### Abstract

This paper presents two results on the complexity of root isolation via Sturm sequences. Both results exploit amortization arguments. For a square-free polynomial A(X) of degree d with L-bit integer coefficients, we use an amortization argument to show that all the roots, real or complex, can be isolated using at most 0(dL + dlgd) Sturm probes. This extends Davenport's result for the case of isolating all real roots. We also show that a relatively straightforward algorithm, based on the classical subresultant PQS, allows us to evaluate the Sturm sequence of A(X) at rational Õ(dL)-bit values in time Õ(d3L); here the Õ-notation means we ignore logarithmic factors. Again, an amortization argument is used. We provide a family of examples to show that such amortization is necessary.

Original language | English (US) |
---|---|

Title of host publication | Symbolic-Numeric Computation |

Publisher | Springer International Publishing |

Pages | 113-129 |

Number of pages | 17 |

Volume | 41 |

ISBN (Print) | 9783764379834 |

State | Published - 2007 |

Event | International Workshop on Symbolic-Numeric Computation, SNC 2005 - Xian, China Duration: Jul 19 2005 → Jul 21 2005 |

### Publication series

Name | Trends in Mathematics |
---|---|

Volume | 41 |

ISSN (Print) | 22970215 |

ISSN (Electronic) | 2297024X |

### Other

Other | International Workshop on Symbolic-Numeric Computation, SNC 2005 |
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Country | China |

City | Xian |

Period | 7/19/05 → 7/21/05 |

### Fingerprint

### Keywords

- Complexity
- Davenport-Mahler bound
- Root isolation
- Separation bound
- Sturm sequence
- Subresultant

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Symbolic-Numeric Computation*(Vol. 41, pp. 113-129). (Trends in Mathematics; Vol. 41). Springer International Publishing.

**Amortized bound for root isolation via sturm sequences.** / Du, Zilin; Sharma, Vikram; Yap, Chee K.

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

*Symbolic-Numeric Computation.*vol. 41, Trends in Mathematics, vol. 41, Springer International Publishing, pp. 113-129, International Workshop on Symbolic-Numeric Computation, SNC 2005, Xian, China, 7/19/05.

}

TY - GEN

T1 - Amortized bound for root isolation via sturm sequences

AU - Du, Zilin

AU - Sharma, Vikram

AU - Yap, Chee K.

PY - 2007

Y1 - 2007

N2 - This paper presents two results on the complexity of root isolation via Sturm sequences. Both results exploit amortization arguments. For a square-free polynomial A(X) of degree d with L-bit integer coefficients, we use an amortization argument to show that all the roots, real or complex, can be isolated using at most 0(dL + dlgd) Sturm probes. This extends Davenport's result for the case of isolating all real roots. We also show that a relatively straightforward algorithm, based on the classical subresultant PQS, allows us to evaluate the Sturm sequence of A(X) at rational Õ(dL)-bit values in time Õ(d3L); here the Õ-notation means we ignore logarithmic factors. Again, an amortization argument is used. We provide a family of examples to show that such amortization is necessary.

AB - This paper presents two results on the complexity of root isolation via Sturm sequences. Both results exploit amortization arguments. For a square-free polynomial A(X) of degree d with L-bit integer coefficients, we use an amortization argument to show that all the roots, real or complex, can be isolated using at most 0(dL + dlgd) Sturm probes. This extends Davenport's result for the case of isolating all real roots. We also show that a relatively straightforward algorithm, based on the classical subresultant PQS, allows us to evaluate the Sturm sequence of A(X) at rational Õ(dL)-bit values in time Õ(d3L); here the Õ-notation means we ignore logarithmic factors. Again, an amortization argument is used. We provide a family of examples to show that such amortization is necessary.

KW - Complexity

KW - Davenport-Mahler bound

KW - Root isolation

KW - Separation bound

KW - Sturm sequence

KW - Subresultant

UR - http://www.scopus.com/inward/record.url?scp=84975698180&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84975698180&partnerID=8YFLogxK

M3 - Conference contribution

SN - 9783764379834

VL - 41

T3 - Trends in Mathematics

SP - 113

EP - 129

BT - Symbolic-Numeric Computation

PB - Springer International Publishing

ER -