One of the flash analytic approximate methods of nowadays for the solution of highly nonlinear algebraic or differential equations arising from the mathematical modelling of industrial and technological applications is the homotopy analysis method (HAM). The success of the HAM is mainly due to a so-called convergence control parameter, $h$, plugged into the system externally which is missing in other competing methods. A simple algorithm to determine this parameter is introduced in this paper, besides the well-known approaches of constant $h$−level curves, the squared residual error and the recent ratio technique. Comparison of the four approaches yields nearly the same convergence control parameters with the advantage of the newly proposed approach in terms of its simplicity and less CPU time requirement. Moreover, a convergence accelerating method is suggested here based on updating the initial guess of the solution at some low-order homotopy series approximation of the solution. It appears to extend the interval of convergence control parameter. The provided examples of real life phenomena in combination with this technique demonstrate a successful improvement over the classical HAM method.