Foundations of Computational Mathematics

, Volume 10, Issue 1, pp 93–114 | Cite as

Parallelization Method for a Continuous Property

Article

Abstract

An automated general purpose method is introduced for computing a rigorous estimate of a bounded region in ℝn whose points satisfy a given property. The method is based on calculations conducted in interval arithmetic and the constructed approximation is built of rectangular boxes of variable sizes. An efficient strategy is proposed, which makes use of parallel computations on multiple machines and refines the estimate gradually. It is proved that under certain assumptions the result of computations converges to the exact result as the precision of calculations increases. The time complexity of the algorithm is analyzed, and the effectiveness of this approach is illustrated by constructing a lower bound on the set of parameters for which an overcompensatory nonlinear Leslie population model exhibits more than one attractor, which is of interest from the biological point of view. This paper is accompanied by efficient and flexible software written in C++ whose source code is freely available at http://www.pawelpilarczyk.com/parallel/.

Keywords

Rigorous numerics Parallel computing Heuristics Algorithm Iterative method Interval arithmetic 

Mathematics Subject Classification (2000)

65D99 68W10 65G40 

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Copyright information

© SFoCM 2009

Authors and Affiliations

  1. 1.Centro de MatemáticaUniversidade do MinhoBragaPortugal

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