Technical Initial Problems and Automatic Transformation

In this paper an outline is given of historical and current developments in the application of recurrent Taylor series to the integration of systems of ordinary differential equations. Then an extremely accurate and fast method for the numerical solution of ordinary differential equations is presented. In general Taylor series method is not included or 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 finding integration algorithms of very high accuracy. For this reason such 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.