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

New family of iterative methods with high order of convergence for solving nonlinear systems

In this paper we present and analyze a set of predictor-corrector iterative methods with increasing order of convergence, for solving systems of nonlinear equations. Our aim is to achieve high order of convergence with few Jacobian and/or functional evaluations. On the other hand, by applying the pseudocomposition technique on each proposed scheme we get to increase their order of convergence, obtaining new high-order and efficient methods. We use the classical efficiency index in order to compare the obtained schemes and make some numerical test.This research was supported by Ministerio de Ciencia y Tecnología MTM2011-28636-C02-02 and by FONDOCYT 2011-1-B1-33, República Dominicana.Cordero Barbero, A.; Torregrosa Sánchez, JR.; Penkova Vassileva, M. (2013). New family of iterative methods with high order of convergence for solving nonlinear systems. En Numerical Analysis and Its Applications. Springer Verlag. 222-230. https://doi.org/10.1007/978-3-642-41515-9_23S222230Cordero, A., Hueso, J.L., Martínez, E., Torregrosa, J.R.: A modified Newton-Jarratt’s composition. Numer. Algor. 55, 87–99 (2010)Cordero, A., Hueso, J.L., Martínez, E., Torregrosa, J.R.: Efficient high-order methods based on golden ratio for nonlinear systems. Applied Mathematics and Computation 217(9), 4548–4556 (2011)Cordero, A., Torregrosa, J.R.: Variants of Newton’s Method using fifth-order quadrature formulas. Applied Mathematics and Computation 190, 686–698 (2007)Cordero, A., Torregrosa, J.R.: On interpolation variants of Newton’s method for functions of several variables. Journal of Computational and Applied Mathematics 234, 34–43 (2010)Cordero, A., Torregrosa, J.R., Vassileva, M.P.: Pseudocomposition: a technique to design predictor-corrector methods for systms of nonlinear equtaions. Applied Mathematics and Computation 218(23), 11496–11504 (2012)Nikkhah-Bahrami, M., Oftadeh, R.: An effective iterative method for computing real and complex roots of systems of nonlinear equations. Applied Mathematics and Computation 215, 1813–1820 (2009)Ostrowski, A.M.: Solutions of equations and systems of equations. Academic Press, New York (1966)Shin, B.-C., Darvishi, M.T., Kim, C.-H.: A comparison of the Newton-Krylov method with high order Newton-like methods to solve nonlinear systems. Applied Mathematics and Computation 217, 3190–3198 (2010

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

An optimal three-point eighth-order iterative method without memory for solving nonlinear equations with its dynamics

We present a three-point iterative method without memory for solving nonlinear equations in one variable. The proposed method provides convergence order eight with four function evaluations per iteration. Hence, it possesses a very high computational efficiency and supports Kung and Traub's conjecture. The construction, the convergence analysis, and the numerical implementation of the method will be presented. Using several test problems, the proposed method will be compared with existing methods of convergence order eight concerning accuracy and basin of attraction. Furthermore, some measures are used to judge methods with respect to their performance in finding the basin of attraction.Comment: arXiv admin note: substantial text overlap with arXiv:1508.0174

Multipoint efficient iterative methods and the dynamics of Ostrowski's method

This is an Author's Accepted Manuscript of an article published in José L. Hueso, Eulalia Martínez & Carles Teruel (2019) Multipoint efficient iterative methods and the dynamics of Ostrowski's method, International Journal of Computer Mathematics, 96:9, 1687-1701, DOI: 10.1080/00207160.2015.1080354 in the International Journal of Computer Mathematics, SEP 2 2019 [copyright Taylor & Francis], available online at: http://www.tandfonline.com/10.1080/00207160.2015.1080354[EN] In this work, we introduce a modification into the technique, presented in A. Cordero, J.L. Hueso, E. Martinez, and J.R. Torregrosa [Increasing the convergence order of an iterative method for nonlinear systems, Appl. Math. Lett. 25 (2012), pp. 2369-2374], that increases by two units the convergence order of an iterative method. The main idea is to compose a given iterative method of order p with a modification of Newton's method that introduces just one evaluation of the function, obtaining a new method of order p+2, avoiding the need to compute more than one derivative, so we improve the efficiency index in the scalar case. This procedure can be repeated n times, with the same approximation to the derivative, obtaining new iterative methods of order p+2n. We perform different numerical tests that confirm the theoretical results. By applying this procedure to Newton's method one obtains the well known fourth order Ostrowski's method. We finally analyse its dynamical behaviour on second and third degree real polynomials.This research was supported by Ministerio de Economia y Competitividad under grant PGC2018-095896-B-C22 and by the project of Generalitat Valenciana Prometeo/2016/089.Hueso, JL.; Martínez Molada, E.; Teruel-Ferragud, C. (2019). Multipoint efficient iterative methods and the dynamics of Ostrowski's method. International Journal of Computer Mathematics. 96(9):1687-1701. https://doi.org/10.1080/00207160.2015.1080354S16871701969Amat, S., Busquier, S., & Plaza, S. (2010). Chaotic dynamics of a third-order Newton-type method. Journal of Mathematical Analysis and Applications, 366(1), 24-32. doi:10.1016/j.jmaa.2010.01.047Cordero, 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., 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.2009.04.015Cordero, A., Hueso, J. L., Martínez, E., & Torregrosa, J. R. (2012). Increasing the convergence order of an iterative method for nonlinear systems. Applied Mathematics Letters, 25(12), 2369-2374. doi:10.1016/j.aml.2012.07.005Jarratt, 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-