Automatically Bounding the Taylor Remainder Series: Tighter Bounds and New Applications

Understand

We present a new algorithm for automatically bounding the Taylor remainder series.

• In the special case of a scalar function $f: \mathbb{R} \to \mathbb{R}$, our algorithm takes as input a reference point $x_0$, trust region $[a, b]$, and integer $k \ge 1$, and returns an interval $I$ such that $f(x) - \sum_{i=0}^{k-1} \frac {1} {i!} f^{(i)}(x_0) (x - x_0)^i \in I (x - x_0)^k$ for all $x \in [a, b]$.
• As in automatic differentiation, the function $f$ is provided to the algorithm in symbolic form, and must be composed of known atomic functions.
• At a high level, our algorithm has two steps.