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

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.

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

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 207 ◽  
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.

scholarly journals Extending the Applicability of Two-Step Solvers for Solving Equations

Mathematics ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 62 ◽  
Author(s):  
Ioannis K. Argyros ◽  
Stepan Shakhno
Keyword(s):  
Nonlinear Operator ◽  
Convergence Criteria ◽  
Operator Equations ◽  
Nonlinear Operator Equations ◽  
First Order ◽  
Special Cases ◽  
Solving Equations ◽  
Original Domain ◽  
Lipschitz Conditions ◽  
Theoretical Results

We present a local convergence of two-step solvers for solving nonlinear operator equations under the generalized Lipschitz conditions for the first- and second-order derivatives and for the first order divided differences. In contrast to earlier works, we use our new idea of 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 better information on the solution. These extensions require the same 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.

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.

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

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.

scholarly journals Extending the Applicability of a Two-Step Chord-Type Method for Non-Differentiable Operators

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 804
Author(s):  
Ioannis K. Argyros ◽  
Neha Gupta ◽  
J. P. Jaiswal
Keyword(s):  
Approximate Solution ◽  
Convergence Analysis ◽  
Banach Spaces ◽  
Recurrence Relation ◽  
Convergence Theorem ◽  
Local Convergence ◽  
Existence And Uniqueness ◽  
Numerical Illustration ◽  
Type Method ◽  
Local Convergence Analysis

The semi-local convergence analysis of a well defined and efficient two-step Chord-type method in Banach spaces is presented in this study. The recurrence relation technique is used under some weak assumptions. The pertinency of the assumed method is extended for nonlinear non-differentiable operators. The convergence theorem is also established to show the existence and uniqueness of the approximate solution. A numerical illustration is quoted to certify the theoretical part which shows that earlier studies fail if the function is non-differentiable.

scholarly journals Extended Local Analysis of Inexact Gauss-Newton-like Method for Least Square Problems using Restricted Convergence Domains

2016 ◽  
Vol 54 (1) ◽  
pp. 17-33
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Convergence Analysis ◽  
Computational Cost ◽  
Nonlinear Least Squares ◽  
Least Square ◽  
Error Tolerance ◽  
Local Analysis ◽  
Large Radius ◽  
Special Cases ◽  
Local Convergence Analysis ◽  
Least Square Problems

Abstract We present a local convergence analysis of inexact Gauss-Newton-like method (IGNLM) for solving nonlinear least-squares problems in a Euclidean space setting. The convergence analysis is based on our new idea of restricted convergence domains. Using this idea, we obtain a more precise information on the location of the iterates than in earlier studies leading to smaller majorizing functions. This way, our approach has the following advantages and under the same computational cost as in earlier studies: A large radius of convergence and more precise estimates on the distances involved to obtain a desired error tolerance. That is, we have a larger choice of initial points and fewer iterations are also needed to achieve the error tolerance. Special cases and numerical examples are also presented to show these advantages.

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

Foundations ◽  
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.

scholarly journals On Local Convergence Analysis of Inexact Newton Method for Singular Systems of Equations under Majorant Condition

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Fangqin Zhou
Keyword(s):  
Convergence Analysis ◽  
Newton Method ◽  
Local Convergence ◽  
Singular Systems ◽  
Inexact Newton Method ◽  
Systems Of Equations ◽  
Convergence Ball ◽  
Special Cases ◽  
Majorant Condition ◽  
Local Convergence Analysis

We present a local convergence analysis of inexact Newton method for solving singular systems of equations. Under the hypothesis that the derivative of the function associated with the singular systems satisfies a majorant condition, we obtain that the method is well defined and converges. Our analysis provides a clear relationship between the majorant function and the function associated with the singular systems. It also allows us to obtain an estimate of convergence ball for inexact Newton method and some important special cases.

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