Accelerated iterative methods for finding solutions of nonlinear equations and their dynamical behavior

In this paper, we present a family of optimal, in the sense of Kung-Traub's conjecture, iterative methods for solving nonlinear equations with eighth-order convergence. Our methods are based on Chun's fourth-order method. We use the Ostrowski's efficiency index and several numerical tests in order to compare the new methods with other known eighth-order ones. We also extend this comparison to the dynamical study of the different methodsThis research was supported by Ministerio de Ciencia y Tecnologia MTM2011-28636-C02-02 and by the Center of Excellence for Mathematics, University of Shahrekord, Iran.Cordero Barbero, A.; Fardi, M.; Ghasemi, M.; Torregrosa Sánchez, JR. (2014). Accelerated iterative methods for finding solutions of nonlinear equations and their dynamical behavior. Calcolo. 51(1):17-30. https://doi.org/10.1007/s10092-012-0073-11730511Bi, W., Ren, H., Wu, Q.: Three-step iterative methods with eighth-order convergence for solving nonlinear equations. J. Comput. Appl. Math. 255, 105–112 (2009)Blanchard, P.: Complex analytic dynamics on the Riemann sphere. Bull. Am. Math. Soc. 11(1), 85–141 (1984)Chun, C.: Some variants of Kings fourth-order family of methods for nonlinear equations. Appl. Math. Comput. 190, 57–62 (2007)Cordero, A., Hueso, J.L., Martínez, E., Torregrosa, J.R.: New modifications of Potra-Pták’s method with optimal fourth and eighth order of convergence. J. Comput. Appl. Math. 234, 2969–2976 (2010)Cordero, A., Torregrosa, J.R.: Variants of Newton’s method using fifth-order quadrature formulas. Appl. Math. Comput. 190, 686–698 (2007)Cordero, A., Torregrosa, J.R., Vassileva, M.P.: A family of modified Ostrowski’s method with optimal eighth order of convergence. Appl. Math. Lett. 24(12), 2082–2086 (2011)Douady, A., Hubbard, J.H.: On the dynamics of polynomials-like mappings. Ann. Sci. Ec. Norm. Sup. (Paris) 18, 287–343 (1985)Kung, H.T., Traub, J.F.: Optimal order of one-point and multi-point iteration. J. Assoc. Comput. Mach. 21, 643–651 (1974)Liu, L., Wang, X.: Eighth-order methods with high efficiency index for solving nonlinear equations. Appl. Math. Comput. 215, 3449–3454 (2010)Ostrowski, A.M.: Solutions of equations and systems of equations. Academic Press, New York (1966)Sharma, J.R., Sharma, R.: A family of modified Ostrowski’s methods with accelerated eighth order convergence. Numer. Algoritms 54, 445–458 (2010)Soleymani, F., Karimi Banani, S., Khan, M., Sharifi, M.: Some modifications of King’s family with optimal eighth order of convergence. Math. Comput. Model. 55, 1373–1380 (2012)Thukral, R., Petkovic, M.S.: A family of three-point methods of optimal order for solving nonlinear equations. J. Comput. Appl. Math. 233, 2278–2284 (2010

Modified Potra-Pták multi-step schemes with accelerated order of convergence for solving sistems of nonlinear equations

[EN] In this study, an iterative scheme of sixth order of convergence for solving systems of nonlinear equations is presented. The scheme is composed of three steps, of which the first two steps are that of third order Potra-Ptak method and last is weighted-Newton step. Furthermore, we generalize our work to derive a family of multi-step iterative methods with order of convergence 3r + 6, r = 0, 1, 2, .... The sixth order method is the special case of this multi-step scheme for r = 0. The family gives a four-step ninth order method for r = 1. As much higher order methods are not used in practice, so we study sixth and ninth order methods in detail. Numerical examples are included to confirm theoretical results and to compare the methods with some existing ones. Different numerical tests, containing academical functions and systems resulting from the discretization of boundary problems, are introduced to show the efficiency and reliability of the proposed methods.This research was partially supported by Ministerio de Economia y Competitividad under grants MTM2014-52016-C2-2-P and Generalitat Valenciana PROMETEO/2016/089.Arora, H.; Torregrosa Sánchez, JR.; Cordero Barbero, A. (2019). Modified Potra-Pták multi-step schemes with accelerated order of convergence for solving sistems of nonlinear equations. Mathematical and Computational Applications (Online). 24(1):1-15. https://doi.org/10.3390/mca24010003S115241Homeier, 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.2003.12.041Darvishi, M. T., & Barati, A. (2007). A fourth-order method from quadrature formulae to solve systems of nonlinear equations. Applied Mathematics and Computation, 188(1), 257-261. doi:10.1016/j.amc.2006.09.115Cordero, 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.2010.11.006Grau-Sánchez, M., Grau, À., & Noguera, M. (2011). On the computational efficiency index and some iterative methods for solving systems of nonlinear equations. Journal of Computational and Applied Mathematics, 236(6), 1259-1266. doi:10.1016/j.cam.2011.08.008Grau-Sánchez, M., Grau, À., & Noguera, M. (2011). Ostrowski type methods for solving systems of nonlinear equations. Applied Mathematics and Computation, 218(6), 2377-2385. doi:10.1016/j.amc.2011.08.011Grau-Sánchez, M., Noguera, M., & Amat, S. (2013). On the approximation of derivatives using divided difference operators preserving the local convergence order of iterative methods. Journal of Computational and Applied Mathematics, 237(1), 363-372. doi:10.1016/j.cam.2012.06.005Sharma, J. R., & Arora, H. (2013). On efficient weighted-Newton methods for solving systems of nonlinear equations. Applied Mathematics and Computation, 222, 497-506. doi:10.1016/j.amc.2013.07.066Cordero, 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.2007.01.062Chicharro, 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

An efficient two-parametric family with memory for nonlinear equations

A new two-parametric family of derivative-free iterative methods for solving nonlinear equations is presented. First, a new biparametric family without memory of optimal order four is proposed. The improvement of the convergence rate of this family is obtained by using two self-accelerating parameters. These varying parameters are calculated in each iterative step employing only information from the current and the previous iteration. The corresponding R-order is 7 and the efficiency index 7(1/3) = 1.913. Numerical examples and comparison with some existing derivative-free optimal eighth-order schemes are included to confirm the theoretical results. In addition, the dynamical behavior of the designed method is analyzed and shows the stability of the scheme.The second author wishes to thank the Islamic Azad University, Hamedan Branch, where the paper was written as a part of the research plan, for financial support.Cordero Barbero, A.; Lotfi, T.; Bakhtiari, P.; Torregrosa Sánchez, JR. (2015). An efficient two-parametric family with memory for nonlinear equations. Numerical Algorithms. 68(2):323-335. doi:10.1007/s11075-014-9846-8S323335682Kung, H.T., Traub, J.F.: Optimal order of one-point and multi-point iteration. J. Assoc. Comput. Math. 21, 643–651 (1974)Cordero, A., Hueso, J.L., Martínez, E., Torregrosa, J.R.: A new technique to obtain derivative-free optimal iterative methods for solving nonlinear equation. J. Comput. Appl. Math. 252, 95–102 (2013)Cordero, A., Torregrosa, J.R., Vassileva, M.P.: Pseudocomposition: a technique to design predictor-corrector methods for systems of nonlinear equations. Appl. Math. Comput. 218, 11496–11508 (2012)Džunić, J.: On efficient two-parameter methods for solving nonlinear equations. Numer. Algorithms. 63(3), 549–569 (2013)Džunić, J., Petković, M.S.: On generalized multipoint root-solvers with memory. J. Comput. Appl. Math. 236, 2909–2920 (2012)Petković, M.S., Neta, B., Petković, L.D., Džunić, J. (ed.).: Multipoint methods for solving nonlinear equations. Elsevier (2013)Sharma, J.R., Sharma, R.: A new family of modified Ostrowski’s methods with accelerated eighth order convergence. Numer. Algorithms 54, 445–458 (2010)Soleymani, F., Shateyi, S.: Two optimal eighth-order derivative-free classes of iterative methods. Abstr. Appl. Anal. 2012(318165), 14 (2012). doi: 10.1155/2012/318165Soleymani, F., Sharma, R., Li, X., Tohidi, E.: An optimized derivative-free form of the Potra-Pták methods. Math. Comput. Model. 56, 97–104 (2012)Thukral, R.: Eighth-order iterative methods without derivatives for solving nonlinear equations. ISRN Appl. Math. 2011(693787), 12 (2011). doi: 10.5402/2011/693787Traub, J.F.: Iterative Methods for the Solution of Equations. Prentice Hall, New York (1964)Wang, X., Džunić, J., Zhang, T.: On an efficient family of derivative free three-point methods for solving nonlinear equations. Appl. Math. Comput. 219, 1749–1760 (2012)Zheng, Q., Li, J., Huang, F.: An optimal Steffensen-type family for solving nonlinear equations. Appl. Math. Comput. 217, 9592–9597 (2011)Ortega, J.M., Rheinboldt, W.G. (ed.).: Iterative Solutions of Nonlinear Equations in Several Variables, Ed. Academic Press, New York (1970)Jay, I.O.: A note on Q-order of convergence. BIT Numer. Math. 41, 422–429 (2001)Blanchard, P.: Complex Analytic Dynamics on the Riemann Sphere. Bull. AMS 11(1), 85–141 (1984)Chicharro, F., Cordero, A., Torregrosa, J.R.: Drawing dynamical and parameters planes of iterative families and methods. arXiv: 1307.6705 [math.NA

On generalization based on Bi et al. Iterative methods with eighth-order convergence for solving nonlinear equations

The primary goal of this work is to provide a general optimal three-step class of iterative methods based on the schemes designed by Bi et al. (2009). Accordingly, it requires four functional evaluations per iteration with eighth-order convergence. Consequently, it satisfies Kung and Traub's conjecture relevant to construction optimal methods without memory. Moreover, some concrete methods of this class are shown and implemented numerically, showing their applicability and efficiency.The authors thank the anonymous referees for their valuable comments and for the suggestions to improve the readability of the paper. This research was supported by Islamic Azad University, Hamedan Branch, and Ministerio de Ciencia y Tecnologia MTM2011-28636-C02-02.Lotfi, T.; Cordero Barbero, A.; Torregrosa Sánchez, JR.; Abadi, MA.; Zadeh, MM. (2014). On generalization based on Bi et al. Iterative methods with eighth-order convergence for solving nonlinear equations. The Scientific World Journal. 2014. https://doi.org/10.1155/2014/272949S2014Behl, R., Kanwar, V., & Sharma, K. K. (2012). Another Simple Way of Deriving Several Iterative Functions to Solve Nonlinear Equations. Journal of Applied Mathematics, 2012, 1-22. doi:10.1155/2012/294086Fernández-Torres, G., & Vásquez-Aquino, J. (2013). Three New Optimal Fourth-Order Iterative Methods to Solve Nonlinear Equations. Advances in Numerical Analysis, 2013, 1-8. doi:10.1155/2013/957496Kang, S. M., Rafiq, A., & Kwun, Y. C. (2013). A New Second-Order Iteration Method for Solving Nonlinear Equations. Abstract and Applied Analysis, 2013, 1-4. doi:10.1155/2013/487062Soleimani, F., Soleymani, F., & Shateyi, S. (2013). Some Iterative Methods Free from Derivatives and Their Basins of Attraction for Nonlinear Equations. Discrete Dynamics in Nature and Society, 2013, 1-10. doi:10.1155/2013/301718Bi, W., Ren, H., & Wu, Q. (2009). Three-step iterative methods with eighth-order convergence for solving nonlinear equations. Journal of Computational and Applied Mathematics, 225(1), 105-112. doi:10.1016/j.cam.2008.07.004Bi, W., Wu, Q., & Ren, H. (2009). A new family of eighth-order iterative methods for solving nonlinear equations. Applied Mathematics and Computation, 214(1), 236-245. doi:10.1016/j.amc.2009.03.077Kung, 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., Hueso, J. L., Martínez, E., & Torregrosa, J. R. (2010). New modifications of Potra–Pták’s method with optimal fourth and eighth orders of convergence. Journal of Computational and Applied Mathematics, 234(10), 2969-2976. doi:10.1016/j.cam.2010.04.009Cordero, A., & Torregrosa, J. R. (2011). A class of Steffensen type methods with optimal order of convergence. Applied Mathematics and Computation, 217(19), 7653-7659. doi:10.1016/j.amc.2011.02.067Cordero, A., Torregrosa, J. R., & Vassileva, M. P. (2011). Three-step iterative methods with optimal eighth-order convergence. Journal of Computational and Applied Mathematics, 235(10), 3189-3194. doi:10.1016/j.cam.2011.01.004Džunić, J., & Petković, M. S. (2012). A Family of Three-Point Methods of Ostrowski’s Type for Solving Nonlinear Equations. Journal of Applied Mathematics, 2012, 1-9. doi:10.1155/2012/425867Džunić, J., Petković, M. S., & Petković, L. D. (2011). A family of optimal three-point methods for solving nonlinear equations using two parametric functions. Applied Mathematics and Computation, 217(19), 7612-7619. doi:10.1016/j.amc.2011.02.055Heydari, M., Hosseini, S. M., & Loghmani, G. B. (2011). On two new families of iterative methods for solving nonlinear equations with optimal order. Applicable Analysis and Discrete Mathematics, 5(1), 93-109. doi:10.2298/aadm110228012hGeum, Y. H., & Kim, Y. I. (2010). A multi-parameter family of three-step eighth-order iterative methods locating a simple root. Applied Mathematics and Computation, 215(9), 3375-3382. doi:10.1016/j.amc.2009.10.030Geum, Y. H., & Kim, Y. I. (2011). A uniparametric family of three-step eighth-order multipoint iterative methods for simple roots. Applied Mathematics Letters, 24(6), 929-935. doi:10.1016/j.aml.2011.01.002Geum, Y. H., & Kim, Y. I. (2011). A biparametric family of eighth-order methods with their third-step weighting function decomposed into a one-variable linear fraction and a two-variable generic function. Computers & Mathematics with Applications, 61(3), 708-714. doi:10.1016/j.camwa.2010.12.020Kou, J., Wang, X., & Li, Y. (2010). Some eighth-order root-finding three-step methods. Communications in Nonlinear Science and Numerical Simulation, 15(3), 536-544. doi:10.1016/j.cnsns.2009.04.013Liu, L., & Wang, X. (2010). Eighth-order methods with high efficiency index for solving nonlinear equations. Applied Mathematics and Computation, 215(9), 3449-3454. doi:10.1016/j.amc.2009.10.040Wang, X., & Liu, L. (2010). New eighth-order iterative methods for solving nonlinear equations. Journal of Computational and Applied Mathematics, 234(5), 1611-1620. doi:10.1016/j.cam.2010.03.002Wang, X., & Liu, L. (2010). Modified Ostrowski’s method with eighth-order convergence and high efficiency index. Applied Mathematics Letters, 23(5), 549-554. doi:10.1016/j.aml.2010.01.009Sharma, J. R., & Sharma, R. (2009). A new family of modified Ostrowski’s methods with accelerated eighth order convergence. Numerical Algorithms, 54(4), 445-458. doi:10.1007/s11075-009-9345-5Soleymani, F. (2011). Novel Computational Iterative Methods with Optimal Order for Nonlinear Equations. Advances in Numerical Analysis, 2011, 1-10. doi:10.1155/2011/270903Soleymani, F., Sharifi, M., & Somayeh Mousavi, B. (2011). An Improvement of Ostrowski’s and King’s Techniques with Optimal Convergence Order Eight. Journal of Optimization Theory and Applications, 153(1), 225-236. doi:10.1007/s10957-011-9929-9Soleymani, F., Karimi Vanani, S., & Afghani, A. (2011). A General Three-Step Class of Optimal Iterations for Nonlinear Equations. Mathematical Problems in Engineering, 2011, 1-10. doi:10.1155/2011/469512Soleymani, F., Vanani, S. K., Khan, M., & Sharifi, M. (2012). Some modifications of King’s family with optimal eighth order of convergence. Mathematical and Computer Modelling, 55(3-4), 1373-1380. doi:10.1016/j.mcm.2011.10.016Soleymani, F., Karimi Vanani, S., & Jamali Paghaleh, M. (2012). A Class of Three-Step Derivative-Free Root Solvers with Optimal Convergence Order. Journal of Applied Mathematics, 2012, 1-15. doi:10.1155/2012/568740Thukral, R. (2010). A new eighth-order iterative method for solving nonlinear equations. Applied Mathematics and Computation, 217(1), 222-229. doi:10.1016/j.amc.2010.05.048Thukral, R. (2011). Eighth-Order Iterative Methods without Derivatives for Solving Nonlinear Equations. ISRN Applied Mathematics, 2011, 1-12. doi:10.5402/2011/693787Thukral, R. (2012). New Eighth-Order Derivative-Free Methods for Solving Nonlinear Equations. International Journal of Mathematics and Mathematical Sciences, 2012, 1-12. doi:10.1155/2012/493456Thukral, R., & Petković, M. S. (2010). A family of three-point methods of optimal order for solving nonlinear equations. Journal of Computational and Applied Mathematics, 233(9), 2278-2284. doi:10.1016/j.cam.2009.10.012Wang, J. (2013). He’s Max-Min Approach for Coupled Cubic Nonlinear Equations Arising in Packaging System. Mathematical Problems in Engineering, 2013, 1-4. doi:10.1155/2013/382509Babajee, 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:10.1155/2012/165452Montazeri, H., Soleymani, F., Shateyi, S., & Motsa, S. S. (2012). On a New Method for Computing the Numerical Solution of Systems of Nonlinear Equations. Journal of Applied Mathematics, 2012, 1-15. doi:10.1155/2012/751975Soleymani, F. (2012). A Rapid Numerical Algorithm to Compute Matrix Inversion. International Journal of Mathematics and Mathematical Sciences, 2012, 1-11. doi:10.1155/2012/134653Soleymani, F. (2013). A new method for solving ill-conditioned linear systems. Opuscula Mathematica, 33(2), 337. doi:10.7494/opmath.2013.33.2.337Thukral, R. (2012). Further Development of Jarratt Method for Solving Nonlinear Equations. Advances in Numerical Analysis, 2012, 1-9. doi:10.1155/2012/493707Cordero, 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

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. Comput. 253, 294–307 (2015)Babajee, D.K.R., Cordero, A., Soleymani, F., Torregrosa, J.R.: On improved three-step schemes with high efficiency index and their dynamics. Numer. Algorithms 65(1), 153–169 (2014)Argyros, I.K., Magreñán, Á.A.: On the convergence of an optimal fourth-order family of methods and its dynamics. Appl. Math. Comput. 252, 336–346 (2015)Petković, M., Neta, B., Petković, L., Džunić, J.: Multipoint Methods for Solving Nonlinear Equations. Academic Press, London (2013)Ostrowski, A.M.: Solution of Equations and System of Equations. Prentice-Hall, Englewood Cliffs, NJ (1964)Kung, H.T., Traub, J.F.: Optimal order of one-point and multipoint iteration. J. ACM 21, 643–651 (1974)Khattri, S.K., Steihaug, T.: Algorithm for forming derivative-free optimal methods. Numer. Algorithms 65(4), 809–824 (2014)Traub, J.F.: Iterative Methods for the Solution of Equations. Prentice Hall, New York (1964)Cordero, A., Soleymani, F., Torregrosa, J.R., Shateyi, S.: Basins of Attraction for Various Steffensen-Type Methods. J. Appl. Math. 2014, 1–17 (2014)Devaney, R.L.: The Mandelbrot Set, the Farey Tree and the Fibonacci sequence. Am. Math. Mon. 106(4), 289–302 (1999)McMullen, C.: Families of rational maps and iterative root-finding algorithms. Ann. Math. 125(3), 467–493 (1987)Chicharro, F., Cordero, A., Gutiérrez, J.M., Torregrosa, J.R.: Complex dynamics of derivative-free methods for nonlinear equations. Appl. Math. Comput. 219, 70237035 (2013)Magreñán, Á.A.: Different anomalies in a Jarratt family of iterative root-finding methods. Appl. Math. Comput. 233, 29–38 (2014)Neta, B., Chun, C., Scott, M.: Basins of attraction for optimal eighth order methods to find simple roots of nonlinear equations. Appl. Math. Comput. 227, 567–592 (2014)Lotfi, T., Magreñán, Á.A., Mahdiani, K., Rainer, J.J.: A variant of Steffensen–King’s type family with accelerated sixth-order convergence and high efficiency index: dynamic study and approach. Appl. Math. Comput. 252, 347–353 (2015)Chicharro, F.I., Cordero, A., Torregrosa, J.R.: Drawing dynamical and parameters planes of iterative families and methods. Sci. World J. 2013, 1–11 (2013)Cordero, A., Lotfi, T., Torregrosa, J.R., Assari, P., Mahdiani, K.: Some new bi-accelerator two-point methods for solving nonlinear equations. Comput. Appl. Math. 35(1), 251–267 (2016)Cordero, A., Lotfi, T., Bakhtiari, P., Torregrosa, J.R.: An efficient two-parametric family with memory for nonlinear equations. Numer. Algorithms 68(2), 323–335 (2015)Lotfi, T., Mahdiani, K., Bakhtiari, P., Soleymani, F.: Constructing two-step iterative methods with and without memory. Comput. Math. Math. Phys. 55(2), 183–193 (2015)Cordero, A., Maimó, J.G., Torregrosa, J.R., Vassileva, M.P.: Solving nonlinear problems by Ostrowski–Chun type parametric families. J. Math. Chem. 53, 430–449 (2015)Abad, M., Cordero, A., Torregrosa, J.R.: A family of seventh-order schemes for solving nonlinear systems. Bull. Math. Soc. Sci. Math. Roum. Tome 57(105), 133–145 (2014)Weerakoon, S., Fernando, T.G.I.: A variant of Newton’s method with accelerated third-order convergence. Appl. Math. Lett. 13, 87–93 (2000)White, F.: Fluid Mechanics. McGraw-Hill, Boston (2003)Zheng, Q., Li, J., Huang, F.: An optimal Steffensen-type family for solving nonlinear equations. Appl. Math. Comput. 217, 9592–9597 (2011)Soleymani, F., Babajee, D.K.R., Shateyi, S., Motsa, S.S.: Construction of optimal derivative-free techniques without memory. J. Appl. Math. (2012). doi: 10.1155/2012/49702

Design of high-order iterative methods for nonlinear systems by using weight-function procedure

We present two classes of iterative methods whose orders of convergence are four and five, respectively, for solving systems of nonlinear equations, by using the technique of weight functions in each step. Moreover, we show an extension to higher order, adding only one functional evaluation of the vectorial nonlinear function. We perform numerical tests to compare the proposed methods with other schemes in the literature and test their effectiveness on specific nonlinear problems. Moreover, some real basins of attraction are analyzed in order to check the relation between the order of convergence and the set of convergent starting points.Artidiello Moreno, SDJ.; Cordero Barbero, A.; Torregrosa Sánchez, JR.; Vassileva, M. (2015). Design of high-order iterative methods for nonlinear systems by using weight-function procedure. Abstract and Applied Analysis. 2015(289029):1-12. doi:10.1155/2015/289029S1122015289029He, Y., & Ding, C. H. Q. (2001). The Journal of Supercomputing, 18(3), 259-277. doi:10.1023/a:1008153532043Gerlach, J. (1994). Accelerated Convergence in Newton’s Method. SIAM Review, 36(2), 272-276. doi:10.1137/1036057Cordero, A., & Torregrosa, J. R. (2006). Variants of Newton’s method for functions of several variables. Applied Mathematics and Computation, 183(1), 199-208. doi:10.1016/j.amc.2006.05.062Cordero, 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.2007.01.062Cordero, A., & Torregrosa, J. R. (2010). On interpolation variants of Newton’s method for functions of several variables. Journal of Computational and Applied Mathematics, 234(1), 34-43. doi:10.1016/j.cam.2009.12.002Frontini, 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)00178-4Cordero, A., Torregrosa, J. R., & Vassileva, M. P. (2012). Pseudocomposition: A technique to design predictor–corrector methods for systems of nonlinear equations. Applied Mathematics and Computation, 218(23), 11496-11504. doi:10.1016/j.amc.2012.04.081Jarratt, P. (1966). Some fourth order multipoint iterative methods for solving equations. Mathematics of Computation, 20(95), 434-434. doi:10.1090/s0025-5718-66-99924-8Sharma, 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/s11075-012-9585-7Sharma, 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., 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., & 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

Preliminary orbit determination of artificial satellites: a vectorial sixth-order approach

A modified classical method for preliminary orbit determination is presented. In our proposal, the spread of the observations is considerably wider than in the original method, as well as the order of convergence of the iterative scheme involved. The numerical approach is made by using matricial weight functions, which will lead us to a class of iterative methods with a sixth local order of convergence. This is a process widely used in the design of iterative methods for solving nonlinear scalar equations, but rarely employed in vectorial cases. The numerical tests confirm the theoretical results, and the analysis of the dynamics of the problem shows the stability of the proposed schemes.The authors thank the anonymous referees for their valuable comments and suggestions. This research was supported by Ministerio de Ciencia y Tecnologia MTM2011-28636-C02-02.Andreu Estellés, C.; Cambil Teba, N.; Cordero Barbero, A.; Torregrosa Sánchez, JR. (2013). Preliminary orbit determination of artificial satellites: a vectorial sixth-order approach. Abstract and Applied Analysis. 2013. https://doi.org/10.1155/2013/960582S2013Fidkowski, K. J., Oliver, T. A., Lu, J., & Darmofal, D. L. (2005). p-Multigrid solution of high-order discontinuous Galerkin discretizations of the compressible Navier–Stokes equations. Journal of Computational Physics, 207(1), 92-113. doi:10.1016/j.jcp.2005.01.005Bruns, D. D., & Bailey, J. E. (1977). Nonlinear feedback control for operating a nonisothermal CSTR near an unstable steady state. Chemical Engineering Science, 32(3), 257-264. doi:10.1016/0009-2509(77)80203-0He, Y., & Ding, C. H. Q. (2001). The Journal of Supercomputing, 18(3), 259-277. doi:10.1023/a:1008153532043Revol, N., & Rouillier, F. (2005). Motivations for an Arbitrary Precision Interval Arithmetic and the MPFI Library. Reliable Computing, 11(4), 275-290. doi:10.1007/s11155-005-6891-yBabajee, D. K. R., Dauhoo, M. Z., Darvishi, M. T., & Barati, A. (2008). A note on the local convergence of iterative methods based on Adomian decomposition method and 3-node quadrature rule. Applied Mathematics and Computation, 200(1), 452-458. doi:10.1016/j.amc.2007.11.009Darvishi, 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.2006.08.080Darvishi, M. T., & Barati, A. (2007). Super cubic iterative methods to solve systems of nonlinear equations. Applied Mathematics and Computation, 188(2), 1678-1685. doi:10.1016/j.amc.2006.11.022Cordero, A., Martínez, E., & Torregrosa, J. R. (2009). Iterative methods of order four and five for systems of nonlinear equations. Journal of Computational and Applied Mathematics, 231(2), 541-551. doi:10.1016/j.cam.2009.04.015Babajee, D. K. R., Dauhoo, M. Z., Darvishi, M. T., Karami, A., & Barati, A. (2010). Analysis of two Chebyshev-like third order methods free from second derivatives for solving systems of nonlinear equations. Journal of Computational and Applied Mathematics, 233(8), 2002-2012. doi:10.1016/j.cam.2009.09.035Soleymani, 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/s11590-013-0617-6Awawdeh, F. (2009). On new iterative method for solving systems of nonlinear equations. Numerical Algorithms, 54(3), 395-409. doi:10.1007/s11075-009-9342-8Babajee, 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:10.1155/2012/165452Cordero, A., Torregrosa, J. R., & Vassileva, M. P. (2012). Pseudocomposition: A technique to design predictor–corrector methods for systems of nonlinear equations. Applied Mathematics and Computation, 218(23), 11496-11504. doi:10.1016/j.amc.2012.04.081Cordero, A., Torregrosa, J. R., & Vassileva, M. P. (2013). Increasing the order of convergence of iterative schemes for solving nonlinear systems. Journal of Computational and Applied Mathematics, 252, 86-94. doi:10.1016/j.cam.2012.11.024Soleymani, F., & Stanimirović, P. S. (2013). A Higher Order Iterative Method for Computing the Drazin Inverse. The Scientific World Journal, 2013, 1-11. doi:10.1155/2013/708647Soleymani, F., Stanimirović, P. S., & Ullah, M. Z. (2013). An accelerated iterative method for computing weighted Moore–Penrose inverse. Applied Mathematics and Computation, 222, 365-371. doi:10.1016/j.amc.2013.07.039Sharma, 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/s11075-012-9585-7Sharma, J. R., & Arora, H. (2013). On efficient weighted-Newton methods for solving systems of nonlinear equations. Applied Mathematics and Computation, 222, 497-506. doi:10.1016/j.amc.2013.07.066Abad, M. F., Cordero, A., & Torregrosa, J. R. (2013). Fourth- and Fifth-Order Methods for Solving Nonlinear Systems of Equations: An Application to the Global Positioning System. Abstract and Applied Analysis, 2013, 1-10. doi:10.1155/2013/586708Cordero, 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-zJarratt, P. (1966). Some fourth order multipoint iterative methods for solving equations. Mathematics of Computation, 20(95), 434-434. doi:10.1090/s0025-5718-66-99924-8Cordero, 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.2007.01.062Chicharro, 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

Semilocal convergence of a family of iterative methods in Banach spaces

[EN] In this work, we prove a third and fourth convergence order result for a family of iterative methods for solving nonlinear systems in Banach spaces. We analyze the semilocal convergence by using recurrence relations, giving the existence and uniqueness theorem that establishes the R-order of the method and the priori error bounds. Finally, we apply the methods to two examples in order to illustrate the presented theory.This work has been supported by Ministerio de Ciencia e Innovaci´on MTM2011-28636-C02-02 and by Vicerrectorado de Investigaci´on. Universitat Polit`ecnica de Val`encia PAID-SP-2012-0498Hueso Pagoaga, JL.; Martínez Molada, E. (2014). Semilocal convergence of a family of iterative methods in Banach spaces. Numerical Algorithms. 67(2):365-384. https://doi.org/10.1007/s11075-013-9795-7S365384672Traub, J.F.: Iterative Methods for the Solution of Nonlinear Equations. Prentice Hall, New York (1964)Kantorovich, L.V.: On the newton method for functional equations. Doklady Akademii Nauk SSSR 59, 1237–1240 (1948)Candela, V., Marquina, A.: Recurrence relations for rational cubic methods, I: The Halley method. Computing 44, 169–184 (1990)Candela, V., Marquina, A.: Recurrence relations for rational cubic methods, II: The Chebyshev method. Computing 45, 355–367 (1990)Hernández, M.A.: Reduced recurrence relations for the Chebyshev method. J. Optim. Theory Appl. 98, 385–397 (1998)Gutiérrez, J.M., Hernández, M.A.: Recurrence relations for super-Halley method. J. Comput. Math. Appl. 7, 1–8 (1998)Ezquerro, J.A., Hernández, M.A.: Recurrence relations for Chebyshev-like methods. Appl. Math. Optim. 41, 227–236 (2000)Ezquerro, J.A., Hernández, M.A.: New iterations of R-order four with reduced computational cost. BIT Numer. Math. 49, 325–342 (2009)Argyros, I., K., Ezquerro, J.A., Gutiérrez, J.M., Hernández, M.A., Hilout, S.: On the semilocal convergence of efficient Chebyshev Secant-type methods. J. Comput. Appl. Math. 235–10, 3195–3206 (2011)Argyros, I.K., Hilout, S.: Weaker conditions for the convergence of Newtons method. J. Complex. 28(3), 364–387 (2012)Wang, X., Gu, C., Kou, J.: Semilocal convergence of a multipoint fourth-order super-Halley method in Banach spaces. Numer. Algoritm. 54, 497–516 (2011)Kou, J., Li, Y., Wang, X.: A variant of super Halley method with accelerated fourth-order convergence. Appl. Math. Comput. 186, 535–539 (2007)Zheng, L., Gu, C.: Recurrence relations for semilocal convergence of a fifth-order method in Banach spaces. Numer. Algoritm. 59, 623–638 (2012)Amat, S., Hernández, M.A., Romero, N.: A modified Chebyshevs iterative method with at least sixth order of convergence. Appl. Math. Comput. 206, 164–174 (2008)Wang, X., Kou, J., Gu, C.: Semilocal convergence of a sixth-order Jarratt method in Banach spaces. Numer. Algoritm. 57, 441–456 (2011)Hernández, M.A.: The newton method for operators with hlder continuous first derivative. J. Optim. Appl. 109, 631–648 (2001)Ye, X., Li, C.: Convergence of the family of the deformed Euler-Halley iterations under the Hlder condition of the second derivative. J. Comput. Appl. Math. 194, 294–308 (2006)Zhao, Y., Wu, Q.: Newton-Kantorovich theorem for a family of modified Halleys method under Hlder continuity conditions in Banach spaces. Appl. Math. Comput. 202, 243–251 (2008)Argyros, I.K.: Improved generalized differentiability conditions for Newton-like methods. J. Complex. 26, 316–333 (2010)Hueso, J.L., Martínez. E., Torregrosa, J.R.: Third and fourth order iterative methods free from second derivative for nonlinear systems. Appl. Math. Comput. 211, 190–197 (2009)Taylor, A.Y., Lay, D.: Introduction to Functional Analysis, 2nd edn.New York, Wiley (1980)Jarrat, P.: Some fourth order multipoint iterative methods for solving equations. Math. Comput. 20, 434–437 (1966)Cordero, A., Torregrosa, J.R.: Variants of Newtons method using fifth-order quadrature formulas. Appl. Math. Comput. 190, 686–698 (2007