Abstract and Applied Analysis

Design of High-Order Iterative Methods for Nonlinear Systems by Using Weight Function Procedure

Santiago Artidiello, Alicia Cordero, Juan R. Torregrosa, and María P. Vassileva

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Abstract

We present two classes of iterative methods whose orders of convergence are four and five, respectively, for solving systems of nonlinear equations, by using the technique of weight functions in each step. Moreover, we show an extension to higher order, adding only one functional evaluation of the vectorial nonlinear function. We perform numerical tests to compare the proposed methods with other schemes in the literature and test their effectiveness on specific nonlinear problems. Moreover, some real basins of attraction are analyzed in order to check the relation between the order of convergence and the set of convergent starting points.

Article information

Source
Abstr. Appl. Anal., Volume 2015, Special Issue (2015), Article ID 289029, 12 pages.

Dates
First available in Project Euclid: 17 August 2015

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1439816309

Digital Object Identifier
doi:10.1155/2015/289029

Mathematical Reviews number (MathSciNet)
MR3384346

Zentralblatt MATH identifier
06929081

Citation

Artidiello, Santiago; Cordero, Alicia; Torregrosa, Juan R.; Vassileva, María P. Design of High-Order Iterative Methods for Nonlinear Systems by Using Weight Function Procedure. Abstr. Appl. Anal. 2015, Special Issue (2015), Article ID 289029, 12 pages. doi:10.1155/2015/289029. https://projecteuclid.org/euclid.aaa/1439816309


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References

  • A. Iliev and N. Kyurkchiev, Nontrivial Methods in Numerical Analysis: Selected Topics in Numerical Analysis, LAP LAMBERT Academic, Saarbrcken, Germany, 2010.
  • Y. Zhang and P. Huang, “High-precision time-interval measurement techniques and methods,” Progress in Astronomy, vol. 24, no. 1, pp. 1–15, 2006.
  • Y. He and C. Ding, “Using accurate arithmetics to improve numerical reproducibility and stability in parallel applications,” The Journal of Supercomputing, vol. 18, no. 3, pp. 259–277, 2001.
  • S. Weerakoon and T. G. Fernando, “A variant of Newton's method with accelerated third-order convergence,” Applied Mathematics Letters, vol. 13, no. 8, pp. 87–93, 2000.
  • A. Y. Özban, “Some new variants of Newton's method,” Applied Mathematics Letters, vol. 17, no. 6, pp. 677–682, 2004.
  • J. Gerlach, “Accelerated convergence in Newton's method,” SIAM Review, vol. 36, no. 2, pp. 272–276, 1994.
  • A. Cordero and J. R. Torregrosa, “Variants of Newton's method for functions of several variables,” Applied Mathematics and Computation, vol. 183, no. 1, pp. 199–208, 2006.
  • A. Cordero and J. R. Torregrosa, “Variants of Newton's method using fifth-order quadrature formulas,” Applied Mathematics and Computation, vol. 190, no. 1, pp. 686–698, 2007.
  • A. Cordero and J. R. Torregrosa, “On interpolation variants of Newton's method for functions of several variables,” Journal of Computational and Applied Mathematics, vol. 234, no. 1, pp. 34–43, 2010.
  • M. Frontini and E. Sormani, “Third-order methods from quadrature formulae for solving systems of nonlinear equations,” Applied Mathematics and Computation, vol. 149, no. 3, pp. 771–782, 2004.
  • A. Cordero, J. R. Torregrosa, and M. P. Vassileva, “Pseudocomposition: a technique to design predictor-corrector methods for systems of nonlinear equations,” Applied Mathematics and Computation, vol. 218, no. 23, pp. 11496–11504, 2012.
  • M. P. Vassileva, Métodos iterativos eficientes para la resolución de sistemas no lineales [Ph.D. thesis], Universidad Politécnica de Valencia, Valencia, Spain, 2011.
  • J. F. Traub, Iterative Methods for the Solution of Equations, Prentice Hall, 1964.
  • P. Jarratt, “Some fourth order multipoint iterative methods for solving equations,” Mathematics of Computation, vol. 20, pp. 434–437, 1966.
  • J. R. Sharma, R. K. Guha, and R. Sharma, “An efficient fourth order weighted-Newton method for systems of nonlinear equations,” Numerical Algorithms, vol. 62, no. 2, pp. 307–323, 2013.
  • J. R. Sharma and P. Gupta, “An efficient fifth order method for solving systems of nonlinear equations,” Computers & Mathematics with Applications, vol. 67, no. 3, pp. 591–601, 2014.
  • A. Cordero, J. L. Hueso, E. Martínez, and J. R. Torregrosa, “A modified Newton-Jarratt's composition,” Numerical Algorithms, vol. 55, no. 1, pp. 87–99, 2010.
  • F. I. Chicharro, A. Cordero, and J. R. Torregrosa, “Drawing dynamical and parameters planes of iterative families and methods,” The Scientific World Journal, vol. 2013, Article ID 780153, 11 pages, 2013.
  • L. B. Rall, Computational Solution of Nonlinear Operator Equations, Robert E. Krieger, New York, NY, USA, 1979. \endinput