scholarly journals Sixteenth-Order Optimal Iterative Scheme Based on Inverse Interpolatory Rational Function for Nonlinear Equations

Symmetry ◽  
2019 ◽  
Vol 11 (5) ◽  
pp. 691 ◽  
Author(s):  
Mehdi Salimi ◽  
Ramandeep Behl
Keyword(s):  
Iterative Scheme ◽  
General Scheme ◽  
Constant Term ◽  
Convergence Order ◽  
Robust Methods ◽  
Eighth Order ◽  
Asymptotic Error ◽  
Error Constant ◽  
Order Scheme ◽  
Asymptotic Error Constant

The principal motivation of this paper is to propose a general scheme that is applicable to every existing multi-point optimal eighth-order method/family of methods to produce a further sixteenth-order scheme. By adopting our technique, we can extend all the existing optimal eighth-order schemes whose first sub-step employs Newton’s method for sixteenth-order convergence. The developed technique has an optimal convergence order regarding classical Kung-Traub conjecture. In addition, we fully investigated the computational and theoretical properties along with a main theorem that demonstrates the convergence order and asymptotic error constant term. By using Mathematica-11 with its high-precision computability, we checked the efficiency of our methods and compared them with existing robust methods with same convergence order.

A General Way to Construct a New Optimal Scheme with Eighth-Order Convergence for Nonlinear Equations

2019 ◽  
Vol 17 (01) ◽  
pp. 1843011
Author(s):  
Ramandeep Behl ◽  
Changbum Chun ◽  
Ali Saleh Alshormani ◽  
S. S. Motsa
Keyword(s):  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Functional Approach ◽  
Eighth Order ◽  
Optimal Scheme ◽  
Asymptotic Error ◽  
Error Constant ◽  
Order Scheme ◽  
Asymptotic Error Constant ◽  
Computational Properties

In this paper, we present a new and interesting optimal scheme of order eight in a general way for solving nonlinear equations, numerically. The beauty of our scheme is that it is capable of producing further new and interesting optimal schemes of order eight from every existing optimal fourth-order scheme whose first substep employs Newton’s method. The construction of this scheme is based on rational functional approach. The theoretical and computational properties of the proposed scheme are fully investigated along with a main theorem which establishes the order of convergence and asymptotic error constant. Several numerical examples are given and analyzed in detail to demonstrate faster convergence and higher computational efficiency of our methods.

scholarly journals A New Biparametric Family of Two-Point Optimal Fourth-Order Multiple-Root Finders

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Young Ik Kim ◽  
Young Hee Geum
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Fourth Order ◽  
Order Of Convergence ◽  
Multiple Root ◽  
Multiple Roots ◽  
Asymptotic Error ◽  
Error Constant ◽  
Asymptotic Error Constant

We construct a biparametric family of fourth-order iterative methods to compute multiple roots of nonlinear equations. This method is verified to be optimally convergent. Various nonlinear equations confirm our proposed method with order of convergence of four and show that the computed asymptotic error constant agrees with the theoretical one.

scholarly journals On a New Iterative Scheme without Memory with Optimal Eighth Order

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
M. Sharifi ◽  
S. Karimi Vanani ◽  
F. Khaksar Haghani ◽  
M. Arab ◽  
S. Shateyi
Keyword(s):  
Nonlinear Equations ◽  
Numerical Experiments ◽  
Iterative Scheme ◽  
Convergence Order ◽  
Free Form ◽  
Eighth Order ◽  
Derivative Free ◽  
Theoretical Results

The purpose of this paper is to derive and discuss a three-step iterative expression for solving nonlinear equations. In fact, we derive a derivative-free form for one of the existing optimal eighth-order methods and preserve its convergence order. Theoretical results will be upheld by numerical experiments.

scholarly journals Ball Convergence of an Efficient Eighth Order Iterative Method Under Weak Conditions

Mathematics ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 260 ◽  
Author(s):  
Janak Sharma ◽  
Ioannis Argyros ◽  
Sunil Kumar
Keyword(s):  
Iterative Method ◽  
Iterative Methods ◽  
Higher Order ◽  
High Order ◽  
Convergence Order ◽  
Eighth Order ◽  
First Derivative ◽  
Ball Convergence ◽  
Derivatives Of

The convergence order of numerous iterative methods is obtained using derivatives of a higher order, although these derivatives are not involved in the methods. Therefore, these methods cannot be used to solve equations with functions that do not have such high-order derivatives, since their convergence is not guaranteed. The convergence in this paper is shown, relying only on the first derivative. That is how we expand the applicability of some popular methods.

scholarly journals New Eighth-Order Derivative-Free Methods for Solving Nonlinear Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Rajinder Thukral
Keyword(s):  
Nonlinear Equations ◽  
Convergence Order ◽  
Order Derivative ◽  
Optimal Order ◽  
Eighth Order ◽  
Optimal Order Of Convergence ◽  
New Family ◽  
Derivative Free ◽  
Derivative Free Methods ◽  
Iteration Methods

A new family of eighth-order derivative-free methods for solving nonlinear equations is presented. It is proved that these methods have the convergence order of eight. These new methods are derivative-free and only use four evaluations of the function per iteration. In fact, we have obtained the optimal order of convergence which supports the Kung and Traub conjecture. Kung and Traub conjectured that the multipoint iteration methods, without memory based onnevaluations could achieve optimal convergence order of . Thus, we present new derivative-free methods which agree with Kung and Traub conjecture for . Numerical comparisons are made to demonstrate the performance of the methods presented.

scholarly journals 6th and 8th Order Hermite Integrator Using Snap and Crackle

2007 ◽  
Vol 3 (S246) ◽  
pp. 473-474
Author(s):  
Keigo Nitadori ◽  
Masaki Iwasawa ◽  
Junichiro Makino
Keyword(s):  
Fourth Order ◽  
Sixth Order ◽  
Floating Point ◽  
Eighth Order ◽  
Third Order ◽  
Order Scheme ◽  
Force Calculation ◽  
Derivatives Of ◽  
Very High ◽  
Better Than

AbstractWe present sixth- and eighth-order Hermite integrators for astrophysical N-body simulations, which use the derivatives of accelerations up to second order (snap) and third order (crackle). These schemes do not require previous values for the corrector, and require only one previous value to construct the predictor. Thus, they are fairly easy to be implemented. The additional cost of the calculation of the higher order derivatives is not very high. Even for the eighth-order scheme, the number of floating-point operations for force calculation is only about two times larger than that for traditional fourth-order Hermite scheme. The sixth order scheme is better than the traditional fourth order scheme for most cases. When the required accuracy is very high, the eighth-order one is the best.

scholarly journals On two new families of iterative methods for solving nonlinear equations with optimal order

2011 ◽  
Vol 5 (1) ◽  
pp. 93-109 ◽  
Author(s):  
M. Heydari ◽  
S.M. Hosseini ◽  
G.B. Loghmani
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Efficiency Index ◽  
Convergence Order ◽  
Optimal Order ◽  
Eighth Order ◽  
Optimal Convergence ◽  
Numerical Comparisons ◽  
First Derivative ◽  
The Third

In this paper, two new families of eighth-order iterative methods for solving nonlinear equations is presented. These methods are developed by combining a class of optimal two-point methods and a modified Newton?s method in the third step. Per iteration the presented methods require three evaluations of the function and one evaluation of its first derivative and therefore have the efficiency index equal to 1:682. Kung and Traub conjectured that a multipoint iteration without memory based on n evaluations could achieve optimal convergence order 2n?1. Thus the new families of eighth-order methods agrees with the conjecture of Kung-Traub for the case n = 4. Numerical comparisons are made with several other existing methods to show the performance of the presented methods.

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