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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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##### References:
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