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A Domain Theoretic Account of Picard’s Theorem

  • A. Edalat
  • D. Pattinson
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3142)

Abstract

We present a domain-theoretic version of Picard’s theorem for solving classical initial value problems in ℝ n . For the case of vector fields that satisfy a Lipschitz condition, we construct an iterative algorithm that gives two sequences of piecewise linear maps with rational coefficients, which converge, respectively from below and above, exponentially fast to the unique solution of the initial value problem. We provide a detailed analysis of the speed of convergence and the complexity of computing the iterates. The algorithm uses proper data types based on rational arithmetic, where no rounding of real numbers is required. Thus, we obtain an implementation framework to solve initial value problems, which is sound and, in contrast to techniques based on interval analysis, also complete: the unique solution can be actually computed within any degree of required accuracy.

Keywords

Lipschitz Condition Piecewise Linear Function Interval Analysis Initial Value Problem Piecewise Constant Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • A. Edalat
    • 1
  • D. Pattinson
    • 2
  1. 1.Department of ComputingImperial CollegeLondonUK
  2. 2.Institut für InformatikLMU MünchenGermany

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