying that a certain function f has a zero z in some interval vector x. For example, Mor in =-=[19]-=- conjectured that a nonlinear function arising in the modelling of chemical destillition columns (the 'methanol-8-problem' ) does have a zero, but was not able to prove it using analytical tools. A co... |

1 |
Rigorous verification in discrete dynamical systems
- Neumaier, Rage
- 1993
(Show Context)
Citation Context ...c operations. The resulting interval is then guaranteed to contain the 'correct' result, and the right end point of the interval is a guaranteed upper bound. Our second sample is by Neumaier and Rage =-=[20]-=- from 1993. The purpose of their work is to prove the existence of (and to accurately compute) a transversal homoclinic point for a given dynamical system with diffeomorphism F. They proceed as follow... |

1 |
Fermat's Enigma: The Quest to Solve the
- Singh
- 1997
(Show Context)
Citation Context ...r conjecture has actually been proved. The work by Hsiang, who claimed a proof in 1993 has been severely criticized by the mathematical community - this even found its way into the popular literature =-=[22]-=-. At the time of writing of this note, a complete, accepted proof has not yet been published as a reviewed Proving Conjectures by Using Interval Arithmetic 5 journal article. But the work of Hales and... |