### Abstract

Smale's notion of an approximate zero of an analytic function f : ℂ → ℂ is extended to take into account the errors incurred in the evaluation of the Newton operator. Call this stronger notion a robust approximate zero. We develop a corresponding robust point estimate for such zeros: we prove that if z_{0} ∈ ℂ satisfies α(f, z _{0}) < 0.02 then z_{0} is a robust approximate zero, with the associated zero z* lying in the closed disc B̄(z_{0}, 0.07/γ(f, z_{0}). Here α(f, z), γ(f, z) are standard functions in point estimates. Suppose f(z) is an L-bit integer square-free polynomial of degree d. Using our new algorithm, we can compute an n-bit absolute approximation of z* ∈ IR starting from a bigfloat Z _{0}, in time O[dM(n + d^{2}(L + lg d) lg(n + L))], where M(n) is the complexity of multiplying n-bit integers.

Original language | English (US) |
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Title of host publication | Lecture Notes in Computer Science |

Editors | G.S. Brodal, S. Leonardi |

Pages | 874-886 |

Number of pages | 13 |

Volume | 3669 |

State | Published - 2005 |

Event | 13th Annual European Symposium on Algorithms, ESA 2005 - Palma de Mallorca, Spain Duration: Oct 3 2005 → Oct 6 2005 |

### Other

Other | 13th Annual European Symposium on Algorithms, ESA 2005 |
---|---|

Country | Spain |

City | Palma de Mallorca |

Period | 10/3/05 → 10/6/05 |

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### ASJC Scopus subject areas

- Computer Science (miscellaneous)

### Cite this

*Lecture Notes in Computer Science*(Vol. 3669, pp. 874-886)

**Robust approximate zeros.** / Sharma, Vikram; Du, Zilin; Yap, Chee.

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

*Lecture Notes in Computer Science.*vol. 3669, pp. 874-886, 13th Annual European Symposium on Algorithms, ESA 2005, Palma de Mallorca, Spain, 10/3/05.

}

TY - GEN

T1 - Robust approximate zeros

AU - Sharma, Vikram

AU - Du, Zilin

AU - Yap, Chee

PY - 2005

Y1 - 2005

N2 - Smale's notion of an approximate zero of an analytic function f : ℂ → ℂ is extended to take into account the errors incurred in the evaluation of the Newton operator. Call this stronger notion a robust approximate zero. We develop a corresponding robust point estimate for such zeros: we prove that if z0 ∈ ℂ satisfies α(f, z 0) < 0.02 then z0 is a robust approximate zero, with the associated zero z* lying in the closed disc B̄(z0, 0.07/γ(f, z0). Here α(f, z), γ(f, z) are standard functions in point estimates. Suppose f(z) is an L-bit integer square-free polynomial of degree d. Using our new algorithm, we can compute an n-bit absolute approximation of z* ∈ IR starting from a bigfloat Z 0, in time O[dM(n + d2(L + lg d) lg(n + L))], where M(n) is the complexity of multiplying n-bit integers.

AB - Smale's notion of an approximate zero of an analytic function f : ℂ → ℂ is extended to take into account the errors incurred in the evaluation of the Newton operator. Call this stronger notion a robust approximate zero. We develop a corresponding robust point estimate for such zeros: we prove that if z0 ∈ ℂ satisfies α(f, z 0) < 0.02 then z0 is a robust approximate zero, with the associated zero z* lying in the closed disc B̄(z0, 0.07/γ(f, z0). Here α(f, z), γ(f, z) are standard functions in point estimates. Suppose f(z) is an L-bit integer square-free polynomial of degree d. Using our new algorithm, we can compute an n-bit absolute approximation of z* ∈ IR starting from a bigfloat Z 0, in time O[dM(n + d2(L + lg d) lg(n + L))], where M(n) is the complexity of multiplying n-bit integers.

UR - http://www.scopus.com/inward/record.url?scp=27144523491&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=27144523491&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:27144523491

VL - 3669

SP - 874

EP - 886

BT - Lecture Notes in Computer Science

A2 - Brodal, G.S.

A2 - Leonardi, S.

ER -