Iterative methods with memory for solving systems of nonlinear equations using a second order approximation

[EN] Iterative methods for solving nonlinear equations are said to have memory when the calculation of the next iterate requires the use of more than one previous iteration. Methods with memory usually have a very stable behavior in the sense of the wideness of the set of convergent initial estimations. With the right choice of parameters, iterative methods without memory can increase their order of convergence significantly, becoming schemes with memory. In this work, starting from a simple method without memory, we increase its order of convergence without adding new functional evaluations by approximating the accelerating parameter with Newton interpolation polynomials of degree one and two. Using this technique in the multidimensional case, we extend the proposed method to systems of nonlinear equations. Numerical tests are presented to verify the theoretical results and a study of the dynamics of the method is applied to different problems to show its stability.This research was supported by PGC2018-095896-B-C22 (MCIU/AEI/FEDER, UE), Generalitat Valenciana PROMETEO/2016/089, and FONDOCYT 2016-2017-212 Republica Dominicana.Cordero Barbero, A.; Maimó, JG.; Torregrosa Sánchez, JR.; Vassileva, MP. (2019). Iterative methods with memory for solving systems of nonlinear equations using a second order approximation. Mathematics. 7(11):1-12. https://doi.org/10.3390/math7111069S112711Soleymani, F., Lotfi, T., Tavakoli, E., & Khaksar Haghani, F. (2015). Several iterative methods with memory using self-accelerators. Applied Mathematics and Computation, 254, 452-458. doi:10.1016/j.amc.2015.01.045Petković, M. S., & Sharma, J. R. (2015). On some efficient derivative-free iterative methods with memory for solving systems of nonlinear equations. Numerical Algorithms, 71(2), 457-474. doi:10.1007/s11075-015-0003-9Narang, M., Bhatia, S., Alshomrani, A. S., & Kanwar, V. (2019). General efficient class of Steffensen type methods with memory for solving systems of nonlinear equations. Journal of Computational and Applied Mathematics, 352, 23-39. doi:10.1016/j.cam.2018.10.048Potra, F. A. (1982). An error analysis for the secant method. Numerische Mathematik, 38(3), 427-445. doi:10.1007/bf01396443Fatou, P. (1919). Sur les équations fonctionnelles. Bulletin de la Société mathématique de France, 2, 161-271. doi:10.24033/bsmf.998Cordero, 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.2007.01.062Campos, B., Cordero, A., Torregrosa, J. R., & Vindel, P. (2015). A multidimensional dynamical approach to iterative methods with memory. Applied Mathematics and Computation, 271, 701-715. doi:10.1016/j.amc.2015.09.056Chicharro, 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

Memory in a new variant of King's family for solving nonlinear systems

[EN] In the recent literature, very few high-order Jacobian-free methods with memory for solving nonlinear systems appear. In this paper, we introduce a new variant of King's family with order four to solve nonlinear systems along with its convergence analysis. The proposed family requires two divided difference operators and to compute only one inverse of a matrix per iteration. Furthermore, we have extended the proposed scheme up to the sixth-order of convergence with two additional functional evaluations. In addition, these schemes are further extended to methods with memory. We illustrate their applicability by performing numerical experiments on a wide variety of practical problems, even big-sized. It is observed that these methods produce approximations of greater accuracy and are more efficient in practice, compared with the existing methods.This research was supported by PGC2018-095896-B-C22 (MCIU/AEI/FEDER, UE).Kansal, M.; Cordero Barbero, A.; Bhalla, S.; Torregrosa Sánchez, JR. (2020). Memory in a new variant of King's family for solving nonlinear systems. Mathematics. 8(8):1-15. https://doi.org/10.3390/math8081251S11588Cordero, 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-zCordero, A., Hueso, J. L., Martínez, E., & Torregrosa, J. R. (2011). Efficient high-order methods based on golden ratio for nonlinear systems. Applied Mathematics and Computation, 217(9), 4548-4556. doi:10.1016/j.amc.2010.11.006Babajee, D. K. R., Cordero, A., Soleymani, F., & Torregrosa, J. R. (2012). On a Novel Fourth-Order Algorithm for Solving Systems of Nonlinear Equations. Journal of Applied Mathematics, 2012, 1-12. doi:10.1155/2012/165452Zheng, Q., Zhao, P., & Huang, F. (2011). A family of fourth-order Steffensen-type methods with the applications on solving nonlinear ODEs. Applied Mathematics and Computation, 217(21), 8196-8203. doi:10.1016/j.amc.2011.01.095Sharma, J., & Arora, H. (2013). An efficient derivative free iterative method for solving systems of nonlinear equations. Applicable Analysis and Discrete Mathematics, 7(2), 390-403. doi:10.2298/aadm130725016sSharma, J. R., Arora, H., & Petković, M. S. (2014). An efficient derivative free family of fourth order methods for solving systems of nonlinear equations. Applied Mathematics and Computation, 235, 383-393. doi:10.1016/j.amc.2014.02.103Wang, X., Zhang, T., Qian, W., & Teng, M. (2015). Seventh-order derivative-free iterative method for solving nonlinear systems. Numerical Algorithms, 70(3), 545-558. doi:10.1007/s11075-015-9960-2Chicharro, F. I., Cordero, A., Garrido, N., & Torregrosa, J. R. (2020). On the improvement of the order of convergence of iterative methods for solving nonlinear systems by means of memory. Applied Mathematics Letters, 104, 106277. doi:10.1016/j.aml.2020.106277Petković, M. S., & Sharma, J. R. (2015). On some efficient derivative-free iterative methods with memory for solving systems of nonlinear equations. Numerical Algorithms, 71(2), 457-474. doi:10.1007/s11075-015-0003-9Narang, M., Bhatia, S., Alshomrani, A. S., & Kanwar, V. (2019). General efficient class of Steffensen type methods with memory for solving systems of nonlinear equations. Journal of Computational and Applied Mathematics, 352, 23-39. doi:10.1016/j.cam.2018.10.048King, R. F. (1973). A Family of Fourth Order Methods for Nonlinear Equations. SIAM Journal on Numerical Analysis, 10(5), 876-879. doi:10.1137/0710072Hermite, M. C., & Borchardt, M. (1878). Sur la formule d’interpolation de Lagrange. Journal für die reine und angewandte Mathematik (Crelles Journal), 1878(84), 70-79. doi:10.1515/crelle-1878-18788405Petkovic, M., Dzunic, J., & Petkovic, L. (2011). A family of two-point methods with memory for solving nonlinear equations. Applicable Analysis and Discrete Mathematics, 5(2), 298-317. doi:10.2298/aadm110905021pCordero, 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.2007.01.062Awawdeh, F. (2009). On new iterative method for solving systems of nonlinear equations. Numerical Algorithms, 54(3), 395-409. doi:10.1007/s11075-009-9342-8Noor, M. A., Waseem, M., & Noor, K. I. (2015). New iterative technique for solving a system of nonlinear equations. Applied Mathematics and Computation, 271, 446-466. doi:10.1016/j.amc.2015.08.125Pramanik, S. (2002). Kinematic Synthesis of a Six-Member Mechanism for Automotive Steering. Journal of Mechanical Design, 124(4), 642-645. doi:10.1115/1.150337

On the improvement of the order of convergence of iterative methods for solving nonlinear systems by means of memory

[EN] Iterative methods with memory for solving nonlinear systems have been designed. For approximating the accelerating parameters the Kurchatov's divided difference is used as an approximation of the derivative of second order. The convergence of the proposed schemes is analyzed by means of Taylor expansions. Numerical examples are shown to compare the performance of the proposed schemes with other known ones, confirming the theoretical results.This research was partially supported by PGC2018-095896-B-C22 (MCIU/AEI/FEDER, UE), and Generalitat Valenciana PROMETEO/2016/089.Chicharro, FI.; Cordero Barbero, A.; Garrido, N.; Torregrosa Sánchez, JR. (2020). On the improvement of the order of convergence of iterative methods for solving nonlinear systems by means of memory. Applied Mathematics Letters. 104:1-8. https://doi.org/10.1016/j.aml.2020.106277S18104Petković, M. S., & Sharma, J. R. (2015). On some efficient derivative-free iterative methods with memory for solving systems of nonlinear equations. Numerical Algorithms, 71(2), 457-474. doi:10.1007/s11075-015-0003-9Narang, M., Bhatia, S., Alshomrani, A. S., & Kanwar, V. (2019). General efficient class of Steffensen type methods with memory for solving systems of nonlinear equations. Journal of Computational and Applied Mathematics, 352, 23-39. doi:10.1016/j.cam.2018.10.048A. Cordero, J.G. Maimó, J.R. Torregrosa, M.P. Vassileva, Iterative methods with memory for solving systems of nonlinear equations using a second order approximation, Mathematics 7 (11). http://dx.doi.org/10.3390/math7111069.Cordero, 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-