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

Venue: | Found. Comput. Math |

Citations: | 16 - 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

### Citations

377 | Complexity and Real Computation - Blum, Cucker, et al. - 1997 |

107 |
Analytic Theory of Polynomials
- Rahman, Schmeisser
- 2002
(Show Context)
Citation Context ...of our unified presentation for any l ∈ {0, . . . , m − 1}. Related Works. Location and approximation of roots of polynomials are classical subjects in numerical analysis. Some general references are =-=[26, 40]-=-. An extended bibliography is collected in [27, 28], and a recent survey on root location can be found in [29]. In contrast to polynomials, few algorithms are known for locating and approximating clus... |

86 | Solving a polynomial equation: Some history and recent progress
- Pan
- 1997
(Show Context)
Citation Context ...p in the computations. In the following paragraphs we briefly discuss the main known strategies to handle clusters of roots of polynomials. A detailed survey on root finder algorithms can be found in =-=[36]-=-. On the one hand, nearly optimal root finders have been well established for a decade [34, 35, 30, 31, 20, 38], following earlier ideas by Schönhage [47]. The fastest algorithms from the theoretical ... |

85 |
Newton’s method estimates from data at one point, The Merging of Disciplines: New
- Smale
- 1986
(Show Context)
Citation Context ...erator N(f; x) := x − f(x) f ′ (x) converges quadratically to ζ, provided the initial point is “sufficiently close” to it. A quantitative analysis of this convergence has been given by Shub and Smale =-=[51, 49, 52, 50]-=-. They relate the convergence of Newton’s method to point estimates — estimates on f and its derivatives at a point. These results extend to the multivariate case and are often referred to as “Smale’s... |

73 |
The fundamental theorem of algebra and complexity
- Smale
- 1981
(Show Context)
Citation Context ...erator N(f; x) := x − f(x) f ′ (x) converges quadratically to ζ, provided the initial point is “sufficiently close” to it. A quantitative analysis of this convergence has been given by Shub and Smale =-=[51, 49, 52, 50]-=-. They relate the convergence of Newton’s method to point estimates — estimates on f and its derivatives at a point. These results extend to the multivariate case and are often referred to as “Smale’s... |

71 |
The fundamental theorem of algebra in terms of computational complexity
- Schönhage
- 1982
(Show Context)
Citation Context ...vey on root finder algorithms can be found in [36]. On the one hand, nearly optimal root finders have been well established for a decade [34, 35, 30, 31, 20, 38], following earlier ideas by Schönhage =-=[47]-=-. The fastest algorithms from the theoretical point of view, namely [31, 38], are based on balanced splittings and make use of the non-trivial generalizations of the Grace-Heawood theorem given in [4]... |

57 |
Solution of Equations and Systems of Equations
- Ostrowski
- 1960
(Show Context)
Citation Context ... The quadratic convergence of Newton’s iteration ceases to hold in the presence of multiple zeros. Instead, the convergence becomes linear and a large amount of works focus on this problem, including =-=[48, 33, 41, 42, 43, 13, 6, 7, 56, 5, 15, 60, 14, 8, 58]-=- (some of them also deal with the more complicated multivariate case). In order to reestablishsCLUSTERS OF ZEROS OF ANALYTIC FUNCTIONS 7 quadratic convergence, Schröder [48] introduced his corrected N... |

48 | Computing multiple roots of inexact polynomials
- Zeng
- 2005
(Show Context)
Citation Context ... a bounded domain. Such a study is yet to be done. Lastly, in the vast literature on polynomial root finding and cluster detection, it is worth mentioning a few other important approaches proposed in =-=[16, 19, 17, 61]-=-, that are less connected to our work. 1. Cluster Location We present point estimate criteria for cluster location that are based on Rouché’s theorem. The first and most general criterion relates the ... |

46 |
Design, analysis, and implementation of a multiprecision polynomial rootfinder. Numerical Algorithms
- Bini, Fiorentino
- 2000
(Show Context)
Citation Context ...l Root Finding. Besides our primary interest in the several variable case [12], we now discuss potential applications of our methods in the field of univariate polynomial root finders. As observed in =-=[1, 37]-=-, univariate polynomials produced by eliminating variables in multivariate polynomial systems (e.g., by means of Gröbner basis computation) are often huge and “ill conditioned”. Thus, clusters and eve... |

46 |
On the worst-case arithmetic complexity of approximating zeros of systems of polynomials
- Renegar
- 1989
(Show Context)
Citation Context ...ds of polynomial root finders have been designed in order to exploit “ill conditioned” situations. They are theoretically slower than the previous methods but they are often efficient in practice. In =-=[44]-=-, Renegar speeds up Weyl’s quad-tree construction (combined to the Schur-Cohn algorithm) thanks to the α-theory for simple roots. Clusters are treated by relating them to clusters of derivatives. This... |

44 |
A bibliography on roots of polynomials
- McNamee
- 1993
(Show Context)
Citation Context ... m − 1}. Related Works. Location and approximation of roots of polynomials are classical subjects in numerical analysis. Some general references are [26, 40]. An extended bibliography is collected in =-=[27, 28]-=-, and a recent survey on root location can be found in [29]. In contrast to polynomials, few algorithms are known for locating and approximating clusters of zeros of analytic functions. Yet such clust... |

42 | Univariate polynomials: nearly optimal algorithms for numerical factorization and root-finding
- Pan
- 2002
(Show Context)
Citation Context ...gies to handle clusters of roots of polynomials. A detailed survey on root finder algorithms can be found in [36]. On the one hand, nearly optimal root finders have been well established for a decade =-=[34, 35, 30, 31, 20, 38]-=-, following earlier ideas by Schönhage [47]. The fastest algorithms from the theoretical point of view, namely [31, 38], are based on balanced splittings and make use of the non-trivial generalization... |

40 |
Detection and validation of clusters of polynomial zeros
- Hribernig, Stetter
- 1995
(Show Context)
Citation Context ... a bounded domain. Such a study is yet to be done. Lastly, in the vast literature on polynomial root finding and cluster detection, it is worth mentioning a few other important approaches proposed in =-=[16, 19, 17, 61]-=-, that are less connected to our work. 1. Cluster Location We present point estimate criteria for cluster location that are based on Rouché’s theorem. The first and most general criterion relates the ... |

38 |
Elementary theory of analytic functions of one or several complex variables
- Cartan
- 1963
(Show Context)
Citation Context ...ally stated in their most general form before being specialized into geometric majorant series. Majorant series techniques belong to the “point de vue de Weierstrass”, as mentioned by Henri Cartan in =-=[3]-=- (see Chapter 1 and pages 218–225). These are a crucial tool for the effective manipulation of power series expansions, that lie at the heart of Smale’s α-theory. From the computational viewpoint, onl... |

31 | Optimal and nearly optimal algorithms for approximating polynomial zeros
- Pan
- 1996
(Show Context)
Citation Context ...gies to handle clusters of roots of polynomials. A detailed survey on root finder algorithms can be found in [36]. On the one hand, nearly optimal root finders have been well established for a decade =-=[34, 35, 30, 31, 20, 38]-=-, following earlier ideas by Schönhage [47]. The fastest algorithms from the theoretical point of view, namely [31, 38], are based on balanced splittings and make use of the non-trivial generalization... |

29 |
Computational complexity: on the geometry of polynomials and the theory of cost
- Shub, Smale
- 1986
(Show Context)
Citation Context ...erator N(f; x) := x − f(x) f ′ (x) converges quadratically to ζ, provided the initial point is “sufficiently close” to it. A quantitative analysis of this convergence has been given by Shub and Smale =-=[51, 49, 52, 50]-=-. They relate the convergence of Newton’s method to point estimates — estimates on f and its derivatives at a point. These results extend to the multivariate case and are often referred to as “Smale’s... |

23 | An efficient algorithm for the complex roots problem
- NEFF, REIF
- 1996
(Show Context)
Citation Context ...gies to handle clusters of roots of polynomials. A detailed survey on root finder algorithms can be found in [36]. On the one hand, nearly optimal root finders have been well established for a decade =-=[34, 35, 30, 31, 20, 38]-=-, following earlier ideas by Schönhage [47]. The fastest algorithms from the theoretical point of view, namely [31, 38], are based on balanced splittings and make use of the non-trivial generalization... |

22 |
Optimal (up to polylog factors) sequential and parallel algorithms for approximating complex polynomial zeros
- Pan
- 1995
(Show Context)
Citation Context |

21 |
Newton’s method at singular points
- Decker, Kelley
- 1980
(Show Context)
Citation Context ... The quadratic convergence of Newton’s iteration ceases to hold in the presence of multiple zeros. Instead, the convergence becomes linear and a large amount of works focus on this problem, including =-=[48, 33, 41, 42, 43, 13, 6, 7, 56, 5, 15, 60, 14, 8, 58]-=- (some of them also deal with the more complicated multivariate case). In order to reestablishsCLUSTERS OF ZEROS OF ANALYTIC FUNCTIONS 7 quadratic convergence, Schröder [48] introduced his corrected N... |

21 |
On solving nonlinear equations with simple singularities or nearly singular solutions
- Griewank
- 1985
(Show Context)
Citation Context ... The quadratic convergence of Newton’s iteration ceases to hold in the presence of multiple zeros. Instead, the convergence becomes linear and a large amount of works focus on this problem, including =-=[48, 33, 41, 42, 43, 13, 6, 7, 56, 5, 15, 60, 14, 8, 58]-=- (some of them also deal with the more complicated multivariate case). In order to reestablishsCLUSTERS OF ZEROS OF ANALYTIC FUNCTIONS 7 quadratic convergence, Schröder [48] introduced his corrected N... |

19 |
Convergence rates for Newton's method at singular points
- Decker, Keller, et al.
- 1983
(Show Context)
Citation Context |

17 | How to find all roots of complex polynomials by Newton’s Method,” Inventiones mathematicae
- Hubbard, Schleicher, et al.
(Show Context)
Citation Context ... a bounded domain. Such a study is yet to be done. Lastly, in the vast literature on polynomial root finding and cluster detection, it is worth mentioning a few other important approaches proposed in =-=[16, 19, 17, 61]-=-, that are less connected to our work. 1. Cluster Location We present point estimate criteria for cluster location that are based on Rouché’s theorem. The first and most general criterion relates the ... |

16 |
Geometry of Polynomials, Second edition
- Marden
- 1966
(Show Context)
Citation Context ...of our unified presentation for any l ∈ {0, . . . , m − 1}. Related Works. Location and approximation of roots of polynomials are classical subjects in numerical analysis. Some general references are =-=[26, 40]-=-. An extended bibliography is collected in [27, 28], and a recent survey on root location can be found in [29]. In contrast to polynomials, few algorithms are known for locating and approximating clus... |

16 |
On Newton’s method for singular problems
- Reddien
- 1978
(Show Context)
Citation Context |

15 |
Estimations for the separation number of a polynomial system
- Dedieu
- 1997
(Show Context)
Citation Context ...s universally (with respect to f) proportional to 1/γ(f; ζ). In particular thiss4 M. GIUSTI, G. LECERF, B. SALVY, AND J.-C. YAKOUBSOHN also provides lower bounds for the distance between simple zeros =-=[9]-=-. As to the so-called α-theorems [2, Chapter 8, Theorem 2], which have given their name to the α-theory, they are more relevant to practical concerns: they show that Newton’s method with initial point... |

14 |
Analysis of Newton’s method at irregular singularities
- Griewank, Osborne
- 1983
(Show Context)
Citation Context |

14 | Approximating Complex Polynomial Zeros: Modified Quadtree (Weyl’s) Construction and Improved Newton’s Iteration
- Pan
(Show Context)
Citation Context ...l Root Finding. Besides our primary interest in the several variable case [12], we now discuss potential applications of our methods in the field of univariate polynomial root finders. As observed in =-=[1, 37]-=-, univariate polynomials produced by eliminating variables in multivariate polynomial systems (e.g., by means of Gröbner basis computation) are often huge and “ill conditioned”. Thus, clusters and eve... |

14 |
Ueber unendlich viele Algorithmen zur Au°õsung der Gleichungen
- Schrõder
(Show Context)
Citation Context ...of analytic functions in the univariate case. In the case of a zero of multiplicity m, the convergence of Newton’s operator is no longer quadratic, but one can use Schröder’s modified Newton operator =-=[48]-=-: Nm(f; x) := x − m f(x) f ′ (x) which has quadratic convergence. Another possibility is to apply Newton’s operator to the (m − 1)th derivative of f. Both methods are covered by our analysis of a fami... |

13 |
On dominating sequence method in the point estimate and Smale’s theorem, Scientia Sinica Ser. A
- Wang, Han
- 1990
(Show Context)
Citation Context ...uadratically, provided α(f; x0) is sufficiently small. Moreover, the distance from x0 to the zero is then bounded by β(f; x0) times a universal constant. The optimal constants are due to Wang and Han =-=[57]-=-. Our Contributions. Overview. In this article we extend the α-theory in order to obtain estimates for clusters of zeros of analytic functions. The text is organized around three central problems: clu... |

12 |
Computing the zeros of analytic functions
- Kravanja, Barel
- 2000
(Show Context)
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... |

12 |
A 2000 Updated Supplementary Bibliography on Roots of Polynomials
- McNamee
(Show Context)
Citation Context ... m − 1}. Related Works. Location and approximation of roots of polynomials are classical subjects in numerical analysis. Some general references are [26, 40]. An extended bibliography is collected in =-=[27, 28]-=-, and a recent survey on root location can be found in [29]. In contrast to polynomials, few algorithms are known for locating and approximating clusters of zeros of analytic functions. Yet such clust... |

11 | Ten methods to bound multiple roots of polynomials
- Rump
(Show Context)
Citation Context ... 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 Approximation. The quadratic convergence of Newton’s iteration ceases to hold in the presenc... |

10 | On location and approximation of clusters of zeroes: case of embedding dimension one
- Giusti, Lecerf, et al.
(Show Context)
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... |

10 |
Finding a cluster of zeros of univariate polynomials
- Yakoubsohn
(Show Context)
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8 |
Starlike domains of convergence for Newton’s method at singularities
- Griewank
- 1980
(Show Context)
Citation Context |

7 |
Roots of a polynomial and its derivatives
- Coppersmith, Neff
- 1994
(Show Context)
Citation Context ...47]. The fastest algorithms from the theoretical point of view, namely [31, 38], are based on balanced splittings and make use of the non-trivial generalizations of the Grace-Heawood theorem given in =-=[4]-=-, which can be seen as a complexification of Rolle’s theorem. More precisely, if k + 1 roots of a polynomial p of degree n are contained in a ball of radius ρ then, for any ℓ ≤ k, at least k + 1 − ℓ r... |

7 |
Convergence acceleration for Newton’s method at singular points
- Decker, Kelley
- 1982
(Show Context)
Citation Context |

6 |
Enclosing clusters of zeros of polynomials
- Neumaier
(Show Context)
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- Kirrinnis
- 1998
(Show Context)
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5 |
A modification of Newton’s method for analytic mappings having multiple zeros. Computing 62
- Kravanja
- 1999
(Show Context)
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5 |
Convergence of the Newton process to multiple solutions
- Rall
- 1966
(Show Context)
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5 |
Approximate zero-points” of real univariate polynomial with large error terms
- Terui, Sasaki
(Show Context)
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4 |
Formulae for high derivatives of composite functions
- Fraenkel
- 1978
(Show Context)
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4 | Barel, On locating clusters of zeros of analytic functions
- Kravanja, Sakurai, et al.
- 1998
(Show Context)
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 |
Newton’s method and high order singularities
- Reddien
- 1979
(Show Context)
Citation Context |

4 |
Finding a multiple zero by transformations and Newton-like methods
- Ypma
- 1983
(Show Context)
Citation Context |

3 |
Modified Newton’s method with third-order convergence and multiple roots
- Frontini, Sormani
(Show Context)
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3 |
Simultaneous Computation of all the zero-cluster of a univariate polynomial
- Yakoubsohn
(Show Context)
Citation Context ...ompute with functions that are implicitly defined, hence generally not polynomial. Cluster Location. Our analysis of Pellet’s criterion in Section 1 follows Yakoubsohn’s approach via Rouché’s theorem =-=[58, 59]-=-. We slightly 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, th... |

2 |
Improving the Van de Vel root-finding method
- King
- 1983
(Show Context)
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... |

2 |
iteration towards a cluster of polynomial zeros
- Kirrinnis, Newton
- 1997
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