The convergence theorem for a family deformed Chebyshev method in Banach space
For a nonlinear operator equation \(F(x)=0\) on an open convex region in a Banach space the Chebyshev method which improves the convergence order of Newton's method is considered. The main disadvantage of the Chebyshev method is that it needs to compute second-order derivatives in each step. Therefore the computing costs will be high. The authors propose a deformation of the Chebyshev method in the form \[ y_n=x_n-F'(x_n)^{-1}F(x_n);\quad H(x_n,y_n)=2F'(x_n)^{-1}(F'(x_n+(y_n-x_n) /2)-F'(x_n)); \] \[ \lambda\in[0,1];\quad x_{n+1}=y_n-H(x_n,y_n)(I-\lambda H(x_n,y_n)) (y_n-x_n)/2, \] to avoid the second derivatives. It is proved that if the \(\gamma\)-condition introduced by \textit{X. Wang} [Chin. Sci. Bull. 42, No.~7, 552--555 (1997; Zbl 0884.30004)] is satisfied, and if the choice of \(\lambda\) is reasonable, then the proposed iteration scheme has the convergence order three.
scientific article; zbMATH DE number 5115408