Approximate integration of the canonical systems of second order ordinary differential equations with the use of Chebyshev series with error estimation for solution and its derivative

An approximate method of solving the Cauchy problem for canonical second-order ordinary differential equations is considered. This method is based on using the shifted Chebyshev series and a Markov quadrature formula. A number of procedures are discussed to estimate the error of the approximate solution and its derivative expressed by partial sums of shifted Chebyshev series of a certain order. The error is estimated using the second approximate solution obtained by a special way and represented by a partial sum of higher order. The proposed procedures are used to develop an algorithm to partition the integration interval into elementary subintervals, which allows one to compute an approximate solution and its derivative with a prescribed accuracy.