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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.
MSC:
65J15 Numerical solutions to equations with nonlinear operators
47J25 Iterative procedures involving nonlinear operators
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References:
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