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<div class="titlepage"><div><div><h3 class="title">
<a name="boost_numeric_odeint.concepts.implicit_system"></a><a class="link" href="implicit_system.html" title="Implicit System">Implicit
      System</a>
</h3></div></div></div>
<h5>
<a name="boost_numeric_odeint.concepts.implicit_system.h0"></a>
        <span class="phrase"><a name="boost_numeric_odeint.concepts.implicit_system.description"></a></span><a class="link" href="implicit_system.html#boost_numeric_odeint.concepts.implicit_system.description">Description</a>
      </h5>
<p>
        This concept describes how to define a ODE that can be solved by an implicit
        routine. Implicit routines need not only the function <span class="emphasis"><em>f(x,t)</em></span>
        but also the Jacobian <span class="emphasis"><em>df/dx = A(x,t)</em></span>. <span class="emphasis"><em>A</em></span>
        is a matrix and implicit routines need to solve the linear problem <span class="emphasis"><em>Ax
        = b</em></span>. In odeint this is implemented with use of <a href="http://www.boost.org/doc/libs/release/libs/numeric/ublas/index.html" target="_top">Boost.uBLAS</a>,
        therefore, the <span class="emphasis"><em>state_type</em></span> implicit routines is <span class="emphasis"><em>ublas::vector</em></span>
        and the matrix is defined as <span class="emphasis"><em>ublas::matrix</em></span>.
      </p>
<h5>
<a name="boost_numeric_odeint.concepts.implicit_system.h1"></a>
        <span class="phrase"><a name="boost_numeric_odeint.concepts.implicit_system.notation"></a></span><a class="link" href="implicit_system.html#boost_numeric_odeint.concepts.implicit_system.notation">Notation</a>
      </h5>
<div class="variablelist">
<p class="title"><b></b></p>
<dl class="variablelist">
<dt><span class="term"><code class="computeroutput"><span class="identifier">System</span></code></span></dt>
<dd><p>
              A type that is a model of <code class="computeroutput"><span class="identifier">Implicit</span>
              <span class="identifier">System</span></code>
            </p></dd>
<dt><span class="term"><code class="computeroutput"><span class="identifier">Time</span></code></span></dt>
<dd><p>
              A type representing the time of the ODE
            </p></dd>
<dt><span class="term"><code class="computeroutput"><span class="identifier">sys</span></code></span></dt>
<dd><p>
              An object of type <code class="computeroutput"><span class="identifier">System</span></code>
            </p></dd>
<dt><span class="term"><code class="computeroutput"><span class="identifier">x</span></code></span></dt>
<dd><p>
              Object of type ublas::vector
            </p></dd>
<dt><span class="term"><code class="computeroutput"><span class="identifier">dxdt</span></code></span></dt>
<dd><p>
              Object of type ublas::vector
            </p></dd>
<dt><span class="term"><code class="computeroutput"><span class="identifier">jacobi</span></code></span></dt>
<dd><p>
              Object of type ublas::matrix
            </p></dd>
<dt><span class="term"><code class="computeroutput"><span class="identifier">t</span></code></span></dt>
<dd><p>
              Object of type <code class="computeroutput"><span class="identifier">Time</span></code>
            </p></dd>
</dl>
</div>
<h5>
<a name="boost_numeric_odeint.concepts.implicit_system.h2"></a>
        <span class="phrase"><a name="boost_numeric_odeint.concepts.implicit_system.valid_expressions"></a></span><a class="link" href="implicit_system.html#boost_numeric_odeint.concepts.implicit_system.valid_expressions">Valid
        Expressions</a>
      </h5>
<div class="informaltable"><table class="table">
<colgroup>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
                <p>
                  Name
                </p>
              </th>
<th>
                <p>
                  Expression
                </p>
              </th>
<th>
                <p>
                  Type
                </p>
              </th>
<th>
                <p>
                  Semantics
                </p>
              </th>
</tr></thead>
<tbody>
<tr>
<td>
                <p>
                  Calculate <span class="emphasis"><em>dx/dt := f(x,t)</em></span>
                </p>
              </td>
<td>
                <p>
                  <code class="computeroutput"><span class="identifier">sys</span><span class="special">.</span><span class="identifier">first</span><span class="special">(</span>
                  <span class="identifier">x</span> <span class="special">,</span>
                  <span class="identifier">dxdt</span> <span class="special">,</span>
                  <span class="identifier">t</span> <span class="special">)</span></code>
                </p>
              </td>
<td>
                <p>
                  <code class="computeroutput"><span class="keyword">void</span></code>
                </p>
              </td>
<td>
                <p>
                  Calculates <code class="computeroutput"><span class="identifier">f</span><span class="special">(</span><span class="identifier">x</span><span class="special">,</span><span class="identifier">t</span><span class="special">)</span></code>,
                  the result is stored into dxdt
                </p>
              </td>
</tr>
<tr>
<td>
                <p>
                  Calculate <span class="emphasis"><em>A := df/dx (x,t)</em></span>
                </p>
              </td>
<td>
                <p>
                  <code class="computeroutput"><span class="identifier">sys</span><span class="special">.</span><span class="identifier">second</span><span class="special">(</span>
                  <span class="identifier">x</span> <span class="special">,</span>
                  <span class="identifier">jacobi</span> <span class="special">,</span>
                  <span class="identifier">t</span> <span class="special">)</span></code>
                </p>
              </td>
<td>
                <p>
                  <code class="computeroutput"><span class="keyword">void</span></code>
                </p>
              </td>
<td>
                <p>
                  Calculates the Jacobian of <span class="emphasis"><em>f</em></span> at <span class="emphasis"><em>x</em></span>,<span class="emphasis"><em>t</em></span>,
                  the result is stored into <code class="computeroutput"><span class="identifier">jacobi</span></code>
                </p>
              </td>
</tr>
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<td align="right"><div class="copyright-footer">Copyright &#169; 2009-2015 Karsten Ahnert and Mario Mulansky<p>
        Distributed under the Boost Software License, Version 1.0. (See accompanying
        file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
      </p>
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