### Abstract

A challenge to current theories of computing in the continua is the proper treatment of the zero test. Such tests are critical for extracting geometric information. Zero tests are expensive and may be uncomputable. So we seek geometric algorithms based on a weak form of such tests, called soft zero tests. Typically, algorithms with such tests can only determine the geometry for "nice" (e.g., non-degenerate, non-singular, smooth, Morse, etc) inputs. Algorithms that avoid such niceness assumptions are said to be complete. Can we design complete algorithms with soft zero tests? We address the basic problem of determining the geometry of the roots of a complex analytic function f. We assume effective box functions for f and its higher derivatives are provided. The problem is formalized as the root clustering problem, and we provide a complete (δ,∈)-exact algorithm based on soft zero tests.

Original language | English (US) |
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Title of host publication | The Nature of Computation |

Subtitle of host publication | Logic, Algorithms, Applications - 9th Conference on Computability in Europe, CiE 2013, Proceedings |

Pages | 434-444 |

Number of pages | 11 |

DOIs | |

State | Published - Jul 26 2013 |

Event | 9th Conference on Computability in Europe - The Nature of Computation: Logic, Algorithms, Applications, CiE 2013 - Milan, Italy Duration: Jul 1 2013 → Jul 5 2013 |

### 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 | 7921 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 9th Conference on Computability in Europe - The Nature of Computation: Logic, Algorithms, Applications, CiE 2013 |
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Country | Italy |

City | Milan |

Period | 7/1/13 → 7/5/13 |

### Fingerprint

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*The Nature of Computation: Logic, Algorithms, Applications - 9th Conference on Computability in Europe, CiE 2013, Proceedings*(pp. 434-444). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 7921 LNCS). https://doi.org/10.1007/978-3-642-39053-1_51