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Chebyshev’s approximation algorithms and applications. (English) Zbl 0985.65058
Let \(X,Y\) be Banach spaces, \(F: X \to Y\) a nonlinear twice Fréchet-differentiable operator such that \(F'(x)\) is continuously invertible. To solve the equation \(F(x)=0\) the author derives the family of multipoint iterations, with \(R\)-order three \[ y_n=x_n-\Gamma_n F(x_n), \qquad z_n=x_n+\theta (y_n-x_n), \] \[ P(x_n,z_n)=\theta^{-1} \Gamma_n [F'(x_n)-F'(z_n)], \] \[ x_{n+1}=y_n+1/2 P(x_n,z_n)(y_n-x_n), \] where \(\theta \in (0,1]\), \(\Gamma_n=[F'(x_n)]^{-1}\). The results of numerical tests with nonlinear integral equations are presented.

MSC:
65J15 Numerical solutions to equations with nonlinear operators (do not use 65Hxx)
47J25 Iterative procedures involving nonlinear operators
45G10 Other nonlinear integral equations
65R20 Numerical methods for integral equations
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