232 lines
28 KiB
HTML
232 lines
28 KiB
HTML
<html>
|
|
<head>
|
|
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
|
|
<title>Naive Monte Carlo Integration</title>
|
|
<link rel="stylesheet" href="../math.css" type="text/css">
|
|
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
|
|
<link rel="home" href="../index.html" title="Math Toolkit 2.11.0">
|
|
<link rel="up" href="../quadrature.html" title="Chapter 13. Quadrature and Differentiation">
|
|
<link rel="prev" href="fourier_integrals.html" title="Fourier Integrals">
|
|
<link rel="next" href="diff.html" title="Numerical Differentiation">
|
|
</head>
|
|
<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
|
|
<table cellpadding="2" width="100%"><tr>
|
|
<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../boost.png"></td>
|
|
<td align="center"><a href="../../../../../index.html">Home</a></td>
|
|
<td align="center"><a href="../../../../../libs/libraries.htm">Libraries</a></td>
|
|
<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td>
|
|
<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td>
|
|
<td align="center"><a href="../../../../../more/index.htm">More</a></td>
|
|
</tr></table>
|
|
<hr>
|
|
<div class="spirit-nav">
|
|
<a accesskey="p" href="fourier_integrals.html"><img src="../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../quadrature.html"><img src="../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../index.html"><img src="../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="diff.html"><img src="../../../../../doc/src/images/next.png" alt="Next"></a>
|
|
</div>
|
|
<div class="section">
|
|
<div class="titlepage"><div><div><h2 class="title" style="clear: both">
|
|
<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>
|
|
</h2></div></div></div>
|
|
<h4>
|
|
<a name="math_toolkit.naive_monte_carlo.h0"></a>
|
|
<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>
|
|
</h4>
|
|
<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>
|
|
<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>
|
|
|
|
<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>
|
|
<span class="keyword">class</span> <span class="identifier">naive_monte_carlo</span>
|
|
<span class="special">{</span>
|
|
<span class="keyword">public</span><span class="special">:</span>
|
|
<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>
|
|
<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>
|
|
<span class="identifier">Real</span> <span class="identifier">error_goal</span><span class="special">,</span>
|
|
<span class="keyword">bool</span> <span class="identifier">singular</span> <span class="special">=</span> <span class="keyword">true</span><span class="special">,</span>
|
|
<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>
|
|
|
|
<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>
|
|
|
|
<span class="keyword">void</span> <span class="identifier">cancel</span><span class="special">();</span>
|
|
|
|
<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>
|
|
|
|
<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>
|
|
|
|
<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>
|
|
|
|
<span class="identifier">Real</span> <span class="identifier">progress</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
|
|
|
|
<span class="identifier">Real</span> <span class="identifier">current_estimate</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
|
|
|
|
<span class="identifier">size_t</span> <span class="identifier">calls</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
|
|
<span class="special">};</span>
|
|
<span class="special">}}}</span> <span class="comment">// namespaces</span>
|
|
</pre>
|
|
<h4>
|
|
<a name="math_toolkit.naive_monte_carlo.h1"></a>
|
|
<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>
|
|
</h4>
|
|
<p>
|
|
The class <code class="computeroutput"><span class="identifier">naive_monte_carlo</span></code>
|
|
performs Monte-Carlo integration on a square integrable function <span class="emphasis"><em>f</em></span>
|
|
on a domain Ω. The theoretical background of Monte-Carlo integration is nicely
|
|
discussed at <a href="https://en.wikipedia.org/wiki/Monte_Carlo_integration" target="_top">Wikipedia</a>,
|
|
and as such will not be discussed here. However, despite being "naive",
|
|
it is a mistake to assume that naive Monte-Carlo integration is not powerful,
|
|
as the simplicity of the method affords a robustness not easily provided by
|
|
more sophisticated tools. The multithreaded nature of the routine allows us
|
|
to compute a large number of sample points with great speed, and hence the
|
|
slow convergence is mitigated by exploiting the full power of modern hardware.
|
|
</p>
|
|
<p>
|
|
The naive Monte-Carlo integration provided by Boost exemplifies the programming
|
|
techniques needed to cope with high-performance computing. For instance, since
|
|
the convergence is only 𝑶(N<sup>-1/2</sup>), the compute time is very sensitive to the
|
|
error goal. Users can easily specify an error goal which causes computation
|
|
to last months-or just a few seconds. Without progress reporting, this situation
|
|
is disorienting and causes the user to behave in a paranoid manner. Even with
|
|
progress reporting, a user might need to cancel a job due to shifting priorities
|
|
of the employing institution, and as such cancellation must be supported. A
|
|
cancelled job which returns no results is wasted, so the cancellation must
|
|
be graceful, returning the best estimate of the result thus far. In addition,
|
|
a task might finish, and the user may well decide to try for a lower error
|
|
bound. Hence restarting without loss of the preceding effort must be supported.
|
|
Finally, on an HPC system, we generally wish to use all available threads.
|
|
But if the computation is performed on a users workstation, employing every
|
|
thread will cause the browser, email, or music apps to become unresponsive,
|
|
so leaving a single thread available for other apps is appreciated.
|
|
</p>
|
|
<p>
|
|
All these use cases are supported, so let's get to the code:
|
|
</p>
|
|
<pre class="programlisting"><span class="comment">// Define a function to integrate:</span>
|
|
<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>
|
|
<span class="special">{</span>
|
|
<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>
|
|
<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>
|
|
<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">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>
|
|
<span class="keyword">double</span> <span class="identifier">error_goal</span> <span class="special">=</span> <span class="number">0.001</span>
|
|
<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="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>
|
|
<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">// The user must decide on a reasonable way to display the progress depending on their environment:</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="keyword">if</span> <span class="special">(</span><span class="identifier">some_signal_heard</span><span class="special">())</span>
|
|
<span class="special">{</span>
|
|
<span class="identifier">mc</span><span class="special">.</span><span class="identifier">cancel</span><span class="special">();</span>
|
|
<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>
|
|
<span class="special">}</span>
|
|
<span class="special">}</span>
|
|
<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>
|
|
</pre>
|
|
<p>
|
|
First off, we define the function we wish to integrate. This function must
|
|
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>,
|
|
and return a <code class="computeroutput"><span class="identifier">Real</span></code>. Next, we
|
|
define the domain of integration. Infinite domains are indicated by the bound
|
|
<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>.
|
|
The call
|
|
</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>
|
|
</pre>
|
|
<p>
|
|
creates an instance of the monte carlo integrator. This is also where the number
|
|
of threads can be set, for instance
|
|
</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="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>
|
|
</pre>
|
|
<p>
|
|
might be more appropriate for running on a user's hardware (the default taking
|
|
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>.
|
|
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>
|
|
<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>
|
|
<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>
|
|
<td align="left"></td>
|
|
<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar
|
|
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
|
|
Hubert Holin, Bruno Lalande, John Maddock, 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>)
|
|
</p>
|
|
</div></td>
|
|
</tr></table>
|
|
<hr>
|
|
<div class="spirit-nav">
|
|
<a accesskey="p" href="fourier_integrals.html"><img src="../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../quadrature.html"><img src="../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../index.html"><img src="../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="diff.html"><img src="../../../../../doc/src/images/next.png" alt="Next"></a>
|
|
</div>
|
|
</body>
|
|
</html>
|