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

Hernández, M. A.; Rubio, M. J.

Comput. Math. Appl. 44, No. 3-4, 277-285 (2002).

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.

MSC:

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

Keywords:

secant method; Recurrence relations; Boundary value problem; nonlinear operator equation; numerical examples; Banach spaces; divided difference

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