On the chord method

Abstract

Let \(X_{1},X_{2}\) be two Banach spaces, \(f:X_{1}\rightarrow X_{2}\) a nonlinear mapping and consider the chord method for solving the equation \(f\left(x\right) =0\): \[x_{n+1}=x_n-[x_{n-1},x_{n};f]^{-1}f(x_n), \quad n=1,…\] Under some simple conditions on the divided differences of order one of \(f\), of the form \[\|[y, u; f] − [x, y; f]\| ≤ l_1 \|x − u\| ^p + l_2 \|x − y\|^p + l_3 \|y − u\|^p\] we show that the chord method converge to the solution. We obtain error estimations and determine the convergence order.

Authors

Ion Păvăloiu

Keywords

chord method; equations in Banach spaces; error estimation; convergence order

References

[1] Argyros, I. K., The secant method and fixed points on nonlinear operators, Mh. Math.,  106 (1988), 85–94.

[2] Balazs, M. ¸si Goldner, G., Observatii asupra diferentelor divizate si asupra metodei coardei. Revista de analiza numerica si teoria aproximatiei vol. 3 (1974) fasc. 1, 19–30.

[3] Pavaloiu, I., Remarks on the secant method for the solution of nonlinear operatorial equations, Research Seminars, Seminar on Mathematical Analysis, Preprint no. 7, (1991), pp. 127 132.

[4] Pavaloiu, I., Introducere in teoria aproximarii solutiilor ecuatiilor. Ed. Dacia, ClujNapoca, 1976.

[5] Schmidt, I. W., Eine Ubertagungen der Regula Falsi auf Gleichungen, in ”Banachraumen” I ZAMM, 48, 1–8 (1963).

[6] Schmidt, I. W., Eine Ubertagungen der Regula Falsi auf Gleichungen, in ”Banachraumen” II 97–110 (1963).

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Cite this paper as:

I. Păvăloiu, On the chord method, Bul. Ştiinţ. Univ. Baia Mare, Seria B. Fasc. Mat.-Fiz., 7 (1991), pp. 61-66.

Journal

Bul. Ştiinţ. Univ. Baia Mare, Seria B. Fasc. Mat.-Fiz., 7 (1991)

Publisher Name

Bul. Ştiinţ. Univ. Baia Mare

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