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  26 <div class="titlepage"><div><div><h3 class="title">
  27 <a name="boost_numeric_odeint.concepts.implicit_system"></a><a class="link" href="implicit_system.html" title="Implicit System">Implicit
  28       System</a>
  29 </h3></div></div></div>
  30 <h5>
  31 <a name="boost_numeric_odeint.concepts.implicit_system.h0"></a>
  32         <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>
  33       </h5>
  34 <p>
  35         This concept describes how to define a ODE that can be solved by an implicit
  36         routine. Implicit routines need not only the function <span class="emphasis"><em>f(x,t)</em></span>
  37         but also the Jacobian <span class="emphasis"><em>df/dx = A(x,t)</em></span>. <span class="emphasis"><em>A</em></span>
  38         is a matrix and implicit routines need to solve the linear problem <span class="emphasis"><em>Ax
  39         = 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>,
  40         therefore, the <span class="emphasis"><em>state_type</em></span> implicit routines is <span class="emphasis"><em>ublas::vector</em></span>
  41         and the matrix is defined as <span class="emphasis"><em>ublas::matrix</em></span>.
  42       </p>
  43 <h5>
  44 <a name="boost_numeric_odeint.concepts.implicit_system.h1"></a>
  45         <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>
  46       </h5>
  47 <div class="variablelist">
  48 <p class="title"><b></b></p>
  49 <dl class="variablelist">
  50 <dt><span class="term"><code class="computeroutput"><span class="identifier">System</span></code></span></dt>
  51 <dd><p>
  52               A type that is a model of <code class="computeroutput"><span class="identifier">Implicit</span>
  53               <span class="identifier">System</span></code>
  54             </p></dd>
  55 <dt><span class="term"><code class="computeroutput"><span class="identifier">Time</span></code></span></dt>
  56 <dd><p>
  57               A type representing the time of the ODE
  58             </p></dd>
  59 <dt><span class="term"><code class="computeroutput"><span class="identifier">sys</span></code></span></dt>
  60 <dd><p>
  61               An object of type <code class="computeroutput"><span class="identifier">System</span></code>
  62             </p></dd>
  63 <dt><span class="term"><code class="computeroutput"><span class="identifier">x</span></code></span></dt>
  64 <dd><p>
  65               Object of type ublas::vector
  66             </p></dd>
  67 <dt><span class="term"><code class="computeroutput"><span class="identifier">dxdt</span></code></span></dt>
  68 <dd><p>
  69               Object of type ublas::vector
  70             </p></dd>
  71 <dt><span class="term"><code class="computeroutput"><span class="identifier">jacobi</span></code></span></dt>
  72 <dd><p>
  73               Object of type ublas::matrix
  74             </p></dd>
  75 <dt><span class="term"><code class="computeroutput"><span class="identifier">t</span></code></span></dt>
  76 <dd><p>
  77               Object of type <code class="computeroutput"><span class="identifier">Time</span></code>
  78             </p></dd>
  79 </dl>
  80 </div>
  81 <h5>
  82 <a name="boost_numeric_odeint.concepts.implicit_system.h2"></a>
  83         <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
  84         Expressions</a>
  85       </h5>
  86 <div class="informaltable"><table class="table">
  87 <colgroup>
  88 <col>
  89 <col>
  90 <col>
  91 <col>
  92 </colgroup>
  93 <thead><tr>
  94 <th>
  95                 <p>
  96                   Name
  97                 </p>
  98               </th>
  99 <th>
 100                 <p>
 101                   Expression
 102                 </p>
 103               </th>
 104 <th>
 105                 <p>
 106                   Type
 107                 </p>
 108               </th>
 109 <th>
 110                 <p>
 111                   Semantics
 112                 </p>
 113               </th>
 114 </tr></thead>
 115 <tbody>
 116 <tr>
 117 <td>
 118                 <p>
 119                   Calculate <span class="emphasis"><em>dx/dt := f(x,t)</em></span>
 120                 </p>
 121               </td>
 122 <td>
 123                 <p>
 124                   <code class="computeroutput"><span class="identifier">sys</span><span class="special">.</span><span class="identifier">first</span><span class="special">(</span>
 125                   <span class="identifier">x</span> <span class="special">,</span>
 126                   <span class="identifier">dxdt</span> <span class="special">,</span>
 127                   <span class="identifier">t</span> <span class="special">)</span></code>
 128                 </p>
 129               </td>
 130 <td>
 131                 <p>
 132                   <code class="computeroutput"><span class="keyword">void</span></code>
 133                 </p>
 134               </td>
 135 <td>
 136                 <p>
 137                   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>,
 138                   the result is stored into dxdt
 139                 </p>
 140               </td>
 141 </tr>
 142 <tr>
 143 <td>
 144                 <p>
 145                   Calculate <span class="emphasis"><em>A := df/dx (x,t)</em></span>
 146                 </p>
 147               </td>
 148 <td>
 149                 <p>
 150                   <code class="computeroutput"><span class="identifier">sys</span><span class="special">.</span><span class="identifier">second</span><span class="special">(</span>
 151                   <span class="identifier">x</span> <span class="special">,</span>
 152                   <span class="identifier">jacobi</span> <span class="special">,</span>
 153                   <span class="identifier">t</span> <span class="special">)</span></code>
 154                 </p>
 155               </td>
 156 <td>
 157                 <p>
 158                   <code class="computeroutput"><span class="keyword">void</span></code>
 159                 </p>
 160               </td>
 161 <td>
 162                 <p>
 163                   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>,
 164                   the result is stored into <code class="computeroutput"><span class="identifier">jacobi</span></code>
 165                 </p>
 166               </td>
 167 </tr>
 168 </tbody>
 169 </table></div>
 170 </div>
 171 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 172 <td align="left"></td>
 173 <td align="right"><div class="copyright-footer">Copyright &#169; 2009-2012 Karsten
 174       Ahnert and Mario Mulansky<p>
 175         Distributed under the Boost Software License, Version 1.0. (See accompanying
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