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The synthesis of classical Computational Complexity Theory with RecursiveAnalysis provides a quantitative foundation to reliable numerics. Here theoperators of maximization, integration, and solving ordinary differentialequations are known to map (even high-order differentiable) polynomial-timecomputable functions to instances which are `hard' for classical complexityclasses NP, #P, and CH; but, restricted to analytic functions, mappolynomial-time computable ones to polynomial-time computable ones --non-uniformly!  We investigate the uniform parameterized complexity of the above operators inthe setting of Weihrauch's TTE and its second-order extension due toKawamura&Cook (2010). That is, we explore which (both continuous and discrete,first and second order) information and parameters on some given f issufficient to obtain similar data on Max(f) and int(f); and within what runningtime, in terms of these parameters and the guaranteed output precision 2^(-n).  It turns out that Gevrey's hierarchy of functions climbing from analytic tosmooth corresponds to the computational complexity of maximization growing frompolytime to NP-hard. Proof techniques involve mainly the Theory of (discrete)Computation, Hard Analysis, and Information-Based Complexity.

Parameterized Uniform Complexity in Numerics: from Smooth to Analytic, from NP-hard to Polytime