Newton method for nonlinear operator equations

Newton?s method is a relatively simple, practical, and widely-used root finding method. It is easy to see that while in some cases the method rapidly converges to a root of the function, in some other cases it may fail to convergeat at all. This is one reason as of why it is so important not only to understand the construction of the method, but also to understand its limitations.Assume that an initial estimate is known for the desired root of f(x) = 0. Newton?s method will produce a sequence of iterates , which we hope will converge to .Since is assumed close to , approximate the graph of the function y = f(x) in the vicinity of its root by constructing its tangent line at ( ,f( )).Then use the root of this tangent line to aproximate ; call this new aproximation x .and this leads to the iteration formulaNewton?s method is the best known procedura for finding the roots of an equation. It had been generalized in many ways for the solution of nonlinear operator equations by L.V.Kantorovich, for example system of nonlinear equations and nonlinear integral and differential equations.Key words: Nonlinear, Operator, Iteration process, Newton method, Equation.