Formally-Verified Round-Off Error Analysis of Runge–Kutta Methods

Abstract

Numerical errors are insidious, difficult to predict and inherent in different levels of critical systems design. Indeed, numerical algorithms generally constitute approximations of an ideal mathematical model, which itself constitutes an approximation of a physical reality which has undergone multiple measurement errors. To this are added rounding errors due to computer arithmetic implementations, often neglected even if they can significantly distort the results obtained. This applies to Runge–Kutta methods used for the numerical integration of ordinary differential equations, that are ubiquitous to model fundamental laws of physics, chemistry, biology or economy. We provide a Coq formalization of the rounding error analysis of Runge–Kutta methods applied to linear systems and implemented in floating-point arithmetic. We propose a generic methodology to build a bound on the error accumulated over the iterations, taking gradual underflow into account. We then instantiate this methodology for two classic Runge–Kutta methods, namely Euler and RK2. The formalization of the results include the definition of matrix norms, the proof of rounding error bounds of matrix operations and the formalization of the generic results and their applications on examples. In order to support the proposed approach, we provide numerical experiments on examples coming from nuclear physics applications.