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Algorithms
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Numerical Algorithms for Solving Nonlinear Equations and Systems 2017
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Numerical Algorithms for Solving Nonlinear Equations and Systems 2017

A special issue of Algorithms (ISSN 1999-4893).

Deadline for manuscript submissions: closed (31 May 2017) | Viewed by 25683

Special Issue Editors

Prof. Dr. Alicia Cordero

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Guest Editor
School of Telecommunications Engineering, Universitat Politècnica de València, 46022 Valencia, Spain
Interests: numerical analysis; iterative methods; nonlinear problems; discrete dynamics, real and complex; fractional iterative methods; chaos and stability in iterative procedures; fractal dimensión in Julia sets
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Juan Ramón Torregrosa Sánchez

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Guest Editor
Institute for Multidisciplinary Mathematics, Universitat Politècnica de València, 46022 València, Spain
Interests: iterative processes; matrix analysis; numerical analysis
Special Issues, Collections and Topics in MDPI journals
Dr. Francisco I. Chicharro

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Guest Editor
Inst. Matemática Multidisciplinar, Universitat Politècnica de València, Camino de Vera, s/n, 46022 Valencia, Spain
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Solving nonlinear equations and systems is a non-trivial task that involves many areas of Science and Technology. Usually, it is not affordable in a direct way, and iterative algorithms play a fundamental role in their approach. This is an area of research that has seen an exponential growth in the last few years.

The main theme of this Special Issue is the design, analysis of convergence and application to practical problems of new iterative schemes for solving nonlinear problems. This includes methods with and without memory, with derivatives or derivative-free, the real or complex dynamics associated to them and an analysis of their convergence that can be local, semi-local or global.

Dr. Alicia Cordero
Dr. Juan R. Torregrosa
Dr. Francisco I. Chicharro
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. Algorithms 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 1600 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

  • Multi-point iterative methods (with or without memory)
  • Iterative methods for singular problems
  • Iterative methods in Banach spaces
  • Dynamical study of iterative methods
  • Nonlinear matrix equations

Published Papers (6 papers)

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Research

282 KiB  
Article
On the Existence of Solutions of Nonlinear Fredholm Integral Equations from Kantorovich’s Technique
by José Antonio Ezquerro and Miguel Ángel Hernández-Verón
Algorithms 2017, 10(3), 89; https://0-doi-org.brum.beds.ac.uk/10.3390/a10030089 - 02 Aug 2017
Cited by 5 | Viewed by 4181
Abstract
The well-known Kantorovich technique based on majorizing sequences is used to analyse the convergence of Newton’s method when it is used to solve nonlinear Fredholm integral equations. In addition, we obtain information about the domains of existence and uniqueness of a solution for [...] Read more.
The well-known Kantorovich technique based on majorizing sequences is used to analyse the convergence of Newton’s method when it is used to solve nonlinear Fredholm integral equations. In addition, we obtain information about the domains of existence and uniqueness of a solution for these equations. Finally, we illustrate the above with two particular Fredholm integral equations. Full article
(This article belongs to the Special Issue Numerical Algorithms for Solving Nonlinear Equations and Systems 2017)
4338 KiB  
Article
On the Lagged Diffusivity Method for the Solution of Nonlinear Finite Difference Systems
by Francesco Mezzadri and Emanuele Galligani
Algorithms 2017, 10(3), 88; https://0-doi-org.brum.beds.ac.uk/10.3390/a10030088 - 02 Aug 2017
Cited by 1 | Viewed by 4673
Abstract
In this paper, we extend the analysis of the Lagged Diffusivity Method for nonlinear, non-steady reaction-convection-diffusion equations. In particular, we describe how the method can be used to solve the systems arising from different discretization schemes, recalling some results on the convergence of [...] Read more.
In this paper, we extend the analysis of the Lagged Diffusivity Method for nonlinear, non-steady reaction-convection-diffusion equations. In particular, we describe how the method can be used to solve the systems arising from different discretization schemes, recalling some results on the convergence of the method itself. Moreover, we also analyze the behavior of the method in case of problems presenting boundary layers or blow-up solutions. Full article
(This article belongs to the Special Issue Numerical Algorithms for Solving Nonlinear Equations and Systems 2017)
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403 KiB  
Article
An Efficient Algorithm for the Separable Nonlinear Least Squares Problem
by Yunqiu Shen and Tjalling J. Ypma
Algorithms 2017, 10(3), 78; https://0-doi-org.brum.beds.ac.uk/10.3390/a10030078 - 10 Jul 2017
Cited by 1 | Viewed by 4016
Abstract
The nonlinear least squares problem m i n y , z ∥ A ( y ) z + b ( y ) ∥ , where A ( y ) is a full-rank ( N + ℓ ) × N matrix, [...] Read more.
The nonlinear least squares problem m i n y , z ∥ A ( y ) z + b ( y ) ∥ , where A ( y ) is a full-rank ( N + ℓ ) × N matrix, y ∈ R n , z ∈ R N and b ( y ) ∈ R N + ℓ with ℓ ≥ n , can be solved by first solving a reduced problem m i n y ∥ f ( y ) ∥ to find the optimal value y * of y, and then solving the resulting linear least squares problem m i n z ∥ A ( y * ) z + b ( y * ) ∥ to find the optimal value z * of z. We have previously justified the use of the reduced function f ( y ) = C T ( y ) b ( y ) , where C ( y ) is a matrix whose columns form an orthonormal basis for the nullspace of A T ( y ) , and presented a quadratically convergent Gauss–Newton type method for solving m i n y ∥ C T ( y ) b ( y ) ∥ based on the use of QR factorization. In this note, we show how LU factorization can replace the QR factorization in those computations, halving the associated computational cost while also providing opportunities to exploit sparsity and thus further enhance computational efficiency. Full article
(This article belongs to the Special Issue Numerical Algorithms for Solving Nonlinear Equations and Systems 2017)
268 KiB  
Article
Expanding the Applicability of Some High Order Househölder-Like Methods
by Sergio Amat, Ioannis K. Argyros, Miguel A. Hernández-Verón and Natalia Romero
Algorithms 2017, 10(2), 64; https://0-doi-org.brum.beds.ac.uk/10.3390/a10020064 - 31 May 2017
Cited by 1 | Viewed by 3958
Abstract
This paper is devoted to the semilocal convergence of a Househölder-like method for nonlinear equations. The method includes many of the studied third order iterative methods. In the present study, we use our new idea of restricted convergence domains leading to smaller [...] Read more.
This paper is devoted to the semilocal convergence of a Househölder-like method for nonlinear equations. The method includes many of the studied third order iterative methods. In the present study, we use our new idea of restricted convergence domains leading to smaller γ -parameters, which in turn lead to the following advantages over earlier works (and under the same computational cost): larger convergence domain; tighter error bounds on the distances involved, and at least as precise information on the location of the solution. Full article
(This article belongs to the Special Issue Numerical Algorithms for Solving Nonlinear Equations and Systems 2017)
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238 KiB  
Article
Extending the Applicability of the MMN-HSS Method for Solving Systems of Nonlinear Equations under Generalized Conditions
by Ioannis K. Argyros, Janak Raj Sharma and Deepak Kumar
Algorithms 2017, 10(2), 54; https://0-doi-org.brum.beds.ac.uk/10.3390/a10020054 - 12 May 2017
Cited by 1 | Viewed by 3653
Abstract
We present the semilocal convergence of a multi-step modified Newton-Hermitian and Skew-Hermitian Splitting method (MMN-HSS method) to approximate a solution of a nonlinear equation. Earlier studies show convergence under only Lipschitz conditions limiting the applicability of this method. The convergence in this study [...] Read more.
We present the semilocal convergence of a multi-step modified Newton-Hermitian and Skew-Hermitian Splitting method (MMN-HSS method) to approximate a solution of a nonlinear equation. Earlier studies show convergence under only Lipschitz conditions limiting the applicability of this method. The convergence in this study is shown under generalized Lipschitz-type conditions and restricted convergence domains. Hence, the applicability of the method is extended. Moreover, numerical examples are also provided to show that our results can be applied to solve equations in cases where earlier study cannot be applied. Furthermore, in the cases where both old and new results are applicable, the latter provides a larger domain of convergence and tighter error bounds on the distances involved. Full article
(This article belongs to the Special Issue Numerical Algorithms for Solving Nonlinear Equations and Systems 2017)
407 KiB  
Article
An Efficient Sixth-Order Newton-Type Method for Solving Nonlinear Systems
by Xiaofeng Wang and Yang Li
Algorithms 2017, 10(2), 45; https://0-doi-org.brum.beds.ac.uk/10.3390/a10020045 - 25 Apr 2017
Cited by 15 | Viewed by 4230
Abstract
In this paper, we present a new sixth-order iterative method for solving nonlinear systems and prove a local convergence result. The new method requires solving five linear systems per iteration. An important feature of the new method is that the LU (lower upper, [...] Read more.
In this paper, we present a new sixth-order iterative method for solving nonlinear systems and prove a local convergence result. The new method requires solving five linear systems per iteration. An important feature of the new method is that the LU (lower upper, also called LU factorization) decomposition of the Jacobian matrix is computed only once in each iteration. The computational efficiency index of the new method is compared to that of some known methods. Numerical results are given to show that the convergence behavior of the new method is similar to the existing methods. The new method can be applied to small- and medium-sized nonlinear systems. Full article
(This article belongs to the Special Issue Numerical Algorithms for Solving Nonlinear Equations and Systems 2017)
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