Abstract
In this paper, we present a simple, and yet powerful and easily applicable scheme in constructing the Newton-like iteration formulae for the computation of the solutions of nonlinear equations. The new scheme is based on the homotopy analysis method applied to equations in general form equivalent to the nonlinear equations. It provides a tool to develop new Newton-like iteration methods or to improve the existing iteration methods which contains the well-known Newton iteration formula in logic; those all improve the Newton method. The orders of convergence and corresponding error equations of the obtained iteration formulae are derived analytically or with the help of Maple. Some numerical tests are given to support the theory developed in this paper.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Abbasbandy S. (2003): Improving Newton–Raphson method for nonlinear equations by modified Adomian decomposition method. Appl. Math. Comput. 145, 887–893
Abbasbandy S. (2006): Modified homotopy perturbation method for nonlinear equations and comparison with Adomian decomposition method. Appl. Math. Comput. 172(1): 431–438
Adomian G. (1976): Nonlinear stochastic differential equations. J. Math. Anal. Appl. 55(3): 441–452
Adomian G., Adomian G.E. (1984): A global method for solution of complex systems. Math. Model. Anal. 5(3): 521–568
Adomain G. (1994): Solving Frontier problems of Physics: The Decomposition Method. Kluwer, Boston and London
Amat S., Busquier S., Gutiérrez J.M. (2003): Geometric constructions of iterative functions to solve nonlinear equations. J. Comput. Appl. Math. 157, 197–205
Argyros I.K. (1996): Results on Newton methods: Part I. A unified approach for constructing perturbed Newton-like methods in Banach space and their applications. Appl. Math. Comput. 74, 119–141
Babolian E., Biazar J. (2002): Solution of nonlinear equations by modified adomian decomposition method. Appl. Math. Comput. 132, 167–172
Basto M., Semiao V., Calheiros F.L. (2006): A new iterative method to compute nonlinear equations. Appl. Math. Comput. 173(1): 468–483
Chun C. (2005): Iterative methods improving Newton’s method by the decomposition method. Comput. Math. Appl. 50, 1559–1568
Chun C., Ham Y. (2006): Newton-like iteration methods for solving nonlinear equations. Commun. Numer. Methods Eng. 22, 475–487
Frontini M., Sormani E. (2003): Some variant of Newton’s method with third-order convergence. Appl. Math. Comput. 140, 419–426
Grau M., Noguera M. (2000): A variant of Cauchy’s method with accelerated fifth-order convergence. Appl. Math. Lett. 17, 87–93
He J.H. (1998): Newton-like iteration method for solving algebraic equations. Commun. Nonlinear Sci. Numer. Simul. 3(2): 106–109
He J.H. (1999): Homotopy perturbation technique for nonlinear problems. Comput. Methods Appl. Mech. Eng. 178, 257–262
He J.H. (2000): A coupling method of homotopy technique and perturbation technique for nonlinear problems. Int. J. Non-linear Mech. 35(1): 37–43
Homeier H.H.H. (2005): On Newton-type methods with cubic convergence. J. Comput. Appl. Math. 176, 425–432
Householder A.S. (1970): The Numerical Treatment of a Single Nonlinear Equation. McGraw-Hill, New York
Huang Z. (1999): On the error estimates of several Newton-like methods. Appl. Math. Comput. 106, 1–16
Liao, S.J.: The proposed homotopy analysis technique for the solution of nonlinear problems. Ph.D. Thesis, Shanghai Jiao Tong university (1992)
Liao S.J. (1995): An approximate solution technique not depending on small parameters: a special example. Int. J. Non-linear Mech. 30(2): 371–380
Liao S.J. (1997): A kind of approximate solution technique which does not depend upon small parameters (II): an application in fluid mechanics. Int. J. Non-linear Mech. 32, 815–822
Liao S.J. (1997): Boundary element method for general nonlinear differential operators. Eng. Anal. Bound. Elem. 20(2): 91–99
Liao S.J. (1997): Homotopy analysis method: an analytic technique not depending on small parameters. Shanghai J. Mech. 18(3): 196–200
Liao S.J. (1999): A simple way to enlarge the convergence region of perturbation approximations. Int. J. Non-linear Dyn. 19(2): 93–110
Liao S.J. (1999): An explicit, totally analytic approximation of Blasius viscous flow problems. Int. J. Non-linear Mech. 34(4): 759–778
Liao S.J. (2003): Beyond Perturbation: Introduction to Homotopy Analysis Method. Chapman & Hall/ CRC Press, Boca Raton
Liao S.J. (2004): On the homotopy analysis method for nonlinear problems. Appl. Math. Comput. 147, 499–513
Liao S.J. (2005): Comparison between the homotopy analysis method and homotopy perturbation method. Appl. Math. Comput. 169(2): 1186–1194
Ostrowski A.M. (1960): Solutions of equations and system of equations. Academic, New York
Petkovic L., et al. (1997): On the construction of simultaneous methods for multiple zeros. Nonlinear Anal. 30, 669–676
Traub J.F. (1982): Iterative methods for the solution of equations. Chelsea Publishing company, New York
Weerakoon S., Fernando G.I. (2000): A variant of Newton’s method with accelerated third-order convergence. Appl. Math. Lett. 17, 87–93
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Chun, C. Construction of Newton-like iteration methods for solving nonlinear equations. Numer. Math. 104, 297–315 (2006). https://doi.org/10.1007/s00211-006-0025-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00211-006-0025-2