Advertisement

New Family of Iterative Methods with High Order of Convergence for Solving Nonlinear Systems

  • Alicia Cordero
  • Juan R. Torregrosa
  • María P. Vassileva
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8236)

Abstract

In this paper we present and analyze a set of predictor-corrector iterative methods with increasing order of convergence, for solving systems of nonlinear equations. Our aim is to achieve high order of convergence with few Jacobian and/or functional evaluations. On the other hand, by applying the pseudocomposition technique on each proposed scheme we get to increase their order of convergence, obtaining new high-order and efficient methods. We use the classical efficiency index in order to compare the obtained schemes and make some numerical test.

Keywords

Nonlinear systems Iterative methods Jacobian matrix Convergence order Efficiency index 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Cordero, A., Hueso, J.L., Martínez, E., Torregrosa, J.R.: A modified Newton-Jarratt’s composition. Numer. Algor. 55, 87–99 (2010)CrossRefzbMATHGoogle Scholar
  2. 2.
    Cordero, A., Hueso, J.L., Martínez, E., Torregrosa, J.R.: Efficient high-order methods based on golden ratio for nonlinear systems. Applied Mathematics and Computation 217(9), 4548–4556 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Cordero, A., Torregrosa, J.R.: Variants of Newton’s Method using fifth-order quadrature formulas. Applied Mathematics and Computation 190, 686–698 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cordero, A., Torregrosa, J.R.: On interpolation variants of Newton’s method for functions of several variables. Journal of Computational and Applied Mathematics 234, 34–43 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cordero, A., Torregrosa, J.R., Vassileva, M.P.: Pseudocomposition: a technique to design predictor-corrector methods for systms of nonlinear equtaions. Applied Mathematics and Computation 218(23), 11496–11504 (2012)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Nikkhah-Bahrami, M., Oftadeh, R.: An effective iterative method for computing real and complex roots of systems of nonlinear equations. Applied Mathematics and Computation 215, 1813–1820 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Ostrowski, A.M.: Solutions of equations and systems of equations. Academic Press, New York (1966)Google Scholar
  8. 8.
    Shin, B.-C., Darvishi, M.T., Kim, C.-H.: A comparison of the Newton-Krylov method with high order Newton-like methods to solve nonlinear systems. Applied Mathematics and Computation 217, 3190–3198 (2010)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Alicia Cordero
    • 1
  • Juan R. Torregrosa
    • 1
  • María P. Vassileva
    • 2
  1. 1.Instituto de Matemática MultidisciplinarUniversitat Politècnica de ValènciaValenciaEspaña
  2. 2.Instituto Tecnológico de Santo Domingo (INTEC)República Dominicana

Personalised recommendations