Halley's Methods for Solving Equations

Bateman, H.

doi:10.1080/00029890.1938.11990759

Cited by 16 publications

(10 citation statements)

References 12 publications

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“…As shown in [13], Halley's method can be obtained by applying Newton's method to the function f √ f . Gerlach [14], gives a generalization of this approach.…”

Section: Muller's Methodmentioning

confidence: 99%

Derivative-free family of higher order root finding methods

Hasan

2009

2009 American Control Conference

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Most higher order root finding methods require evaluation of a function and/or its derivatives at one or multiple points. There are cases where the derivatives of a given function are costly to compute. In this paper, higher order methods which do not require computation of any derivatives are derived. Asymptotic analysis has shown that these methods are approximations of root iterations. One of the main features of the proposed approaches is that one can develop multi-point derivative-free methods of any desired order. For lower order methods, these correspond to the Newton, and Ostrowski iterations. Several examples involving polynomials and entire functions have shown that the proposed methods can be applied to polynomial and nonpolynomial equations.

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“…As shown in [13], Halley's method can be obtained by applying Newton's method to the function f √ f . Gerlach [14], gives a generalization of this approach.…”

Section: Muller's Methodmentioning

confidence: 99%

Derivative-free family of higher order root finding methods

Hasan

2009

2009 American Control Conference

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“…There are many approaches for the derivation of Halley's method. One of these is by applying Newton's method to the function f √ f [9]. Convergence analysis of Halley's method is given in [10].…”

Section: Halley's Methodsmentioning

confidence: 99%

New Families of Higher Order Iterative Methods for Solving Equations

Hasan

2006

Proceedings of the 45th IEEE Conference on Decision and Control

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In this paper, several one-parameter families of root-finding algorithms that have higher order convergence to simple and/or multiple roots have been derived. Specifically, the rth root iterations for simple and multiple zeros are analyzed. The rth root iteration family is an infinite family of rth order methods for every positive integer r, and uses only the first r − 1 derivatives. This family includes Newton's method and the square root iteration as the first and second member, respectively. In addition, this work provides analyses and generalizations of Halley's and Laguerre's iterations, and develops a procedure of deriving higher order methods of any desired order. Many important properties of the rth root iteration family and its variants are established. Some of these variants maintain a high order of convergence for multiple roots whether the multiplicity is known or not. Based on individual methods, disks containing at least one zero are derived.

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“…According to Traub [12], this iteration function (IF) "must share with the secant IF the distinction of being the most frequently rediscovered IF in the literature." We refer to [1], [2], [3], [4], and [12] to name but a few.…”

Section: Halley's Methodmentioning

confidence: 99%

“…One way to obtain (3.5), apparently first pointed out in [1], is to apply Newton's method to the equivalent problem f/ √ f = 0 (for f > 0). It is a relatively easy exercise to obtain our sufficient conditions for convergence from Theorem 1.1 when applied to the function ff −1/2 .…”

Section: 2mentioning

confidence: 99%