Recurrence relations for Chebyshev-type methods (Q1964695)

The main aim of this paper is to construct a new iterative method to solve the equation \(F(x)= 0\). This iteration, defined by: \[ \Gamma_n= F'(x_n)^{-1},\quad T(x_n)= \textstyle{{1\over 2}} \Gamma_nA\Gamma_n F(x_n),\quad x_{n+1}= x_n- [I+ T(x_n)]\Gamma_nF(x_n),\quad n\geq 0 \] (where \(I\) is the identity operator on \(X\) and \(A: X\times X\to Y\) is a bilinear operator which satisfies \(\|A\|= \alpha\) \((\alpha\geq 0)\)) has similar operatorial costs and works under the same conditions as Newton's method. Two examples are also given, proving that the velocity of the convergence increases and better error estimates are provided.