Dormand-Prince method

The Dormand-Prince method, also called RKDP or DOPRI, is an embedded method for solving ordinary differential equations and a member of the Runge-Kutta family of solvers. It uses six function evaluations to calculate fourth-order and fifth-order accurate solutions, and the difference between the two is taken as the error estimate of the fourth-order solution, which is convenient for adaptive step-size integration. The method has seven stages but needs only six evaluations per step because of the First Same As Last (FSAL) property: the last stage is evaluated at the same point as the first stage of the next step. Dormand and Prince chose the coefficients to minimize the error of the fifth-order solution, unlike the Fehlberg method, which was built so that the fourth-order solution has a small error, so Dormand-Prince is better suited to local extrapolation. It is the default method in the ode45 solver of MATLAB and GNU Octave, and an option in the SciPy and Julia ODE libraries.