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Error bounds for the secant method. (English) Zbl 0757.65070
The paper studies an iterative procedure of the secant method in the form \(x_{n+1}=x_ n-\delta f(x_{n-1},x_ n)^{-1}f(x_ n)\) for the solution of the equation \(f(x)=0\), where \(f\) is a nonlinear operator between Banach spaces \(E\) and \(\hat E\), \(x_{-1}\) and \(x_ 0\) are two points in the domain of \(f\) and \(\delta f\) is a consistent approximation of \(\nabla f\).
A priori and a posteriori error estimates for the iterative procedure are provided. The estimates are eventually better than those presented in literature, under the same assumptions. A simple example is provided to show that the results are compared favourably with the available corresponding results.
65J15 Numerical solutions to equations with nonlinear operators
47J25 Iterative procedures involving nonlinear operators
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