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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 Ψ / dt = ( 1 + i η ) Ψ + ( 1 + i α ) | Ψ |<sup>2</sup> Ψ </em></span>
- </p>
- <p>
- where <span class="emphasis"><em>Ψ</em></span> and <span class="emphasis"><em>i</em></span> is a complex variable.
- The definition of this ODE in C++ using complex< double > 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"><</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">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">&</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="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"><</span> <span class="keyword">double</span> <span class="special">></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">&</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="identifier">t</span> <span class="special">)</span>
- <span class="special">{</span>
- <span class="keyword">const</span> <span class="identifier">complex</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">></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"><</span><span class="keyword">double</span><span class="special">></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"><</span> <span class="keyword">double</span> <span class="special">>(</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"><</span> <span class="identifier">state_type</span> <span class="special">></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/blob/master/examples/stuart_landau.cpp" target="_top">stuart_landau.cpp</a>
- </p>
- </div>
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- <td align="right"><div class="copyright-footer">Copyright © 2009-2015 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>)
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