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| <div class="titlepage"><div><div><h3 class="title"> |
| <a name="boost_numeric_odeint.tutorial.using_matrices_as_state_types"></a><a class="link" href="using_matrices_as_state_types.html" title="Using matrices as state types">Using |
| matrices as state types</a> |
| </h3></div></div></div> |
| <p> |
| odeint works well with a variety of different state types. It is not restricted |
| to pure vector-wise types, like <code class="computeroutput"><span class="identifier">vector</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">></span></code>, <code class="computeroutput"><span class="identifier">array</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">,</span> <span class="identifier">N</span> <span class="special">></span></code>, |
| <code class="computeroutput"><span class="identifier">fusion</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">,</span> <span class="keyword">double</span> <span class="special">></span></code>, |
| etc. but also works with types having a different topology then simple vectors. |
| Here, we show how odeint can be used with matrices as states type, in the |
| next section we will show how can be used to solve ODEs defined on complex |
| networks. |
| </p> |
| <p> |
| By default, odeint can be used with <code class="computeroutput"><span class="identifier">ublas</span><span class="special">::</span><span class="identifier">matrix</span><span class="special"><</span> <span class="identifier">T</span> <span class="special">></span></code> as state type for matrices. A simple |
| example is a two-dimensional lattice of coupled phase oscillators. Other |
| matrix types like <code class="computeroutput"><span class="identifier">mtl</span><span class="special">::</span><span class="identifier">dense_matrix</span></code> or blitz arrays and matrices |
| can used as well but need some kind of activation in order to work with odeint. |
| This activation is described in following sections, |
| </p> |
| <p> |
| The definition of the system is |
| </p> |
| <p> |
| </p> |
| <pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">numeric</span><span class="special">::</span><span class="identifier">ublas</span><span class="special">::</span><span class="identifier">matrix</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">></span> <span class="identifier">state_type</span><span class="special">;</span> |
| |
| <span class="keyword">struct</span> <span class="identifier">two_dimensional_phase_lattice</span> |
| <span class="special">{</span> |
| <span class="identifier">two_dimensional_phase_lattice</span><span class="special">(</span> <span class="keyword">double</span> <span class="identifier">gamma</span> <span class="special">=</span> <span class="number">0.5</span> <span class="special">)</span> |
| <span class="special">:</span> <span class="identifier">m_gamma</span><span class="special">(</span> <span class="identifier">gamma</span> <span class="special">)</span> <span class="special">{</span> <span class="special">}</span> |
| |
| <span class="keyword">void</span> <span class="keyword">operator</span><span class="special">()(</span> <span class="keyword">const</span> <span class="identifier">state_type</span> <span class="special">&</span><span class="identifier">x</span> <span class="special">,</span> <span class="identifier">state_type</span> <span class="special">&</span><span class="identifier">dxdt</span> <span class="special">,</span> <span class="keyword">double</span> <span class="comment">/* t */</span> <span class="special">)</span> <span class="keyword">const</span> |
| <span class="special">{</span> |
| <span class="identifier">size_t</span> <span class="identifier">size1</span> <span class="special">=</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">size1</span><span class="special">()</span> <span class="special">,</span> <span class="identifier">size2</span> <span class="special">=</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">size2</span><span class="special">();</span> |
| |
| <span class="keyword">for</span><span class="special">(</span> <span class="identifier">size_t</span> <span class="identifier">i</span><span class="special">=</span><span class="number">1</span> <span class="special">;</span> <span class="identifier">i</span><span class="special"><</span><span class="identifier">size1</span><span class="special">-</span><span class="number">1</span> <span class="special">;</span> <span class="special">++</span><span class="identifier">i</span> <span class="special">)</span> |
| <span class="special">{</span> |
| <span class="keyword">for</span><span class="special">(</span> <span class="identifier">size_t</span> <span class="identifier">j</span><span class="special">=</span><span class="number">1</span> <span class="special">;</span> <span class="identifier">j</span><span class="special"><</span><span class="identifier">size2</span><span class="special">-</span><span class="number">1</span> <span class="special">;</span> <span class="special">++</span><span class="identifier">j</span> <span class="special">)</span> |
| <span class="special">{</span> |
| <span class="identifier">dxdt</span><span class="special">(</span> <span class="identifier">i</span> <span class="special">,</span> <span class="identifier">j</span> <span class="special">)</span> <span class="special">=</span> |
| <span class="identifier">coupling_func</span><span class="special">(</span> <span class="identifier">x</span><span class="special">(</span> <span class="identifier">i</span> <span class="special">+</span> <span class="number">1</span> <span class="special">,</span> <span class="identifier">j</span> <span class="special">)</span> <span class="special">-</span> <span class="identifier">x</span><span class="special">(</span> <span class="identifier">i</span> <span class="special">,</span> <span class="identifier">j</span> <span class="special">)</span> <span class="special">)</span> <span class="special">+</span> |
| <span class="identifier">coupling_func</span><span class="special">(</span> <span class="identifier">x</span><span class="special">(</span> <span class="identifier">i</span> <span class="special">-</span> <span class="number">1</span> <span class="special">,</span> <span class="identifier">j</span> <span class="special">)</span> <span class="special">-</span> <span class="identifier">x</span><span class="special">(</span> <span class="identifier">i</span> <span class="special">,</span> <span class="identifier">j</span> <span class="special">)</span> <span class="special">)</span> <span class="special">+</span> |
| <span class="identifier">coupling_func</span><span class="special">(</span> <span class="identifier">x</span><span class="special">(</span> <span class="identifier">i</span> <span class="special">,</span> <span class="identifier">j</span> <span class="special">+</span> <span class="number">1</span> <span class="special">)</span> <span class="special">-</span> <span class="identifier">x</span><span class="special">(</span> <span class="identifier">i</span> <span class="special">,</span> <span class="identifier">j</span> <span class="special">)</span> <span class="special">)</span> <span class="special">+</span> |
| <span class="identifier">coupling_func</span><span class="special">(</span> <span class="identifier">x</span><span class="special">(</span> <span class="identifier">i</span> <span class="special">,</span> <span class="identifier">j</span> <span class="special">-</span> <span class="number">1</span> <span class="special">)</span> <span class="special">-</span> <span class="identifier">x</span><span class="special">(</span> <span class="identifier">i</span> <span class="special">,</span> <span class="identifier">j</span> <span class="special">)</span> <span class="special">);</span> |
| <span class="special">}</span> |
| <span class="special">}</span> |
| |
| <span class="keyword">for</span><span class="special">(</span> <span class="identifier">size_t</span> <span class="identifier">i</span><span class="special">=</span><span class="number">0</span> <span class="special">;</span> <span class="identifier">i</span><span class="special"><</span><span class="identifier">x</span><span class="special">.</span><span class="identifier">size1</span><span class="special">()</span> <span class="special">;</span> <span class="special">++</span><span class="identifier">i</span> <span class="special">)</span> <span class="identifier">dxdt</span><span class="special">(</span> <span class="identifier">i</span> <span class="special">,</span> <span class="number">0</span> <span class="special">)</span> <span class="special">=</span> <span class="identifier">dxdt</span><span class="special">(</span> <span class="identifier">i</span> <span class="special">,</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">size2</span><span class="special">()</span> <span class="special">-</span><span class="number">1</span> <span class="special">)</span> <span class="special">=</span> <span class="number">0.0</span><span class="special">;</span> |
| <span class="keyword">for</span><span class="special">(</span> <span class="identifier">size_t</span> <span class="identifier">j</span><span class="special">=</span><span class="number">0</span> <span class="special">;</span> <span class="identifier">j</span><span class="special"><</span><span class="identifier">x</span><span class="special">.</span><span class="identifier">size2</span><span class="special">()</span> <span class="special">;</span> <span class="special">++</span><span class="identifier">j</span> <span class="special">)</span> <span class="identifier">dxdt</span><span class="special">(</span> <span class="number">0</span> <span class="special">,</span> <span class="identifier">j</span> <span class="special">)</span> <span class="special">=</span> <span class="identifier">dxdt</span><span class="special">(</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">size1</span><span class="special">()</span> <span class="special">-</span><span class="number">1</span> <span class="special">,</span> <span class="identifier">j</span> <span class="special">)</span> <span class="special">=</span> <span class="number">0.0</span><span class="special">;</span> |
| <span class="special">}</span> |
| |
| <span class="keyword">double</span> <span class="identifier">coupling_func</span><span class="special">(</span> <span class="keyword">double</span> <span class="identifier">x</span> <span class="special">)</span> <span class="keyword">const</span> |
| <span class="special">{</span> |
| <span class="keyword">return</span> <span class="identifier">sin</span><span class="special">(</span> <span class="identifier">x</span> <span class="special">)</span> <span class="special">-</span> <span class="identifier">m_gamma</span> <span class="special">*</span> <span class="special">(</span> <span class="number">1.0</span> <span class="special">-</span> <span class="identifier">cos</span><span class="special">(</span> <span class="identifier">x</span> <span class="special">)</span> <span class="special">);</span> |
| <span class="special">}</span> |
| |
| <span class="keyword">double</span> <span class="identifier">m_gamma</span><span class="special">;</span> |
| <span class="special">};</span> |
| </pre> |
| <p> |
| </p> |
| <p> |
| In principle this is all. Please note, that the above code is far from being |
| optimal. Better performance can be achieved if every interaction is only |
| calculated once and iterators for columns and rows are used. Below are some |
| visualizations of the evolution of this lattice equation. |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../phase_lattice_2d_0000.jpg" alt="phase_lattice_2d_0000"></span> <span class="inlinemediaobject"><img src="../../phase_lattice_2d_0100.jpg" alt="phase_lattice_2d_0100"></span> <span class="inlinemediaobject"><img src="../../phase_lattice_2d_1000.jpg" alt="phase_lattice_2d_1000"></span> |
| </p> |
| <p> |
| The full cpp for this example can be found here <a href="https://github.com/headmyshoulder/odeint-v2/blob/master/examples/two_dimensional_phase_lattice.cpp" target="_top">two_dimensional_phase_lattice.cpp</a>. |
| </p> |
| </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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