## 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).

## About this paper

##### 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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