Semilocal convergence of the secant method under mild convergence conditions of differentiability. (English) Zbl 1055.65069

Summary: We obtain a semilocal convergence result for the secant method in Banach spaces under mild convergence conditions. We consider a condition for divided differences which generalizes those usual ones, i.e., Lipschitz continuous and Hölder continuous conditions. Also, we obtain a result for uniqueness of solutions.


65J15 Numerical solutions to equations with nonlinear operators
47J25 Iterative procedures involving nonlinear operators
65L10 Numerical solution of boundary value problems involving ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
Full Text: DOI


[1] Sergeev, A., On the method of chords, Sibirsk. mat. Ẑ., 2, 282-289, (1961)
[2] Schmidt, J.W., Regula-falsi verfahren mit konsistenter steigung und majoranten prinzip, Periodica Mathematica hungarica, 5, 187-193, (1974) · Zbl 0291.65017
[3] Argyros, I.K., On the secant method, Publ. math. debrecen, 43, 3/4, 223-238, (1993) · Zbl 0796.65075
[4] Dennis, J.R., Toward a unified convergence theory for Newton-like methods, (), 425-472
[5] Rheinboldt, W.C., A unified convergence theory for a class of iterative processes, SIAM J. numer. anal., 5, 1, 42-63, (1968) · Zbl 0155.46701
[6] Potra, F.A., An application of the induction method of V. pták to the study of regula falsi, Aplikace matematiky, 26, 111-120, (1981) · Zbl 0486.65038
[7] Potra, F.A.; Pták, V., Nondiscrete induction and iterative processes, (1984), Pitman · Zbl 0549.41001
[8] Rokne, J., Newton’s method under mild differentiability conditions with error analysis, Numer. math., 18, 401-412, (1972) · Zbl 0221.65084
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.