Polynomial Inclusion Functions

When using interval analysis, the bounds of an inclusion function are often non-tight due to dependency effects. The benefit of Taylor Models (TMs) or Verified Taylor Series (VTSs) is the use of higher order derivatives terms, significantly reducing the dependency effect. In this paper, it is assumed that the required information to derive these inclusion functions is obtained using automatic differentiation. The drawback of TMs and VTSs is that not all available information is used, resulting in non-optimal inclusion functions. In this paper the Polynomial Inclusion Function (PIF) is presented, which is guaranteed to form equal or shaper enclosures than any (combination of) Taylor Model(s) defined using the same set of information. The PIF is derived for the one dimensional case. Extension to n-dimensional functions is performed via application of the PIF to every dimension independently. The performance of the PIF is compared to that of Verified Taylor Series for multiple (non-linear) functions and is shown to yield to superior inclusions. Moreover, unlike with TMs or VTSs, increasing the order of the PIF will always sharpen its bounds.