Uni- and Bi-Parametric Two-Step Iterative Method with Memory for Solving Nonlinear Equations
Download PDF
Authors: N. KUMAR AND J. P. JAISWAL
DOI: 10.46793/KgJMat2604.513K
Abstract:
In this paper, we have suggested a two-step with memory method for solving nonlinear equations by transforming an extant optimal fourth-order without memory method. The acceleration of the order of convergence is attained by employing a single and two self-accelerating parameters. These parameters are estimated by a Hermite interpolating polynomial to enhance the convergence order of iterative method without memory. This order of convergence acceleration is achieved without the use of any additional functional evaluations, precisely the convergence order of the suggested two-step with memory method is reached from 4 to 5.70156. The rate of convergence is also verified by Herzberger’s matrix method. Finally, various examples are taken into consideration to support the theoretical outcomes.
Keywords:
Iterative method with memory, Hermite interpolating polynomial, R-order of convergence, nonlinear equation, root finding, computational efficiency.
References:
[1] A. Cordero, J. R. Torregrosa and M. P. Vassileva, A family of modified Ostrowski’s methods with optimal eighth order of convergence, Appl. Math. Lett. 24(12) (2011), 2082–2086. https://doi.org/10.1016/j.aml.2011.06.002
[2] A. Cordero, T. Lotfi, K. Mahdiani and J. R. Torregrosa, Two optimal general classes of iterative methods with eighth-order, Acta Appl. Math. 134(1) (2014), 61–74. https://doi.org/10.1007/s10440-014-9869-0
[3] J. P. Jaiswal, Some class of third-and fourth-order iterative methods for solving nonlinear equations, J. Appl. Math. 2014(1) (2014), 1–17. https://doi.org/10.1155/2014/817656
[4] T. Lotfi, F. Soleymani, M. Ghorbanzadeh and P. Assari, On the construction of some tri-parametric iterative methods with memory, Numer. Algorithms 70 (2015), 835–845. https://doi.org/10.1007/s11075-015-9976-7
[5] S. Kumar, V. Kanwar, S. K. Tomar and S. Singh, Geometrically constructed families of Newton’s method for unconstrained optimization and nonlinear equations, Int. J. Math. Math. Sci. 2011 (2011), 1–9. https://doi.org/10.1155/2011/972537
[6] J. F. Traub, Iterative Methods for the Solution of Equations, Prentice-Hall, Englewood Cliffs, New Jersey, 1964.
[7] M. Petković, B. Neta, L. Petković and J. Džunić, Multipoint Methods for Solving Nonlinear Equations, Academic Press, New York, 2012.
[8] J. Džunić and M. S. Petković, On generalized multipoint root-solvers with memory, J. Comput. Appl. Math. 236(11) (2012), 2909–2920. https://doi.org/10.1016/j.cam.2012.01.035
[9] J. Džunić, Modified Newton’s method with memory, Facta Univ. Ser. Math. Inform. 28(4) (2013), 429–441.
[10] J. Džunić and M. S. Petković, On generalized biparametric multipoint root finding methods with memory, J. Comput. Appl. Math. 255 (2014), 362–375. https://doi.org/10.1016/j.cam.2013.05.013
[11] T. Lotfi and P. Assari, New three-and four-parametric iterative with memory methods with efficiency index near 2, Appl. Math. Comput. 270 (2015), 1004–1010. https://doi.org/10.1016/j.amc.2015.08.017
[12] J. P. Jaiswal, Improved bi-accelerator derivative free with memory family for solving nonlinear equations, J. Appl. Anal. Comput. 6(1) (2016), 196–206. https://doi.org/10.11948/2016016
[13] J. Džunić, L. D. Petković and M. S. Petković, On an application of Herzberger’s matrix method to multipoint families of root-solvers, Filomat 32(11) (2018), 3815–3829. https://doi.org/10.2298/FIL1811815D
[14] S. K. Khattri, Quadrature based optimal iterative methods with applications in high-precision computing, Numer. Math. Theory Methods Appl. 5(4) (2012), 592–601. https://doi.org/10.4208/nmtma.2012.m1114
[15] J. Herzberger, Über matrixdarstellungen für iterationverfahren bei nichtlinearen gleichungen, Computing 12(3) (1974), 215–222.
[16] X. Wang and T. Zhang, A new family of Newton-type iterative methods with and without memory for solving nonlinear equations, Calcolo 51(1) (2014), 1–15. https://doi.org/10.1007/s10092-012-0072-2
[17] N. Choubey and J. P. Jaiswal, Two-and three-point with memory methods for solving nonlinear equations, Numer. Anal. Appl. 10(1) (2017), 74–89. https://doi.org/10.1134/S1995423917010086
[18] X. Wang and T. Zhang, High-order Newton-type iterative methods with memory for solving nonlinear equations, Math. Commun. 19(1) (2014), 91–109.
[19] N. Choubey, B. Panday and J. P. Jaiswal, Several two-point with memory iterative methods for solving nonlinear equations, Afr. Mat. 29 (2018), 435–449. https://doi.org/10.1007/s13370-018-0552-x
[20] S. Weerakoon and T. Fernando, A variant of Newton’s method with accelerated third-order convergence, Appl. Math. Lett. 13(8) (2000), 87–93. https://doi.org/10.1016/S0893-9659(00)00100-2
Login:
