Optimization by the convergence control parameter in iterative methods

The classical iterative methods, such as the fixed point iteration, the Adomian decomposition method and the homotopy analysis method are discussed in the present paper. It is proven that adding a convergence control parameter into these makes them powerful and rapidly converging to the true solution, whilst the classical correspondences may fail to or slowly converge to the desired solution. The key is to demonstrate the presence of a continuous interval of the convergence control parameter for the considered problem. This allows convergence of such modified iterative methods with an optimum convergence control parameter obtained from squared residual errors of either the original equation or the derivative of iterative solution with respect to the convergence control parameter.