Stability anomalies of some jacobian-free iterative methods of high order of convergence

[EN] In this manuscript, we design two classes of parametric iterative schemes to solve nonlinear problems that do not need to evaluate Jacobian matrices and need to solve three linear systems per iteration with the same divided difference operator as the coefficient matrix. The stability performance of the classes is analyzed on a quadratic polynomial system, and it is shown that for many values of the parameter, only convergence to the roots of the problem exists. Finally, we check the performance of these methods on some test problems to confirm the theoretical results.This research was partially supported by Ministerio de Economia y Competitividad under grants PGC2018-095896-B-C22, Generalitat Valenciana PROMETEO/2016/089 and FONDOCYT 027-2018 and 029-2018, Dominican Republic.Cordero Barbero, A.; García-Maimo, J.; Torregrosa Sánchez, JR.; Vassileva, MP. (2019). Stability anomalies of some jacobian-free iterative methods of high order of convergence. Axioms. 8(2):1-15. https://doi.org/10.3390/axioms8020051S11582Frontini, M., & Sormani, E. (2004). Third-order methods from quadrature formulae for solving systems of nonlinear equations. Applied Mathematics and Computation, 149(3), 771-782. doi:10.1016/s0096-3003(03)00178-4Homeier, H. H. . (2004). A modified Newton method with cubic convergence: the multivariate case. Journal of Computational and Applied Mathematics, 169(1), 161-169. doi:10.1016/j.cam.2003.12.041Aslam Noor, M., & Waseem, M. (2009). Some iterative methods for solving a system of nonlinear equations. Computers & Mathematics with Applications, 57(1), 101-106. doi:10.1016/j.camwa.2008.10.067Xiao, X., & Yin, H. (2015). A new class of methods with higher order of convergence for solving systems of nonlinear equations. Applied Mathematics and Computation, 264, 300-309. doi:10.1016/j.amc.2015.04.094Cordero, 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.2007.01.062Darvishi, M. T., & Barati, A. (2007). A third-order Newton-type method to solve systems of nonlinear equations. Applied Mathematics and Computation, 187(2), 630-635. doi:10.1016/j.amc.2006.08.080Sharma, J. R., Guha, R. K., & Sharma, R. (2012). An efficient fourth order weighted-Newton method for systems of nonlinear equations. Numerical Algorithms, 62(2), 307-323. doi:10.1007/s11075-012-9585-7Narang, M., Bhatia, S., & Kanwar, V. (2016). New two-parameter Chebyshev–Halley-like family of fourth and sixth-order methods for systems of nonlinear equations. Applied Mathematics and Computation, 275, 394-403. doi:10.1016/j.amc.2015.11.063Behl, R., Sarría, Í., González, R., & Magreñán, Á. A. (2019). Highly efficient family of iterative methods for solving nonlinear models. Journal of Computational and Applied Mathematics, 346, 110-132. doi:10.1016/j.cam.2018.06.042Amorós, C., Argyros, I., González, R., Magreñán, Á., Orcos, L., & Sarría, Í. (2019). Study of a High Order Family: Local Convergence and Dynamics. Mathematics, 7(3), 225. doi:10.3390/math7030225Argyros, I., & González, D. (2015). Local Convergence for an Improved Jarratt-type Method in Banach Space. International Journal of Interactive Multimedia and Artificial Intelligence, 3(4), 20. doi:10.9781/ijimai.2015.344Sharma, J. R., & Gupta, P. (2014). An efficient fifth order method for solving systems of nonlinear equations. Computers & Mathematics with Applications, 67(3), 591-601. doi:10.1016/j.camwa.2013.12.004Cordero, A., Gutiérrez, J. M., Magreñán, Á. A., & Torregrosa, J. R. (2016). Stability analysis of a parametric family of iterative methods for solving nonlinear models. Applied Mathematics and Computation, 285, 26-40. doi:10.1016/j.amc.2016.03.021Cordero, A., Soleymani, F., & Torregrosa, J. R. (2014). Dynamical analysis of iterative methods for nonlinear systems or how to deal with the dimension? Applied Mathematics and Computation, 244, 398-412. doi:10.1016/j.amc.2014.07.010Cordero, A., Hueso, J. L., Martínez, E., & Torregrosa, J. R. (2009). A modified Newton-Jarratt’s composition. Numerical Algorithms, 55(1), 87-99. doi:10.1007/s11075-009-9359-zArgyros, I., & George, S. (2015). Ball Convergence for Steffensen-type Fourth-order Methods. International Journal of Interactive Multimedia and Artificial Intelligence, 3(4), 37. doi:10.9781/ijimai.2015.347Chicharro, F. I., Cordero, A., & Torregrosa, J. R. (2013). Drawing Dynamical and Parameters Planes of Iterative Families and Methods. The Scientific World Journal, 2013, 1-11. doi:10.1155/2013/78015