Computer based methods Runge-Kutta

In this flow chart is the end-point of integration, is the start-point of integration and NSTEP is the number of steps. The computation of qh zu i = 1(1)3 is based on the Runge-Kutta-Nystrom method of Dormand and Prince 8(7) (see 9-10). 

Based on the results presented in the relative papers and based on some numerical tests made for this review, the most efficient Runge-Kutta method for specific Schrodinger equations is the one developed by Simos and Williams106 with seven stages while the Runge-Kutta-Nystrom method developed by Simos, Dimas and Sideridis107 gives similar results in accuracy and computational efficiency. 

The KPS calculation was based on integrating Hamilton s equations of motion for the time evolution of the Cartesian components of the Jacobi coordinates that describe the three-atom system. A fourth-order Runge-Kutta method was used for the numerical integration, and the computations were done using an IBM 7090-4 computer at the IBM Watson Research Center and the Columbia Computing Center. The computation time per trajectory was listed as 10 s using a time step of 0.025 fs. 

The differential equation has been resolved with an algorithm which makes use of an explicit Runge-Kutta(4,5) formula, the so-called Dormand-Prince pair. The least square minimization was implemented by a large-scale algorithm based on the interior-reflective Newton method. All the computations have been achieved by MATLAB software. 

A. Higinio Ramos and A. Jesus Vigo-Aguiar, A fourth-order Runge-Kutta method based on BDF-type Chebyshev approximations, Journal of Computational and Applied Mathematics, 2007, 204, 124 136. 

---