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

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

Some Classes of third and Fourth-order iterative methods for solving nonlinear equations

The object of the present work is to present the new classes of third-order and fourth-order iterative methods for solving nonlinear equations. Our third-order method includes methods of Weerakoon \cite{Weerakoon}, Homeier \cite{Homeier2}, Chun \cite{Chun} e.t.c. as particular cases. After that we make this third-order method to fourth-order (optimal) by using a single weight function rather than using two different weight functions in \cite{Soleymani}. Finally some examples are given to illustrate the performance of the our method by comparing with new existing third and fourth-order methods.Comment: arXiv admin note: substantial text overlap with arXiv:1307.733

Memory in a new variant of King's family for solving nonlinear systems

[EN] In the recent literature, very few high-order Jacobian-free methods with memory for solving nonlinear systems appear. In this paper, we introduce a new variant of King's family with order four to solve nonlinear systems along with its convergence analysis. The proposed family requires two divided difference operators and to compute only one inverse of a matrix per iteration. Furthermore, we have extended the proposed scheme up to the sixth-order of convergence with two additional functional evaluations. In addition, these schemes are further extended to methods with memory. We illustrate their applicability by performing numerical experiments on a wide variety of practical problems, even big-sized. It is observed that these methods produce approximations of greater accuracy and are more efficient in practice, compared with the existing methods.This research was supported by PGC2018-095896-B-C22 (MCIU/AEI/FEDER, UE).Kansal, M.; Cordero Barbero, A.; Bhalla, S.; Torregrosa Sánchez, JR. (2020). Memory in a new variant of King's family for solving nonlinear systems. Mathematics. 8(8):1-15. https://doi.org/10.3390/math8081251S11588Cordero, 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.2010.11.006Babajee, 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:10.1155/2012/165452Zheng, Q., Zhao, P., & Huang, F. (2011). A family of fourth-order Steffensen-type methods with the applications on solving nonlinear ODEs. Applied Mathematics and Computation, 217(21), 8196-8203. doi:10.1016/j.amc.2011.01.095Sharma, J., & Arora, H. (2013). An efficient derivative free iterative method for solving systems of nonlinear equations. Applicable Analysis and Discrete Mathematics, 7(2), 390-403. doi:10.2298/aadm130725016sSharma, J. R., Arora, H., & Petković, M. S. (2014). An efficient derivative free family of fourth order methods for solving systems of nonlinear equations. Applied Mathematics and Computation, 235, 383-393. doi:10.1016/j.amc.2014.02.103Wang, X., Zhang, T., Qian, W., & Teng, M. (2015). Seventh-order derivative-free iterative method for solving nonlinear systems. Numerical Algorithms, 70(3), 545-558. doi:10.1007/s11075-015-9960-2Chicharro, F. I., Cordero, A., Garrido, N., & Torregrosa, J. R. (2020). On the improvement of the order of convergence of iterative methods for solving nonlinear systems by means of memory. Applied Mathematics Letters, 104, 106277. doi:10.1016/j.aml.2020.106277Petković, M. S., & Sharma, J. R. (2015). On some efficient derivative-free iterative methods with memory for solving systems of nonlinear equations. Numerical Algorithms, 71(2), 457-474. doi:10.1007/s11075-015-0003-9Narang, M., Bhatia, S., Alshomrani, A. S., & Kanwar, V. (2019). General efficient class of Steffensen type methods with memory for solving systems of nonlinear equations. Journal of Computational and Applied Mathematics, 352, 23-39. doi:10.1016/j.cam.2018.10.048King, R. F. (1973). A Family of Fourth Order Methods for Nonlinear Equations. SIAM Journal on Numerical Analysis, 10(5), 876-879. doi:10.1137/0710072Hermite, M. C., & Borchardt, M. (1878). Sur la formule d’interpolation de Lagrange. Journal für die reine und angewandte Mathematik (Crelles Journal), 1878(84), 70-79. doi:10.1515/crelle-1878-18788405Petkovic, M., Dzunic, J., & Petkovic, L. (2011). A family of two-point methods with memory for solving nonlinear equations. Applicable Analysis and Discrete Mathematics, 5(2), 298-317. doi:10.2298/aadm110905021pCordero, 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.2007.01.062Awawdeh, F. (2009). On new iterative method for solving systems of nonlinear equations. Numerical Algorithms, 54(3), 395-409. doi:10.1007/s11075-009-9342-8Noor, M. A., Waseem, M., & Noor, K. I. (2015). New iterative technique for solving a system of nonlinear equations. Applied Mathematics and Computation, 271, 446-466. doi:10.1016/j.amc.2015.08.125Pramanik, S. (2002). Kinematic Synthesis of a Six-Member Mechanism for Automotive Steering. Journal of Mechanical Design, 124(4), 642-645. doi:10.1115/1.150337

A family of iterative methods with accelerated eighth-order convergence

We propose a family of eighth-order iterative methods without memory for solving nonlinear equations. The new iterative methods are developed by using weight function method and using an approximation for the last derivative, which reduces the required number of functional evaluations per step. Their efficiency indices are all found to be 1.682. Several examples allow us to compare our algorithms with known ones and confirm the theoretical results.The authors would like to thank the referee for the valuable comments and for the suggestions to improve the readability of the paper. This research was supported by Ministerio de Ciencia y Tecnologia MTM2011-28636-C02-02 and by Vicerrectorado de Investigacion, Universitat Politecnica de Valencia PAID-06-2010-2285.Cordero Barbero, A.; Fardi, M.; Ghasemi, M.; Torregrosa Sánchez, JR. (2012). A family of iterative methods with accelerated eighth-order convergence. Journal of Applied Mathematics. 2012. https://doi.org/10.1155/2012/2825612012Jarratt, P. (1966). Some fourth order multipoint iterative methods for solving equations. Mathematics of Computation, 20(95), 434-434. doi:10.1090/s0025-5718-66-99924-8Homeier, H. H. H. (2005). On Newton-type methods with cubic convergence. Journal of Computational and Applied Mathematics, 176(2), 425-432. doi:10.1016/j.cam.2004.07.027Kung, H. T., & Traub, J. F. (1974). Optimal Order of One-Point and Multipoint Iteration. Journal of the ACM, 21(4), 643-651. doi:10.1145/321850.321860King, R. F. (1973). A Family of Fourth Order Methods for Nonlinear Equations. SIAM Journal on Numerical Analysis, 10(5), 876-879. doi:10.1137/0710072Chun, C. (2007). Some variants of King’s fourth-order family of methods for nonlinear equations. Applied Mathematics and Computation, 190(1), 57-62. doi:10.1016/j.amc.2007.01.006Chun, C. (2008). Some fourth-order iterative methods for solving nonlinear equations. Applied Mathematics and Computation, 195(2), 454-459. doi:10.1016/j.amc.2007.04.105Chun, C., Lee, M. Y., Neta, B., & Džunić, J. (2012). On optimal fourth-order iterative methods free from second derivative and their dynamics. Applied Mathematics and Computation, 218(11), 6427-6438. doi:10.1016/j.amc.2011.12.013Maheshwari, A. K. (2009). A fourth order iterative method for solving nonlinear equations. Applied Mathematics and Computation, 211(2), 383-391. doi:10.1016/j.amc.2009.01.047Neta, B. (1981). On a family of multipoint methods for non-linear equations. International Journal of Computer Mathematics, 9(4), 353-361. doi:10.1080/00207168108803257Bi, 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.2008.07.004Cordero, 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.2011.01.004Liu, 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.2009.10.040Cordero, 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

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

Multidimensional Homeier's generalized class and its application to planar 1D Bratu problem

[EN] In this paper, a parametric family of iterative methods for solving nonlinear systems, including Homeier’s scheme is presented, proving its third-order of convergence. The numerical section is devoted to obtain an estimation of the solution of the classical Bratu problem by transforming it in a nonlinear system by using finite differences, and solving it with different elements of the iterative family.This research was supported by Ministerio de Economía y Competitividad MTM2014-52016-C02-02.Cordero Barbero, A.; Franqués García, AM.; Torregrosa Sánchez, JR. (2015). Multidimensional Homeier's generalized class and its application to planar 1D Bratu problem. Journal of the Spanish Society of Applied Mathematics. 70(1):1-10. https://doi.org/10.1007/s40324-015-0037-xS110701Abad, M. F., Cordero, A., Torregrosa, J. R.: Fourth-and fifth-order for solving nonlinear systems of equations: an application to the global positioning system, Abstr. Appl. Anal. (2013) (Article ID 586708)Andreu, C., Cambil, N., Cordero, A., Torregrosa, J.R.: Preliminary orbit determination of artificial satellites: a vectorial sixth-order approach, Abstr. Appl. Anal. (2013) (Article ID 960582)Awawdeh, F.: On new iterative method for solving systems of nonlinear equations. Numer. Algorithms 54, 395–409 (2010)Boyd, J.P.: One-point pseudospectral collocation for the one-dimensional Bratu equation. Appl. Math. Comput. 217, 5553–5565 (2011)Bratu, G.: Sur les equation integrals non-lineaires. Bull. Math. Soc. France 42, 113–142 (1914)Buckmire, R.: Applications of Mickens finite differences to several related boundary value problems. In: Mickens, R.E. (ed.) Advances in the Applications of Nonstandard Finite Difference Schemes, pp. 47–87. World Scientific Publishing, Singapore (2005)Cordero, A., Hueso, J.L., Martínez, E., Torregrosa, J.R.: A modified Newton-Jarratt’s composition. Numer. Algorithms 55, 87–99 (2010)Gelfand, I.M.: Some problems in the theory of quasi-linear equations. Trans. Am. Math. Soc. Ser. 2, 295–381 (1963)Homeier, H.H.H.: On Newton-tyoe methods with cubic convergence. J. Comput. Appl. Math. 176, 425–432 (2005)Jacobsen, J., Schmitt, K.: The Liouville-Bratu-Gelfand problem for radial operators. J. Differ. Equ. 184, 283–298 (2002)Jalilian, R.: Non-polynomial spline method for solving Bratu’s problem. Comput. Phys. Comm. 181, 1868–1872 (2010)Kanwar, V., Kumar, S., Behl, R.: Several new families of Jarratts method for solving systems of nonlinear equations. Appl. Appl. Math. 8(2), 701–716 (2013)Mohsen, A.: A simple solution of the Bratu problem. Comput. Math. with Appl. 67, 26–33 (2014)Petković, M., Neta, B., Petković, L., Džunić, J.: Multipoint Methods for Solving Nonlinear Equations. Academic Press, Amsterdam (2013)Sharma, J.R., Guna, R.K., Sharma, R.: An efficient fourth order weighted-Newton method for systems of nonlinear equations. Numer. Algorithms 62, 307–323 (2013)Sharma, J.R., Arora, H.: On efficient weighted-Newton methods for solving systems of nonlinear equations. Appl. Math. Comput. 222, 497–506 (2013)Traub, J.F.: Iterative Methods for the Solution of Equations. Chelsea Publishing Company, New York (1982)Wan, Y.Q., Guo, Q., Pan, N.: Thermo-electro-hydrodynamic model for electrospinning process. Int. J. Nonlinear Sci. Numer. Simul. 5, 5–8 (2004

A family of parametric schemes of arbitrary even order for solving nonlinear models

[EN] Many problems related to gas dynamics, heat transfer or chemical reactions are modeled by means of partial differential equations that usually are solved by using approximation techniques. When they are transformed in nonlinear systems of equations via a discretization process, this system is big-sized and high-order iterative methods are specially useful. In this paper, we construct a new family of parametric iterative methods with arbitrary even order, based on the extension of Ostrowski' and Chun's methods for solving nonlinear systems. Some elements of the proposed class are known methods meanwhile others are new schemes with good properties. Some numerical tests confirm the theoretical results and allow us to compare the numerical results obtained by applying new methods and known ones on academical examples. In addition, we apply one of our methods for approximating the solution of a heat conduction problem described by a parabolic partial differential equation.This research was partially supported by Ministerio de Economia y Competitividad MTM2014-52016-C02-2-P and FONDOCYT 2014-1C1-088 Republica Dominicana.Cordero Barbero, A.; Torregrosa Sánchez, JR.; Vassileva, MP. (2017). A family of parametric schemes of arbitrary even order for solving nonlinear models. Journal of Mathematical Chemistry. 55(7):1443-1460. https://doi.org/10.1007/s10910-016-0723-7S14431460557R. Escobedo, L.L. Bonilla, Numerical methods for quantum drift-diffusion equation in semiconductor phisics. Math. Chem. 40(1), 3–13 (2006)S.J. Preece, J. Villingham, A.C. King, Chemical clock reactions: the effect of precursor consumtion. Math. Chem. 26, 47–73 (1999)H. Montazeri, F. Soleymani, S. Shateyi, S.S. Motsa, On a new method for computing the numerical solution of systems of nonlinear equations. J. Appl. Math. 2012 ID. 751975, 15 pages (2012)J.L. Hueso, E. Martínez, C. Teruel, Convergence, effiency and dinamimics of new fourth and sixth order families of iterative methods for nonlinear systems. J. Comput. Appl. Math. 275, 412–420 (2015)J.R. Sharma, H. Arora, Efficient Jarratt-like methods for solving systems of nonlinear equations. Calcolo 51, 193–210 (2014)X. Wang, T. Zhang, W. Qian, M. Teng, Seventh-order derivative-free iterative method for solving nonlinear systems. Numer. Algor. 70, 545–558 (2015)J.R. Sharma, H. Arora, On efficient weighted-Newton methods for solving systems of nonlinear equations. Appl. Math. Comput. 222, 497–506 (2013)A. Cordero, J.G. Maimó, J.R. Torregrosa, M.P. Vassileva, Solving nonlinear problems by Ostrowski-Chun type parametric families. J. Math. Chem. 53, 430–449 (2015)A.M. Ostrowski, Solution of equations and systems of equations (Prentice-Hall, Englewood Cliffs, New York, 1964)C. Chun, Construction of Newton-like iterative methods for solving nonlinear equations. Numer. Math. 104, 297–315 (2006)A. Cordero, J.L. Hueso, E. Martínez, J.R. Torregrosa, A modified Newton-Jarratt’s composition. Numer. Algor. 55, 87–99 (2010)J.M. Ortega, W.C. Rheinboldt, Iterative solution of nonlinear equations in several variables (Academic, New York, 1970)C. Hermite, Sur la formule dinterpolation de Lagrange. Reine Angew. Math. 84, 70–79 (1878)A. Cordero, J.R. Torregrosa, Variants of Newton’s method using fifth-order quadrature formulas. Appl. Math. Comput. 190, 686–698 (2007

Stability anomalies of some jacobian-free iterative methods of high order of convergence

[EN] In this manuscript, we design two classes of parametric iterative schemes to solve nonlinear problems that do not need to evaluate Jacobian matrices and need to solve three linear systems per iteration with the same divided difference operator as the coefficient matrix. The stability performance of the classes is analyzed on a quadratic polynomial system, and it is shown that for many values of the parameter, only convergence to the roots of the problem exists. Finally, we check the performance of these methods on some test problems to confirm the theoretical results.This research was partially supported by Ministerio de Economia y Competitividad under grants PGC2018-095896-B-C22, Generalitat Valenciana PROMETEO/2016/089 and FONDOCYT 027-2018 and 029-2018, Dominican Republic.Cordero Barbero, A.; García-Maimo, J.; Torregrosa Sánchez, JR.; Vassileva, MP. (2019). Stability anomalies of some jacobian-free iterative methods of high order of convergence. Axioms. 8(2):1-15. https://doi.org/10.3390/axioms8020051S11582Frontini, 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)00178-4Homeier, 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.2003.12.041Aslam Noor, M., & Waseem, M. (2009). Some iterative methods for solving a system of nonlinear equations. Computers & Mathematics with Applications, 57(1), 101-106. doi:10.1016/j.camwa.2008.10.067Xiao, X., & Yin, H. (2015). A new class of methods with higher order of convergence for solving systems of nonlinear equations. Applied Mathematics and Computation, 264, 300-309. doi:10.1016/j.amc.2015.04.094Cordero, 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.2007.01.062Darvishi, 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.2006.08.080Sharma, 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/s11075-012-9585-7Narang, M., Bhatia, S., & Kanwar, V. (2016). New two-parameter Chebyshev–Halley-like family of fourth and sixth-order methods for systems of nonlinear equations. Applied Mathematics and Computation, 275, 394-403. doi:10.1016/j.amc.2015.11.063Behl, R., Sarría, Í., González, R., & Magreñán, Á. A. (2019). Highly efficient family of iterative methods for solving nonlinear models. Journal of Computational and Applied Mathematics, 346, 110-132. doi:10.1016/j.cam.2018.06.042Amorós, C., Argyros, I., González, R., Magreñán, Á., Orcos, L., & Sarría, Í. (2019). Study of a High Order Family: Local Convergence and Dynamics. Mathematics, 7(3), 225. doi:10.3390/math7030225Argyros, I., & González, D. (2015). Local Convergence for an Improved Jarratt-type Method in Banach Space. International Journal of Interactive Multimedia and Artificial Intelligence, 3(4), 20. doi:10.9781/ijimai.2015.344Sharma, 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., Gutiérrez, J. M., Magreñán, Á. A., & Torregrosa, J. R. (2016). Stability analysis of a parametric family of iterative methods for solving nonlinear models. Applied Mathematics and Computation, 285, 26-40. doi:10.1016/j.amc.2016.03.021Cordero, A., Soleymani, F., & Torregrosa, J. R. (2014). Dynamical analysis of iterative methods for nonlinear systems or how to deal with the dimension? Applied Mathematics and Computation, 244, 398-412. doi:10.1016/j.amc.2014.07.010Cordero, 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-zArgyros, I., & George, S. (2015). Ball Convergence for Steffensen-type Fourth-order Methods. International Journal of Interactive Multimedia and Artificial Intelligence, 3(4), 37. doi:10.9781/ijimai.2015.347Chicharro, 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

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