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<div class="titlepage"><div><div><h3 class="title">
<a name="boost_numeric_odeint.tutorial.complex_state_types"></a><a class="link" href="complex_state_types.html" title="Complex state types">Complex
      state types</a>
</h3></div></div></div>
<p>
        Thus far we have seen several examples defined for real values. odeint can
        handle complex state types, hence ODEs which are defined on complex vector
        spaces, as well. An example is the Stuart-Landau oscillator
      </p>
<p>
        <span class="emphasis"><em>d &#936; / dt = ( 1 + i &#951; ) &#936; + ( 1 + i &#945; ) | &#936; |<sup>2</sup> &#936; </em></span>
      </p>
<p>
        where <span class="emphasis"><em>&#936;</em></span> and <span class="emphasis"><em>i</em></span> is a complex variable.
        The definition of this ODE in C++ using complex&lt; double &gt; as a state
        type may look as follows
      </p>
<p>
</p>
<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">complex</span><span class="special">&lt;</span> <span class="keyword">double</span> <span class="special">&gt;</span> <span class="identifier">state_type</span><span class="special">;</span>

<span class="keyword">struct</span> <span class="identifier">stuart_landau</span>
<span class="special">{</span>
    <span class="keyword">double</span> <span class="identifier">m_eta</span><span class="special">;</span>
    <span class="keyword">double</span> <span class="identifier">m_alpha</span><span class="special">;</span>

    <span class="identifier">stuart_landau</span><span class="special">(</span> <span class="keyword">double</span> <span class="identifier">eta</span> <span class="special">=</span> <span class="number">1.0</span> <span class="special">,</span> <span class="keyword">double</span> <span class="identifier">alpha</span> <span class="special">=</span> <span class="number">1.0</span> <span class="special">)</span>
    <span class="special">:</span> <span class="identifier">m_eta</span><span class="special">(</span> <span class="identifier">eta</span> <span class="special">)</span> <span class="special">,</span> <span class="identifier">m_alpha</span><span class="special">(</span> <span class="identifier">alpha</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">state_type</span> <span class="special">&amp;</span><span class="identifier">dxdt</span> <span class="special">,</span> <span class="keyword">double</span> <span class="identifier">t</span> <span class="special">)</span> <span class="keyword">const</span>
    <span class="special">{</span>
        <span class="keyword">const</span> <span class="identifier">complex</span><span class="special">&lt;</span> <span class="keyword">double</span> <span class="special">&gt;</span> <span class="identifier">I</span><span class="special">(</span> <span class="number">0.0</span> <span class="special">,</span> <span class="number">1.0</span> <span class="special">);</span>
        <span class="identifier">dxdt</span> <span class="special">=</span> <span class="special">(</span> <span class="number">1.0</span> <span class="special">+</span> <span class="identifier">m_eta</span> <span class="special">*</span> <span class="identifier">I</span> <span class="special">)</span> <span class="special">*</span> <span class="identifier">x</span> <span class="special">-</span> <span class="special">(</span> <span class="number">1.0</span> <span class="special">+</span> <span class="identifier">m_alpha</span> <span class="special">*</span> <span class="identifier">I</span> <span class="special">)</span> <span class="special">*</span> <span class="identifier">norm</span><span class="special">(</span> <span class="identifier">x</span> <span class="special">)</span> <span class="special">*</span> <span class="identifier">x</span><span class="special">;</span>
    <span class="special">}</span>
<span class="special">};</span>
</pre>
<p>
      </p>
<p>
        One can also use a function instead of a functor to implement it
      </p>
<p>
</p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">eta</span> <span class="special">=</span> <span class="number">1.0</span><span class="special">;</span>
<span class="keyword">double</span> <span class="identifier">alpha</span> <span class="special">=</span> <span class="number">1.0</span><span class="special">;</span>

<span class="keyword">void</span> <span class="identifier">stuart_landau</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">state_type</span> <span class="special">&amp;</span><span class="identifier">dxdt</span> <span class="special">,</span> <span class="keyword">double</span> <span class="identifier">t</span> <span class="special">)</span>
<span class="special">{</span>
    <span class="keyword">const</span> <span class="identifier">complex</span><span class="special">&lt;</span> <span class="keyword">double</span> <span class="special">&gt;</span> <span class="identifier">I</span><span class="special">(</span> <span class="number">0.0</span> <span class="special">,</span> <span class="number">1.0</span> <span class="special">);</span>
    <span class="identifier">dxdt</span> <span class="special">=</span> <span class="special">(</span> <span class="number">1.0</span> <span class="special">+</span> <span class="identifier">m_eta</span> <span class="special">*</span> <span class="identifier">I</span> <span class="special">)</span> <span class="special">*</span> <span class="identifier">x</span> <span class="special">-</span> <span class="special">(</span> <span class="number">1.0</span> <span class="special">+</span> <span class="identifier">m_alpha</span> <span class="special">*</span> <span class="identifier">I</span> <span class="special">)</span> <span class="special">*</span> <span class="identifier">norm</span><span class="special">(</span> <span class="identifier">x</span> <span class="special">)</span> <span class="special">*</span> <span class="identifier">x</span><span class="special">;</span>
<span class="special">}</span>
</pre>
<p>
      </p>
<p>
        We strongly recommend to use the first ansatz. In this case you have explicit
        control over the parameters of the system and are not restricted to use global
        variables to parametrize the oscillator.
      </p>
<p>
        When choosing the stepper type one has to account for the "unusual"
        state type: it is a single <code class="computeroutput"><span class="identifier">complex</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span></code> opposed to the vector types used in the
        previous examples. This means that no iterations over vector elements have
        to be performed inside the stepper algorithm. Odeint already detects that
        and automatically uses the <code class="computeroutput"><span class="identifier">vector_space_algebra</span></code>
        for computation. You can enforce this by supplying additional template arguments
        to the stepper including the <code class="computeroutput"><span class="identifier">vector_space_algebra</span></code>.
        Details on the usage of algebras can be found in the section <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>
<p>
</p>
<pre class="programlisting"><span class="identifier">state_type</span> <span class="identifier">x</span> <span class="special">=</span> <span class="identifier">complex</span><span class="special">&lt;</span> <span class="keyword">double</span> <span class="special">&gt;(</span> <span class="number">1.0</span> <span class="special">,</span> <span class="number">0.0</span> <span class="special">);</span>

<span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">dt</span> <span class="special">=</span> <span class="number">0.1</span><span class="special">;</span>

<span class="keyword">typedef</span> <span class="identifier">runge_kutta4</span><span class="special">&lt;</span> <span class="identifier">state_type</span> <span class="special">&gt;</span> <span class="identifier">stepper_type</span><span class="special">;</span>

<span class="identifier">integrate_const</span><span class="special">(</span> <span class="identifier">stepper_type</span><span class="special">()</span> <span class="special">,</span> <span class="identifier">stuart_landau</span><span class="special">(</span> <span class="number">2.0</span> <span class="special">,</span> <span class="number">1.0</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="number">10.0</span> <span class="special">,</span> <span class="identifier">dt</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>
<p>
        The full cpp file for the Stuart-Landau example can be found here <a href="https://github.com/headmyshoulder/odeint-v2/tree/master/libs/numeric/odeint/examples/stuart_landau.cpp" target="_top">stuart_landau.cpp</a>
      </p>
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