1. MMC
  2. All Volumes and Issues
  3. Volume 9, Number 3, 2022
  4. Enlarging the radius of convergence for Newton–like method in which the derivative is re-evaluated after certain steps

Enlarging the radius of convergence for Newton–like method in which the derivative is re-evaluated after certain steps

MMC.
2022;
Volume 9, Number 3
: pp. 594–598
https://doi.org/10.23939/mmc2022.03.594
Received: January 13, 2022
Revised: June 17, 2022
Accepted: June 24, 2022

Mathematical Modeling and Computing, Vol. 9, No. 3, pp. 594–598 (2022)

Authors:
  1. C. I. Argyros
  2. I. K. Argyros
  3. S. M. Shakhno
  4. H. P. Yarmola
1
Cameron University, Lawton, OK 73505, USA
2
Cameron University, Lawton, OK 73505, USA
3
Ivan Franko National University of Lviv
4
Ivan Franko National University of Lviv

Numerous attempts have been made to enlarge the radius of convergence for Newton–like method under the same set of conditions.  It turns out that not only the radius of convergence but the error bounds on the distances involved and the uniqueness of the solution ball  can more accurately be defined.

radius of convergence
Newton–like method
Hilbert space
  1. Măruşter Ş.  Estimating local radius of convergence.  Symposium "Symbolic and Numeric Algorithm for Scientific Computation" (SYNASC),  Workshop Iteratime Approximation of Fixed Points, 24–27 Sept. 2016.  West University of Timisoara, Timisoara, Romania (2016).
  2. Măruşter Ş.  On the local convergence of the Modified Newton method.  Annals of West University of Timisoara – Mathematics and Computer Science. 57 (1), 13–22 (2019).
  3. Potra F. A., Pták V.  Nondiscrete induction and iterative processes.  Pitman Publ., London (1984).
  4. Traub J. F.  Iterative methods for the solution of equations.  Chelsea Publishing Company, New York (1982).
  5. Ortega J. M., Rheinboldt W. C.  Iterative solution of nonlinear equation in several variables.  Acad. Press, New York (1970).
  6. Ezquerro J. A, Hernández M. A.  An improvement of the region of accessibility of Chebyshev's method from Newton's method.  Mathematics of Computation. 78 (267), 1613–1627 (2009).
  7. Ezquerro J. A, Hernández M. A.  An optimization of Chebyshev's method.  Journal of Complexity. 25 (4), 343–361 (2009).
  8. Hernández-Veron M. A., Romero N.  On the local convergence of a third order family of iterative processes.  Algorithms. 8, 1121–1128 (2015).
  9. Cătinaş E.  Estimating the radius of an attraction ball.  Applied Mathematics Letters. 22, 712–714 (2009).
  10. Iakymchuk R. P., Shakhno S. M., Yarmola H. P.  Convergence analysis of a two-step modification of the Gauss–Newton method and its applications.  Journal of Numerical and Applied Mathematics. 126, 61–74 (2017).
  11. Magreñán Á. A., Argyros I. K.  Two-step Newton methods.  Journal of Complexity. 30 (4), 533–553 (2014).
  12. Argyros I. K., Shakhno S.  Extending the Applicability of Two-Step Solvers for Solving Equations.  Mathematics. 7 (1), 62 (2019).
  13. Argyros I. K., Magreñán Á. A.  A Contemporary Study of Iterative Methods.  Convergence, Dynamics and Applications. 333–346 (2018).
  14. Shakhno S. M.  On an iterative algorithm with superquadratic convergence for solving nonlinear operator equations.  Journal of Computational and Applied Mathematics. 231 (1), 222–235 (2009).
  15. Argyros I. K., Shakhno S., Yarmola H.  Two-Step Solver for Nonlinear Equations.  Symmetry. 11 (2), 128 (2019).
  16. Kantorovich L. V., Akilov G. P.  Functional Analysis. Oxford, Pergamon (1982).