Robust approximate zeros

Vikram Sharma, Zilin Du, Chee Yap

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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 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.

Original languageEnglish (US)
Title of host publicationLecture Notes in Computer Science
EditorsG.S. Brodal, S. Leonardi
Pages874-886
Number of pages13
Volume3669
StatePublished - 2005
Event13th Annual European Symposium on Algorithms, ESA 2005 - Palma de Mallorca, Spain
Duration: Oct 3 2005Oct 6 2005

Other

Other13th Annual European Symposium on Algorithms, ESA 2005
CountrySpain
CityPalma de Mallorca
Period10/3/0510/6/05

Fingerprint

Polynomials

ASJC Scopus subject areas

  • Computer Science (miscellaneous)

Cite this

Sharma, V., Du, Z., & Yap, C. (2005). Robust approximate zeros. In G. S. Brodal, & S. Leonardi (Eds.), Lecture Notes in Computer Science (Vol. 3669, pp. 874-886)

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

Lecture Notes in Computer Science. ed. / G.S. Brodal; S. Leonardi. Vol. 3669 2005. p. 874-886.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Sharma, V, Du, Z & Yap, C 2005, Robust approximate zeros. in GS Brodal & S Leonardi (eds), 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.
Sharma V, Du Z, Yap C. Robust approximate zeros. In Brodal GS, Leonardi S, editors, Lecture Notes in Computer Science. Vol. 3669. 2005. p. 874-886
Sharma, Vikram ; Du, Zilin ; Yap, Chee. / Robust approximate zeros. Lecture Notes in Computer Science. editor / G.S. Brodal ; S. Leonardi. Vol. 3669 2005. pp. 874-886
@inproceedings{87800e957fde43f6aa156cc83bdcc9c8,
title = "Robust approximate zeros",
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 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.",
author = "Vikram Sharma and Zilin Du and Chee Yap",
year = "2005",
language = "English (US)",
volume = "3669",
pages = "874--886",
editor = "G.S. Brodal and S. Leonardi",
booktitle = "Lecture Notes in Computer Science",

}

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 -