A Variant of Sharma-Arora's Optimal Eighth-Order Family of Methods for Finding A Simple Root of Nonlinear Equation

Ćebić, Dejan; Paunović, Marija; Ralević, Nebojša

doi:10.1109/sisy.2018.8524857

Cited by 3 publications

(2 citation statements)

References 23 publications

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“…In the modern age, the researchers improved the order and efficiency of the existing methods and introduced multistep algorithms. For more details, see [2][3][4][5][6][7][8][9][10][11] and the references cited therein. Usually, multistep algorithms possess greater convergence with higher computational cost due to the involvement of higher derivatives which is the downside of these algorithms.…”

Section: Introductionmentioning

confidence: 99%

A New Root‐Finding Algorithm for Solving Real‐World Problems and Its Complex Dynamics via Computer Technology

2021

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Nowadays, the use of computers is becoming very important in various fields of mathematics and engineering sciences. Many complex statistics can be sorted out easily with the help of different computer programs in seconds, especially in computational and applied Mathematics. With the help of different computer tools and languages, a variety of iterative algorithms can be operated in computers for solving different nonlinear problems. The most important factor of an iterative algorithm is its efficiency that relies upon the convergence rate and computational cost per iteration. Taking these facts into account, this article aims to design a new iterative algorithm that is derivative-free and performs better. We construct this algorithm by applying the forward- and finite-difference schemes on Golbabai–Javidi’s method which yields us an efficient and derivative-free algorithm whose computational cost is low as per iteration. We also study the convergence criterion of the designed algorithm and prove its quartic-order convergence. To analyze it numerically, we consider nine different types of numerical test examples and solve them for demonstrating its accuracy, validity, and applicability. The considered problems also involve some real-life applications of civil and chemical engineering. The obtained numerical results of the test examples show that the newly designed algorithm is working better against the other similar algorithms in the literature. For the graphical analysis, we consider some different degrees’ complex polynomials and draw the polynomiographs of the designed quartic-order algorithm and compare it with the other similar existing methods with the help of a computer program. The graphical results reveal the better convergence speed and the other graphical characteristics of the designed algorithm over the other comparable ones.

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Section: Introductionmentioning

confidence: 99%

A New Root‐Finding Algorithm for Solving Real‐World Problems and Its Complex Dynamics via Computer Technology

2021

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“…which is cubically convergent Golbabai and Javidi's method for finding zeros of non-linear equations in one dimension In recent years, many mathematicians tried to modify existing methods using different mathematical techniques and proposed some new multi-step iterative methods, having higher order of convergence [1]- [4], [7], [10], [28], [29], [32], [35], [39].…”

Section: Introductionmentioning

confidence: 99%

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

et al. 2021

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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.

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A Novel Root-Finding Algorithm With Engineering Applications and its Dynamics via Computer Technology

2022

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Root-finding of non-linear equations is one of the most appearing problems in engineering sciences. Most of the complicated engineering problems can be modeled easily by means of non-linear functions. The role of iterative algorithms via computers for solving such functions is much important and cannot be denied in the modern age. In an iterative algorithm, the convergence order and the computational cost per iteration are the main characteristics that depict its efficiency and performance i.e., a method with higher-order and lower computational cost will be more efficient and vice versa. Keeping these facts into consideration, the main goal of this paper is to introduce a new derivative-free iterative method that performs better. We develop this algorithm by utilizing the forward-and finite-difference schemes on well-known Househ"older's method, resulting in an efficient and derivative-free algorithm with a low per iteration computing cost. We also look at the developed algorithm's convergence criterion and show that it is quartic-order convergent. We investigate nine test-examples and solve them to demonstrate its correctness, validity, and efficiency numerically. Some real-world engineering problems in civil and chemical engineering are also included in these examples. The numerical results of the test-examples reveal that the newly constructed method outperforms the existing similar algorithms found in the literature. We consider various different-degrees complex polynomials for the graphical analysis and used a computer tool to create the polynomiographs of the proposed quartic-order algorithm and compare it to other comparable existing approaches. The graphical findings show that the developed method has a faster convergence speed than the other comparable algorithms.

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A Variant of Sharma-Arora's Optimal Eighth-Order Family of Methods for Finding A Simple Root of Nonlinear Equation

Cited by 3 publications

References 23 publications

A New Root‐Finding Algorithm for Solving Real‐World Problems and Its Complex Dynamics via Computer Technology

A New Root‐Finding Algorithm for Solving Real‐World Problems and Its Complex Dynamics via Computer Technology

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

A Novel Root-Finding Algorithm With Engineering Applications and its Dynamics via Computer Technology

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