Optimal iterative methods for finding multiple roots of nonlinear equations using free parameters

[EN] In this paper, we propose a family of optimal eighth order convergent iterative methods for multiple roots with known multiplicity with the introduction of two free parameters and three univariate weight functions. Also numerical experiments have applied to a number of academical test functions and chemical problems for different special schemes from this family that satisfies the conditions given in convergence result.This research was partially supported by Ministerio de Economia y Competitividad MTM2014-52016-C02-2-P and Generalitat Valenciana PROMETEO/2016/089.Zafar, F.; Cordero Barbero, A.; Quratulain, R.; Torregrosa Sánchez, JR. (2018). Optimal iterative methods for finding multiple roots of nonlinear equations using free parameters. Journal of Mathematical Chemistry. 56(7):1884-1901. https://doi.org/10.1007/s10910-017-0813-1S18841901567R. Behl, A. Cordero, S.S. Motsa, J.R. Torregrosa, On developing fourth-order optimal families of methods for multiple roots and their dynamics. Appl. Math. Comput. 265(15), 520–532 (2015)R. Behl, A. Cordero, S.S. Motsa, J.R. Torregrosa, V. Kanwar, An optimal fourth-order family of methods for multiple roots and its dynamics. Numer. Algor. 71(4), 775–796 (2016)R. Behl, A. Cordero, S.S. Motsa, J.R. Torregrosa, An eighth-order family of optimal multiple root finders and its dynamics. Numer. Algor. (2017). doi: 10.1007/s11075-017-0361-6F.I. Chicharro, A. Cordero, J. R. Torregrosa, Drawing dynamical and parameters planes of iterative families and methods. Sci. World J. ID 780153 (2013)A. Constantinides, N. Mostoufi, Numerical Methods for Chemical Engineers with MATLAB Applications (Prentice Hall PTR, New Jersey, 1999)J.M. Douglas, Process Dynamics and Control, vol. 2 (Prentice Hall, Englewood Cliffs, 1972)Y.H. Geum, Y.I. Kim, B. Neta, A class of two-point sixth-order multiple-zero finders of modified double-Newton type and their dynamics. Appl. Math. Comput. 270, 387–400 (2015)Y.H. Geum, Y.I. Kim, B. Neta, A sixth-order family of three-point modified Newton-like multiple-root finders and the dynamics behind their extraneous fixed points. Appl. Math. Comput. 283, 120–140 (2016)J.L. Hueso, E. Martınez, C. Teruel, Determination of multiple roots of nonlinear equations and applications. J. Math. Chem. 53, 880–892 (2015)L.O. Jay, A note on Q-order of convergence. BIT Numer. Math. 41, 422–429 (2001)S. Li, X. Liao, L. Cheng, A new fourth-order iterative method for finding multiple roots of nonlinear equations. Appl. Math. Comput. 215, 1288–1292 (2009)S.G. Li, L.Z. Cheng, B. Neta, Some fourth-order nonlinear solvers with closed formulae for multiple roots. Comput. Math. Appl. 59, 126–135 (2010)B. Liu, X. Zhou, A new family of fourth-order methods for multiple roots of nonlinear equations. Nonlinear Anal. Model. Control 18(2), 143–152 (2013)M. Shacham, Numerical solution of constrained nonlinear algebraic equations. Int. J. Numer. Method Eng. 23, 1455–1481 (1986)M. Sharifi, D.K.R. Babajee, F. Soleymani, Finding the solution of nonlinear equations by a class of optimal methods. Comput. Math. Appl. 63, 764–774 (2012)J.R. Sharma, R. Sharma, Modified Jarratt method for computing multiple roots. Appl. Math. Comput. 217, 878–881 (2010)F. Soleymani, D.K.R. Babajee, T. Lofti, On a numerical technique forfinding multiple zeros and its dynamic. J. Egypt. Math. Soc. 21, 346–353 (2013)F. Soleymani, D.K.R. Babajee, Computing multiple zeros using a class of quartically convergent methods. Alex. Eng. J. 52, 531–541 (2013)R. Thukral, A new family of fourth-order iterative methods for solving nonlinear equations with multiple roots. J. Numer. Math. Stoch. 6(1), 37–44 (2014)R. Thukral, Introduction to higher-order iterative methods for finding multiple roots of nonlinear equations. J. Math. Article ID 404635 (2013)X. Zhou, X. Chen, Y. Song, Constructing higher-order methods for obtaining the muliplte roots of nonlinear equations. J. Comput. Math. Appl. 235, 4199–4206 (2011)X. Zhou, X. Chen, Y. Song, Families of third and fourth order methods for multiple roots of nonlinear equations. Appl. Math. Comput. 219, 6030–6038 (2013

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-

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

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

CMMSE2017: On two classes of fourth- and seventh-order vectorial methods with stable behavior

[EN] A family of fourth-order iterative methods without memory, for solving nonlinear systems, and its seventh-order extension, are analyzed. By using complex dynamics tools, their stability and reliability are studied by means of the properties of the rational function obtained when they are applied on quadratic polynomials. The stability of their fixed points, in terms of the value of the parameter, its critical points and their associated parameter planes, etc. give us important information about which members of the family have good properties of stability and whether in any of them appear chaos in the iterative process. The conclusions obtained in this dynamical analysis are used in the numerical section, where an academical problem and also the chemical problem of predicting the diffusion and reaction in a porous catalyst pellet are solved.This research was partially supported by Ministerio de Economia y Competitividad MTM2014-52016-C02-2-P and Generalitat Valenciana PROMETEO/2016/089.Cordero Barbero, A.; Guasp, L.; Torregrosa Sánchez, JR. (2018). CMMSE2017: On two classes of fourth- and seventh-order vectorial methods with stable behavior. Journal of Mathematical Chemistry. 56(7):1902-1923. https://doi.org/10.1007/s10910-017-0814-0S19021923567S. Amat, S. Busquier, Advances in Iterative Methods for Nonlinear Equations (Springer, Berlin, 2016)S. Amat, S. Busquier, S. Plaza, Review of some iterative root-finding methods from a dynamical point of view. Sci. Ser. A Math. Sci. 10, 3–35 (2004)S. Amat, S. Busquier, S. Plaza, A construction of attracting periodic orbits for some classical third-order iterative methods. Comput. Appl. Math. 189, 22–33 (2006)I.K. Argyros, Á.A. Magreñn, On the convergence of an optimal fourth-order family of methods and its dynamics. Appl. Math. Comput. 252, 336–346 (2015)D.K.R. Babajee, A. Cordero, J.R. Torregrosa, Study of multipoint iterative methods through the Cayley quadratic test. Comput. Appl. Math. 291, 358–369 (2016). doi: 10.1016/J.CAM.2014.09.020P. Blanchard, The dynamics of Newton’s method. Proc. Symp. Appl. Math. 49, 139–154 (1994)F.I. Chicharro, A. Cordero, J.R. Torregrosa, Drawing dynamical and parameters planes of iterative families and methods. Sci. World J. 2013, Article ID 780153 (2013)C. Chun, M.Y. Lee, B. Neta, J. Džunić, On optimal fourth-order iterative methods free from second derivative and their dynamics. Appl. Math. Comput. 218, 6427–6438 (2012)A. Cordero, E. Gómez, J.R. Torregrosa, Efficient high-order iterative methods for solving nonlinear systems and their application on heat conduction problems. Complexity 2017, Article ID 6457532 (2017)A. Cordero, J.R. Torregrosa, Variants of Newton’s method using fifth-order quadrature formulas. Appl. Math. Comput. 190, 686–698 (2007)R.L. Devaney, An Introduction to Chaotic Dynamical Systems (Addison-Wesley Publishing Company, Reading, 1989)P.G. Logrado, J.D.M. Vianna, Partitioning technique procedure revisited: formalism and first application to atomic problems. Math. Chem. 22, 107–116 (1997)C.G. Jesudason, I. Numerical nonlinear analysis: differential methods and optimization applied to chemical reaction rate determination. Math. Chem. 49, 1384–1415 (2011)Á.A. Magreñán, Different anomalies in a Jarratt family of iterative root-finding methods. Appl. Math. Comput. 233, 29–38 (2014)M. Mahalakshmi, G. Hariharan, K. Kannan, The wavelet methods to linear and nonlinear reaction-diffusion model arising in mathematical chemistry. Math. Chem. 51(9), 2361–2385 (2013)K. Maleknejad, M. Alizadeh, An efficient numerical scheme for solving Hammerstein integral equation arisen in chemical phenomenon. Proc. Comput. Sci. 3, 361–364 (2011)B. Neta, C. Chun, M. Scott, Basins of attraction for optimal eighth-order methods to find simple roots of nonlinear equations. Appl. Math. Comput. 227, 567–592 (2014)M.S. Petković, B. Neta, L.D. Petković, J. Džunić, Multipoint Methods for Solving Nonlinear Equations (Elsevier, Amsterdam, 2013)R.C. Rach, J.S. Duan, A.M. Wazwaz, Solving coupled Lane–Emden boundary value problems in catalytic diffusion reactions by the Adomian decomposition method. Math. Chem. 52(1), 255–267 (2014)R. Singh, G. Nelakanti, J. Kumar, A new effcient technique for solving two-point boundary value problems for integro-differential equations. Math. Chem. 52, 2030–2051 (2014

Widening basins of attraction of optimal iterative methods

[EN] In this work, we analyze the dynamical behavior on quadratic polynomials of a class of derivative-free optimal parametric iterative methods, designed by Khattri and Steihaug. By using their parameter as an accelerator, we develop different methods with memory of orders three, six and twelve, without adding new functional evaluations. Then a dynamical approach is made, comparing each of the proposed methods with the original ones without memory, with the following empiric conclusion: Basins of attraction of iterative schemes with memory are wider and the behavior is more stable. This has been numerically checked by estimating the solution of a practical problem, as the friction factor of a pipe and also of other nonlinear academic problems.This research was supported by Islamic Azad University, Hamedan Branch, Ministerio de Economia y Competitividad MTM2014-52016-C02-2-P and Generalitat Valenciana PROMETEO/2016/089.Bakhtiari, P.; Cordero Barbero, A.; Lotfi, T.; Mahdiani, K.; Torregrosa Sánchez, JR. (2017). Widening basins of attraction of optimal iterative methods. Nonlinear Dynamics. 87(2):913-938. https://doi.org/10.1007/s11071-016-3089-2S913938872Amat, S., Busquier, S., Bermúdez, C., Plaza, S.: On two families of high order Newton type methods. Appl. Math. Lett. 25, 2209–2217 (2012)Amat, S., Busquier, S., Bermúdez, C., Magreñán, Á.A.: On the election of the damped parameter of a two-step relaxed Newton-type method. Nonlinear Dyn. 84(1), 9–18 (2016)Chun, C., Neta, B.: An analysis of a family of Maheshwari-based optimal eighth order methods. Appl. Math. 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Basins of attraction for various Steffensen-Type methods

The dynamical behavior of different Steffensen-type methods is analyzed. We check the chaotic behaviors alongside the convergence radii (understood as the wideness of the basin of attraction) needed by Steffensen-type methods, that is, derivative-free iteration functions, to converge to a root and compare the results using different numerical tests. We will conclude that the convergence radii (and the stability) of Steffensen-type methods are improved by increasing the convergence order. The computer programming package MATHEMATICA provides a powerful but easy environment for all aspects of numerics. This paper puts on show one of the application of this computer algebra system in finding fixed points of iteration functions.The authors are indebted to the referees for some interesting comments and suggestions. This research was supported by Ministerio de Ciencia y Tecnologia MTM2011-28636-C02-02.Cordero Barbero, A.; Soleymani, F.; Torregrosa Sánchez, JR.; Shateyi, S. (2014). 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A new fourth-order family for solving nonlinear problems and its dynamics

In this manuscript, a new parametric class of iterative methods for solving nonlinear systems of equations is proposed. Its fourth-order of convergence is proved and a dynamical analysis on low-degree polynomials is made in order to choose those elements of the family with better conditions of stability. These results are checked by solving the nonlinear system that arises from the partial differential equation of molecular interaction.This research was supported by Ministerio de Ciencia y Tecnologia MTM2011-28636-C02-{01, 02} and Universitat Politecnica de Valencia SP20120474.Cordero Barbero, A.; Feng, L.; Magrenan, A.; Torregrosa Sánchez, JR. (2015). A new fourth-order family for solving nonlinear problems and its dynamics. Journal of Mathematical Chemistry. 53(3):893-910. https://doi.org/10.1007/s10910-014-0464-4S893910533R.C. Rach, J.S. Duan, A.M. Wazwaz, Solving coupled Lane–Emden boundary value problems in catalytic diffusion reactions by the Adomian decomposition method. J. 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