Finding a Multiple Zero by Transformations and Newton-Like Methods

Ypma, Tjalling

doi:10.1137/1025077

Cited by 20 publications

(16 citation statements)

References 10 publications

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“…Following the ideas of Stewart [29] and Ypma [32], there is a region, where the function values can be considered zero among machine numbers…”

Section: Accuracy Domain Of Convergencementioning

confidence: 99%

“…Observe that h can be given by −2u [1] /(1 + 1 − 2u [1] /u [2] ) after comparing with (32). Only the real roots are sought in the first phase.…”

Section: The Algorithm Of Crouse and Putt For Polynomial Rootsmentioning

confidence: 99%

“…Such algorithms were suggested by Crouse and Putt [3], Krishnamurty and Sen [17] (see also [2]). A third possible solution of the multiplicity problem is the classical Schröder approach [27] that replaces f by a function g that is constructed from f that x * is a simple zero of g. Stewart [29] and Ypma [32] gave a thorough analysis of such transformation methods showing that they can fail to achieve the maximum accuracy that is attainable at a fixed precision, domain of convergence is restricted, and if two zeros are not well separated then the methods have difficulty in finding either of the zeros.…”

Section: Introductionmentioning

confidence: 99%

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A study of accelerated Newton methods for multiple polynomial roots

Galántai

Hegedűs

2009

Numer Algor

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We analyze and compare several accelerated Newton methods with built in multiplicity estimates. We also introduce the concept of indicator functions and discuss the Crouse-Putt method. It is shown that many of the accelerated Newton methods not only derive from Schröder's classic approach but are equivalent. The related computational experiments show that the built in multiplicity estimates can significantly decrease the number of Newton iterations, while the error of these estimates may significantly increase.

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“…Following the ideas of Stewart [29] and Ypma [32], there is a region, where the function values can be considered zero among machine numbers…”

Section: Accuracy Domain Of Convergencementioning

confidence: 99%

“…Observe that h can be given by −2u [1] /(1 + 1 − 2u [1] /u [2] ) after comparing with (32). Only the real roots are sought in the first phase.…”

Section: The Algorithm Of Crouse and Putt For Polynomial Rootsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A study of accelerated Newton methods for multiple polynomial roots

Galántai

Hegedűs

2009

Numer Algor

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“…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 reestablish quadratic convergence, Schröder [48] introduced his corrected Newton operator.…”

Section: Related Workmentioning

confidence: 99%

On Location and Approximation of Clusters of Zeros of Analytic Functions

et al. 2005

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At the beginning of the 1980s, 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 Date initial point of the iteration. Generalizing this theory to multiple zeros and clusters of zeros is still a challenging problem. In this paper we focus on one complex variable function. 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 is of the order of its diameter.

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“…The iteration function 03 is found by applying Newton's method to the function g(x) = f (x)/f'(x) (see [15] for example) . These methods do have some limitations as noted in [15] :…”

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confidence: 99%

An interval iteration for multiple roots of transcendental equations

Lin

Rokne

1995

Bit Numer Math

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.An iteration method for roots of algebraic functions with roots of multiplicity greater than one is established using tools and techniques from interval arithmetic . The method is based on an interval iteration functions for multiple roots and it retains the convergence order of the underlying iteration method while preserving global convergence over an initial interval . A number of simple examples are provided to show that the method is feasible and that it produces reasonable results .

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Finding a Multiple Zero by Transformations and Newton-Like Methods

Cited by 20 publications

References 10 publications

A study of accelerated Newton methods for multiple polynomial roots

A study of accelerated Newton methods for multiple polynomial roots

On Location and Approximation of Clusters of Zeros of Analytic Functions

An interval iteration for multiple roots of transcendental equations

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