Methodology of the Taylor Series Based Computations

In this paper an outline is presented of historical and current developments in the application of recurrent Taylor series to the integration of systems of ordinary differential equations. The idea of an extremely accurate and fast method for numerical solutions of differential equations is presented in the paper. In general Taylor series are not included or not even mentioned in surveys on numerical integration techniques as the programs were written by mathematicians with the main objective of demonstrating the feasibility of the concept and with the goal of providing integration algorithms of very high accuracy. For this reason the programs should be looked upon as a stimulus for writing more advanced software employing Taylor series better able to compete with programs using other methods. An attempt in this direction is TKSL, a program the results of which will be dealt with.