A New Newton-Type Method with Third-Order for Solving Systems of Nonlinear Equations

Abstract

In this paper, a new two-step Newton-type method with third-order convergence for solving systems of nonlinear equations is proposed. We construct the new method based on the integral interpolation of Newton’s method. Its cubic convergence and error equation are proved theoretically, and demonstrated numerically. Its application to systems of nonlinear equations and boundary-value problems of nonlinear ODEs are shown as well in the numerical examples.

Share and Cite:

Liu, Z. and Fang, Q. (2015) A New Newton-Type Method with Third-Order for Solving Systems of Nonlinear Equations. Journal of Applied Mathematics and Physics, 3, 1256-1261. doi: 10.4236/jamp.2015.310154.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] Ortega, J.M. and Rheinboldt, W.G. (1970) Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York.
[2] Traub, J.F. (1964) Iterative Methods for the Solution of Equations. Prentice-Hall, Englewood Cliffs, New Jer-sey.
[3] Darvish, M.T. and Barati, A. (2007) A Third-Order Newton-Type Method to Solve Systems of Nonlinear Equations. Applied Mathematics and Computation, 187, 630-635.
http://dx.doi.org/10.1016/j.amc.2006.08.080
[4] Frontini, M. and Sormani, E. (2004) Third-Order Methods from Quadrature Formulae for Solving Systems of Nonlinear Equations. Applied Mathematics and Computation, 149, 771-782.
http://dx.doi.org/10.1016/S0096-3003(03)00178-4
[5] Weerakoon, S. and Fernando, T.G.I. (2000) A Variant of Newton’s Method with Accelerated Third-Order Convergence. Applied Mathematics Letters, 13, 87-93.
http://dx.doi.org/10.1016/S0893-9659(00)00100-2
[6] Hafiz, M.A. and Bahgat, M.S.M. (2012) An Efficient Two-Step Iterative Method for Solving System of Nonlinear Equations. Journal of Mathematics Research, 4, 28-34.
[7] Darvish, M.T. and Barati, A. (2007) A Fourth-Order Method from Quadrature Formulae to Solve Systems of Nonlinear Equations. Applied Mathematics and Computation, 188, 1678-1685.
http://dx.doi.org/10.1016/j.amc.2006.11.022
[8] Noor, M.A. and Wasteem, M. (2009) Some Iterative Methods for Solving a System of Nonlinear Equations. Computers and Mathematics with Applications, 57, 101-106.
http://dx.doi.org/10.1016/j.camwa.2008.10.067
[9] Zhang, X. and Tan, J.Q. (2013) The Fifth-Order of Three-Step Iterative Methods for Solving Systems of Nonlinear Equations. Mathematica Numerica Sinica, 35, 297-304.

Copyright © 2021 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.