Table of Contents
International Journal of Analysis
Volume 2017, Article ID 7364236, 9 pages
https://doi.org/10.1155/2017/7364236
Research Article

Influence of the Center Condition on the Two-Step Secant Method

1Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, India
2Department of Mathematics, School of Arts and Sciences, Amrita Vishwa Vidyapeetham (Amrita University), Amritapuri, India

Correspondence should be addressed to Shwetabh Srivastava; moc.liamg@tiihbatewhs

Received 23 June 2017; Accepted 8 August 2017; Published 24 September 2017

Academic Editor: Shamsul Qamar

Copyright © 2017 Abhimanyu Kumar et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. A. Margeñán and I. K. Argyros, “New improved convergence analysis for the secant method,” Mathematics and Computers in Simulation, vol. 119, pp. 161–170, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  2. S. Singh, D. K. Gupta, E. Martínez, and J. L. Hueso, “Semilocal and local convergence of a fifth order iteration with Fréchet derivative satisfying Hölder condition,” Applied Mathematics and Computation, vol. 276, pp. 266–277, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  3. P. K. Parida and D. K. Gupta, “Recurrence relations for semilocal convergence of a Newton-like method in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 345, no. 1, pp. 350–361, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, NY, USA, 1970. View at MathSciNet
  5. I. K. Argyros, A. Cordero, A. Margeñán, and J. R. Torregrosa, “On the convergence of a damped Newton-like method with modified right hand side vector,” Applied Mathematics and Computation, vol. 266, Article ID 21280, pp. 927–936, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  6. P. Maroju, R. Behl, and S. S. Motsa, “Convergence of a parameter based iterative method for solving nonlinear equations in Banach spaces,” in S.S. Convergence of a parameter based iterative method for solving nonlinear equations in Banach spaces, pp. 10–1007, II. Ser, Rend. Circ. Mat. Palermo, 2016. View at Google Scholar
  7. I. K. Argyros and S. George, “Local convergence of deformed Halley method in Banach space under HOLder continuity conditions,” Journal of Nonlinear Science and its Applications. JNSA, vol. 8, no. 3, pp. 246–254, 2015. View at Google Scholar · View at MathSciNet
  8. S. Amat, S. Busquier, and J. M. Gutiérrez, “On the local convergence of secant-type methods,” International Journal of Computer Mathematics, vol. 81, no. 9, pp. 1153–1161, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  9. J. A. Ezquerro and M. Á. Hernández-Verón, “On the accessibility of Newton's method under a Hölder condition on the first derivative,” Algorithms (Basel), vol. 8, no. 3, pp. 514–528, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  10. J. A. Ezquerro and M. Á. Hernández-Verón, “Enlarging the domain of starting points for Newton's method under center conditions on the first Fréchet-derivative,” Journal of Complexity, vol. 33, pp. 89–106, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  11. L. V. Kantorovich and G. P. Akilov, Functional Analysis, Pergamon Press, Nauka, 1982. View at MathSciNet
  12. A. Margeñán and I. K. Argyros, “New semilocal and local convergence analysis for the secant method,” Applied Mathematics and Computation, vol. 262, pp. 298–307, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  13. I. K. Argyros and A. Margeñán, “Expanding the applicability of the secant method under weaker conditions,” Applied Mathematics and Computation, vol. 266, Article ID 21308, pp. 1000–1012, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  14. R. F. King, “Tangent methods for nonlinear equations,” Numerische Mathematik, vol. 18, pp. 298–304, 1971/72. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. I. K. Argyros and H. Ren, “On the convergence of efficient King–Werner-type methods of order 1 + 2,” Journal of Computational and Applied Mathematics, vol. 285, pp. 169–180, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  16. W. Werner, “Some supplementary results on the 1 + √2 order method for the solution of nonlinear equations,” Numerische Mathematik, vol. 38, no. 3, pp. 383–392, 1981/82. View at Publisher · View at Google Scholar · View at MathSciNet
  17. H. Ren and I. K. Argyros, “On the convergence of King-Werner-type methods of order 1 + √2 free of derivatives,” Applied Mathematics and Computation, vol. 256, pp. 148–159, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  18. A. Kumar, D. Gupta, E. Martínez, and S. Singh, “Semilocal convergence of a Secant-type method under weak Lipschitz conditions in Banach spaces,” Journal of Computational and Applied Mathematics, 2017. View at Publisher · View at Google Scholar
  19. L. B. Rall, Computational Solution of Nonlinear Operator Equations, With an appendix by Ramon E. Moore, John Wiley & Sons, Inc, New York, NY, USA, 1969. View at MathSciNet