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<div class="titlepage"><div><div><h3 class="title">
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<a name="boost_numeric_odeint.getting_started.short_example"></a><a class="link" href="short_example.html" title="Short Example">Short
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Example</a>
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</h3></div></div></div>
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<p>
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Imaging, you want to numerically integrate a harmonic oscillator with friction.
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The equations of motion are given by <span class="emphasis"><em>x'' = -x + γ x'</em></span>.
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Odeint only deals with first order ODEs that have no higher derivatives than
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x' involved. However, any higher order ODE can be transformed to a system
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of first order ODEs by introducing the new variables <span class="emphasis"><em>q=x</em></span>
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and <span class="emphasis"><em>p=x'</em></span> such that <span class="emphasis"><em>w=(q,p)</em></span>. To
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apply numerical integration one first has to design the right hand side of
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the equation <span class="emphasis"><em>w' = f(w) = (p,-q+γ p)</em></span>:
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</p>
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<p>
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</p>
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<pre class="programlisting"><span class="comment">/* The type of container used to hold the state vector */</span>
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<span class="keyword">typedef</span> <span class="identifier">std</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">state_type</span><span class="special">;</span>
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<span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">gam</span> <span class="special">=</span> <span class="number">0.15</span><span class="special">;</span>
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<span class="comment">/* The rhs of x' = f(x) */</span>
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<span class="keyword">void</span> <span class="identifier">harmonic_oscillator</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">const</span> <span class="keyword">double</span> <span class="comment">/* t */</span> <span class="special">)</span>
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<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="identifier">x</span><span class="special">[</span><span class="number">1</span><span class="special">];</span>
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<span class="identifier">dxdt</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">x</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">-</span> <span class="identifier">gam</span><span class="special">*</span><span class="identifier">x</span><span class="special">[</span><span class="number">1</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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Here we chose <code class="computeroutput"><span class="identifier">vector</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span></code>
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as the state type, but others are also possible, for example <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="number">2</span><span class="special">></span></code>. odeint is designed in such a way that
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you can easily use your own state types. Next, the ODE is defined which is
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in this case a simple function calculating <span class="emphasis"><em>f(x)'</em></span>. The
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parameter signature of this function is crucial: the integration methods
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will always call them in the form <code class="computeroutput"><span class="identifier">f</span><span class="special">(</span><span class="identifier">x</span><span class="special">,</span>
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<span class="identifier">dxdt</span><span class="special">,</span>
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<span class="identifier">t</span><span class="special">)</span></code>
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(there are exceptions for some special routines). So, even if there is no
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explicit time dependence, one has to define <code class="computeroutput"><span class="identifier">t</span></code>
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as a function parameter.
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</p>
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<p>
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Now, we have to define the initial state from which the integration should
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start:
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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="number">2</span><span class="special">);</span>
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<span class="identifier">x</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">=</span> <span class="number">1.0</span><span class="special">;</span> <span class="comment">// start at x=1.0, p=0.0</span>
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<span class="identifier">x</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>
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</pre>
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<p>
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</p>
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<p>
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For the integration itself we'll use the <code class="computeroutput">integrate</code>
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function, which is a convenient way to get quick results. It is based on
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the error-controlled <code class="computeroutput">runge_kutta_rk5_ck</code>
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stepper (5th order) and uses adaptive step-size.
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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">size_t</span> <span class="identifier">steps</span> <span class="special">=</span> <span class="identifier">integrate</span><span class="special">(</span> <span class="identifier">harmonic_oscillator</span> <span class="special">,</span>
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<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="number">0.1</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 integrate function expects as parameters the rhs of the ode as defined
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above, the initial state <code class="computeroutput"><span class="identifier">x</span></code>,
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the start-and end-time of the integration as well as the initial time step=size.
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Note, that <code class="computeroutput">integrate</code>
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uses an adaptive step-size during the integration steps so the time points
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will not be equally spaced. The integration returns the number of steps that
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were applied and updates x which is set to the approximate solution of the
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ODE at the end of integration.
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</p>
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<p>
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It is, of course, also possible to represent the ode system as a class. The
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rhs must then be implemented as a functor having defined the ()-operator:
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</p>
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<p>
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</p>
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<pre class="programlisting"><span class="comment">/* The rhs of x' = f(x) defined as a class */</span>
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<span class="keyword">class</span> <span class="identifier">harm_osc</span> <span class="special">{</span>
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<span class="keyword">double</span> <span class="identifier">m_gam</span><span class="special">;</span>
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<span class="keyword">public</span><span class="special">:</span>
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<span class="identifier">harm_osc</span><span class="special">(</span> <span class="keyword">double</span> <span class="identifier">gam</span> <span class="special">)</span> <span class="special">:</span> <span class="identifier">m_gam</span><span class="special">(</span><span class="identifier">gam</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="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">const</span> <span class="keyword">double</span> <span class="comment">/* t */</span> <span class="special">)</span>
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<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="identifier">x</span><span class="special">[</span><span class="number">1</span><span class="special">];</span>
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<span class="identifier">dxdt</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">x</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">-</span> <span class="identifier">m_gam</span><span class="special">*</span><span class="identifier">x</span><span class="special">[</span><span class="number">1</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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which can be used via
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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">harm_osc</span> <span class="identifier">ho</span><span class="special">(</span><span class="number">0.15</span><span class="special">);</span>
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<span class="identifier">steps</span> <span class="special">=</span> <span class="identifier">integrate</span><span class="special">(</span> <span class="identifier">ho</span> <span class="special">,</span>
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<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="number">0.1</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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You surely have already noticed that during the integration a lot of steps
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had to be done. You might wonder if you could access them do observe the
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solution during the iteration. Yes, this is possible, of course. All you
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have to do is to provide a reasonable observer. An example is
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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">struct</span> <span class="identifier">push_back_state_and_time</span>
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<span class="special">{</span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span> <span class="identifier">state_type</span> <span class="special">>&</span> <span class="identifier">m_states</span><span class="special">;</span>
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<span class="identifier">std</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_times</span><span class="special">;</span>
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<span class="identifier">push_back_state_and_time</span><span class="special">(</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span> <span class="identifier">state_type</span> <span class="special">></span> <span class="special">&</span><span class="identifier">states</span> <span class="special">,</span> <span class="identifier">std</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="special">&</span><span class="identifier">times</span> <span class="special">)</span>
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<span class="special">:</span> <span class="identifier">m_states</span><span class="special">(</span> <span class="identifier">states</span> <span class="special">)</span> <span class="special">,</span> <span class="identifier">m_times</span><span class="special">(</span> <span class="identifier">times</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="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="identifier">m_states</span><span class="special">.</span><span class="identifier">push_back</span><span class="special">(</span> <span class="identifier">x</span> <span class="special">);</span>
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<span class="identifier">m_times</span><span class="special">.</span><span class="identifier">push_back</span><span class="special">(</span> <span class="identifier">t</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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which stores the intermediate steps in a container. Note, the argument structure
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of the ()-operator: odeint calls the observer exactly in this way, providing
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the current state and time. Now, you only have to pass this container to
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the integration function:
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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">vector</span><span class="special"><</span><span class="identifier">state_type</span><span class="special">></span> <span class="identifier">x_vec</span><span class="special">;</span>
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<span class="identifier">vector</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">times</span><span class="special">;</span>
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<span class="identifier">steps</span> <span class="special">=</span> <span class="identifier">integrate</span><span class="special">(</span> <span class="identifier">harmonic_oscillator</span> <span class="special">,</span>
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<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="number">0.1</span> <span class="special">,</span>
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<span class="identifier">push_back_state_and_time</span><span class="special">(</span> <span class="identifier">x_vec</span> <span class="special">,</span> <span class="identifier">times</span> <span class="special">)</span> <span class="special">);</span>
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<span class="comment">/* output */</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">steps</span><span class="special">;</span> <span class="identifier">i</span><span class="special">++</span> <span class="special">)</span>
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<span class="special">{</span>
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<span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">times</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special"><<</span> <span class="char">'\t'</span> <span class="special"><<</span> <span class="identifier">x_vec</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="char">'\t'</span> <span class="special"><<</span> <span class="identifier">x_vec</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="char">'\n'</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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That is all. Of course, you can use functional libraries like <a href="http://www.boost.org/doc/libs/release/doc/html/lambda.html" target="_top">Boost.Lambda</a>
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or <a href="http://www.boost.org/doc/libs/1_46_1/libs/spirit/phoenix/doc/html/index.html" target="_top">Boost.Phoenix</a>
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to ease the creation of observer functions.
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</p>
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<p>
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The full cpp file for this example can be found here: <a href="../../../../examples/harmonic_oscillator.cpp" target="_top">../../examples/harmonic_oscillator.cpp</a>
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</p>
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</div>
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<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
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<td align="left"></td>
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<td align="right"><div class="copyright-footer">Copyright © 2009-2011 Karsten Ahnert
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and Mario Mulansky<p>
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Distributed under the Boost Software License, Version 1.0. (See accompanying
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file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
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</p>
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</div></td>
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</tr></table>
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