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<div class="titlepage"><div><div><h2 class="title" style="clear: both">
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<a name="math_toolkit.naive_monte_carlo"></a><a class="link" href="naive_monte_carlo.html" title="Naive Monte Carlo Integration">Naive Monte Carlo Integration</a>
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</h2></div></div></div>
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<h4>
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<a name="math_toolkit.naive_monte_carlo.h0"></a>
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<span class="phrase"><a name="math_toolkit.naive_monte_carlo.synopsis"></a></span><a class="link" href="naive_monte_carlo.html#math_toolkit.naive_monte_carlo.synopsis">Synopsis</a>
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</h4>
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<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">quadrature</span><span class="special">/</span><span class="identifier">naive_monte_carlo</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
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<span class="keyword">namespace</span> <span class="identifier">boost</span> <span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span> <span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">quadrature</span> <span class="special">{</span>
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<span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">Real</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">RNG</span> <span class="special">=</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">mt19937_64</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Policy</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">policies</span><span class="special">::</span><span class="identifier">policy</span><span class="special"><>></span>
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<span class="keyword">class</span> <span class="identifier">naive_monte_carlo</span>
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<span class="special">{</span>
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<span class="keyword">public</span><span class="special">:</span>
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<span class="identifier">naive_monte_carlo</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">F</span><span class="special">&</span> <span class="identifier">integrand</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="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">Real</span><span class="special">,</span> <span class="identifier">Real</span><span class="special">>></span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">bounds</span><span class="special">,</span>
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<span class="identifier">Real</span> <span class="identifier">error_goal</span><span class="special">,</span>
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<span class="keyword">bool</span> <span class="identifier">singular</span> <span class="special">=</span> <span class="keyword">true</span><span class="special">,</span>
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<span class="identifier">size_t</span> <span class="identifier">threads</span> <span class="special">=</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">thread</span><span class="special">::</span><span class="identifier">hardware_concurrency</span><span class="special">());</span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">future</span><span class="special"><</span><span class="identifier">Real</span><span class="special">></span> <span class="identifier">integrate</span><span class="special">();</span>
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<span class="keyword">void</span> <span class="identifier">cancel</span><span class="special">();</span>
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<span class="identifier">Real</span> <span class="identifier">current_error_estimate</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">chrono</span><span class="special">::</span><span class="identifier">duration</span><span class="special"><</span><span class="identifier">Real</span><span class="special">></span> <span class="identifier">estimated_time_to_completion</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
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<span class="keyword">void</span> <span class="identifier">update_target_error</span><span class="special">(</span><span class="identifier">Real</span> <span class="identifier">new_target_error</span><span class="special">);</span>
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<span class="identifier">Real</span> <span class="identifier">progress</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
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<span class="identifier">Real</span> <span class="identifier">current_estimate</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
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<span class="identifier">size_t</span> <span class="identifier">calls</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
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<span class="special">};</span>
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<span class="special">}}}</span> <span class="comment">// namespaces</span>
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</pre>
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<h4>
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<a name="math_toolkit.naive_monte_carlo.h1"></a>
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<span class="phrase"><a name="math_toolkit.naive_monte_carlo.description"></a></span><a class="link" href="naive_monte_carlo.html#math_toolkit.naive_monte_carlo.description">Description</a>
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</h4>
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<p>
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The class <code class="computeroutput"><span class="identifier">naive_monte_carlo</span></code>
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performs Monte-Carlo integration on a square integrable function <span class="emphasis"><em>f</em></span>
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on a domain Ω. The theoretical background of Monte-Carlo integration is nicely
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discussed at <a href="https://en.wikipedia.org/wiki/Monte_Carlo_integration" target="_top">Wikipedia</a>,
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and as such will not be discussed here. However, despite being "naive",
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it is a mistake to assume that naive Monte-Carlo integration is not powerful,
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as the simplicity of the method affords a robustness not easily provided by
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more sophisticated tools. The multithreaded nature of the routine allows us
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to compute a large number of sample points with great speed, and hence the
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slow convergence is mitigated by exploiting the full power of modern hardware.
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</p>
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<p>
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The naive Monte-Carlo integration provided by Boost exemplifies the programming
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techniques needed to cope with high-performance computing. For instance, since
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the convergence is only 𝑶(N<sup>-1/2</sup>), the compute time is very sensitive to the
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error goal. Users can easily specify an error goal which causes computation
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to last months-or just a few seconds. Without progress reporting, this situation
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is disorienting and causes the user to behave in a paranoid manner. Even with
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progress reporting, a user might need to cancel a job due to shifting priorities
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of the employing institution, and as such cancellation must be supported. A
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cancelled job which returns no results is wasted, so the cancellation must
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||
be graceful, returning the best estimate of the result thus far. In addition,
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a task might finish, and the user may well decide to try for a lower error
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bound. Hence restarting without loss of the preceding effort must be supported.
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Finally, on an HPC system, we generally wish to use all available threads.
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But if the computation is performed on a users workstation, employing every
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thread will cause the browser, email, or music apps to become unresponsive,
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so leaving a single thread available for other apps is appreciated.
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</p>
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<p>
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All these use cases are supported, so let's get to the code:
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</p>
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<pre class="programlisting"><span class="comment">// Define a function to integrate:</span>
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<span class="keyword">auto</span> <span class="identifier">g</span> <span class="special">=</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="keyword">const</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="keyword">constexpr</span> <span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">A</span> <span class="special">=</span> <span class="number">1.0</span> <span class="special">/</span> <span class="special">(</span><span class="identifier">M_PI</span> <span class="special">*</span> <span class="identifier">M_PI</span> <span class="special">*</span> <span class="identifier">M_PI</span><span class="special">);</span>
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<span class="keyword">return</span> <span class="identifier">A</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="number">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="number">1</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="number">2</span><span class="special">]));</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">std</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">bounds</span><span class="special">{{</span><span class="number">0</span><span class="special">,</span> <span class="identifier">M_PI</span><span class="special">},</span> <span class="special">{</span><span class="number">0</span><span class="special">,</span> <span class="identifier">M_PI</span><span class="special">},</span> <span class="special">{</span><span class="number">0</span><span class="special">,</span> <span class="identifier">M_PI</span><span class="special">}};</span>
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<span class="keyword">double</span> <span class="identifier">error_goal</span> <span class="special">=</span> <span class="number">0.001</span>
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<span class="identifier">naive_monte_carlo</span><span class="special"><</span><span class="keyword">double</span><span class="special">,</span> <span class="keyword">decltype</span><span class="special">(</span><span class="identifier">g</span><span class="special">)></span> <span class="identifier">mc</span><span class="special">(</span><span class="identifier">g</span><span class="special">,</span> <span class="identifier">bounds</span><span class="special">,</span> <span class="identifier">error_goal</span><span class="special">);</span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">future</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">task</span> <span class="special">=</span> <span class="identifier">mc</span><span class="special">.</span><span class="identifier">integrate</span><span class="special">();</span>
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<span class="keyword">while</span> <span class="special">(</span><span class="identifier">task</span><span class="special">.</span><span class="identifier">wait_for</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">chrono</span><span class="special">::</span><span class="identifier">seconds</span><span class="special">(</span><span class="number">1</span><span class="special">))</span> <span class="special">!=</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">future_status</span><span class="special">::</span><span class="identifier">ready</span><span class="special">)</span>
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||
<span class="special">{</span>
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<span class="comment">// The user must decide on a reasonable way to display the progress depending on their environment:</span>
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||
<span class="identifier">display_progress</span><span class="special">(</span><span class="identifier">mc</span><span class="special">.</span><span class="identifier">progress</span><span class="special">(),</span>
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||
<span class="identifier">mc</span><span class="special">.</span><span class="identifier">current_error_estimate</span><span class="special">(),</span>
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||
<span class="identifier">mc</span><span class="special">.</span><span class="identifier">current_estimate</span><span class="special">(),</span>
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<span class="identifier">mc</span><span class="special">.</span><span class="identifier">estimated_time_to_completion</span><span class="special">());</span>
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||
<span class="keyword">if</span> <span class="special">(</span><span class="identifier">some_signal_heard</span><span class="special">())</span>
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<span class="special">{</span>
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<span class="identifier">mc</span><span class="special">.</span><span class="identifier">cancel</span><span class="special">();</span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"\nCancelling because this is too slow!\n"</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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<span class="keyword">double</span> <span class="identifier">y</span> <span class="special">=</span> <span class="identifier">task</span><span class="special">.</span><span class="identifier">get</span><span class="special">();</span>
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</pre>
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<p>
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First off, we define the function we wish to integrate. This function must
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accept a <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="identifier">Real</span><span class="special">></span> <span class="keyword">const</span> <span class="special">&</span></code>,
|
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and return a <code class="computeroutput"><span class="identifier">Real</span></code>. Next, we
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||
define the domain of integration. Infinite domains are indicated by the bound
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<code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">Real</span><span class="special">>::</span><span class="identifier">infinity</span><span class="special">()</span></code>.
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The call
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</p>
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<pre class="programlisting"><span class="identifier">naive_monte_carlo</span><span class="special"><</span><span class="keyword">double</span><span class="special">,</span> <span class="keyword">decltype</span><span class="special">(</span><span class="identifier">g</span><span class="special">)></span> <span class="identifier">mc</span><span class="special">(</span><span class="identifier">g</span><span class="special">,</span> <span class="identifier">bounds</span><span class="special">,</span> <span class="identifier">error_goal</span><span class="special">);</span>
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</pre>
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<p>
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creates an instance of the monte carlo integrator. This is also where the number
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of threads can be set, for instance
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</p>
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<pre class="programlisting"><span class="identifier">naive_monte_carlo</span><span class="special"><</span><span class="keyword">double</span><span class="special">,</span> <span class="keyword">decltype</span><span class="special">(</span><span class="identifier">g</span><span class="special">)></span> <span class="identifier">mc</span><span class="special">(</span><span class="identifier">g</span><span class="special">,</span> <span class="identifier">bounds</span><span class="special">,</span> <span class="identifier">error_goal</span><span class="special">,</span> <span class="keyword">true</span><span class="special">,</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">thread</span><span class="special">::</span><span class="identifier">hardware_concurrency</span><span class="special">()</span> <span class="special">-</span> <span class="number">1</span><span class="special">);</span>
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</pre>
|
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<p>
|
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might be more appropriate for running on a user's hardware (the default taking
|
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all the threads). The call to <code class="computeroutput"><span class="identifier">integrate</span><span class="special">()</span></code> does not return the value of the integral,
|
||
but rather a <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">future</span><span class="special"><</span><span class="identifier">Real</span><span class="special">></span></code>.
|
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This allows us to do progress reporting from the master thread via
|
||
</p>
|
||
<pre class="programlisting"><span class="keyword">while</span> <span class="special">(</span><span class="identifier">task</span><span class="special">.</span><span class="identifier">wait_for</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">chrono</span><span class="special">::</span><span class="identifier">seconds</span><span class="special">(</span><span class="number">1</span><span class="special">))</span> <span class="special">!=</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">future_status</span><span class="special">::</span><span class="identifier">ready</span><span class="special">)</span>
|
||
<span class="special">{</span>
|
||
<span class="comment">// some reasonable method of displaying progress, based on the requirements of your app.</span>
|
||
<span class="identifier">display_progress</span><span class="special">(</span><span class="identifier">mc</span><span class="special">.</span><span class="identifier">progress</span><span class="special">(),</span>
|
||
<span class="identifier">mc</span><span class="special">.</span><span class="identifier">current_error_estimate</span><span class="special">(),</span>
|
||
<span class="identifier">mc</span><span class="special">.</span><span class="identifier">current_estimate</span><span class="special">(),</span>
|
||
<span class="identifier">mc</span><span class="special">.</span><span class="identifier">estimated_time_to_completion</span><span class="special">());</span>
|
||
<span class="special">}</span>
|
||
</pre>
|
||
<p>
|
||
The file <code class="computeroutput"><span class="identifier">example</span><span class="special">/</span><span class="identifier">naive_monte_carlo_example</span><span class="special">.</span><span class="identifier">cpp</span></code> has an implementation of <code class="computeroutput"><span class="identifier">display_progress</span></code> which is reasonable for
|
||
command line apps. In addition, we can call <code class="computeroutput"><span class="identifier">mc</span><span class="special">.</span><span class="identifier">cancel</span><span class="special">()</span></code>
|
||
in this loop to stop the integration. Progress reporting is especially useful
|
||
if you accidentally pass in an integrand which is not square integrable-the
|
||
variance increases without bound, and the progress decreases from some noisy
|
||
initial value down to zero with time. Calling <code class="computeroutput"><span class="identifier">task</span><span class="special">.</span><span class="identifier">get</span><span class="special">()</span></code>
|
||
returns the current estimate. Once the future is ready, we can get the value
|
||
of the integral via
|
||
</p>
|
||
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">result</span> <span class="special">=</span> <span class="identifier">task</span><span class="special">.</span><span class="identifier">get</span><span class="special">();</span>
|
||
</pre>
|
||
<p>
|
||
At this point, the user may wish to reduce the error goal. This is achieved
|
||
by
|
||
</p>
|
||
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">new_target_error</span> <span class="special">=</span> <span class="number">0.0005</span><span class="special">;</span>
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||
<span class="identifier">mc</span><span class="special">.</span><span class="identifier">update_target_error</span><span class="special">(</span><span class="identifier">new_target_error</span><span class="special">);</span>
|
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<span class="identifier">task</span> <span class="special">=</span> <span class="identifier">mc</span><span class="special">.</span><span class="identifier">integrate</span><span class="special">();</span>
|
||
<span class="identifier">y</span> <span class="special">=</span> <span class="identifier">task</span><span class="special">.</span><span class="identifier">get</span><span class="special">();</span>
|
||
</pre>
|
||
<p>
|
||
There is one additional "advanced" parameter: Whether or not the
|
||
integrand is singular on the boundary. If the integrand is <span class="bold"><strong>not</strong></span>
|
||
singular on the boundary, then the integrand is evaluated over the closed set
|
||
∏<sub>i</sub> [ <span class="emphasis"><em>a</em></span><sub><span class="emphasis"><em>i</em></span></sub>, <span class="emphasis"><em>b</em></span><sub><span class="emphasis"><em>i</em></span></sub> ].
|
||
If the integrand is singular (the default) then the integrand is evaluated
|
||
over the closed set ∏<sub>i</sub> [ /a(1+ε)/, /b(1-ε)/ ]. (Note that there is sadly
|
||
no such thing as an open set in floating point arithmetic.) When does the difference
|
||
matter? Recall the stricture to never peel a high-dimensional orange, because
|
||
when you do, nothing is left. The same idea applied here. The fraction of the
|
||
volume within a distance ε of the boundary is approximately ε<span class="emphasis"><em>d</em></span>,
|
||
where <span class="emphasis"><em>d</em></span> is the number of dimensions. If the number of
|
||
dimensions is large and the precision of the type is low, then it is possible
|
||
that no correct digits will be obtained. If the integrand is singular on the
|
||
boundary, you have no options; you simply must resort to higher precision computations.
|
||
If the integrand is not singular on the boundary, then you can tell this to
|
||
the integration routine via
|
||
</p>
|
||
<pre class="programlisting"><span class="identifier">naive_monte_carlo</span><span class="special"><</span><span class="keyword">double</span><span class="special">,</span> <span class="keyword">decltype</span><span class="special">(</span><span class="identifier">g</span><span class="special">)></span> <span class="identifier">mc</span><span class="special">(</span><span class="identifier">g</span><span class="special">,</span> <span class="identifier">bounds</span><span class="special">,</span> <span class="identifier">error_goal</span><span class="special">,</span> <span class="comment">/*singular = */</span> <span class="keyword">false</span><span class="special">);</span>
|
||
</pre>
|
||
<p>
|
||
and this problem will not be encountered. In practice, you will need ~1,000
|
||
dimensions for this to be relevant in 16 bit floating point, ~100,000 dimensions
|
||
in 32 bit floating point, and an astronomical number of dimensions in double
|
||
precision.
|
||
</p>
|
||
<p>
|
||
Finally, alternative random number generators may be provided to the class.
|
||
The default random number generator is the standard library <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">mt19937_64</span></code>.
|
||
However, here is an example which uses the 32-bit Mersenne twister random number
|
||
generator instead:
|
||
</p>
|
||
<pre class="programlisting"><span class="identifier">naive_monte_carlo</span><span class="special"><</span><span class="identifier">Real</span><span class="special">,</span> <span class="keyword">decltype</span><span class="special">(</span><span class="identifier">g</span><span class="special">),</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">mt19937</span><span class="special">></span> <span class="identifier">mc</span><span class="special">(</span><span class="identifier">g</span><span class="special">,</span> <span class="identifier">bounds</span><span class="special">,</span> <span class="special">(</span><span class="identifier">Real</span><span class="special">)</span> <span class="number">0.001</span><span class="special">);</span>
|
||
</pre>
|
||
</div>
|
||
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
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<td align="right"><div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
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||
Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
|
||
Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
|
||
Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
|
||
Walker and Xiaogang Zhang<p>
|
||
Distributed under the Boost Software License, Version 1.0. (See accompanying
|
||
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
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</p>
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