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 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.

scholarly journals Convergence and Dynamics of a Higher-Order Method

Symmetry ◽  
2020 ◽  
Vol 12 (3) ◽  
pp. 420 ◽  
Author(s):  
Alejandro Moysi ◽  
Ioannis K. Argyros ◽  
Samundra Regmi ◽  
Daniel González ◽  
Á. Alberto Magreñán ◽  
...  
Keyword(s):  
Nonlinear Equation ◽  
Iterative Method ◽  
Higher Order ◽  
High Order ◽  
Local Form ◽  
Order Method ◽  
First Derivative ◽  
The Family ◽  
Higher Order Method

Solving problems in various disciplines such as biology, chemistry, economics, medicine, physics, and engineering, to mention a few, reduces to solving an equation. Its solution is one of the greatest challenges. It involves some iterative method generating a sequence approximating the solution. That is why, in this work, we analyze the convergence in a local form for an iterative method with a high order to find the solution of a nonlinear equation. We extend the applicability of previous results using only the first derivative that actually appears in the method. This is in contrast to either works using a derivative higher than one, or ones not in this method. Moreover, we consider the dynamics of some members of the family in order to see the existing differences between them.

scholarly journals Extended High Order Algorithms for Equations under the Same Set of Conditions

Algorithms ◽  
2021 ◽  
Vol 14 (7) ◽  
pp. 207
Author(s):  
Ioannis K. Argyros ◽  
Debasis Sharma ◽  
Christopher I. Argyros ◽  
Sanjaya Kumar Parhi ◽  
Shanta Kumari Sunanda ◽  
...  
Keyword(s):  
Higher Order ◽  
High Order ◽  
Sixth Order ◽  
Convergence Order ◽  
First Derivative ◽  
Solving Equations

A variety of strategies are used to construct algorithms for solving equations. However, higher order derivatives are usually assumed to calculate the convergence order. More importantly, bounds on error and uniqueness regions for the solution are also not derived. Therefore, the benefits of these algorithms are limited. We simply use the first derivative to tackle all these issues and study the ball analysis for two sixth order algorithms under the same set of conditions. In addition, we present a calculable ball comparison between these algorithms. In this manner, we enhance the utility of these algorithms. Our idea is very general. That is why it can also be used to extend other algorithms as well in the same way.

scholarly journals Convergence Ball and Complex Geometry of an Iteration Function of Higher Order

Mathematics ◽  
2018 ◽  
Vol 7 (1) ◽  
pp. 28 ◽  
Author(s):  
Deepak Kumar ◽  
Ioannis Argyros ◽  
Janak Sharma
Keyword(s):  
Complex Geometry ◽  
Basins Of Attraction ◽  
Higher Order ◽  
Order Method ◽  
Eighth Order ◽  
Science And Engineering ◽  
Convergence Ball ◽  
Convergence Results ◽  
Iteration Function ◽  
Derivatives Of

Higher-order derivatives are used to determine the convergence order of iterative methods. However, such derivatives are not present in the formulas. Therefore, the assumptions on the higher-order derivatives of the function restrict the applicability of methods. Our convergence analysis of an eighth-order method uses only the derivative of order one. The convergence results so obtained are applied to some real problems, which arise in science and engineering. Finally, stability of the method is checked through complex geometry shown by drawing basins of attraction of the solutions.

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.

Differentiation of Mass and Stiffness Matrices for High Order Sensitivity Calculations in Finite Element-Based Equilibrium Problems

1993 ◽  
Vol 115 (4) ◽  
pp. 829-832 ◽  
Author(s):  
J. E. Bernard ◽  
S. K. Kwon ◽  
J. A. Wilson
Keyword(s):  
Finite Element ◽  
Design Variable ◽  
Equilibrium Problems ◽  
Higher Order ◽  
High Order ◽  
Stiffness Matrices ◽  
Higher Order Terms ◽  
Fixed Boundaries ◽  
Cubic Polynomials ◽  
Derivatives Of

Extension of sensitivity methods to include higher order terms depends on the ability to compute higher order derivatives of the mass and stiffness matrices. This paper presents a method based on the use of cubic polynomials to fit mass and stiffness matrices across a range of interest of the design variable. The method is illustrated through an example which uses Pade´ approximants to expand the solution to a statics problem. The design variable is the thickness of one part of a plate with fixed boundaries. The solution gives a very good approximation over fivefold change in the value of the design variable.

scholarly journals A High-Order Iterate Method for ComputingAT,S(2)

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Xiaoji Liu ◽  
Zemeng Zuo
Keyword(s):  
Iterative Method ◽  
Generalized Inverse ◽  
Higher Order ◽  
High Order ◽  
New Method ◽  
Order Convergence ◽  
Approximate Inverses

We investigate a new higher order iterative method for computing the generalized inverseAT,S(2)for a given matrixA. We also discuss how the new method could be applied for finding approximate inverses of nonsingular square matrices. Analysis of convergence is included to show that the proposed scheme has at least fifteenth-order convergence. Some tests are also presented to show the superiority of the new method.

scholarly journals Local Convergence of an Efficient Multipoint Iterative Method in Banach Space

Algorithms ◽  
2020 ◽  
Vol 13 (1) ◽  
pp. 25
Author(s):  
Janak Raj Sharma ◽  
Sunil Kumar ◽  
Ioannis K. Argyros
Keyword(s):  
Banach Space ◽  
Iterative Method ◽  
Local Convergence ◽  
Numerical Experiments ◽  
Order Method ◽  
Eighth Order ◽  
First Derivative ◽  
Taylor Expansions ◽  
Derivative Free ◽  
Class Of Functions

We discuss the local convergence of a derivative-free eighth order method in a Banach space setting. The present study provides the radius of convergence and bounds on errors under the hypothesis based on the first Fréchet-derivative only. The approaches of using Taylor expansions, containing higher order derivatives, do not provide such estimates since the derivatives may be nonexistent or costly to compute. By using only first derivative, the method can be applied to a wider class of functions and hence its applications are expanded. Numerical experiments show that the present results are applicable to the cases wherein previous results cannot be applied.

A comparison between two competing sixth convergence order algorithms under the same set of conditions

2021 ◽  
Vol 30 (1) ◽  
pp. 19-28
Author(s):  
GUS ARGYROS ◽  
MICHAEL ARGYROS ◽  
IOANNIS K. ARGYROS ◽  
GEORGE SANTHOSH
Keyword(s):  
Banach Space ◽  
Error Estimates ◽  
Taylor Series ◽  
High Order ◽  
Convergence Order ◽  
Convergence Criteria ◽  
Convergence Conditions ◽  
First Derivative ◽  
Banach Space Operators

There is a plethora of algorithms of the same convergence order for generating a sequence approximating a solution of an equation involving Banach space operators. But the set of convergence criteria is not the same in general. This makes the comparison between them hard and only numerically. Moreover, the convergence is established using Taylor series and by assuming the existence of high order derivatives not even appearing on these algorithms. Furthermore, no computable error estimates, uniqueness for the solution results or a ball of convergence is given. We address all these problems by using only the first derivative that actually appears on these algorithms and under the same set of convergence conditions. Our technique is so general that it can be used to extend the applicability of other algorithms along the same lines.

Tangential derivatives and higher-order regularizing properties of the double layer heat potential

Analysis ◽  
2019 ◽  
Vol 38 (4) ◽  
pp. 167-193 ◽  
Author(s):  
Massimo Lanza de Cristoforis ◽  
Paolo Luzzini
Keyword(s):  
Integral Operator ◽  
Explicit Formula ◽  
Double Layer ◽  
Higher Order ◽  
Time Variable ◽  
High Order ◽  
Hölder Spaces ◽  
Heat Potential ◽  
Holder Spaces ◽  
Derivatives Of

Abstract We prove an explicit formula for the tangential derivatives of the double layer heat potential. By exploiting such a formula, we prove the validity of a regularizing property for the integral operator associated to the double layer heat potential in spaces of functions with high-order derivatives in parabolic Hölder spaces defined on the boundary of parabolic cylinders which are unbounded in the time variable.

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