A family of Steffensen type methods with seventh-order convergence
The authors develop a new family of seventh-order derivative-free methods for finding the root of nonlinear equations \(f(x)=0\). They consider a new iteration scheme \(y_n=x_n-f(x_n)/f[x_n,z_n]\), \(u_n=g_4(x_n)\), \(x_{n+1}=u_n-f(u_n)/(f[u_n,x_n]+f[u_n,y_n]-f[x_n,y_n])\), where \(g_4(x_n)\) is an optimal fourth-order Steffensen type method, and the third step is constructed by using suitable Taylor expansions and divided difference approximations. The new method is proven to be seventh-order. Finally, the authors give some numerical examples to illustrate the convergence of the method.
scientific article; zbMATH DE number 6149002