Constructing an efficient multi-step iterative scheme for nonlinear system of equations. (English) Zbl 07468460

Summary: The objective of this research is to propose a new multi-step method in tackling a system of nonlinear equations. The constructed iterative scheme achieves a higher rate of convergence whereas only one LU decomposition per cycle is required to proceed. This makes the efficiency index to be high as well in contrast to the existing solvers. The usefulness of the presented approach for tackling differential equations of nonlinear type with partial derivatives is also given.


65-XX Numerical analysis
41A15 Spline approximation
65H10 Numerical computation of solutions to systems of equations


Full Text: DOI


[1] F. Ahmad, E. Tohidi, M. Zaka Ullah, and J. A. Carrasco,Higher order multi-step Jarratt-like method for solving systems of nonlinear equations: Application to PDEs and ODEs, Comput. Math. Appl.70(2015), 624-636. · Zbl 1443.65070
[2] F. Ahmad, S. ur R´ehman, M. Zaka Ullah, H. Moaiteq Aljahdali, S. Ahmad, A. S. Alshomrani, J. A. Carrasco, S. Ahmad, and S. Sivasankaran,Frozen Jacobian multistep iterative method for solving nonlinear IVPs and BVPs, Complexity,2017(2017), 1-30. · Zbl 1367.65077
[3] F. Ahmad, T. A. Bhutta, U. Sohaib, M. Zaka Ullah, A. S. Alshomrani, S. Ahmad, and S. Ahmad,A preconditioned iterative method for solving systems of nonlinear equations having unknown multiplicity, Algorithms,10(2017), 1-9. · Zbl 1461.65082
[4] D. K. R. Babajee, M. Z. Dauhoo, M. T. Darvishi, and A. Barati,A note on the local convergence of iterative methods based on Adomian decomposition method and 3-node quadrature rule, Appl. Math. Comput.,200(2008), 452-458. · Zbl 1160.65018
[5] R. Behl, E. Mart´ınez, F. Cevallos, and D. Alarc´on,A higher order Chebyshev-Halley-type family of iterative methods for multiple roots, Math.,7(2019), 1-12.
[6] A. Cordero, J. L. Hueso, E. Mart´ınez, and J. R. Torregrosa,A modified Newton-Jarratt’s composition, Numer. Algor.,55(2010), 87-99. · Zbl 1251.65074
[7] P. Darania,Superconvergence analysis of multistep collocation method for delay Volterra integral equations, Comput. Meth. Diff. Equ.,4(2016), 205-216. · Zbl 1424.65099
[8] I. G. Ivanov and T. Mateva,Interval methods with fifth order of convergence for solving nonlinear scalar equations, Axioms,8(2019), 1-11. · Zbl 1432.65058
[9] M. Grau-S´anchez, ‘A. Grau, and M. Noguera,On the computational efficiency index and some iterative methods for solving systems of nonlinear equations, J. Comput. Appl. Math.,236 (2011), 1259-1266. · Zbl 1231.65090
[10] F. W. Khdhr, F. Soleymani, R. K. Saeed, and A. Akg¨ul,An optimized Steffensen-type iterative method with memory associated with annuity calculation, Euro. Phys. J. Plus,134(2019), Article ID: 146, 1-12.
[11] F. W. Khdhr, R. K. Saeed, and F. Soleymani,Improving the computational efficiency of a variant of Steffensen’s method for nonlinear equations, Math.,7(2019), Article ID: 306, 1-9.
[12] Y. I. Kim, R. Behl, and S. S. Motsa,An optimal family of eighth-order iterative methods with an inverse interpolatory rational function error corrector for nonlinear equations, Math. Model. Anal.,22(2017), 321-336.
[13] C. Lin, C. W. Hsu, T. E. Simos, and Ch. Tsitouras,Explicit, semi-symmetric, hybrid, six-step, eighth order methods for solvingY00=F(X;Y), Appl. Comput. Math.,18(2019), 296-304. · Zbl 1431.65096
[14] S. Mangano,Mathematica Cookbook, O’Reilly Media, USA, 2010.
[15] H. Montazeri, F. Soleymani, S. Shateyi, and S. S. Motsa,On a new method for computing the numerical solution of systems of nonlinear equations, J. Appl. Math.,2012(2012), Article ID: 751975, 15 pages. · Zbl 1268.65075
[16] O. Ogbereyivwe and K. Obiajulu Muka,Multistep quadrature based methods for nonlinear system of equations with singular Jacobian, J. Appl. Math. Phys.,7(2019), 702-725.
[17] J. M. Ortega and W. C. Rheinboldt,Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970. · Zbl 0241.65046
[18] T. Sauer,Numerical Analysis, Pearson, 2nd edition, USA, 2012.
[19] K. Shah and K. Rahmat,Iterative scheme to a coupled system of highly nonlinear fractional order differential equations, Computational Methods for Differential Equations,3(3) (2015), 163-176. · Zbl 1412.34041
[20] J. R. Sharma, R. K. Guha, and R. Sharma,An efficient fourth order weighted-Newton method for systems of nonlinear equations, Numer. Algor.,62(2013), 307-323. · Zbl 1283.65051
[21] T. E. Simos and Ch. Tsitouras,High phase lag-order, four-step methods for solvingy00=f(x, y), Appl. Comput. Math.,17(2018), 307-316. · Zbl 1432.65094
[22] F. Soleymani, M. Barfeie, and F. Khaksar Haghani,Inverse multi-quadric RBF for computing the weights of FD method: Application to American options, Commun. Nonlinear Sci. Numer. Simul.,64(2018), 74-88. · Zbl 07265261
[23] F. Soleymani,Pricing multi-asset option problems: A Chebyshev pseudo-spectral method, BIT, 59(2019), 243-270. · Zbl 1419.91654
[24] F. Soleymani and B. N. Saray,Pricing the financial Heston-Hull-White model with arbitrary correlation factors via an adaptive FDM, Comput. Math. Appl,77(2019), 1107-1123. · Zbl 1442.91105
[25] J. F. Traub,Iterative Methods for the Solution of Equations, Prentice Hall, New York, 1964. · Zbl 0121.11204
[26] Wolfram Research, Inc., Mathematica, Version 11.0, Champaign, IL (2016).
[27] M. Zaka Ullah, F. Soleymani, and A. S. Al-Fhaid,Numerical solution of nonlinear systems by a general class of iterative methods with application to nonlinear PDEs, Numer. Algor,67(2014), 223-242 · Zbl 1316.65053
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