Approximate radical for clusters: A global approach using Gaussian elimination or SVD

Itnuit Janovitz-Freireich, L. Rónyai, Ágnes Szántó

Research output: Contribution to journalArticle

5 Citations (Scopus)

Abstract

We introduce a matrix of traces, attached to a zero dimensional ideal Ĩ. We show that the matrix of traces can be a useful tool in handling systems of polynomial equations with clustered roots. We present a method based on Dickson's lemma to compute the "approximate radical" of Ĩ in ℂ[χ1,...,χm] which has zero clusters: the approximate radical ideal has exactly one root in each cluster for sufficiently small clusters. Our method is "global" in the sense that it works simultaneously for all clusters: the problem is reduced to the computation of the numerical nullspace of the matrix of traces, a matrix efficiently computable from the generating polynomials of Ĩ. To compute the numerical nullspace of the matrix of traces we propose to use Gaussian elimination with pivoting or singular value decomposition. We prove that if Ĩ has k distinct zero clusters each of radius at most ε in the ∞-norm, then k steps of Gaussian elimination on the matrix of traces yields a submatrix with all entries asymptotically equal to ε2. We also show that the (k + 1)-th singular value of the matrix of traces is proportional to ε2. The resulting approximate radical has one root in each cluster with coordinates which are the arithmetic mean of the cluster, up to an error term asymptotically equal to ε2. In the univariate case our method gives an alternative to known approximate square-free factorization algorithms which is simpler and its accuracy is better understood.

Original languageEnglish
Pages (from-to)393-425
Number of pages33
JournalMathematics in Computer Science
Volume1
Issue number2
DOIs
Publication statusPublished - Dec 2007

Fingerprint

Gaussian elimination
Singular value decomposition
Trace
Roots
Polynomials
Square free
Pivoting
Zero-dimensional
Polynomial equation
Zero
Singular Values
Error term
Factorization
Univariate
Lemma
Directly proportional
Radius
Distinct
Norm
Polynomial

Keywords

  • Clusters
  • Matrix of traces
  • Radical ideal
  • Symbolic-numeric computation

ASJC Scopus subject areas

  • Applied Mathematics
  • Computational Mathematics
  • Computational Theory and Mathematics

Cite this

Approximate radical for clusters : A global approach using Gaussian elimination or SVD. / Janovitz-Freireich, Itnuit; Rónyai, L.; Szántó, Ágnes.

In: Mathematics in Computer Science, Vol. 1, No. 2, 12.2007, p. 393-425.

Research output: Contribution to journalArticle

Janovitz-Freireich, Itnuit ; Rónyai, L. ; Szántó, Ágnes. / Approximate radical for clusters : A global approach using Gaussian elimination or SVD. In: Mathematics in Computer Science. 2007 ; Vol. 1, No. 2. pp. 393-425.
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