Special Issue "Numerical Algorithms for Solving Nonlinear Equations and Systems"

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

Deadline for manuscript submissions: closed (31 October 2015)

Special Issue Editors

Guest Editor
Prof. Dr. Alicia Cordero

Inst. Matemática Multidisciplinar, Universitat Politècnica de València, Camino de Vera, s/n, 46022-Valencia, Spain
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Interests: numerical analysis
Guest Editor
Prof. Dr. Juan R. Torregrosa

Inst. Matemática Multidisciplinar, Universitat Politècnica de València, Camino de Vera, s/n, 46022-Valencia, Spain
Website | E-Mail
Interests: numerical analysis; solution of nonlinear equations and systems; matrix equations and dynamical analysis of rational functions
Guest Editor
Dr. Francisco I. Chicharro

Inst. Matemática Multidisciplinar, Universitat Politècnica de València, Camino de Vera, s/n, 46022-Valencia, Spain
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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

Published Papers (26 papers)

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Research

Open AccessArticle The Iterative Solution to Discrete-Time H Control Problems for Periodic Systems
Algorithms 2016, 9(1), 20; https://doi.org/10.3390/a9010020
Received: 22 October 2015 / Revised: 25 February 2016 / Accepted: 9 March 2016 / Published: 14 March 2016
Cited by 1 | PDF Full-text (235 KB) | HTML Full-text | XML Full-text
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
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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)
Open AccessArticle Constructing Frozen Jacobian Iterative Methods for Solving Systems of Nonlinear Equations, Associated with ODEs and PDEs Using the Homotopy Method
Algorithms 2016, 9(1), 18; https://doi.org/10.3390/a9010018
Received: 25 December 2015 / Revised: 3 March 2016 / Accepted: 4 March 2016 / Published: 11 March 2016
Cited by 2 | PDF Full-text (263 KB) | HTML Full-text | XML Full-text
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)
Open AccessArticle A Geometric Orthogonal Projection Strategy for Computing the Minimum Distance Between a Point and a Spatial Parametric Curve
Algorithms 2016, 9(1), 15; https://doi.org/10.3390/a9010015
Received: 25 November 2015 / Accepted: 2 February 2016 / Published: 6 February 2016
Cited by 1 | PDF Full-text (252 KB) | HTML Full-text | XML Full-text
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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Open AccessArticle Two Efficient Derivative-Free Iterative Methods for Solving Nonlinear Systems
Algorithms 2016, 9(1), 14; https://doi.org/10.3390/a9010014
Received: 16 October 2015 / Revised: 26 January 2016 / Accepted: 27 January 2016 / Published: 1 February 2016
Cited by 2 | PDF Full-text (277 KB) | HTML Full-text | XML Full-text
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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Open AccessArticle An Optimal Order Method for Multiple Roots in Case of Unknown Multiplicity
Algorithms 2016, 9(1), 10; https://doi.org/10.3390/a9010010
Received: 20 October 2015 / Revised: 7 January 2016 / Accepted: 18 January 2016 / Published: 22 January 2016
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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)
Open AccessArticle A Family of Iterative Methods for Solving Systems of Nonlinear Equations Having Unknown Multiplicity
Algorithms 2016, 9(1), 5; https://doi.org/10.3390/a9010005
Received: 8 December 2015 / Revised: 19 December 2015 / Accepted: 22 December 2015 / Published: 31 December 2015
Cited by 1 | PDF Full-text (230 KB) | HTML Full-text | XML Full-text
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)
Open AccessArticle On the Kung-Traub Conjecture for Iterative Methods for Solving Quadratic Equations
Algorithms 2016, 9(1), 1; https://doi.org/10.3390/a9010001
Received: 26 October 2015 / Revised: 13 December 2015 / Accepted: 16 December 2015 / Published: 24 December 2015
Cited by 2 | PDF Full-text (295 KB) | HTML Full-text | XML Full-text
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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Open AccessArticle Offset-Assisted Factored Solution of Nonlinear Systems
Algorithms 2016, 9(1), 2; https://doi.org/10.3390/a9010002
Received: 31 October 2015 / Revised: 11 December 2015 / Accepted: 14 December 2015 / Published: 23 December 2015
PDF Full-text (1775 KB) | HTML Full-text | XML Full-text
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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Open AccessArticle Numerical Properties of Different Root-Finding Algorithms Obtained for Approximating Continuous Newton’s Method
Algorithms 2015, 8(4), 1210-1218; https://doi.org/10.3390/a8041210
Received: 28 October 2015 / Revised: 10 December 2015 / Accepted: 14 December 2015 / Published: 17 December 2015
Cited by 3 | PDF Full-text (374 KB) | HTML Full-text | XML Full-text
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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Open AccessArticle On the Local Convergence of a Third Order Family of Iterative Processes
Algorithms 2015, 8(4), 1121-1128; https://doi.org/10.3390/a8041121
Received: 7 September 2015 / Revised: 24 November 2015 / Accepted: 26 November 2015 / Published: 1 December 2015
Cited by 7 | PDF Full-text (315 KB) | HTML Full-text | XML Full-text
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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Open AccessArticle An Optimal Biparametric Multipoint Family and Its Self-Acceleration with Memory for Solving Nonlinear Equations
Algorithms 2015, 8(4), 1111-1120; https://doi.org/10.3390/a8041111
Received: 8 October 2015 / Revised: 22 November 2015 / Accepted: 24 November 2015 / Published: 1 December 2015
Cited by 1 | PDF Full-text (210 KB) | HTML Full-text | XML Full-text
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)
Open AccessArticle Local Convergence of an Efficient High Convergence Order Method Using Hypothesis Only on the First Derivative
Algorithms 2015, 8(4), 1076-1087; https://doi.org/10.3390/a8041076
Received: 25 September 2015 / Accepted: 11 November 2015 / Published: 20 November 2015
Cited by 1 | PDF Full-text (234 KB) | HTML Full-text | XML Full-text
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)
Open AccessArticle Some Matrix Iterations for Computing Generalized Inverses and Balancing Chemical Equations
Algorithms 2015, 8(4), 982-998; https://doi.org/10.3390/a8040982
Received: 25 June 2015 / Revised: 24 October 2015 / Accepted: 26 October 2015 / Published: 3 November 2015
Cited by 1 | PDF Full-text (407 KB) | HTML Full-text | XML Full-text
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
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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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Open AccessArticle On Some Improved Harmonic Mean Newton-Like Methods for Solving Systems of Nonlinear Equations
Algorithms 2015, 8(4), 895-909; https://doi.org/10.3390/a8040895
Received: 5 September 2015 / Revised: 23 September 2015 / Accepted: 24 September 2015 / Published: 9 October 2015
Cited by 2 | PDF Full-text (342 KB) | HTML Full-text | XML Full-text
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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Open AccessArticle Newton-Type Methods on Generalized Banach Spaces and Applications in Fractional Calculus
Algorithms 2015, 8(4), 832-849; https://doi.org/10.3390/a8040832
Received: 23 June 2015 / Revised: 13 September 2015 / Accepted: 29 September 2015 / Published: 9 October 2015
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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)
Open AccessArticle A Family of Newton Type Iterative Methods for Solving Nonlinear Equations
Algorithms 2015, 8(3), 786-798; https://doi.org/10.3390/a8030786
Received: 9 July 2015 / Revised: 14 September 2015 / Accepted: 15 September 2015 / Published: 22 September 2015
Cited by 6 | PDF Full-text (249 KB) | HTML Full-text | XML Full-text
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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Open AccessArticle Parallel Variants of Broyden’s Method
Algorithms 2015, 8(3), 774-785; https://doi.org/10.3390/a8030774
Received: 23 June 2015 / Revised: 27 August 2015 / Accepted: 1 September 2015 / Published: 15 September 2015
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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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Open AccessArticle Gradient-Based Iterative Identification for Wiener Nonlinear Dynamic Systems with Moving Average Noises
Algorithms 2015, 8(3), 712-722; https://doi.org/10.3390/a8030712
Received: 30 June 2015 / Revised: 20 July 2015 / Accepted: 20 August 2015 / Published: 26 August 2015
Cited by 2 | PDF Full-text (242 KB) | HTML Full-text | XML Full-text
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
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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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Open AccessArticle Expanding the Applicability of a Third Order Newton-Type Method Free of Bilinear Operators
Algorithms 2015, 8(3), 669-679; https://doi.org/10.3390/a8030669
Received: 26 May 2015 / Revised: 9 August 2015 / Accepted: 14 August 2015 / Published: 21 August 2015
Cited by 1 | PDF Full-text (212 KB) | HTML Full-text | XML Full-text
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)
Open AccessArticle Fifth-Order Iterative Method for Solving Multiple Roots of the Highest Multiplicity of Nonlinear Equation
Algorithms 2015, 8(3), 656-668; https://doi.org/10.3390/a8030656
Received: 8 June 2015 / Revised: 27 July 2015 / Accepted: 14 August 2015 / Published: 20 August 2015
Cited by 1 | PDF Full-text (306 KB) | HTML Full-text | XML Full-text
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)
Open AccessArticle Local Convergence of an Optimal Eighth Order Method under Weak Conditions
Algorithms 2015, 8(3), 645-655; https://doi.org/10.3390/a8030645
Received: 9 June 2015 / Revised: 31 July 2015 / Accepted: 5 August 2015 / Published: 19 August 2015
Cited by 1 | PDF Full-text (236 KB) | HTML Full-text | XML Full-text
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)
Open AccessArticle Some Improvements to a Third Order Variant of Newton’s Method from Simpson’s Rule
Algorithms 2015, 8(3), 552-561; https://doi.org/10.3390/a8030552
Received: 11 May 2015 / Revised: 20 July 2015 / Accepted: 21 July 2015 / Published: 29 July 2015
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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
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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)
Open AccessArticle On the Accessibility of Newton’s Method under a Hölder Condition on the First Derivative
Algorithms 2015, 8(3), 514-528; https://doi.org/10.3390/a8030514
Received: 22 June 2015 / Revised: 15 July 2015 / Accepted: 17 July 2015 / Published: 23 July 2015
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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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Open AccessArticle A Quartically Convergent Jarratt-Type Method for Nonlinear System of Equations
Algorithms 2015, 8(3), 415-423; https://doi.org/10.3390/a8030415
Received: 28 May 2015 / Revised: 1 July 2015 / Accepted: 2 July 2015 / Published: 6 July 2015
Cited by 3 | PDF Full-text (248 KB) | HTML Full-text | XML Full-text
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.
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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)
Open AccessArticle An Optimal Eighth-Order Derivative-Free Family of Potra-Pták’s Method
Algorithms 2015, 8(2), 309-320; https://doi.org/10.3390/a8020309
Received: 25 April 2015 / Accepted: 8 June 2015 / Published: 15 June 2015
Cited by 3 | PDF Full-text (295 KB) | HTML Full-text | XML Full-text
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.
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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)
Open AccessArticle Numerical Solution of Turbulence Problems by Solving Burgers’ Equation
Algorithms 2015, 8(2), 224-233; https://doi.org/10.3390/a8020224
Received: 4 April 2015 / Revised: 27 April 2015 / Accepted: 30 April 2015 / Published: 8 May 2015
Cited by 3 | PDF Full-text (431 KB) | HTML Full-text | XML Full-text
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
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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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