scholarly journals A 4th-Order Optimal Extension of Ostrowski’s Method for Multiple Zeros of Univariate Nonlinear Functions

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 803 ◽  
Author(s):  
Ramandeep Behl ◽  
Waleed M. Al-Hamdan
Keyword(s):  
Nonlinear Functions ◽  
Numerical Examples ◽  
Multiple Zeros ◽  
New Methods ◽  
Design Structure ◽  
Optimal Extension ◽  
Robust Techniques ◽  
New Strategy ◽  
Ostrowski’S Method ◽  
Good Number

We present a new optimal class of Ostrowski’s method for obtaining multiple zeros of univariate nonlinear functions. Several researchers tried to construct an optimal family of Ostrowski’s method for multiple zeros, but they did not have success in this direction. The new strategy adopts a weight function approach. The design structure of new families of Ostrowski’s technique is simpler than the existing classical families of the same order for multiple zeros. The classical Ostrowski’s method of fourth-order can obtain a particular form for the simple root. Their efficiency is checked on a good number of relevant numerical examples. These results demonstrate the performance of our methods. We find that the new methods are just as competent as other existing robust techniques available in the literature.

An efficient class of fourth-order derivative-free method for multiple-roots

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Sunil Kumar ◽  
Deepak Kumar ◽  
Janak Raj Sharma ◽  
Ioannis K. Argyros
Keyword(s):  
Multiple Root ◽  
Second Step ◽  
Optimal Order ◽  
Multiple Roots ◽  
Nonlinear Functions ◽  
New Methods ◽  
Derivative Free ◽  
Derivative Free Methods ◽  
Derivative Free Method ◽  
Good Number

Abstract Many optimal order multiple root techniques, which use derivatives in the algorithm, have been proposed in literature. Many researchers tried to construct an optimal family of derivative-free methods for multiple roots, but they did not get success in this direction. With this as a motivation factor, here, we present a new optimal class of derivative-free methods for obtaining multiple roots of nonlinear functions. This procedure involves Traub–Steffensen iteration in the first step and Traub–Steffensen-like iteration in the second step. Efficacy is checked on a good number of relevant numerical problems that verifies the efficient convergent nature of the new methods. Moreover, we find that the new derivative-free methods are just as competent as the other existing robust methods that use derivatives.

scholarly journals Laguerre-like methods for the simultaneous approximation of polynomial multiple zeros

2006 ◽  
Vol 16 (1) ◽  
pp. 31-44
Author(s):  
Miodrag Petkovic ◽  
Lidija Rancic ◽  
Dusan Milosevic
Keyword(s):  
Fixed Point ◽  
Convergence Rate ◽  
Fourth Order ◽  
Interval Arithmetic ◽  
Simultaneous Approximation ◽  
Numerical Examples ◽  
Multiple Zeros ◽  
Complex Arithmetic ◽  
New Methods

Two new methods of the fourth order for the simultaneous determination of multiple zeros of a polynomial are proposed. The presented methods are based on the fixed point relation of Laguerre's type and realized in ordinary complex arithmetic as well as circular complex interval arithmetic. The derived iterative formulas are suitable for the construction of modified methods with improved convergence rate with negligible additional operations. Very fast convergence of the considered methods is illustrated by two numerical examples.

Extended conjugate gradient squared and conjugate residual squared methods for solving the generalized coupled Sylvester tensor equations

2020 ◽  
pp. 014233122093238
Author(s):  
Eisa Khosravi Dehdezi ◽  
Saeed Karimi
Keyword(s):  
Iterative Methods ◽  
Conjugate Gradient ◽  
Numerical Examples ◽  
New Methods ◽  
Conjugate Gradient Squared ◽  
And Performance ◽  
Tensor Computations ◽  
Conjugate Residual

In this paper, two attractive iterative methods – conjugate gradient squared (CGS) and conjugate residual squared (CRS) – are extended to solve the generalized coupled Sylvester tensor equations [Formula: see text]. The proposed methods use tensor computations with no maricizations involved. Also, some properties of the new methods are presented. Finally, several numerical examples are given to compare the efficiency and performance of the proposed methods with some existing algorithms.

A higher order method for multiple zeros of nonlinear functions

1983 ◽  
Vol 12 (3-4) ◽  
pp. 329-335 ◽  
Author(s):  
H. D. Victory ◽  
B. Neta
Keyword(s):  
Higher Order ◽  
Order Method ◽  
Nonlinear Functions ◽  
Multiple Zeros ◽  
Higher Order Method

scholarly journals Some New Iterative Methods for Nonlinear Equations

2010 ◽  
Vol 2010 ◽  
pp. 1-12 ◽  
Author(s):  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor ◽  
Eisa Al-Said ◽  
Muhammad Waseem
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Fourth Order ◽  
System Of Equations ◽  
Decomposition Technique ◽  
Methods Comparison ◽  
Numerical Examples ◽  
New Methods ◽  
And Performance

We suggest and analyze some new iterative methods for solving the nonlinear equationsf(x)=0using the decomposition technique coupled with the system of equations. We prove that new methods have convergence of fourth order. Several numerical examples are given to illustrate the efficiency and performance of the new methods. Comparison with other similar methods is given.

scholarly journals A Family of Fourteenth-Order Convergent Iterative Methods for Solving Nonlinear Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Fiza Zafar ◽  
Gulshan Bibi
Keyword(s):  
Iterative Method ◽  
Iterative Methods ◽  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Efficiency Index ◽  
Numerical Examples ◽  
New Class ◽  
First Derivative ◽  
New Methods ◽  
Specific Step

We present a family of fourteenth-order convergent iterative methods for solving nonlinear equations involving a specific step which when combined with any two-step iterative method raises the convergence order by n+10, if n is the order of convergence of the two-step iterative method. This new class include four evaluations of function and one evaluation of the first derivative per iteration. Therefore, the efficiency index of this family is 141/5 =1.695218203. Several numerical examples are given to show that the new methods of this family are comparable with the existing methods.

Efficient Halley-like Methods for the Inclusion of Multiple Zeros of Polynomials

2012 ◽  
Vol 12 (3) ◽  
pp. 351-366 ◽  
Author(s):  
Miodrag S. Petković ◽  
Mimica R. Milošević
Keyword(s):  
Fixed Point ◽  
Convergence Rate ◽  
Fourth Order ◽  
Interval Arithmetic ◽  
Order Of Convergence ◽  
Single Step ◽  
Zeros Of Polynomials ◽  
Numerical Examples ◽  
Multiple Zeros ◽  
Inclusion Methods

AbstractStarting from suitable zero-relation, we derive higher-order iterative methods for the simultaneous inclusion of polynomial multiple zeros in circular complex interval arithmetic. The convergence rate is increased using a family of two-point methods of the fourth order for solving nonlinear equations as a predictor. The methods are more efficient compared to existing inclusion methods for multiple zeros, based on fixed point relations. Using the concept of the R-order of convergence of mutually dependent sequences, we present the convergence analysis of the total-step and the single-step methods. The proposed self-validated methods possess a great computational efficiency since the acceleration of the convergence rate from four to seven is achieved only by a few additional calculations. To demonstrate convergence behavior of the presented methods, two numerical examples are given.

scholarly journals Modified King’s Family for Multiple Zeros of Scalar Nonlinear Functions

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 827
Author(s):  
Ramandeep Behl ◽  
Munish Kansal ◽  
Mehdi Salimi
Keyword(s):  
Fourth Order ◽  
Numerical Experiments ◽  
Scientific Literature ◽  
Numerical Comparison ◽  
Nonlinear Functions ◽  
Multiple Zeros ◽  
Cpu Time ◽  
Order Scheme ◽  
Function Strategy ◽  
Exhaustive Study

There is no doubt that there is plethora of optimal fourth-order iterative approaches available to estimate the simple zeros of nonlinear functions. We can extend these method/methods for multiple zeros but the main issue is to preserve the same convergence order. Therefore, numerous optimal and non-optimal modifications have been introduced in the literature to preserve the order of convergence. Such count of methods that can estimate the multiple zeros are limited in the scientific literature. With this point, a new optimal fourth-order scheme is presented for multiple zeros with known multiplicity. The proposed scheme is based on the weight function strategy involving functions in ratio. Moreover, the scheme is optimal as it satisfies the hypothesis of Kung–Traub conjecture. An exhaustive study of the convergence is shown to determine the fourth order of the methods under certain conditions. To demonstrate the validity and appropriateness for the proposed family, several numerical experiments have been performed. The numerical comparison highlights the effectiveness of scheme in terms of accuracy, stability, and CPU time.

scholarly journals A stable class of modified Newton-like methods for multiple roots and their dynamics

2020 ◽  
Vol 21 (6) ◽  
pp. 603-621
Author(s):  
Munish Kansal ◽  
Alicia Cordero ◽  
Juan R. Torregrosa ◽  
Sonia Bhalla
Keyword(s):  
Real Life ◽  
Chemical Reactor ◽  
Function Technique ◽  
Continuous Stirred Tank Reactor ◽  
Parallel Plates ◽  
Nonlinear Functions ◽  
Eighth Order ◽  
Multiple Zeros ◽  
Life Problems ◽  
Global Carbon

AbstractThere have appeared in the literature a lot of optimal eighth-order iterative methods for approximating simple zeros of nonlinear functions. Although, the similar ideas can be extended for the case of multiple zeros but the main drawback is that the order of convergence and computational efficiency reduce dramatically. Therefore, in order to retain the accuracy and convergence order, several optimal and non-optimal modifications have been proposed in the literature. But, as far as we know, there are limited number of optimal eighth-order methods that can handle the case of multiple zeros. With this aim, a wide general class of optimal eighth-order methods for multiple zeros with known multiplicity is brought forward, which is based on weight function technique involving function-to-function ratio. An extensive convergence analysis is demonstrated to establish the eighth-order of the developed methods. The numerical experiments considered the superiority of the new methods for solving concrete variety of real life problems coming from different disciplines such as trajectory of an electron in the air gap between two parallel plates, the fractional conversion in a chemical reactor, continuous stirred tank reactor problem, Planck’s radiation law problem, which calculates the energy density within an isothermal blackbody and the problem arising from global carbon dioxide model in ocean chemistry, in comparison with methods of similar characteristics appeared in the literature.

scholarly journals Modified Homotopy Perturbation Technique for the Approximate Solution of Nonlinear Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Farooq Ahmed Shah
Keyword(s):  
Approximate Solution ◽  
Iterative Methods ◽  
Nonlinear Equations ◽  
Homotopy Perturbation Method ◽  
Perturbation Technique ◽  
Numerical Examples ◽  
Homotopy Perturbation ◽  
New Methods ◽  
And Performance ◽  
Homotopy Perturbation Technique

We use a new modified homotopy perturbation method to suggest and analyze some new iterative methods for solving nonlinear equations. This new modification of the homotopy method is quite flexible. Various numerical examples are given to illustrate the efficiency and performance of the new methods. These new iterative methods may be viewed as an addition and generalization of the existing methods for solving nonlinear equations.

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