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<div class="section">
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<div class="titlepage"><div><div><h3 class="title">
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<a name="boost_numeric_odeint.tutorial.special_topics"></a><a class="link" href="special_topics.html" title="Special topics">Special
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topics</a>
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</h3></div></div></div>
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<div class="toc"><dl>
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<dt><span class="section"><a href="special_topics.html#boost_numeric_odeint.tutorial.special_topics.complex_state_types">Complex
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state types</a></span></dt>
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<dt><span class="section"><a href="special_topics.html#boost_numeric_odeint.tutorial.special_topics.lattice_systems">Lattice
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systems</a></span></dt>
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<dt><span class="section"><a href="special_topics.html#boost_numeric_odeint.tutorial.special_topics.ensembles_of_oscillators">Ensembles
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of oscillators</a></span></dt>
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<dt><span class="section"><a href="special_topics.html#boost_numeric_odeint.tutorial.special_topics.using_boost__units">Using
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boost::units</a></span></dt>
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<dt><span class="section"><a href="special_topics.html#boost_numeric_odeint.tutorial.special_topics.using_matrices_as_state_types">Using
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matrices as state types</a></span></dt>
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<dt><span class="section"><a href="special_topics.html#boost_numeric_odeint.tutorial.special_topics.using_arbitrary_precision_floating_point_types">Using
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arbitrary precision floating point types</a></span></dt>
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<dt><span class="section"><a href="special_topics.html#boost_numeric_odeint.tutorial.special_topics.self_expanding_lattices">Self
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expanding lattices</a></span></dt>
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</dl></div>
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<div class="section">
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<div class="titlepage"><div><div><h4 class="title">
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<a name="boost_numeric_odeint.tutorial.special_topics.complex_state_types"></a><a class="link" href="special_topics.html#boost_numeric_odeint.tutorial.special_topics.complex_state_types" title="Complex state types">Complex
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state types</a>
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</h4></div></div></div>
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<p>
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Thus far we have seen several examples defined for real values. Of course,
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odeint can handle complex state types, hence ODEs which are defined on
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complex vector spaces, as well. An example is the Stuart-Landau oscillator
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</p>
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<p>
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<span class="emphasis"><em>d Ψ / dt = ( 1 + i η ) Ψ + ( 1 + i α ) | Ψ |<sup>2</sup> Ψ </em></span>
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</p>
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<p>
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where <span class="emphasis"><em>Ψ</em></span> and <span class="emphasis"><em>i</em></span> is a complex variable.
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Hence the state type of this system is complex< double >. The definition
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of this ODE in C++ code is very simple
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</p>
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<p>
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</p>
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<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>
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<span class="keyword">struct</span> <span class="identifier">stuart_landau</span>
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<span class="special">{</span>
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<span class="keyword">double</span> <span class="identifier">m_eta</span><span class="special">;</span>
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<span class="keyword">double</span> <span class="identifier">m_alpha</span><span class="special">;</span>
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<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>
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<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>
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<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>
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<span class="special">{</span>
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<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>
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<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>
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<span class="special">}</span>
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<span class="special">};</span>
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</pre>
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<p>
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</p>
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<p>
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Of course, one can also use classical functions to implement the state
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function. In this cast the Stuart-Landau oscillator looks like
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</p>
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<p>
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</p>
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<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>
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<span class="keyword">double</span> <span class="identifier">alpha</span> <span class="special">=</span> <span class="number">1.0</span><span class="special">;</span>
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<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>
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<span class="special">{</span>
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<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>
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<span class="identifier">dxdt</span><span class="special">[</span><span class="number">0</span><span class="special">]</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="number">0</span><span class="special">]</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="number">0</span><span class="special">]</span> <span class="special">)</span> <span class="special">*</span> <span class="identifier">x</span><span class="special">[</span><span class="number">0</span><span class="special">];</span>
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<span class="special">}</span>
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</pre>
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<p>
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</p>
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<p>
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We strongly recommend to use the first ansatz. In this case you have explicit
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control over the parameters of the system and are not restricted to use
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global variables to parametrize the oscillator.
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</p>
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<p>
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When chosing the stepper type one has to account for the "unusual"
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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
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the previous examples. This means that no iterations over vector elements
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have to be performed inside the stepper algorithm. You can enforce this
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by supplying additional template arguments to the stepper including the
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<code class="computeroutput"><span class="identifier">vector_space_algebra</span></code>. Details
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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
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you own state types</a>.
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</p>
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<p>
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</p>
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<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>
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<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>
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<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="keyword">double</span> <span class="special">,</span> <span class="identifier">state_type</span> <span class="special">,</span> <span class="keyword">double</span> <span class="special">,</span>
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<span class="identifier">vector_space_algebra</span> <span class="special">></span> <span class="identifier">stepper_type</span><span class="special">;</span>
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<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>
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</pre>
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<p>
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</p>
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<p>
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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>
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</p>
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<div class="note"><table border="0" summary="Note">
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<tr>
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<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../images/note.png"></td>
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<th align="left">Note</th>
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</tr>
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<tr><td align="left" valign="top"><p>
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The fact that we have to configure a different algebra is solely due
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to the fact that we use a non-vector state type and not to the usage
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of complex values. So for, e.g. <code class="computeroutput"><span class="identifier">vector</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="special">></span></code>,
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this would not be required.
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</p></td></tr>
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</table></div>
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</div>
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<div class="section">
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<div class="titlepage"><div><div><h4 class="title">
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<a name="boost_numeric_odeint.tutorial.special_topics.lattice_systems"></a><a class="link" href="special_topics.html#boost_numeric_odeint.tutorial.special_topics.lattice_systems" title="Lattice systems">Lattice
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systems</a>
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</h4></div></div></div>
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<p>
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odeint can also be used to solve ordinary differential equations defined
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on lattices. A prominent example is the Fermi-Pasta-Ulam system [8]. It
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is a Hamiltonian system of nonlinear coupled harmonic oscillators. The
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Hamiltonian is
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</p>
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<p>
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<span class="emphasis"><em>H = Σ<sub>​i</sub> p<sub>​i</sub><sup>2</sup>/2 + 1/2 ( q<sub>​i+1</sub> - q<sub>​i</sub> )^2 + β / 4 ( q<sub>​i+1</sub> - q<sub>​i</sub> )^4 </em></span>
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</p>
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<p>
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Remarkably, the Fermi-Pasta-Ulam system was the first numerical experiment
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which has been implemented on a computer in 1953. It was studied at Los
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Alamos on one of the first computer (a MANIAC I) and it triggered a whole
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new tree of mathematical and physical science.
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</p>
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<p>
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Like the <a class="link" href="solar_system.html" title="Solar system">Solar
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System</a>, the FPU is solved again by a symplectic solver, but in this
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case we can speed up the computation because the <span class="emphasis"><em>q</em></span>
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components trivially reduce to <span class="emphasis"><em>dq<sub>​i</sub> / dt = p<sub>​i</sub></em></span>. odeint
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is capable of doing this performance improvement. All you have to do is
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to call the symplectic solver with an state function for the <span class="emphasis"><em>p</em></span>
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components. Here is how this function looks like
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</p>
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<p>
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</p>
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<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">vector</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">></span> <span class="identifier">container_type</span><span class="special">;</span>
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<span class="keyword">struct</span> <span class="identifier">fpu</span>
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<span class="special">{</span>
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<span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">m_beta</span><span class="special">;</span>
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<span class="identifier">fpu</span><span class="special">(</span> <span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">beta</span> <span class="special">=</span> <span class="number">1.0</span> <span class="special">)</span> <span class="special">:</span> <span class="identifier">m_beta</span><span class="special">(</span> <span class="identifier">beta</span> <span class="special">)</span> <span class="special">{</span> <span class="special">}</span>
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<span class="comment">// system function defining the ODE</span>
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<span class="keyword">void</span> <span class="keyword">operator</span><span class="special">()(</span> <span class="keyword">const</span> <span class="identifier">container_type</span> <span class="special">&</span><span class="identifier">q</span> <span class="special">,</span> <span class="identifier">container_type</span> <span class="special">&</span><span class="identifier">dpdt</span> <span class="special">)</span> <span class="keyword">const</span>
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<span class="special">{</span>
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<span class="identifier">size_t</span> <span class="identifier">n</span> <span class="special">=</span> <span class="identifier">q</span><span class="special">.</span><span class="identifier">size</span><span class="special">();</span>
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<span class="keyword">double</span> <span class="identifier">tmp</span> <span class="special">=</span> <span class="identifier">q</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">-</span> <span class="number">0.0</span><span class="special">;</span>
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<span class="keyword">double</span> <span class="identifier">tmp2</span> <span class="special">=</span> <span class="identifier">tmp</span> <span class="special">+</span> <span class="identifier">m_beta</span> <span class="special">*</span> <span class="identifier">tmp</span> <span class="special">*</span> <span class="identifier">tmp</span> <span class="special">*</span> <span class="identifier">tmp</span><span class="special">;</span>
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<span class="identifier">dpdt</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">=</span> <span class="special">-</span><span class="identifier">tmp2</span><span class="special">;</span>
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<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">n</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>
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<span class="special">{</span>
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<span class="identifier">tmp</span> <span class="special">=</span> <span class="identifier">q</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="special">-</span> <span class="identifier">q</span><span class="special">[</span><span class="identifier">i</span><span class="special">];</span>
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<span class="identifier">tmp2</span> <span class="special">=</span> <span class="identifier">tmp</span> <span class="special">+</span> <span class="identifier">m_beta</span> <span class="special">*</span> <span class="identifier">tmp</span> <span class="special">*</span> <span class="identifier">tmp</span> <span class="special">*</span> <span class="identifier">tmp</span><span class="special">;</span>
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<span class="identifier">dpdt</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special">+=</span> <span class="identifier">tmp2</span><span class="special">;</span>
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<span class="identifier">dpdt</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="special">=</span> <span class="special">-</span><span class="identifier">tmp2</span><span class="special">;</span>
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<span class="special">}</span>
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<span class="identifier">tmp</span> <span class="special">=</span> <span class="special">-</span> <span class="identifier">q</span><span class="special">[</span><span class="identifier">n</span><span class="special">-</span><span class="number">1</span><span class="special">];</span>
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<span class="identifier">tmp2</span> <span class="special">=</span> <span class="identifier">tmp</span> <span class="special">+</span> <span class="identifier">m_beta</span> <span class="special">*</span> <span class="identifier">tmp</span> <span class="special">*</span> <span class="identifier">tmp</span> <span class="special">*</span> <span class="identifier">tmp</span><span class="special">;</span>
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<span class="identifier">dpdt</span><span class="special">[</span><span class="identifier">n</span><span class="special">-</span><span class="number">1</span><span class="special">]</span> <span class="special">+=</span> <span class="identifier">tmp2</span><span class="special">;</span>
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<span class="special">}</span>
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<span class="comment">// calculates the energy of the system</span>
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<span class="keyword">double</span> <span class="identifier">energy</span><span class="special">(</span> <span class="keyword">const</span> <span class="identifier">container_type</span> <span class="special">&</span><span class="identifier">q</span> <span class="special">,</span> <span class="keyword">const</span> <span class="identifier">container_type</span> <span class="special">&</span><span class="identifier">p</span> <span class="special">)</span> <span class="keyword">const</span>
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<span class="special">{</span>
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<span class="comment">// ...</span>
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<span class="special">}</span>
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<span class="comment">// calculates the local energy of the system</span>
|
|
<span class="keyword">void</span> <span class="identifier">local_energy</span><span class="special">(</span> <span class="keyword">const</span> <span class="identifier">container_type</span> <span class="special">&</span><span class="identifier">q</span> <span class="special">,</span> <span class="keyword">const</span> <span class="identifier">container_type</span> <span class="special">&</span><span class="identifier">p</span> <span class="special">,</span> <span class="identifier">container_type</span> <span class="special">&</span><span class="identifier">e</span> <span class="special">)</span> <span class="keyword">const</span>
|
|
<span class="special">{</span>
|
|
<span class="comment">// ...</span>
|
|
<span class="special">}</span>
|
|
<span class="special">};</span>
|
|
</pre>
|
|
<p>
|
|
</p>
|
|
<p>
|
|
Of course, you can also use <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><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> for the state type.
|
|
</p>
|
|
<p>
|
|
Now, you have to define your initial values and perform the integration.
|
|
All this can be easily done with the following piece of code:
|
|
</p>
|
|
<p>
|
|
</p>
|
|
<pre class="programlisting"><span class="keyword">const</span> <span class="identifier">size_t</span> <span class="identifier">n</span> <span class="special">=</span> <span class="number">64</span><span class="special">;</span>
|
|
<span class="identifier">container_type</span> <span class="identifier">q</span><span class="special">(</span> <span class="identifier">n</span> <span class="special">,</span> <span class="number">0.0</span> <span class="special">)</span> <span class="special">,</span> <span class="identifier">p</span><span class="special">(</span> <span class="identifier">n</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">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">n</span> <span class="special">;</span> <span class="special">++</span><span class="identifier">i</span> <span class="special">)</span>
|
|
<span class="special">{</span>
|
|
<span class="identifier">p</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special">=</span> <span class="number">0.0</span><span class="special">;</span>
|
|
<span class="identifier">q</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special">=</span> <span class="number">32.0</span> <span class="special">*</span> <span class="identifier">sin</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">1</span> <span class="special">)</span> <span class="special">/</span> <span class="keyword">double</span><span class="special">(</span> <span class="identifier">n</span> <span class="special">+</span> <span class="number">1</span> <span class="special">)</span> <span class="special">*</span> <span class="identifier">M_PI</span> <span class="special">);</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">symplectic_rkn_sb3a_mclachlan</span><span class="special"><</span> <span class="identifier">container_type</span> <span class="special">></span> <span class="identifier">stepper_type</span><span class="special">;</span>
|
|
<span class="identifier">fpu</span> <span class="identifier">fpu_instance</span><span class="special">(</span> <span class="number">8.0</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">fpu_instance</span> <span class="special">,</span>
|
|
<span class="identifier">make_pair</span><span class="special">(</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">ref</span><span class="special">(</span> <span class="identifier">q</span> <span class="special">)</span> <span class="special">,</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">ref</span><span class="special">(</span> <span class="identifier">p</span> <span class="special">)</span> <span class="special">)</span> <span class="special">,</span>
|
|
<span class="number">0.0</span> <span class="special">,</span> <span class="number">1000.0</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="identifier">fpu_instance</span> <span class="special">,</span> <span class="number">10</span> <span class="special">)</span> <span class="special">);</span>
|
|
</pre>
|
|
<p>
|
|
</p>
|
|
<p>
|
|
The observer uses a reference to the system object to calculate the local
|
|
energies:
|
|
</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">&</span> <span class="identifier">m_out</span><span class="special">;</span>
|
|
<span class="keyword">const</span> <span class="identifier">fpu</span> <span class="special">&</span><span class="identifier">m_fpu</span><span class="special">;</span>
|
|
<span class="identifier">size_t</span> <span class="identifier">m_write_every</span><span class="special">;</span>
|
|
<span class="identifier">size_t</span> <span class="identifier">m_count</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">&</span><span class="identifier">out</span> <span class="special">,</span> <span class="keyword">const</span> <span class="identifier">fpu</span> <span class="special">&</span><span class="identifier">f</span> <span class="special">,</span> <span class="identifier">size_t</span> <span class="identifier">write_every</span> <span class="special">=</span> <span class="number">100</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="identifier">m_fpu</span><span class="special">(</span> <span class="identifier">f</span> <span class="special">)</span> <span class="special">,</span> <span class="identifier">m_write_every</span><span class="special">(</span> <span class="identifier">write_every</span> <span class="special">)</span> <span class="special">,</span> <span class="identifier">m_count</span><span class="special">(</span> <span class="number">0</span> <span class="special">)</span> <span class="special">{</span> <span class="special">}</span>
|
|
|
|
<span class="keyword">template</span><span class="special"><</span> <span class="keyword">class</span> <span class="identifier">State</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</span> <span class="special">&</span><span class="identifier">x</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">if</span><span class="special">(</span> <span class="special">(</span> <span class="identifier">m_count</span> <span class="special">%</span> <span class="identifier">m_write_every</span> <span class="special">)</span> <span class="special">==</span> <span class="number">0</span> <span class="special">)</span>
|
|
<span class="special">{</span>
|
|
<span class="identifier">container_type</span> <span class="special">&</span><span class="identifier">q</span> <span class="special">=</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">first</span><span class="special">;</span>
|
|
<span class="identifier">container_type</span> <span class="special">&</span><span class="identifier">p</span> <span class="special">=</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">second</span><span class="special">;</span>
|
|
<span class="identifier">container_type</span> <span class="identifier">energy</span><span class="special">(</span> <span class="identifier">q</span><span class="special">.</span><span class="identifier">size</span><span class="special">()</span> <span class="special">);</span>
|
|
<span class="identifier">m_fpu</span><span class="special">.</span><span class="identifier">local_energy</span><span class="special">(</span> <span class="identifier">q</span> <span class="special">,</span> <span class="identifier">p</span> <span class="special">,</span> <span class="identifier">energy</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">q</span><span class="special">.</span><span class="identifier">size</span><span class="special">()</span> <span class="special">;</span> <span class="special">++</span><span class="identifier">i</span> <span class="special">)</span>
|
|
<span class="special">{</span>
|
|
<span class="identifier">m_out</span> <span class="special"><<</span> <span class="identifier">t</span> <span class="special"><<</span> <span class="string">"\t"</span> <span class="special"><<</span> <span class="identifier">i</span> <span class="special"><<</span> <span class="string">"\t"</span> <span class="special"><<</span> <span class="identifier">q</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special"><<</span> <span class="string">"\t"</span> <span class="special"><<</span> <span class="identifier">p</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special"><<</span> <span class="string">"\t"</span> <span class="special"><<</span> <span class="identifier">energy</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special"><<</span> <span class="string">"\n"</span><span class="special">;</span>
|
|
<span class="special">}</span>
|
|
<span class="identifier">m_out</span> <span class="special"><<</span> <span class="string">"\n"</span><span class="special">;</span>
|
|
<span class="identifier">clog</span> <span class="special"><<</span> <span class="identifier">t</span> <span class="special"><<</span> <span class="string">"\t"</span> <span class="special"><<</span> <span class="identifier">accumulate</span><span class="special">(</span> <span class="identifier">energy</span><span class="special">.</span><span class="identifier">begin</span><span class="special">()</span> <span class="special">,</span> <span class="identifier">energy</span><span class="special">.</span><span class="identifier">end</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="string">"\n"</span><span class="special">;</span>
|
|
<span class="special">}</span>
|
|
<span class="special">++</span><span class="identifier">m_count</span><span class="special">;</span>
|
|
<span class="special">}</span>
|
|
<span class="special">};</span>
|
|
</pre>
|
|
<p>
|
|
</p>
|
|
<p>
|
|
The full cpp file for this FPU example can be found here <a href="https://github.com/headmyshoulder/odeint-v2/tree/master/libs/numeric/odeint/examples/fpu.cpp" target="_top">fpu.cpp</a>
|
|
</p>
|
|
</div>
|
|
<div class="section">
|
|
<div class="titlepage"><div><div><h4 class="title">
|
|
<a name="boost_numeric_odeint.tutorial.special_topics.ensembles_of_oscillators"></a><a class="link" href="special_topics.html#boost_numeric_odeint.tutorial.special_topics.ensembles_of_oscillators" title="Ensembles of oscillators">Ensembles
|
|
of oscillators</a>
|
|
</h4></div></div></div>
|
|
<p>
|
|
Another import high dimensional system of coupled ordinary differential
|
|
equations is an ensemble of <span class="emphasis"><em>N</em></span> all-to-all coupled phase
|
|
oscillators [9]. It is defined as
|
|
</p>
|
|
<p>
|
|
<span class="emphasis"><em>dφ<sub>​k</sub> / dt = ω<sub>​k</sub> + ε / N Σ<sub>​j</sub> sin( φ<sub>​j</sub> - φ<sub>​k</sub> )</em></span>
|
|
</p>
|
|
<p>
|
|
The natural frequencies <span class="emphasis"><em>ω<sub>​i</sub></em></span> of each oscillator follow
|
|
some distribution and <span class="emphasis"><em>ε</em></span> is the coupling strength. We
|
|
choose here a Lorentzian distribution for <span class="emphasis"><em>ω<sub>​i</sub></em></span>. Interestingly
|
|
a phase transition can be observed if the coupling strength exceeds a critical
|
|
value. Above this value synchronization sets in and some of the oscillators
|
|
oscillate with the same frequency despite their different natural frequencies.
|
|
The transition is also called Kuramoto transition. Its behavior can be
|
|
analyzed by employing the mean field of the phase
|
|
</p>
|
|
<p>
|
|
<span class="emphasis"><em>Z = K e<sup>i Θ</sup> = 1 / N Σ<sub>​k</sub>e<sup>i φ<sub>​k</sub></sup></em></span>
|
|
</p>
|
|
<p>
|
|
The definition of the system function is now a bit more complex since we
|
|
also need to store the individual frequencies of each oscillator.
|
|
</p>
|
|
<p>
|
|
</p>
|
|
<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">vector</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">></span> <span class="identifier">container_type</span><span class="special">;</span>
|
|
|
|
|
|
<span class="identifier">pair</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">,</span> <span class="keyword">double</span> <span class="special">></span> <span class="identifier">calc_mean_field</span><span class="special">(</span> <span class="keyword">const</span> <span class="identifier">container_type</span> <span class="special">&</span><span class="identifier">x</span> <span class="special">)</span>
|
|
<span class="special">{</span>
|
|
<span class="identifier">size_t</span> <span class="identifier">n</span> <span class="special">=</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">size</span><span class="special">();</span>
|
|
<span class="keyword">double</span> <span class="identifier">cos_sum</span> <span class="special">=</span> <span class="number">0.0</span> <span class="special">,</span> <span class="identifier">sin_sum</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">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">n</span> <span class="special">;</span> <span class="special">++</span><span class="identifier">i</span> <span class="special">)</span>
|
|
<span class="special">{</span>
|
|
<span class="identifier">cos_sum</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="identifier">i</span><span class="special">]</span> <span class="special">);</span>
|
|
<span class="identifier">sin_sum</span> <span class="special">+=</span> <span class="identifier">sin</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="special">);</span>
|
|
<span class="special">}</span>
|
|
<span class="identifier">cos_sum</span> <span class="special">/=</span> <span class="keyword">double</span><span class="special">(</span> <span class="identifier">n</span> <span class="special">);</span>
|
|
<span class="identifier">sin_sum</span> <span class="special">/=</span> <span class="keyword">double</span><span class="special">(</span> <span class="identifier">n</span> <span class="special">);</span>
|
|
|
|
<span class="keyword">double</span> <span class="identifier">K</span> <span class="special">=</span> <span class="identifier">sqrt</span><span class="special">(</span> <span class="identifier">cos_sum</span> <span class="special">*</span> <span class="identifier">cos_sum</span> <span class="special">+</span> <span class="identifier">sin_sum</span> <span class="special">*</span> <span class="identifier">sin_sum</span> <span class="special">);</span>
|
|
<span class="keyword">double</span> <span class="identifier">Theta</span> <span class="special">=</span> <span class="identifier">atan2</span><span class="special">(</span> <span class="identifier">sin_sum</span> <span class="special">,</span> <span class="identifier">cos_sum</span> <span class="special">);</span>
|
|
|
|
<span class="keyword">return</span> <span class="identifier">make_pair</span><span class="special">(</span> <span class="identifier">K</span> <span class="special">,</span> <span class="identifier">Theta</span> <span class="special">);</span>
|
|
<span class="special">}</span>
|
|
|
|
|
|
<span class="keyword">struct</span> <span class="identifier">phase_ensemble</span>
|
|
<span class="special">{</span>
|
|
<span class="identifier">container_type</span> <span class="identifier">m_omega</span><span class="special">;</span>
|
|
<span class="keyword">double</span> <span class="identifier">m_epsilon</span><span class="special">;</span>
|
|
|
|
<span class="identifier">phase_ensemble</span><span class="special">(</span> <span class="keyword">const</span> <span class="identifier">size_t</span> <span class="identifier">n</span> <span class="special">,</span> <span class="keyword">double</span> <span class="identifier">g</span> <span class="special">=</span> <span class="number">1.0</span> <span class="special">,</span> <span class="keyword">double</span> <span class="identifier">epsilon</span> <span class="special">=</span> <span class="number">1.0</span> <span class="special">)</span>
|
|
<span class="special">:</span> <span class="identifier">m_omega</span><span class="special">(</span> <span class="identifier">n</span> <span class="special">,</span> <span class="number">0.0</span> <span class="special">)</span> <span class="special">,</span> <span class="identifier">m_epsilon</span><span class="special">(</span> <span class="identifier">epsilon</span> <span class="special">)</span>
|
|
<span class="special">{</span>
|
|
<span class="identifier">create_frequencies</span><span class="special">(</span> <span class="identifier">g</span> <span class="special">);</span>
|
|
<span class="special">}</span>
|
|
|
|
<span class="keyword">void</span> <span class="identifier">create_frequencies</span><span class="special">(</span> <span class="keyword">double</span> <span class="identifier">g</span> <span class="special">)</span>
|
|
<span class="special">{</span>
|
|
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">mt19937</span> <span class="identifier">rng</span><span class="special">;</span>
|
|
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">cauchy_distribution</span><span class="special"><></span> <span class="identifier">cauchy</span><span class="special">(</span> <span class="number">0.0</span> <span class="special">,</span> <span class="identifier">g</span> <span class="special">);</span>
|
|
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">variate_generator</span><span class="special"><</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">mt19937</span><span class="special">&,</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">cauchy_distribution</span><span class="special"><></span> <span class="special">></span> <span class="identifier">gen</span><span class="special">(</span> <span class="identifier">rng</span> <span class="special">,</span> <span class="identifier">cauchy</span> <span class="special">);</span>
|
|
<span class="identifier">generate</span><span class="special">(</span> <span class="identifier">m_omega</span><span class="special">.</span><span class="identifier">begin</span><span class="special">()</span> <span class="special">,</span> <span class="identifier">m_omega</span><span class="special">.</span><span class="identifier">end</span><span class="special">()</span> <span class="special">,</span> <span class="identifier">gen</span> <span class="special">);</span>
|
|
<span class="special">}</span>
|
|
|
|
<span class="keyword">void</span> <span class="identifier">set_epsilon</span><span class="special">(</span> <span class="keyword">double</span> <span class="identifier">epsilon</span> <span class="special">)</span> <span class="special">{</span> <span class="identifier">m_epsilon</span> <span class="special">=</span> <span class="identifier">epsilon</span><span class="special">;</span> <span class="special">}</span>
|
|
|
|
<span class="keyword">double</span> <span class="identifier">get_epsilon</span><span class="special">(</span> <span class="keyword">void</span> <span class="special">)</span> <span class="keyword">const</span> <span class="special">{</span> <span class="keyword">return</span> <span class="identifier">m_epsilon</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">container_type</span> <span class="special">&</span><span class="identifier">x</span> <span class="special">,</span> <span class="identifier">container_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">pair</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">,</span> <span class="keyword">double</span> <span class="special">></span> <span class="identifier">mean</span> <span class="special">=</span> <span class="identifier">calc_mean_field</span><span class="special">(</span> <span class="identifier">x</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">size</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="special">=</span> <span class="identifier">m_omega</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special">+</span> <span class="identifier">m_epsilon</span> <span class="special">*</span> <span class="identifier">mean</span><span class="special">.</span><span class="identifier">first</span> <span class="special">*</span> <span class="identifier">sin</span><span class="special">(</span> <span class="identifier">mean</span><span class="special">.</span><span class="identifier">second</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="special">);</span>
|
|
<span class="special">}</span>
|
|
<span class="special">};</span>
|
|
</pre>
|
|
<p>
|
|
</p>
|
|
<p>
|
|
Note, that we have used <span class="emphasis"><em>Z</em></span> to simplify the equations
|
|
of motion. Next, we create an observer which computes the value of <span class="emphasis"><em>Z</em></span>
|
|
and we record <span class="emphasis"><em>Z</em></span> for different values of <span class="emphasis"><em>ε</em></span>.
|
|
</p>
|
|
<p>
|
|
</p>
|
|
<pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">statistics_observer</span>
|
|
<span class="special">{</span>
|
|
<span class="keyword">double</span> <span class="identifier">m_K_mean</span><span class="special">;</span>
|
|
<span class="identifier">size_t</span> <span class="identifier">m_count</span><span class="special">;</span>
|
|
|
|
<span class="identifier">statistics_observer</span><span class="special">(</span> <span class="keyword">void</span> <span class="special">)</span>
|
|
<span class="special">:</span> <span class="identifier">m_K_mean</span><span class="special">(</span> <span class="number">0.0</span> <span class="special">)</span> <span class="special">,</span> <span class="identifier">m_count</span><span class="special">(</span> <span class="number">0</span> <span class="special">)</span> <span class="special">{</span> <span class="special">}</span>
|
|
|
|
<span class="keyword">template</span><span class="special"><</span> <span class="keyword">class</span> <span class="identifier">State</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</span> <span class="special">&</span><span class="identifier">x</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="identifier">pair</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">,</span> <span class="keyword">double</span> <span class="special">></span> <span class="identifier">mean</span> <span class="special">=</span> <span class="identifier">calc_mean_field</span><span class="special">(</span> <span class="identifier">x</span> <span class="special">);</span>
|
|
<span class="identifier">m_K_mean</span> <span class="special">+=</span> <span class="identifier">mean</span><span class="special">.</span><span class="identifier">first</span><span class="special">;</span>
|
|
<span class="special">++</span><span class="identifier">m_count</span><span class="special">;</span>
|
|
<span class="special">}</span>
|
|
|
|
<span class="keyword">double</span> <span class="identifier">get_K_mean</span><span class="special">(</span> <span class="keyword">void</span> <span class="special">)</span> <span class="keyword">const</span> <span class="special">{</span> <span class="keyword">return</span> <span class="special">(</span> <span class="identifier">m_count</span> <span class="special">!=</span> <span class="number">0</span> <span class="special">)</span> <span class="special">?</span> <span class="identifier">m_K_mean</span> <span class="special">/</span> <span class="keyword">double</span><span class="special">(</span> <span class="identifier">m_count</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">void</span> <span class="identifier">reset</span><span class="special">(</span> <span class="keyword">void</span> <span class="special">)</span> <span class="special">{</span> <span class="identifier">m_K_mean</span> <span class="special">=</span> <span class="number">0.0</span><span class="special">;</span> <span class="identifier">m_count</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span> <span class="special">}</span>
|
|
<span class="special">};</span>
|
|
</pre>
|
|
<p>
|
|
</p>
|
|
<p>
|
|
Now, we do several integrations for different values of <span class="emphasis"><em>ε</em></span>
|
|
and record <span class="emphasis"><em>Z</em></span>. The result nicely confirms the analytical
|
|
result of the phase transition, i.e. in our example the standard deviation
|
|
of the Lorentzian is 1 such that the transition will be observed at <span class="emphasis"><em>ε =
|
|
2</em></span>.
|
|
</p>
|
|
<p>
|
|
</p>
|
|
<pre class="programlisting"><span class="keyword">const</span> <span class="identifier">size_t</span> <span class="identifier">n</span> <span class="special">=</span> <span class="number">16384</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="identifier">container_type</span> <span class="identifier">x</span><span class="special">(</span> <span class="identifier">n</span> <span class="special">);</span>
|
|
|
|
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">mt19937</span> <span class="identifier">rng</span><span class="special">;</span>
|
|
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">uniform_real</span><span class="special"><></span> <span class="identifier">unif</span><span class="special">(</span> <span class="number">0.0</span> <span class="special">,</span> <span class="number">2.0</span> <span class="special">*</span> <span class="identifier">M_PI</span> <span class="special">);</span>
|
|
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">variate_generator</span><span class="special"><</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">mt19937</span><span class="special">&,</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">uniform_real</span><span class="special"><></span> <span class="special">></span> <span class="identifier">gen</span><span class="special">(</span> <span class="identifier">rng</span> <span class="special">,</span> <span class="identifier">unif</span> <span class="special">);</span>
|
|
|
|
<span class="comment">// gamma = 1, the phase transition occurs at epsilon = 2</span>
|
|
<span class="identifier">phase_ensemble</span> <span class="identifier">ensemble</span><span class="special">(</span> <span class="identifier">n</span> <span class="special">,</span> <span class="number">1.0</span> <span class="special">);</span>
|
|
<span class="identifier">statistics_observer</span> <span class="identifier">obs</span><span class="special">;</span>
|
|
|
|
<span class="keyword">for</span><span class="special">(</span> <span class="keyword">double</span> <span class="identifier">epsilon</span> <span class="special">=</span> <span class="number">0.0</span> <span class="special">;</span> <span class="identifier">epsilon</span> <span class="special"><</span> <span class="number">5.0</span> <span class="special">;</span> <span class="identifier">epsilon</span> <span class="special">+=</span> <span class="number">0.1</span> <span class="special">)</span>
|
|
<span class="special">{</span>
|
|
<span class="identifier">ensemble</span><span class="special">.</span><span class="identifier">set_epsilon</span><span class="special">(</span> <span class="identifier">epsilon</span> <span class="special">);</span>
|
|
<span class="identifier">obs</span><span class="special">.</span><span class="identifier">reset</span><span class="special">();</span>
|
|
|
|
<span class="comment">// start with random initial conditions</span>
|
|
<span class="identifier">generate</span><span class="special">(</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">begin</span><span class="special">()</span> <span class="special">,</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">end</span><span class="special">()</span> <span class="special">,</span> <span class="identifier">gen</span> <span class="special">);</span>
|
|
|
|
<span class="comment">// calculate some transients steps</span>
|
|
<span class="identifier">integrate_const</span><span class="special">(</span> <span class="identifier">runge_kutta4</span><span class="special"><</span> <span class="identifier">container_type</span> <span class="special">>()</span> <span class="special">,</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">ref</span><span class="special">(</span> <span class="identifier">ensemble</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="comment">// integrate and compute the statistics</span>
|
|
<span class="identifier">integrate_const</span><span class="special">(</span> <span class="identifier">runge_kutta4</span><span class="special"><</span> <span class="identifier">container_type</span> <span class="special">>()</span> <span class="special">,</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">ref</span><span class="special">(</span> <span class="identifier">ensemble</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">100.0</span> <span class="special">,</span> <span class="identifier">dt</span> <span class="special">,</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">ref</span><span class="special">(</span> <span class="identifier">obs</span> <span class="special">)</span> <span class="special">);</span>
|
|
<span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">epsilon</span> <span class="special"><<</span> <span class="string">"\t"</span> <span class="special"><<</span> <span class="identifier">obs</span><span class="special">.</span><span class="identifier">get_K_mean</span><span class="special">()</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span>
|
|
<span class="special">}</span>
|
|
|
|
|
|
</pre>
|
|
<p>
|
|
</p>
|
|
<p>
|
|
The full cpp file for this example can be found here <a href="https://github.com/headmyshoulder/odeint-v2/tree/master/libs/numeric/odeint/examples/phase_oscillator_ensemble.cpp" target="_top">phase_oscillator_ensemble.cpp</a>
|
|
</p>
|
|
</div>
|
|
<div class="section">
|
|
<div class="titlepage"><div><div><h4 class="title">
|
|
<a name="boost_numeric_odeint.tutorial.special_topics.using_boost__units"></a><a class="link" href="special_topics.html#boost_numeric_odeint.tutorial.special_topics.using_boost__units" title="Using boost::units">Using
|
|
boost::units</a>
|
|
</h4></div></div></div>
|
|
<p>
|
|
odeint also works well with <a href="http://www.boost.org/doc/libs/release/libs/units/index.html" target="_top">Boost.Units</a>
|
|
- a library for compile time unit and dimension analysis. Ot 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"><</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">></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"><</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">></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"><</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">></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"><</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">></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/index.html" 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/index.html" 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"><</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">></span>
|
|
<span class="preprocessor">#include</span> <span class="special"><</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">></span>
|
|
|
|
<span class="preprocessor">#include</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">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">></span>
|
|
<span class="preprocessor">#include</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">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">></span>
|
|
<span class="preprocessor">#include</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">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">></span>
|
|
<span class="preprocessor">#include</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">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">></span>
|
|
<span class="preprocessor">#include</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">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">></span>
|
|
|
|
<span class="preprocessor">#include</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="identifier">container</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></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"><</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">></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"><</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">></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"><</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">></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"><</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">></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"><</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">></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"><</span> <span class="identifier">length_type</span> <span class="special">,</span> <span class="identifier">velocity_type</span> <span class="special">></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"><</span> <span class="identifier">velocity_type</span> <span class="special">,</span> <span class="identifier">acceleration_type</span> <span class="special">></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">&</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">&</span><span class="identifier">x</span> <span class="special">,</span> <span class="identifier">deriv_type</span> <span class="special">&</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"><</span> <span class="number">0</span> <span class="special">>(</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"><</span> <span class="number">1</span> <span class="special">>(</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"><</span> <span class="number">1</span> <span class="special">>(</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"><</span> <span class="number">0</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>
|
|
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>. Furthermore, the iteration
|
|
over vector elements is now done by the <code class="computeroutput"><span class="identifier">fusion_algebra</span></code>
|
|
which must also be given. 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
|
|
you own state types</a>.
|
|
</p>
|
|
<p>
|
|
</p>
|
|
<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">runge_kutta_dopri5</span><span class="special"><</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">,</span> <span class="identifier">fusion_algebra</span> <span class="special">></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>
|
|
<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">&</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">&</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">&</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">&</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"><</span> <span class="keyword">class</span> <span class="identifier">T</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">T</span> <span class="special">&</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"><<</span> <span class="string">"\t"</span> <span class="special"><<</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"><</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">></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">&</span><span class="identifier">x</span> <span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Time</span> <span class="special">&</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"><<</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"><<</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="../../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/index.html" target="_top">Boost.Units</a>
|
|
works nicely but compilation can be very time and memory consuming. For
|
|
example the unit test for <a href="http://www.boost.org/doc/libs/release/libs/units/index.html" target="_top">Boost.Units</a>
|
|
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/tree/master/libs/numeric/odeint/examples/harmonic_oscillator_units.cpp" target="_top">harmonic_oscillator_units.cpp</a>.
|
|
</p>
|
|
</div>
|
|
<div class="section">
|
|
<div class="titlepage"><div><div><h4 class="title">
|
|
<a name="boost_numeric_odeint.tutorial.special_topics.using_matrices_as_state_types"></a><a class="link" href="special_topics.html#boost_numeric_odeint.tutorial.special_topics.using_matrices_as_state_types" title="Using matrices as state types">Using
|
|
matrices as state types</a>
|
|
</h4></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. We like
|
|
phase oscillators, they are extremely easy and might serve for different
|
|
demonstration purposes. 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="number">0</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="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>
|
|
This is in principle 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/tree/master/libs/numeric/odeint/examples/two_dimensional_phase_lattice.cpp" target="_top">two_dimensional_phase_lattice.cpp</a>.
|
|
</p>
|
|
</div>
|
|
<div class="orderedlist"><ol class="orderedlist" type="1">
|
|
<li class="listitem">
|
|
[section Partial differential equations]
|
|
</li>
|
|
<li class="listitem">
|
|
</li>
|
|
<li class="listitem">
|
|
To be continued:
|
|
</li>
|
|
<li class="listitem">
|
|
</li>
|
|
<li class="listitem">
|
|
*Wave equation
|
|
</li>
|
|
<li class="listitem">
|
|
</li>
|
|
<li class="listitem">
|
|
*KdV
|
|
</li>
|
|
<li class="listitem">
|
|
</li>
|
|
<li class="listitem">
|
|
*Ginzburg-Landau
|
|
</li>
|
|
<li class="listitem">
|
|
</li>
|
|
<li class="listitem">
|
|
[endsect]
|
|
</li>
|
|
<li class="listitem">
|
|
</li>
|
|
<li class="listitem">
|
|
[section Ordinary differential equations on networks]
|
|
</li>
|
|
<li class="listitem">
|
|
</li>
|
|
<li class="listitem">
|
|
to be continued
|
|
</li>
|
|
<li class="listitem">
|
|
</li>
|
|
<li class="listitem">
|
|
[endsect]
|
|
</li>
|
|
</ol></div>
|
|
<div class="section">
|
|
<div class="titlepage"><div><div><h4 class="title">
|
|
<a name="boost_numeric_odeint.tutorial.special_topics.using_arbitrary_precision_floating_point_types"></a><a class="link" href="special_topics.html#boost_numeric_odeint.tutorial.special_topics.using_arbitrary_precision_floating_point_types" title="Using arbitrary precision floating point types">Using
|
|
arbitrary precision floating point types</a>
|
|
</h4></div></div></div>
|
|
<p>
|
|
Besides the classical floating point number like <code class="computeroutput"><span class="keyword">float</span></code>,
|
|
<code class="computeroutput"><span class="keyword">double</span></code>, <code class="computeroutput"><span class="identifier">complex</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">></span></code> you can also use arbitrary precision
|
|
types, like the types from <a href="http://gmplib.org/" target="_top">gmp</a>
|
|
and <a href="http://www.mpfr.org/" target="_top">mpfr</a>. But you have to be
|
|
careful about instantiating any numbers.
|
|
</p>
|
|
<p>
|
|
For gmp types you have to set the default precision before any number is
|
|
instantiated. This can be easily done by calling <code class="computeroutput"><span class="identifier">mpf_set_default_prec</span><span class="special">(</span> <span class="identifier">precision</span>
|
|
<span class="special">)</span></code> as the first function in your
|
|
main program. Secondly, you can not use any global constant variables since
|
|
they will not be set with the default precision you have already set.
|
|
</p>
|
|
<p>
|
|
Here is a simple example:
|
|
</p>
|
|
<p>
|
|
</p>
|
|
<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">mpf_class</span> <span class="identifier">value_type</span><span class="special">;</span>
|
|
<span class="keyword">typedef</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">array</span><span class="special"><</span> <span class="identifier">value_type</span> <span class="special">,</span> <span class="number">3</span> <span class="special">></span> <span class="identifier">state_type</span><span class="special">;</span>
|
|
|
|
<span class="keyword">struct</span> <span class="identifier">lorenz</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="identifier">value_type</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">value_type</span> <span class="identifier">sigma</span><span class="special">(</span> <span class="number">10.0</span> <span class="special">);</span>
|
|
<span class="keyword">const</span> <span class="identifier">value_type</span> <span class="identifier">R</span><span class="special">(</span> <span class="number">28.0</span> <span class="special">);</span>
|
|
<span class="keyword">const</span> <span class="identifier">value_type</span> <span class="identifier">b</span><span class="special">(</span> <span class="identifier">value_type</span><span class="special">(</span> <span class="number">8.0</span> <span class="special">)</span> <span class="special">/</span> <span class="identifier">value_type</span><span class="special">(</span> <span class="number">3.0</span> <span class="special">)</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="special">=</span> <span class="identifier">sigma</span> <span class="special">*</span> <span class="special">(</span> <span class="identifier">x</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="number">0</span><span class="special">]</span> <span class="special">);</span>
|
|
<span class="identifier">dxdt</span><span class="special">[</span><span class="number">1</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">R</span> <span class="special">*</span> <span class="identifier">x</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">-</span> <span class="identifier">x</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="number">0</span><span class="special">]</span> <span class="special">*</span> <span class="identifier">x</span><span class="special">[</span><span class="number">2</span><span class="special">];</span>
|
|
<span class="identifier">dxdt</span><span class="special">[</span><span class="number">2</span><span class="special">]</span> <span class="special">=</span> <span class="special">-</span><span class="identifier">b</span> <span class="special">*</span> <span class="identifier">x</span><span class="special">[</span><span class="number">2</span><span class="special">]</span> <span class="special">+</span> <span class="identifier">x</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">*</span> <span class="identifier">x</span><span class="special">[</span><span class="number">1</span><span class="special">];</span>
|
|
<span class="special">}</span>
|
|
<span class="special">};</span>
|
|
</pre>
|
|
<p>
|
|
</p>
|
|
<p>
|
|
which can be integrated:
|
|
</p>
|
|
<p>
|
|
</p>
|
|
<pre class="programlisting"><span class="keyword">const</span> <span class="keyword">int</span> <span class="identifier">precision</span> <span class="special">=</span> <span class="number">1024</span><span class="special">;</span>
|
|
<span class="identifier">mpf_set_default_prec</span><span class="special">(</span> <span class="identifier">precision</span> <span class="special">);</span>
|
|
|
|
<span class="identifier">state_type</span> <span class="identifier">x</span> <span class="special">=</span> <span class="special">{{</span> <span class="identifier">value_type</span><span class="special">(</span> <span class="number">10.0</span> <span class="special">)</span> <span class="special">,</span> <span class="identifier">value_type</span><span class="special">(</span> <span class="number">10.0</span> <span class="special">)</span> <span class="special">,</span> <span class="identifier">value_type</span><span class="special">(</span> <span class="number">10.0</span> <span class="special">)</span> <span class="special">}};</span>
|
|
|
|
<span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</span><span class="special">(</span> <span class="number">1000</span> <span class="special">);</span>
|
|
<span class="identifier">integrate_const</span><span class="special">(</span> <span class="identifier">runge_kutta4</span><span class="special"><</span> <span class="identifier">state_type</span> <span class="special">,</span> <span class="identifier">value_type</span> <span class="special">>()</span> <span class="special">,</span>
|
|
<span class="identifier">lorenz</span><span class="special">()</span> <span class="special">,</span> <span class="identifier">x</span> <span class="special">,</span> <span class="identifier">value_type</span><span class="special">(</span> <span class="number">0.0</span> <span class="special">)</span> <span class="special">,</span> <span class="identifier">value_type</span><span class="special">(</span> <span class="number">10.0</span> <span class="special">)</span> <span class="special">,</span> <span class="identifier">value_type</span><span class="special">(</span> <span class="identifier">value_type</span><span class="special">(</span> <span class="number">1.0</span> <span class="special">)</span> <span class="special">/</span> <span class="identifier">value_type</span><span class="special">(</span> <span class="number">10.0</span> <span class="special">)</span> <span class="special">)</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="caution"><table border="0" summary="Caution">
|
|
<tr>
|
|
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../images/caution.png"></td>
|
|
<th align="left">Caution</th>
|
|
</tr>
|
|
<tr><td align="left" valign="top"><p>
|
|
The full support of arbitrary precision types depends on the functionality
|
|
they provide. For example, the types from gmp are lacking of functions
|
|
for calculating the power and arbitrary roots, hence they can not be
|
|
used with the controlled steppers. In detail, for full support the <code class="computeroutput"><span class="identifier">min</span><span class="special">(</span> <span class="identifier">x</span> <span class="special">,</span> <span class="identifier">y</span> <span class="special">)</span></code>,
|
|
<code class="computeroutput"><span class="identifier">max</span><span class="special">(</span>
|
|
<span class="identifier">x</span> <span class="special">,</span>
|
|
<span class="identifier">y</span> <span class="special">)</span></code>,
|
|
<code class="computeroutput"><span class="identifier">pow</span><span class="special">(</span>
|
|
<span class="identifier">y</span> <span class="special">,</span>
|
|
<span class="identifier">y</span> <span class="special">)</span></code>
|
|
must be callable.
|
|
</p></td></tr>
|
|
</table></div>
|
|
<p>
|
|
The full example can be found at <a href="https://github.com/headmyshoulder/odeint-v2/tree/master/libs/numeric/odeint/examples/lorenz_gmpxx.cpp" target="_top">lorenz_gmpxx.cpp</a>.
|
|
</p>
|
|
</div>
|
|
<div class="section">
|
|
<div class="titlepage"><div><div><h4 class="title">
|
|
<a name="boost_numeric_odeint.tutorial.special_topics.self_expanding_lattices"></a><a class="link" href="special_topics.html#boost_numeric_odeint.tutorial.special_topics.self_expanding_lattices" title="Self expanding lattices">Self
|
|
expanding lattices</a>
|
|
</h4></div></div></div>
|
|
<p>
|
|
odeint supports changes of the state size during integration if a state_type
|
|
is used which can be resized, like <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span></code>.
|
|
The adjustment of the state's size has to be done from outside and the
|
|
stepper has to be instantiated with <code class="computeroutput"><span class="identifier">always_resizer</span></code>
|
|
as the template argument for the <code class="computeroutput"><span class="identifier">resizer_type</span></code>.
|
|
In this configuration, the stepper checks for changes in the state size
|
|
and adjust it's internal storage accordingly.
|
|
</p>
|
|
<p>
|
|
We exemplary show this for a Hamiltonian system of nonlinear, disordered
|
|
oscillators with nonlinear nearest neighbor coupling.
|
|
</p>
|
|
<p>
|
|
The system function is implemented in terms of a class that also provides
|
|
functions for calculating the energy. Note, that this class stores the
|
|
random potential internally which is not resized, but rather a start index
|
|
is kept which should be changed whenever the states' size change.
|
|
</p>
|
|
<p>
|
|
</p>
|
|
<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">vector</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">></span> <span class="identifier">coord_type</span><span class="special">;</span>
|
|
<span class="keyword">typedef</span> <span class="identifier">pair</span><span class="special"><</span> <span class="identifier">coord_type</span> <span class="special">,</span> <span class="identifier">coord_type</span> <span class="special">></span> <span class="identifier">state_type</span><span class="special">;</span>
|
|
|
|
<span class="keyword">struct</span> <span class="identifier">compacton_lattice</span>
|
|
<span class="special">{</span>
|
|
<span class="keyword">const</span> <span class="keyword">int</span> <span class="identifier">m_max_N</span><span class="special">;</span>
|
|
<span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">m_beta</span><span class="special">;</span>
|
|
<span class="keyword">int</span> <span class="identifier">m_pot_start_index</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="identifier">m_pot</span><span class="special">;</span>
|
|
|
|
<span class="identifier">compacton_lattice</span><span class="special">(</span> <span class="keyword">int</span> <span class="identifier">max_N</span> <span class="special">,</span> <span class="keyword">double</span> <span class="identifier">beta</span> <span class="special">,</span> <span class="keyword">int</span> <span class="identifier">pot_start_index</span> <span class="special">)</span>
|
|
<span class="special">:</span> <span class="identifier">m_max_N</span><span class="special">(</span> <span class="identifier">max_N</span> <span class="special">)</span> <span class="special">,</span> <span class="identifier">m_beta</span><span class="special">(</span> <span class="identifier">beta</span> <span class="special">)</span> <span class="special">,</span> <span class="identifier">m_pot_start_index</span><span class="special">(</span> <span class="identifier">pot_start_index</span> <span class="special">)</span> <span class="special">,</span> <span class="identifier">m_pot</span><span class="special">(</span> <span class="identifier">max_N</span> <span class="special">)</span>
|
|
<span class="special">{</span>
|
|
<span class="identifier">srand</span><span class="special">(</span> <span class="identifier">time</span><span class="special">(</span> <span class="identifier">NULL</span> <span class="special">)</span> <span class="special">);</span>
|
|
<span class="comment">// fill random potential with iid values from [0,1]</span>
|
|
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">mt19937</span> <span class="identifier">rng</span><span class="special">;</span>
|
|
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">uniform_real</span><span class="special"><></span> <span class="identifier">unif</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">boost</span><span class="special">::</span><span class="identifier">variate_generator</span><span class="special"><</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">mt19937</span><span class="special">&,</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">uniform_real</span><span class="special"><></span> <span class="special">></span> <span class="identifier">gen</span><span class="special">(</span> <span class="identifier">rng</span> <span class="special">,</span> <span class="identifier">unif</span> <span class="special">);</span>
|
|
<span class="identifier">generate</span><span class="special">(</span> <span class="identifier">m_pot</span><span class="special">.</span><span class="identifier">begin</span><span class="special">()</span> <span class="special">,</span> <span class="identifier">m_pot</span><span class="special">.</span><span class="identifier">end</span><span class="special">()</span> <span class="special">,</span> <span class="identifier">gen</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">coord_type</span> <span class="special">&</span><span class="identifier">q</span> <span class="special">,</span> <span class="identifier">coord_type</span> <span class="special">&</span><span class="identifier">dpdt</span> <span class="special">)</span>
|
|
<span class="special">{</span>
|
|
<span class="comment">// calculate dpdt = -dH/dq of this hamiltonian system</span>
|
|
<span class="comment">// dp_i/dt = - V_i * q_i^3 - beta*(q_i - q_{i-1})^3 + beta*(q_{i+1} - q_i)^3</span>
|
|
<span class="keyword">const</span> <span class="keyword">int</span> <span class="identifier">N</span> <span class="special">=</span> <span class="identifier">q</span><span class="special">.</span><span class="identifier">size</span><span class="special">();</span>
|
|
<span class="keyword">double</span> <span class="identifier">diff</span> <span class="special">=</span> <span class="identifier">q</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">-</span> <span class="identifier">q</span><span class="special">[</span><span class="identifier">N</span><span class="special">-</span><span class="number">1</span><span class="special">];</span>
|
|
<span class="keyword">for</span><span class="special">(</span> <span class="keyword">int</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">N</span> <span class="special">;</span> <span class="special">++</span><span class="identifier">i</span> <span class="special">)</span>
|
|
<span class="special">{</span>
|
|
<span class="identifier">dpdt</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special">=</span> <span class="special">-</span> <span class="identifier">m_pot</span><span class="special">[</span><span class="identifier">m_pot_start_index</span><span class="special">+</span><span class="identifier">i</span><span class="special">]</span> <span class="special">*</span> <span class="identifier">q</span><span class="special">[</span><span class="identifier">i</span><span class="special">]*</span><span class="identifier">q</span><span class="special">[</span><span class="identifier">i</span><span class="special">]*</span><span class="identifier">q</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special">-</span>
|
|
<span class="identifier">m_beta</span> <span class="special">*</span> <span class="identifier">diff</span><span class="special">*</span><span class="identifier">diff</span><span class="special">*</span><span class="identifier">diff</span><span class="special">;</span>
|
|
<span class="identifier">diff</span> <span class="special">=</span> <span class="identifier">q</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="special">%</span> <span class="identifier">N</span><span class="special">]</span> <span class="special">-</span> <span class="identifier">q</span><span class="special">[</span><span class="identifier">i</span><span class="special">];</span>
|
|
<span class="identifier">dpdt</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special">+=</span> <span class="identifier">m_beta</span> <span class="special">*</span> <span class="identifier">diff</span><span class="special">*</span><span class="identifier">diff</span><span class="special">*</span><span class="identifier">diff</span><span class="special">;</span>
|
|
<span class="special">}</span>
|
|
<span class="special">}</span>
|
|
|
|
<span class="keyword">void</span> <span class="identifier">energy_distribution</span><span class="special">(</span> <span class="keyword">const</span> <span class="identifier">coord_type</span> <span class="special">&</span><span class="identifier">q</span> <span class="special">,</span> <span class="keyword">const</span> <span class="identifier">coord_type</span> <span class="special">&</span><span class="identifier">p</span> <span class="special">,</span> <span class="identifier">coord_type</span> <span class="special">&</span><span class="identifier">energies</span> <span class="special">)</span>
|
|
<span class="special">{</span>
|
|
<span class="comment">// computes the energy per lattice site normalized by total energy</span>
|
|
<span class="keyword">const</span> <span class="identifier">size_t</span> <span class="identifier">N</span> <span class="special">=</span> <span class="identifier">q</span><span class="special">.</span><span class="identifier">size</span><span class="special">();</span>
|
|
<span class="keyword">double</span> <span class="identifier">en</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">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">N</span> <span class="special">;</span> <span class="identifier">i</span><span class="special">++</span> <span class="special">)</span>
|
|
<span class="special">{</span>
|
|
<span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">diff</span> <span class="special">=</span> <span class="identifier">q</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="special">%</span> <span class="identifier">N</span><span class="special">]</span> <span class="special">-</span> <span class="identifier">q</span><span class="special">[</span><span class="identifier">i</span><span class="special">];</span>
|
|
<span class="identifier">energies</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">p</span><span class="special">[</span><span class="identifier">i</span><span class="special">]*</span><span class="identifier">p</span><span class="special">[</span><span class="identifier">i</span><span class="special">]/</span><span class="number">2.0</span>
|
|
<span class="special">+</span> <span class="identifier">m_pot</span><span class="special">[</span><span class="identifier">m_pot_start_index</span><span class="special">+</span><span class="identifier">i</span><span class="special">]*</span><span class="identifier">q</span><span class="special">[</span><span class="identifier">i</span><span class="special">]*</span><span class="identifier">q</span><span class="special">[</span><span class="identifier">i</span><span class="special">]*</span><span class="identifier">q</span><span class="special">[</span><span class="identifier">i</span><span class="special">]*</span><span class="identifier">q</span><span class="special">[</span><span class="identifier">i</span><span class="special">]/</span><span class="number">4.0</span>
|
|
<span class="special">+</span> <span class="identifier">m_beta</span><span class="special">/</span><span class="number">4.0</span> <span class="special">*</span> <span class="identifier">diff</span><span class="special">*</span><span class="identifier">diff</span><span class="special">*</span><span class="identifier">diff</span><span class="special">*</span><span class="identifier">diff</span><span class="special">;</span>
|
|
<span class="identifier">en</span> <span class="special">+=</span> <span class="identifier">energies</span><span class="special">[</span><span class="identifier">i</span><span class="special">];</span>
|
|
<span class="special">}</span>
|
|
<span class="identifier">en</span> <span class="special">=</span> <span class="number">1.0</span><span class="special">/</span><span class="identifier">en</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">N</span> <span class="special">;</span> <span class="identifier">i</span><span class="special">++</span> <span class="special">)</span>
|
|
<span class="special">{</span>
|
|
<span class="identifier">energies</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special">*=</span> <span class="identifier">en</span><span class="special">;</span>
|
|
<span class="special">}</span>
|
|
<span class="special">}</span>
|
|
|
|
<span class="keyword">double</span> <span class="identifier">energy</span><span class="special">(</span> <span class="keyword">const</span> <span class="identifier">coord_type</span> <span class="special">&</span><span class="identifier">q</span> <span class="special">,</span> <span class="keyword">const</span> <span class="identifier">coord_type</span> <span class="special">&</span><span class="identifier">p</span> <span class="special">)</span>
|
|
<span class="special">{</span>
|
|
<span class="comment">// calculates the total energy of the excitation</span>
|
|
<span class="keyword">const</span> <span class="identifier">size_t</span> <span class="identifier">N</span> <span class="special">=</span> <span class="identifier">q</span><span class="special">.</span><span class="identifier">size</span><span class="special">();</span>
|
|
<span class="keyword">double</span> <span class="identifier">en</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">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">N</span> <span class="special">;</span> <span class="identifier">i</span><span class="special">++</span> <span class="special">)</span>
|
|
<span class="special">{</span>
|
|
<span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">diff</span> <span class="special">=</span> <span class="identifier">q</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="special">%</span> <span class="identifier">N</span><span class="special">]</span> <span class="special">-</span> <span class="identifier">q</span><span class="special">[</span><span class="identifier">i</span><span class="special">];</span>
|
|
<span class="identifier">en</span> <span class="special">+=</span> <span class="identifier">p</span><span class="special">[</span><span class="identifier">i</span><span class="special">]*</span><span class="identifier">p</span><span class="special">[</span><span class="identifier">i</span><span class="special">]/</span><span class="number">2.0</span>
|
|
<span class="special">+</span> <span class="identifier">m_pot</span><span class="special">[</span><span class="identifier">m_pot_start_index</span><span class="special">+</span><span class="identifier">i</span><span class="special">]*</span><span class="identifier">q</span><span class="special">[</span><span class="identifier">i</span><span class="special">]*</span><span class="identifier">q</span><span class="special">[</span><span class="identifier">i</span><span class="special">]*</span><span class="identifier">q</span><span class="special">[</span><span class="identifier">i</span><span class="special">]*</span><span class="identifier">q</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special">/</span> <span class="number">4.0</span>
|
|
<span class="special">+</span> <span class="identifier">m_beta</span><span class="special">/</span><span class="number">4.0</span> <span class="special">*</span> <span class="identifier">diff</span><span class="special">*</span><span class="identifier">diff</span><span class="special">*</span><span class="identifier">diff</span><span class="special">*</span><span class="identifier">diff</span><span class="special">;</span>
|
|
<span class="special">}</span>
|
|
<span class="keyword">return</span> <span class="identifier">en</span><span class="special">;</span>
|
|
<span class="special">}</span>
|
|
|
|
<span class="keyword">void</span> <span class="identifier">change_pot_start</span><span class="special">(</span> <span class="keyword">const</span> <span class="keyword">int</span> <span class="identifier">delta</span> <span class="special">)</span>
|
|
<span class="special">{</span>
|
|
<span class="identifier">m_pot_start_index</span> <span class="special">+=</span> <span class="identifier">delta</span><span class="special">;</span>
|
|
<span class="special">}</span>
|
|
<span class="special">};</span>
|
|
</pre>
|
|
<p>
|
|
</p>
|
|
<p>
|
|
The total size we allow is 1024 and we start with an initial state size
|
|
of 60.
|
|
</p>
|
|
<p>
|
|
</p>
|
|
<pre class="programlisting"><span class="comment">//start with 60 sites</span>
|
|
<span class="keyword">const</span> <span class="keyword">int</span> <span class="identifier">N_start</span> <span class="special">=</span> <span class="number">60</span><span class="special">;</span>
|
|
<span class="identifier">coord_type</span> <span class="identifier">q</span><span class="special">(</span> <span class="identifier">N_start</span> <span class="special">,</span> <span class="number">0.0</span> <span class="special">);</span>
|
|
<span class="identifier">q</span><span class="special">.</span><span class="identifier">reserve</span><span class="special">(</span> <span class="identifier">max_N</span> <span class="special">);</span>
|
|
<span class="identifier">coord_type</span> <span class="identifier">p</span><span class="special">(</span> <span class="identifier">N_start</span> <span class="special">,</span> <span class="number">0.0</span> <span class="special">);</span>
|
|
<span class="identifier">p</span><span class="special">.</span><span class="identifier">reserve</span><span class="special">(</span> <span class="identifier">max_N</span> <span class="special">);</span>
|
|
<span class="comment">// start with uniform momentum distribution over 20 sites</span>
|
|
<span class="identifier">fill</span><span class="special">(</span> <span class="identifier">p</span><span class="special">.</span><span class="identifier">begin</span><span class="special">()+</span><span class="number">20</span> <span class="special">,</span> <span class="identifier">p</span><span class="special">.</span><span class="identifier">end</span><span class="special">()-</span><span class="number">20</span> <span class="special">,</span> <span class="number">1.0</span><span class="special">/</span><span class="identifier">sqrt</span><span class="special">(</span><span class="number">20.0</span><span class="special">)</span> <span class="special">);</span>
|
|
|
|
<span class="identifier">coord_type</span> <span class="identifier">distr</span><span class="special">(</span> <span class="identifier">N_start</span> <span class="special">,</span> <span class="number">0.0</span> <span class="special">);</span>
|
|
<span class="identifier">distr</span><span class="special">.</span><span class="identifier">reserve</span><span class="special">(</span> <span class="identifier">max_N</span> <span class="special">);</span>
|
|
|
|
<span class="comment">// create the system</span>
|
|
<span class="identifier">compacton_lattice</span> <span class="identifier">lattice</span><span class="special">(</span> <span class="identifier">max_N</span> <span class="special">,</span> <span class="identifier">beta</span> <span class="special">,</span> <span class="special">(</span><span class="identifier">max_N</span><span class="special">-</span><span class="identifier">N_start</span><span class="special">)/</span><span class="number">2</span> <span class="special">);</span>
|
|
|
|
<span class="comment">//create the stepper, note that we use an always_resizer because state size might change during steps</span>
|
|
<span class="keyword">typedef</span> <span class="identifier">symplectic_rkn_sb3a_mclachlan</span><span class="special"><</span> <span class="identifier">coord_type</span> <span class="special">,</span> <span class="identifier">coord_type</span> <span class="special">,</span> <span class="keyword">double</span> <span class="special">,</span> <span class="identifier">coord_type</span> <span class="special">,</span> <span class="identifier">coord_type</span> <span class="special">,</span> <span class="keyword">double</span> <span class="special">,</span>
|
|
<span class="identifier">range_algebra</span> <span class="special">,</span> <span class="identifier">default_operations</span> <span class="special">,</span> <span class="identifier">always_resizer</span> <span class="special">></span> <span class="identifier">hamiltonian_stepper</span><span class="special">;</span>
|
|
<span class="identifier">hamiltonian_stepper</span> <span class="identifier">stepper</span><span class="special">;</span>
|
|
<span class="identifier">hamiltonian_stepper</span><span class="special">::</span><span class="identifier">state_type</span> <span class="identifier">state</span> <span class="special">=</span> <span class="identifier">make_pair</span><span class="special">(</span> <span class="identifier">q</span> <span class="special">,</span> <span class="identifier">p</span> <span class="special">);</span>
|
|
</pre>
|
|
<p>
|
|
</p>
|
|
<p>
|
|
The lattice gets resized whenever the energy distribution comes close to
|
|
the borders <code class="computeroutput"><span class="identifier">distr</span><span class="special">[</span><span class="number">10</span><span class="special">]</span> <span class="special">></span>
|
|
<span class="number">1E-150</span></code>, <code class="computeroutput"><span class="identifier">distr</span><span class="special">[</span><span class="identifier">distr</span><span class="special">.</span><span class="identifier">size</span><span class="special">()-</span><span class="number">10</span><span class="special">]</span>
|
|
<span class="special">></span> <span class="number">1E-150</span></code>.
|
|
If we increase to the left, <code class="computeroutput"><span class="identifier">q</span></code>
|
|
and <code class="computeroutput"><span class="identifier">p</span></code> have to be rotated
|
|
because their resize function always appends at the end. Additionally,
|
|
the start index of the potential changes in this case.
|
|
</p>
|
|
<p>
|
|
</p>
|
|
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">t</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">const</span> <span class="keyword">int</span> <span class="identifier">steps</span> <span class="special">=</span> <span class="number">10000</span><span class="special">;</span>
|
|
<span class="keyword">for</span><span class="special">(</span> <span class="keyword">int</span> <span class="identifier">step</span> <span class="special">=</span> <span class="number">0</span> <span class="special">;</span> <span class="identifier">step</span> <span class="special"><</span> <span class="identifier">steps</span> <span class="special">;</span> <span class="special">++</span><span class="identifier">step</span> <span class="special">)</span>
|
|
<span class="special">{</span>
|
|
<span class="identifier">stepper</span><span class="special">.</span><span class="identifier">do_step</span><span class="special">(</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">ref</span><span class="special">(</span><span class="identifier">lattice</span><span class="special">)</span> <span class="special">,</span> <span class="identifier">state</span> <span class="special">,</span> <span class="identifier">t</span> <span class="special">,</span> <span class="identifier">dt</span> <span class="special">);</span>
|
|
<span class="identifier">lattice</span><span class="special">.</span><span class="identifier">energy_distribution</span><span class="special">(</span> <span class="identifier">state</span><span class="special">.</span><span class="identifier">first</span> <span class="special">,</span> <span class="identifier">state</span><span class="special">.</span><span class="identifier">second</span> <span class="special">,</span> <span class="identifier">distr</span> <span class="special">);</span>
|
|
<span class="keyword">if</span><span class="special">(</span> <span class="identifier">distr</span><span class="special">[</span><span class="number">10</span><span class="special">]</span> <span class="special">></span> <span class="number">1E-150</span> <span class="special">)</span>
|
|
<span class="special">{</span>
|
|
<span class="identifier">do_resize</span><span class="special">(</span> <span class="identifier">state</span><span class="special">.</span><span class="identifier">first</span> <span class="special">,</span> <span class="identifier">state</span><span class="special">.</span><span class="identifier">second</span> <span class="special">,</span> <span class="identifier">distr</span> <span class="special">,</span> <span class="identifier">state</span><span class="special">.</span><span class="identifier">first</span><span class="special">.</span><span class="identifier">size</span><span class="special">()+</span><span class="number">20</span> <span class="special">);</span>
|
|
<span class="identifier">rotate</span><span class="special">(</span> <span class="identifier">state</span><span class="special">.</span><span class="identifier">first</span><span class="special">.</span><span class="identifier">begin</span><span class="special">()</span> <span class="special">,</span> <span class="identifier">state</span><span class="special">.</span><span class="identifier">first</span><span class="special">.</span><span class="identifier">end</span><span class="special">()-</span><span class="number">20</span> <span class="special">,</span> <span class="identifier">state</span><span class="special">.</span><span class="identifier">first</span><span class="special">.</span><span class="identifier">end</span><span class="special">()</span> <span class="special">);</span>
|
|
<span class="identifier">rotate</span><span class="special">(</span> <span class="identifier">state</span><span class="special">.</span><span class="identifier">second</span><span class="special">.</span><span class="identifier">begin</span><span class="special">()</span> <span class="special">,</span> <span class="identifier">state</span><span class="special">.</span><span class="identifier">second</span><span class="special">.</span><span class="identifier">end</span><span class="special">()-</span><span class="number">20</span> <span class="special">,</span> <span class="identifier">state</span><span class="special">.</span><span class="identifier">second</span><span class="special">.</span><span class="identifier">end</span><span class="special">()</span> <span class="special">);</span>
|
|
<span class="identifier">lattice</span><span class="special">.</span><span class="identifier">change_pot_start</span><span class="special">(</span> <span class="special">-</span><span class="number">20</span> <span class="special">);</span>
|
|
<span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">t</span> <span class="special"><<</span> <span class="string">": resized left to "</span> <span class="special"><<</span> <span class="identifier">distr</span><span class="special">.</span><span class="identifier">size</span><span class="special">()</span> <span class="special"><<</span> <span class="string">", energy = "</span> <span class="special"><<</span> <span class="identifier">lattice</span><span class="special">.</span><span class="identifier">energy</span><span class="special">(</span> <span class="identifier">state</span><span class="special">.</span><span class="identifier">first</span> <span class="special">,</span> <span class="identifier">state</span><span class="special">.</span><span class="identifier">second</span> <span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span>
|
|
<span class="special">}</span>
|
|
<span class="keyword">if</span><span class="special">(</span> <span class="identifier">distr</span><span class="special">[</span><span class="identifier">distr</span><span class="special">.</span><span class="identifier">size</span><span class="special">()-</span><span class="number">10</span><span class="special">]</span> <span class="special">></span> <span class="number">1E-150</span> <span class="special">)</span>
|
|
<span class="special">{</span>
|
|
<span class="identifier">do_resize</span><span class="special">(</span> <span class="identifier">state</span><span class="special">.</span><span class="identifier">first</span> <span class="special">,</span> <span class="identifier">state</span><span class="special">.</span><span class="identifier">second</span> <span class="special">,</span> <span class="identifier">distr</span> <span class="special">,</span> <span class="identifier">state</span><span class="special">.</span><span class="identifier">first</span><span class="special">.</span><span class="identifier">size</span><span class="special">()+</span><span class="number">20</span> <span class="special">);</span>
|
|
<span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">t</span> <span class="special"><<</span> <span class="string">": resized right to "</span> <span class="special"><<</span> <span class="identifier">distr</span><span class="special">.</span><span class="identifier">size</span><span class="special">()</span> <span class="special"><<</span> <span class="string">", energy = "</span> <span class="special"><<</span> <span class="identifier">lattice</span><span class="special">.</span><span class="identifier">energy</span><span class="special">(</span> <span class="identifier">state</span><span class="special">.</span><span class="identifier">first</span> <span class="special">,</span> <span class="identifier">state</span><span class="special">.</span><span class="identifier">second</span> <span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span>
|
|
<span class="special">}</span>
|
|
<span class="identifier">t</span> <span class="special">+=</span> <span class="identifier">dt</span><span class="special">;</span>
|
|
<span class="special">}</span>
|
|
</pre>
|
|
<p>
|
|
</p>
|
|
<p>
|
|
The <code class="computeroutput"><span class="identifier">do_resize</span></code> function
|
|
simply calls <code class="computeroutput"><span class="identifier">vector</span><span class="special">.</span><span class="identifier">resize</span></code> of <code class="computeroutput"><span class="identifier">q</span></code>
|
|
, <code class="computeroutput"><span class="identifier">p</span></code> and <code class="computeroutput"><span class="identifier">distr</span></code>.
|
|
</p>
|
|
<p>
|
|
</p>
|
|
<pre class="programlisting"><span class="keyword">void</span> <span class="identifier">do_resize</span><span class="special">(</span> <span class="identifier">coord_type</span> <span class="special">&</span><span class="identifier">q</span> <span class="special">,</span> <span class="identifier">coord_type</span> <span class="special">&</span><span class="identifier">p</span> <span class="special">,</span> <span class="identifier">coord_type</span> <span class="special">&</span><span class="identifier">distr</span> <span class="special">,</span> <span class="keyword">const</span> <span class="keyword">int</span> <span class="identifier">N</span> <span class="special">)</span>
|
|
<span class="special">{</span>
|
|
<span class="identifier">q</span><span class="special">.</span><span class="identifier">resize</span><span class="special">(</span> <span class="identifier">N</span> <span class="special">);</span>
|
|
<span class="identifier">p</span><span class="special">.</span><span class="identifier">resize</span><span class="special">(</span> <span class="identifier">N</span> <span class="special">);</span>
|
|
<span class="identifier">distr</span><span class="special">.</span><span class="identifier">resize</span><span class="special">(</span> <span class="identifier">N</span> <span class="special">);</span>
|
|
<span class="special">}</span>
|
|
</pre>
|
|
<p>
|
|
</p>
|
|
<p>
|
|
The full example can be found in <a href="https://github.com/headmyshoulder/odeint-v2/tree/master/libs/numeric/odeint/examples/resizing_lattice.cpp" target="_top">resizing_lattice.cpp</a>
|
|
</p>
|
|
</div>
|
|
</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 © 2009-2011 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>
|
|
</div></td>
|
|
</tr></table>
|
|
<hr>
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</body>
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</html>
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