# Two Novel With and Without Memory Multi-Point Iterative Methods for Solving Non-Linear Equations

## DOI:

https://doi.org/10.26713/cma.v15i1.2432## Keywords:

Iterative method, Non-linear equation, With memory scheme, Hermite interpolation polynomial, Gauss quadrature approach, Efficiency index## Abstract

In this work, two new iterative methods are proposed for finding simple roots of non-linear equations. The new methods are the modifications of the existing work proposed by Rafiullah and Jabeen (New eighth and sixteenth order iterative methods to solve nonlinear equations, *International Journal of Applied and Computational Mathematics* 3 (2017), 2467 – 2476). The first method obtained is of fifth-order two-step with memory method while the second scheme is three-point eight-order optimal without memory method. Firstly, the Hermite interpolation polynomial is employed to eliminate the first derivative. To maintain order, the conversion to a memory scheme was accomplished by introducing self-accelerated parameters, all without requiring any new function evaluations. Additionally, the Gauss quadrature approach was incorporated for the first derivative, aiming to attain optimal eighth-order convergence. In particular, the efficiency index is increased from 1.4953 to 1.7099 and 1.5157 to 1.6817 for fifth- and eighth-orders respectively. Some real-life application based problems, such as Kepler’s equation, an ocean engineering problem, Planck’s radiation law, a blood rheology model, and the charge between two parallel plates were presented to validate and demonstrate the superiority of the proposed scheme. Another benefit of the proposed scheme is on the restriction of the Newton’s method that \(f '(v)\neq 0\) can be eliminated close to the root.

### Downloads

## References

S. Abdullah, N. Choubey and S. Dara, An efficient two-point iterative method with memory for solving non-linear equations and its dynamics, Journal of Applied Mathematics and Computing 70 (2024), 285 – 315, DOI: 10.1007/s12190-023-01953-w.

G. Alefeld and J. Herzberger, Introduction to Interval Computation, 1st edition, Academic Press Inc., UK (1984).

E. A. Avallone and T. Baumeister III, Marks’ Standard Handbook for Mechanical Engineers, 10th edition, McGraw-Hill, New York (1987).

F. Awawdeh, On new iterative method for solving systems of nonlinear equations, Numerical Algorithms 54(3) (2010), 395 – 409, DOI: 10.1007/s11075-009-9342-8.

S. U. Chaudhary, S.-Y. Shin, J.-K.Won and K.-H. Cho, Multiscale modeling of tumorigenesis induced by mitochondrial incapacitation in cell death, IEEE Transactions on Biomedical Engineering 58(10) (2011), 3028 – 3032, DOI: 10.1109/TBME.2011.2159713.

J. Chen, Some new iterative methods with three-order convergence, Applied Mathematics and Computation 181(2) (2006), 1519 – 1522, DOI: 10.1016/j.amc.2006.02.037.

N. Choubey and J. Jaiswal, Two-and three-point with memory methods for solving nonlinear equations, Numerical Analysis and Applications 10(1) (2017), 74 – 89, DOI: 10.1134/S1995423917010086.

N. Choubey, J. Jaiswal and A. Choubey, Family of multipoint with memory iterative schemes for solving nonlinear equations, International Journal of Applied and Computational Mathematics 8 (2022), Article number 83, DOI: 10.1007/s40819-022-01283-8.

N. Choubey and J. P. Jaiswal, An improved optimal eighth-order iterative scheme with its dynamical behaviour, International Journal of Computing Science and Mathematics 7(4) (2016), 361 – 370, DOI: 10.1504/IJCSM.2016.078685.

N. Choubey, B. Panday and J. P. Jaiswal, Several two-point with memory iterative methods for solving nonlinear equations, Afrika Matematika 29(3) (2018), 435 – 449, DOI: 10.1007/s13370-018-0552-x.

C. Chun and M. Y. Lee, A new optimal eighth-order family of iterative methods for the solution of nonlinear equations, Applied Mathematics and Computation 223 (2013), 506 – 519, DOI: 10.1016/j.amc.2013.08.033.

A. Cordero, J. L. Hueso, E. Martínez and J. R. Torregrosa, A modified Newton-Jarratt’s composition, Numerical Algorithms 55(1) (2010), 87 – 99, DOI: 10.1007/s11075-009-9359-z.

H.-F. Ding, Y.-X. Zhang, S.-F. Wang and X.-Y. Yang, A note on some quadrature based three-step iterative methods for non-linear equations, Applied Mathematics and Computation 215(1) (2009), 53 – 57, DOI: 10.1016/j.amc.2009.04.036.

J. Džuni´c, M. Petkovi´c and L. Petkovi´c, Three-point methods with and without memory for solving nonlinear equations, Applied Mathematics and Computation 218(9) (2012), 4917 – 4927, DOI: 10.1016/j.amc.2011.10.057.

L. Fang, L. Sun and G. He, An efficient Newton-type method with fifth-order convergence for solving nonlinear equations, Computational & Applied Mathematics 27(3) (2008), 269 – 274, DOI: 10.1590/s1807-03022008000300003.

C. Grosan and A. Abraham, A new approach for solving nonlinear equations systems, IEEE Transactions on Systems, Man, and Cybernetics-Part A: Systems and Humans 38(3) (2008), 698 – 714, DOI: 10.1109/TSMCA.2008.918599.

M. Hafiz, An efficient three-step tenth-order method without second order derivative, Palestine Journal of Mathematics 3(2) (2014), 198 – 203, URL: https://pjm.ppu.edu/sites/default/files/papers/7_1.pdf.

Y. Ham and C. Chun, A fifth-order iterative method for solving nonlinear equations, Applied Mathematics and Computation 194(1) (2007), 287 – 290, DOI: 10.1016/j.amc.2007.04.005.

S. Jaturonglumlert and T. Kiatsiriroat, Heat and mass transfer in combined convective and far-infrared drying of fruit leather, Journal of Food Engineering 100(2) (2010), 254 – 260, DOI: 10.1016/j.jfoodeng.2010.04.007.

M. Kansal, V. Kanwar and S. Bhatia, Efficient derivative-free variants of Hansen-Patrick’s family with memory for solving nonlinear equations, Numerical Algorithms 73 (2016), 1017 – 1036, DOI: 10.1007/s11075-016-0127-6.

E. D. Kaplan and C. J. Hegarty, Understanding GPS/GNSS: Principles and Applications, 3rd edition, Artech House, UK, xxi + 993 pages (2017).

J. Kiusalaas, Numerical Methods in Engineering with MATLAB®, 2nd edition, Cambridge University Press, Cambridge, xii + 431 pages (2009), DOI: 10.1017/cbo9780511812200.

J. Kou and Y. Li, The improvements of Chebyshev-Halley methods with fifth-order convergence, Applied Mathematics and Computation 188(1) (2007), 143 – 147, DOI: 10.1016/j.amc.2006.09.097.

J. Kou, Y. Li and X. Wang, A family of fifth-order iterations composed of Newton and third-order methods, Applied Mathematics and Computation 186(2) (2007), 1258 – 1262, DOI: 10.1016/j.amc.2006.07.150.

S. Kumar, V. Kanwar, S. K. Tomar and S. Singh, Geometrically constructed families of Newton’s method for unconstrained optimization and nonlinear equations, International Journal of Mathematics and Mathematical Sciences 2011 (2011), Article ID 972537, 9 pages, DOI: 10.1155/2011/972537.

H. Kung and J. F. Traub, Optimal order of one-point and multipoint iteration, Journal of the ACM 21(4) (1974), 643 – 651, DOI: 10.1145/321850.321860.

Y. Lin, L. Bao and X. Jia, Convergence analysis of a variant of the newton method for solving nonlinear equations, Computers & Mathematics with Applications 59(6) (2010), 2121 – 2127, DOI: 10.1016/j.camwa.2009.12.017.

L. Liu and X. Wang, Eighth-order methods with high efficiency index for solving nonlinear equations, Applied Mathematics and Computation 215(9) (2010), 3449 – 3454, DOI: 10.1016/j.amc.2009.10.040.

W. Nazeer, A. R. Nizami, M. Tanveer and I. Sarfaraz, A ninth-order iterative method for non-linear equations along with polynomiography, Journal of Prime Research in Mathematics 13(1) (2017), 41 – 55, URL: https://jprm.sms.edu.pk/media/pdf/jprm/volume_13/05.pdf.

M. A. Noor, W. A. Khan and A. Hussain, A new modified Halley method without second derivatives for nonlinear equation, Applied Mathematics and Computation 189(2) (2007), 1268 – 1273, DOI: 10.1016/j.amc.2006.12.011.

M. A. Noor and K. I. Noor, Fifth-order iterative methods for solving nonlinear equations, Applied Mathematics and Computation 188(1) (2007), 406 – 410, DOI: 10.1016/j.amc.2006.10.007.

R. Nourgaliev, P. Greene, B. Weston, R. Barney, A. Anderson, S. Khairallah and J.-P. Delplanque, High-order fully implicit solver for all-speed fluid dynamics, Shock Waves 29(5) (2019), 651 – 689, DOI: 10.1007/s00193-018-0871-8.

J. M. Ortega, Numerical Analysis: A Second Course, Classics in Applied Mathematics, SIAM, xv + 201 pages (1990), DOI: 10.1137/1.9781611971323.

J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Classics in Applied Mathematics, SIAM, xxvi + 572 pages (2000), DOI: 10.1137/1.9780898719468.

S. Panday, A. Sharma and G. Thangkhenpau, Optimal fourth and eighth-order iterative methods for non-linear equations, Journal of Applied Mathematics and Computing 69(1) (2023), 953 – 971, DOI: 10.1007/s12190-022-01775-2.

S. Parimala, K. Madhu and J. Jayaraman, A new class of optimal eighth order method with two weight functions for solving nonlinear equation, Journal Nonlinear Analysis and Application 2018(2) (2018), 83 – 94.

M. S. Petkovi´c, B. Neta, L. D. Petkovi´c and J. Džuni´c, Multipoint methods for solving nonlinear equations: A survey, Applied Mathematics and Computation 226 (2014), 635 – 660, DOI: 10.1016/j.amc.2013.10.072.

S. Qureshi, H. Ramos and A. K. Soomro, A new nonlinear ninth-order root-finding method with error analysis and basins of attraction, Mathematics 9(16) (2021), 1996, DOI: 10.3390/math9161996.

M. Rafiullah, Multi-step Higher Order Iterative Methods for Solving Nonlinear Equations, Master’s Thesis, Higher Education Commission of Pakistan, Spring (2013).

M. Rafiullah and D. Jabeen, New eighth and sixteenth order iterative methods to solve nonlinear equations, International Journal of Applied and Computational Mathematics 3 (2017), 2467 – 2476, DOI: 10.1007/s40819-016-0245-9.

R. Rouzbar and S. Eyi, Reacting flow analysis of a cavity-based scramjet combustor using a Jacobian-free Newton-Krylov method, The Aeronautical Journal 122(1258) (2018), 1884 – 1915, DOI: 10.1017/aer.2018.110.

M. Salimi, T. Lotfi, S. Sharifi and S. Siegmund, Optimal Newton-Secant like methods without memory for solving nonlinear equations with its dynamics, International Journal of Computer Mathematics 94(9) (2017), 1759 – 1777, DOI: 10.1080/00207160.2016.1227800.

M. Shams, N. Rafiq, N. Kausar, N. A. Mir and A. Alalyani, Computer oriented numerical scheme for solving engineering problems., Computer Systems Science and Engineering 42(2) (2022), 689 – 701, DOI: 10.32604/csse.2022.022269.

H. Sharma, M. Kansal and R. Behl, An efficient two-step iterative family adaptive with memory for solving nonlinear equations and their applications, Mathematical and Computational Applications 27(6) (2022), 97, DOI: 10.3390/mca27060097.

J. R. Sharma and H. Arora, An efficient family of weighted-Newton methods with optimal eighth order convergence, Applied Mathematics Letters 29 (2014), 1 – 6, DOI: 10.1016/j.aml.2013.10.002.

F. Soleymani, T. Lotfi, E. Tavakoli and F. K. Haghani, Several iterative methods with memory using self-accelerators, Applied Mathematics and Computation 254 (2015), 452 – 458, DOI: 10.1016/j.amc.2015.01.045.

B. D. Stewart, Attractor Basins of Various Root-Finding Methods, Master’s thesis, Naval Postgraduate School, Monterey, California (2001).

A. Tassaddiq, S. Qureshi, A. Soomro, E. Hincal, D. Baleanu and A. A. Shaikh, A new three-step root-finding numerical method and its fractal global behavior, Fractal and Fractional 5(4) (2021), 204, DOI: 10.3390/fractalfract5040204.

V. Torkashvand, A two-step method adaptive with memory with eighth-order for solving nonlinear equations and its dynamic, Computational Methods for Differential Equations 10(4) (2022), 1007 – 1026, DOI: 10.22034/cmde.2022.46651.1961.

J. F. Traub, Iterative Methods for the Solution of Equations, Vol. 312, American Mathematical Society Chelsea Publishing, (1964).

I. Tsoulos and A. Stavrakoudis, On locating all roots of systems of nonlinear equations inside bounded domain using global optimization methods, Nonlinear Analysis: Real World Applications 11(4) (2010), 2465 – 2471, DOI: 10.1016/j.nonrwa.2009.08.003.

E. Vrscay and W. Gilbert, Extraneous fixed points, basin boundaries and chaotic dynamics for Schröder and König rational iteration functions, Numericshe Mathematik 52 (1988), 1 – 16, DOI: 10.1007/BF01401018.

X. Wang, An Ostrowski-type method with memory using a novel self-accelerating parameter, Journal of Computational and Applied Mathematics 330 (2018), 710 – 720, DOI: 10.1016/j.cam.2017.04.021.

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, DOI: 10.1007/s10092-012-0072-2.

X. Wang and T. Zhang, Some Newton-type iterative methods with and without memory for solving nonlinear equations, International Journal of Computational Methods 11(05) (2014), 1350078, DOI: 10.1142/S0219876213500783.

X. Wang and T. Zhang, Efficient n-point iterative methods with memory for solving nonlinear equations, Numerical Algorithms 70(2) (2015), 357 – 375, DOI: 10.1007/s11075-014-9951-8.

S. Weerakoon and T. Fernando, A variant of Newton’s method with accelerated third-order convergence, Applied Mathematics Letters 13(8) (2000), 87 – 93, DOI: 10.1016/S0893-9659(00)00100-2.

J. Zachary, Introduction to Scientific Programming: Computational Problem Solving Using Maple and C, Springer-Verlag, Berlin – Heidelberg, 380 pages (1996).

H. Zhang, J. Guo, J. Lu, J. Niu, F. Li and Y. Xu, The comparison between nonlinear and linear preconditioning JFNK method for transient neutronics/thermal-hydraulics coupling problem, Annals of Nuclear Energy 132 (2019), 357 – 368, DOI: 10.1016/j.anucene.2019.04.053.

## Downloads

## Published

## How to Cite

*Communications in Mathematics and Applications*,

*15*(1), 9–31. https://doi.org/10.26713/cma.v15i1.2432

## Issue

## Section

## License

Authors who publish with this journal agree to the following terms:

- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a CCAL that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work.