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Time Complexity and Convergence Analysis of Domain Theoretic Picard Method

  • Amin Farjudian
  • Michal Konečný
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5110)

Abstract

We present an implementation of the domain-theoretic Picard method for solving initial value problems (IVPs) introduced by Edalat and Pattinson [1]. Compared to Edalat and Pattinson’s implementation, our algorithm uses a more efficient arithmetic based on an arbitrary precision floating-point library. Despite the additional overestimations due to floating-point rounding, we obtain a similar bound on the convergence rate of the produced approximations. Moreover, our convergence analysis is detailed enough to allow a static optimisation in the growth of the precision used in successive Picard iterations. Such optimisation greatly improves the efficiency of the solving process. Although a similar optimisation could be performed dynamically without our analysis, a static one gives us a significant advantage: we are able to predict the time it will take the solver to obtain an approximation of a certain (arbitrarily high) quality.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Amin Farjudian
    • 1
  • Michal Konečný
    • 1
  1. 1.Computer ScienceAston UniversityBirminghamUK

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