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<html>
<head>
<title>
RKF45 - Runge-Kutta-Fehlberg ODE Solver
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
RKF45 <br> Runge-Kutta-Fehlberg ODE Solver
</h1>
<hr>
<p>
<b>RKF45</b>
is a MATLAB library which
implements the Watt and Shampine RKF45 ODE solver.
</p>
<p>
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.
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files described and made available on this web page
are distributed under
<a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>RKF45</b> is available in
<a href = "../../c_src/rkf45/rkf45.html">a C version</a> and
<a href = "../../cpp_src/rkf45/rkf45.html">a C++ version</a> and
<a href = "../../f77_src/rkf45/rkf45.html">a FORTRAN77 version</a> and
<a href = "../../f_src/rkf45/rkf45.html">a FORTRAN90 version</a> and
<a href = "../../m_src/rkf45/rkf45.html">a MATLAB version</a> and
<a href = "../../py_src/rkf45/rkf45.html">a PYTHON version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../m_src/dfield8/dfield8.html">
DFIELD8</a>,
a MATLAB program which
interactively displays the direction field of a differential equation,
by John Polking.
</p>
<p>
<a href = "../../m_src/flame_ode/flame_ode.html">
FLAME_ODE</a>,
a MATLAB library which
considers an ordinary differential equation (ODE) which models
the growth of a ball of flame in a combustion process.
</p>
<p>
<a href = "../../m_src/ode_predator_prey/ode_predator_prey.html">
ODE_PREDATOR_PREY</a>,
a MATLAB program which
solves a time-dependent predator-prey system using MATLAB's ODE23 solver.
</p>
<p>
<a href = "../../m_src/pplane8/pplane8.html">
PPLANE8</a>,
a MATLAB program which
interactively displays the phase plane of a differential equation,
by John Polking.
</p>
<p>
<a href = "../../m_src/test_ode/test_ode.html">
TEST_ODE</a>,
a MATLAB library which
defines some sample ODE's for testing initial value
problem solvers;
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Erwin Fehlberg,<br>
Low-order Classical Runge-Kutta Formulas with Stepsize Control,<br>
NASA Technical Report R-315, 1969.
</li>
<li>
Lawrence Shampine, Herman Watts, S Davenport,<br>
Solving Non-stiff Ordinary Differential Equations -
The State of the Art,<br>
SIAM Review,<br>
Volume 18, pages 376-411, 1976.
</li>
<li>
The source code for Shampine and Watt's original FORTRAN77
routine is available at
<a href = "http://www.netlib.org/ode/">
http://www.netlib.org/ode/</a>
the NETLIB ODE web site.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "r8_fehl.m">r8_fehl.m</a>
takes one Fehlberg fourth-fifth order step (double precision).
</li>
<li>
<a href = "r8_rkf45.m">r8_rkf45.m</a>
carries out the Runge-Kutta-Fehlberg method (double precision).
</li>
<li>
<a href = "timestamp.m">timestamp.m</a>
prints the current YMDHMS date as a time stamp.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "rkf45_test.m">rkf45_test.m</a>, calls all the tests;
</li>
<li>
<a href = "rkf45_test_output.txt">rkf45_test_output.txt</a>,
the output file;
</li>
<li>
<a href = "rkf45_test04.m">rkf45_test04.m</a>,
solves a scalar ODE.
</li>
<li>
<a href = "rkf45_test05.m">rkf45_test05.m</a>,
solves a vector ODE.
</li>
<li>
<a href = "rkf45_test06.m">rkf45_test06.m</a>,
solves a scalar ODE, using one-step integration.
</li>
<li>
<a href = "r8_f1.m">r8_f1.m</a>,
evaluates the derivative for the ODE.
</li>
<li>
<a href = "r8_f2.m">r8_f2.m</a>,
evaluates the derivative for the ODE.
</li>
<li>
<a href = "r8_y1x.m">r8_y1x.m</a>,
evaluates the exact solution of the ODE.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../m_src.html">
the MATLAB source codes</a>.
</p>
<hr>
<i>
Last revised on 30 June 2011.
</i>
<!-- John Burkardt -->
</body>
</html>