Further development of the new algorithms for accelerating the convergence of functional-type sequences

There are two aims of this paper. Firstly, we shall introduce the determinantal formulae for the new functional-type algorithms and secondly we shall demonstrate further development of the new functional-type algorithms namely, ρ-algorithms and ϕ-algorithms. We consider the use of the new algorithms to accelerate the convergence of functional-type sequences and their effectiveness for approximating the solution of a given power series is illustrated. In process, we shall demonstrate the convergence of each of the methods considered. The estimates derived using the ρ-algorithms and ϕ-algorithms are found to be very comparative with other similar type of algorithms.