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<title>Implicit System</title>
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<link rel="home" href="../../index.html" title="Chapter 1. Boost.Numeric.Odeint">
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<link rel="prev" href="simple_symplectic_system.html" title="Simple Symplectic System">
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<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
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<a accesskey="p" href="simple_symplectic_system.html"><img src="../../images/prev.png" alt="Prev"></a><a accesskey="u" href="../concepts.html"><img src="../../images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../images/home.png" alt="Home"></a><a accesskey="n" href="stepper.html"><img src="../../images/next.png" alt="Next"></a>
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<div class="section">
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<div class="titlepage"><div><div><h3 class="title">
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<a name="boost_numeric_odeint.concepts.implicit_system"></a><a class="link" href="implicit_system.html" title="Implicit System">Implicit
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System</a>
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</h3></div></div></div>
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<h5>
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<a name="boost_numeric_odeint.concepts.implicit_system.h0"></a>
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<span><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>
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</h5>
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<p>
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This concept describes how to define a ODE that can be solved by an implicit
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routine. Implicit routines need not only the function <span class="emphasis"><em>f(x,t)</em></span>
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but also the Jacobian <span class="emphasis"><em>df/dx = A(x,t)</em></span>. <span class="emphasis"><em>A</em></span>
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is a matrix and implicit routines need to solve the linear problem <span class="emphasis"><em>Ax
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= 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>,
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therefore, the <span class="emphasis"><em>state_type</em></span> implicit routines is <span class="emphasis"><em>ublas::vector</em></span>
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and the matrix is defined as <span class="emphasis"><em>ublas::matrix</em></span>.
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</p>
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<h5>
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<a name="boost_numeric_odeint.concepts.implicit_system.h1"></a>
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<span><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>
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</h5>
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<div class="variablelist">
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<p class="title"><b></b></p>
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<dl>
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<dt><span class="term"><code class="computeroutput"><span class="identifier">System</span></code></span></dt>
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<dd><p>
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A type that is a model of <code class="computeroutput"><span class="identifier">Implicit_System</span></code>
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</p></dd>
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<dt><span class="term"><code class="computeroutput"><span class="identifier">Time</span></code></span></dt>
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<dd><p>
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A type representing the time of the ODE
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</p></dd>
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<dt><span class="term"><code class="computeroutput"><span class="identifier">sys</span></code></span></dt>
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<dd><p>
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An object of type <code class="computeroutput"><span class="identifier">System</span></code>
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</p></dd>
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<dt><span class="term"><code class="computeroutput"><span class="identifier">x</span></code></span></dt>
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<dd><p>
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Object of type ublas::vector
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</p></dd>
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<dt><span class="term"><code class="computeroutput"><span class="identifier">dxdt</span></code></span></dt>
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<dd><p>
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Object of type ublas::vector
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</p></dd>
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<dt><span class="term"><code class="computeroutput"><span class="identifier">jacobi</span></code></span></dt>
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<dd><p>
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Object of type ublas::matrix
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</p></dd>
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<dt><span class="term"><code class="computeroutput"><span class="identifier">t</span></code></span></dt>
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<dd><p>
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Object of type <code class="computeroutput"><span class="identifier">Time</span></code>
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</p></dd>
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</dl>
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</div>
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<h5>
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<a name="boost_numeric_odeint.concepts.implicit_system.h2"></a>
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<span><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
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Expressions</a>
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</h5>
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<div class="informaltable"><table class="table">
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<colgroup>
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<col>
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<col>
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<col>
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<col>
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</colgroup>
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<thead><tr>
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<th>
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<p>
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Name
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</p>
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</th>
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<th>
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<p>
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Expression
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</p>
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</th>
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<th>
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<p>
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Type
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</p>
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</th>
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<th>
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<p>
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Semantics
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</p>
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</th>
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</tr></thead>
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<tbody>
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<tr>
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<td>
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<p>
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Calculate <span class="emphasis"><em>dx/dt := f(x,t)</em></span>
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</p>
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</td>
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<td>
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<p>
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<code class="computeroutput"><span class="identifier">sys</span><span class="special">.</span><span class="identifier">first</span><span class="special">(</span>
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<span class="identifier">x</span> <span class="special">,</span>
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<span class="identifier">dxdt</span> <span class="special">,</span>
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<span class="identifier">t</span> <span class="special">)</span></code>
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</p>
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</td>
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<td>
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<p>
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<code class="computeroutput"><span class="keyword">void</span></code>
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</p>
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</td>
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<td>
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<p>
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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>,
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the result is stored into dxdt
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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Calculate <span class="emphasis"><em>A := df/dx (x,t)</em></span>
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</p>
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</td>
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<td>
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<p>
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<code class="computeroutput"><span class="identifier">sys</span><span class="special">.</span><span class="identifier">second</span><span class="special">(</span>
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<span class="identifier">x</span> <span class="special">,</span>
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<span class="identifier">jacobi</span> <span class="special">,</span>
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<span class="identifier">t</span> <span class="special">)</span></code>
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</p>
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</td>
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<td>
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<p>
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<code class="computeroutput"><span class="keyword">void</span></code>
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</p>
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</td>
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<td>
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<p>
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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>,
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the result is stored into <code class="computeroutput"><span class="identifier">jacobi</span></code>
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</p>
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</td>
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</tr>
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</tbody>
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</table></div>
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</div>
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<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
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<td align="left"></td>
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<td align="right"><div class="copyright-footer">Copyright © 2009-2011 Karsten Ahnert
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and Mario Mulansky<p>
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Distributed under the Boost Software License, Version 1.0. (See accompanying
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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>)
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</p>
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</div></td>
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</tr></table>
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<hr>
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