Runge-Kutta-Fehlberg ODE Solver

RKF45 is a Python library which implements the Watt and Shampine RKF45 solver for systems of ordinary differential equations (ODE's).

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:

LORENZ_ODE, a Python program which approximates solutions to the Lorenz system of ordinary differential equations (ODE's) creating graphics output using matplotlib.

RK4, a Python library which applies the fourth order Runge-Kutta (RK) algorithm to estimate the solution of an ordinary differential equation (ODE) at the next time step.


This Python implementation was written by Peter Monk.


  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:

You can go up one level to the Python source codes.

Last revised on 08 May 2012.