New Higher Order Iterative Methods for Solving Nonlinear Equations

Huang, Shuliang; Rafiq, Arif; Shahzad, M. R.; Ali, Faisal

doi:10.15672/hjms.2017.449

Cited by 8 publications

(8 citation statements)

References 20 publications

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“…Solving nonlinear equations is one of the most significant problems in the fields of science and engineering. There is a vast literature available for finding the solution of nonlinear equations and almost all analytical and numerical methods depend upon the iteration efficiency [2,3,30,37,42]. Though some one-step methods for nonlinear equations are much attractive [19,21,22,31,36,52], an iteration process is still needed to find an even accurate solution.…”

Section: Introductionmentioning

confidence: 99%

New optimal fourth-order iterative method based on linear combination technique

Nadeem

Ali

2021

Hacettepe Journal of Mathematics and Statistics

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Newton's iteration method is widely used in numerical methods, but its convergence is low. Though a higher order iteration algorithm leads to a fast convergence, it is always complex. An optimal iteration formulation is much needed for both fast convergence and simple calculation. Here, we develop a two-step optimal fourth-order iterative method based on linear combination of two iterative schemes for nonlinear equations, and we explore the convergence criteria of the proposed method and also demonstrate its validity and efficiency by considering some test problems. We present both numerical as well as graphical comparisons. Further, the dynamical behavior of the proposed method is revealed.

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Section: Introductionmentioning

confidence: 99%

New optimal fourth-order iterative method based on linear combination technique

Nadeem

Ali

2021

Hacettepe Journal of Mathematics and Statistics

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“…Numerical analysis has interesting applications in several branches of pure and applied science that can be studied in the general framework of the non-linear equations [2,12,17,26,32]. Searching out a solution for non-linear equations is highly significant.…”

Section: Introductionmentioning

confidence: 99%

“…Searching out a solution for non-linear equations is highly significant. Due to their importance, several numerical methods have been suggested and analyzed under different conditions [1,6,9,12,21,23,25,30]. These numerical methods have been constructed by using different techniques for solving the non-linear equations such as Taylor series, quadrature formula, the variational iteration method, and the decomposition method [3,7,11,15,18,20,21,33].…”

Section: Introductionmentioning

confidence: 99%

An optimal fourth order method for solving nonlinear equations

Hafiz¹,

Khirallah²

2020

J. Math. Computer Sci.

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In this paper, we use both weight functions and composition techniques together for solving non-linear equations. We designed a new fourth order iterative method to increase the order of convergence without increasing the functional evaluations in a drastic way. This method uses one evaluation of the function and two evaluations of the first derivative. The new method attains the optimality with efficiency index 1.587. The convergence analysis of our new methods is discussed. Furthermore, the correlations between the attracting domains and the corresponding required number of iterations have also been illustrated and discussed. The comparison with several numerical methods and the use of complex dynamics and basins of attraction show that the new method gives good results.

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“…Most of the problems in science and engineering involves nonlinear equation of the form f(x) � 0, where f: D ⊆ R ⟶ R is a sufficiently smooth function in the neighborhood of a simple zero α ∈ D. Many physical problems related to diverse areas such as biological applications in population dynamics and genetics where impulses arise naturally, motion of a particle on an inclined plane and projectile motion in physics, Van der Waals problem, and continuous stirred tank reactor equation in chemistry etc., can be modelled by nonlinear equations. Consequently, many numerical methods based on different techniques have been developed for solving nonlinear equations, see for example [1][2][3][4][5][6][7][8][9][10][11][12] and references therein.…”

Section: Introductionmentioning

confidence: 99%

Iteration Methods with an Auxiliary Function for Nonlinear Equations

Ali

Aslam

Khalid

et al. 2020

Journal of Mathematics

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Various iterative methods have been introduced by involving Taylor’s series on the auxiliary function g x to solve the nonlinear equation f x = 0 . In this paper, we introduce the expansion of g x with the inclusion of weights w i such that ∑ i = 1 p w i = 1 and knots τ i ∈ 0,1 in order to develop a new family of iterative methods. The methods proposed in the present paper are applicable for different choices of auxiliary function g x , and some already known methods can be viewed as the special cases of these methods. We consider the diverse scientific/engineering models to demonstrate the efficiency of the proposed methods.

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New Higher Order Iterative Methods for Solving Nonlinear Equations

Cited by 8 publications

References 20 publications

New optimal fourth-order iterative method based on linear combination technique

New optimal fourth-order iterative method based on linear combination technique

An optimal fourth order method for solving nonlinear equations

Iteration Methods with an Auxiliary Function for Nonlinear Equations

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