Convergence of a continuous analog of Newton's method for solving nonlinear equations

The influence of the parameter in the continuous analog of Newton's method (CANM) on the convergence and on the convergence rate is studied. A $\tau$-region of convergence of CANM for both scalar equations and equations in a Banach space is obtained. Some almost optimal choices of the parameter are proposed. It is also shown that the well-known higher order convergent iterative methods lead to the CANM with an almost optimal parameter. Several sufficient convergence conditions for these methods are obtained.