Generalized high-order classes for solving nonlinear systems and their applications

[EN] A generalized high-order class for approximating the solution of nonlinear systems of equations is introduced. First, from a fourth-order iterative family for solving nonlinear equations, we propose an extension to nonlinear systems of equations holding the same order of convergence but replacing the Jacobian by a divided difference in the weight functions for systems. The proposed GH family of methods is designed from this fourth-order family using both the composition and the weight functions technique. The resulting family has order of convergence 9. The performance of a particular iterative method of both families is analyzed for solving different test systems and also for the Fisher's problem, showing the good performance of the new methods.This research was partially supported by both Ministerio de Ciencia, Innovacion y Universidades and Generalitat Valenciana, under grants PGC2018-095896-B-C22 (MCIU/AEI/FEDER/UE) and PROMETEO/2016/089, respectively.Chicharro, FI.; Cordero Barbero, A.; Garrido-Saez, N.; Torregrosa Sánchez, JR. (2019). Generalized high-order classes for solving nonlinear systems and their applications. Mathematics. 7(12):1-14. https://doi.org/10.3390/math7121194S114712Petković, M. S., Neta, B., Petković, L. D., & Džunić, J. (2014). Multipoint methods for solving nonlinear equations: A survey. Applied Mathematics and Computation, 226, 635-660. doi:10.1016/j.amc.2013.10.072Kung, 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., Gómez, E., & Torregrosa, J. R. (2017). Efficient High-Order Iterative Methods for Solving Nonlinear Systems and Their Application on Heat Conduction Problems. Complexity, 2017, 1-11. doi:10.1155/2017/6457532Sharma, J. R., & Arora, H. (2016). Improved Newton-like methods for solving systems of nonlinear equations. SeMA Journal, 74(2), 147-163. doi:10.1007/s40324-016-0085-xAmiri, A., Cordero, A., Taghi Darvishi, M., & Torregrosa, J. R. (2018). Stability analysis of a parametric family of seventh-order iterative methods for solving nonlinear systems. Applied Mathematics and Computation, 323, 43-57. doi:10.1016/j.amc.2017.11.040Cordero, 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., Garrido, N., & Torregrosa, J. R. (2019). Wide stability in a new family of optimal fourth‐order iterative methods. Computational and Mathematical Methods, 1(2), e1023. doi:10.1002/cmm4.1023FISHER, R. A. (1937). THE WAVE OF ADVANCE OF ADVANTAGEOUS GENES. Annals of Eugenics, 7(4), 355-369. doi:10.1111/j.1469-1809.1937.tb02153.xSharma, 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/s11075-012-9585-7Soleymani, 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/s11590-013-0617-6Cordero, 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

Solving nonlinear problems by Ostrowski Chun type parametric families

In this paper, by using a generalization of Ostrowski' and Chun's methods two bi-parametric families of predictor-corrector iterative schemes, with order of convergence four for solving system of nonlinear equations, are presented. The predictor of the first family is Newton's method, and the predictor of the second one is Steffensen's scheme. One of them is extended to the multidimensional case. Some numerical tests are performed to compare proposed methods with existing ones and to confirm the theoretical results. We check the obtained results by solving the molecular interaction problem.This research was supported by Ministerio de Ciencia y Tecnologia MTM2011-28636-C02-02 and FONDOCYT, Republica Dominicana.Cordero Barbero, A.; Maimo, J.; Torregrosa Sánchez, JR.; Vassileva, M. (2015). Solving nonlinear problems by Ostrowski Chun type parametric families. Journal of Mathematical Chemistry. 53(1):430-449. https://doi.org/10.1007/s10910-014-0432-zS430449531M.S. Petkovic̀, B. Neta, L.D. Petkovic̀, J. Dz̆unic̀, Multipoint Methods for Solving Nonlinear Equations (Academic, New York, 2013)M. Mahalakshmi, G. Hariharan, K. Kannan, The wavelet methods to linear and nonlinear reaction–diffusion model arising in mathematical chemistry. J. Math. Chem. 51(9), 2361–2385 (2013)P.G. Logrado, J.D.M. Vianna, Partitioning technique procedure revisited: Formalism and first application to atomic problems. J. Math. Chem. 22, 107–116 (1997)C.G. Jesudason, I. Numerical nonlinear analysis: differential methods and optimization applied to chemical reaction rate determination. J. Math. Chem. 49, 1384–1415 (2011)K. Maleknejad, M. Alizadeh, An efficient numerical scheme for solving hammerstein integral equation arisen in chemical phenomenon. Procedia Comput. Sci. 3, 361–364 (2011)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. J. Math. Chem. 52, 255–267 (2014)J.F. Steffensen, Remarks on iteration. Skand. Aktuar Tidskr. 16, 64–72 (1933)J.M. Ortega, W.C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables (Academic, New York, 1970)H.T. Kung, J.F. Traub, Optimal order of one-point and multipoint iteration. J. ACM 21, 643–651 (1974)J.R. Sharma, R.K. Guha, R. Sharma, An efficient fourth order weighted-Newton method for systems of nonlinear equations. Numer. Algorithms 62, 307–323 (2013)J.R. Sharma, H. Arora, On efficient weighted-Newton methods for solving systems of nonlinear equations. Appl. Math. Comput. 222, 497–506 (2013)M. Abad, A. Cordero, J.R. Torregrosa, Fourth- and fifth-order methods for solving nonlinear systems of equations: an application to the Global positioning system. Abstr. Appl. Anal.(2013) Article ID:586708. doi: 10.1155/2013/586708F. Soleymani, T. Lotfi, P. Bakhtiari, A multi-step class of iterative methods for nonlinear systems. Optim. Lett. 8, 1001–1015 (2014)M.T. Darvishi, N. Darvishi, SOR-Steffensen-Newton method to solve systems of nonlinear equations. Appl. Math. 2(2), 21–27 (2012). doi: 10.5923/j.am.20120202.05F. Awawdeh, On new iterative method for solving systems of nonlinear equations. Numer. Algorithms 5(3), 395–409 (2010)D.K.R. Babajee, A. Cordero, F. Soleymani, J.R. Torregrosa, On a novel fourth-order algorithm for solving systems of nonlinear equations. J. Appl. Math. (2012) Article ID:165452. doi: 10.1155/2012/165452A. Cordero, J.R. Torregrosa, M.P. Vassileva, Pseudocomposition: a technique to design predictor–corrector methods for systems of nonlinear equations. Appl. Math. Comput. 218(23), 1496–1504 (2012)A. Cordero, J.R. Torregrosa, M.P. Vassileva, Increasing the order of convergence of iterative schemes for solving nonlinear systems. J. Comput. Appl. Math. 252, 86–94 (2013)A.M. Ostrowski, Solution of Equations and System of Equations (Academic, New York, 1966)C. Chun, Construction of Newton-like iterative methods for solving nonlinear equations. Numer. Math. 104, 297–315 (2006)R. King, A family of fourth order methods for nonlinear equations. SIAM J. Numer. Anal. 10, 876–879 (1973)A. Cordero, J.R. Torregrosa, Low-complexity root-finding iteration functions with no derivatives of any order of convergence. J. Comput. Appl. Math. (2014). doi: 10.1016/j.cam.2014.01.024A. Cordero, J.L. Hueso, E. Martínez, J.R. Torregrosa, A modified Newton Jarratts composition. Numer. Algorithms 55, 87–99 (2010)P. Jarratt, Some fourth order multipoint methods for solving equations. Math. Comput. 20, 434–437 (1966)A. Cordero, J.R. Torregrosa, Variants of Newtons method using fifth-order quadrature formulas. Appl. Math. Comput. 190, 686–698 (2007)Z. Liu, Q. Zheng, P. Zhao, A variant of Steffensens method of fourth-order convergence and its applications. Appl. Math. Comput. 216, 1978–1983 (2010)A. Cordero, J.R. Torregrosa, A class of Steffensen type methods with optimal order of convergence. Appl. Math. Comput. 217, 7653–7659 (2011)L.B. Rall, New York, Computational Solution of Nonlinear Operator Equations (Robert E. Krieger Publishing Company Inc, New York, 1969

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

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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Comput. 190, 686–698 (2007)H.T. Kung, J.F. Traub, Optimal order of one-point and multi-point iterations. J. Assoc. Comput. Math. 21, 643–651 (1974)A.M. Ostrowski, Solution of Equations and Systems of Equations (Prentice-Hall, Englewood Cliffs, 1964)P. Jarratt, Some fourth order multipoint iterative methods for solving equations. Math. Comput. 20, 434–437 (1966)R.F. King, A family of fourth order methods for nonlinear equations. SIAM J. Numer. Anal. 10, 876–879 (1973)A. Cordero, J.L. Hueso, E. Martínez, J.R. Torregrosa, A modified Newton Jarratt’s composition. Numer. Algorithms 55, 87–99 (2010)S. Amat, S. Busquier, Á.A. Magreñán, Reducing Chaos and Bifurcations in Newton-Type Methods. Abstract and Applied Analysis Volume 2013 (2013), Article ID 726701, 10 pages, doi: 10.1155/2013/726701S. 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)F. Chicharro, A. Cordero, J.M. Gutiérrez, J.R. 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The Scientific World J. 2013 (Article ID 780153) (2013)L.B. Rall, Computational Solution of Nonlinear Operator Equations (Robert E. Krieger Publishing Company Inc., New York, 1969)J.R. Sharma, R.K. Guna, R. Sharma, An efficient fourth order weighted-Newton method for systems of nonlinear equations. Numer. Algorithms 62, 307–323 (2013

Determination of multiple roots of nonlinear equations and applications

The final publication is available at Springer via https://dx.doi.org/10.1007/s10910-014-0460-8[EN] In this work we focus on the problem of approximating multiple roots of nonlinear equations. Multiple roots appear in some applications such as the compression of band-limited signals and the multipactor effect in electronic devices. We present a new family of iterative methods for multiple roots whose multiplicity is known. The methods are optimal in Kung-Traub's sense (Kung and Traub in J Assoc Comput Mach 21:643-651, [1]), because only three functional values per iteration are computed. By adding just one more function evaluation we make this family derivative free while preserving the convergence order. To check the theoretical results, we codify the new algorithms and apply them to different numerical examples.This research was supported by Ministerio de Ciencia y Tecnologia MTM2011-28636-C02-02 and by Vicerrectorado de Investigacion, Universitat Politecnica de Valencia PAID-SP-2012-0474.Hueso Pagoaga, JL.; Martínez Molada, E.; Teruel Ferragud, C. (2015). Determination of multiple roots of nonlinear equations and applications. Journal of Mathematical Chemistry. 53(3):880-892. https://doi.org/10.1007/s10910-014-0460-8S880892533H.T. Kung, J.F. Traub, Optimal order of one-point and multi-point iteration. J. Assoc. Comput. Mach. 21, 643–651 (1974)W. Bi, H. Ren, Q. Wu, Three-step iterative methods with eighth-order convergence for solving nonlinear equations. J. Comput. Appl. Math. 255, 105–112 (2009)W. Bi, Q. Wu, H. Ren, A new family of eighth-order iterative methods for solving nonlinear equations. Appl. Math. Comput. 214, 236–245 (2009)A. Cordero, J.L. Hueso, E. Martínez, J.R. Torregrosa, New modifications of Potra-Pták’s method with optimal fourth and eighth order of convergence. J. Comput. Appl. Math. 234, 2969–2976 (2010)E. Schröder, Über unendlich viele Algorithmen zur Auflösung der Gleichungen. Math. Ann. 2, 317–365 (1870)C. Chun, B. Neta, A third-order modification of Newtons method for multiple roots. Appl. Math. Comput. 211, 474–479 (2009)Y.I. Kim, S.D. Lee, A third-order variant of NewtonSecant method finding a multiple zero. J. Chungcheong Math. Soc. 23(4), 845–852 (2010)B. Neta, Extension of Murakamis high-order nonlinear solver to multiple roots. Int. J. Comput. Math. 8, 1023–1031 (2010)H. Ren, Q. Wu, W. Bi, A class of two-step Steffensen type methods with fourth-order convergence. Appl. Math. Comput. 209, 206–210 (2009)Q. Zheng, J. Wang, P. Zhao, L. Zhang, A Steffensen-like method and its higher-order variants. Appl. Math. Comput. 214, 10–16 (2009)S. Amat, S. Busquier, On a Steffensen’s type method and its behavior for semismooth equations. Appl. Math. Comput. 177, 819–823 (2006)X. Feng, Y. He, High order iterative methods without derivatives for solving nonlinear equations. Appl. Math. Comput. 186, 1617–1623 (2007)A. Cordero, J.R. Torregrosa, A class of Steffensen type methods with optimal order of convergence. Appl. Math. Comput. doi: 10.1016/j.amc.2011.02.067F. Marvasti, A. Jain, Zero crossings, bandwidth compression, and restoration of nonlinearly distorted band-limited signals. J. Opt. Soc. Am. A 3, 651–654 (1986)S. Anza, C. Vicente, B. Gimeno, V.E. Boria, J. Armendáriz, Long-term multipactor discharge in multicarrier systems. Physics of Plasmas 14(8), 082–112 (2007)J.L. Hueso, E. Martínez, C. Teruel, New families of iterative methods with fourth and sixth order of convergence and their dynamics, in Proceedings of the 13th International Conference on Computational and Mathematical Methods in Science and Engineering, CMMSE 2013, 24–27 June 2013A. Cordero, J.R. Torregrosa, Low-complexity root-finding iteration functions with no derivatives of any order of convergence. J. Comput. Appl. Math. doi: 10.10016/j.cam.2014.01.024 (2014)J.R. Sharma, R. Sharma, Modified Jarratt method for computing multiple roots. Appl. Math. Comput. 217, 878–881 (2010

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

A Parameterized multi-step Newton method for solving systems of nonlinear equations

We construct a novel multi-step iterative method for solving systems of nonlinear equations by introducing a parameter. to generalize the multi-step Newton method while keeping its order of convergence and computational cost. By an appropriate selection of theta, the new method can both have faster convergence and have larger radius of convergence. The new iterative method only requires one Jacobian inversion per iteration, and therefore, can be efficiently implemented using Krylov subspace methods. The new method can be used to solve nonlinear systems of partial differential equations, such as complex generalized Zakharov systems of partial differential equations, by transforming them into systems of nonlinear equations by discretizing approaches in both spatial and temporal independent variables such as, for instance, the Chebyshev pseudo-spectral discretizing method. Quite extensive tests show that the new method can have significantly faster convergence and significantly larger radius of convergence than the multi-step Newton method.Peer ReviewedPostprint (author's final draft