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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 © 2009-2012 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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