Parameterized Complexity for Uniform Operators on Multidimensional Analytic Functions and ODE Solving

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10944)


Real complexity theory is a resource-bounded refinement of computable analysis and provides a realistic notion of running time of computations over real numbers, sequences, and functions by relying on Turing machines to handle approximations of arbitrary but guaranteed absolute error. Classical results in real complexity show that important numerical operators can map polynomial time computable functions to functions that are hard for some higher complexity class like \(\mathsf {NP}\) or \(\mathsf {\# P}\). Restricted to analytic functions, however, those operators map polynomial time computable functions again to polynomial time computable functions. Recent work by Kawamura, Müller, Rösnick and Ziegler discusses how to extend this to uniform algorithms on one-dimensional analytic functions over simple compact domains using second-order and parameterized complexity. In this paper, we extend some of their results to the case of multidimensional analytic functions. We further use this to show that the operator mapping an analytic ordinary differential equations to its solution is computable in parameterized polynomial time. Finally, we discuss how the theory can be used as a basis for verified exact numerical computation with analytic functions and provide a prototypical implementation in the iRRAM C++ framework for exact real arithmetic.



This work was supported by JSPS KAKENHI Grant Numbers JP18H03203 and JP18J10407 and by the Japan Society for the Promotion of Science (JSPS), Core-to-Core Program (A. Advanced Research Networks).


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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Kyushu UniversityFukuokaJapan
  2. 2.InriaRocquencourtFrance
  3. 3.University of TokyoTokyoJapan

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