scholarly journals On the Solution of Equations by Extended Discretization

Computation ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 69
Author(s):  
Gus I. Argyros ◽  
Michael I. Argyros ◽  
Samundra Regmi ◽  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Banach Space ◽  
Nonlinear Equations ◽  
Error Bounds ◽  
Hölder Continuity ◽  
Fréchet Derivative ◽  
Holder Continuity ◽  
Frechet Derivative ◽  
Solution Of Equations ◽  
Discretization Technique

The method of discretization is used to solve nonlinear equations involving Banach space valued operators using Lipschitz or Hölder constants. But these constants cannot always be found. That is why we present results using ω− continuity conditions on the Fréchet derivative of the operator involved. This way, we extend the applicability of the discretization technique. It turns out that if we specialize ω− continuity our new results improve those in the literature too in the case of Lipschitz or Hölder continuity. Our analysis includes tighter upper error bounds on the distances involved.

Recurrence relations for a family of iterations assuming Hölder continuous second order Fréchet derivative

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Dharmendra Kumar Gupta ◽  
Eulalia Martínez ◽  
Sukhjit Singh ◽  
Jose Luis Hueso ◽  
Shwetabh Srivastava ◽  
...  
Keyword(s):  
Recurrence Relations ◽  
Lipschitz Continuity ◽  
Hölder Continuity ◽  
Continuity Condition ◽  
Fréchet Derivative ◽  
Second Order ◽  
Holder Continuity ◽  
Hölder Continuous ◽  
Dynamical Study ◽  
Frechet Derivative

Abstract The semilocal convergence using recurrence relations of a family of iterations for solving nonlinear equations in Banach spaces is established. It is done under the assumption that the second order Fréchet derivative satisfies the Hölder continuity condition. This condition is more general than the usual Lipschitz continuity condition used for this purpose. Examples can be given for which the Lipschitz continuity condition fails but the Hölder continuity condition works on the second order Fréchet derivative. Recurrence relations based on three parameters are derived. A theorem for existence and uniqueness along with the error bounds for the solution is provided. The R-order of convergence is shown to be equal to 3 + q when θ = ±1; otherwise it is 2 + q, where q ∈ (0, 1]. Numerical examples involving nonlinear integral equations and boundary value problems are solved and improved convergence balls are found for them. Finally, the dynamical study of the family of iterations is also carried out.

scholarly journals Local Convergence for Multi-Step High Order Solvers under Weak Conditions

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 179
Author(s):  
Ramandeep Behl ◽  
Ioannis K. Argyros
Keyword(s):  
Banach Space ◽  
Nonlinear Equations ◽  
Local Convergence ◽  
Nonlinear Problems ◽  
Fréchet Derivative ◽  
High Order ◽  
Order Derivative ◽  
First Order ◽  
Lipschitz Constants ◽  
Convergence Study

Our aim in this article is to suggest an extended local convergence study for a class of multi-step solvers for nonlinear equations valued in a Banach space. In comparison to previous studies, where they adopt hypotheses up to 7th Fŕechet-derivative, we restrict the hypotheses to only first-order derivative of considered operators and Lipschitz constants. Hence, we enlarge the suitability region of these solvers along with computable radii of convergence. In the end of this study, we choose a variety of numerical problems which illustrate that our works are applicable but not earlier to solve nonlinear problems.

scholarly journals On Newton's method and nondiscrete mathematical induction

1988 ◽  
Vol 38 (1) ◽  
pp. 131-140 ◽  
Author(s):  
Ioannis K. Argyros
Keyword(s):  
Lipschitz Condition ◽  
Error Bounds ◽  
Newton’S Method ◽  
Newton's Method ◽  
Mathematical Induction ◽  
Fréchet Derivative ◽  
Hölder Continuous ◽  
Frechet Derivative ◽  
Nondiscrete Mathematical Induction ◽  
Sharp Error

The method of nondiscrete mathematical induction is used to find sharp error bounds for Newton's method. We assume only that the operator has Hölder continuous derivatives. In the case when the Fréchet-derivative of the operator satisfies a Lipschitz condition, our results reduce to the ones obtained by Ptak and Potra in 1972.

Local convergence of deformed Euler–Halley-type methods in Banach space under weak conditions

2017 ◽  
Vol 10 (02) ◽  
pp. 1750086
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Banach Space ◽  
Nonlinear Equation ◽  
Convergence Analysis ◽  
Error Estimates ◽  
Local Convergence ◽  
Fréchet Derivative ◽  
Convergence Ball ◽  
Numerical Examples ◽  
Frechet Derivative ◽  
Local Convergence Analysis

We present a unified local convergence analysis for deformed Euler–Halley-type methods in order to approximate a solution of a nonlinear equation in a Banach space setting. Our methods include the Euler, Halley and other high order methods. The convergence ball and error estimates are given for these methods under hypotheses up to the first Fréchet derivative in contrast to earlier studies using hypotheses up to the second Fréchet derivative. Numerical examples are also provided in this study.

scholarly journals Ball convergence for Traub-Steffensen like methods in Banach space

2015 ◽  
Vol 53 (2) ◽  
pp. 3-16
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Banach Space ◽  
Fréchet Derivative ◽  
Not Given ◽  
Frechet Derivative ◽  
The Third ◽  
Taylor Expansions ◽  
Lipschitz Constants ◽  
Ball Convergence ◽  
Computable Error Bounds ◽  
Local Convergence Analysis

Abstract We present a local convergence analysis for two Traub-Steffensen-like methods in order to approximate a locally unique solution of an equation in a Banach space setting. In earlier studies such as [16, 23] Taylor expansions and hypotheses up to the third Fréchet-derivative are used. We expand the applicability of these methods using only hypotheses on the first Fréchet derivative. Moreover, we obtain a radius of convergence and computable error bounds using Lipschitz constants not given before. Numerical examples are also presented in this study.

Ball Convergence for a Family of Quadrature-Based Methods for Solving Equations in Banach Space

2017 ◽  
Vol 14 (02) ◽  
pp. 1750017 ◽  
Author(s):  
Ioannis K. Argyros ◽  
Ramandeep Behl ◽  
S. S. Motsa
Keyword(s):  
Banach Space ◽  
Taylor Series Expansion ◽  
Fréchet Derivative ◽  
Dimensional Euclidean Space ◽  
Frechet Derivative ◽  
Space Setting ◽  
Lipschitz Constants ◽  
Solving Equations ◽  
Ball Convergence ◽  
Predictor Corrector

We present a local convergence analysis for a family of quadrature-based predictor–corrector methods in order to approximate a locally unique solution of an equation in a Banach space setting. In earlier studies such as Howk [2016] [Howk, C. L. [2016] “A classs of efficient quadrature-based predictor–corrector methods for solving nonlinear systems, Appl. Math. Comput. 276, 394–406] the [Formula: see text] order of convergence was shown on the [Formula: see text]-dimensional Euclidean space using Taylor series expansion and hypotheses reaching up to the third-order Fréchet-derivative of the operator involved although only the first-order Fréchet-derivative appears in these methods, which restrict the applicability of these methods. In this paper, we expand the applicability of these methods in a Banach space setting and using hypotheses only on the first Fréchet-derivative. Moreover, we provide computable radii of convergence as well as error bounds on the distances involved using Lipschitz constants. Numerical examples are also presented to solve equations in cases where earlier results cannot apply.

scholarly journals Sharp error bounds for Newton-like methods under weak smoothness assumptions

1992 ◽  
Vol 45 (3) ◽  
pp. 415-422 ◽  
Author(s):  
Ioannis K. Argyros
Keyword(s):  
Banach Space ◽  
Nonlinear Equations ◽  
Error Bounds ◽  
Wide Class ◽  
Convergence Conditions ◽  
Sharp Error

We provide sufficient convergence conditions as well as sharp error bounds for Newton-like iterations which generalise a wide class of known methods for solving nonlinear equations in Banach space.

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