A q-VARIANT OF STEFFENSEN'S METHOD OF FOURTH-ORDER CONVERGENCE

Starting from q-Taylor formula, we suggest a new q-variant of Stef-fensen's method of fourth-order convergence for solving non-linear equations

Stochastic Steffensen method

Is it possible for a first-order method, i.e., only first derivatives allowed, to be quadratically convergent? For univariate loss functions, the answer is yes -- the Steffensen method avoids second derivatives and is still quadratically convergent like Newton method. By incorporating an optimal step size we can even push its convergence order beyond quadratic to $1+\sqrt{2} \approx 2.414$. While such high convergence orders are a pointless overkill for a deterministic algorithm, they become rewarding when the algorithm is randomized for problems of massive sizes, as randomization invariably compromises convergence speed. We will introduce two adaptive learning rates inspired by the Steffensen method, intended for use in a stochastic optimization setting and requires no hyperparameter tuning aside from batch size. Extensive experiments show that they compare favorably with several existing first-order methods. When restricted to a quadratic objective, our stochastic Steffensen methods reduce to randomized Kaczmarz method -- note that this is not true for SGD or SLBFGS -- and thus we may also view our methods as a generalization of randomized Kaczmarz to arbitrary objectives.Comment: 22 pages, 3 figure

Steffensen type methods for solving nonlinear equations

[EN] In the present paper, by approximating the derivatives in the well known fourth-order Ostrowski's method and in a sixth-order improved Ostrowski's method by central-difference quotients, we obtain new modifications of these methods free from derivatives. We prove the important fact that the methods obtained preserve their convergence orders 4 and 6, respectively, without calculating any derivatives. Finally, numerical tests confirm the theoretical results and allow us to compare these variants with the corresponding methods that make use of derivatives and with the classical Newton's method. (C) 2010 Elsevier B.V. All rights reserved.This research was supported by Ministerio de Ciencia y Tecnología MTM2010-18539Cordero Barbero, A.; Hueso Pagoaga, JL.; Martínez Molada, E.; Torregrosa Sánchez, JR. (2012). Steffensen type methods for solving nonlinear equations. Journal of Computational and Applied Mathematics. 236(12):3058-3064. https://doi.org/10.1016/j.cam.2010.08.043S305830642361

Derivative-free high-order methods applied to preliminary orbit determination

From position and velocity coordinates for several given instants, it is possible to determine the orbital elements of the preliminary orbit, taking only into account mutual gravitational attraction forces between the Earth and the satellite. Nevertheless, it should be refined with later observations from ground stations, whose geographic coordinates are previously known. Different methods developed for this purpose need to find a solution of a nonlinear function. In some classical methods it is usual to employ fixed point or secant methods. The second iterative scheme is often used when it is not possible to obtain the derivative of the nonlinear function. Nowadays, there exist efficient numerical methods that are able to highly improve the results obtained by the classical schemes. We will focus our attention on the method of iteration of the true anomaly, in which the secant method is replaced by more efficient methods, such as the second-order Steffensen's method, as well as other high-order derivative-free methods. (C) 2011 Elsevier Ltd. All rights reserved.This research was supported by Ministerio de Ciencia y Tecnologia MTM2011-28636-C02-02 and by Vicerrectorado de Investigacion, Universitat Politecnica de Valencia PAID-06-2010-2285.Chicharro López, FI.; Cordero Barbero, A.; Torregrosa Sánchez, JR. (2013). Derivative-free high-order methods applied to preliminary orbit determination. Mathematical and Computer Modelling. 57(7-8):1795-1799. https://doi.org/10.1016/j.mcm.2011.11.045S17951799577-

A new technique to obtain derivative-free optimal iterative methods for solving nonlinear equations

A new technique to obtain derivative-free methods with optimal order of convergence in the sense of the Kung-Traub conjecture for solving nonlinear smooth equations is described. The procedure uses Steffensen-like methods and Pade approximants. Some numerical examples are provided to show the good performance of the new methods. (c) 2012 Elsevier B.V. All rights reserved.This research was supported by Ministerio de Ciencia y Tecnologia MTM2011-28636-C02-02 and by Vicerrectorado de Investigacion, Universitat Politecnica de Valencia PAID-06-2010-2285.Cordero Barbero, A.; Hueso Pagoaga, JL.; Martínez Molada, E.; Torregrosa Sánchez, JR. (2013). A new technique to obtain derivative-free optimal iterative methods for solving nonlinear equations. Journal of Computational and Applied Mathematics. 252:95-102. https://doi.org/10.1016/j.cam.2012.03.030S9510225

Semilocal Convergence of the Extension of Chun's Method

[EN] In this work, we use the technique of recurrence relations to prove the semilocal convergence in Banach spaces of the multidimensional extension of Chun's iterative method. This is an iterative method of fourth order, that can be transferred to the multivariable case by using the divided difference operator. We obtain the domain of existence and uniqueness by taking a suitable starting point and imposing a Lipschitz condition to the first Frechet derivative in the whole domain. Moreover, we apply the theoretical results obtained to a nonlinear integral equation of Hammerstein type, showing the applicability of our results.This research was supported by PGC2018-095896-B-C22 (MCIU/AEI/FEDER, UE) and FONDOCYT 027-2018 Republica Dominicana.Cordero Barbero, A.; Maimó, JG.; Martínez Molada, E.; Torregrosa Sánchez, JR.; Vassileva, MP. (2021). Semilocal Convergence of the Extension of Chun's Method. Axioms. 10(3):1-11. https://doi.org/10.3390/axioms10030161S11110

A family of Kurchatov-type methods and its stability

[EN] We present a parametric family of iterative methods with memory for solving of nonlinear problems including Kurchatov¿s scheme, preserving its second-order of convergence. By using the tools of multidimensional real dynamics, the stability of members of this family is analyzed on low-degree polynomials, showing some elements of this class more stable behavior than the original Kurchatov¿s method. The iteration is extended for multi-dimensional case. Computational efficiencies of proposed technique is discussed and compared with the existing methods. A couple of numerical examples are considered to test the performance of the new family of iterations.The authors thank to the anonymous referees for their valuable comments and for the suggestions that have improved the final version of the paper. This research was partially supported by Ministerio de Economia y Competitividad MTM2014-52016-C2-2-P and Generalitat Valenciana PROMETEO/2016/089.Cordero Barbero, A.; Soleymani, F.; Torregrosa Sánchez, JR.; Haghani, FK. (2017). A family of Kurchatov-type methods and its stability. Applied Mathematics and Computation. 294:264-279. https://doi.org/10.1016/j.amc.2016.09.021S26427929

Minimization of Nonlinear Functions by Certain Numerical Algorithms

Abstract: In this paper, we propose few new algorithms, for minimization of nonlinear functions. Then comparative study among the new algorithms and Newton's algorithm is established by means of various examples