Exponentially fitted variants of Newton's method with quadratic and cubic convergence

In this paper, we present some new families of Newton-type iterative methods, in which f'(x)=0 is permitted at some points. The presented approach of deriving these iterative methods is different. They have well-known geometric interpretation and admit their geometric derivation from an exponential fitted osculating parabola. Cubically convergent methods require the use of the first and second derivatives of the function as Euler's, Halley's, Chebyshev's and other classical methods do. Furthermore, new classes of third-order multipoint iterative methods free from second derivative are derived by semi-discrete modifications of cubically convergent iterative methods. Further, the approach has been extended to solve a system of non-linear equations.