### Abstract

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 NP or # 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.

Original language | English |
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Title of host publication | Logic, Language, Information, and Computation - 25th International Workshop, WoLLIC 2018, Proceedings |

Editors | Ruy de Queiroz, Maricarmen Martinez, Lawrence S. Moss |

Publisher | Springer Verlag |

Pages | 223-236 |

Number of pages | 14 |

ISBN (Print) | 9783662576687 |

DOIs | |

Publication status | Published - Jan 1 2018 |

Event | 25th International Workshop on Logic, Language, Information, and Computation, WoLLIC 2018 - Bogota, Colombia Duration: Jul 24 2018 → Jul 27 2018 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 10944 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 25th International Workshop on Logic, Language, Information, and Computation, WoLLIC 2018 |
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Country | Colombia |

City | Bogota |

Period | 7/24/18 → 7/27/18 |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

*Logic, Language, Information, and Computation - 25th International Workshop, WoLLIC 2018, Proceedings*(pp. 223-236). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 10944 LNCS). Springer Verlag. https://doi.org/10.1007/978-3-662-57669-4_13