An iterative method of global convergence without derivatives in the class of smooth functions (Q1195737)

Iterative methods for solving the nonlinear equation \(f(x)=0\), only using first order derivatives or values of \(f\) are established introducing a parameter function and making the global estimate for the remainder. These methods are globally convergent and contain a real parameter \(\lambda\) (\(\geq 0\)). When \(0\leq\lambda\leq 1\), the order of convergence of the methods is \(1+\lambda\) for a simple real zero of \(f(x)\) and 1 for a multiple real zero. When \(1<\lambda\), these methods are only linearly convergent for any real zero of \(f(x)\).