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Although machine learning is increasingly applied in control approaches, only few methods guarantee certifiable safety, which is necessary for real world applications. These approaches typically rely on well-understood learning algorithms, which allow formal theoretical analysis. Gaussian process regression is a prominent example among those methods, which attracts growing attention due to its strong Bayesian foundations. Even though many problems regarding the analysis of Gaussian processes have a similar structure, specific approaches are typically tailored for them individually, without strong focus on computational efficiency. Thereby, the practical applicability and performance of these approaches is limited. In order to overcome this issue, we propose a novel framework called GP3, general purpose computation on graphics processing units for Gaussian processes, which allows to solve many of the existing problems efficiently. By employing interval analysis, local Lipschitz constants are computed in order to extend properties verified on a grid to continuous state spaces. Since the computation is completely parallelizable, the computational benefits of GPU processing are exploited in combination with multi-resolution sampling in order to allow high resolution analysis.