On the global convergence of Halley's iteration formula

Davies, M.G.; Dawson, B.

doi:10.1007/bf01400962

Cited by 38 publications

(17 citation statements)

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“…For Halley iteration (2), Davies and Dawson [Dav75] have given a proof of the monotonic convergence to the zeros of the entire function h(x) of the following form:…”

Section: Rewrite the Modified Halley Iteration Function Asmentioning

confidence: 99%

On Halley iteration

Yao

1999

Numerische Mathematik

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“…For Halley iteration (2), Davies and Dawson [Dav75] have given a proof of the monotonic convergence to the zeros of the entire function h(x) of the following form:…”

Section: Rewrite the Modified Halley Iteration Function Asmentioning

confidence: 99%

On Halley iteration

Yao

1999

Numerische Mathematik

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“…This yields -f (xo) (7) T hen, we use N.M. (2) to approximate t he factor Xl -Xo on t he right-hand side of (7) to obtain, aft er some algebra,…”

Section: Halley's Methods (Hm)mentioning

confidence: 99%

Some Derivatives of Newton's Method

Thoo

2002

PRIMUS

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Every first-year calc ulus st udent learns Newton's met hod as part of a repertoire of numerical root-finding methods. We present two ot her numer ical root-finding met hods t hat we der ive using Newton's method: a simplified Newton's met hod , and Ha lley's mcthod . These two methods are not new, yet they seldom find th eir way into a first calculus course even though th ey would not be out of pla ce in suc h a course .

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“…We denote a system of nonlinear equations by There are also some other classical iterative methods that have convergence order three. For instance, the Halley [7,10] and Chebyshev [17] y n = initial guess, F (y n ) φ φ φ = F(y n ), F (y n ) L(y n ) = F (y n )φ φ φ, y n+1 = y n − I + 1 2 L(y n ) φ φ φ.…”

Section: Introductionmentioning

confidence: 99%

Frozen jacobian iterative method for solving systems of nonlinear equations application to nonlinear IVPs and BVPs

Ullah¹,

Ahmad²,

Alzahrani³

et al. 2016

J. Nonlinear Sci. Appl.

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Frozen Jacobian iterative methods are of practical interest to solve the system of nonlinear equations. A frozen Jacobian multi-step iterative method is presented. We divide the multi-step iterative method into two parts namely base method and multi-step part. The convergence order of the constructed frozen Jacobian iterative method is three, and we design the base method in a way that we can maximize the convergence order in the multi-step part. In the multi-step part, we utilize a single evaluation of the function, solve four systems of lower and upper triangular systems and a second frozen Jacobian. The attained convergence order per multi-step is four. Hence, the general formula for the convergence order is 3 + 4(m − 2) for m ≥ 2 and m is the number of multi-steps. In a single instance of the iterative method, we employ only single inversion of the Jacobian in the form of LU factors that makes the method computationally cheaper because the LU factors are used to solve four system of lower and upper triangular systems repeatedly. The claimed convergence order is verified by computing the computational order of convergence for a system of nonlinear equations. The efficiency and validity of the proposed iterative method are narrated by solving many nonlinear initial and boundary value problems. c 2016 All rights reserved.Keywords: Frozen Jacobian iterative methods, multi-step iterative methods, systems of nonlinear equations, nonlinear initial value problems, nonlinear boundary value problems. 2010 MSC: 65H10, 65N22.

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On the global convergence of Halley's iteration formula

Cited by 38 publications

References 3 publications

On Halley iteration

On Halley iteration

Some Derivatives of Newton's Method

Frozen jacobian iterative method for solving systems of nonlinear equations application to nonlinear IVPs and BVPs

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