rkf45, a FORTRAN90 code which implements the Watt and Shampine RKF45 solver for systems of ordinary differential equations (ODEs).

The 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 MIT license


rkf45 is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version and an Octave version and a Python version.

Related Data and Programs:

midpoint, a FORTRAN90 code which solves one or more ordinary differential equations (ODE) using the midpoint method.

NMS, a FORTRAN90 code which includes a wide variety of numerical software.

ODE, a FORTRAN90 code which solves a system of ordinary differential equations (ODE), by Shampine and Gordon.


TEST_ODE, a FORTRAN90 code 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 https://www.netlib.org/ode/ the NETLIB ODE web site.

Source Code:

Last revised on 25 August 2020.