Uncertainty propagation or box propagation

This paper discusses the use of recently developed techniques and software in the numerical propagation of uncertainties in initial coordinates and/or parameters for initial value problems. We present an approach based on several validated numerical integration techniques but focusing on the propagation of boxes. The procedure uses a multivariable high order Taylor series development of the solution of the system whose Taylor coefficients are calculated via extended automatic differentiation rules for all the basic operations. These techniques are implemented in the recent free-software TIDES. The classical two-body and Lorenz problems are chosen as examples to show the benefits of the approach. The results show that the solution of uncertainties can be approximated in an analytic form by means of a Taylor series and that these techniques can be extremely useful in different practical applications.