Convergence analysis for a fast class of multi-step Chebyshev-Halley-type methods under weak conditions

In this study a convergence analysis for a fast multi-step Chebyshe-Halley-type method for solving nonlinear equations involving Banach space valued operator is presented. We introduce a more precise convergence region containing the iterates leading to tighter Lipschitz constants and functions. This way advantages are obtained in both the local as well as the semi-local convergence case under the same computational cost such as: extended convergence domain, tighter error bounds on the distances involved and a more precise information on the location of the solution. The new technique can be used to extend the applicability of other iterative methods. The numerical examples further validate the theoretical results.

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Extended Convergence of a Two-Step-Secant-Type Method Under a Restricted Convergence Domain

Kragujevac Journal of Mathematics ◽

10.46793/kgjmat2101.155a ◽

2020 ◽

Vol 45 (01) ◽

pp. 155-164

Author(s):

IOANNIS K. ARGYROS ◽

GEORGE SANTHOSH

Keyword(s):

Computational Effort ◽

Convergence Domain ◽

Type Method ◽

Precise Information ◽

Numerical Examples ◽

Special Cases ◽

Lipschitz Constants ◽

Local Convergence Analysis ◽

Lipschitz Conditions ◽

Theoretical Results

We present a local as well as a semi-local convergence analysis of a two-step secant-type method for solving nonlinear equations involving Banach space valued operators. By using weakened Lipschitz and center Lipschitz conditions in combination with a more precise domain containing the iterates, we obtain tighter Lipschitz constants than in earlier studies. This technique lead to an extended convergence domain, more precise information on the location of the solution and tighter error bounds on the distances involved. These advantages are obtained under the same computational eﬀort, since the new constants are special cases of the old ones used in earlier studies. The new technique can be used on other iterative methods. The numerical examples further illustrate the theoretical results.

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Unified Local Convergence for Newton’s Method and Uniqueness of the Solution of Equations under Generalized Conditions in a Banach Space

Mathematics ◽

10.3390/math7050463 ◽

2019 ◽

Vol 7 (5) ◽

pp. 463 ◽

Cited By ~ 2

Author(s):

Ioannis K. Argyros ◽

Ángel Alberto Magreñán ◽

Lara Orcos ◽

Íñigo Sarría

Keyword(s):

Banach Space ◽

Newton’S Method ◽

Newton's Method ◽

Convergence Region ◽

Precise Information ◽

Solution Of Equations ◽

Lipschitz Constants ◽

Uniqueness Of The Solution ◽

Generalized Newton ◽

Theoretical Results

Under the hypotheses that a function and its Fréchet derivative satisfy some generalized Newton–Mysovskii conditions, precise estimates on the radii of the convergence balls of Newton’s method, and of the uniqueness ball for the solution of the equations, are given for Banach space-valued operators. Some of the existing results are improved with the advantages of larger convergence region, tighter error estimates on the distances involved, and at-least-as-precise information on the location of the solution. These advantages are obtained using the same functions and Lipschitz constants as in earlier studies. Numerical examples are used to test the theoretical results.

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A Semilocal Convergence for a Uniparametric Family of Efficient Secant-Like Methods

Journal of Function Spaces ◽

10.1155/2014/467980 ◽

2014 ◽

Vol 2014 ◽

pp. 1-10 ◽

Cited By ~ 4

Author(s):

I. K. Argyros ◽

D. González ◽

Á. A. Magreñán

Keyword(s):

Banach Space ◽

Computational Cost ◽

Semilocal Convergence ◽

Recurrent Functions ◽

Convergence Conditions ◽

Precise Information ◽

Numerical Examples ◽

Convergence Results ◽

Special Cases ◽

Theoretical Results

We present a semilocal convergence analysis for a uniparametric family of efficient secant-like methods (including the secant and Kurchatov method as special cases) in a Banach space setting (Ezquerro et al., 2000–2012). Using our idea of recurrent functions and tighter majorizing sequences, we provide convergence results under the same or less computational cost than the ones of Ezquerro et al., (2013, 2010, and 2012) and Hernández et al., (2000, 2005, and 2002) and with the following advantages: weaker sufficient convergence conditions, tighter error estimates on the distances involved, and at least as precise information on the location of the solution. Numerical examples validating our theoretical results are also provided 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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On the Ostrowski Method for Solving Equations

European Journal of Mathematical Analysis ◽

10.28924/ada/ma.2.3 ◽

2021 ◽

Vol 2 ◽

pp. 3

Author(s):

Ioannis K. Argyros ◽

Santhosh George ◽

Christopher I. Argyros

Keyword(s):

Banach Space ◽

Convergence Analysis ◽

Local Convergence ◽

Numerical Experiments ◽

Convergence Domain ◽

Lipschitz Constants ◽

Solving Equations ◽

Ostrowski’S Method

In this paper, we revisited the Ostrowski's method for solving Banach space valued equations. We developed a technique to determine a subset of the original convergence domain and using this new Lipschitz constants derived. These constants are at least as tight as the earlier ones leading to a finer convergence analysis in both the semi-local and the local convergence case. These techniques are very general, so they can be used to extend the applicability of other methods without additional hypotheses. Numerical experiments complete this study.

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On an improved convergence analysis of a two-step Gauss-Newton type method under generalized Lipschitz conditions

Carpathian Journal of Mathematics ◽

10.37193/cjm.2020.03.04 ◽

2020 ◽

Vol 36 (3) ◽

pp. 365-372

Author(s):

I. K. ARGYROS ◽

R. P. IAKYMCHUK ◽

S. M. SHAKHNO ◽

H. P. YARMOLA

Keyword(s):

Convergence Analysis ◽

Computational Effort ◽

Convergence Criteria ◽

Type Method ◽

Precise Information ◽

Special Cases ◽

Original Domain ◽

Local Convergence Analysis ◽

Lipschitz Conditions ◽

Theoretical Results

We present a local convergence analysis of a two-step Gauss-Newton method under the generalized and classical Lipschitz conditions for the first- and second-order derivatives. In contrast to earlier works, we use our new idea using a center average Lipschitz conditions through which, we define a subset of the original domain that also contains the iterates. Then, the remaining average Lipschitz conditions are at least as tight as the corresponding ones in earlier works. This way, we obtain: weaker sufficient convergence criteria, larger radius of convergence, tighter error estimates and more precise information on the location of the solution. These advantages are obtained under the same computational effort, since the new Lipschitz functions are special cases of the ones in earlier works. Finally, we give a numerical example that confirms the theoretical results, and compares favorably to the results from previous works.

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Extending the Applicability of the Super-Halley-Like Method Using ω-Continuous Derivatives and Restricted Convergence Domains

Annales Mathematicae Silesianae ◽

10.2478/amsil-2018-0008 ◽

2019 ◽

Vol 33 (1) ◽

pp. 21-40

Author(s):

Ioannis K. Argyros ◽

Santhosh George

Keyword(s):

Banach Space ◽

Convergence Analysis ◽

Local Convergence ◽

Uniqueness Result ◽

Second Derivative ◽

Numerical Examples ◽

The Third ◽

Lipschitz Constants ◽

Local Convergence Analysis ◽

Third Derivative

AbstractWe present a local convergence analysis of the super-Halley-like method in order to approximate a locally unique solution of an equation in a Banach space setting. The convergence analysis in earlier studies was based on hypotheses reaching up to the third derivative of the operator. In the present study we expand the applicability of the super-Halley-like method by using hypotheses up to the second derivative. We also provide: a computable error on the distances involved and a uniqueness result based on Lipschitz constants. Numerical examples are also presented in this study.

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Extended Local Convergence for the Combined Newton-Kurchatov Method Under the Generalized Lipschitz Conditions

Mathematics ◽

10.3390/math7020207 ◽

2019 ◽

Vol 7 (2) ◽

pp. 207 ◽

Cited By ~ 2

Author(s):

Ioannis Argyros ◽

Stepan Shakhno

Keyword(s):

Banach Space ◽

Convergence Analysis ◽

Local Convergence ◽

Computational Effort ◽

Convergence Criteria ◽

Type Condition ◽

Convergence Region ◽

Precise Information ◽

Lipschitz Conditions

We present a local convergence of the combined Newton-Kurchatov method for solving Banach space valued equations. The convergence criteria involve derivatives until the second and Lipschitz-type conditions are satisfied, as well as a new center-Lipschitz-type condition and the notion of the restricted convergence region. These modifications of earlier conditions result in a tighter convergence analysis and more precise information on the location of the solution. These advantages are obtained under the same computational effort. Using illuminating examples, we further justify the superiority of our new results over earlier ones.

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Local convergence of deformed Jarratt-type methods in Banach space without inverses

Asian-European Journal of Mathematics ◽

10.1142/s1793557116500157 ◽

2016 ◽

Vol 09 (01) ◽

pp. 1650015

Author(s):

Ioannis K. Argyros ◽

Santhosh George

Keyword(s):

Banach Space ◽

Convergence Analysis ◽

Local Convergence ◽

Computational Cost ◽

Fréchet Derivative ◽

Convergence Conditions ◽

Type Method ◽

Frechet Derivative ◽

The Third ◽

Local Convergence Analysis

We present a local convergence analysis for the Jarratt-type method of high convergence order in order to approximate a solution of a nonlinear equation in a Banach space. Our sufficient convergence conditions involve only hypotheses on the first Fréchet-derivative of the operator involved. Earlier studies use hypotheses up to the third Fréchet-derivative. Hence, the applicability of these methods is expanded under weaker hypotheses and less computational cost for the constants involved in the convergence analysis. Numerical examples are also provided in this study.

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On the Convergence Region of Multi-step Chebyshev-Halley-type Schemes for Solving Equations

Earthline Journal of Mathematical Sciences ◽

10.34198/ejms.1219.187207 ◽

2019 ◽

pp. 187-207

Author(s):

Ioannis K. Argyros ◽

Santhosh George

Keyword(s):

Banach Space ◽

Convergence Analysis ◽

Local Convergence ◽

Small Region ◽

Computational Effort ◽

Convergence Region ◽

Numerical Examples ◽

Solving Equations ◽

Original Region ◽

Local Convergence Analysis

The aim of this article is to extend the convergence region of certain multi-step Chebyshev-Halley-type schemes for solving Banach space valued nonlinear equations. In particular, we find an at least as small region as the region of the operator involved containing the iterates. This way the majorant functions are tighter than the ones related to the original region, leading to a finer local as well as a semi-local convergence analysis under the same computational effort. Numerical examples complete this article.

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