Stepan Shakhno - Academia.edu (original) (raw)
Papers by Stepan Shakhno
Journal of Complexity, Dec 1, 2023
Математичні методи та фізико-механічні поля, Nov 14, 2018
Algorithms, Dec 20, 2022
This article is an open access article distributed under the terms and conditions of the Creative... more This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY
Mathematical modeling and computing, 2022
Numerous attempts have been made to enlarge the radius of convergence for Newtonlike method under... more Numerous attempts have been made to enlarge the radius of convergence for Newtonlike method under the same set of conditions. It turns out that not only the radius of convergence but the error bounds on the distances involved and the uniqueness of the solution ball can more accurately be defined.
Journal of Mathematical Sciences, Feb 21, 2022
We consider the problem of contact interaction of several elastic bodies with nonlinear Winkler s... more We consider the problem of contact interaction of several elastic bodies with nonlinear Winkler surface layers. To solve a nonlinear variational equation with nondifferentiable operator corresponding to this contact problem, we propose to use implicit two-point combined differential-difference parallel iterative Robin-type domain decomposition algorithms. We propose software realization of these algorithms for the case of plane contact problems based on finite-element approximations. We compare the numerical efficiency of the two-point and one-point iterative domain decomposition methods for the problem of contact of two elastic bodies containing a groove through a nonlinear Winkler layer.
Математичні методи та фізико-механічні поля, Apr 2, 2020
Journal of Mathematical Sciences, Apr 4, 2020
We propose and study an iterative method for the solution of a nonlinear least-squares problem wi... more We propose and study an iterative method for the solution of a nonlinear least-squares problem with nondifferentiable operator. In this method, instead of the Jacobian, we use the sum of derivative of the differentiable part of the operator and divided difference with specially chosen points of the nondifferentiable part of operator. We prove a theorem substantiating the process of convergence of the proposed method and establish its rate. We also present the results of numerical experiments carried out for the test problems with nondifferentiable operators.
In the paper we study a combined differential-difference method for solving nonlinear equations w... more In the paper we study a combined differential-difference method for solving nonlinear equations with non-differentiable operator. The semilocal convergence of the method is investigated and the order of convergence is established.
Journal of Mathematical Sciences, Nov 13, 2015
We study the problem of local convergence of the accelerated Newton method for the solution of no... more We study the problem of local convergence of the accelerated Newton method for the solution of nonlinear functional equations under generalized Lipschitz conditions for the first-and second-order Fréchet derivatives. We show that the accelerated method is characterized by the quadratic order of convergence and compare it with the classical Newton method.
Symmetry
Symmetries play an important role in the study of a plethora of physical phenomena, including the... more Symmetries play an important role in the study of a plethora of physical phenomena, including the study of microworlds. These phenomena reduce to solving nonlinear equations in abstract spaces. Therefore, it is important to design iterative methods for approximating the solutions, since closed forms of them can be found only in special cases. Several iterative methods were developed whose convergence was established under very general conditions. Numerous applications are also provided to solve systems of nonlinear equations and differential equations appearing in the aforementioned areas. The ball convergence analysis was developed for the King-like and Jarratt-like families of methods to solve equations under the same set of conditions. Earlier studies have used conditions up to the fifth derivative, but they failed to show the fourth convergence order. Moreover, no error distances or results on the uniqueness of the solution were given either. However, we provide such results inv...
HAL (Le Centre pour la Communication Scientifique Directe), Feb 15, 2018
Computation
A local and semi-local convergence is developed of a class of iterative methods without derivativ... more A local and semi-local convergence is developed of a class of iterative methods without derivatives for solving nonlinear Banach space valued operator equations under the classical Lipschitz conditions for first-order divided differences. Special cases of this method are well-known iterative algorithms, in particular, the Secant, Kurchatov, and Steffensen methods as well as the Newton method. For the semi-local convergence analysis, we use a technique of recurrent functions and majorizing scalar sequences. First, the convergence of the scalar sequence is proved and its limit is determined. It is then shown that the sequence obtained by the proposed method is bounded by this scalar sequence. In the local convergence analysis, a computable radius of convergence is determined. Finally, the results of the numerical experiments are given that confirm obtained theoretical estimates.
Application Mathematics and Informatics
Symmetry
The study of the microworld, quantum physics including the fundamental standard models are closel... more The study of the microworld, quantum physics including the fundamental standard models are closely related to the basis of symmetry principles. These phenomena are reduced to solving nonlinear equations in suitable abstract spaces. Such equations are solved mostly iteratively. That is why two-step iterative methods of the Kurchatov type for solving nonlinear operator equations are investigated using approximation by the Fréchet derivative of an operator of a nonlinear equation by divided differences. Local and semi-local convergence of the methods is studied under conditions that the first-order divided differences satisfy the generalized Lipschitz conditions. The conditions and speed of convergence of these methods are determined. Moreover, the domain of uniqueness is found for the solution. The results of numerical experiments validate the theoretical results. The new idea can be used on other iterative methods utilizing inverses of divided differences of order one.
Matematychni Studii
We study a local and semi-local convergence of Kurchatov's method and its two-step modificati... more We study a local and semi-local convergence of Kurchatov's method and its two-step modification for solving nonlinear equations under the classical Lipschitz conditions for the first-order divided differences. To develop a convergence analysis we use the approach of restricted convergence regions in a combination to our technique of recurrent functions. The semi-local convergence is based on the majorizing scalar sequences. Also, the results of the numerical experiment are given.
Application Mathematics and Informatics, 2019
Symmetry
A plethora of quantum physics problems are related to symmetry principles. Moreover, by using sym... more A plethora of quantum physics problems are related to symmetry principles. Moreover, by using symmetry theory and mathematical modeling, these problems reduce to solving iteratively finite differences and systems of nonlinear equations. In particular, Newton-type methods are introduced to generate sequences approximating simple solutions of nonlinear equations in the setting of Banach spaces. Specializations of these methods include the modified Newton method, Newton’s method, and other single-step methods. The convergence of these methods is established with similar conditions. However, the convergence region is not large in general. That is why a unified semilocal convergence analysis is developed that can be used to handle these methods under even weaker conditions that are not previously considered. The approach leads to the extension of the applicability of these methods in cases not covered before but without new conditions. The idea is to replace the Lipschitz parameters or o...
Journal of Numerical Analysis and Approximation Theory
In the paper a local and a semi-local convergence of combined iterative process for solving nonli... more In the paper a local and a semi-local convergence of combined iterative process for solving nonlinear operator equations is investigated. This solver is built based on Newton solver and has R-convergence order 1.839.... The radius of the convergence ball and convergence order of the investigated solver are determined in an earlier paper. Modifications of previous conditions leads to extended convergence domain. These advantages are obtained under the same computational effort. Numerical experiments are carried out on the test examples with nondifferentiable operator.
HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific r... more HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Convergence analysis of a two-step method for the nonlinear least squares problem with decomposition of operator Stepan Shakhno, Roman Iakymchuk, Halyna Yarmola
Journal of Complexity, Dec 1, 2023
Математичні методи та фізико-механічні поля, Nov 14, 2018
Algorithms, Dec 20, 2022
This article is an open access article distributed under the terms and conditions of the Creative... more This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY
Mathematical modeling and computing, 2022
Numerous attempts have been made to enlarge the radius of convergence for Newtonlike method under... more Numerous attempts have been made to enlarge the radius of convergence for Newtonlike method under the same set of conditions. It turns out that not only the radius of convergence but the error bounds on the distances involved and the uniqueness of the solution ball can more accurately be defined.
Journal of Mathematical Sciences, Feb 21, 2022
We consider the problem of contact interaction of several elastic bodies with nonlinear Winkler s... more We consider the problem of contact interaction of several elastic bodies with nonlinear Winkler surface layers. To solve a nonlinear variational equation with nondifferentiable operator corresponding to this contact problem, we propose to use implicit two-point combined differential-difference parallel iterative Robin-type domain decomposition algorithms. We propose software realization of these algorithms for the case of plane contact problems based on finite-element approximations. We compare the numerical efficiency of the two-point and one-point iterative domain decomposition methods for the problem of contact of two elastic bodies containing a groove through a nonlinear Winkler layer.
Математичні методи та фізико-механічні поля, Apr 2, 2020
Journal of Mathematical Sciences, Apr 4, 2020
We propose and study an iterative method for the solution of a nonlinear least-squares problem wi... more We propose and study an iterative method for the solution of a nonlinear least-squares problem with nondifferentiable operator. In this method, instead of the Jacobian, we use the sum of derivative of the differentiable part of the operator and divided difference with specially chosen points of the nondifferentiable part of operator. We prove a theorem substantiating the process of convergence of the proposed method and establish its rate. We also present the results of numerical experiments carried out for the test problems with nondifferentiable operators.
In the paper we study a combined differential-difference method for solving nonlinear equations w... more In the paper we study a combined differential-difference method for solving nonlinear equations with non-differentiable operator. The semilocal convergence of the method is investigated and the order of convergence is established.
Journal of Mathematical Sciences, Nov 13, 2015
We study the problem of local convergence of the accelerated Newton method for the solution of no... more We study the problem of local convergence of the accelerated Newton method for the solution of nonlinear functional equations under generalized Lipschitz conditions for the first-and second-order Fréchet derivatives. We show that the accelerated method is characterized by the quadratic order of convergence and compare it with the classical Newton method.
Symmetry
Symmetries play an important role in the study of a plethora of physical phenomena, including the... more Symmetries play an important role in the study of a plethora of physical phenomena, including the study of microworlds. These phenomena reduce to solving nonlinear equations in abstract spaces. Therefore, it is important to design iterative methods for approximating the solutions, since closed forms of them can be found only in special cases. Several iterative methods were developed whose convergence was established under very general conditions. Numerous applications are also provided to solve systems of nonlinear equations and differential equations appearing in the aforementioned areas. The ball convergence analysis was developed for the King-like and Jarratt-like families of methods to solve equations under the same set of conditions. Earlier studies have used conditions up to the fifth derivative, but they failed to show the fourth convergence order. Moreover, no error distances or results on the uniqueness of the solution were given either. However, we provide such results inv...
HAL (Le Centre pour la Communication Scientifique Directe), Feb 15, 2018
Computation
A local and semi-local convergence is developed of a class of iterative methods without derivativ... more A local and semi-local convergence is developed of a class of iterative methods without derivatives for solving nonlinear Banach space valued operator equations under the classical Lipschitz conditions for first-order divided differences. Special cases of this method are well-known iterative algorithms, in particular, the Secant, Kurchatov, and Steffensen methods as well as the Newton method. For the semi-local convergence analysis, we use a technique of recurrent functions and majorizing scalar sequences. First, the convergence of the scalar sequence is proved and its limit is determined. It is then shown that the sequence obtained by the proposed method is bounded by this scalar sequence. In the local convergence analysis, a computable radius of convergence is determined. Finally, the results of the numerical experiments are given that confirm obtained theoretical estimates.
Application Mathematics and Informatics
Symmetry
The study of the microworld, quantum physics including the fundamental standard models are closel... more The study of the microworld, quantum physics including the fundamental standard models are closely related to the basis of symmetry principles. These phenomena are reduced to solving nonlinear equations in suitable abstract spaces. Such equations are solved mostly iteratively. That is why two-step iterative methods of the Kurchatov type for solving nonlinear operator equations are investigated using approximation by the Fréchet derivative of an operator of a nonlinear equation by divided differences. Local and semi-local convergence of the methods is studied under conditions that the first-order divided differences satisfy the generalized Lipschitz conditions. The conditions and speed of convergence of these methods are determined. Moreover, the domain of uniqueness is found for the solution. The results of numerical experiments validate the theoretical results. The new idea can be used on other iterative methods utilizing inverses of divided differences of order one.
Matematychni Studii
We study a local and semi-local convergence of Kurchatov's method and its two-step modificati... more We study a local and semi-local convergence of Kurchatov's method and its two-step modification for solving nonlinear equations under the classical Lipschitz conditions for the first-order divided differences. To develop a convergence analysis we use the approach of restricted convergence regions in a combination to our technique of recurrent functions. The semi-local convergence is based on the majorizing scalar sequences. Also, the results of the numerical experiment are given.
Application Mathematics and Informatics, 2019
Symmetry
A plethora of quantum physics problems are related to symmetry principles. Moreover, by using sym... more A plethora of quantum physics problems are related to symmetry principles. Moreover, by using symmetry theory and mathematical modeling, these problems reduce to solving iteratively finite differences and systems of nonlinear equations. In particular, Newton-type methods are introduced to generate sequences approximating simple solutions of nonlinear equations in the setting of Banach spaces. Specializations of these methods include the modified Newton method, Newton’s method, and other single-step methods. The convergence of these methods is established with similar conditions. However, the convergence region is not large in general. That is why a unified semilocal convergence analysis is developed that can be used to handle these methods under even weaker conditions that are not previously considered. The approach leads to the extension of the applicability of these methods in cases not covered before but without new conditions. The idea is to replace the Lipschitz parameters or o...
Journal of Numerical Analysis and Approximation Theory
In the paper a local and a semi-local convergence of combined iterative process for solving nonli... more In the paper a local and a semi-local convergence of combined iterative process for solving nonlinear operator equations is investigated. This solver is built based on Newton solver and has R-convergence order 1.839.... The radius of the convergence ball and convergence order of the investigated solver are determined in an earlier paper. Modifications of previous conditions leads to extended convergence domain. These advantages are obtained under the same computational effort. Numerical experiments are carried out on the test examples with nondifferentiable operator.
HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific r... more HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Convergence analysis of a two-step method for the nonlinear least squares problem with decomposition of operator Stepan Shakhno, Roman Iakymchuk, Halyna Yarmola