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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="../../db/d40/integral__approximation2_8cpp.html#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>
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<tr class="memdesc:a71249ee535f16f8ed2e9cc8f0199a2cf"><td class="mdescLeft">&#160;</td><td class="mdescRight">short-hand for std::functions used in this implementation <br /></td></tr>
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<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>
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<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>
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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>
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<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"> 117</span> <a class="code hl_function" href="../../db/d40/integral__approximation2_8cpp.html#a71249ee535f16f8ed2e9cc8f0199a2cf">generate_samples</a>(start_point, pdf, num_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.<a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/vector/size.html">size</a>());</div>
<div class="line"><span class="lineno"> 124</span>}</div>
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<div class="ttc" id="asize_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/container/vector/size.html">std::vector::size</a></div><div class="ttdeci">T size(T... args)</div></div>
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<p>Main function. </p>
<dl class="section return"><dt>Returns</dt><dd>0 on exit </dd></dl>
<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="../../db/d40/integral__approximation2_8cpp.html#aa8dca7b867074164d5f45b0f3851269d">test</a>(); <span class="comment">// run self-test implementations</span></div>
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<div class="line"><span class="lineno"> 162</span> <span class="keywordflow">return</span> 0.0;</div>
<div class="line"><span class="lineno"> 163</span> };</div>
<div class="line"><span class="lineno"> 164</span> </div>
<div class="line"><span class="lineno"> 165</span> integral = <a class="code hl_function" href="../../db/d40/integral__approximation2_8cpp.html#af7da9ba8932f1f48b9bbc2d80471af51">math::monte_carlo::integral_monte_carlo</a>(</div>
<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>
@@ -435,7 +435,7 @@ Here is the call graph for this function:</div>
<div class="line"><span class="lineno"> 186</span> <span class="keywordflow">return</span> 0.0;</div>
<div class="line"><span class="lineno"> 187</span> };</div>
<div class="line"><span class="lineno"> 188</span> </div>
<div class="line"><span class="lineno"> 189</span> integral = <a class="code hl_function" href="../../db/d40/integral__approximation2_8cpp.html#af7da9ba8932f1f48b9bbc2d80471af51">math::monte_carlo::integral_monte_carlo</a>(</div>
<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>
@@ -451,7 +451,7 @@ Here is the call graph for this function:</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"> 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="../../db/d40/integral__approximation2_8cpp.html#af7da9ba8932f1f48b9bbc2d80471af51">math::monte_carlo::integral_monte_carlo</a>(0.0, f, pdf, 10000000);</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>
@@ -460,8 +460,7 @@ Here is the call graph for this function:</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="afunction_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/utility/functional/function.html">std::function</a></div></div>
<div class="ttc" id="aintegral__approximation2_8cpp_html_af7da9ba8932f1f48b9bbc2d80471af51"><div class="ttname"><a href="../../db/d40/integral__approximation2_8cpp.html#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="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>
@@ -480,7 +479,7 @@ Here is the call graph for this function:</div>
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