Behavior of the Parallel Code Method

Parallel Code Method (PCM) is one of iterative methods for solving nonlinear equations. The method uses an initial constant matrix instead of Jacobi matrix at every iteration. First, the notion of local convergence for PCM is defined and a sufficient condition for local convergence is given. Secondly, it is shown that the approximations generated by PCM are represented by an infinite power series under the condition that a root of the nonlinear equation is not multiple. Next, based on this property, the sequence of approximations of PCM is accelerated and a better approximation of a root is estimated. Finally, it is verified by numerical examples that the sequence is accelerated according to the property