Approximate integration of ordinary differential equations using Chebyshev series with precision control

An approximate method of solving the Cauchy problem for canonical systems of second order ordinary differential equations is considered. The method is based on using the shifted Chebyshev series and a Markov quadrature formula. Some approaches are given to estimate the errors of an approximate solution and its derivative expressed by partial sums of a certain order shifted Chebyshev series. The errors are estimated using the second approximation of the solution calculated in a special way and expressed by a partial sum of a higher order series. An algorithm of partitioning the integration interval into elementary subintervals to ensure the computation of the solution and its derivative with prescribed accuracy is discussed on the basis of proposed approaches to error estimation.