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On a higher order secant method. (English) Zbl 1035.65057
A modified secant method is introduced for solving a nonlinear equation $f(x)=0$. The secant of $f(x)$ at $x_{k}$ is replaced by a higher order approximation. It is show that this method can have quadratic convergence as the Newton method. Global monotonic convergence is proven for real functions. In Banach space, the convergence is established with relaxed Kantorovich-Ostrowski type conditions. Numerical examples confirm the conclusions.

65J15Equations with nonlinear operators (numerical methods)
47J25Iterative procedures (nonlinear operator equations)
Full Text: DOI
[1] Argyros, I. K.: A convergence theorem for Newton-like methods under generalized Chen--yamamoto-type assumptions. Appl. math. Comput. 61, 25-36 (1994) · Zbl 0796.65077
[2] Amat, S.; Busquier, S.; Candela, V. F.: Geometry and convergence of some third order methods. Southwest, J. Pure appl. Math. 2, 61-72 (2001) · Zbl 0992.65047
[3] Hernández, M. A.; Rubio, M. J.: A modification of the kantarovich conditions for the secant method. Southwest J. Pure appl. Math. 1, 13-21 (2001) · Zbl 1008.47064
[4] M.A. Hernández, M.J. Rubio, Las Diferencias divididas en la Aproximación de Races para Operadores No Diferenciables. Proceeding CEDYA VII-CMA IV, Salamanca, 2001
[5] Ortega, J. M.; Rheinboldt, W. C.: Iterative solution of nonlinear equations in Banach spaces. (1970) · Zbl 0241.65046
[6] Zabrejko, P. P.; Nguen, D. F.: The majorant method in the theory of Newton--Kantorovich approximation and the ptak error estimates. Numer. func. Anal. opt. 9, 671-684 (1987) · Zbl 0627.65069