Polynomial-Time Solution of Initial Value Problems Using Polynomial Enclosures

Domain theory has been used with great success in providing a semantic framework for Turing computability, over both discrete and continuous spaces. On the other hand, classical approximation theory provides a rich set of tools for computations over real functions with (mainly) polynomial and rational function approximations. We present a semantic model for computations over real functions based on polynomial enclosures. As an important case study, we analyse the convergence and complexity of Picard’s method of initial value problem solving in our framework. We obtain a geometric rate of convergence over Lipschitz fields and then, by using Chebyshev truncations, we modify Picard’s algorithm into one which runs in polynomial-time over a set of polynomial-space representable fields, thus achieving a reduction in complexity which would be impossible in the step-function based domain models.