A modified Newton method with cubic convergence: the multivariate case

Homeier, Herbert H. H.

doi:10.1016/j.cam.2003.12.041

Cited by 142 publications

(91 citation statements)

References 6 publications

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“…It converges quadratically when an initial guess value (0) x is close to the roots ξ of the system of nonlinear equations. In order to improve the order of convergence, a few two-step variants of Newton's methods with cubic convergence have been proposed in some literature [3][4][5][6][7][8][9] and references therein. S.…”

Section: F Xmentioning

confidence: 99%

A Family Third-order Methods with One Parameter for Solving Systems of Nonlinear Equations and BVP-ODEs

Liu¹,

Zhang²

2017

dtetr

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Abstract. Efficient numerical solutions for systems of nonlinear equations have always appealed greatly to people in scientific computation and engineering fields. And how to make the two iterative approximate values as precise as possible is an important issue in actual computation. In this paper, we use the linear-combination method to construct a family third-order scheme with one parameter for solving systems of nonlinear equations. Their cubic convergence and corresponding error equation are proved theoretically, and numerical examples are demonstrated as well to show the efficiency and feasibility of the suggested iterative methods.

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Section: F Xmentioning

confidence: 99%

A Family Third-order Methods with One Parameter for Solving Systems of Nonlinear Equations and BVP-ODEs

Liu¹,

Zhang²

2017

dtetr

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“…For multiple roots cases, some of these mentioned methods are studied in [13,14]. In addition to these, for multivariate cases some variants of Newton method are investigated in [15,16]. Here, we give some basic definitions and properties of the multiplicative derivative theory which can be found in [17,18].…”

Section: Introductionmentioning

confidence: 99%

Multiplicative Newton s Methods with Cubic Convergence

Cumhur¹,

Gökdoğan²,

Ünal³

2017

ntmsci

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Abstract:In this paper, we develop some modifications of the multiplicative Newton method which are third-order convergence. We use the multiplicative Newton Theorem and Newton Cotes quadrature formulas to present these new modifications of the multiplicative Newton method. Using the multiplicative Taylor expansion, we give also the convergence analysis of these new methods. Furthermore, we compare the multiplicative Newton methods with the classical Newton methods in details.

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“…A further multivariate version of this method has been discussed in [4,6]. By applying Newton's method to the inverse function x = f (y) instead y = f (x), in [7], Homeier derived the following cubically convergent iteration scheme:…”

Section: Introductionmentioning

confidence: 99%

A new Householders method free from second derivatives for solving nonlinear equations and polynomiography

Nazeer¹,

Tanveer²,

Kang³

et al. 2016

J. Nonlinear Sci. Appl.

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In this paper, we describe the new Husehölder's method free from second derivatives for solving nonlinear equations. The new Husehölder's method has convergence of order five and efficiency index 5 1 3 ≈ 1.70998, which converges faster than the Newton's method, the Halley's method and the Husehölder's method. The comparison table demonstrate the faster convergence of our method. Polynomiography via the new Husehölder's method is also presented.

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A modified Newton method with cubic convergence: the multivariate case

Cited by 142 publications

References 6 publications

A Family Third-order Methods with One Parameter for Solving Systems of Nonlinear Equations and BVP-ODEs

A Family Third-order Methods with One Parameter for Solving Systems of Nonlinear Equations and BVP-ODEs

Multiplicative Newton s Methods with Cubic Convergence

A new Householders method free from second derivatives for solving nonlinear equations and polynomiography

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