A family of higher order iterations free from second derivative for nonlinear equations in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="mml224" display="inline" overflow="scroll" altimg="si224.gif"><mml:mi mathvariant="double-struck">R</mml:mi></mml:math>

Kumar, Akhilesh; Prashanth, M.; Behl, Ramandeep; Gupta, Dharmendra K.; Motsa, S. S.

doi:10.1016/j.cam.2017.07.005

Cited by 14 publications

(7 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Let us note that the iterative methods BB2-BB5, MM1-MM4, and BB1 satisfy Cayley's test [28][29][30][31] for all parameter values of β. It can be observed from Figure 8 that iterative methods BB1-BB5 and MM1-MM4 verifying Cayley's test [28] have the same dynamical properties as Newton's method.…”

Section: Dynamical Planesmentioning

confidence: 99%

Study of Dynamical Behavior and Stability of Iterative Methods for Nonlinear Equation with Applications in Engineering

Rafiq

Akram

Mir

et al. 2020

Mathematical Problems in Engineering

View full text Add to dashboard Cite

In this article, we first construct a family of optimal 2-step iterative methods for finding a single root of the nonlinear equation using the procedure of weight function. We then extend these methods for determining all roots simultaneously. Convergence analysis is presented for both cases to show that the order of convergence is 4 in case of the single-root finding method and is 6 for simultaneous determination of all distinct as well as multiple roots of a nonlinear equation. The dynamical behavior is presented to analyze the stability of fixed and critical points of the rational operator of one-point iterative methods. The computational cost, basins of attraction, efficiency, log of the residual, and numerical test examples show that the newly constructed methods are more efficient as compared with the existing methods in the literature.

show abstract

Section: Dynamical Planesmentioning

confidence: 99%

Study of Dynamical Behavior and Stability of Iterative Methods for Nonlinear Equation with Applications in Engineering

Rafiq

Akram

Mir

et al. 2020

Mathematical Problems in Engineering

View full text Add to dashboard Cite

show abstract

“…Nazeer et al [24] in 2016 proposed a new second derivative free generalized Newton-Raphson's method with convergence of order five by means of finite difference scheme. In 2017, Kumar et al [25] suggested a sixth-order parameter-based family of algorithms for solving nonlinear equations. In the same year, Salimi et al [26] proposed an optimal class of eighth-order methods by using weight functions and Newton interpolation technique.…”

Section: Introductionmentioning

confidence: 99%

Some Novel Sixth-Order Iteration Schemes for Computing Zeros of Nonlinear Scalar Equations and Their Applications in Engineering

Rehman

Naseem

Abdeljawad

2021

Journal of Function Spaces

View full text Add to dashboard Cite

In this paper, we propose two novel iteration schemes for computing zeros of nonlinear equations in one dimension. We develop these iteration schemes with the help of Taylor’s series expansion, generalized Newton-Raphson’s method, and interpolation technique. The convergence analysis of the proposed iteration schemes is discussed. It is established that the newly developed iteration schemes have sixth order of convergence. Several numerical examples have been solved to illustrate the applicability and validity of the suggested schemes. These problems also include some real-life applications associated with the chemical and civil engineering such as adiabatic flame temperature equation, conversion of nitrogen-hydrogen feed to ammonia, the van der Wall’s equation, and the open channel flow problem whose numerical results prove the better efficiency of these methods as compared to other well-known existing iterative methods of the same kind.

show abstract

“…Nazeer et al [30], in (2016), proposed a new second derivative free generalized Newton-Raphson's method with convergence of order five by means of finite difference scheme. After that Kumar et al [23], in 2018, established a new sixth-order parameter based family of algorithm for solving non-linear equations. In 2019, Solaiman et al [38] suggested derivative free optimal fourth-order and eighthorder modifications of King's method by implementing the composition technique combined with rational interpolation, and the idea of Padé approximation.…”

Section: Introductionmentioning

confidence: 99%

Novel Iteration Schemes for Computing Zeros of Non-Linear Equations With Engineering Applications and Their Dynamics

et al. 2021

View full text Add to dashboard Cite

The task of root-finding of the non-linear equations is perhaps, one of the most complicated problems in applied mathematics especially in a diverse range of engineering applications. The characteristics of the root-finding methods such as convergence rate, performance, efficiency, etc., is directly rely upon the initial guess of the solution to execute the process in most of the systems of non-linear equations. Keeping these facts into mind, based on Taylor's series expansion, we present some new modifications of Halley, Househölder and Golbabai and Javidi's methods and then making them second derivative free by applying Taylor's series. The convergence analysis of the suggested methods is discussed. It is established that the proposed methods possess convergence of orders five and six. Several numerical problems have been tested to demonstrate the validity and applicability of the proposed methods. These test examples also include some real-life problems associated with the chemical and civil engineering such as open channel flow problem, the adiabatic flame temperature equation, conversion of nitrogen-hydrogen feed to ammonia and the van der Wall's equation whose numerical results prove the better performance of the suggested method as compared to other well known existing methods of the same kind in literature. Finally, the dynamics of the presented algorithms in the form of polynomiographs have been shown with the aid of computer program by considering some complex polynomials and compared them with the other wellknown iterative algorithms that revealed the convergence speed and other dynamical aspects of the presented methods.

show abstract

A family of higher order iterations free from second derivative for nonlinear equations inR

Cited by 14 publications

References 26 publications

Study of Dynamical Behavior and Stability of Iterative Methods for Nonlinear Equation with Applications in Engineering

Study of Dynamical Behavior and Stability of Iterative Methods for Nonlinear Equation with Applications in Engineering

Some Novel Sixth-Order Iteration Schemes for Computing Zeros of Nonlinear Scalar Equations and Their Applications in Engineering

Novel Iteration Schemes for Computing Zeros of Non-Linear Equations With Engineering Applications and Their Dynamics

Contact Info

Product

Resources

About