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A domain theoretic account of Picard's theorem

Abbas Edalat, Dirk Pattinson

Conference or Workshop Paper
31st international colloquium on automata, languages and programming (ICALP 2004), Turku University, Maths Department, Turku, Finland
Lecture Notes in Computer Science
Volume 3142
ISBN 3-5402-2849-7

We present a domain-theoretic version of Picardrsquos theorem for solving classical initial value problems in $\mathbb{R}^n$. 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.

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