Introduction to the improved functional epsilon algorithm

This paper introduces the improved functional epsilon algorithm. We have defined this new method in principle of the modified Aitken Δ2 algorithm. Moreover, we have found that the improved functional epsilon algorithm has remarkable precision of the approximation of the exact solution and there exists a relationship with the integral Padé approximant. The use of the improved functional epsilon algorithm for accelerating the convergence of sequence of functions is demonstrated. The relationship of the improved functional epsilon algorithm with the integral Padé approximant is also demonstrated. Moreover, we illustrate the similarity between the integral Padé approximant and the modified Aitken Δ2 algorithm; thus we have shown that the integral Padé approximant is a natural generalisation of modified Aitken Δ2 algorithm.