On the chord method

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.

Ion Păvăloiu
(Tiberiu Popoviciu Institute of Numerical Analysis)

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

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

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