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<div class="titlepage"><div><div><h3 class="title">
<a name="boost_numeric_odeint.tutorial.using_boost__units"></a><a class="link" href="using_boost__units.html" title="Using boost::units">Using
      boost::units</a>
</h3></div></div></div>
<p>
        odeint also works well with <a href="http://www.boost.org/doc/libs/release/libs/units/" target="_top">Boost.Units</a>
        - a library for compile time unit and dimension analysis. It works by decoding
        unit information into the types of values. For a one-dimensional unit you
        can just use the Boost.Unit types as state type, deriv type and time type
        and hand the <code class="computeroutput"><span class="identifier">vector_space_algebra</span></code>
        to the stepper definition and everything works just fine:
      </p>
<p>
</p>
<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">units</span><span class="special">::</span><span class="identifier">quantity</span><span class="special">&lt;</span> <span class="identifier">si</span><span class="special">::</span><span class="identifier">time</span> <span class="special">,</span> <span class="keyword">double</span> <span class="special">&gt;</span> <span class="identifier">time_type</span><span class="special">;</span>
<span class="keyword">typedef</span> <span class="identifier">units</span><span class="special">::</span><span class="identifier">quantity</span><span class="special">&lt;</span> <span class="identifier">si</span><span class="special">::</span><span class="identifier">length</span> <span class="special">,</span> <span class="keyword">double</span> <span class="special">&gt;</span> <span class="identifier">length_type</span><span class="special">;</span>
<span class="keyword">typedef</span> <span class="identifier">units</span><span class="special">::</span><span class="identifier">quantity</span><span class="special">&lt;</span> <span class="identifier">si</span><span class="special">::</span><span class="identifier">velocity</span> <span class="special">,</span> <span class="keyword">double</span> <span class="special">&gt;</span> <span class="identifier">velocity_type</span><span class="special">;</span>

<span class="keyword">typedef</span> <span class="identifier">runge_kutta4</span><span class="special">&lt;</span> <span class="identifier">length_type</span> <span class="special">,</span> <span class="keyword">double</span> <span class="special">,</span> <span class="identifier">velocity_type</span> <span class="special">,</span> <span class="identifier">time_type</span> <span class="special">,</span>
                      <span class="identifier">vector_space_algebra</span> <span class="special">&gt;</span> <span class="identifier">stepper_type</span><span class="special">;</span>
</pre>
<p>
      </p>
<p>
        If you want to solve more-dimensional problems the individual entries typically
        have different units. That means that the <code class="computeroutput"><span class="identifier">state_type</span></code>
        is now possibly heterogeneous, meaning that every entry might have a different
        type. To solve this problem, compile-time sequences from <a href="http://www.boost.org/doc/libs/release/libs/fusion/" target="_top">Boost.Fusion</a>
        can be used.
      </p>
<p>
        To illustrate how odeint works with <a href="http://www.boost.org/doc/libs/release/libs/units/" target="_top">Boost.Units</a>
        we use the harmonic oscillator as primary example. We start with defining
        all quantities
      </p>
<p>
</p>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">numeric</span><span class="special">/</span><span class="identifier">odeint</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">numeric</span><span class="special">/</span><span class="identifier">odeint</span><span class="special">/</span><span class="identifier">algebra</span><span class="special">/</span><span class="identifier">fusion_algebra</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">numeric</span><span class="special">/</span><span class="identifier">odeint</span><span class="special">/</span><span class="identifier">algebra</span><span class="special">/</span><span class="identifier">fusion_algebra_dispatcher</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>

<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">units</span><span class="special">/</span><span class="identifier">systems</span><span class="special">/</span><span class="identifier">si</span><span class="special">/</span><span class="identifier">length</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">units</span><span class="special">/</span><span class="identifier">systems</span><span class="special">/</span><span class="identifier">si</span><span class="special">/</span><span class="identifier">time</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">units</span><span class="special">/</span><span class="identifier">systems</span><span class="special">/</span><span class="identifier">si</span><span class="special">/</span><span class="identifier">velocity</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">units</span><span class="special">/</span><span class="identifier">systems</span><span class="special">/</span><span class="identifier">si</span><span class="special">/</span><span class="identifier">acceleration</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">units</span><span class="special">/</span><span class="identifier">systems</span><span class="special">/</span><span class="identifier">si</span><span class="special">/</span><span class="identifier">io</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>

<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">fusion</span><span class="special">/</span><span class="identifier">container</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>

<span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">std</span><span class="special">;</span>
<span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">numeric</span><span class="special">::</span><span class="identifier">odeint</span><span class="special">;</span>
<span class="keyword">namespace</span> <span class="identifier">fusion</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">fusion</span><span class="special">;</span>
<span class="keyword">namespace</span> <span class="identifier">units</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">units</span><span class="special">;</span>
<span class="keyword">namespace</span> <span class="identifier">si</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">units</span><span class="special">::</span><span class="identifier">si</span><span class="special">;</span>

<span class="keyword">typedef</span> <span class="identifier">units</span><span class="special">::</span><span class="identifier">quantity</span><span class="special">&lt;</span> <span class="identifier">si</span><span class="special">::</span><span class="identifier">time</span> <span class="special">,</span> <span class="keyword">double</span> <span class="special">&gt;</span> <span class="identifier">time_type</span><span class="special">;</span>
<span class="keyword">typedef</span> <span class="identifier">units</span><span class="special">::</span><span class="identifier">quantity</span><span class="special">&lt;</span> <span class="identifier">si</span><span class="special">::</span><span class="identifier">length</span> <span class="special">,</span> <span class="keyword">double</span> <span class="special">&gt;</span> <span class="identifier">length_type</span><span class="special">;</span>
<span class="keyword">typedef</span> <span class="identifier">units</span><span class="special">::</span><span class="identifier">quantity</span><span class="special">&lt;</span> <span class="identifier">si</span><span class="special">::</span><span class="identifier">velocity</span> <span class="special">,</span> <span class="keyword">double</span> <span class="special">&gt;</span> <span class="identifier">velocity_type</span><span class="special">;</span>
<span class="keyword">typedef</span> <span class="identifier">units</span><span class="special">::</span><span class="identifier">quantity</span><span class="special">&lt;</span> <span class="identifier">si</span><span class="special">::</span><span class="identifier">acceleration</span> <span class="special">,</span> <span class="keyword">double</span> <span class="special">&gt;</span> <span class="identifier">acceleration_type</span><span class="special">;</span>
<span class="keyword">typedef</span> <span class="identifier">units</span><span class="special">::</span><span class="identifier">quantity</span><span class="special">&lt;</span> <span class="identifier">si</span><span class="special">::</span><span class="identifier">frequency</span> <span class="special">,</span> <span class="keyword">double</span> <span class="special">&gt;</span> <span class="identifier">frequency_type</span><span class="special">;</span>

<span class="keyword">typedef</span> <span class="identifier">fusion</span><span class="special">::</span><span class="identifier">vector</span><span class="special">&lt;</span> <span class="identifier">length_type</span> <span class="special">,</span> <span class="identifier">velocity_type</span> <span class="special">&gt;</span> <span class="identifier">state_type</span><span class="special">;</span>
<span class="keyword">typedef</span> <span class="identifier">fusion</span><span class="special">::</span><span class="identifier">vector</span><span class="special">&lt;</span> <span class="identifier">velocity_type</span> <span class="special">,</span> <span class="identifier">acceleration_type</span> <span class="special">&gt;</span> <span class="identifier">deriv_type</span><span class="special">;</span>
</pre>
<p>
      </p>
<p>
        Note, that the <code class="computeroutput"><span class="identifier">state_type</span></code>
        and the <code class="computeroutput"><span class="identifier">deriv_type</span></code> are now
        a compile-time fusion sequences. <code class="computeroutput"><span class="identifier">deriv_type</span></code>
        represents <span class="emphasis"><em>x'</em></span> and is now different from the state type
        as it has different unit definitions. Next, we define the ordinary differential
        equation which is completely equivalent to the example in <a class="link" href="harmonic_oscillator.html" title="Harmonic oscillator">Harmonic
        Oscillator</a>:
      </p>
<p>
</p>
<pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">oscillator</span>
<span class="special">{</span>
    <span class="identifier">frequency_type</span> <span class="identifier">m_omega</span><span class="special">;</span>

    <span class="identifier">oscillator</span><span class="special">(</span> <span class="keyword">const</span> <span class="identifier">frequency_type</span> <span class="special">&amp;</span><span class="identifier">omega</span> <span class="special">=</span> <span class="number">1.0</span> <span class="special">*</span> <span class="identifier">si</span><span class="special">::</span><span class="identifier">hertz</span> <span class="special">)</span> <span class="special">:</span> <span class="identifier">m_omega</span><span class="special">(</span> <span class="identifier">omega</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">&amp;</span><span class="identifier">x</span> <span class="special">,</span> <span class="identifier">deriv_type</span> <span class="special">&amp;</span><span class="identifier">dxdt</span> <span class="special">,</span> <span class="identifier">time_type</span> <span class="identifier">t</span> <span class="special">)</span> <span class="keyword">const</span>
    <span class="special">{</span>
        <span class="identifier">fusion</span><span class="special">::</span><span class="identifier">at_c</span><span class="special">&lt;</span> <span class="number">0</span> <span class="special">&gt;(</span> <span class="identifier">dxdt</span> <span class="special">)</span> <span class="special">=</span> <span class="identifier">fusion</span><span class="special">::</span><span class="identifier">at_c</span><span class="special">&lt;</span> <span class="number">1</span> <span class="special">&gt;(</span> <span class="identifier">x</span> <span class="special">);</span>
        <span class="identifier">fusion</span><span class="special">::</span><span class="identifier">at_c</span><span class="special">&lt;</span> <span class="number">1</span> <span class="special">&gt;(</span> <span class="identifier">dxdt</span> <span class="special">)</span> <span class="special">=</span> <span class="special">-</span> <span class="identifier">m_omega</span> <span class="special">*</span> <span class="identifier">m_omega</span> <span class="special">*</span> <span class="identifier">fusion</span><span class="special">::</span><span class="identifier">at_c</span><span class="special">&lt;</span> <span class="number">0</span> <span class="special">&gt;(</span> <span class="identifier">x</span> <span class="special">);</span>
    <span class="special">}</span>
<span class="special">};</span>
</pre>
<p>
      </p>
<p>
        Next, we instantiate an appropriate stepper. We must explicitly parametrize
        the stepper with the <code class="computeroutput"><span class="identifier">state_type</span></code>,
        <code class="computeroutput"><span class="identifier">deriv_type</span></code>, <code class="computeroutput"><span class="identifier">time_type</span></code>.
      </p>
<p>
</p>
<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">runge_kutta_dopri5</span><span class="special">&lt;</span> <span class="identifier">state_type</span> <span class="special">,</span> <span class="keyword">double</span> <span class="special">,</span> <span class="identifier">deriv_type</span> <span class="special">,</span> <span class="identifier">time_type</span> <span class="special">&gt;</span> <span class="identifier">stepper_type</span><span class="special">;</span>

<span class="identifier">state_type</span> <span class="identifier">x</span><span class="special">(</span> <span class="number">1.0</span> <span class="special">*</span> <span class="identifier">si</span><span class="special">::</span><span class="identifier">meter</span> <span class="special">,</span> <span class="number">0.0</span> <span class="special">*</span> <span class="identifier">si</span><span class="special">::</span><span class="identifier">meter_per_second</span> <span class="special">);</span>

<span class="identifier">integrate_const</span><span class="special">(</span> <span class="identifier">make_dense_output</span><span class="special">(</span> <span class="number">1.0e-6</span> <span class="special">,</span> <span class="number">1.0e-6</span> <span class="special">,</span> <span class="identifier">stepper_type</span><span class="special">()</span> <span class="special">)</span> <span class="special">,</span> <span class="identifier">oscillator</span><span class="special">(</span> <span class="number">2.0</span> <span class="special">*</span> <span class="identifier">si</span><span class="special">::</span><span class="identifier">hertz</span> <span class="special">)</span> <span class="special">,</span>
                 <span class="identifier">x</span> <span class="special">,</span> <span class="number">0.0</span> <span class="special">*</span> <span class="identifier">si</span><span class="special">::</span><span class="identifier">second</span> <span class="special">,</span> <span class="number">100.0</span> <span class="special">*</span> <span class="identifier">si</span><span class="special">::</span><span class="identifier">second</span> <span class="special">,</span> <span class="number">0.1</span> <span class="special">*</span> <span class="identifier">si</span><span class="special">::</span><span class="identifier">second</span> <span class="special">,</span> <span class="identifier">streaming_observer</span><span class="special">(</span> <span class="identifier">cout</span> <span class="special">)</span> <span class="special">);</span>
</pre>
<p>
      </p>
<div class="note"><table border="0" summary="Note">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td>
<th align="left">Note</th>
</tr>
<tr><td align="left" valign="top"><p>
          When using compile-time sequences, the iteration over vector elements is
          done by the <code class="computeroutput"><span class="identifier">fusion_algebra</span></code>,
          which is automatically chosen by odeint. For more on the state types /
          algebras see chapter <a class="link" href="../odeint_in_detail/state_types__algebras_and_operations.html" title="State types, algebras and operations">Adapt
          your own state types</a>.
        </p></td></tr>
</table></div>
<p>
        It is quite easy but the compilation time might take very long. Furthermore,
        the observer is defined a bit different
      </p>
<p>
</p>
<pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">streaming_observer</span>
<span class="special">{</span>
    <span class="identifier">std</span><span class="special">::</span><span class="identifier">ostream</span><span class="special">&amp;</span> <span class="identifier">m_out</span><span class="special">;</span>

    <span class="identifier">streaming_observer</span><span class="special">(</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">ostream</span> <span class="special">&amp;</span><span class="identifier">out</span> <span class="special">)</span> <span class="special">:</span> <span class="identifier">m_out</span><span class="special">(</span> <span class="identifier">out</span> <span class="special">)</span> <span class="special">{</span> <span class="special">}</span>

    <span class="keyword">struct</span> <span class="identifier">write_element</span>
    <span class="special">{</span>
        <span class="identifier">std</span><span class="special">::</span><span class="identifier">ostream</span> <span class="special">&amp;</span><span class="identifier">m_out</span><span class="special">;</span>
        <span class="identifier">write_element</span><span class="special">(</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">ostream</span> <span class="special">&amp;</span><span class="identifier">out</span> <span class="special">)</span> <span class="special">:</span> <span class="identifier">m_out</span><span class="special">(</span> <span class="identifier">out</span> <span class="special">)</span> <span class="special">{</span> <span class="special">};</span>

        <span class="keyword">template</span><span class="special">&lt;</span> <span class="keyword">class</span> <span class="identifier">T</span> <span class="special">&gt;</span>
        <span class="keyword">void</span> <span class="keyword">operator</span><span class="special">()(</span> <span class="keyword">const</span> <span class="identifier">T</span> <span class="special">&amp;</span><span class="identifier">t</span> <span class="special">)</span> <span class="keyword">const</span>
        <span class="special">{</span>
            <span class="identifier">m_out</span> <span class="special">&lt;&lt;</span> <span class="string">"\t"</span> <span class="special">&lt;&lt;</span> <span class="identifier">t</span><span class="special">;</span>
        <span class="special">}</span>
    <span class="special">};</span>

    <span class="keyword">template</span><span class="special">&lt;</span> <span class="keyword">class</span> <span class="identifier">State</span> <span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Time</span> <span class="special">&gt;</span>
    <span class="keyword">void</span> <span class="keyword">operator</span><span class="special">()(</span> <span class="keyword">const</span> <span class="identifier">State</span> <span class="special">&amp;</span><span class="identifier">x</span> <span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Time</span> <span class="special">&amp;</span><span class="identifier">t</span> <span class="special">)</span> <span class="keyword">const</span>
    <span class="special">{</span>
        <span class="identifier">m_out</span> <span class="special">&lt;&lt;</span> <span class="identifier">t</span><span class="special">;</span>
        <span class="identifier">fusion</span><span class="special">::</span><span class="identifier">for_each</span><span class="special">(</span> <span class="identifier">x</span> <span class="special">,</span> <span class="identifier">write_element</span><span class="special">(</span> <span class="identifier">m_out</span> <span class="special">)</span> <span class="special">);</span>
        <span class="identifier">m_out</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
    <span class="special">}</span>
<span class="special">};</span>
</pre>
<p>
      </p>
<div class="caution"><table border="0" summary="Caution">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../../../../../../doc/src/images/caution.png"></td>
<th align="left">Caution</th>
</tr>
<tr><td align="left" valign="top"><p>
          Using <a href="http://www.boost.org/doc/libs/release/libs/units/" target="_top">Boost.Units</a>
          works nicely but compilation can be very time and memory consuming. For
          example the unit test for the usage of <a href="http://www.boost.org/doc/libs/release/libs/units/" target="_top">Boost.Units</a>
          in odeint take up to 4 GB of memory at compilation.
        </p></td></tr>
</table></div>
<p>
        The full cpp file for this example can be found here <a href="https://github.com/headmyshoulder/odeint-v2/blob/master/examples/harmonic_oscillator_units.cpp" target="_top">harmonic_oscillator_units.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 &#169; 2009-2015 Karsten Ahnert and Mario Mulansky<p>
        Distributed under the Boost Software License, Version 1.0. (See accompanying
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