Preconditioning of Taylor models, implementation and test cases

Abstract

Makino and Berz introduced the Taylor model approach for validated integration of initial value problems (IVPs) for ordinary differential equations (ODEs). Especially, they invented preconditioning of Taylor models for stabilizing the integration and proposed the following different types: parallelepiped preconditioning (with and without blunting), QR preconditioning, and curvilinear preconditioning. We review these types of preconditioning and show how they are implemented in INTLAB's verified ODE solver verifyode by stating explicit MATLAB code. Finally, we test our implementation with several examples.