M. A. Hernández

Universidad de La Rioja (Spain), Logroño, La Rioja, Spain

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Publications (136)119.4 Total impact

  • S. Amat, M.A. Hernández, N. Romero
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    ABSTRACT: We study a class of at least third order iterative methods for nonlinear equations on Banach spaces. A characterization of the convergence under Gamma-type conditions is presented. Though, in general, these methods are not very extended due to their computational costs, we can find examples in which they are competitive and even cheaper than other simpler methods. Indeed, we propose a new nonlinear mathematical model for the denoising of digital images, where the best method in the family has fourth order of convergence. Moreover, our family includes two-step Newton type methods with good numerical behavior in general. We center our analysis in both, analytic and computational, aspects.
    Numerische Mathematik 01/2014; 127(2). · 1.55 Impact Factor
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    J.A. Ezquerro, M.A. Hernández, N. Romero, A.I. Velasco
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    ABSTRACT: We present a modification of Steffensen’s method as a predictor–corrector iterative method, so that we can use Steffensen’s method to approximate a solution of a nonlinear equation in Banach spaces from the same starting points from which Newton’s method converges. We study the semilocal convergence of the predictor–corrector method by using the majorant principle. We illustrate the method with an application to a discrete problem.
    Journal of Computational and Applied Mathematics 09/2013; 249:9–23. · 1.08 Impact Factor
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    J.A. Ezquerro, D. González, M.A. Hernández
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    ABSTRACT: Following an idea similar to that given by Dennis and Schnabel (1996) in [2], we prove a local convergence result for Newton’s method under generalized conditions of Kantorovich type.
    Applied Mathematics Letters 05/2013; 26(5):566–570. · 1.48 Impact Factor
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    ABSTRACT: From some modifications of Chebyshev's method, we consider a uniparametric family of iterative methods that are more efficient than Newton's method, and we then construct two iterative methods in a similar way to the Secant method from Newton's method. These iterative methods do not use derivatives in their algorithms and one of them is more efficient than the Secant method, which is the classical method with this feature.
    Analysis and Applications 05/2013; 11(03). · 1.50 Impact Factor
  • J.A. Ezquerro, D. González, M.A. Hernández
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    ABSTRACT: We study the semilocal convergence of Newton’s method in Banach spaces under a modification of the classic conditions of Kantorovich, which leads to a generalization of Kantorovich’s theory. We illustrate this study with two Hammerstein integral equations of the second kind, where the classic conditions of Kantorovich cannot be applied, but our modification of them can.
    Mathematical and Computer Modelling 02/2013; 57(s 3–4):584–594. · 2.02 Impact Factor
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    ABSTRACT: From well-known secant-like methods, we observe that we can construct a new family of secant-like methods that includes the secant method and Kurchatov’s method. We analyse the local orders of convergence and the efficiencies of the methods of the family and study the semilocal convergence for differentiable and nondifferentiable operators. Finally, we apply our results to conservative problems.
    Journal of Mathematical Analysis and Applications 02/2013; 398(1):100–112. · 1.12 Impact Factor
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    ABSTRACT: 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-differentiable problems.
    Milan Journal of Mathematics 01/2013; 81(1). · 0.82 Impact Factor
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    J.A. Ezquerro, M.A. Hernández, N. Romero, A.I. Velasco
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    ABSTRACT: From the simplified secant method and a family of secant-like methods, we construct a family of predictor–corrector iterative methods for solving a nonlinear equation in Banach spaces.
    Applied Mathematics and Computation 12/2012; 219(8):3677–3692. · 1.60 Impact Factor
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    ABSTRACT: Some modifications of the secant method for solving nonlinear equations are revisited and the local order of convergence is found in a direct symbolic computation. To do this, a development of the inverse of the first order divided differences of a function of several variables in two points is presented. A generalisation of the efficiency index used in the scalar case to several variables is also analysed in order to use the most competitive algorithm.
    Computers & Mathematics with Applications 09/2012; 64(6):2066–2073. · 2.00 Impact Factor
  • S. Amat, M.A. Hernández, N. Romero
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    ABSTRACT: In this paper the modification of Chebyshevʼs iterative method constructed in Amat et al. (2008) [1] is revisited. The behavior of this method when considering quadratic nonlinear operators is analyzed. In this case, the iterative method has a competitive behavior due to its computational efficiency. Moreover, a new result of semilocal convergence assuming only a pointwise condition is obtained, improving the result given in Amat et al. (2008) [1]. The domain of uniqueness of the solution is also improved. The new technique used in the proof of these results allows us to achieve all these improvements. Finally, some theoretical and numerical applications for a quadratic system of equations are presented.
    Applied Numerical Mathematics 07/2012; 62(7):833–841. · 1.04 Impact Factor
  • J.A. Ezquerro, D. González, M.A. Hernández
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    ABSTRACT: We study nonlinear integral equations of mixed Hammerstein type using Newton’s method as follows. We investigate the theoretical significance of Newton’s method to draw conclusions about the existence and uniqueness of solutions of these equations. After that, we approximate the solutions of a particular nonlinear integral equation by Newton’s method. For this, we use the majorant principle, which is based on the concept of majorizing sequence given by Kantorovich, and milder convergence conditions than those of Kantorovich. Actually, we prove a semilocal convergence theorem which is applicable to situations where Kantorovich’s theorem is not.
    Applied Mathematics and Computation 05/2012; 218(18):9536–9546. · 1.60 Impact Factor
  • J. A. Ezquerro, D. González, M. A. Hernández
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    ABSTRACT: The most restrictive condition used by Kantorovich for proving the semilocal convergence of Newton’s method in Banach spaces is relaxed in this paper, providing we can guarantee the semilocal convergence in situations that Kantorovich cannot. To achieve this, we use Kantorovich’s technique based on majorizing sequences, but our majorizing sequences are obtained differently, by solving initial value problems.
    Journal of Computational and Applied Mathematics 03/2012; 236:2246-2258. · 1.08 Impact Factor
  • J.A. Ezquerro, M.A. Hernández
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    ABSTRACT: A new iterative method of third-order of convergence is constructed with the interesting feature that the calculation of inverse operators is not needed if the inverse of an operator is approximated. The semilocal convergence of the method is studied under classical Kantorovich-type conditions for iterative methods of second-order. Some applications are given, where the most important features of the method are shown.
    Nonlinear Analysis Real World Applications 02/2012; 13(1):14–26. · 2.34 Impact Factor
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    ABSTRACT: We construct a new iterative method for approximating the solutions of nonlinear operator equations, where the operator involved is not differentiable. The algorithm proposed does not need to evaluate derivatives and is more efficient than the secant method. For this, we extend a result of Traub for one-point iterative methods to one-point iterative methods with memory.
    Journal of Complexity 02/2012; 28:48-58. · 1.19 Impact Factor
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    ABSTRACT: We present a modified method for solving nonlinear systems of equations with order of convergence higher than other competitive methods. We generalize also the efficiency index used in the one-dimensional case to several variables. Finally, we show some numerical examples, where the theoretical results obtained in this paper are applied.
    Journal of Optimization Theory and Applications 10/2011; 151(1):163-174. · 1.41 Impact Factor
  • J. A. Ezquerro, M. A. Hernández, N. Romero
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    ABSTRACT: The application of high order iterative methods for solving nonlinear integral equations is not usual in mathematics. But, in this paper, we show that high order iterative methods can be used to solve a special case of nonlinear integral equations of Fredholm type and second kind. In particular, those that have the property of the second derivative of the corresponding operator have associated with them a vector of diagonal matrices once a process of discretization has been done.
    Journal of Computational and Applied Mathematics 10/2011; 236:1449-1463. · 1.08 Impact Factor
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    ABSTRACT: 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.
    Journal of Computational and Applied Mathematics 03/2011; 235:3195-3206. · 1.08 Impact Factor
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    ABSTRACT: We present a generalization of a modification of a classic Traub’s result, which allows constructing, from a given iterative method with order of convergence p, efficient iterative methods with order of convergence at least pq+1 (q≤p−1, q∈N).
    Computers & Mathematics with Applications 12/2010; 60:2899-2908. · 2.00 Impact Factor
  • J. A. Ezquerro, M. A. Hernández, N. Romero
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    ABSTRACT: In the study of iterative methods with high order of convergence, Gander provides a general expression for iterative methods with order of convergence at least three in the scalar case. Taking into account an extension of this result, we define a family of iterations in Banach spaces with R-order of convergence at least four for quadratic equations.
    Journal of Computational and Applied Mathematics 06/2010; 234:960-971. · 1.08 Impact Factor
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    J. M. Gutiérrez, M. A. Hernández, N. Romero
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    ABSTRACT: 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 quadratic polynomials with different roots. Newton’s and Chebyshev’s methods appear as particular choices of the family appear for m=0 and m=1, respectively. We make both analytical and graphical studies of these methods, which give rise to rational functions defined in the extended complex plane. Firstly, we prove that the coefficients of the aforementioned family of iterative processes can be written in terms of the Catalan numbers. Secondly, we make an incursion into its dynamical behavior. In fact, we show that the rational maps related to these methods can be written in terms of the entries of the Catalan triangle. Next we analyze its general convergence, by including some computer plots showing the intricate structure of the Universal Julia sets associated with the methods.
    Journal of Computational and Applied Mathematics 03/2010; 233:2688-2695. · 1.08 Impact Factor