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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>
- </tbody>
- </table></div>
- </div>
- <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
- <td align="left"></td>
- <td align="right"><div class="copyright-footer">Copyright © 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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