### 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 ε^{2}. We also show that the (k + 1)-th singular value of the matrix of traces is proportional to ε^{2}. 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 ε^{2}. 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 language | English |
---|---|

Pages (from-to) | 393-425 |

Number of pages | 33 |

Journal | Mathematics in Computer Science |

Volume | 1 |

Issue number | 2 |

DOIs | |

Publication status | Published - Dec 2007 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Applied Mathematics
- Computational Mathematics
- Computational Theory and Mathematics

### Cite this

*Mathematics in Computer Science*,

*1*(2), 393-425. https://doi.org/10.1007/s11786-007-0013-7

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

Research output: Contribution to journal › Article

*Mathematics in Computer Science*, vol. 1, no. 2, pp. 393-425. https://doi.org/10.1007/s11786-007-0013-7

}

TY - JOUR

T1 - Approximate radical for clusters

T2 - A global approach using Gaussian elimination or SVD

AU - Janovitz-Freireich, Itnuit

AU - Rónyai, L.

AU - Szántó, Ágnes

PY - 2007/12

Y1 - 2007/12

N2 - 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.

AB - 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.

KW - Clusters

KW - Matrix of traces

KW - Radical ideal

KW - Symbolic-numeric computation

UR - http://www.scopus.com/inward/record.url?scp=49249122549&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=49249122549&partnerID=8YFLogxK

U2 - 10.1007/s11786-007-0013-7

DO - 10.1007/s11786-007-0013-7

M3 - Article

AN - SCOPUS:49249122549

VL - 1

SP - 393

EP - 425

JO - Mathematics in Computer Science

JF - Mathematics in Computer Science

SN - 1661-8270

IS - 2

ER -