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

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

Deadline for manuscript submissions: closed (31 October 2015) | Viewed by 135978

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
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Juan Ramón Torregrosa Sánchez

E-Mail Website
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, directly generating solutions to such equations and systems is not affordable. Thus, iterative algorithms play a fundamental role. This is an area of research that has experienced exponential growth in recent years.

The main theme of this Special Issue (but not the exclusive one) is the design and analysis of convergence and the applications 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 with such methods, and an analysis of their convergence, which can be local, semi-local or global.

Dr. Alicia Cordero
Dr. Juan R. Torregrosa
Dr. Francisco I. Chicharro
Guest Editors

Keywords

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

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Published Papers (26 papers)

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Research

235 KiB  
Article
The Iterative Solution to Discrete-Time H∞ Control Problems for Periodic Systems
by Ivan G. Ivanov and Boryana C. Bogdanova
Algorithms 2016, 9(1), 20; https://doi.org/10.3390/a9010020 - 14 Mar 2016
Cited by 2 | Viewed by 4305
Abstract
This paper addresses the problem of solving discrete-time H ∞ control problems for periodic systems. The approach for solving such a type of equations is well known in the literature. However, the focus of our research is set on the numerical computation of [...] Read more.
This paper addresses the problem of solving discrete-time H ∞ control problems for periodic systems. The approach for solving such a type of equations is well known in the literature. However, the focus of our research is set on the numerical computation of the stabilizing solution. In particular, two effective methods for practical realization of the known iterative processes are described. Furthermore, a new iterative approach is investigated and applied. On the basis of numerical experiments, we compare the presented methods. A major conclusion is that the new iterative approach is faster than rest of the methods and it uses less RAM memory than other methods. Full article
(This article belongs to the Special Issue Numerical Algorithms for Solving Nonlinear Equations and Systems)
263 KiB  
Article
Constructing Frozen Jacobian Iterative Methods for Solving Systems of Nonlinear Equations, Associated with ODEs and PDEs Using the Homotopy Method
by Uswah Qasim, Zulifqar Ali, Fayyaz Ahmad, Stefano Serra-Capizzano, Malik Zaka Ullah and Mir Asma
Algorithms 2016, 9(1), 18; https://doi.org/10.3390/a9010018 - 11 Mar 2016
Cited by 8 | Viewed by 5716
Abstract
A homotopy method is presented for the construction of frozen Jacobian iterative methods. The frozen Jacobian iterative methods are attractive because the inversion of the Jacobian is performed in terms of LUfactorization only once, for a single instance of the iterative method. We [...] Read more.
A homotopy method is presented for the construction of frozen Jacobian iterative methods. The frozen Jacobian iterative methods are attractive because the inversion of the Jacobian is performed in terms of LUfactorization only once, for a single instance of the iterative method. We embedded parameters in the iterative methods with the help of the homotopy method: the values of the parameters are determined in such a way that a better convergence rate is achieved. The proposed homotopy technique is general and has the ability to construct different families of iterative methods, for solving weakly nonlinear systems of equations. Further iterative methods are also proposed for solving general systems of nonlinear equations. Full article
(This article belongs to the Special Issue Numerical Algorithms for Solving Nonlinear Equations and Systems)
252 KiB  
Article
A Geometric Orthogonal Projection Strategy for Computing the Minimum Distance Between a Point and a Spatial Parametric Curve
by Xiaowu Li, Zhinan Wu, Linke Hou, Lin Wang, Chunguang Yue and Qiao Xin
Algorithms 2016, 9(1), 15; https://doi.org/10.3390/a9010015 - 6 Feb 2016
Cited by 6 | Viewed by 6124
Abstract
A new orthogonal projection method for computing the minimum distance between a point and a spatial parametric curve is presented. It consists of a geometric iteration which converges faster than the existing Newton’s method, and it is insensitive to the choice of initial [...] Read more.
A new orthogonal projection method for computing the minimum distance between a point and a spatial parametric curve is presented. It consists of a geometric iteration which converges faster than the existing Newton’s method, and it is insensitive to the choice of initial values. We prove that projecting a point onto a spatial parametric curve under the method is globally second-order convergence. Full article
(This article belongs to the Special Issue Numerical Algorithms for Solving Nonlinear Equations and Systems)
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277 KiB  
Article
Two Efficient Derivative-Free Iterative Methods for Solving Nonlinear Systems
by Xiaofeng Wang and Xiaodong Fan
Algorithms 2016, 9(1), 14; https://doi.org/10.3390/a9010014 - 1 Feb 2016
Cited by 10 | Viewed by 4974
Abstract
In this work, two multi-step derivative-free iterative methods are presented for solving system of nonlinear equations. The new methods have high computational efficiency and low computational cost. The order of convergence of the new methods is proved by a development of an inverse [...] Read more.
In this work, two multi-step derivative-free iterative methods are presented for solving system of nonlinear equations. The new methods have high computational efficiency and low computational cost. The order of convergence of the new methods is proved by a development of an inverse first-order divided difference operator. The computational efficiency is compared with the existing methods. Numerical experiments support the theoretical results. Experimental results show that the new methods remarkably reduce the computing time in the process of high-precision computing. Full article
(This article belongs to the Special Issue Numerical Algorithms for Solving Nonlinear Equations and Systems)
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387 KiB  
Article
An Optimal Order Method for Multiple Roots in Case of Unknown Multiplicity
by Jai Prakash Jaiswal
Algorithms 2016, 9(1), 10; https://doi.org/10.3390/a9010010 - 22 Jan 2016
Cited by 4 | Viewed by 4269
Abstract
In the literature, recently, some three-step schemes involving four function evaluations for the solution of multiple roots of nonlinear equations, whose multiplicity is not known in advance, are considered, but they do not agree with Kung–Traub’s conjecture. The present article is devoted to [...] Read more.
In the literature, recently, some three-step schemes involving four function evaluations for the solution of multiple roots of nonlinear equations, whose multiplicity is not known in advance, are considered, but they do not agree with Kung–Traub’s conjecture. The present article is devoted to the study of an iterative scheme for approximating multiple roots with a convergence rate of eight, when the multiplicity is hidden, which agrees with Kung–Traub’s conjecture. The theoretical study of the convergence rate is investigated and demonstrated. A few nonlinear problems are presented to justify the theoretical study. Full article
(This article belongs to the Special Issue Numerical Algorithms for Solving Nonlinear Equations and Systems)
230 KiB  
Article
A Family of Iterative Methods for Solving Systems of Nonlinear Equations Having Unknown Multiplicity
by Fayyaz Ahmad, S. Serra-Capizzano, Malik Zaka Ullah and A. S. Al-Fhaid
Algorithms 2016, 9(1), 5; https://doi.org/10.3390/a9010005 - 31 Dec 2015
Cited by 1 | Viewed by 5349
Abstract
The singularity of Jacobian happens when we are looking for a root, with multiplicity greater than one, of a system of nonlinear equations. The purpose of this article is two-fold. Firstly, we will present a modification of an existing method that computes roots [...] Read more.
The singularity of Jacobian happens when we are looking for a root, with multiplicity greater than one, of a system of nonlinear equations. The purpose of this article is two-fold. Firstly, we will present a modification of an existing method that computes roots with known multiplicities. Secondly, will propose the generalization of a family of methods for solving nonlinear equations with unknown multiplicities, to the system of nonlinear equations. The inclusion of a nonzero multi-variable auxiliary function is the key idea. Different choices of the auxiliary function give different families of the iterative method to find roots with unknown multiplicities. Few illustrative numerical experiments and a critical discussion end the paper. Full article
(This article belongs to the Special Issue Numerical Algorithms for Solving Nonlinear Equations and Systems)
295 KiB  
Article
On the Kung-Traub Conjecture for Iterative Methods for Solving Quadratic Equations
by Diyashvir Kreetee Rajiv Babajee
Algorithms 2016, 9(1), 1; https://doi.org/10.3390/a9010001 - 24 Dec 2015
Cited by 7 | Viewed by 5491
Abstract
Kung-Traub’s conjecture states that an optimal iterative method based on d function evaluations for finding a simple zero of a nonlinear function could achieve a maximum convergence order of 2 d−1. During the last years, many attempts have been made to prove [...] Read more.
Kung-Traub’s conjecture states that an optimal iterative method based on d function evaluations for finding a simple zero of a nonlinear function could achieve a maximum convergence order of 2 d−1. During the last years, many attempts have been made to prove this conjecture or develop optimal methods which satisfy the conjecture. We understand from the conjecture that the maximum order reached by a method with three function evaluations is four, even for quadratic functions. In this paper, we show that the conjecture fails for quadratic functions. In fact, we can find a 2-point method with three function evaluations reaching fifth order convergence. We also develop 2-point 3rd to 8th order methods with one function and two first derivative evaluations using weight functions. Furthermore, we show that with the same number of function evaluations we can develop higher order 2-point methods of order r + 2 , where r is a positive integer, ≥ 1 . We also show that we can develop a higher order method with the same number of function evaluations if we know the asymptotic error constant of the previous method. We prove the local convergence of these methods which we term as Babajee’s Quadratic Iterative Methods and we extend these methods to systems involving quadratic equations. We test our methods with some numerical experiments including an application to Chandrasekhar’s integral equation arising in radiative heat transfer theory. Full article
(This article belongs to the Special Issue Numerical Algorithms for Solving Nonlinear Equations and Systems)
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1775 KiB  
Article
Offset-Assisted Factored Solution of Nonlinear Systems
by José M. Ruiz-Oltra, Catalina Gómez-Quiles and Antonio Gómez-Expósito
Algorithms 2016, 9(1), 2; https://doi.org/10.3390/a9010002 - 23 Dec 2015
Viewed by 5286
Abstract
This paper presents an improvement to the recently-introduced factored method for the solution of nonlinear equations. The basic idea consists of transforming the original system by adding an offset to all unknowns. When searching for real solutions, a real offset prevents the intermediate [...] Read more.
This paper presents an improvement to the recently-introduced factored method for the solution of nonlinear equations. The basic idea consists of transforming the original system by adding an offset to all unknowns. When searching for real solutions, a real offset prevents the intermediate values of unknowns from becoming complex. Reciprocally, when searching for complex solutions, a complex offset is advisable to allow the iterative process to quickly abandon the real domain. Several examples are used to illustrate the performance of the proposed algorithm, when compared to Newton’s method. Full article
(This article belongs to the Special Issue Numerical Algorithms for Solving Nonlinear Equations and Systems)
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374 KiB  
Article
Numerical Properties of Different Root-Finding Algorithms Obtained for Approximating Continuous Newton’s Method
by José M. Gutiérrez
Algorithms 2015, 8(4), 1210-1218; https://doi.org/10.3390/a8041210 - 17 Dec 2015
Cited by 6 | Viewed by 4758
Abstract
This paper is dedicated to the study of continuous Newton’s method, which is a generic differential equation whose associated flow tends to the zeros of a given polynomial. Firstly, we analyze some numerical features related to the root-finding methods obtained after applying different [...] Read more.
This paper is dedicated to the study of continuous Newton’s method, which is a generic differential equation whose associated flow tends to the zeros of a given polynomial. Firstly, we analyze some numerical features related to the root-finding methods obtained after applying different numerical methods for solving initial value problems. The relationship between the step size and the order of convergence is particularly considered. We have analyzed both the cases of a constant and non-constant step size in the procedure of integration. We show that working with a non-constant step, the well-known Chebyshev-Halley family of iterative methods for solving nonlinear scalar equations is obtained. Full article
(This article belongs to the Special Issue Numerical Algorithms for Solving Nonlinear Equations and Systems)
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315 KiB  
Article
On the Local Convergence of a Third Order Family of Iterative Processes
by M. A. Hernández-Verón and N. Romero
Algorithms 2015, 8(4), 1121-1128; https://doi.org/10.3390/a8041121 - 1 Dec 2015
Cited by 11 | Viewed by 4556
Abstract
Efficiency is generally the most important aspect to take into account when choosing an iterative method to approximate a solution of an equation, but is not the only aspect to consider in the iterative process. Another important aspect to consider is the accessibility [...] Read more.
Efficiency is generally the most important aspect to take into account when choosing an iterative method to approximate a solution of an equation, but is not the only aspect to consider in the iterative process. Another important aspect to consider is the accessibility of the iterative process, which shows the domain of starting points from which the iterative process converges to a solution of the equation. So, we consider a family of iterative processes with a higher efficiency index than Newton’s method. However, this family of proecsses presents problems of accessibility to the solution x * . From a local study of the convergence of this family, we perform an optimization study of the accessibility and obtain iterative processes with better accessibility than Newton’s method. Full article
(This article belongs to the Special Issue Numerical Algorithms for Solving Nonlinear Equations and Systems)
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210 KiB  
Article
An Optimal Biparametric Multipoint Family and Its Self-Acceleration with Memory for Solving Nonlinear Equations
by Quan Zheng, Xin Zhao and Yufeng Liu
Algorithms 2015, 8(4), 1111-1120; https://doi.org/10.3390/a8041111 - 1 Dec 2015
Cited by 5 | Viewed by 4290
Abstract
In this paper, a family of Steffensen-type methods of optimal order of convergence with two parameters is constructed by direct Newtonian interpolation. It satisfies the conjecture proposed by Kung and Traub (J. Assoc. Comput. Math. 1974, 21, 634–651) that an iterative method based [...] Read more.
In this paper, a family of Steffensen-type methods of optimal order of convergence with two parameters is constructed by direct Newtonian interpolation. It satisfies the conjecture proposed by Kung and Traub (J. Assoc. Comput. Math. 1974, 21, 634–651) that an iterative method based on m evaluations per iteration without memory would arrive at the optimal convergence of order 2m-1 . Furthermore, the family of Steffensen-type methods of super convergence is suggested by using arithmetic expressions for the parameters with memory but no additional new evaluation of the function. Their error equations, asymptotic convergence constants and convergence orders are obtained. Finally, they are compared with related root-finding methods in the numerical examples. Full article
(This article belongs to the Special Issue Numerical Algorithms for Solving Nonlinear Equations and Systems)
234 KiB  
Article
Local Convergence of an Efficient High Convergence Order Method Using Hypothesis Only on the First Derivative
by Ioannis K. Argyros, Ramandeep Behl and S.S. Motsa
Algorithms 2015, 8(4), 1076-1087; https://doi.org/10.3390/a8041076 - 20 Nov 2015
Cited by 2 | Viewed by 4222
Abstract
We present a local convergence analysis of an eighth order three step methodin order to approximate a locally unique solution of nonlinear equation in a Banach spacesetting. In an earlier study by Sharma and Arora (2015), the order of convergence wasshown using Taylor [...] Read more.
We present a local convergence analysis of an eighth order three step methodin order to approximate a locally unique solution of nonlinear equation in a Banach spacesetting. In an earlier study by Sharma and Arora (2015), the order of convergence wasshown using Taylor series expansions and hypotheses up to the fourth order derivative oreven higher of the function involved which restrict the applicability of the proposed scheme. However, only first order derivative appears in the proposed scheme. In order to overcomethis problem, we proposed the hypotheses up to only the first order derivative. In this way,we not only expand the applicability of the methods but also propose convergence domain. Finally, where earlier studies cannot be applied, a variety of concrete numerical examplesare proposed to obtain the solutions of nonlinear equations. Our study does not exhibit thistype of problem/restriction. Full article
(This article belongs to the Special Issue Numerical Algorithms for Solving Nonlinear Equations and Systems)
407 KiB  
Article
Some Matrix Iterations for Computing Generalized Inverses and Balancing Chemical Equations
by Farahnaz Soleimani, Predrag S. Stanimirovi´c and Fazlollah Soleymani
Algorithms 2015, 8(4), 982-998; https://doi.org/10.3390/a8040982 - 3 Nov 2015
Cited by 21 | Viewed by 6076
Abstract
An application of iterative methods for computing the Moore–Penrose inverse in balancing chemical equations is considered. With the aim to illustrate proposed algorithms, an improved high order hyper-power matrix iterative method for computing generalized inverses is introduced and applied. The improvements of the [...] Read more.
An application of iterative methods for computing the Moore–Penrose inverse in balancing chemical equations is considered. With the aim to illustrate proposed algorithms, an improved high order hyper-power matrix iterative method for computing generalized inverses is introduced and applied. The improvements of the hyper-power iterative scheme are based on its proper factorization, as well as on the possibility to accelerate the iterations in the initial phase of the convergence. Although the effectiveness of our approach is confirmed on the basis of the theoretical point of view, some numerical comparisons in balancing chemical equations, as well as on randomly-generated matrices are furnished. Full article
(This article belongs to the Special Issue Numerical Algorithms for Solving Nonlinear Equations and Systems)
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342 KiB  
Article
On Some Improved Harmonic Mean Newton-Like Methods for Solving Systems of Nonlinear Equations
by Diyashvir Kreetee Rajiv Babajee, Kalyanasundaram Madhu and Jayakumar Jayaraman
Algorithms 2015, 8(4), 895-909; https://doi.org/10.3390/a8040895 - 9 Oct 2015
Cited by 14 | Viewed by 5270
Abstract
In this work, we have developed a fourth order Newton-like method based on harmonic mean and its multi-step version for solving system of nonlinear equations. The new fourth order method requires evaluation of one function and two first order Fréchet derivatives for each [...] Read more.
In this work, we have developed a fourth order Newton-like method based on harmonic mean and its multi-step version for solving system of nonlinear equations. The new fourth order method requires evaluation of one function and two first order Fréchet derivatives for each iteration. The multi-step version requires one more function evaluation for each iteration. The proposed new scheme does not require the evaluation of second or higher order Fréchet derivatives and still reaches fourth order convergence. The multi-step version converges with order 2r+4, where r is a positive integer and r ≥ 1. We have proved that the root α is a point of attraction for a general iterative function, whereas the proposed new schemes also satisfy this result. Numerical experiments including an application to 1-D Bratu problem are given to illustrate the efficiency of the new methods. Also, the new methods are compared with some existing methods. Full article
(This article belongs to the Special Issue Numerical Algorithms for Solving Nonlinear Equations and Systems)
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247 KiB  
Article
Newton-Type Methods on Generalized Banach Spaces and Applications in Fractional Calculus
by George A. Anastassiou and Ioannis K. Argyros
Algorithms 2015, 8(4), 832-849; https://doi.org/10.3390/a8040832 - 9 Oct 2015
Viewed by 4282
Abstract
We present a semilocal convergence study of Newton-type methods on a generalized Banach space setting to approximate a locally unique zero of an operator. Earlier studies require that the operator involved is Fréchet differentiable. In the present study we assume that the operator [...] Read more.
We present a semilocal convergence study of Newton-type methods on a generalized Banach space setting to approximate a locally unique zero of an operator. Earlier studies require that the operator involved is Fréchet differentiable. In the present study we assume that the operator is only continuous. This way we extend the applicability of Newton-type methods to include fractional calculus and problems from other areas. Moreover, under the same or weaker conditions, we obtain weaker sufficient convergence criteria, tighter error bounds on the distances involved and an at least as precise information on the location of the solution. Special cases are provided where the old convergence criteria cannot apply but the new criteria can apply to locate zeros of operators. Some applications include fractional calculus involving the Riemann-Liouville fractional integral and the Caputo fractional derivative. Fractional calculus is very important for its applications in many applied sciences. Full article
(This article belongs to the Special Issue Numerical Algorithms for Solving Nonlinear Equations and Systems)
249 KiB  
Article
A Family of Newton Type Iterative Methods for Solving Nonlinear Equations
by Xiaofeng Wang, Yuping Qin, Weiyi Qian, Sheng Zhang and Xiaodong Fan
Algorithms 2015, 8(3), 786-798; https://doi.org/10.3390/a8030786 - 22 Sep 2015
Cited by 13 | Viewed by 5816
Abstract
In this paper, a general family of n-point Newton type iterative methods for solving nonlinear equations is constructed by using direct Hermite interpolation. The order of convergence of the new n-point iterative methods without memory is 2n requiring the evaluations of n functions [...] Read more.
In this paper, a general family of n-point Newton type iterative methods for solving nonlinear equations is constructed by using direct Hermite interpolation. The order of convergence of the new n-point iterative methods without memory is 2n requiring the evaluations of n functions and one first-order derivative in per full iteration, which implies that this family is optimal according to Kung and Traub’s conjecture (1974). Its error equations and asymptotic convergence constants are obtained. The n-point iterative methods with memory are obtained by using a self-accelerating parameter, which achieve much faster convergence than the corresponding n-point methods without memory. The increase of convergence order is attained without any additional calculations so that the n-point Newton type iterative methods with memory possess a very high computational efficiency. Numerical examples are demonstrated to confirm theoretical results. Full article
(This article belongs to the Special Issue Numerical Algorithms for Solving Nonlinear Equations and Systems)
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251 KiB  
Article
Parallel Variants of Broyden’s Method
by Ioan Bistran, Stefan Maruster and Liviu Octavian Mafteiu-Scai
Algorithms 2015, 8(3), 774-785; https://doi.org/10.3390/a8030774 - 15 Sep 2015
Cited by 1 | Viewed by 5663
Abstract
In this paper we investigate some parallel variants of Broyden’s method and, for the basic variant, we present its convergence properties. The main result is that the behavior of the considered parallel Broyden’s variants is comparable with the classical parallel Newton method, and [...] Read more.
In this paper we investigate some parallel variants of Broyden’s method and, for the basic variant, we present its convergence properties. The main result is that the behavior of the considered parallel Broyden’s variants is comparable with the classical parallel Newton method, and significantly better than the parallel Cimmino method, both for linear and nonlinear cases. The considered variants are also compared with two more recently proposed parallel Broyden’s method. Some numerical experiments are presented to illustrate the advantages and limits of the proposed algorithms. Full article
(This article belongs to the Special Issue Numerical Algorithms for Solving Nonlinear Equations and Systems)
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242 KiB  
Article
Gradient-Based Iterative Identification for Wiener Nonlinear Dynamic Systems with Moving Average Noises
by Lincheng Zhou, Xiangli Li, Huigang Xu and Peiyi Zhu
Algorithms 2015, 8(3), 712-722; https://doi.org/10.3390/a8030712 - 26 Aug 2015
Cited by 5 | Viewed by 4935
Abstract
This paper focuses on the parameter identification problem for Wiener nonlinear dynamic systems with moving average noises. In order to improve the convergence rate, the gradient-based iterative algorithm is presented by replacing the unmeasurable variables with their corresponding iterative estimates, and to compute [...] Read more.
This paper focuses on the parameter identification problem for Wiener nonlinear dynamic systems with moving average noises. In order to improve the convergence rate, the gradient-based iterative algorithm is presented by replacing the unmeasurable variables with their corresponding iterative estimates, and to compute iteratively the noise estimates based on the obtained parameter estimates. The simulation results show that the proposed algorithm can effectively estimate the parameters of Wiener systems with moving average noises. Full article
(This article belongs to the Special Issue Numerical Algorithms for Solving Nonlinear Equations and Systems)
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212 KiB  
Article
Expanding the Applicability of a Third Order Newton-Type Method Free of Bilinear Operators
by Sergio Amat, Sonia Busquier, Concepción Bermúdez and Ángel Alberto Magreñán
Algorithms 2015, 8(3), 669-679; https://doi.org/10.3390/a8030669 - 21 Aug 2015
Cited by 5 | Viewed by 4942
Abstract
This paper is devoted to the semilocal convergence, using centered hypotheses, of a third order Newton-type method in a Banach space setting. The method is free of bilinear operators and then interesting for the solution of systems of equations. Without imposing any type [...] Read more.
This paper is devoted to the semilocal convergence, using centered hypotheses, of a third order Newton-type method in a Banach space setting. The method is free of bilinear operators and then interesting for the solution of systems of equations. Without imposing any type of Fréchet differentiability on the operator, a variant using divided differences is also analyzed. A variant of the method using only divided differences is also presented. Full article
(This article belongs to the Special Issue Numerical Algorithms for Solving Nonlinear Equations and Systems)
306 KiB  
Article
Fifth-Order Iterative Method for Solving Multiple Roots of the Highest Multiplicity of Nonlinear Equation
by Juan Liang, Xiaowu Li, Zhinan Wu, Mingsheng Zhang, Lin Wang and Feng Pan
Algorithms 2015, 8(3), 656-668; https://doi.org/10.3390/a8030656 - 20 Aug 2015
Cited by 2 | Viewed by 5483
Abstract
A three-step iterative method with fifth-order convergence as a new modification of Newton’s method was presented. This method is for finding multiple roots of nonlinear equation with unknown multiplicity m whose multiplicity m is the highest multiplicity. Its order of convergence is analyzed [...] Read more.
A three-step iterative method with fifth-order convergence as a new modification of Newton’s method was presented. This method is for finding multiple roots of nonlinear equation with unknown multiplicity m whose multiplicity m is the highest multiplicity. Its order of convergence is analyzed and proved. Results for some numerical examples show the efficiency of the new method. Full article
(This article belongs to the Special Issue Numerical Algorithms for Solving Nonlinear Equations and Systems)
236 KiB  
Article
Local Convergence of an Optimal Eighth Order Method under Weak Conditions
by Ioannis K. Argyros, Ramandeep Behl and S.S. Motsa
Algorithms 2015, 8(3), 645-655; https://doi.org/10.3390/a8030645 - 19 Aug 2015
Cited by 5 | Viewed by 4813
Abstract
We study the local convergence of an eighth order Newton-like method to approximate a locally-unique solution of a nonlinear equation. Earlier studies, such as Chen et al. (2015) show convergence under hypotheses on the seventh derivative or even higher, although only the first [...] Read more.
We study the local convergence of an eighth order Newton-like method to approximate a locally-unique solution of a nonlinear equation. Earlier studies, such as Chen et al. (2015) show convergence under hypotheses on the seventh derivative or even higher, although only the first derivative and the divided difference appear in these methods. The convergence in this study is shown under hypotheses only on the first derivative. Hence, the applicability of the method is expanded. Finally, numerical examples are also provided to show that our results apply to solve equations in cases where earlier studies cannot apply. Full article
(This article belongs to the Special Issue Numerical Algorithms for Solving Nonlinear Equations and Systems)
203 KiB  
Article
Some Improvements to a Third Order Variant of Newton’s Method from Simpson’s Rule
by Diyashvir Kreetee Rajiv Babajee
Algorithms 2015, 8(3), 552-561; https://doi.org/10.3390/a8030552 - 29 Jul 2015
Cited by 2 | Viewed by 5133
Abstract
In this paper, we present three improvements to a three-point third order variant of Newton’s method derived from the Simpson rule. The first one is a fifth order method using the same number of functional evaluations as the third order method, the second [...] Read more.
In this paper, we present three improvements to a three-point third order variant of Newton’s method derived from the Simpson rule. The first one is a fifth order method using the same number of functional evaluations as the third order method, the second one is a four-point 10th order method and the last one is a five-point 20th order method. In terms of computational point of view, our methods require four evaluations (one function and three first derivatives) to get fifth order, five evaluations (two functions and three derivatives) to get 10th order and six evaluations (three functions and three derivatives) to get 20th order. Hence, these methods have efficiency indexes of 1.495, 1.585 and 1.648, respectively which are better than the efficiency index of 1.316 of the third order method. We test the methods through some numerical experiments which show that the 20th order method is very efficient. Full article
(This article belongs to the Special Issue Numerical Algorithms for Solving Nonlinear Equations and Systems)
579 KiB  
Article
On the Accessibility of Newton’s Method under a Hölder Condition on the First Derivative
by José Antonio Ezquerro and Miguel Ángel Hernández-Verón
Algorithms 2015, 8(3), 514-528; https://doi.org/10.3390/a8030514 - 23 Jul 2015
Cited by 2 | Viewed by 5156
Abstract
We see how we can improve the accessibility of Newton’s method for approximating a solution of a nonlinear equation in Banach spaces when a center Hölder condition on the first derivative is used to prove its semi-local convergence. Full article
(This article belongs to the Special Issue Numerical Algorithms for Solving Nonlinear Equations and Systems)
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248 KiB  
Article
A Quartically Convergent Jarratt-Type Method for Nonlinear System of Equations
by Mohammad Ghorbanzadeh and Fazlollah Soleymani
Algorithms 2015, 8(3), 415-423; https://doi.org/10.3390/a8030415 - 6 Jul 2015
Cited by 5 | Viewed by 5324
Abstract
In this work, we propose a new fourth-order Jarratt-type method for solving systems of nonlinear equations. The local convergence order of the method is proven analytically. Finally, we validate our results via some numerical experiments including an application to the Chandrashekar integral equations. [...] Read more.
In this work, we propose a new fourth-order Jarratt-type method for solving systems of nonlinear equations. The local convergence order of the method is proven analytically. Finally, we validate our results via some numerical experiments including an application to the Chandrashekar integral equations. Full article
(This article belongs to the Special Issue Numerical Algorithms for Solving Nonlinear Equations and Systems)
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Article
An Optimal Eighth-Order Derivative-Free Family of Potra-Pták’s Method
by Munish Kansal, Vinay Kanwar and Saurabh Bhatia
Algorithms 2015, 8(2), 309-320; https://doi.org/10.3390/a8020309 - 15 Jun 2015
Cited by 8 | Viewed by 5126
Abstract
In this paper, we present a new three-step derivative-free family based on Potra-Pták’s method for solving nonlinear equations numerically. In terms of computational cost, each member of the proposed family requires only four functional evaluations per full iteration to achieve optimal eighth-order convergence. [...] Read more.
In this paper, we present a new three-step derivative-free family based on Potra-Pták’s method for solving nonlinear equations numerically. In terms of computational cost, each member of the proposed family requires only four functional evaluations per full iteration to achieve optimal eighth-order convergence. Further, computational results demonstrate that the proposed methods are highly efficient as compared with many well-known methods. Full article
(This article belongs to the Special Issue Numerical Algorithms for Solving Nonlinear Equations and Systems)
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Article
Numerical Solution of Turbulence Problems by Solving Burgers’ Equation
by Alicia Cordero, Antonio Franques and Juan R. Torregrosa
Algorithms 2015, 8(2), 224-233; https://doi.org/10.3390/a8020224 - 8 May 2015
Cited by 11 | Viewed by 6388
Abstract
In this work we generate the numerical solutions of Burgers’ equation by applying the Crank-Nicholson method and different schemes for solving nonlinear systems, instead of using Hopf-Cole transformation to reduce Burgers’ equation into the linear heat equation. The method is analyzed on two [...] Read more.
In this work we generate the numerical solutions of Burgers’ equation by applying the Crank-Nicholson method and different schemes for solving nonlinear systems, instead of using Hopf-Cole transformation to reduce Burgers’ equation into the linear heat equation. The method is analyzed on two test problems in order to check its efficiency on different kinds of initial conditions. Numerical solutions as well as exact solutions for different values of viscosity are calculated, concluding that the numerical results are very close to the exact solution. Full article
(This article belongs to the Special Issue Numerical Algorithms for Solving Nonlinear Equations and Systems)
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