Semilocal convergence conditions for the secant method, using recurrent functions

Ioannis K. Argyros; Saïd Hilout

doi:10.33993/jnaat402-1041

Authors

Ioannis K. Argyros Cameron University, USA
Saïd Hilout Poitiers University, France

DOI:

https://doi.org/10.33993/jnaat402-1041

Keywords:

recurrent functions, semilocal convergence, secant method, Banach space, majorizing sequence, divided difference, Fréchet derivative

Abstract views: 250

Abstract

Using our new concept of recurrent functions, we present new sufficient convergence conditions for the secant method to a locally unique solution of a nonlinear equation in a Banach space. We combine Lipschitz and center-Lipschitz conditions on the divided difference operator to obtain the semilocal convergence analysis of the secant method. Our error bounds are tighter than earlier ones. Moreover, under our convergence hypotheses, we can expand the applicability of the secant method in cases not covered before [8], [9], [12]-[14], [16], [19]-[21]. Application and examples are also provided in this study.

Downloads

Download data is not yet available.

References

I.K. Argyros, On the Newton-Kantorovich hypothesis for solving equations, J. Comput. Appl. Math., 169, pp. 315-332, 2004, https://doi.org/10.1016/j.cam.2004.01.029 DOI: https://doi.org/10.1016/j.cam.2004.01.029

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, pp. 374-397, 2004, https://doi.org/10.1016/j.jmaa.2004.04.008 DOI: https://doi.org/10.1016/j.jmaa.2004.04.008

I.K. Argyros, New sufficient convergence conditions for the Secant method, Chechoslovak Math. J., 55, pp. 175-187, 2005, https://doi.org/10.1007/s10587-005-0013-1 DOI: https://doi.org/10.1007/s10587-005-0013-1

I.K. Argyros, Convergence and applications of Newton-type iterations, Springer-Verlag Publ., New-York, pp.1-68, 2008. DOI: https://doi.org/10.1007/978-0-387-72743-1_4

I.K. Argyros and S. Hilout, Efficient methods for solving equations and variational inequalities, Polimetrica Publisher, Milano, Italy, 2009.

I.K. Argyros and S. Hilout, Convergence conditions for Secant--type methods, Chechoslovak Math. J., 60, pp. 253-272. 2010, https://doi.org/10.1007/s10587-010-0014-6 DOI: https://doi.org/10.1007/s10587-010-0014-6

W.E. Bosarge and P.L. Falb, A multipoint method of third order, J. Optimiz. Th. Appl., 4, pp. 156-166, 1969, https://doi.org/10.1007/bf00930576 DOI: https://doi.org/10.1007/BF00930576

J.E. Dennis, Toward a unified convergence theory for Newton--like methods, in Nonlinear Functional Analysis and Application, (L.B. Rall, ed.), Academic Press, New York, pp. 425-472, 1971), https://doi.org/10.1016/b978-0-12-576350-9.50010-2 DOI: https://doi.org/10.1016/B978-0-12-576350-9.50010-2

M.A. Hernández, M.J. Rubio and J.A. Ezquerro, Solving a special case of conservative problems by Secant-like method, Appl. Math. Cmput., 169, pp. 926-942, 2005, https://doi.org/10.1016/j.amc.2004.09.070 DOI: https://doi.org/10.1016/j.amc.2004.09.070

M.A. Hernández, M.J. Rubio and J.A. Ezquerro, Secant--like methods for solving nonlinear integral equations of the Hammerstein type, J. Comput. Appl. Math., 115, pp. 245-254, 2000, https://doi.org/10.1016/s0377-0427(99)00116-8 DOI: https://doi.org/10.1016/S0377-0427(99)00116-8

L.V. Kantorovich and G.P. Akilov, Functional Analysis, Pergamon Press, Oxford, 1982.

P. Laasonen, Ein überquadratisch konvergenter iterativer algorithmus, Ann. Acad. Sci. Fenn. Ser I, 450, pp. 1-10, 1969, https://doi.org/10.5186/aasfm.1969.450 DOI: https://doi.org/10.5186/aasfm.1969.450

J.M. Ortega and W.C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970.

F.A. Potra, On the convergence of a class of Newton-like methods. Iterative solution of nonlinear systems of equations, (Oberwolfach, 1982), Lecture Notes in Math., 953, Springer, Berlin-New York, pp. 125-137, 1982. https://doi.org/10.1007/bfb0069378 DOI: https://doi.org/10.1007/BFb0069378

F.A. Potra, On an iterative algorithm of order 1.839⋯ for solving nonlinear operator equations, Numer. Funct. Anal. Optim., 7(1), pp. 75-106, 1984/85, https://doi.org/10.1080/01630568508816182 DOI: https://doi.org/10.1080/01630568508816182

F.A. Potra, Sharp error bounds for a class of Newton-like methods, Libertas Mathematica, 5, pp. 71-84, 1985.

P.D. Proinov, General local convergence theory for a class of iterative processes and its applications to Newton's method, J. Complexity, 25, pp. 38-62, 2009, https://doi.org/10.1016/j.jco.2008.05.006 DOI: https://doi.org/10.1016/j.jco.2008.05.006

P.D. Proinov, New general convergence theory for iterative processes and its applications to Newton-Kantorovich type theorems, J. Complexity, 26, pp. 3-42, 2010, https://doi.org/10.1016/j.jco.2009.05.001 DOI: https://doi.org/10.1016/j.jco.2009.05.001

J.W. Schmidt, Untere Fehlerschranken fur Regula-Falsi Verhafren, Period. Hungar., 9, pp. 241-247, 1978, https://doi.org/10.1007/bf02018090 DOI: https://doi.org/10.1007/BF02018090

T. Yamamoto, A convergence theorem for Newton--like methods in Banach spaces, Numer. Math., 51, pp. 545-557, 1987, https://doi.org/10.1007/bf01400355 DOI: https://doi.org/10.1007/BF01400355

M.A. Wolfe, Extended iterative methods for the solution of operator equations, Numer. Math., 31, pp. 153-174, 1978, https://doi.org/10.1007/bf01397473 DOI: https://doi.org/10.1007/BF01397473