scholarly journals An Efficient Class of Traub–Steffensen-Type Methods for Computing Multiple Zeros

Axioms ◽  
2019 ◽  
Vol 8 (2) ◽  
pp. 65 ◽  
Author(s):  
Deepak Kumar ◽  
Janak Raj Sharma ◽  
Clemente Cesarano
Keyword(s):  
Iterative Methods ◽  
Basins Of Attraction ◽  
Higher Order ◽  
Order Derivative ◽  
Third Order ◽  
Higher Order Methods ◽  
Multiple Zeros ◽  
Derivative Free ◽  
Graphical Tool ◽  
Numerical Experimentation

Numerous higher-order methods with derivative evaluations are accessible in the literature for computing multiple zeros. However, higher-order methods without derivatives are very rare for multiple zeros. Encouraged by this fact, we present a family of third-order derivative-free iterative methods for multiple zeros that require only evaluations of three functions per iteration. Convergence of the proposed class is demonstrated by means of using a graphical tool, namely basins of attraction. Applicability of the methods is demonstrated through numerical experimentation on different functions that illustrates the efficient behavior. Comparison of numerical results shows that the presented iterative methods are good competitors to the existing techniques.

scholarly journals Development of Optimal Eighth Order Derivative-Free Methods for Multiple Roots of Nonlinear Equations

Symmetry ◽  
2019 ◽  
Vol 11 (6) ◽  
pp. 766 ◽  
Author(s):  
Janak Raj Sharma ◽  
Sunil Kumar ◽  
Ioannis K. Argyros
Keyword(s):  
Basins Of Attraction ◽  
Higher Order ◽  
Order Derivative ◽  
Multiple Roots ◽  
Nonlinear Functions ◽  
Eighth Order ◽  
Multiple Zeros ◽  
Derivative Free ◽  
Graphical Tool ◽  
Derivative Free Methods

A number of higher order iterative methods with derivative evaluations are developed in literature for computing multiple zeros. However, higher order methods without derivative for multiple zeros are difficult to obtain and hence such methods are rare in literature. Motivated by this fact, we present a family of eighth order derivative-free methods for computing multiple zeros. Per iteration the methods require only four function evaluations, therefore, these are optimal in the sense of Kung-Traub conjecture. Stability of the proposed class is demonstrated by means of using a graphical tool, namely, basins of attraction. Boundaries of the basins are fractal like shapes through which basins are symmetric. Applicability of the methods is demonstrated on different nonlinear functions which illustrates the efficient convergence behavior. Comparison of the numerical results shows that the new derivative-free methods are good competitors to the existing optimal eighth-order techniques which require derivative evaluations.

scholarly journals An Optimal Fourth Order Derivative-Free Numerical Algorithm for Multiple Roots

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 1038 ◽  
Author(s):  
Sunil Kumar ◽  
Deepak Kumar ◽  
Janak Raj Sharma ◽  
Clemente Cesarano ◽  
Praveen Agarwal ◽  
...  
Keyword(s):  
Iterative Methods ◽  
Numerical Algorithm ◽  
Fourth Order ◽  
Iterative Scheme ◽  
Higher Order ◽  
New Technique ◽  
Multiple Roots ◽  
Order Convergence ◽  
Higher Order Methods ◽  
Derivative Free

A plethora of higher order iterative methods, involving derivatives in algorithms, are available in the literature for finding multiple roots. Contrary to this fact, the higher order methods without derivatives in the iteration are difficult to construct, and hence, such methods are almost non-existent. This motivated us to explore a derivative-free iterative scheme with optimal fourth order convergence. The applicability of the new scheme is shown by testing on different functions, which illustrates the excellent convergence. Moreover, the comparison of the performance shows that the new technique is a good competitor to existing optimal fourth order Newton-like techniques.

A Class of Higher-Order Newton-Like Methods for Systems of Nonlinear Equations

2021 ◽  
pp. 2150059
Author(s):  
Janak Raj Sharma ◽  
Sunil Kumar ◽  
Ioannis K. Argyros
Keyword(s):  
Nonlinear Equations ◽  
Inverse Operator ◽  
General Setting ◽  
Basins Of Attraction ◽  
Function Evaluation ◽  
Systems Of Nonlinear Equations ◽  
Third Order ◽  
Additional Function ◽  
Numerical Experimentation ◽  
The Cost

In this paper, a class of efficient iterative methods with increasing order of convergence for solving systems of nonlinear equations is developed and analyzed. The methodology uses well-known third-order Potra–Pták iteration in the first step and Newton-like iterations in the subsequent steps. Novelty of the methods is the increase in convergence order by an amount three per step at the cost of only one additional function evaluation. In addition, the algorithm uses a single inverse operator in each iteration, which makes it computationally more efficient and attractive. Local convergence is studied in the more general setting of a Banach space under suitable assumptions. Theoretical results of convergence and computational efficiency are verified through numerical experimentation. Comparison of numerical results indicates that the developed algorithms outperform the other similar algorithms available in the literature, particularly when applied to solve the large systems of equations. The basins of attraction of some of the existing methods along with the proposed method are given to exhibit their performance.

scholarly journals On a Reduced Cost Higher Order Traub-Steffensen-Like Method for Nonlinear Systems

Symmetry ◽  
2019 ◽  
Vol 11 (7) ◽  
pp. 891 ◽  
Author(s):  
Janak Raj Sharma ◽  
Deepak Kumar ◽  
Lorentz Jäntschi
Keyword(s):  
Order Of Convergence ◽  
Higher Order ◽  
New Method ◽  
Systems Of Nonlinear Equations ◽  
Third Order ◽  
Type Method ◽  
Cpu Time ◽  
Chebyshev’S Method ◽  
Derivative Free ◽  
Fifth Order

We propose a derivative-free iterative method with fifth order of convergence for solving systems of nonlinear equations. The scheme is composed of three steps, of which the first two steps are that of third order Traub-Steffensen-type method and the last is derivative-free modification of Chebyshev’s method. Computational efficiency is examined and comparison between the efficiencies of presented technique with existing techniques is performed. It is proved that, in general, the new method is more efficient. Numerical problems, including those resulting from practical problems viz. integral equations and boundary value problems, are considered to compare the performance of the proposed method with existing methods. Calculation of computational order of convergence shows that the order of convergence of the new method is preserved in all the numerical examples, which is not so in the case of some of the existing higher order methods. Moreover, the numerical results, including the CPU-time consumed in the execution of program, confirm the accurate and efficient behavior of the new technique.

scholarly journals Some Real-Life Applications of a Newly Constructed Derivative Free Iterative Scheme

Symmetry ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 239 ◽  
Author(s):  
Ramandeep Behl ◽  
M. Salimi ◽  
M. Ferrara ◽  
S. Sharifi ◽  
Samaher Alharbi
Keyword(s):  
Iterative Methods ◽  
Chemical Engineering ◽  
Flame Temperature ◽  
Real Life ◽  
Chemical Reactor ◽  
Basins Of Attraction ◽  
Eighth Order ◽  
Present Scheme ◽  
New Family ◽  
Derivative Free

In this study, we present a new higher-order scheme without memory for simple zeros which has two major advantages. The first one is that each member of our scheme is derivative free and the second one is that the present scheme is capable of producing many new optimal family of eighth-order methods from every 4-order optimal derivative free scheme (available in the literature) whose first substep employs a Steffensen or a Steffensen-like method. In addition, the theoretical and computational properties of the present scheme are fully investigated along with the main theorem, which demonstrates the convergence order and asymptotic error constant. Moreover, the effectiveness of our scheme is tested on several real-life problems like Van der Waal’s, fractional transformation in a chemical reactor, chemical engineering, adiabatic flame temperature, etc. In comparison with the existing robust techniques, the iterative methods in the new family perform better in the considered test examples. The study of dynamics on the proposed iterative methods also confirms this fact via basins of attraction applied to a number of test functions.

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