Abstract
Point of attraction theory is an important tool to analyze the local convergence of iterative methods for solving systems of nonlinear equations. In this work, we prove a generalized form of Ortega-Rheinbolt result based on point of attraction theory. The new result guarantees that the solution of the nonlinear system is a point of attraction of iterative scheme, especially multipoint iterations. We then apply it to study the attraction theorem of the Frontini-Sormani family of multipoint third order methods from Quadrature Rule. Error estimates are given and compared with existing ones. We also obtain the radius of convergence of the special members of the family. Two numerical examples are provided to illustrate the theory. Further, a spectral analysis of the Discrete Fourier Transform of the numerical errors is conducted in order to find the best method of the family. The convergence and the spectral analysis of a multistep version of one of the special member of the family are studied.
Similar content being viewed by others
References
Aslam Noor, M., Waseem, M.: Some iterative methods for solving a system of nonlinear equations. Comput. Math. Appl. 57, 101–106 (2009)
Babajee, D.K.R., Dauhoo, M.Z.: An analysis of the properties of the variants of Newton’s method with third order convergence. Appl. Math. Comput. 183, 659–684 (2006)
Babajee, D.K.R., Dauhoo, M.Z.: Analysis of a family of two-point iterative methods with third order convergence. In: Simos, T.E., Psihoyios, G., Tsitouras, Ch. (eds.) International Conference on Numerical Analysis and Applied Mathematics 2006, pp. 658–661. Wiley-VCH Verlag GmbH & Co. KGaA, Greece (2006)
Babajee, D.K.R., Dauhoo, M.Z.: Spectral analysis of the errors of some families of multi-step Newton-like methods. Numer. Algorithms (2008). doi:10.1007/s11075-008-9256-x
Ezquerro, J.A., Gutierrez, J.M., Hernandez, M.A, Salnova, M.A.: A biparametric family of inverse-free multipoint iterations. Comput. Appl. Math. 19(1), 109–124 (2000)
Ezquerro, J.A., Hernandez, M.A: A uniparametric Halley-type iteration with free second derivative. Int. J. Pure Appl. Math. 6(1), 103–114 (2003)
Frontini, M., Sormani, E.: Some variant of Newton’s method with third order convergence. Appl. Math. Comput. 140(2–3), 419–426 (2003)
Frontini, M., Sormani, E.: Modified Newton’s method with third-order convergence and multiple roots. Comput. Appl. Math. 156(2), 345–354 (2003)
Frontini, M., Sormani, E.: Third order methods from quadrature formulae for solving systems of nonlinear equations. Appl. Math. Comp. 149, 771–782 (2004)
Grau-Sanchez, M., Peris, J.M., Gutierrez, J.M.: Accelerated iterative methods for finding solutions of a system of nonlinear equations. Appl. Math. Comput. 190, 1815–1823 (2007)
Hasanov, V.I., Ivanov, I.G., Nedzhibov, G.: A new modification of Newton method. Appl. Math. Eng. 27, 278–286 (2002)
Homeier, H.H.H.: Modified Newton method with cubic convergence: the multivariate case. Comput. Appl. Math. 169, 161–169 (2004)
Kou, J., Li, Y., Wang, X.: Some modifications of Newton’s method with fifth-order convergence. Comput. Appl. Math. 209, 146–152 (2007)
Nedzhibov, G.: On a few iterative methods for solving nonlinear equations. In: Application of Mathematics in Engineering and Economics’28, Bulvest–2000, Sofia, pp. 56–64 (2003)
Ortega, J.M., Rheinbolt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic, New York (1970)
Ozban, A.: Some new variants of Newton’s method. Appl. Math. Lett. 17, 677–682 (2004)
Petkovic, L.D., Petkovic, M.S.: A note on some recent methods for solving nonlinear equations. Appl. Math. Comput. 185, 368–374 (2007)
Ren, H.: A note on a paper by D.K.R. Babajee and M.Z. Dauhoo. Appl. Math. Comput. 200(1–2), 830–833 (2008)
Traub, J.F.: Iterative Methods for the Solution of Equations, 2nd edn. Chelsea, New York (1964)
Wait, R.: The Numerical Solution of Algebraic Equations. Wiley, New York (1979)
Weerakoon, S., Fernando, T.G.I.: A variant of Newton’s method with accelerated third order convergence. Appl. Math. Lett. 13, 87–93 (2000)
Author information
Authors and Affiliations
Corresponding author
Additional information
The work of the first author is partly supported by Tertiary Education Commission and University of Mauritius.
Rights and permissions
About this article
Cite this article
Babajee, D.K.R., Dauhoo, M.Z. Convergence and spectral analysis of the Frontini-Sormani family of multipoint third order methods from quadrature rule. Numer Algor 53, 467–484 (2010). https://doi.org/10.1007/s11075-009-9314-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-009-9314-z
Keywords
- Point of attraction
- Systems of nonlinear equations
- Ortega-Rheinbolt result
- Multipoint third order methods
- Quadrature rule
- Error estimates
- Radius of convergence
- Power Spectrum
- Amplitude of the Power Spectrum
- Mean Power Spectrum