A domain-theoretic account of Picard's theorem
Abbas Edalat and Dirk Pattinson
Abstract: The authors 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, they
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. They further 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. The authors thus obtain a sound implementation framework to
solve initial value problems. In particular, the use of rational
arithmetic guarantees that implementations of their technique will
adhere to the bounds on convergence speed and algebraic
complexity.
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