Taylor Tube Method for Validated IVP

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 $\mathbf{\text{Euler}}\mathbf{\text{Tube}}$: 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 $\mathbf{\text{Taylor}}\mathbf{\text{Tube}}$ 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.