Iterative Algorithms for Nonlinear Equations and Dynamical Behaviors: Applications

Numerical iteration methods for solving the roots of nonlinear transcendental or algebraic model equations (in 1D, 2D or 3D) are useful in most applied sciences (Biology, physics, mathematics, Chemistry…) and in engineering, for example, problems of beam deflections. This article presents new iterative algorithms for finding roots of nonlinear equations applying some fixed point transformation and interpolation. A method for solving nonlinear systems (in higher dimensions, for multi-variables) is also considered. Our main focus is on methods not involving the equation f(x) in problem and or its derivatives. These new algorithm can be considered as the acceleration convergence of several existing methods. For convergence and efficiency proofs and applications, we solve deflection of a beam differential equation and some test experiments in in Matlab.  Different (real & complex) dynamical (convergence plane) analyzes are also shown graphically. Keywords: nonlinear equations, deflection of beam, iterations, dynamical analysis, applications, 2

Multiplicity anomalies of an optimal fourth-order class of iterative methods for solving nonlinear equations

[EN] There is a few number of optimal fourth-order iterative methods for obtaining the multiple roots of nonlinear equations. But, in most of the earlier studies, scholars gave the flexibility in their proposed schemes only at the second step (not at the first step) in order to explore new schemes. Unlike what happens in existing methods, the main aim of this manuscript is to construct a new fourth-order optimal scheme which will give the flexibility to the researchers at both steps as well as faster convergence, smaller residual errors and asymptotic error constants. The construction of the proposed scheme is based on the mid-point formula and weight function approach. From the computational point of view, the stability of the resulting class of iterative methods is studied by means of the conjugacy maps and the analysis of strange fixed points. Their basins of attractions and parameter planes are also given to show their dynamical behavior around the multiple roots. Finally, we consider a real-life problem and a concrete variety of standard test functions for numerical experiments and relevant results are extensively treated to confirm the theoretical development.This research was partially supported by Ministerio de Economia y Competitividad MTM2014-52016-C2-2-P and Generalitat Valenciana PROMETEO/2016/089.Behl, R.; Cordero Barbero, A.; Motsa, SS.; Torregrosa Sánchez, JR. (2018). Multiplicity anomalies of an optimal fourth-order class of iterative methods for solving nonlinear equations. Nonlinear Dynamics. 91(1):81-112. https://doi.org/10.1007/s11071-017-3858-6S81112911Behl, R., Cordero, A., Motsa, S.S., Torregrosa, J.R., Kanwar, V.: An optimal fourth-order family of methods for multiple roots and its dynamics. Numer. Algorithms 71(4), 775–796 (2016)Blanchard, P.: Complex analytic dynamics on the Riemann sphere. Bull. Am. Math. Soc. 11(1), 85–141 (1984)Chicharro, F., Cordero, A., Torregrosa, J.R.: Drawing dynamical and parameter planes of iterative families and methods. Sci. World J. 2013(2013), 1–11 (2013)Devaney, R.L.: The Mandelbrot Set, the Farey Tree and the Fibonacci sequence. Am. Math. Mon. 106(4), 289–302 (1999)Dong, C.: A family of multipoint iterative functions for finding multiple roots of equations. Int. J. Comput. Math. 21, 363–367 (1987)Hueso, J.L., Martínez, E., Teruel, C.: Determination of multiple roots of nonlinear equations and applications. J. Math. Chem. 53, 880–892 (2015)Kung, H.T., Traub, J.F.: Optimal order of one-point and multipoint iteration. J. Assoc. Comput. Mach. 21, 643–651 (1974)Li, S.G., Cheng, L.Z., Neta, B.: Some fourth-order nonlinear solvers with closed formulae for multiple roots. Comput. Math. Appl. 59, 126–135 (2010)Li, S., Liao, X., Cheng, L.: A new fourth-order iterative method for finding multiple roots of nonlinear equations. Appl. Math. Comput. 215, 1288–1292 (2009)Petković, M.S., Neta, B., Petković, L.D., Dz̆unić, J.: Multipoint Methods for Solving Nonlinear Equations. Academic Press, New York (2013)Sbibih, D., Serghini, A., Tijini, A., Zidna, A.: A general family of third order method for finding multiple roots. AMC 233, 338–350 (2014)Schröder, E.: Über unendlichviele Algorithm zur Auffosung der Gleichungen. Math. Ann. 2, 317–365 (1870)Sharifi, M., Babajee, D.K.R., Soleymani, F.: Finding the solution of nonlinear equations by a class of optimal methods. Comput. Math. Appl. 63, 764–774 (2012)Soleymani, F., Babajee, D.K.R.: Computing multiple zeros using a class of quartically convergent methods. Alex. Eng. J. 52, 531–541 (2013)Soleymani, F., Babajee, D.K.R., Lofti, T.: On a numerical technique forfinding multiple zeros and its dynamic. J. Egypt. Math. Soc. 21, 346–353 (2013)Traub, J.F.: Iterative Methods for the Solution of Equations. Prentice-Hall, Englewood Cliffs (1964)Zhou, X., Chen, X., Song, Y.: Families of third and fourth order methods for multiple roots of nonlinear equations. Appl. Math. Comput. 219, 6030–6038 (2013)Zhou, X., Chen, X., Song, Y.: Constructing higher-order methods for obtaining the muliplte roots of nonlinear equations. J. Comput. Math. Appl. 235, 4199–4206 (2011

Numerical Analysis

Acknowledgements: This article will appear in the forthcoming Princeton Companion to Mathematics, edited by Timothy Gowers with June Barrow-Green, to be published by Princeton University Press.\ud \ud In preparing this essay I have benefitted from the advice of many colleagues who corrected a number of errors of fact and emphasis. I have not always followed their advice, however, preferring as one friend put it, to "put my head above the parapet". So I must take full responsibility for errors and omissions here.\ud \ud With thanks to: Aurelio Arranz, Alexander Barnett, Carl de Boor, David Bindel, Jean-Marc Blanc, Mike Bochev, Folkmar Bornemann, Richard Brent, Martin Campbell-Kelly, Sam Clark, Tim Davis, Iain Duff, Stan Eisenstat, Don Estep, Janice Giudice, Gene Golub, Nick Gould, Tim Gowers, Anne Greenbaum, Leslie Greengard, Martin Gutknecht, Raphael Hauser, Des Higham, Nick Higham, Ilse Ipsen, Arieh Iserles, David Kincaid, Louis Komzsik, David Knezevic, Dirk Laurie, Randy LeVeque, Bill Morton, John C Nash, Michael Overton, Yoshio Oyanagi, Beresford Parlett, Linda Petzold, Bill Phillips, Mike Powell, Alex Prideaux, Siegfried Rump, Thomas Schmelzer, Thomas Sonar, Hans Stetter, Gil Strang, Endre Süli, Defeng Sun, Mike Sussman, Daniel Szyld, Garry Tee, Dmitry Vasilyev, Andy Wathen, Margaret Wright and Steve Wright