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      RK4 - Runge-Kutta 4th Order ODE Solver
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      RK4 <br> Runge-Kutta 4th Order ODE Solver
    </h1>

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    <p>
      <b>RK4</b>
      is a FORTRAN77 library which
      implements a simple Runge-Kutta solver for an initial value problem.
    </p>

    <p>
      The <b>rk4()</b> routine does not include any error estimator.  It
      takes a single step at a time, requiring the user to control the step size.
    </p>

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      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>

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      Languages:
    </h3>

    <p>
      <b>RK4</b> is available in
      <a href = "../../c_src/rk4/rk4.html">a C version</a> and
      <a href = "../../cpp_src/rk4/rk4.html">a C++ version</a> and
      <a href = "../../f77_src/rk4/rk4.html">a FORTRAN77 version</a> and
      <a href = "../../f_src/rk4/rk4.html">a FORTRAN90 version</a> and
      <a href = "../../m_src/rk4/rk4.html">a MATLAB version</a>.
    </p>

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      Related Data and Programs:
    </h3>

    <p>
      <a href = "../../f77_src/driv/driv.html">
      CWG_ODE</a>,
      a FORTRAN77 library which
      implements three ODE solvers by C William Gear.
    </p>

    <p>
      <a href = "../../f77_src/driv/driv.html">
      DRIV</a>,
      a FORTRAN77 library which
      solves real or complex systems of ordinary differential equations;
    </p>

    <p>
      <a href = "../../f77_src/nms/nms.html">
      NMS</a>,
      a FORTRAN77 library which
      includes the <b>DDRIV</b> package of ODE solvers.
    </p>

    <p>
      <a href = "../../f77_src/ode/ode.html">
      ODE</a>,
      a FORTRAN77 library which
      solves a system of ordinary differential equations,
      by Shampine and Gordon.
    </p>

    <p>
      <a href = "../../f77_src/odepack/odepack.html">
      ODEPACK</a>,
      a FORTRAN77 library which
      contains nine ODE solvers, including LSODE, LSODES, LSODA,
      LSODAR, LSODPK, LSODKR, LSODI, LSOIBT, and LSODIS,
      by Alan Hindmarsh.
    </p>

    <p>
      <a href = "../../f77_src/rkf45/rkf45.html">
      RKF45</a>,
      a FORTRAN77 library which
      implements the Runge-Kutta-Fehlberg ODE solver.
    </p>

    <p>
      <a href = "../../f_src/test_ode/test_ode.html">
      TEST_ODE</a>,
      a FORTRAN90 library which
      contains routines which define some test problems for ODE solvers.
    </p>

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      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>
      </ol>
    </p>

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      Source Code:
    </h3>

    <p>
      <ul>
        <li>
          <a href = "rk4.f">rk4.f</a>, the source code.
        </li>
        <li>
          <a href = "rk4.sh">rk4.sh</a>,
          BASH commands to compile the source code.
        </li>
      </ul>
    </p>

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      Examples and Tests:
    </h3>

    <p>
      <ul>
        <li>
          <a href = "rk4_prb.f">rk4_prb.f</a>,
          a sample calling program.
        </li>
        <li>
          <a href = "rk4_prb.sh">rk4_prb.sh</a>,
          BASH commands to compile and run the sample program.
        </li>
        <li>
          <a href = "rk4_prb_output.txt">rk4_prb_output.txt</a>,
          the output file.
        </li>
        <li>
          <a href = "rk4_prb.png">rk4_prb.png</a>,
          an image of the solution.
        </li>
      </ul>
    </p>

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      List of Routines:
    </h3>

    <p>
      <ul>
        <li>
          <b>RK4</b> takes one Runge Kutta step.
        </li>
      </ul>
    </p>

    <p>
      You can go up one level to <a href = "../f77_src.html">
      the FORTRAN77 source codes</a>.
    </p>

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    <i>
      Last revised on 31 January 2012.
    </i>

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