### Abstract

The problem of isolating all real roots of a square-free integer polynomial f(X) inside any given interval I_{0} is a fundamental problem. EVAL is a simple and practical exact numerical algorithm for this problem: it recursively bisects I0, and any sub-interval I ⊆ I0, until a certain numerical predicate C_{0}(I) ⊆ C_{1}(I) holds on each I. We prove that the size of the recursion tree is O(d(L + r + log d)) where f has degree d, its coefficients have absolute values < 2^{L}, and I0 contains r roots of f. In the range L ≥ d, our bound is the sharpest known, and provably optimal. Our results are closely paralleled by recent bounds on EVAL by Sagraloff-Yap (ISSAC 2011) and Burr-Krahmer (2012). In the range L ≤ d, our bound is incomparable with those of Sagraloff-Yap or Burr-Krahmer. Similar to the Burr-Krahmer proof, we exploit the technique of "continuous amortization" from Burr-Krahmer-Yap (2009), namely to bound the tree size by an integral I_{0} G(x)dx over a suitable "charging function" G(x). We give an application of this feature to the problem of ray-shooting (i.e., finding smallest root in a given interval).

Original language | English (US) |
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Title of host publication | ISSAC 2012 - Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation |

Pages | 319-326 |

Number of pages | 8 |

DOIs | |

State | Published - Dec 1 2012 |

Event | 37th International Symposium on Symbolic and Algebraic Computation, ISSAC 2012 - Grenoble, France Duration: Jul 22 2012 → Jul 25 2012 |

### Publication series

Name | Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC |
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### Other

Other | 37th International Symposium on Symbolic and Algebraic Computation, ISSAC 2012 |
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Country | France |

City | Grenoble |

Period | 7/22/12 → 7/25/12 |

### Fingerprint

### Keywords

- Continuous amortization
- Integral analysis
- Real root isolation
- Subdivision algorithm

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*ISSAC 2012 - Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation*(pp. 319-326). (Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC). https://doi.org/10.1145/2442829.2442875