Newton-Type Iterations, Convergence and Accelerations

The damped Newton method or continuous analogue of Newton’s method (CANM) for solving nonlinear equations is considered. The influence of the parameter \(\tau _k\) on the convergence of the method is studied, and the \(\tau \) -region of convergence is given. The semilocal convergence theorem for CANM is proved using a technique consisting of a new system of recurrence relations. A close relationship between the inexact Newton’s method and CANM is revealed. Based on this relationship, it is concluded that the choice of \(\tau _k\) from (0,2) can guarantee the convergence of the method. We developed an accelerating procedure for Newton-type iterations, which leads to speeding up the convergence by certain units.