José Manuel Gutiérrez

Universidad de La Rioja (Spain) | UNIRIOJA · Mathematics and Computation

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 o...

The aim of this paper is to study, from a topological and geometrical point of view, the iteration map obtained by the application of iterative methods (Newton or relaxed Newton’s method) to a polynomial equation. In fact, we present a collection of algorithms that avoid the problem of overflows caused by denominators close to zero and the problem...

The purpose of this work is to give a first approach to the dynamical behavior of Schröder’s method, a well-known iterative process for solving nonlinear equations. In this context, we consider equations defined in the complex plane. By using topological conjugations, we characterize the basins of attraction of Schröder’s method applied to polynomi...

In this work, we present an application of Newton’s method for solving nonlinear equations in Banach spaces to a particular problem: the approximation of the inverse operators that appear in the solution of Fredholm integral equations. Therefore, we construct an iterative method with quadratic convergence that does not use either derivatives or inv...

The purpose of this work is to give a first approach to the dynamical behavior of Schr\"oder's method, a well known iterative process for solving nonlinear equations. In this context we consider equations defined in the complex plane. By using topological conjugations, we characterize the basins of attraction of Schr\"oder's method applied to polyn...

We use Newton’s method to approximate locally unique solutions for a class of boundary value problems by applying the shooting method. The utilized operator is Fréchet-differentiable between Banach spaces. These conditions are more general than those that appear in previous works. In particular, we show that the old semilocal and local convergence...

This work is devoted to Fredholm integral equations of second kind with non-separable kernels. Our strategy is to approximate the non-separable kernel by using an adequate Taylor’s development. Then, we adapt an already known technique used for separable kernels to our case. First, we study the local convergence of the proposed iterative scheme, so...

In this work we provide analytic and graphic arguments to explain the behaviour of Chebyshev’s method applied to cubic polynomials in the complex plane. In particular, we study the parameter plane related to this method and we compare it with other previously known, such as Newton’s or Halley’s methods. Our specific interest is to characterize “bad...

The main aim of this manuscript is to propose two new schemes having three and four substeps of order eight and sixteen, respectively. Both families are optimal in the sense to Kung-Traub conjecture. The derivation of them are based on the weight function approach. In addition, theoretical and computational properties are fully investigated along w...

The goal of this paper is to present a new project "A Julia Implementation for the Iteration of Univariate Rational Functions" which allows us to iterate a rational function on the Riemann sphere by using its geometry and complex structure. We have developed and implemented in Julia language a collection of algorithms for the iteration of a rationa...

In this work, we study some numerical properties of the continuous Newton's method, the continuous version of the classical Newton's method for solving nonlinear equations p(z)=0. In fact, continuous Newton's method is an initial value problem whose solutions flow to a root of the equation. We show the influence of the multiplicity of the roots of...

This paper deals with the enlargement of the region of convergence of Newton’s method for solving nonlinear equations defined in Banach spaces. We have used an homotopy method to obtain approximate zeros of the considered function. The novelty in our approach is the establishment of new convergence results based on a Lipschitz condition with a L-av...

A one-parametric family of fourth-order iterative methods for solving nonlinear systems is presented, proving the fourth-order of convergence of all members in this family, except one of them whose order is five. The methods in our family are numerically compared with other known methods in terms of the classical efficiency index (order of converge...

The relaxed Newton's method modifies the classical Newton's method with a parameter h in such a way that when it is applied to a polynomial with multiple roots and we take as parameter one of these multiplicities, it is increased the order of convergence to the related multiple root. For polynomials of degree three or higher, the relaxed Newton's m...

In this paper we extend to the multidimensional case some iterative methods that are known in their scalar version. All the schemes considered here are two-step methods with fourth-order local convergence, where the first step is Newton’s method. We analyze the efficiency of these new four algorithms and compare them in terms of the elapsed time ne...

In this paper we explore some properties of the well known root-finding Chebyshev’s method applied to polynomials defined on the real field. In particular we are interested in showing the existence of extraneous fixed points, that is fixed points of the iteration map that are not root of the considered polynomial. The existence of such extraneous f...

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 relat...

In this communication we propose a new and effecfive strategy to apply Newton's method to the problem of finding the intersections of two real algebraic curves, that is, the roots of a pair of real bivariate polynomials. The use of adequate homogeneous coordinates and the extension of the domain where the iteration function is defined allow us to a...

In this paper we propose a new and effective strategy to apply Newton's method to the problem of finding the intersections of two real algebraic curves, that is, the roots of a pair of real bivariate polynomials. The use of adequate homogeneous coordinates and the extension of the domain where the iteration function is defined allow us to avoid som...

The aim of this paper is to investigate the iterative root-finding Chebyshev’s method from a dynamical perspective. We analyze the behavior of the method applied to low degree polynomials. In this work we focus on the complex case. Actually, we show the existence of extraneous fixed points for Chebyshev’s, that is fixed points of the iterative meth...

The real dynamics of a family of fourth-order iterative methods is studied when it is applied on quadratic polynomials. A Scaling Theorem is obtained and the conjugacy classes are analyzed. The convergence plane is used to obtain the same kind of information as from the parameter space, and even more, in complex dynamics.

The aim of this paper is to show the existence of extraneous fixed points for Cheby-shev's method applied to complex polynomials. This fact is a distinguishing feature in the dynamical study of Chebyshev's method compared with other known iterative methods as Newton's or Halley's methods. In addition, in this work we consider other dynamical aspect...

In this work, we develop and implement two algorithms for plotting and computing the measure of the basins of attraction of rational maps defined on the Riemann sphere. These algorithms are based on the subdivisions of a cubical decomposition of a sphere and they have been made by using different computational environments. As an application, we st...

The main goal of this paper is to study the order of convergence and the efficiency of four families of iterative methods using frozen divided differences. The first two families correspond to a generalization of the secant method and the implementation made by Schmidt and Schwetlick. The other two frozen schemes consist of a generalization of Kurc...

In this work we study the local and semilocal convergence of the relaxed Newton's method, that is Newton's method with a relaxation parameter 0 < < 2. We give a Kantorovich-like theorem that can be applied for operators defined between two Banach spaces. In fact, we obtain the recurrent sequence that majorizes the one given by the method and we cha...

In this work we study Newton's method for solving nonlinear equations with operators defined between two Banach spaces. Together with the classical Kantorovich theory, we consider a center-Lipschitz condition for the Frechet derivative of the involved operator. This fact allow us to obtain a majorizing sequence for the sequence defined in Banach sp...

In this article we characterize the existence of attractive cycles for the Newton method applied to Kepler's equation. In fact, we take the conditions imposed by Charles and Tatum, ([2]), who have shown that E0 = M is not a good choice as a starting point for Newton's method applied to Kepler's equation and that there are pairs (e, M) which fails t...

In this paper we develop a Kantorovich-like theory for Chebyshev’s method, a well-known iterative method for solving nonlinear equations in Banach spaces. We improve the results obtained previously by considering Chebyshev’s method as an element of a family of iterative processes.

In this paper we give a semilocal convergence theorem for a family of iterative methods for solving nonlinear equations defined between two Banach spaces. This family is obtained as a combination of the well known Secant method and Chebyshev method. We give a very general convergence result that allow the application of these methods to non-differe...

The dynamical behavior of two iterative derivative-free schemes, Steffensen and M4 methods, is studied in case of quadratic and cubic polynomials. The parameter plane is analyzed for both procedures on quadratic polynomials. Different dynamical planes are showed when the mentioned methods are applied on particular cubic polynomials with real or com...

In this paper we study the real dynamics of the damped Newton’s methods applied to cubic polynomials, but instead of taking a value of the damping factor λ∈(0,1), we consider all values of λ∈ℝ. The method for unusual values of λ presents different behaviors such as convergence to n-cycles or even chaos.

In this paper, we study the dynamical behaviour of a two-point iterative method with order of convergence five to solve nonlinear equations in the complex plane. In fact, we complement the dynamical study started in previous works with a more systematic analysis for polynomials with at most two different roots and different multiplicities. In addit...

In this work we study the dynamics of damped Newton's method defined by Nλ,p(z) = z - λ p(z)/p'(z) , λ ∈ C applied to different polynomial equations. For polynomials with two different roots we analyze the influence of the damping factor λ in the character of the fixed points as well as in the behavior of the critical points. It is shown that, for...

In this paper we introduce a process we have called “Gauss-Seidelization” for solving nonlinear equations. We have used this name because the process is inspired by the well-known Gauss–Seidel method to numerically solve a system of linear equations. Together with some convergence results, we present several numerical experiments in order to emphas...

The concept of universal Julia set introduced in [5] allows us to conclude that the dynamics of a root‐finding algorithm applied to any quadratic polynomial can be understood through the analysis of a particular rational map. In this study we go a step beyond in this direction. In particular, we can define the universal fractal dimension of the afo...

We introduce a three-step Chebyshev–Secant-type method (CSTM) with high efficiency index for solving nonlinear equations in a Banach space setting. We provide a semilocal convergence analysis for (CSTM) using recurrence relations. Numerical examples validating our theoretical results are also provided in this study.

The geometrical interpretation of a family of higher order iterative methods for solving nonlinear scalar equations was presented in [S. Amat, S. Busquier, J.M. Gutiérrez, Geometric constructions of iterative functions to solve nonlinear equations. J. Comput. Appl. Math. 157(1) (2003) 197–205]. This family includes, as particular cases, some of the...

In this chapter we consider different semilocal convergence results applied to Newton's method to numerically solve nonlinear equations. Let {xn}¥ n=0 be the Newton sequence to approximate the root x of a nonlinear equation f (x) = 0. A semilocal convergence result imposes conditions on the starting point x0 in order to guarantee the convergence of...

In this chapter we consider different semilocal convergence results applied to Newton’s method to numerically solve nonlinear equations. Let {xn}1 n=0 be the Newton sequence to approximate the root x� of a nonlinear equation f(x) = 0. A semilocal convergence result imposes conditions on the starting point x0 in order to guarantee the convergence of...

Two variants of the Computational Order of Convergence (COC) of an iterative method for solving nonlinear equations are presented. Furthermore, the way to approximate the COC and the new variants to the local order of convergence is analyzed. The new definitions given here does not involve the unknown root. Numerical experiments using adaptive arit...

In this work we show the presence of the well-known Catalan numbers in the study of the convergence and the dynamical behavior of a family of iterative methods for solving nonlinear equations. In fact, we introduce a family of methods, depending on a parameter m∈N∪{0}. These methods reach the order of convergence m+2 when they are applied to quadra...

In this paper we present some techniques for constructing high-order iterative methods in order to approximate the zeros of a non-linear equation f(x)=0, starting from a well-known family of cubic iterative processes. The first technique is based on an additional functional evaluation that allows us to increase the order of convergence from three t...

This work has two targets. First we study the damped Newton's method: x n+1 = x n - λF′(x n) -1F(x n) 0 < λ≤ 1 , n ≥ 0 where F is an operator defined between two Banach spaces X and Y . We study the semilocal convergence of the method under the Kantorovich-like conditions: 1. x 0 ∈ X is a point where the operator Γ 0 = F′(x 0) -1 is defined. 2. ∥Γ...

A comparison of different third order methods is performed when these schemes are used to solve geometric problems. Sufficient hypotheses that ensure their convergence to the solution of the problem are derived. Explicit, implicit and parametric representations are considered.

In this paper two families of zero-finding iterative methods for solving nonlinear equations f(x)=0 are presented. The key idea to derive them is to solve an initial value problem applying Obreshkov-like techniques. More explicitly, Obreshkov’s methods have been used to numerically solve an initial value problem that involves the inverse of the fun...

In [A. Melman, Geometry and convergence of Euler's and Halley's methods, SIAM Rev. 39(4) (1997) 728–735] the geometry and global convergence of Euler's and Halley's methods was studied. Now we complete Melman's paper by considering other classical third-order method: Chebyshev's method. By using the geometric interpretation of this method a global...

In this paper we prove new identities in the Catalan triangle whose (n, p) entry is defined by B(n,p) := p/n (2n n - p), n,p is an element of N, p <= n. In fact, we show some new identities involving the well-known Catalan numbers, and specially the identity Sigma(i)(p=1) B(n,p)B(n,n+p-i) (n + 2p - i) = (n + 1)C(n) (2(n - 1) i - 1), i <= n, that ap...

We use a recurrence technique to obtain semilocal convergence results for Ulm's iterative method to approximate a solution of a nonlinear equation F(x)=0This method does not contain inverse operators in its expression and we prove it converges with the Newton rate. We also use this method to approximate a solution of integral equations of Fredholm-...

This paper presents a numerical method to solve the forward position problem in spatial mechanisms. The method may be incorporated in a software for the kinematic analysis of mechanisms, where the procedure is systematic and can be easily implemented, achieving a high degree of automation in simulation. The procedure presents high computational eff...

In this paper, we present a technique to construct iterative methods to approximate the zeros of a nonlinear equation F(x)=0, where F is a function of several variables. This technique is based on the approximation of the inverse function of F and on the use of a fixed point iteration. Depending on the number of steps considered in the fixed point...

A fourth-order iterative method for quadratic equations is presented. A semilocal convergence theorem is performed. A multiresolution transform corresponding to interpolatory technique is used for fast application of the method. In designing this algorithm we apply data compression to the linear and the bilinear forms that appear on the method. Fin...

Local convergence theorems are provided for Newton's method in a Banach space setting. Our conditions are very general and in special cases provide larger convergence radius than before, which in turn allows a wider choice of initial guesses for Newton's method. This observations is important in numerical analysis and finds applications in step len...

In this article, we carry out a local convergence study for Secant-type methods. Our goal is to enlarge the radius of convergence, without increasing the necessary hypothesis. Finally, some numerical tests and comparisons with early results are analyzed.E-mail: sonia.busquier@upct.es E-mail: sergio.amat@upct.es

In this paper we present the geometrical interpretation of several iterative methods to solve a nonlinear scalar equation. In addition, we also review the extension to general Banach spaces and some computational aspects of these methods.

We apply a family of iterative methods to the problem of extracting the nth root of a positive number R, that is, to solve the nonlinear equation t(n) - R = 0. For each value of n we obtain the method in the family for which the highest order of convergence is reached.

From a study of the convexity we give an acceleration for Newton's method and obtain a new third order method. Then we use this method for solving non-linear equations in Banach spaces, establishing conditions on convergence, existence and uniqueness of solution, as well as error estimates

Newton's method is applied to an operator that satisfies stronger conditions than those of Kantorovich. Convergence and error estimates are compared in the two situations. As an application, we obtain information on the existence and uniqueness of a solution for differential and integral equations.

The classical Kantorovich theorem for Newton's method assumes that the derivative of the involved operator satisfies a Lipschitz condition \\F'(x(0))(-1) [F'(x) - F'(y)] \\ less than or equal to L\\x - y\\ In this communication, we analyse the different modifications of this condition, with a special emphasis in the center- Lipschitz condition: \\F...

The classical Kantorovich theorem on Newton's method assumes that the derivative of the operator involved satisfies a Lipschitz condition ∥F′(x) - F′(y)∥ ≤ L∥x - y∥. In this paper we weaken this condition, assuming that ∥F′(x) - F′(x0)∥ ≤ L∥x - x0∥ for a given point x0.

The classical Kantorovich theorem for Newton’s method assumes that the derivative of the involved operator satisfies a Lipschitz condition ∥;F’(x 0)-1 [F’(x) -’’(y)] ∥≤ L∥x - y∥ In this communication, we analyse the different modifications of this condition, with a special emphasis in the center-Lipschitz condition: ∥F’(x0)-1 [F’(x) - F-(x0)]∥≤ω(∥x...

We introduce a new biparametric family of multipoint methods with R-order of at least three, to approximate a solution of nonlinear equations in Banach spaces. An existence-uniqueness theorem and error estimates are provided for this family of iterations using Newton-Kantorovich-type assumptions and a new technique based on a system of recurrence r...

The super-Halley method is, in general, an iterative process with order of convergence three. In this paper we study this method in Banach spaces and we prove that the method converges with order four when it is applied to quadratic equations. Consequently, for this type of equations, the application of the super-Halley iteration could be of practi...

We use the classical Kantorovich technique to study the convergence of a new uniparametric family of third order iterative process defined in Banach spaces. We obtain information about this family by the study of the same family in the real case. Moreover, for a value of the parameter, we obtain a method which has order four when applied to quadrat...

We show the classical Kantorovich technique to study the convergence of a new uniparametric family of third order iterative processes defined in Banach spaces. We obtain information about this family from the study of the same family in the real case. Besides we obtain, for a value of the parameter, a method which has order four when it is applied...

In this paper we give sufficient conditions in order to assure the convergence of the super-Halley method in Banach spaces. We use a system of recurrence relations analogous to those given in the classical Newton-Kantorovich theorem, or those given for Chebyshev and Halley methods by different authors.

We consider an inverse-free Jarratt-type approximation, whose order of convergence is four, for solving nonlinear equations. The convergence of this method is analysed under two different types of conditions. We use a new technique based on constructing a system of real sequences. Finally, this method is applied to the study of Hammerstein's integr...

We extend the analysis of convergence of the iterations considered in Ezquerro et al. [Appl. Math. Comput. 85 (1997) 181] for solving nonlinear operator equations in Banach spaces. We establish a different Kantorovich-type convergence theorem for this family and give some error estimates in terms of a real parameter α ϵ [−5, 1).

In this study, we introduce a family of third-order iterative methods, cited in [4], for solving a nonlinear equation f(x) = 0. First we provide a convergence analysis for a real function depending of one real parameter α . Next it is proved that we can always apply a method of this family to solve f(x) = 0.

We study the Kantorovich convergence for parameter-based methods for solving nonlinear operator equations in Banach spaces. We also derive a closed form of error bounds in terms of a real parameter α ϵ [0,1].

We analyse the classical third-order methods (Chebyshev, Halley, super-Halley) to solve a nonlnnear equation F(x) = 0, where F is an operator defined between two Banach spaces. Until now the convergence of these methods is established assuming that the second derivative F″ satisfies a Lipschitz condition. In this paper we prove, by using recurrence...

A new semilocal convergence theorem for Newton's method is established for solving a nonlinear equation F(x) = 0, defined in Banach spaces. It is assumed that the operator F is twice Fréchet differentiable, and F″ satisfies a Lipschitz type condition. Results on uniqueness of solution and error estimates are also given. Finally, these results are c...

New conditions on the convergence of the Chebyshev method in Banach spaces are stated by using recurrence relations. The results are compared with the ones obtained by other authors.

A family of third-order iterative processes (that includes Chebyshev and Halley's methods) is studied in Banach spaces. Results on convergence and uniqueness of solution are given, as well as error estimates. This study allows us to compare the most famous third-order iterative processes.

A new convergence theorem is established for the super-Halley method. This method has, in general, order three, but when it is applied to quadratic equations, its order is four.

We give new conditions for the convergence of Newton's method in Banach spaces, in terms of the degree of logarithmic convexity. These conditions guarantee the convergence of Newton sequence in cases where the hypothesis of Kantorovich theorem are not verified, as we show in some examples.

Newton's method is a well known iterative method to solve a nonlinear equation F(x) = 0. We analyze the convergence of this method for operators defined between two Banach spaces, so our results can be applied in a wide range of problems, such as real or complex equations, nonlinear systems of equations, differential or integral equations. Firstly...

SUMMARY

INTRODUCCIÓN El espacio de trabajo en manipuladores paralelos es generalmente complejo, dividido internamente por superficies de singularidad del Problema Cinemático Directo (PCD) y estando definida su frontera por las singularidades del Problema Cinemático Inverso (PCI). Por ello, diversas investigaciones se centran en la obtención y caracterizaci...