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

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
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.

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

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.

On the quadratic convergence of Newton’s method under center-Lipschitz but not necessarily Lipschitz hypotheses

2013 ◽  
Vol 63 (3) ◽  
Author(s):  
Ioannis Argyros ◽  
Saïd Hilout
Keyword(s):  
Banach Space ◽  
Newton’S Method ◽  
Newton's Method ◽  
Quadratic Convergence ◽  
Lipschitz Constant ◽  
Semilocal Convergence ◽  
Convergence Conditions ◽  
Numerical Examples ◽  
Convergence Results

AbstractWe provide new local and semilocal convergence results for Newton’s method in a Banach space. The sufficient convergence conditions do not include the Lipschitz constant usually associated with Newton’s method. Numerical examples demonstrating the expansion of Newton’s method are also provided in this study.

scholarly journals Extending the applicability of Newton’s method using Wang’s– Smale’s α–theory

2017 ◽  
Vol 33 (1) ◽  
pp. 27-33
Author(s):  
IOANNIS K. ARGYROS ◽  
◽  
SANTHOSH GEORGE ◽  
Keyword(s):  
Newton’S Method ◽  
Newton's Method ◽  
Computational Cost ◽  
Semilocal Convergence ◽  
Numerical Examples ◽  
Convergence Results

We improve semilocal convergence results for Newton’s method by defining a more precise domain where the Newton iterate lies than in earlier studies using the Smale’s α– theory. These improvements are obtained under the same computational cost. Numerical examples are also presented in this study to show that the earlier results cannot apply but the new results can apply to solve equations.

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

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.

scholarly journals Edge detection with trigonometric polynomial shearlets

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.

Convergence of derivative free iterative methods

2019 ◽  
Vol 28 (1) ◽  
pp. 19-26
Author(s):  
IOANNIS K. ARGYROS ◽  
◽  
SANTHOSH GEORGE ◽  
Keyword(s):  
Banach Space ◽  
Iterative Methods ◽  
Local Convergence ◽  
Practical Interest ◽  
Systems Of Equations ◽  
Hölder Continuous ◽  
Numerical Examples ◽  
Lipschitz Continuous ◽  
Derivative Free ◽  
Special Cases

We present the local as well as the semi-local convergence of some iterative methods free of derivatives for Banach space valued operators. These methods contain the secant and the Kurchatov method as special cases. The convergence is based on weak hypotheses specializing to Lipschitz continuous or Holder continuous hypotheses. The results are of theoretical and practical interest. In particular the method is compared favorably ¨ to other methods using concrete numerical examples to solve systems of equations containing a nondifferentiable term.

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

Mathematics ◽  
2019 ◽  
Vol 7 (5) ◽  
pp. 463 ◽  
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.

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

Mathematics ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 299 ◽  
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.

EXTENDING THE APPLICABILITY OF NEWTON'S METHOD ON RIEMANNIAN MANIFOLDS WITH VALUES IN A CONE

2013 ◽  
Vol 06 (03) ◽  
pp. 1350041
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Error Analysis ◽  
Convergence Analysis ◽  
Lipschitz Condition ◽  
Riemannian Manifolds ◽  
Newton’S Method ◽  
Newton's Method ◽  
Semilocal Convergence ◽  
Inclusion Problem ◽  
Convergence Conditions ◽  
Precise Information

We present a new semilocal convergence analysis of Newton's method on Riemannian manifolds with values in a cone in order to solve the inclusion problem. Using more precise majorizing sequences than in earlier studies such as [J. H. Wang, S. Huang and C. Li, Extended Newton's method for mappings on Riemannian manifolds with values in a cone, Taiwanese J. Math.13(2B) (2009) 633–656] and the concept of L-average Lipschitz condition we provide: weaker sufficient convergence conditions; tighter error analysis on the distances involved and an at least as precise information on the solutions. These advantages are obtained using the same parameters and functions. Applications include the celebrated Newton–Kantorovich theorem.

scholarly journals Approximation of the Solution of Delay Fractional Differential Equation Using AA-Iterative Scheme

Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 273
Author(s):  
Mujahid Abbas ◽  
Muhammad Waseem Asghar ◽  
Manuel De la Sen
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Contraction Mapping ◽  
Iterative Scheme ◽  
Convex Banach Space ◽  
Uniformly Convex ◽  
Numerical Examples ◽  
Strong And Weak Convergence ◽  
Convergence Results ◽  
Fractional Differential

The aim of this paper is to propose a new faster iterative scheme (called AA-iteration) to approximate the fixed point of (b,η)-enriched contraction mapping in the framework of Banach spaces. It is also proved that our iteration is stable and converges faster than many iterations existing in the literature. For validity of our proposed scheme, we presented some numerical examples. Further, we proved some strong and weak convergence results for b-enriched nonexpansive mapping in the uniformly convex Banach space. Finally, we approximate the solution of delay fractional differential equations using AA-iterative scheme.

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