### Abstract

We present a new exact subdivision algorithm CEVAL for isolating the complex roots of a square-free polynomial in any given box. It is a generalization of a previous real root isolation algorithm called EVAL. Under suitable conditions, our approach is applicable for general analytic functions. CEVAL is based on the simple Bolzano Principle and is easy to implement exactly. Preliminary experiments have shown its competitiveness. We further show that, for the "benchmark problem" of isolating all roots of a square-free polynomial with integer coefficients, the asymptotic complexity of both algorithms EVAL and CEVAL matches (up a logarithmic term) that of more sophisticated real root isolation methods which are based on Descartes' Rule of Signs, Continued Fraction or Sturm sequence. In particular, we show that the tree size of EVAL matches that of other algorithms. Our analysis is based on a novel technique called Δ-clusters from which we expect to see further applications.

Original language | English (US) |
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Title of host publication | ISSAC 2011 - Proceedings of the 36th International Symposium on Symbolic and Algebraic Computation |

Pages | 353-360 |

Number of pages | 8 |

DOIs | |

State | Published - 2011 |

Event | 36th International Symposium on Symbolic and Algebraic Computation, ISSAC 2011 - San Jose, CA, United States Duration: Jun 8 2011 → Jun 11 2011 |

### Other

Other | 36th International Symposium on Symbolic and Algebraic Computation, ISSAC 2011 |
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Country | United States |

City | San Jose, CA |

Period | 6/8/11 → 6/11/11 |

### Fingerprint

### Keywords

- bolzano methods
- complexity of complex root isolation
- evaluation-based root isolation
- exact root isolation
- subdivision algorithms

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*ISSAC 2011 - Proceedings of the 36th International Symposium on Symbolic and Algebraic Computation*(pp. 353-360) https://doi.org/10.1145/1993886.1993938

**A simple but exact and efficient algorithm for complex root isolation.** / Yap, Chee; Sagraloff, Michael.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*ISSAC 2011 - Proceedings of the 36th International Symposium on Symbolic and Algebraic Computation.*pp. 353-360, 36th International Symposium on Symbolic and Algebraic Computation, ISSAC 2011, San Jose, CA, United States, 6/8/11. https://doi.org/10.1145/1993886.1993938

}

TY - GEN

T1 - A simple but exact and efficient algorithm for complex root isolation

AU - Yap, Chee

AU - Sagraloff, Michael

PY - 2011

Y1 - 2011

N2 - We present a new exact subdivision algorithm CEVAL for isolating the complex roots of a square-free polynomial in any given box. It is a generalization of a previous real root isolation algorithm called EVAL. Under suitable conditions, our approach is applicable for general analytic functions. CEVAL is based on the simple Bolzano Principle and is easy to implement exactly. Preliminary experiments have shown its competitiveness. We further show that, for the "benchmark problem" of isolating all roots of a square-free polynomial with integer coefficients, the asymptotic complexity of both algorithms EVAL and CEVAL matches (up a logarithmic term) that of more sophisticated real root isolation methods which are based on Descartes' Rule of Signs, Continued Fraction or Sturm sequence. In particular, we show that the tree size of EVAL matches that of other algorithms. Our analysis is based on a novel technique called Δ-clusters from which we expect to see further applications.

AB - We present a new exact subdivision algorithm CEVAL for isolating the complex roots of a square-free polynomial in any given box. It is a generalization of a previous real root isolation algorithm called EVAL. Under suitable conditions, our approach is applicable for general analytic functions. CEVAL is based on the simple Bolzano Principle and is easy to implement exactly. Preliminary experiments have shown its competitiveness. We further show that, for the "benchmark problem" of isolating all roots of a square-free polynomial with integer coefficients, the asymptotic complexity of both algorithms EVAL and CEVAL matches (up a logarithmic term) that of more sophisticated real root isolation methods which are based on Descartes' Rule of Signs, Continued Fraction or Sturm sequence. In particular, we show that the tree size of EVAL matches that of other algorithms. Our analysis is based on a novel technique called Δ-clusters from which we expect to see further applications.

KW - bolzano methods

KW - complexity of complex root isolation

KW - evaluation-based root isolation

KW - exact root isolation

KW - subdivision algorithms

UR - http://www.scopus.com/inward/record.url?scp=79959628862&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79959628862&partnerID=8YFLogxK

U2 - 10.1145/1993886.1993938

DO - 10.1145/1993886.1993938

M3 - Conference contribution

AN - SCOPUS:79959628862

SN - 9781450306751

SP - 353

EP - 360

BT - ISSAC 2011 - Proceedings of the 36th International Symposium on Symbolic and Algebraic Computation

ER -