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Weaker convergence conditions for the secant method. (English) Zbl 1340.65109
Summary: We use tighter majorizing sequences than in earlier studies to provide a semilocal convergence analysis for the secant method. Our sufficient convergence conditions are also weaker. Numerical examples are provided where earlier conditions do not hold but for which the new conditions are satisfied.

MSC:
65J15 Numerical solutions to equations with nonlinear operators (do not use 65Hxx)
47J25 Iterative procedures involving nonlinear operators
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[1] Argyros, I. K., A unifying local-semilocal convergence analysis and applications for two-point Newton-like methods in Banach space, J. Math. Anal. Appl., 298, 374-397, (2004) · Zbl 1057.65029
[2] I. K. Argyros: Convergence and Applications of Newton-Type Iterations. Springer, New York, 2008. · Zbl 1153.65057
[3] Argyros, I. K., New sufficient convergence conditions for the secant method, Czech. Math. J., 55, 175-187, (2005) · Zbl 1081.65043
[4] Argyros, I. K., On the Newton-Kantorovich hypothesis for solving equations, J. Comput. Appl. Math., 169, 315-332, (2004) · Zbl 1055.65066
[5] I. K. Argyros, Y. J. Cho, S. Hilout: Numerical Methods for Equations and Its Applications. CRC Press, Boca Raton, 2012. · Zbl 1254.65068
[6] I. K. Argyros, S. Hilout: Computational Methods in Nonlinear Analysis. Efficient Algorithms, Fixed Point Theory and Applications. World Scientific, Hackensack, 2013. · Zbl 1279.65062
[7] Argyros, I. K.; Hilout, S., Convergence conditions for secant-type methods, Czech. Math. J., 60, 253-272, (2010) · Zbl 1224.65141
[8] Argyros, I. K.; Hilout, S., Semilocal convergence conditions for the secant method using recurrent functions, Rev. Anal. Numér. Théor. Approx., 40, 107-119, (2011) · Zbl 1289.65136
[9] Argyros, I. K.; Hilout, S., Weaker conditions for the convergence of newton’s method, J. Complexity, 28, 364-387, (2012) · Zbl 1245.65058
[10] Bosarge, W. E.; Falb, P. L., A multipoint method of third order, J. Optimization Theory Appl., 4, 155-166, (1969) · Zbl 0172.18703
[11] Dennis, J. E.; Rall, L.B. (ed.), Toward a unified convergence theory for Newton-like methods. nonlinear functional analysis and applications, 425-472, (1971), New York
[12] Hernández, M. A.; Rubio, M. J.; Ezquerro, J.A., Secant-like methods for solving nonlinear integral equations of the Hammerstein type, J. Comput. Appl. Math., 115, 245-254, (2000) · Zbl 0944.65146
[13] Hernández, M. A.; Rubio, M. J.; Ezquerro, J.A., Solving a special case of conservative problems by secant-like methods, Appl. Math. Comput., 169, 926-942, (2005) · Zbl 1080.65044
[14] L. V. Kantorovich, G. P. Akilov: Functional Analysis. Pergamon Press. Transl. from the Russian by Howard L. Silcock. 2nd ed, Oxford, 1982. · Zbl 0375.65030
[15] Laasonen, P., Ein überquadratisch konvergenter iterativer algorithmus, Ann. Acad. Sci. Fenn., Ser. A I, 450, 10, (1969) · Zbl 0193.11704
[16] J. M. Ortega, W. C. Rheinboldt: Iterative Solution of Nonlinear Equations in Several Variables. Computer Science and Applied Mathematics, Academic Press, New York, 1970. · Zbl 0241.65046
[17] Potra, F.-A., On the convergence of a class of Newton-like methods. iiterative solution of nonlinear systems of equations, No. 953, 125-137, (1982), Berlin
[18] Potra, F.-A., Sharp error bounds for a class of Newton-like methods, Libertas Math., 5, 71-84, (1985) · Zbl 0581.47050
[19] F.-A. Potra, V. Pták: Nondiscrete Induction and Iterative Processes. Research Notes in Mathematics 103, Pitman Advanced Publishing Program, Boston, 1984. · Zbl 0549.41001
[20] Potra, F.-A.; Pták, V., Sharp error bounds for newton’s process, Numer. Math., 34, 63-72, (1980) · Zbl 0434.65034
[21] Proinov, P. D., New general convergence theory for iterative processes and its applications to Newton-Kantorovich type theorems, J. Complexity, 26, 3-42, (2010) · Zbl 1185.65095
[22] Schmidt, J. W., Untere fehlerschranken für regula-falsi-verfahren, Period. Math. Hung., 9, 241-247, (1978) · Zbl 0401.65036
[23] Wolfe, M. A., Extended iterative methods for the solution of operator equations, Numer. Math., 31, 153-174, (1978) · Zbl 0375.65030
[24] Yamamoto, T., A convergence theorem for Newton-like methods in Banach spaces, Numer. Math., 51, 545-557, (1987) · Zbl 0633.65049
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