MODIFIKASI METODE HANSEN-PATRICK DENGAN ORDE KONVERGENSI OPTIMAL UNTUK MENYELESAIKAN PERSAMAAN NONLINIER

The Hansen-Patrick method is a third-order  iterative  method used to solve nonlinear equation. The method requires three evaluation of functions and has an efficiency index 31/3 » 1,4224. This study discusses a modification of the Hansen-Patrick method using the second order Taylor series. The second derivative is reduced using hyperbolic function with one parameter h. The aim of modification is to improve the convergence order of the Hansen-Patrick’s method. Based on the convergence analysis, the method has a fourth-order of convergence and envolve three evaluation of functions. So, its efficiency index is 41/3 » 1,5874. Numerical simulation is given to illustrate performance  of the iterative method using six real functions. The performance of the iterative method include : a computational order of convergence, the number of iteration, evaluation of function, absolute error, and value of function, will be compared with Newton’s method, Halley’s method, Newton-Steffensen’s method, and Hansen-Patrick method. The numerical simulation shows that the performance of the method better than other

Multipoint efficient iterative methods and the dynamics of Ostrowski's method

This is an Author's Accepted Manuscript of an article published in José L. Hueso, Eulalia Martínez & Carles Teruel (2019) Multipoint efficient iterative methods and the dynamics of Ostrowski's method, International Journal of Computer Mathematics, 96:9, 1687-1701, DOI: 10.1080/00207160.2015.1080354 in the International Journal of Computer Mathematics, SEP 2 2019 [copyright Taylor & Francis], available online at: http://www.tandfonline.com/10.1080/00207160.2015.1080354[EN] In this work, we introduce a modification into the technique, presented in A. Cordero, J.L. Hueso, E. Martinez, and J.R. Torregrosa [Increasing the convergence order of an iterative method for nonlinear systems, Appl. Math. Lett. 25 (2012), pp. 2369-2374], that increases by two units the convergence order of an iterative method. The main idea is to compose a given iterative method of order p with a modification of Newton's method that introduces just one evaluation of the function, obtaining a new method of order p+2, avoiding the need to compute more than one derivative, so we improve the efficiency index in the scalar case. This procedure can be repeated n times, with the same approximation to the derivative, obtaining new iterative methods of order p+2n. We perform different numerical tests that confirm the theoretical results. By applying this procedure to Newton's method one obtains the well known fourth order Ostrowski's method. We finally analyse its dynamical behaviour on second and third degree real polynomials.This research was supported by Ministerio de Economia y Competitividad under grant PGC2018-095896-B-C22 and by the project of Generalitat Valenciana Prometeo/2016/089.Hueso, JL.; Martínez Molada, E.; Teruel-Ferragud, C. (2019). Multipoint efficient iterative methods and the dynamics of Ostrowski's method. International Journal of Computer Mathematics. 96(9):1687-1701. https://doi.org/10.1080/00207160.2015.1080354S16871701969Amat, S., Busquier, S., & Plaza, S. (2010). Chaotic dynamics of a third-order Newton-type method. Journal of Mathematical Analysis and Applications, 366(1), 24-32. doi:10.1016/j.jmaa.2010.01.047Cordero, A., & Torregrosa, J. R. (2007). Variants of Newton’s Method using fifth-order quadrature formulas. Applied Mathematics and Computation, 190(1), 686-698. doi:10.1016/j.amc.2007.01.062Cordero, A., Martínez, E., & Torregrosa, J. R. (2009). Iterative methods of order four and five for systems of nonlinear equations. Journal of Computational and Applied Mathematics, 231(2), 541-551. doi:10.1016/j.cam.2009.04.015Cordero, A., Hueso, J. L., Martínez, E., & Torregrosa, J. R. (2012). Increasing the convergence order of an iterative method for nonlinear systems. Applied Mathematics Letters, 25(12), 2369-2374. doi:10.1016/j.aml.2012.07.005Jarratt, P. (1966). Some fourth order multipoint iterative methods for solving equations. Mathematics of Computation, 20(95), 434-434. doi:10.1090/s0025-5718-66-99924-

An improved method for solving quasilinear convection diffusion problems on a coarse mesh

A method is developed for solving quasilinear convection diffusion problems starting on a coarse mesh where the data and solution-dependent coefficients are unresolved, the problem is unstable and approximation properties do not hold. The Newton-like iterations of the solver are based on the framework of regularized pseudo-transient continuation where the proposed time integrator is a variation on the Newmark strategy, designed to introduce controllable numerical dissipation and to reduce the fluctuation between the iterates in the coarse mesh regime where the data is rough and the linearized problems are badly conditioned and possibly indefinite. An algorithm and updated marking strategy is presented to produce a stable sequence of iterates as boundary and internal layers in the data are captured by adaptive mesh partitioning. The method is suitable for use in an adaptive framework making use of local error indicators to determine mesh refinement and targeted regularization. Derivation and q-linear local convergence of the method is established, and numerical examples demonstrate the theory including the predicted rate of convergence of the iterations.Comment: 21 pages, 8 figures, 1 tabl