many cubes near where the solution curve was bisected, resulting in the variably sized cubes noticeable in Figure \[fig:Verified\]. To address this we recombine the cubes in ${\mathcal{R}}$ which have the same $\alpha$ values, then subsequently use the Krawczyk operator to show that $(a)$ holds on the recombined cubes (see Algorithm \[alg:Recombine\]). In this fashion, we prove Theorem \[prop:Jones\].

Krawczyk Operator {#sec:KrawczykBanach}
-----------------

In numerical analysis there are many variations on the theme of Newton’s method: $
    x_{n+1} \mapsto x_n - Df(x_n)^{-1} f(x_n). 
$ As inverting a matrix is computationally expensive, one alternative method is to replace $ DF(x_n)^{-1}$ with a fixed matrix $A^\dagger \approx Df(x_0)^{-1}$. If $f(x_0) \approx 0$, then the Newton-Kantorovich theorem gives conditions for when the map $ T(x) = x - A^\dagger f(x)$ defines a contraction map in a neighborhood about $x_0$. The Krawczyk operator may be thought of as a way of bounding the image of $T$, itself being defined on rectangular sets $ X \subseteq {\mathbb{R}}^n$ and having the property that $T(X) \subseteq K(X,x_0)$. Rectangular, in the sense that $X$ can be given as the product of intervals in the coordinate directions of ${\mathbb{R}}^n$. Here we generalize the Krawczyk operator to non-rectangular subsets of 