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

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

Open Journal of Mathematical Sciences ◽

10.30538/oms2021.0143 ◽

2021 ◽

Vol 4 (1) ◽

pp. 34-43

Author(s):

Samundra Regmi ◽

◽

Ioannis K. Argyros ◽

Santhosh George ◽

◽

...

Keyword(s):

Banach Space ◽

Convergence Analysis ◽

Computational Cost ◽

Convergence Domain ◽

Convergence Region ◽

Type Method ◽

Precise Information ◽

Numerical Examples ◽

Lipschitz Constants ◽

Theoretical Results

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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Advances in the Semilocal Convergence of Newton’s Method with Real-World Applications

Mathematics ◽

10.3390/math7030299 ◽

2019 ◽

Vol 7 (3) ◽

pp. 299 ◽

Cited By ~ 4

Author(s):

Ioannis Argyros ◽

Á. Magreñán ◽

Lara Orcos ◽

Íñigo Sarría

Keyword(s):

Newton’S Method ◽

Newton's Method ◽

Lipschitz Constant ◽

Semilocal Convergence ◽

Convergence Domain ◽

Special Cases ◽

Real World Applications ◽

Lipschitz Constants ◽

Local Convergence Analysis ◽

Theoretical Results

The aim of this paper is to present a new semi-local convergence analysis for Newton’s method in a Banach space setting. The novelty of this paper is that by using more precise Lipschitz constants than in earlier studies and our new idea of restricted convergence domains, we extend the applicability of Newton’s method as follows: The convergence domain is extended; the error estimates are tighter and the information on the location of the solution is at least as precise as before. These advantages are obtained using the same information as before, since new Lipschitz constant are tighter and special cases of the ones used before. Numerical examples and applications are used to test favorable the theoretical results to earlier ones.

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Design and analysis of a faster King-Werner-type derivative free method

Boletim da Sociedade Paranaense de Matemática ◽

10.5269/bspm.44132 ◽

2022 ◽

Vol 40 ◽

pp. 1-18

Author(s):

J. R. Sharma ◽

Ioannis K. Argyros ◽

Deepak Kumar

Keyword(s):

Convergence Analysis ◽

Local Convergence ◽

Numerical Examples ◽

Weak Center ◽

Derivative Free ◽

Derivative Free Method ◽

Local Convergence Analysis ◽

Lipschitz Conditions ◽

Theoretical Results ◽

Methods Numerical

We introduce a new faster King-Werner-type derivative-free method for solving nonlinear equations. The local as well as semi-local convergence analysis is presented under weak center Lipschitz and Lipschitz conditions. The convergence order as well as the convergence radii are also provided. The radii are compared to the corresponding ones from similar methods. Numerical examples further validate 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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Edge detection with trigonometric polynomial shearlets

Advances in Computational Mathematics ◽

10.1007/s10444-020-09838-3 ◽

2021 ◽

Vol 47 (1) ◽

Author(s):

Kevin Schober ◽

Jürgen Prestin ◽

Serhii A. Stasyuk

Keyword(s):

Edge Detection ◽

Trigonometric Polynomial ◽

Two Dimensions ◽

Characteristic Functions ◽

Numerical Examples ◽

Special Cases ◽

Periodic Characteristic ◽

Boundary Curves ◽

Theoretical Results ◽

De La Vallée Poussin

AbstractIn this paper, we show that certain trigonometric polynomial shearlets which are special cases of directional de la Vallée Poussin-type wavelets are able to detect step discontinuities along boundary curves of periodic characteristic functions. Motivated by recent results for discrete shearlets in two dimensions, we provide lower and upper estimates for the magnitude of the corresponding inner products. In the proof, we use localization properties of trigonometric polynomial shearlets in the time and frequency domain and, among other things, bounds for certain Fresnel integrals. Moreover, we give numerical examples which underline the theoretical results.

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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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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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New Improvement of the Domain of Parameters for Newton’s Method

Mathematics ◽

10.3390/math8010103 ◽

2020 ◽

Vol 8 (1) ◽

pp. 103 ◽

Cited By ~ 1

Author(s):

Cristina Amorós ◽

Ioannis K. Argyros ◽

Daniel González ◽

Ángel Alberto Magreñán ◽

Samundra Regmi ◽

...

Keyword(s):

Newton’S Method ◽

Newton's Method ◽

Convergence Domain ◽

Operator Equations ◽

Numerical Examples ◽

New Information ◽

Analogous Manner ◽

The One ◽

Domain Of Parameters ◽

Local Convergence Analysis

There is a need to extend the convergence domain of iterative methods for computing a locally unique solution of Banach space valued operator equations. This is because the domain is small in general, limiting the applicability of the methods. The new idea involves the construction of a tighter set than the ones used before also containing the iterates leading to at least as tight Lipschitz parameters and consequently a finer local as well as a semi-local convergence analysis. We used Newton’s method to demonstrate our technique. However, our technique can be used to extend the applicability of other methods too in an analogous manner. In particular, the new information related to the location of the solution improves the one in previous studies. This work also includes numerical examples that validate the proven results.

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On the Influence of Center-Lipschitz Conditions in the Convergence Analysis of Multi-Point Iterative Methods

10.20944/preprints201812.0277.v1 ◽

2018 ◽

Author(s):

Ioannis K Argyros ◽

Santhosh George

Keyword(s):

Convergence Analysis ◽

Iterative Methods ◽

Local Convergence ◽

Convergence Region ◽

Numerical Examples ◽

Local Convergence Analysis ◽

Lipschitz Conditions

The aim of this article is to extend the local as well as the semi-local convergence analysis of multi-point iterative methods using center Lipschitz conditions in combination with our idea, of the restricted convergence region. It turns out that this way a finer convergence analysis for these methods is obtained than in earlier works and without additional hypotheses. Numerical examples favoring our technique over earlier ones completes this article.

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