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A Domain-Theoretic Account of Picard's Theorem

Published online by Cambridge University Press:  01 February 2010

Abbas Edalat  and
Dirk Pattinson
Abbas Edalat
Affiliation:
Department of Computing, Imperial College London, United Kingdom, ae@doc.ic.ac.uk
Dirk Pattinson
Affiliation:
Department of Computing, Imperial College London, United Kingdom, dirk@doc.ic.ac.uk

Abstract

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We present a domain-theoretic version of Picard's theorem for solving classical initial value problems in ℝn. 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, from below and above respectively, 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 a sound implementation framework to solve initial value problems. In particular, the use of rational arithmetic guarantees that implementations of our technique will adhere to the bounds on convergence speed and algebraic complexity.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 10 , 2007 , pp. 83 - 118
Copyright
Copyright © London Mathematical Society 2007

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