Generalized high-order classes for solving nonlinear systems and their applications

[EN] A generalized high-order class for approximating the solution of nonlinear systems of equations is introduced. First, from a fourth-order iterative family for solving nonlinear equations, we propose an extension to nonlinear systems of equations holding the same order of convergence but replacing the Jacobian by a divided difference in the weight functions for systems. The proposed GH family of methods is designed from this fourth-order family using both the composition and the weight functions technique. The resulting family has order of convergence 9. The performance of a particular iterative method of both families is analyzed for solving different test systems and also for the Fisher's problem, showing the good performance of the new methods.This research was partially supported by both Ministerio de Ciencia, Innovacion y Universidades and Generalitat Valenciana, under grants PGC2018-095896-B-C22 (MCIU/AEI/FEDER/UE) and PROMETEO/2016/089, respectively.Chicharro, FI.; Cordero Barbero, A.; Garrido-Saez, N.; Torregrosa Sánchez, JR. (2019). Generalized high-order classes for solving nonlinear systems and their applications. Mathematics. 7(12):1-14. https://doi.org/10.3390/math7121194114712Petković, M. S., Neta, B., Petković, L. D., & Džunić, J. (2014). Multipoint methods for solving nonlinear equations: A survey. Applied Mathematics and Computation, 226, 635-660. doi:10.1016/j.amc.2013.10.072Kung, H. T., & Traub, J. F. (1974). Optimal Order of One-Point and Multipoint Iteration. Journal of the ACM, 21(4), 643-651. doi:10.1145/321850.321860Cordero, A., Gómez, E., & Torregrosa, J. R. (2017). Efficient High-Order Iterative Methods for Solving Nonlinear Systems and Their Application on Heat Conduction Problems. Complexity, 2017, 1-11. doi:10.1155/2017/6457532Sharma, J. R., & Arora, H. (2016). Improved Newton-like methods for solving systems of nonlinear equations. SeMA Journal, 74(2), 147-163. doi:10.1007/s40324-016-0085-xAmiri, A., Cordero, A., Taghi Darvishi, M., & Torregrosa, J. R. (2018). Stability analysis of a parametric family of seventh-order iterative methods for solving nonlinear systems. Applied Mathematics and Computation, 323, 43-57. doi:10.1016/j.amc.2017.11.040Cordero, 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-zChicharro, F. I., Cordero, A., Garrido, N., & Torregrosa, J. R. (2019). Wide stability in a new family of optimal fourth‐order iterative methods. Computational and Mathematical Methods, 1(2), e1023. doi:10.1002/cmm4.1023FISHER, R. A. (1937). THE WAVE OF ADVANCE OF ADVANTAGEOUS GENES. Annals of Eugenics, 7(4), 355-369. doi:10.1111/j.1469-1809.1937.tb02153.xSharma, 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/s11075-012-9585-7Soleymani, F., Lotfi, T., & Bakhtiari, P. (2013). A multi-step class of iterative methods for nonlinear systems. Optimization Letters, 8(3), 1001-1015. doi:10.1007/s11590-013-0617-6Cordero, 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.2007.01.06