A Domain-Theoretic Account of Picard's Theorem

Abstract

AbstractWe 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.