SciELO - Scientific Electronic Library Online

 
vol.22 número1A sufficiently complicated noded Schottky group of rank threeCertain results on the conharmonic curvature tensor of (κ, μ)-contact metric manifolds índice de autoresíndice de materiabúsqueda de artículos
Home Pagelista alfabética de revistas  

Servicios Personalizados

Revista

Articulo

Indicadores

Links relacionados

  • En proceso de indezaciónCitado por Google
  • No hay articulos similaresSimilares en SciELO
  • En proceso de indezaciónSimilares en Google

Compartir


Cubo (Temuco)

versión On-line ISSN 0719-0646

Cubo vol.22 no.1 Temuco abr. 2020

http://dx.doi.org/10.4067/S0719-06462020000100055 

Articles

Super-Halley method under majorant conditions in Banach spaces

Shwet Nisha1 

P. K. Parida2 

1Department of Applied Mathematics, Central University of Jharkhand, Ranchi-835205, India. shwetnisha1988@gmail.com

2Department of Applied Mathematics, Central University of Jharkhand, Ranchi-835205, India. pkparida@cuj.ac.in

Abstract

In this paper, we have studied local convergence of Super-Halley method in Banach spaces under the assumption of second order majorant conditions. This approach allows us to obtain generalization of earlier convergence analysis under majorizing sequences. Two important special cases of the convergence analysis based on the premises of Kantorovich and Smale type conditions have also been concluded. To show efficacy of our approach we have given three numerical examples.

Keywords and Phrases: Nonlinear equations; Super-Halley method; Majorant conditions; Local Convergence; Semilocal Convergnce; Smale-type conditions; Kantorovich-type conditions

Resumen

En este artículo, hemos estudiado la convergencia local del método Super-Halley en espacios de Banach, asumiendo condiciones mayorantes de segundo orden. Este punto de vista nos permite obtener generalizaciones de análisis de convergencia bajo sucesiones mayorantes obtenidos anteriormente. También se han concluido dos casos especiales del análisis de convergencia basados en las premisas de condiciones tipo Kantorovich y Smale. Para mostrar la eficacia de nuestro enfoque, damos tres ejemplos numéricos.

Texto completo disponible sólo en PDF.

Full text available only in PDF format.

References

[1] I. K. Argyros andH. Ren. Ball convergence theorem for Halley’s method in banach spaces. J. Appl. Math. Comp. 38(2012)453-465. [ Links ]

[2] D. Chen, I. K. Argyros and Q. Qian. A local Convergence theorem for the Super-Halleymethod in Banach space. Appl. Math. 7(1994)49-52. [ Links ]

[3] P. Deuflhard. Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithms. Springer, Berlin Heindelberg, 2004. [ Links ]

[4] P. Deuflhard and G. Heindl. Affine invariant convergent theorems for Newtons method and extensions to related methods. SIAM J. Numer. Anal. 16(1979) 1-10. [ Links ]

[5] J. A. Ezquerro and M. A. Hernández. On a convex acceleration of Newton’s method, J. Optim. Theory Appl. 100 (1999)311-326. [ Links ]

[6] O.P. Ferreira. Local convergence of Newton’s method in Banach space from the viewpoint of the majorant principle. IMA J. Numer. Anal. 29(2009)746-759. [ Links ]

[7] O.P. Ferreira and B.F. Svaiter. Kantorovich’s majorants principle for Newton’s method. Comput. Optim. Appl. 42(2009)213-229. [ Links ]

[8] W.B. Gragg and R.A. Tapia. Optimal error bounds for the Newton-Kantorovich theorem. SIAM J. Numer. Anal. 11(1974)10-13. [ Links ]

[9] J. M. Gutiérrez andM. A. Hernández . Recurrence relations for the Super-Halley method, Comput. Math. Appl. 36(1998)1-8. [ Links ]

[10] J. M. Gutiérrez andM. A. Hernández . Newton’s method under weak Kantorovich conditions. IMA J. Numer. Anal. 20(2000)521-532. [ Links ]

[11] J. M. Gutiérrez andM. A. Hernández. An acceleration of Newton’s method: super-Halley method. Appl. Math. Comput. 117(2001)223-239. [ Links ]

[12] D.Han , X.Wang. The error estimates of Halley’s method. Numer. Math. JCU Engl. Ser. 6(1997)231-240. [ Links ]

[13] M. A. Hernández and N. Romero. On the characterization of some Newton like methods of R-order at least three. J. Comput. Appl. Math . 183(2005)53-66. [ Links ]

[14] M. A. Hernández andN. Romero . Towards a unified theory for third R-order iterative methods for operators with unbounded second derivative, Appl. Math. Comput. 215(2009)2248-2261. [ Links ]

[15] L. O. Jay. A note on Q-order of convergence, BIT Numer. Math. 41(2001)422-429. [ Links ]

[16] L. V. Kantorovich and G. P. Akilov. Functional Analysis. Pergamon Press, Oxford, 1982. [ Links ]

[17] Y. Ling and X. Xu. On the semilocal convergence behaviour of Halley’s method, Comput. Optim. Appl. 58(2014)597-61. [ Links ]

[18] F. A. Potra. On Q-order and R-order of convergence, J. Optim. Theory Appl. 63(1989)415-431. [ Links ]

[19] M. Prashanth and D. K. Gupta. Recurrence relation for Super-Halley’s method with hölder continuous second derivative in Banach spaces, Kodai Math. J. 36(2013)119-136. [ Links ]

[20] M. Prashanth , D. K. Gupta and S. Singh. Semilocal convergence for the Super-Halley method. Numer. Anal. Appl., 7(2014)70-84. [ Links ]

[21] S. Smale. Newton’s method estimates from data at one point. In: Ewing, R., Gross, K., Martin, C.(eds.) The Merging of Disciplines: New Directions in Pure, Applied and computational Mathematics, 185-196. Springer, New York, 1986. [ Links ]

[22] X. Wang. Convergence of Newton’s method and inverse functions theorem in Banach space. Math. Comput. 68(1999)169-186. [ Links ]

[23] X. Wang. Convergence of Newton’s method and uniqueness of the solution of equations in Banach space, IMA J.Numer. Anal. 20(2000)123-134. [ Links ]

[24] X. Wang and D. Han. On the dominating sequence method in the point estimates and smale’s, theorem. Scientia Sinica Ser. A. 33(1990)135-144. [ Links ]

[25] X. Wang andD. Han . Criterion _ and Newton’s method in the weak conditions (in Chinese), Math.Numer. Sinica 19(1997)103-112. [ Links ]

[26] X. Xu and C. Li. Convergence of Newton’s method for systems of equations with constant rank derivatives. J. Comput. Math. 25(2007)705-718. [ Links ]

[27] X. Xu and C. Li. Convergence criterion of Newton’s method for singular systems of equations with constant rank derivatives. J. Math. Anal. Appl. 245(2008)689-701. [ Links ]

[28] T. Yamamoto. On the method of tangent hyperbolas in Banach spaces, J. Comput. Appl. Math . 21(1988)75-86. [ Links ]

[29] T.J. Ypma. Affine invariant convergence results for Newton’s method. BIT Numer. Math. 22(1982)108-118. [ Links ]

Creative Commons License This is an open-access article distributed under the terms of the Creative Commons Attribution License