Taylor Tube Method for Validated IVP
This work provides an incremental improvement to validated IVP solvers by extending a known technique to higher orders, offering potential efficiency gains for practitioners.
The paper generalizes the Euler Tube method to Taylor Tubes of arbitrary degree for validated initial value problems, showing that higher-degree Taylor Tubes improve accuracy and, surprisingly, can also speed up computation when combined with bisection.
We recently introduced a novel architecture for the design of validated IVP algorithms. This architecture forms the basis of our complete validated algorithm for IVP. A key subroutine in our algorithm is the \textbf{Euler Tube}: it gave a technique for refining end- and full-enclosures and is also key to deriving a complexity bound of our IVP solver. In this paper, we generalize it to \textbf{Taylor Tube} of degree $p\ge 1$. As expected, higher-degree Taylor Tubes improve accuracy. But surprisingly, our experiments show that it can also lead to an overall speedup when combined with bisection.