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112 Juergen Garloff and Andrew P. Smith−0.3−0.4−0.5−0.6−0.7−0.8−0.9−10 0.2 0.4 0.6 0.8 1−0.6−0.7−0.8−0.9−10 0.2 0.4 0.6 0.8 1−0.6−0.7−0.8−0.9−10 0.2 0.4 0.6 0.8 1−0.3−0.4−0.5−0.6−0.7−0.8−0.9−10 0.2 0.4 0.6 0.8 1Figure 2. The polynomial of Example 1 for l = 3, 8, 13, 17, with its control points (circles), and correspondinglower bound function.Figure 3 shows the graphs of p 1 and p 2 over the unit box, their control points, and affinelower bound functions. This example, a model for hydrogen combustion with excess fuel, istaken from [7].5. Rigorous Bound FunctionsDue to rounding errors, inaccuracies may be introduced into the calculation of the boundfunctions. As a result, the computed affine function may not stay below the given polynomialover the box of interest. We also wish to consider the case of uncertain (interval) input data. Inour talk, we focus on a method by which an affine lower bound functions based on Bernsteinexpansion can be computed such that it can be guaranteed to stay below the given polynomial.The aforementioned methods can be adjusted to work with interval data, and the safe interpolationof interval control points can be facilitated by a method similar to that introduced in[2].6. Future WorkWe report on the integration of our software into the environment of the COCONUT project[1] funded by the European Union which aims at the solution of global optimisation andcontinuous constraint satisfaction problems.We conclude with some suggestions for the extension of our approach to the constructionof affine lower bound function for arbitrary sufficiently differentiable functions.