Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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EXPANDING THE APPLICABILITY OF SECANT METHOD WITH APPLICATIONS

  • Magrenan, A. Alberto (Escuela De Ingenieria Universidad Internacional De La Rioja) ;
  • Argyros, Ioannis K. (Department of Mathematics Sciences Cameron University)
  • Received : 2014.03.24
  • Published : 2015.05.31
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Abstract

We present new sufficient convergence criteria for the convergence of the secant-method to a locally unique solution of a nonlinear equation in a Banach space. Our idea uses Lipschitz and center-Lipschitz instead of just Lipschitz conditions in the convergence analysis. The new convergence criteria can always be weaker than the corresponding ones in earlier studies. Numerical examples are also provided in this study to solve equations in cases not possible before.

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References

  1. S. Amat, S. Busquier, and J. M. Gutierrez, On the local convergence of secant-type methods, Int. J. Comput. Math. 81 (2004), no. 9, 1153-1161. https://doi.org/10.1080/00207160412331284123
  2. S. Amat, S. Busquier, and A. A. Magrenan, Reducing chaos and bifurcations in Newtontype methods, Abstr. Appl. Anal. 2013 (2013), Article ID 726701, 10 pages.
  3. S. Amat, S. Busquier, and M. Negra, Adaptive approximation of nonlinear operators, Numer. Funct. Anal. Optim. 25 (2004), no. 5-6, 397-405. https://doi.org/10.1081/NFA-200042628
  4. S. Amat, A. A. Magrenan, and N. Romero, On a two step relaxed Newton-type methods, Appl. Math. Comput. 219 (2013), no. 24, 11341-11347. https://doi.org/10.1016/j.amc.2013.04.061
  5. I. K. Argyros, A unifying local-semilocal convergence analysis and applications for two-point Newton-like methods in Banach space, J. Math. Anal. Appl. 298 (2004), no. 2, 374-397. https://doi.org/10.1016/j.jmaa.2004.04.008
  6. I. K. Argyros, New sufficient convergence conditions for the Secant method, Chechoslovak Math. J. 55(130) (2005), no. 1, 175-187. https://doi.org/10.1007/s10587-005-0013-1
  7. I. K. Argyros, Convergence and Applications of Newton-Type Iterations, Springer-Verlag Publ., New-York, 2008.
  8. I. K. Argyros, Weaker conditions for the convergence of Newton's method, J. Complexity 28 (2012), no. 3, 364-387. https://doi.org/10.1016/j.jco.2011.12.003
  9. I. K. Argyros, D. Gonzalez, and A. A. Magrenan, A semilocal convergence for a uniparametric family of efficient secant-like methods, J. Function Spaces 2014 (2014), Article ID 467980, 10 pages.
  10. W. E. Bosarge and P. L. Falb, A multipoint method of third order, J. Optimization Theory Appl. 4 (1969), 156-166. https://doi.org/10.1007/BF00930576
  11. J. E. Dennis, Toward a unified convergence theory for Newton-like methods, in Nonlinear Functional Anal. and Appl. (Proc. Advanced Sem., Math. Res. Center, Univ. of Wisconsin, Madison, Wis., 1970) pp. 425-472 Academic Press, New York, 1971.
  12. M. Garcia-Olivo, El metodo de Chebyshev para el calculo de las raices de ecuaciones no lineales, (PhD Thesis), Servicio de Publicaciones, Universidad de La Rioja, 2013.
  13. J. M. Gutierrez, A. A. Magrenan, and N. Romero, On the semilocal convergence of Newton-Kantorovich method under center-Lipschitz conditions, Appl. Math. Comput. 221 (2013), 79-88. https://doi.org/10.1016/j.amc.2013.05.078
  14. M. A. Hernandez, M. J. Rubio, and J. A. Ezquerro, Secant-like methods for solving nonlinear integral equations of the Hammerstein type, J. Comput. Appl. Math. 115 (2000), 245-254. https://doi.org/10.1016/S0377-0427(99)00116-8
  15. M. A. Hernandez, M. J. Rubio, and J. A. Ezquerro, Solving a special case of conservative problems by Secant-like method, Appl. Math. Comput. 169 (2005), no. 2, 926-942. https://doi.org/10.1016/j.amc.2004.09.070
  16. L. V. Kantorovich and G. P. Akilov, Functional Analysis, Pergamon Press, Oxford, 1982.
  17. P. Laasonen, Ein uberquadratisch konvergenter iterativer algorithmus, Ann. Acad. Sci. Fenn. Ser I 450 (1969), 1-10.
  18. A. A. Magrenan, Estudio de la dinamica del metodo de Newton amortiguado, PhD Thesis, Servicio de Publicaciones, Universidad de La Rioja, 2013.
  19. J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970.
  20. A. M. Ostrowski, Solution of equations in Euclidian and Banach Spaces, Academic Press, New York, 1972.
  21. F. A. Potra, An error analysis for the secant method, Numer. Math. 38 (1982), no. 3, 427-445. https://doi.org/10.1007/BF01396443
  22. F. A. Potra, Sharp error bounds for a class of Newton-like methods, Libertas Math. 5 (1985), 71-84.
  23. F. A. Potra and V. Ptak, Nondiscrete Induction and Iterative Processes, Pitman, New York, 1984.
  24. J. W. Schmidt, Untere Fehlerschranken fur Regula-Falsi Verhafren, Period. Math. Hungar. 9 (1978), no. 3, 241-247. https://doi.org/10.1007/BF02018090
  25. A. S. Sergeev, The method of Chords, Sibirsk, Mat. Z. 11 (1961), 282-289.
  26. S. Ulm, A majorant principle and the method of secants, Izv. Akad. Nauk Eston. SSR, Ser. Fiz.-Mat. 13 (1964), 217-227.
  27. T. Yamamoto, A convergence theorem for Newton-like methods in Banach spaces, Numer. Math. 51 (1987), no. 5, 545-557. https://doi.org/10.1007/BF01400355
  28. M. A. Wolfe, Extended iterative methods for the solution of operator equations, Numer. Math. 31 (1978), no. 2, 153-174. https://doi.org/10.1007/BF01397473