A unified approach for the convergence of certain numerical algorithms, using recurrent functions

1. Argyros IK (1998) The theory and application of abstract polynomial equations. St.Lucie/CRC/Lewis Publisher, Mathematics series, Boca Raton Florida, U.S.A.

2. Argyros IK (2004) On the Newton–Kantorovich hypothesis for solving equations. J Comput Appl Math 169: 315–332

   Article  MATH  MathSciNet  Google Scholar 

3. Argyros IK (2005) Concerning the “terra incognita” between convergence regions of two Newton methods. Nonlinear Anal 62: 179–194

   Article  MATH  MathSciNet  Google Scholar 

4. Argyros IK (2008) Convergence and applications of Newton-type iterations. Springer-Verlag, New York

   MATH  Google Scholar 

5. Argyros IK (2009) On a class of Newton-like methods for solving nonlinear equations. J Comput Appl Math 228: 115–122

   Article  MATH  MathSciNet  Google Scholar 

6. Argyros IK, Hilout S (2009) Efficient methods for solving equations and variational inequalities. Polimetrica Publisher, Milano

   Google Scholar 

7. Cătinaş E (1994) On some iterative methods for solving nonlinear equations. Revue Anal Num Théo Appr 23: 47–53

   MATH  Google Scholar 

8. Chen X, Nashed MZ (1993) Convergence of Newton-like methods for singular operator equations using outer inverses. Numer Math 66: 235–257

   Article  MATH  MathSciNet  Google Scholar 

9. Chen X, Yamamoto T (1989) Convergence domains of certain iterative methods for solving nonlinear equations. Numer Funct Anal Optim 10: 37–48

   Article  MATH  MathSciNet  Google Scholar 

10. Cianciaruso F (2009) A further journey in the “terra incognita” of the Newton–Kantorovich method. Nonlinear Funct Anal Appl, pp 1–11

11. Dennis JE (1971) Toward a unified convergence theory for Newton-like methods. In: Rall LB (eds) Nonlinear functional analysis and applications. Academic Press, New York, pp 425–472

    Google Scholar 

12. Deuflhard P (2004) Newton methods for nonlinear problems. Affine invariance and adaptive algorithms. In: Springer series in computational mathematics, vol 35. Springer-Verlag, Berlin

13. Deuflhard P, Heindl G (1979) Affine invariant convergence theorems for Newton’s method and extensions to related methods. SIAM J Numer Anal 16: 1–10

    Article  MATH  MathSciNet  Google Scholar 

14. Gutiérrez JM, Hernández MA, Salanova MA (2000) A discretization scheme for some conservative problems. Proceedings of the 8th international Congress on computer and application mathematics, ICCAM-98 (Leuven). J Comput Appl Math 115: 181–192

    Google Scholar 

15. Kantorovich LV, Akilov GP (1982) Functional analysis. Pergamon Press, Oxford

    MATH  Google Scholar 

16. Miel GJ (1979) Unified error analysis for Newton-type methods. Numer Math 33: 391–396

    Article  MATH  MathSciNet  Google Scholar 

17. Miel GJ (1980) Majorizing sequences and error bounds for iterative methods. Math Comput 34: 185–202

    MATH  MathSciNet  Google Scholar 

18. Moret I (1984) A note on Newton type iterative methods. Computing 33: 65–73

    Article  MATH  MathSciNet  Google Scholar 

19. Păvăloiu I (1968) Sur la méthode de Steffensen pour la résolution des équations opérationnelles non linéaires. Rev Roumaine Math Pures Appl 13(6): 857–861

    MATH  MathSciNet  Google Scholar 

20. Păvăloiu I (1976) Introduction in the theory of approximation of equations solutions. Dacia Ed., Cluj-Napoca

21. Păvăloiu I (1991) On the convergency of a Steffensen-type method. In: Seminar on mathematical analysis. Preprint, 91-7, “Babeş-Bolyai” University, Cluj-Napoca, pp 121–126

22. Potra FA (1982) On the convergence of a class of Newton-like methods. In: Iterative solution of nonlinear systems of equations (Oberwolfach, 1982). Lecture notes in mathematics, vol 953. Springer, Berlin, pp 125–137

23. Potra FA (1984) On an iterative algorithm of order 1.839... for solving nonlinear operator equations. Numer Funct Anal Optim 7(1): 75–106

    Article  MATH  MathSciNet  Google Scholar 

24. Potra FA (1985) Sharp error bounds for a class of Newton-like methods. Libertas Math 5: 71–84

    MATH  MathSciNet  Google Scholar 

25. Ulm SJ (1964) A generalization of Steffensen’s method for solving non-linear operator equations.. Z̆ Vyc̆isl Mat i Mat Fiz 4: 1093–1097 (in Russian)

    MathSciNet  Google Scholar 

26. Rheinboldt WC (1968) A unified convergence theory for a class of iterative processes. SIAM J Numer Anal 5: 42–63

    Article  MATH  MathSciNet  Google Scholar 

27. Yamamoto T (1987) A convergence theorem for Newton-like methods in Banach spaces. Numer Math 51: 545–557

    Article  MATH  MathSciNet  Google Scholar 

28. Zabrejko PP, Nguen DF (1987) The majorant method in the theory of Newton–Kantorovich approximations and the Pták error estimates. Numer Funct Anal Optim 9: 671–684

    Article  MATH  MathSciNet  Google Scholar 

29. Zinc̆enko AI (1963) Some approximate methods of solving equations with non-differentiable operators. Dopovidi Akad. Nauk Ukraïn. RSR, pp 156–161 (in Ukrainian)

Download references