Extended Kung–Traub Methods for Solving Equations with Applications

Kung and Traub (1974) proposed an iterative method for solving equations defined on the real line. The convergence order four was shown using Taylor expansions, requiring the existence of the fifth derivative not in this method. However, these hypotheses limit the utilization of it to functions that are at least five times differentiable, although the methods may converge. As far as we know, no semi-local convergence has been given in this setting. Our goal is to extend the applicability of this method in both the local and semi-local convergence case and in the more general setting of Banach space valued operators. Moreover, we use our idea of recurrent functions and conditions only on the first derivative and divided difference, which appear in the method. This idea can be used to extend other high convergence multipoint and multistep methods. Numerical experiments testing the convergence criteria complement this study.

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Unified Convergence Analysis of Two-Step Iterative Methods for Solving Equations

European Journal of Mathematical Analysis ◽

10.28924/ada/ma.1.68 ◽

2021 ◽

Vol 1 (2) ◽

pp. 68-85

Author(s):

Ioannis K. Argyros

Keyword(s):

Banach Space ◽

Convergence Analysis ◽

Iterative Methods ◽

Local Convergence ◽

General Setting ◽

Convergence Criteria ◽

Recurrent Functions ◽

Solving Equations ◽

Order Four ◽

Methods Numerical

In this paper we consider unified convergence analysis of two-step iterative methods for solving equations in the Banach space setting. The convergence order four was shown using Taylor expansions requiring the existence of the fifth derivative not on this method. But these hypotheses limit the utilization of it to functions which are at least five times differentiable although the method may converge. As far as we know no semi-local convergence has been given in this setting. Our goal is to extend the applicability of this method in both the local and semi-local convergence case and in the more general setting of Banach space valued operators. Moreover, we use our idea of recurrent functions and conditions only on the first derivative and divided differences which appear on the method. This idea can be used to extend other high convergence multipoint and multistep methods. Numerical experiments testing the convergence criteria complement this study.

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Convergence Criteria of Three Step Schemes for Solving Equations

Mathematics ◽

10.3390/math9233106 ◽

2021 ◽

Vol 9 (23) ◽

pp. 3106

Author(s):

Samundra Regmi ◽

Christopher I. Argyros ◽

Ioannis K. Argyros ◽

Santhosh George

Keyword(s):

Banach Space ◽

Euclidean Space ◽

Convergence Analysis ◽

Local Convergence ◽

Dimensional Euclidean Space ◽

Convergence Criteria ◽

Iterative Schemes ◽

Finite Dimensional ◽

Taylor Expansions ◽

Solving Equations

We develop a unified convergence analysis of three-step iterative schemes for solving nonlinear Banach space valued equations. The local convergence order has been shown before to be five on the finite dimensional Euclidean space assuming Taylor expansions and the existence of the sixth derivative not on these schemes. So, the usage of them is restricted six or higher differentiable mappings. But in our paper only the first Frèchet derivative is utilized to show convergence. Consequently, the scheme is expanded. Numerical applications are also given to test convergence.

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

Annales Universitatis Paedagogicae Cracoviensis Studia Mathematica ◽

10.2478/aupcsm-2019-0001 ◽

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.

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Local convergence for an eighth order method for solving equations and systems of equations

Nonlinear Engineering ◽

10.1515/nleng-2017-0105 ◽

2019 ◽

Vol 8 (1) ◽

pp. 74-79

Author(s):

Ioannis K. Argyros ◽

Santhosh George

Keyword(s):

Banach Space ◽

Local Convergence ◽

Divided Difference ◽

Dimensional Euclidean Space ◽

Order Method ◽

Systems Of Equations ◽

Eighth Order ◽

First Derivative ◽

Solving Equations ◽

Computable Error Bounds

AbstractThe aim of this study is to extend the applicability of an eighth convergence order method from thek−dimensional Euclidean space to a Banach space setting. We use hypotheses only on the first derivative to show the local convergence of the method. Earlier studies use hypotheses up to the eighth derivative although only the first derivative and a divided difference of order one appear in the method. Moreover, we provide computable error bounds based on Lipschitz-type functions.

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High Order Semi-implicit Multistep Methods for Time-Dependent Partial Differential Equations

Communications on Applied Mathematics and Computation ◽

10.1007/s42967-020-00110-5 ◽

2021 ◽

Author(s):

Giacomo Albi ◽

Lorenzo Pareschi

Keyword(s):

Multistep Methods ◽

General Setting ◽

Reaction Diffusion ◽

Strong Stability ◽

High Order ◽

Iterative Solvers ◽

Time Dependent ◽

Linear Multistep Methods ◽

Advection Diffusion Equation ◽

Separation Of Scales

AbstractWe consider the construction of semi-implicit linear multistep methods that can be applied to time-dependent PDEs where the separation of scales in additive form, typically used in implicit-explicit (IMEX) methods, is not possible. As shown in Boscarino et al. (J. Sci. Comput. 68: 975–1001, 2016) for Runge-Kutta methods, these semi-implicit techniques give a great flexibility, and allow, in many cases, the construction of simple linearly implicit schemes with no need of iterative solvers. In this work, we develop a general setting for the construction of high order semi-implicit linear multistep methods and analyze their stability properties for a prototype linear advection-diffusion equation and in the setting of strong stability preserving (SSP) methods. Our findings are demonstrated on several examples, including nonlinear reaction-diffusion and convection-diffusion problems.

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Semi-local convergence of a Newton-like method for solving equations with a singular derivative

Creative Mathematics and Informatics ◽

10.37193/cmi.2018.01.01 ◽

2018 ◽

Vol 27 (1) ◽

pp. 01-08

Author(s):

IOANNIS K. ARGYROS ◽

◽

GEORGE SANTHOSH ◽

Keyword(s):

Banach Space ◽

Convergence Analysis ◽

Real Line ◽

Local Convergence ◽

Approximate Solutions ◽

Convergence Domain ◽

Solving Equations ◽

Solutions Of Equations ◽

Local Convergence Analysis ◽

Special Case

We present a semi-local convergence analysis for a Newton-like method to approximate solutions of equations when the derivative is not necessarily non-singular in a Banach space setting. In the special case when the equation is defined on the real line the convergence domain is improved for this method when compared to earlier results. Numerical results where earlier results cannot apply but the new results can apply to solve nonlinear equations are also presented in this study.

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On the Semi-Local Convergence of a Traub-Type Method for Solving Equations

Foundations ◽

10.3390/foundations2010006 ◽

2022 ◽

Vol 2 (1) ◽

pp. 114-127

Author(s):

Samundra Regmi ◽

Christopher I. Argyros ◽

Ioannis K. Argyros ◽

Santhosh George

Keyword(s):

Banach Space ◽

Local Convergence ◽

Type Method ◽

Lipschitz Constants ◽

Solving Equations ◽

Original Domain ◽

Theoretical Results

The celebrated Traub’s method involving Banach space-defined operators is extended. The main feature in this study involves the determination of a subset of the original domain that also contains the Traub iterates. In the smaller domain, the Lipschitz constants are smaller too. Hence, a finer analysis is developed without the usage of additional conditions. This methodology applies to other methods. The examples justify the theoretical results.

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Memory in a New Variant of King’s Family for Solving Nonlinear Systems

Mathematics ◽

10.3390/math8081251 ◽

2020 ◽

Vol 8 (8) ◽

pp. 1251

Author(s):

Munish Kansal ◽

Alicia Cordero ◽

Sonia Bhalla ◽

Juan R. Torregrosa

Keyword(s):

Nonlinear Systems ◽

Convergence Analysis ◽

Numerical Experiments ◽

Order Of Convergence ◽

Recent Literature ◽

Divided Difference ◽

Difference Operators ◽

New Variant ◽

Order Four ◽

With Memory

In the recent literature, very few high-order Jacobian-free methods with memory for solving nonlinear systems appear. In this paper, we introduce a new variant of King’s family with order four to solve nonlinear systems along with its convergence analysis. The proposed family requires two divided difference operators and to compute only one inverse of a matrix per iteration. Furthermore, we have extended the proposed scheme up to the sixth-order of convergence with two additional functional evaluations. In addition, these schemes are further extended to methods with memory. We illustrate their applicability by performing numerical experiments on a wide variety of practical problems, even big-sized. It is observed that these methods produce approximations of greater accuracy and are more efficient in practice, compared with the existing methods.

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Local Convergence and Attraction Basins of Higher Order, Jarratt-Like Iterations

Mathematics ◽

10.3390/math7121203 ◽

2019 ◽

Vol 7 (12) ◽

pp. 1203

Author(s):

Janak Raj Sharma ◽

Deepak Kumar ◽

Ioannis K. Argyros

Keyword(s):

Banach Space ◽

Local Convergence ◽

Basins Of Attraction ◽

Higher Order ◽

Geometrical Approach ◽

Stable Convergence ◽

Numerical Testing ◽

Taylor Expansions ◽

Attraction Basins ◽

Class Of Functions

We studied the local convergence of a family of sixth order Jarratt-like methods in Banach space setting. The procedure so applied provides the radius of convergence and bounds on errors under the conditions based on the first Fréchet-derivative only. Such estimates are not proposed in the approaches using Taylor expansions of higher order derivatives which may be nonexistent or costly to compute. In this sense we can extend usage of the methods considered, since the methods can be applied to a wider class of functions. Numerical testing on examples show that the present results can be applied to the cases where earlier results are not applicable. Finally, the convergence domains are assessed by means of a geometrical approach; namely, the basins of attraction that allow us to find members of family with stable convergence behavior and with unstable behavior.

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Estimation of Longest Stability Interval for a Kind of Explicit Linear Multistep Methods

Discrete Dynamics in Nature and Society ◽

10.1155/2010/912691 ◽

2010 ◽

Vol 2010 ◽

pp. 1-18 ◽

Cited By ~ 1

Author(s):

Y. Xu ◽

J. J. Zhao

Keyword(s):

Absolute Stability ◽

Numerical Experiments ◽

Multistep Methods ◽

Linear Multistep Methods ◽

Stability Intervals ◽

The Stability ◽

Order Four

The new explicit linear three-order four-step methods with longest interval of absolute stability are proposed. Some numerical experiments are made for comparing different kinds of linear multistep methods. It is shown that the stability intervals of proposed methods can be longer than that of known explicit linear multistep methods.

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