Runge-Kutta-Fehlberg ODE Solver

RKF45 is a C++ library which implements the Watt and Shampine RKF45 ODE solver.

The RKF45 ODE solver is a Runge-Kutta-Fehlberg algorithm for solving an ordinary differential equation, with automatic error estimation using rules of order 4 and 5.


The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.


RKF45 is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version and a Python version.

Related Data and Programs:

GSL, a C++ library which includes routines for solving differential equations.

NMS, a FORTRAN90 library which includes the DDRIV package of ODE solvers.

ODE, a C++ library which solves a system of ordinary differential equations, by Shampine and Gordon.

ODEPACK, a FORTRAN77 library which contains nine ODE solvers, including LSODE, LSODES, LSODA, LSODAR, LSODPK, LSODKR, LSODI, LSOIBT, and LSODIS, by Alan Hindmarsh.

RK4, a C++ library which applies the fourth order Runge-Kutta algorithm to estimate the solution of an ordinary differential equation at the next time step.

TEST_ODE, a FORTRAN90 library which contains routines which define some test problems for ODE solvers.


  1. Erwin Fehlberg,
    Low-order Classical Runge-Kutta Formulas with Stepsize Control,
    NASA Technical Report R-315, 1969.
  2. Lawrence Shampine, Herman Watts, S Davenport,
    Solving Non-stiff Ordinary Differential Equations - The State of the Art,
    SIAM Review,
    Volume 18, pages 376-411, 1976.
  3. The source code for Shampine and Watt's original FORTRAN77 routine is available at http://www.netlib.org/ode/ the NETLIB ODE web site.

Source Code:

Examples and Tests:

RKF45_test includes a number of examples of how to use RKF45.

RKF45_test2 includes an example in which the ODE includes parameters ALPHA, BETA, and GAMMA, which the user wants to set at run time. The example shows one way in which these values can be shared with the derivative routine.

List of Routines:

You can go up one level to the C++ source codes.

Last revised on 20 November 2007.