## On location and approximation of clusters of zeroes of analytic functions (2005)

Venue: | Found. Comput. Math |

Citations: | 14 - 4 self |

### BibTeX

@ARTICLE{Giusti05onlocation,

author = {M. Giusti and G. Lecerf and B. Salvy and J. -c. Yakoubsohn},

title = {On location and approximation of clusters of zeroes of analytic functions},

journal = {Found. Comput. Math},

year = {2005},

volume = {5},

pages = {257--311}

}

### OpenURL

### Abstract

Abstract. In the beginning of the eighties, M. Shub and S. Smale developed a quantitative analysis of Newton’s method for multivariate analytic maps. In particular, their α-theory gives an effective criterion that ensures safe convergence to a simple isolated zero. This criterion requires only information concerning the map at the initial point of the iteration. Generalizing this theory to multiple zeros and clusters of zeros is still a challenging problem. In this article we focus on one complex variable functions. We study general criteria for detecting clusters and analyze the convergence of Schröder’s iteration to a cluster. In the case of a multiple root, it is well-known that this convergence is quadratic. In the case of a cluster with positive diameter, the convergence is still quadratic provided the iteration is stopped sufficiently early. We propose a criterion for stopping this iteration at a distance from the cluster which

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Citation Context ...e known for locating and approximating clusters of zeros of analytic functions. Yet such clusters naturally arise in many theoretical and practical situations. The results of this article are used in =-=[12]-=-, which deals with location and approximation of special types of clusters of multivariate analytic maps: even when starting with polynomial maps, the algorithm of [12] needs to compute with functions... |

9 |
Finding cluster of zeros of univariate polynomial
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Citation Context ...roduced his corrected Newton operator. The correction requires prior knowledge of the multiplicity. This multiplicity may also be approximated dynamically at the price of slowing down the convergence =-=[54, 18, 55, 21, 58]-=-. Higher order operators have also been adapted to multiple zeros [11]. In this article we deal with clusters, not only with multiples zeros. We assume that the multiplicity is known in advance. Our t... |

5 |
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4 | Barel, On locating clusters of zeros of analytic functions
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Citation Context ...tly improve the criteria of [58, 53, 59], and generalize them to clusters of roots of derivatives. Other cluster location algorithms have been proposed in the analytic case. For instance, the ones of =-=[22, 24, 23]-=- rely on numerical path integration: they are more powerful albeit more expensive. In [46], Pellet’s criterion is compared to nine other location methods on several families of polynomials. Cluster Ap... |

4 |
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4 |
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4 |
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3 |
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2 |
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2 |
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iteration towards a cluster of polynomial zeros
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