Several New Families of Jarratt’s Method for Solving Systems of Nonlinear Equations

In this study, we suggest and analyze a new and wide general class of Jarratt’s method for solving systems of nonlinear equations. These methods have fourth-order convergence and do not require the evaluation of any second or higher-order Fréchet derivatives. In terms of computational cost, all these methods require evaluations of one function and two first-order Fréchet derivatives. The performance of proposed methods is compared with their closest competitors in a series of numerical experiments. It is worth mentioning that all the methods considered here are found to be effective and comparable to the robust methods available in the literature

Sixth-Order Two-Point Efficient Family of Super-Halley Type Methods

The main focus of this manuscript is to provide a highly efficient two-point sixth-order family of super-Halley type methods that do not require any second-order derivative evaluation for obtaining simple roots of nonlinear equations, numerically. Each member of the proposed family requires two evaluations of the given function and two evaluations of the first-order derivative per iteration. By using Mathematica-9 with its high precision compatibility, a variety of concrete numerical experiments and relevant results are extensively treated to confirm t he t heoretical d evelopment. From their basins of attraction, it has been observed that the proposed methods have better stability and robustness as compared to the other sixth-order methods available in the literature

A study of the local convergence of a fifth order iterative method

[EN] We present a local convergence study of a fifth order iterative method to approximate a locally unique root of nonlinear equations. The analysis is discussed under the assumption that first order Frechet derivative satisfies the Lipschitz continuity condition. Moreover, we consider the derivative free method that obtained through approximating the derivative with divided difference along with the local convergence study. Finally, we provide computable radii and error bounds based on the Lipschitz constant for both cases. Some of the numerical examples are worked out and compared the results with existing methods.This research was partially supported by Ministerio de Economia y Competitividad under grant PGC2018-095896-B-C21-C22.Singh, S.; Martínez Molada, E.; Maroju, P.; Behl, R. (2020). A study of the local convergence of a fifth order iterative method. Indian Journal of Pure and Applied Mathematics. 51(2):439-455. https://doi.org/10.1007/s13226-020-0409-5S439455512A. Constantinides and N. Mostoufi, Numerical Methods for Chemical Engineers with MATLAB Applications, Prentice Hall PTR, New Jersey, (1999).J. M. Douglas, Process Dynamics and Control, Prentice Hall, Englewood Cliffs, (1972).M. Shacham, An improved memory method for the solution of a nonlinear equation, Chem. Eng. Sci., 44 (1989), 1495–1501.J. M. Ortega and W. C. Rheinboldt, Iterative solution of nonlinear equations in several variables, Academic Press, New-York, (1970).J. R. Sharma and H. Arora, A novel derivative free algorithm with seventh order convergence for solving systems of nonlinear equations, Numer. Algorithms, 67 (2014), 917–933.I. K. Argyros, A. A. Magreńan, and L. Orcos, Local convergence and a chemical application of derivative free root finding methods with one parameter based on interpolation, J. Math. Chem., 54 (2016), 1404–1416.E. L. Allgower and K. Georg, Lectures in Applied Mathematics, American Mathematical Society (Providence, RI) 26, 723–762.A. V. Rangan, D. Cai, and L. Tao, Numerical methods for solving moment equations in kinetic theory of neuronal network dynamics, J. Comput. Phys., 221 (2007), 781–798.A. Nejat and C. Ollivier-Gooch, Effect of discretization order on preconditioning and convergence of a high-order unstructured Newton-GMRES solver for the Euler equations, J. Comput. Phys., 227 (2008), 2366–2386.C. Grosan and A. Abraham, A new approach for solving nonlinear equations systems, IEEE Trans. Syst. Man Cybernet Part A: System Humans, 38 (2008), 698–714.F. Awawdeh, On new iterative method for solving systems of nonlinear equations, Numer. Algorithms, 54 (2010), 395–409.I. G. Tsoulos and A. Stavrakoudis, On locating all roots of systems of nonlinear equations inside bounded domain using global optimization methods, Nonlinear Anal. Real World Appl., 11 (2010), 2465–2471.E. Martínez, S. Singh, J. L. Hueso, and D. K. Gupta, Enlarging the convergence domain in local convergence studies for iterative methods in Banach spaces, Appl. Math. Comput., 281 (2016), 252–265.S. Singh, D. K. Gupta, E. Martínez, and J. L. Hueso, Semi local and local convergence of a fifth order iteration with Fréchet derivative satisfying Hölder condition, Appl. Math. Comput., 276 (2016), 266–277.I. K. Argyros and S. George, Local convergence of modified Halley-like methods with less computation of inversion, Novi. Sad.J. Math., 45 (2015), 47–58.I. K. Argyros, R. Behl, and S. S. Motsa, Local Convergence of an Efficient High Convergence Order Method Using Hypothesis Only on the First Derivative Algorithms 2015, 8, 1076–1087; doi:https://doi.org/10.3390/a8041076.A. Cordero, J. L. Hueso, E. Martínez, and J. R. Torregrosa, Increasing the convergence order of an iterative method for nonlinear systems, Appl. Math. Lett., 25 (2012), 2369–2374.I. K. Argyros and A. A. Magreñán, A study on the local convergence and dynamics of Chebyshev- Halley-type methods free from second derivative, Numer. Algorithms71 (2016), 1–23.M. Grau-Sánchez, Á Grau, asnd M. Noguera, Frozen divided difference scheme for solving systems of nonlinear equations, J. Comput. Appl. Math., 235 (2011), 1739–1743.M. Grau-Sánchez, M. Noguera, and S. Amat, On the approximation of derivatives using divided difference operators preserving the local convergence order of iterative methods, J. Comput. Appl. Math., 237 (2013), 363–372

Ball Comparison between Three Sixth Order Methods for Banach Space Valued Operators

Three methods of sixth order convergence are tackled for approximating the solution of an equation defined on the finitely dimensional Euclidean space. This convergence requires the existence of derivatives of, at least, order seven. However, only derivatives of order one are involved in such methods. Moreover, we have no estimates on the error distances, conclusions about the uniqueness of the solution in any domain, and the convergence domain is not sufficiently large. Hence, these methods have limited usage. This paper introduces a new technique on a general Banach space setting based only the first derivative and Lipschitz type conditions that allow the study of the convergence. In addition, we find usable error distances as well as uniqueness of the solution. A comparison between the convergence balls of three methods, not possible to drive with the previous approaches, is also given. The technique is possible to use with methods available in literature improving, consequently, their applicability. Several numerical examples compare these methods and illustrate the convergence criteria.This project was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah, Saudi Arabia, under Grant No. D-540-130-1441. The authors, therefore, acknowledge with thanks DSR for technical and financial support.info:eu-repo/semantics/publishedVersio

A New High-Order Jacobian-Free Iterative Method with Memory for Solving Nonlinear Systems

[EN] We used a Kurchatov-type accelerator to construct an iterative method with memory for solving nonlinear systems, with sixth-order convergence. It was developed from an initial scheme without memory, with order of convergence four. There exist few multidimensional schemes using more than one previous iterate in the very recent literature, mostly with low orders of convergence. The proposed scheme showed its efficiency and robustness in several numerical tests, where it was also compared with the existing procedures with high orders of convergence. These numerical tests included large nonlinear systems. In addition, we show that the proposed scheme has very stable qualitative behavior, by means of the analysis of an associated multidimensional, real rational function and also by means of a comparison of its basin of attraction with those of comparison methods.This research was supported by PGC2018-095896-B-C22 (MCIU/AEI/FEDER, UE).Behl, R.; Cordero Barbero, A.; Torregrosa Sánchez, JR.; Bhalla, S. (2021). A New High-Order Jacobian-Free Iterative Method with Memory for Solving Nonlinear Systems. Mathematics. 9(17):1-16. https://doi.org/10.3390/math9172122S11691

A Higher Order Chebyshev-Halley-Type Family of Iterative Methods for Multiple Roots

[EN] The aim of this paper is to introduce new high order iterative methods for multiple roots of the nonlinear scalar equation; this is a demanding task in the area of computational mathematics and numerical analysis. Specifically, we present a new Chebyshev¿Halley-type iteration function having at least sixth-order convergence and eighth-order convergence for a particular value in the case of multiple roots. With regard to computational cost, each member of our scheme needs four functional evaluations each step. Therefore, the maximum efficiency index of our scheme is 1.6818 for ¿ = 2,which corresponds to an optimal method in the sense of Kung and Traub¿s conjecture. We obtain the theoretical convergence order by using Taylor developments. Finally, we consider some real-life situations for establishing some numerical experiments to corroborate the theoretical results.This research was partially supported by Ministerio de Economia y Competitividad under Grant MTM2014-52016-C2-1-2-P and by the project of Generalitat Valenciana Prometeo/2016/089Behl, R.; Martínez Molada, E.; Cevallos-Alarcon, FA.; Alarcon-Correa, D. (2019). A Higher Order Chebyshev-Halley-Type Family of Iterative Methods for Multiple Roots. Mathematics. 7(4):1-12. https://doi.org/10.3390/math7040339S11274Gutiérrez, J. M., & Hernández, M. A. (1997). A family of Chebyshev-Halley type methods in Banach spaces. Bulletin of the Australian Mathematical Society, 55(1), 113-130. doi:10.1017/s0004972700030586Kanwar, V., Singh, S., & Bakshi, S. (2008). Simple geometric constructions of quadratically and cubically convergent iterative functions to solve nonlinear equations. Numerical Algorithms, 47(1), 95-107. doi:10.1007/s11075-007-9149-4Argyros, I. K., Ezquerro, J. A., Gutiérrez, J. M., Hernández, M. A., & Hilout, S. (2011). On the semilocal convergence of efficient Chebyshev–Secant-type methods. Journal of Computational and Applied Mathematics, 235(10), 3195-3206. doi:10.1016/j.cam.2011.01.005Xiaojian, Z. (2008). Modified Chebyshev–Halley methods free from second derivative. Applied Mathematics and Computation, 203(2), 824-827. doi:10.1016/j.amc.2008.05.092Amat, S., Hernández, M. A., & Romero, N. (2008). A modified Chebyshev’s iterative method with at least sixth order of convergence. Applied Mathematics and Computation, 206(1), 164-174. doi:10.1016/j.amc.2008.08.050Kou, J., & Li, Y. (2007). Modified Chebyshev–Halley methods with sixth-order convergence. Applied Mathematics and Computation, 188(1), 681-685. doi:10.1016/j.amc.2006.10.018Li, D., Liu, P., & Kou, J. (2014). An improvement of Chebyshev–Halley methods free from second derivative. Applied Mathematics and Computation, 235, 221-225. doi:10.1016/j.amc.2014.02.083Sharma, J. R. (2015). Improved Chebyshev–Halley methods with sixth and eighth order convergence. Applied Mathematics and Computation, 256, 119-124. doi:10.1016/j.amc.2015.01.002Neta, B. (2010). Extension of Murakami’s high-order non-linear solver to multiple roots. International Journal of Computer Mathematics, 87(5), 1023-1031. doi:10.1080/00207160802272263Zhou, X., Chen, X., & Song, Y. (2011). Constructing higher-order methods for obtaining the multiple roots of nonlinear equations. Journal of Computational and Applied Mathematics, 235(14), 4199-4206. doi:10.1016/j.cam.2011.03.014Hueso, J. L., Martínez, E., & Teruel, C. (2014). Determination of multiple roots of nonlinear equations and applications. Journal of Mathematical Chemistry, 53(3), 880-892. doi:10.1007/s10910-014-0460-8Behl, R., Cordero, A., Motsa, S. S., & Torregrosa, J. R. (2015). On developing fourth-order optimal families of methods for multiple roots and their dynamics. Applied Mathematics and Computation, 265, 520-532. doi:10.1016/j.amc.2015.05.004Behl, R., Cordero, A., Motsa, S. S., Torregrosa, J. R., & Kanwar, V. (2015). An optimal fourth-order family of methods for multiple roots and its dynamics. Numerical Algorithms, 71(4), 775-796. doi:10.1007/s11075-015-0023-5Geum, Y. H., Kim, Y. I., & Neta, B. (2015). A class of two-point sixth-order multiple-zero finders of modified double-Newton type and their dynamics. Applied Mathematics and Computation, 270, 387-400. doi:10.1016/j.amc.2015.08.039Geum, Y. H., Kim, Y. I., & Neta, B. (2016). A sixth-order family of three-point modified Newton-like multiple-root finders and the dynamics behind their extraneous fixed points. Applied Mathematics and Computation, 283, 120-140. doi:10.1016/j.amc.2016.02.029Behl, R., Alshomrani, A. S., & Motsa, S. S. (2018). An optimal scheme for multiple roots of nonlinear equations with eighth-order convergence. Journal of Mathematical Chemistry, 56(7), 2069-2084. doi:10.1007/s10910-018-0857-xMcNamee, J. M. (1998). A comparison of methods for accelerating convergence of Newton’s method for multiple polynomial roots. ACM SIGNUM Newsletter, 33(2), 17-22. doi:10.1145/290590.290592Cordero, 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

Higher-order families of multiple root finding methods suitable for non-convergent cases and their dynamics

In this paper, we present many new one-parameter families of classical Rall’s method (modified Newton’s method), Schröder’s method, Halley’s method and super-Halley method for the first time which will converge even though the guess is far away from the desired root or the derivative is small in the vicinity of the root and have the same error equations as those of their original methods respectively, for multiple roots. Further, we also propose an optimal family of iterative methods of fourth-order convergence and converging to a required root in a stable manner without divergence, oscillation or jumping problems. All the methods considered here are found to be more effective than the similar robust methods available in the literature. In their dynamical study, it has been observed that the proposed methods have equal or better stability and robustness as compared to the other methods

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

Local convergence balls for nonlinear problems with multiplicity and their extension to eight-order of convergence

[EN] The main contribution of this study is to present a new optimal eighth-order scheme for locating zeros with multiplicity m > 1. An extensive convergence analysis is presented with the main theorem in order to demonstrate the optimal eighth-order convergence of the proposed scheme. Moreover, a local convergence study for the optimal fourth-order method defined by the first two steps of the new method is presented, allowing us to obtain the radius of the local convergence ball. Finally, numerical tests on some real-life problems, such as a Van der Waals equation of state, a conversion Chemical engineering problem and two standard academic test problems are presented, which confirm the theoretical results established in this paper and the efficiency of this proposed iterative method. We observed from the numerical experiments that our proposed iterative methods have good values for convergence radii. Further, they have not only faster convergence towards the desired zero of the involved function but they also have both smaller residual error and a smaller difference between two consecutive iterations than current existing techniques.This research was partially supported by Ministerio de Economia y Competitividad under grant MTM2014-52016-C2-2-P and by the project of Generalitat Valenciana Prometeo/2016/089.Behl, R.; Martínez Molada, E.; Cevallos-Alarcon, FA.; Alshomrani, AS. (2019). Local convergence balls for nonlinear problems with multiplicity and their extension to eight-order of convergence. Mathematical Problems in Engineering. 2019:1-18. https://doi.org/10.1155/2019/1427809S1182019Petković, M. S., Neta, B., Petković, L. D., & Džunić, J. (2013). Basic concepts. Multipoint Methods, 1-26. doi:10.1016/b978-0-12-397013-8.00001-7Shengguo, L., Xiangke, L., & Lizhi, C. (2009). A new fourth-order iterative method for finding multiple roots of nonlinear equations. Applied Mathematics and Computation, 215(3), 1288-1292. doi:10.1016/j.amc.2009.06.065Neta, B. (2010). Extension of Murakami’s high-order non-linear solver to multiple roots. International Journal of Computer Mathematics, 87(5), 1023-1031. doi:10.1080/00207160802272263Li, S. G., Cheng, L. Z., & Neta, B. (2010). Some fourth-order nonlinear solvers with closed formulae for multiple roots. Computers & Mathematics with Applications, 59(1), 126-135. doi:10.1016/j.camwa.2009.08.066Zhou, X., Chen, X., & Song, Y. (2011). Constructing higher-order methods for obtaining the multiple roots of nonlinear equations. Journal of Computational and Applied Mathematics, 235(14), 4199-4206. doi:10.1016/j.cam.2011.03.014Sharifi, M., Babajee, D. K. R., & Soleymani, F. (2012). Finding the solution of nonlinear equations by a class of optimal methods. Computers & Mathematics with Applications, 63(4), 764-774. doi:10.1016/j.camwa.2011.11.040Soleymani, F., & Babajee, D. K. R. (2013). Computing multiple zeros using a class of quartically convergent methods. Alexandria Engineering Journal, 52(3), 531-541. doi:10.1016/j.aej.2013.05.001Soleymani, F., Babajee, D. K. R., & Lotfi, T. (2013). On a numerical technique for finding multiple zeros and its dynamic. Journal of the Egyptian Mathematical Society, 21(3), 346-353. doi:10.1016/j.joems.2013.03.011Zhou, X., Chen, X., & Song, Y. (2013). Families of third and fourth order methods for multiple roots of nonlinear equations. Applied Mathematics and Computation, 219(11), 6030-6038. doi:10.1016/j.amc.2012.12.041Hueso, J. L., Martínez, E., & Teruel, C. (2014). Determination of multiple roots of nonlinear equations and applications. Journal of Mathematical Chemistry, 53(3), 880-892. doi:10.1007/s10910-014-0460-8Behl, R., Cordero, A., Motsa, S. S., & Torregrosa, J. R. (2015). On developing fourth-order optimal families of methods for multiple roots and their dynamics. Applied Mathematics and Computation, 265, 520-532. doi:10.1016/j.amc.2015.05.004Zafar, F., Cordero, A., Quratulain, R., & Torregrosa, J. R. (2017). Optimal iterative methods for finding multiple roots of nonlinear equations using free parameters. Journal of Mathematical Chemistry, 56(7), 1884-1901. doi:10.1007/s10910-017-0813-1Geum, Y. H., Kim, Y. I., & Neta, B. (2018). Constructing a family of optimal eighth-order modified Newton-type multiple-zero finders along with the dynamics behind their purely imaginary extraneous fixed points. Journal of Computational and Applied Mathematics, 333, 131-156. doi:10.1016/j.cam.2017.10.033Geum, Y. H., Kim, Y. I., & Magreñán, Á. A. (2018). A study of dynamics via Möbius conjugacy map on a family of sixth-order modified Newton-like multiple-zero finders with bivariate polynomial weight functions. Journal of Computational and Applied Mathematics, 344, 608-623. doi:10.1016/j.cam.2018.06.006Chun, C., & Neta, B. (2015). An analysis of a family of Maheshwari-based optimal eighth order methods. Applied Mathematics and Computation, 253, 294-307. doi:10.1016/j.amc.2014.12.064Thukral, R. (2013). Introduction to Higher-Order Iterative Methods for Finding Multiple Roots of Nonlinear Equations. Journal of Mathematics, 2013, 1-3. doi:10.1155/2013/404635Geum, Y. H., Kim, Y. I., & Neta, B. (2016). A sixth-order family of three-point modified Newton-like multiple-root finders and the dynamics behind their extraneous fixed points. Applied Mathematics and Computation, 283, 120-140. doi:10.1016/j.amc.2016.02.029Argyros, I. (2003). On The Convergence And Application Of Newton’s Method Under Weak HÖlder Continuity Assumptions. International Journal of Computer Mathematics, 80(6), 767-780. doi:10.1080/0020716021000059160Zhou, X., Chen, X., & Song, Y. (2013). On the convergence radius of the modified Newton method for multiple roots under the center–Hölder condition. Numerical Algorithms, 65(2), 221-232. doi:10.1007/s11075-013-9702-2Bi, W., Ren, H., & Wu, Q. (2011). Convergence of the modified Halley’s method for multiple zeros under Hölder continuous derivative. Numerical Algorithms, 58(4), 497-512. doi:10.1007/s11075-011-9466-5Zhou, X., & Song, Y. (2014). Convergence radius of Osada’s method under center-Hölder continuous condition. Applied Mathematics and Computation, 243, 809-816. doi:10.1016/j.amc.2014.06.068Cordero, 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.2007.01.062Balaji, G. V., & Seader, J. D. (1995). Application of interval Newton’s method to chemical engineering problems. Reliable Computing, 1(3), 215-223. doi:10.1007/bf02385253Shacham, M. (1989). An improved memory method for the solution of a nonlinear equation. Chemical Engineering Science, 44(7), 1495-1501. doi:10.1016/0009-2509(89)80026-

Stable high-order iterative methods for solving nonlinear models

[EN] There are several problems of pure and applied science which can be studied in the unified framework of the scalar and vectorial nonlinear equations. In this paper, we propose a sixth-order family of Jarratt type methods for solving nonlinear equations. Further, we extend this family to the multidimensional case preserving the order of convergence. Their theoretical and computational properties are fully investigated along with two main theorems describing the order of convergence. We use complex dynamics techniques in order to select, among the elements of this class of iterative methods, those more stable. This process is made by analyzing the conjugacy class, calculating the fixed and critical points and getting conclusions from parameter and dynamical planes. For the implementation of the proposed schemes for system of nonlinear equations, we consider some applied science problems namely, Van der Pol problem, kinematic syntheses, etc. Further, we compare them with existing sixth-order methods to check the validity of the theoretical results. From the numerical experiments, we find that our proposed schemes perform better than the existing ones. Further, we also consider a variety of nonlinear equations to check the performance of the proposed methods for scalar equations.This research was partially supported by Ministerio de Economia y Competitividad MTM2014-52016-C2-2-P and by Generalitat Valenciana PROMETEO/2016/089.Behl, R.; Cordero Barbero, A.; Motsa, SS.; Torregrosa Sánchez, JR. (2017). Stable high-order iterative methods for solving nonlinear models. Applied Mathematics and Computation. 303:70-88. https://doi.org/10.1016/j.amc.2017.01.029S708830