Previous |  Up |  Next


secant method; Banach space; majorizing sequence; divided difference; Fréchet-derivative
We provide new sufficient conditions for the convergence of the secant method to a locally unique solution of a nonlinear equation in a Banach space. Our new idea uses “Lipschitz-type” and center-“Lipschitz-type” instead of just “Lipschitz-type” conditions on the divided difference of the operator involved. It turns out that this way our error bounds are more precise than the earlier ones and under our convergence hypotheses we can cover cases where the earlier conditions are violated.
[1] I. K.  Argyros: A convergence theorem for Newton-like methods under generalized Chen-Yamamoto-type assumptions. Appl. Math. Comput. 61 (1994), 25–37. DOI 10.1016/0096-3003(94)90144-9 | MR 1274298 | Zbl 0796.65077
[2] I. K.  Argyros: On a new Newton-Mysovskii-type theorem with applications to inexact-Newton-like methods and their discretizations. IMA J.  Numer. Anal. 18 (1998), 37–56. DOI 10.1093/imanum/18.1.37 | MR 1492047
[3] I. K.  Argyros and F.  Szidarovszky: The Theory and Application of Iteration Methods. CRC Press, Boca Raton, 1993. MR 1272012
[4] W. E.  Bosarge and P. L.  Falb: A multipoint method of third order. J.  Optimiz. Th. Applic. 4 (1969), 156–166. DOI 10.1007/BF00930576 | MR 0248581
[5] J. E.  Dennis: Toward a unified convergence theory for Newton-like methods. In: Nonlinear Functional Analysis, L. B.  Rall (ed.), Academic Press, New York, 1971. MR 0278556 | Zbl 0276.65029
[6] J. M.  Gutiérrez: A new semilocal convergence theorem for Newton’s method. J.  Comput. Appl. Math. 79 (1997), 131–145. DOI 10.1016/S0377-0427(97)81611-1 | MR 1437974
[7] J. M.  Gutiérrez, M. A.  Hernandez and M. A. Salanova: Accessibility of solutions by Newton’s method. Intern. J.  Comput. Math. 57 (1995), 239–247. DOI 10.1080/00207169508804427
[8] 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. DOI 10.1016/S0377-0427(99)00116-8 | MR 1747223
[9] L. V.  Kantorovich and G. R.  Akilov: Functional Analysis in Normed Spaces. Pergamon Press, Oxford Press, , 1982. MR 0664597
[10] J. M.  Ortega and W. C.  Rheinboldt: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York, 1970. MR 0273810
[11] F. A.  Potra: Sharp error bounds for a class of Newton-like methods. Libertas Mathematica 5 (1985), 71–84. MR 0816258 | Zbl 0581.47050
[12] T.  Yamamoto: A convergence theorem for Newton-like methods in Banach spaces. Numer. Math. 51 (1987), 545–557. DOI 10.1007/BF01400355 | MR 0910864 | Zbl 0633.65049
Partner of
EuDML logo