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<div id="projectbrief">Set of algorithms implemented in C++.</div>
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@@ -127,7 +144,9 @@ Include dependency graph for integral_approximation2.cpp:</div>
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Namespaces</h2></td></tr>
<tr class="memitem:"><td class="memItemLeft" align="right" valign="top">namespace &#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../dd/d47/namespacemath.html">math</a></td></tr>
@@ -144,10 +163,10 @@ Typedefs</h2></td></tr>
</table><table class="memberdecls">
<tr class="heading"><td colspan="2"><h2 class="groupheader"><a id="func-members" name="func-members"></a>
Functions</h2></td></tr>
<tr class="memitem:a71249ee535f16f8ed2e9cc8f0199a2cf" id="r_a71249ee535f16f8ed2e9cc8f0199a2cf"><td class="memItemLeft" align="right" valign="top"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/vector.html">std::vector</a>&lt; double &gt;&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="#a71249ee535f16f8ed2e9cc8f0199a2cf">math::monte_carlo::generate_samples</a> (const double &amp;start_point, const <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/utility/functional/function.html">Function</a> &amp;pdf, const uint32_t &amp;num_samples, const uint32_t &amp;discard=100000)</td></tr>
<tr class="memitem:a71249ee535f16f8ed2e9cc8f0199a2cf" id="r_a71249ee535f16f8ed2e9cc8f0199a2cf"><td class="memItemLeft" align="right" valign="top">std::vector&lt; double &gt;&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="#a71249ee535f16f8ed2e9cc8f0199a2cf">math::monte_carlo::generate_samples</a> (const double &amp;start_point, const Function &amp;pdf, const uint32_t &amp;num_samples, const uint32_t &amp;discard=100000)</td></tr>
<tr class="memdesc:a71249ee535f16f8ed2e9cc8f0199a2cf"><td class="mdescLeft">&#160;</td><td class="mdescRight">short-hand for std::functions used in this implementation <br /></td></tr>
<tr class="separator:a71249ee535f16f8ed2e9cc8f0199a2cf"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:af7da9ba8932f1f48b9bbc2d80471af51" id="r_af7da9ba8932f1f48b9bbc2d80471af51"><td class="memItemLeft" align="right" valign="top">double&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="#af7da9ba8932f1f48b9bbc2d80471af51">math::monte_carlo::integral_monte_carlo</a> (const double &amp;start_point, const <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/utility/functional/function.html">Function</a> &amp;function, const <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/utility/functional/function.html">Function</a> &amp;pdf, const uint32_t &amp;num_samples=1000000)</td></tr>
<tr class="memitem:af7da9ba8932f1f48b9bbc2d80471af51" id="r_af7da9ba8932f1f48b9bbc2d80471af51"><td class="memItemLeft" align="right" valign="top">double&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="#af7da9ba8932f1f48b9bbc2d80471af51">math::monte_carlo::integral_monte_carlo</a> (const double &amp;start_point, const Function &amp;function, const Function &amp;pdf, const uint32_t &amp;num_samples=1000000)</td></tr>
<tr class="memdesc:af7da9ba8932f1f48b9bbc2d80471af51"><td class="mdescLeft">&#160;</td><td class="mdescRight">Compute an approximation of an integral using Monte Carlo integration. <br /></td></tr>
<tr class="separator:af7da9ba8932f1f48b9bbc2d80471af51"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:aa8dca7b867074164d5f45b0f3851269d" id="r_aa8dca7b867074164d5f45b0f3851269d"><td class="memItemLeft" align="right" valign="top">static void&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="#aa8dca7b867074164d5f45b0f3851269d">test</a> ()</td></tr>
@@ -162,7 +181,26 @@ Functions</h2></td></tr>
<p>In mathematics, Monte Carlo integration is a technique for numerical integration using random numbers. It is a particular Monte Carlo method that numerically computes a definite integral. While other algorithms usually evaluate the integrand at a regular grid, Monte Carlo randomly chooses points at which the integrand is evaluated. This method is particularly useful for higher-dimensional integrals.</p>
<p>This implementation supports arbitrary pdfs. These pdfs are sampled using the <a href="https://en.wikipedia.org/wiki/Metropolis–Hastings_algorithm" target="_blank">Metropolis-Hastings algorithm</a>. This can be swapped out by every other sampling techniques for example the inverse method. Metropolis-Hastings was chosen because it is the most general and can also be extended for a higher dimensional sampling space.</p>
<dl class="section author"><dt>Author</dt><dd><a href="https://github.com/DerAndereDomenic" target="_blank">Domenic Zingsheim</a> </dd></dl>
</div><h2 class="groupheader">Typedef Documentation</h2>
<p class="definition">Definition in file <a class="el" href="../../db/d40/integral__approximation2_8cpp_source.html">integral_approximation2.cpp</a>.</p>
</div><h2 class="groupheader">Macro Definition Documentation</h2>
<a id="a525335710b53cb064ca56b936120431e" name="a525335710b53cb064ca56b936120431e"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a525335710b53cb064ca56b936120431e">&#9670;&#160;</a></span>_USE_MATH_DEFINES</h2>
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<td class="memname">#define _USE_MATH_DEFINES</td>
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<p class="definition">Definition at line <a class="el" href="../../db/d40/integral__approximation2_8cpp_source.html#l00024">24</a> of file <a class="el" href="../../db/d40/integral__approximation2_8cpp_source.html">integral_approximation2.cpp</a>.</p>
</div>
</div>
<h2 class="groupheader">Typedef Documentation</h2>
<a id="aa546b0e9315b7d9e9a8f6b71dd643de7" name="aa546b0e9315b7d9e9a8f6b71dd643de7"></a>
<h2 class="memtitle"><span class="permalink"><a href="#aa546b0e9315b7d9e9a8f6b71dd643de7">&#9670;&#160;</a></span>Function</h2>
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<div class="memproto">
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<td class="memname">using <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/utility/functional/function.html">math::monte_carlo::Function</a></td>
<td class="memname">using math::monte_carlo::Function</td>
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<b>Initial value:</b><div class="fragment"><div class="line"> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/utility/functional/function.html">std::function</a>&lt;double(</div>
<b>Initial value:</b><div class="fragment"><div class="line"> std::function&lt;double(</div>
<div class="line"> <span class="keywordtype">double</span>&amp;)&gt;</div>
<div class="ttc" id="afunction_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/utility/functional/function.html">std::function</a></div></div>
</div><!-- fragment -->
<p class="definition">Definition at line <a class="el" href="../../db/d40/integral__approximation2_8cpp_source.html#l00046">46</a> of file <a class="el" href="../../db/d40/integral__approximation2_8cpp_source.html">integral_approximation2.cpp</a>.</p>
</div>
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<h2 class="groupheader">Function Documentation</h2>
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<div class="memproto">
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<td class="memname"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/vector.html">std::vector</a>&lt; double &gt; math::monte_carlo::generate_samples </td>
<td class="memname">std::vector&lt; double &gt; math::monte_carlo::generate_samples </td>
<td>(</td>
<td class="paramtype">const double &amp;</td> <td class="paramname"><span class="paramname"><em>start_point</em></span>, </td>
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<td class="paramtype">const <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/utility/functional/function.html">Function</a> &amp;</td> <td class="paramname"><span class="paramname"><em>pdf</em></span>, </td>
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@@ -222,22 +261,24 @@ Functions</h2></td></tr>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>A vector of size num_samples with samples distributed according to the pdf </dd></dl>
<p class="definition">Definition at line <a class="el" href="../../db/d40/integral__approximation2_8cpp_source.html#l00064">64</a> of file <a class="el" href="../../db/d40/integral__approximation2_8cpp_source.html">integral_approximation2.cpp</a>.</p>
<div class="fragment"><div class="line"><span class="lineno"> 67</span> {</div>
<div class="line"><span class="lineno"> 68</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/vector.html">std::vector&lt;double&gt;</a> samples;</div>
<div class="line"><span class="lineno"> 69</span> samples.<a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/vector/reserve.html">reserve</a>(num_samples);</div>
<div class="line"><span class="lineno"> 68</span> std::vector&lt;double&gt; samples;</div>
<div class="line"><span class="lineno"> 69</span> samples.reserve(num_samples);</div>
<div class="line"><span class="lineno"> 70</span> </div>
<div class="line"><span class="lineno"> 71</span> <span class="keywordtype">double</span> x_t = start_point;</div>
<div class="line"><span class="lineno"> 72</span> </div>
<div class="line"><span class="lineno"> 73</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/random.html">std::default_random_engine</a> generator;</div>
<div class="line"><span class="lineno"> 74</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/random/uniform_real_distribution.html">std::uniform_real_distribution&lt;double&gt;</a> uniform(0.0, 1.0);</div>
<div class="line"><span class="lineno"> 75</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/random/normal_distribution.html">std::normal_distribution&lt;double&gt;</a> normal(0.0, 1.0);</div>
<div class="line"><span class="lineno"> 76</span> generator.seed(<a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/chrono/c/time.html">time</a>(<span class="keyword">nullptr</span>));</div>
<div class="line"><span class="lineno"> 73</span> std::default_random_engine generator;</div>
<div class="line"><span class="lineno"> 74</span> std::uniform_real_distribution&lt;double&gt; uniform(0.0, 1.0);</div>
<div class="line"><span class="lineno"> 75</span> std::normal_distribution&lt;double&gt; normal(0.0, 1.0);</div>
<div class="line"><span class="lineno"> 76</span> generator.seed(time(<span class="keyword">nullptr</span>));</div>
<div class="line"><span class="lineno"> 77</span> </div>
<div class="line"><span class="lineno"> 78</span> <span class="keywordflow">for</span> (uint32_t t = 0; t &lt; num_samples + discard; ++t) {</div>
<div class="line"><span class="lineno"> 79</span> <span class="comment">// Generate a new proposal according to some mutation strategy.</span></div>
<div class="line"><span class="lineno"> 80</span> <span class="comment">// This is arbitrary and can be swapped.</span></div>
<div class="line"><span class="lineno"> 81</span> <span class="keywordtype">double</span> x_dash = normal(generator) + x_t;</div>
<div class="line"><span class="lineno"> 82</span> <span class="keywordtype">double</span> acceptance_probability = <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/algorithm/min.html">std::min</a>(pdf(x_dash) / pdf(x_t), 1.0);</div>
<div class="line"><span class="lineno"> 82</span> <span class="keywordtype">double</span> acceptance_probability = std::min(pdf(x_dash) / pdf(x_t), 1.0);</div>
<div class="line"><span class="lineno"> 83</span> <span class="keywordtype">double</span> u = uniform(generator);</div>
<div class="line"><span class="lineno"> 84</span> </div>
<div class="line"><span class="lineno"> 85</span> <span class="comment">// Accept &quot;new state&quot; according to the acceptance_probability</span></div>
@@ -246,26 +287,13 @@ Functions</h2></td></tr>
<div class="line"><span class="lineno"> 88</span> }</div>
<div class="line"><span class="lineno"> 89</span> </div>
<div class="line"><span class="lineno"> 90</span> <span class="keywordflow">if</span> (t &gt;= discard) {</div>
<div class="line"><span class="lineno"> 91</span> samples.<a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/vector/push_back.html">push_back</a>(x_t);</div>
<div class="line"><span class="lineno"> 91</span> samples.push_back(x_t);</div>
<div class="line"><span class="lineno"> 92</span> }</div>
<div class="line"><span class="lineno"> 93</span> }</div>
<div class="line"><span class="lineno"> 94</span> </div>
<div class="line"><span class="lineno"> 95</span> <span class="keywordflow">return</span> samples;</div>
<div class="line"><span class="lineno"> 96</span>}</div>
<div class="ttc" id="amin_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/algorithm/min.html">std::min</a></div><div class="ttdeci">T min(T... args)</div></div>
<div class="ttc" id="anormal_distribution_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/numeric/random/normal_distribution.html">std::normal_distribution</a></div></div>
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<div class="ttc" id="auniform_real_distribution_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/numeric/random/uniform_real_distribution.html">std::uniform_real_distribution</a></div></div>
<div class="ttc" id="avector_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/container/vector.html">std::vector</a></div></div>
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<dl class="section return"><dt>Returns</dt><dd>The approximation of the integral according to 1/N \sum_{i}^N f(x_i) / p(x_i) </dd></dl>
<p class="definition">Definition at line <a class="el" href="../../db/d40/integral__approximation2_8cpp_source.html#l00112">112</a> of file <a class="el" href="../../db/d40/integral__approximation2_8cpp_source.html">integral_approximation2.cpp</a>.</p>
<div class="fragment"><div class="line"><span class="lineno"> 114</span> {</div>
<div class="line"><span class="lineno"> 115</span> <span class="keywordtype">double</span> integral = 0.0;</div>
<div class="line"><span class="lineno"> 116</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/vector.html">std::vector&lt;double&gt;</a> samples =</div>
<div class="line"><span class="lineno"> 116</span> std::vector&lt;double&gt; samples =</div>
<div class="line"><span class="lineno"> 117</span> <a class="code hl_function" href="#a71249ee535f16f8ed2e9cc8f0199a2cf">generate_samples</a>(start_point, pdf, num_samples);</div>
<div class="line"><span class="lineno"> 118</span> </div>
<div class="line"><span class="lineno"> 119</span> <span class="keywordflow">for</span> (<span class="keywordtype">double</span> sample : samples) {</div>
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<div class="line"><span class="lineno"> 122</span> </div>
<div class="line"><span class="lineno"> 123</span> <span class="keywordflow">return</span> integral / <span class="keyword">static_cast&lt;</span><span class="keywordtype">double</span><span class="keyword">&gt;</span>(samples.size());</div>
<div class="line"><span class="lineno"> 124</span>}</div>
<div class="ttc" id="aintegral__approximation2_8cpp_html_a71249ee535f16f8ed2e9cc8f0199a2cf"><div class="ttname"><a href="#a71249ee535f16f8ed2e9cc8f0199a2cf">math::monte_carlo::generate_samples</a></div><div class="ttdeci">std::vector&lt; double &gt; generate_samples(const double &amp;start_point, const Function &amp;pdf, const uint32_t &amp;num_samples, const uint32_t &amp;discard=100000)</div><div class="ttdoc">short-hand for std::functions used in this implementation</div><div class="ttdef"><b>Definition</b> integral_approximation2.cpp:64</div></div>
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<div class="ttc" id="aintegral__approximation2_8cpp_html_a71249ee535f16f8ed2e9cc8f0199a2cf"><div class="ttname"><a href="#a71249ee535f16f8ed2e9cc8f0199a2cf">math::monte_carlo::generate_samples</a></div><div class="ttdeci">std::vector&lt; double &gt; generate_samples(const double &amp;start_point, const Function &amp;pdf, const uint32_t &amp;num_samples, const uint32_t &amp;discard=100000)</div><div class="ttdoc">short-hand for std::functions used in this implementation</div><div class="ttdef"><b>Definition</b> <a href="../../db/d40/integral__approximation2_8cpp_source.html#l00064">integral_approximation2.cpp:64</a></div></div>
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@@ -345,17 +370,14 @@ Here is the call graph for this function:</div>
<p>Main function. </p>
<dl class="section return"><dt>Returns</dt><dd>0 on exit </dd></dl>
<p class="definition">Definition at line <a class="el" href="../../db/d40/integral__approximation2_8cpp_source.html#l00215">215</a> of file <a class="el" href="../../db/d40/integral__approximation2_8cpp_source.html">integral_approximation2.cpp</a>.</p>
<div class="fragment"><div class="line"><span class="lineno"> 215</span> {</div>
<div class="line"><span class="lineno"> 216</span> <a class="code hl_function" href="#aa8dca7b867074164d5f45b0f3851269d">test</a>(); <span class="comment">// run self-test implementations</span></div>
<div class="line"><span class="lineno"> 217</span> <span class="keywordflow">return</span> 0;</div>
<div class="line"><span class="lineno"> 218</span>}</div>
<div class="ttc" id="aintegral__approximation2_8cpp_html_aa8dca7b867074164d5f45b0f3851269d"><div class="ttname"><a href="#aa8dca7b867074164d5f45b0f3851269d">test</a></div><div class="ttdeci">static void test()</div><div class="ttdoc">Self-test implementations.</div><div class="ttdef"><b>Definition</b> integral_approximation2.cpp:133</div></div>
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<p>Self-test implementations. </p>
<dl class="section return"><dt>Returns</dt><dd>void </dd></dl>
<p class="definition">Definition at line <a class="el" href="../../db/d40/integral__approximation2_8cpp_source.html#l00133">133</a> of file <a class="el" href="../../db/d40/integral__approximation2_8cpp_source.html">integral_approximation2.cpp</a>.</p>
<div class="fragment"><div class="line"><span class="lineno"> 133</span> {</div>
<div class="line"><span class="lineno"> 134</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; <span class="stringliteral">&quot;Disclaimer: Because this is a randomized algorithm,&quot;</span></div>
<div class="line"><span class="lineno"> 135</span> &lt;&lt; <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/endl.html">std::endl</a>;</div>
<div class="line"><span class="lineno"> 136</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a></div>
<div class="line"><span class="lineno"> 134</span> std::cout &lt;&lt; <span class="stringliteral">&quot;Disclaimer: Because this is a randomized algorithm,&quot;</span></div>
<div class="line"><span class="lineno"> 135</span> &lt;&lt; std::endl;</div>
<div class="line"><span class="lineno"> 136</span> std::cout</div>
<div class="line"><span class="lineno"> 137</span> &lt;&lt; <span class="stringliteral">&quot;it may happen that singular samples deviate from the true result.&quot;</span></div>
<div class="line"><span class="lineno"> 138</span> &lt;&lt; <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/endl.html">std::endl</a></div>
<div class="line"><span class="lineno"> 139</span> &lt;&lt; <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/endl.html">std::endl</a>;</div>
<div class="line"><span class="lineno"> 138</span> &lt;&lt; std::endl</div>
<div class="line"><span class="lineno"> 139</span> &lt;&lt; std::endl;</div>
<div class="line"><span class="lineno"> 140</span> ;</div>
<div class="line"><span class="lineno"> 141</span> </div>
<div class="line"><span class="lineno"> 142</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/utility/functional/function.html">math::monte_carlo::Function</a> <a class="code hl_function" href="../../d4/d18/composite__simpson__rule_8cpp.html#a4251b4df4748a0b9c43a48f61bdd2397">f</a>;</div>
<div class="line"><span class="lineno"> 143</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/utility/functional/function.html">math::monte_carlo::Function</a> pdf;</div>
<div class="line"><span class="lineno"> 142</span> math::monte_carlo::Function <a class="code hl_function" href="../../d4/d18/composite__simpson__rule_8cpp.html#a4251b4df4748a0b9c43a48f61bdd2397">f</a>;</div>
<div class="line"><span class="lineno"> 143</span> math::monte_carlo::Function pdf;</div>
<div class="line"><span class="lineno"> 144</span> <span class="keywordtype">double</span> integral = 0;</div>
<div class="line"><span class="lineno"> 145</span> <span class="keywordtype">double</span> <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/algorithm/lower_bound.html">lower_bound</a> = 0, <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/algorithm/upper_bound.html">upper_bound</a> = 0;</div>
<div class="line"><span class="lineno"> 145</span> <span class="keywordtype">double</span> lower_bound = 0, upper_bound = 0;</div>
<div class="line"><span class="lineno"> 146</span> </div>
<div class="line"><span class="lineno"> 147</span> <span class="comment">/* \int_{-2}^{2} -x^2 + 4 dx */</span></div>
<div class="line"><span class="lineno"> 148</span> <a class="code hl_function" href="../../d4/d18/composite__simpson__rule_8cpp.html#a4251b4df4748a0b9c43a48f61bdd2397">f</a> = [&amp;](<span class="keywordtype">double</span>&amp; x) { <span class="keywordflow">return</span> -x * x + 4.0; };</div>
<div class="line"><span class="lineno"> 149</span> </div>
<div class="line"><span class="lineno"> 150</span> <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/algorithm/lower_bound.html">lower_bound</a> = -2.0;</div>
<div class="line"><span class="lineno"> 151</span> <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/algorithm/upper_bound.html">upper_bound</a> = 2.0;</div>
<div class="line"><span class="lineno"> 150</span> lower_bound = -2.0;</div>
<div class="line"><span class="lineno"> 151</span> upper_bound = 2.0;</div>
<div class="line"><span class="lineno"> 152</span> pdf = [&amp;](<span class="keywordtype">double</span>&amp; x) {</div>
<div class="line"><span class="lineno"> 153</span> <span class="keywordflow">if</span> (x &gt;= lower_bound &amp;&amp; x &lt;= -1.0) {</div>
<div class="line"><span class="lineno"> 154</span> <span class="keywordflow">return</span> 0.1;</div>
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<div class="line"><span class="lineno"> 165</span> integral = <a class="code hl_function" href="#af7da9ba8932f1f48b9bbc2d80471af51">math::monte_carlo::integral_monte_carlo</a>(</div>
<div class="line"><span class="lineno"> 166</span> (upper_bound - lower_bound) / 2.0, f, pdf);</div>
<div class="line"><span class="lineno"> 167</span> </div>
<div class="line"><span class="lineno"> 168</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; <span class="stringliteral">&quot;This number should be close to 10.666666: &quot;</span> &lt;&lt; integral</div>
<div class="line"><span class="lineno"> 169</span> &lt;&lt; <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/endl.html">std::endl</a>;</div>
<div class="line"><span class="lineno"> 168</span> std::cout &lt;&lt; <span class="stringliteral">&quot;This number should be close to 10.666666: &quot;</span> &lt;&lt; integral</div>
<div class="line"><span class="lineno"> 169</span> &lt;&lt; std::endl;</div>
<div class="line"><span class="lineno"> 170</span> </div>
<div class="line"><span class="lineno"> 171</span> <span class="comment">/* \int_{0}^{1} e^x dx */</span></div>
<div class="line"><span class="lineno"> 172</span> <a class="code hl_function" href="../../d4/d18/composite__simpson__rule_8cpp.html#a4251b4df4748a0b9c43a48f61bdd2397">f</a> = [&amp;](<span class="keywordtype">double</span>&amp; x) { <span class="keywordflow">return</span> <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/math/exp.html">std::exp</a>(x); };</div>
<div class="line"><span class="lineno"> 172</span> <a class="code hl_function" href="../../d4/d18/composite__simpson__rule_8cpp.html#a4251b4df4748a0b9c43a48f61bdd2397">f</a> = [&amp;](<span class="keywordtype">double</span>&amp; x) { <span class="keywordflow">return</span> std::exp(x); };</div>
<div class="line"><span class="lineno"> 173</span> </div>
<div class="line"><span class="lineno"> 174</span> <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/algorithm/lower_bound.html">lower_bound</a> = 0.0;</div>
<div class="line"><span class="lineno"> 175</span> <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/algorithm/upper_bound.html">upper_bound</a> = 1.0;</div>
<div class="line"><span class="lineno"> 174</span> lower_bound = 0.0;</div>
<div class="line"><span class="lineno"> 175</span> upper_bound = 1.0;</div>
<div class="line"><span class="lineno"> 176</span> pdf = [&amp;](<span class="keywordtype">double</span>&amp; x) {</div>
<div class="line"><span class="lineno"> 177</span> <span class="keywordflow">if</span> (x &gt;= lower_bound &amp;&amp; x &lt;= 0.2) {</div>
<div class="line"><span class="lineno"> 178</span> <span class="keywordflow">return</span> 0.1;</div>
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<div class="line"><span class="lineno"> 189</span> integral = <a class="code hl_function" href="#af7da9ba8932f1f48b9bbc2d80471af51">math::monte_carlo::integral_monte_carlo</a>(</div>
<div class="line"><span class="lineno"> 190</span> (upper_bound - lower_bound) / 2.0, f, pdf);</div>
<div class="line"><span class="lineno"> 191</span> </div>
<div class="line"><span class="lineno"> 192</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; <span class="stringliteral">&quot;This number should be close to 1.7182818: &quot;</span> &lt;&lt; integral</div>
<div class="line"><span class="lineno"> 193</span> &lt;&lt; <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/endl.html">std::endl</a>;</div>
<div class="line"><span class="lineno"> 192</span> std::cout &lt;&lt; <span class="stringliteral">&quot;This number should be close to 1.7182818: &quot;</span> &lt;&lt; integral</div>
<div class="line"><span class="lineno"> 193</span> &lt;&lt; std::endl;</div>
<div class="line"><span class="lineno"> 194</span> </div>
<div class="line"><span class="lineno"> 195</span> <span class="comment">/* \int_{-\infty}^{\infty} sinc(x) dx, sinc(x) = sin(pi * x) / (pi * x)</span></div>
<div class="line"><span class="lineno"> 196</span><span class="comment"> This is a difficult integral because of its infinite domain.</span></div>
<div class="line"><span class="lineno"> 197</span><span class="comment"> Therefore, it may deviate largely from the expected result.</span></div>
<div class="line"><span class="lineno"> 198</span><span class="comment"> */</span></div>
<div class="line"><span class="lineno"> 199</span> <a class="code hl_function" href="../../d4/d18/composite__simpson__rule_8cpp.html#a4251b4df4748a0b9c43a48f61bdd2397">f</a> = [&amp;](<span class="keywordtype">double</span>&amp; x) { <span class="keywordflow">return</span> <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/math/sin.html">std::sin</a>(M_PI * x) / (M_PI * x); };</div>
<div class="line"><span class="lineno"> 199</span> <a class="code hl_function" href="../../d4/d18/composite__simpson__rule_8cpp.html#a4251b4df4748a0b9c43a48f61bdd2397">f</a> = [&amp;](<span class="keywordtype">double</span>&amp; x) { <span class="keywordflow">return</span> std::sin(M_PI * x) / (M_PI * x); };</div>
<div class="line"><span class="lineno"> 200</span> </div>
<div class="line"><span class="lineno"> 201</span> pdf = [&amp;](<span class="keywordtype">double</span>&amp; x) {</div>
<div class="line"><span class="lineno"> 202</span> <span class="keywordflow">return</span> 1.0 / <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/math/sqrt.html">std::sqrt</a>(2.0 * M_PI) * <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/math/exp.html">std::exp</a>(-x * x / 2.0);</div>
<div class="line"><span class="lineno"> 202</span> <span class="keywordflow">return</span> 1.0 / std::sqrt(2.0 * M_PI) * std::exp(-x * x / 2.0);</div>
<div class="line"><span class="lineno"> 203</span> };</div>
<div class="line"><span class="lineno"> 204</span> </div>
<div class="line"><span class="lineno"> 205</span> integral = <a class="code hl_function" href="#af7da9ba8932f1f48b9bbc2d80471af51">math::monte_carlo::integral_monte_carlo</a>(0.0, f, pdf, 10000000);</div>
<div class="line"><span class="lineno"> 206</span> </div>
<div class="line"><span class="lineno"> 207</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; <span class="stringliteral">&quot;This number should be close to 1.0: &quot;</span> &lt;&lt; integral</div>
<div class="line"><span class="lineno"> 208</span> &lt;&lt; <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/endl.html">std::endl</a>;</div>
<div class="line"><span class="lineno"> 207</span> std::cout &lt;&lt; <span class="stringliteral">&quot;This number should be close to 1.0: &quot;</span> &lt;&lt; integral</div>
<div class="line"><span class="lineno"> 208</span> &lt;&lt; std::endl;</div>
<div class="line"><span class="lineno"> 209</span>}</div>
<div class="ttc" id="abasic_ostream_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a></div></div>
<div class="ttc" id="acomposite__simpson__rule_8cpp_html_a4251b4df4748a0b9c43a48f61bdd2397"><div class="ttname"><a href="../../d4/d18/composite__simpson__rule_8cpp.html#a4251b4df4748a0b9c43a48f61bdd2397">numerical_methods::simpson_method::f</a></div><div class="ttdeci">double f(double x)</div><div class="ttdoc">A function f(x) that will be used to test the method.</div><div class="ttdef"><b>Definition</b> composite_simpson_rule.cpp:113</div></div>
<div class="ttc" id="aendl_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/io/manip/endl.html">std::endl</a></div><div class="ttdeci">T endl(T... args)</div></div>
<div class="ttc" id="aexp_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/numeric/math/exp.html">std::exp</a></div><div class="ttdeci">T exp(T... args)</div></div>
<div class="ttc" id="aintegral__approximation2_8cpp_html_af7da9ba8932f1f48b9bbc2d80471af51"><div class="ttname"><a href="#af7da9ba8932f1f48b9bbc2d80471af51">math::monte_carlo::integral_monte_carlo</a></div><div class="ttdeci">double integral_monte_carlo(const double &amp;start_point, const Function &amp;function, const Function &amp;pdf, const uint32_t &amp;num_samples=1000000)</div><div class="ttdoc">Compute an approximation of an integral using Monte Carlo integration.</div><div class="ttdef"><b>Definition</b> integral_approximation2.cpp:112</div></div>
<div class="ttc" id="alower_bound_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/algorithm/lower_bound.html">std::lower_bound</a></div><div class="ttdeci">T lower_bound(T... args)</div></div>
<div class="ttc" id="asin_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/numeric/math/sin.html">std::sin</a></div><div class="ttdeci">T sin(T... args)</div></div>
<div class="ttc" id="asqrt_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/numeric/math/sqrt.html">std::sqrt</a></div><div class="ttdeci">T sqrt(T... args)</div></div>
<div class="ttc" id="aupper_bound_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/algorithm/upper_bound.html">std::upper_bound</a></div><div class="ttdeci">T upper_bound(T... args)</div></div>
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<div class="ttc" id="acomposite__simpson__rule_8cpp_html_a4251b4df4748a0b9c43a48f61bdd2397"><div class="ttname"><a href="../../d4/d18/composite__simpson__rule_8cpp.html#a4251b4df4748a0b9c43a48f61bdd2397">numerical_methods::simpson_method::f</a></div><div class="ttdeci">double f(double x)</div><div class="ttdoc">A function f(x) that will be used to test the method.</div><div class="ttdef"><b>Definition</b> <a href="../../d4/d18/composite__simpson__rule_8cpp_source.html#l00113">composite_simpson_rule.cpp:113</a></div></div>
<div class="ttc" id="aintegral__approximation2_8cpp_html_af7da9ba8932f1f48b9bbc2d80471af51"><div class="ttname"><a href="#af7da9ba8932f1f48b9bbc2d80471af51">math::monte_carlo::integral_monte_carlo</a></div><div class="ttdeci">double integral_monte_carlo(const double &amp;start_point, const Function &amp;function, const Function &amp;pdf, const uint32_t &amp;num_samples=1000000)</div><div class="ttdoc">Compute an approximation of an integral using Monte Carlo integration.</div><div class="ttdef"><b>Definition</b> <a href="../../db/d40/integral__approximation2_8cpp_source.html#l00112">integral_approximation2.cpp:112</a></div></div>
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