A fourth-order method from quadrature formulae to solve systems of nonlinear equations

Darvishi, M. T.; Barati, Ali

doi:10.1016/j.amc.2006.09.115

Cited by 79 publications

(59 citation statements)

References 2 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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“…In general, few papers for the multidimensional case introduce methods with high order of convergence. The authors design in [3] a modified Newton-Jarrat scheme of sixth-order; in [4] a third-order method is presented for computing real and complex roots of nonlinear systems; Darvishi et al in [5] improve the order of convergence of known methods from quadrature formulae; Shin et. al.…”

Section: Introductionmentioning

confidence: 99%

Increasing the order of convergence of iterative schemes for solving nonlinear systems

Cordero

Torregrosa

Vassileva

2013

Journal of Computational and Applied Mathematics

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A set of multistep iterative methods with increasing order of convergence is presented, for solving systems of nonlinear equations. One of the main advantages of these schemes is to achieve high order of convergence with few Jacobian and functional evaluations, joint with the use of the same matrix of coefficients in the most of the linear systems involved in the process. Indeed, the application of the pseudocomposition technique on these proposed schemes allows us to increase their order of convergence, obtaining new high-order, efficient methods. Finally, some numerical tests are performed in order to check their practical behavior.

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“…In order to improve the order of convergence of Newton's method, many modifications have been proposed in literature, for example, see [6][7][8][9][10][11][12][13][14][15][16][17][18] and references therein. For a system of equations in unknowns, the first Fréchet derivative is a matrix with 2 evaluations whereas the second Fréchet derivative has 2 ( + 1)/2 evaluations.…”

Section: Introductionmentioning

confidence: 99%

On Some Efficient Techniques for Solving Systems of Nonlinear Equations

Sharma

Gupta²

2013

Advances in Numerical Analysis

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We present iterative methods of convergence order three, five, and six for solving systems of nonlinear equations. Third-order method is composed of two steps, namely, Newton iteration as the first step and weighted-Newton iteration as the second step. Fifth and sixth-order methods are composed of three steps of which the first two steps are same as that of the third-order method whereas the third is again a weighted-Newton step. Computational efficiency in its general form is discussed and a comparison between the efficiencies of proposed techniques with existing ones is made. The performance is tested through numerical examples. Moreover, theoretical results concerning order of convergence and computational efficiency are verified in the examples. It is shown that the present methods have an edge over similar existing methods, particularly when applied to large systems of equations.

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A fourth-order method from quadrature formulae to solve systems of nonlinear equations

Cited by 79 publications

References 2 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

Increasing the order of convergence of iterative schemes for solving nonlinear systems

On Some Efficient Techniques for Solving Systems of Nonlinear Equations

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