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Symmetry
Special Issues
Recent Advances and Application of Iterative Methods
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Recent Advances and Application of Iterative Methods

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (15 October 2022) | Viewed by 8478

Special Issue Editors

Prof. Dr. Petko D. Proinov

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Guest Editor
Faculty of Mathematics and Informatics, University of Plovdiv Paisii Hilendarski, 4000 Plovdiv, Bulgaria
Interests: iterative methods; fixed point theory
Dr. Stoil I. Ivanov

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Guest Editor
Faculty of Physics and Technology, University of Plovdiv Paisii Hilendarski, 24 Tzar Asen, 4000 Plovdiv, Bulgaria
Interests: iterative methods; numerical algorithms; convergence analysis; polynomial zeros; phase transitions
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Iterative methods are among the most powerful tools for solving various kinds of complex symmetric linear and nonlinear problems in mathematics and the other sciences. They play an important role in fixed point theory and many branches of mathematics. Iterative methods keep getting more and more attractive because of their easy computer implementations and vast applicability.

This Special Issue is an advanced forum for high-value scientific studies on iterative methods and their applications. In particular, works on fixed point theory and other areas of mathematics that use iterative techniques are welcome.

A limited number of expository and survey articles will also be published.

Prof. Dr. Petko D. Proinov
Dr. Stoil I. Ivanov
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • iterative methods
  • convergence analysis
  • error estimates
  • stability analysis
  • dynamical analysis
  • computational efficiency
  • attraction basin
  • operator equations
  • dynamical systems
  • fixed point theory

Published Papers (5 papers)

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Research

13 pages, 583 KiB  
Article
An Ulm-Type Inverse-Free Iterative Scheme for Fredholm Integral Equations of Second Kind
by José M. Gutiérrez and Miguel Á. Hernández-Verón
Symmetry 2021, 13(10), 1957; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13101957 - 17 Oct 2021
Viewed by 1001
Abstract
In this paper, we present an iterative method based on the well-known Ulm’s method to numerically solve Fredholm integral equations of the second kind. We support our strategy in the symmetry between two well-known problems in Numerical Analysis: the solution of linear integral [...] Read more.
In this paper, we present an iterative method based on the well-known Ulm’s method to numerically solve Fredholm integral equations of the second kind. We support our strategy in the symmetry between two well-known problems in Numerical Analysis: the solution of linear integral equations and the approximation of inverse operators. In this way, we obtain a two-folded algorithm that allows us to approximate, with quadratic order of convergence, the solution of the integral equation as well as the inverses at the solution of the derivative of the operator related to the problem. We have studied the semilocal convergence of the method and we have obtained the expression of the method in a particular case, given by some adequate initial choices. The theoretical results are illustrated with two applications to integral equations, given by symmetric non-separable kernels. Full article
(This article belongs to the Special Issue Recent Advances and Application of Iterative Methods)
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12 pages, 747 KiB  
Article
Derivative-Free Iterative Methods with Some Kurchatov-Type Accelerating Parameters for Solving Nonlinear Systems
by Xiaofeng Wang, Yingfanghua Jin and Yali Zhao
Symmetry 2021, 13(6), 943; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13060943 - 26 May 2021
Cited by 6 | Viewed by 1715
Abstract
Some Kurchatov-type accelerating parameters are used to construct some derivative-free iterative methods with memory for solving nonlinear systems. New iterative methods are developed from an initial scheme without memory with order of convergence three. New methods have the convergence order [...] Read more.
Some Kurchatov-type accelerating parameters are used to construct some derivative-free iterative methods with memory for solving nonlinear systems. New iterative methods are developed from an initial scheme without memory with order of convergence three. New methods have the convergence order 2+5≈4.236 and 5, respectively. The application of new methods can solve standard nonlinear systems and nonlinear ordinary differential equations (ODEs) in numerical experiments. Numerical results support the theoretical results. Full article
(This article belongs to the Special Issue Recent Advances and Application of Iterative Methods)
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17 pages, 1943 KiB  
Article
A General Optimal Iterative Scheme with Arbitrary Order of Convergence
by Alicia Cordero, Juan R. Torregrosa and Paula Triguero-Navarro
Symmetry 2021, 13(5), 884; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13050884 - 16 May 2021
Cited by 9 | Viewed by 1796
Abstract
A general optimal iterative method, for approximating the solution of nonlinear equations, of (n+1) steps with 2n+1 order of convergence is presented. Cases n=0 and n=1 correspond to Newton’s and Ostrowski’s schemes, [...] Read more.
A general optimal iterative method, for approximating the solution of nonlinear equations, of (n+1) steps with 2n+1 order of convergence is presented. Cases n=0 and n=1 correspond to Newton’s and Ostrowski’s schemes, respectively. The basins of attraction of the proposed schemes on different test functions are analyzed and compared with the corresponding to other known methods. The dynamical planes showing the different symmetries of the basins of attraction of new and known methods are presented. The performance of different methods on some test functions is shown. Full article
(This article belongs to the Special Issue Recent Advances and Application of Iterative Methods)
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29 pages, 441 KiB  
Article
Two Classes of Iteration Functions and Q-Convergence of Two Iterative Methods for Polynomial Zeros
by Petko D. Proinov
Symmetry 2021, 13(3), 371; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13030371 - 25 Feb 2021
Cited by 6 | Viewed by 1274
Abstract
In this work, two broad classes of iteration functions in n-dimensional vector spaces are introduced. They are called iteration functions of the first and second kind at a fixed point of the corresponding iteration function. Two general local convergence theorems are presented for [...] Read more.
In this work, two broad classes of iteration functions in n-dimensional vector spaces are introduced. They are called iteration functions of the first and second kind at a fixed point of the corresponding iteration function. Two general local convergence theorems are presented for Picard-type iterative methods with high Q-order of convergence. In particular, it is shown that if an iterative method is generated by an iteration function of first or second kind, then it is Q-convergent under each initial approximation that is sufficiently close to the fixed point. As an application, a detailed local convergence analysis of two fourth-order iterative methods is provided for finding all zeros of a polynomial simultaneously. The new results improve the previous ones for these methods in several directions. Full article
(This article belongs to the Special Issue Recent Advances and Application of Iterative Methods)
14 pages, 471 KiB  
Article
Gradient Iterative Method with Optimal Convergent Factor for Solving a Generalized Sylvester Matrix Equation with Applications to Diffusion Equations
by Nunthakarn Boonruangkan and Pattrawut Chansangiam
Symmetry 2020, 12(10), 1732; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12101732 - 20 Oct 2020
Cited by 2 | Viewed by 1483
Abstract
We introduce a gradient iterative scheme with an optimal convergent factor for solving a generalized Sylvester matrix equation ∑i=1pAiXBi=F, where Ai,Bi and F are conformable rectangular matrices. [...] Read more.
We introduce a gradient iterative scheme with an optimal convergent factor for solving a generalized Sylvester matrix equation ∑i=1pAiXBi=F, where Ai,Bi and F are conformable rectangular matrices. The iterative scheme is derived from the gradients of the squared norm-errors of the associated subsystems for the equation. The convergence analysis reveals that the sequence of approximated solutions converge to the exact solution for any initial value if and only if the convergent factor is chosen properly in terms of the spectral radius of the associated iteration matrix. We also discuss the convergent rate and error estimations. Moreover, we determine the fastest convergent factor so that the associated iteration matrix has the smallest spectral radius. Furthermore, we provide numerical examples to illustrate the capability and efficiency of this method. Finally, we apply the proposed scheme to discretized equations for boundary value problems involving convection and diffusion. Full article
(This article belongs to the Special Issue Recent Advances and Application of Iterative Methods)
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