Some improvements of Ostrowski's method
The authors present some new variants of Ostrowski's method for finding a simple root of a non-linear equation \(f(x) = 0\), where \(f\) is a scalar function on an open interval \(D \subset\mathbb R\). One family of methods with eighth-order convergence is given by \[ H_\beta (y_n,z_n)= \frac{f(z_n)}{f(y_n)- \beta f(z_n)}, \] \[ x_{n+1}= z_n- [(1+H_2 (x_n,y_n))^2+ (1+4H_2 (x_n,y_n))H_\beta (y_n,z_n)] \frac{f(z_n)}{{f}'(x_n)}, \] where \(\beta \in\mathbb R\). The second is given by \[ u_n= z_n- (1+H_2 (x_n,y_n))^2\frac{f(z_n)}{{f}'(x_n)}, \] \[ x_{n+1}= u_n- (1+4H_2 (x_n,y_n))\frac{z_n-u_n}{y_n-u_n-\beta (z_n-u_n)}\frac{f(z_n)}{{f}'(x_n)}. \] In both methods \(y_n, z_n\) and \(H_2\) are defined by \[ y_n= x_n- \frac{f(x_n)}{{f}'(x_n)}, \] \[ H_2 (x_n,y_n)= \frac{f(y_n)}{f(x_n)- 2f(y_n)}, \] \[ z_n= y_n- H_2 (x_n,y_n)(x_n-y_n). \] For the each iteration these methods require three evaluation of the function and one evaluation of its first derivative. The methods have the efficiency index equal to 1.682. Numerical tests verifying the theoretical results are also given in this paper.
scientific article; zbMATH DE number 5655254