On the convergence of a novel seventh convergence order schemes for solving equations

2022 ◽  
Author(s):  
Samundra Regmi ◽  
Ioannis K. Argyros ◽  
Santhosh George ◽  
Christopher I. Argyros
Keyword(s):  
Convergence Order ◽  
Solving Equations

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 Local convergence comparison between two novel sixth order methods for solving equations

2019 ◽  
Vol 18 (1) ◽  
pp. 5-19
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Banach Space ◽  
Convergence Analysis ◽  
Local Convergence ◽  
Sixth Order ◽  
Convergence Order ◽  
Convergence Criteria ◽  
Numerical Examples ◽  
First Derivative ◽  
Solving Equations ◽  
Local Convergence Analysis

Abstract The aim of this article is to provide the local convergence analysis of two novel competing sixth convergence order methods for solving equations involving Banach space valued operators. Earlier studies have used hypotheses reaching up to the sixth derivative but only the first derivative appears in these methods. These hypotheses limit the applicability of the methods. That is why we are motivated to present convergence analysis based only on the first derivative. Numerical examples where the convergence criteria are tested are provided. It turns out that in these examples the criteria in the earlier works are not satisfied, so these results cannot be used to solve equations but our results can be used.

scholarly journals Extended convergence of a sixth order scheme for solving equations under ω–continuity conditions

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Samundra Regmi ◽  
Christopher I. Argyros ◽  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Banach Space ◽  
Sixth Order ◽  
Convergence Order ◽  
Not Given ◽  
Numerical Examples ◽  
First Derivative ◽  
Uniqueness Of The Solution ◽  
Order Scheme ◽  
Solving Equations

Abstract The applicability of an efficient sixth convergence order scheme is extended for solving Banach space valued equations. In previous works, the seventh derivative has been used not appearing on the scheme. But we use only the first derivative that appears on the scheme. Moreover, bounds on the error distances and results on the uniqueness of the solution are provided (not given in earlier works) based on ω–continuity conditions. Numerical examples complete this article.

scholarly journals Extending the Convergence of Two Similar Sixth Order Schemes for Solving Equations under Generalized Conditions

2021 ◽  
pp. 246-257
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George ◽  
Christopher I. Argyros
Keyword(s):  
Banach Space ◽  
Sixth Order ◽  
Convergence Order ◽  
Not Given ◽  
Numerical Examples ◽  
First Derivative ◽  
Uniqueness Of The Solution ◽  
Solving Equations

The applicability of two competing efficient sixth convergence order schemes is extended for solving Banach space valued equations. In previous works, the seventh derivative has been used not appearing on the schemes. But we use only the first derivative that appears on the scheme. Moreover, bounds on the error distances and results on the uniqueness of the solution are provided not given in the earlier works based on ω-continuity conditions. Our technique extends other schemes analogously, since it is so general. Numerical examples complete this work.

A fifth-order iterative method for solving equations

1987 ◽  
Vol 49 (1-2) ◽  
pp. 129-137 ◽  
Author(s):  
Duong Thuy Vy
Keyword(s):  
Iterative Method ◽  
Solving Equations ◽  
Fifth Order

scholarly journals Superconvergence of interpolated collocation solutions for weakly singular Volterra integral equations of the second kind

2021 ◽  
Vol 40 (3) ◽  
Author(s):  
Qiumei Huang ◽  
Min Wang
Keyword(s):  
Integral Equations ◽  
Numerical Experiments ◽  
Volterra Integral Equations ◽  
Least Square ◽  
Convergence Order ◽  
Collocation Points ◽  
Weakly Singular ◽  
Interpolation Postprocessing ◽  
Collocation Solution

AbstractIn this paper, we discuss the superconvergence of the “interpolated” collocation solutions for weakly singular Volterra integral equations of the second kind. Based on the collocation solution $$u_h$$ u h , two different interpolation postprocessing approximations of higher accuracy: $$I_{2h}^{2m-1}u_h$$ I 2 h 2 m - 1 u h based on the collocation points and $$I_{2h}^{m}u_h$$ I 2 h m u h based on the least square scheme are constructed, whose convergence order are the same as that of the iterated collocation solution. Such interpolation postprocessing methods are much simpler in computation. We further apply this interpolation postprocessing technique to hybrid collocation solutions and similar results are obtained. Numerical experiments are shown to demonstrate the efficiency of the interpolation postprocessing methods.

Method of solving equations of motion of viscoelastic media

1975 ◽  
Vol 9 (3) ◽  
pp. 382-388
Author(s):  
I. G. Filippov
Keyword(s):  
Equations Of Motion ◽  
Viscoelastic Media ◽  
Solving Equations
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