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<title>Algorithms_in_C++: math/integral_approximation2.cpp File Reference</title>
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<div id="projectbrief">Set of algorithms implemented in C++.</div>
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@@ -201,36 +201,36 @@ Functions</h2></td></tr>
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</dd>
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</dl>
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<dl class="section return"><dt>Returns</dt><dd>A vector of size num_samples with samples distributed according to the pdf </dd></dl>
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<div class="fragment"><div class="line"><a id="l00067" name="l00067"></a><span class="lineno"> 67</span> {</div>
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<div class="line"><a id="l00068" name="l00068"></a><span class="lineno"> 68</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/vector.html">std::vector<double></a> samples;</div>
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<div class="line"><a id="l00069" name="l00069"></a><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>
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<div class="line"><a id="l00070" name="l00070"></a><span class="lineno"> 70</span> </div>
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<div class="line"><a id="l00071" name="l00071"></a><span class="lineno"> 71</span> <span class="keywordtype">double</span> x_t = start_point;</div>
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<div class="line"><a id="l00072" name="l00072"></a><span class="lineno"> 72</span> </div>
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<div class="line"><a id="l00073" name="l00073"></a><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>
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<div class="line"><a id="l00074" name="l00074"></a><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<double></a> uniform(0.0, 1.0);</div>
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<div class="line"><a id="l00075" name="l00075"></a><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<double></a> normal(0.0, 1.0);</div>
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<div class="line"><a id="l00076" name="l00076"></a><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>
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<div class="line"><a id="l00077" name="l00077"></a><span class="lineno"> 77</span> </div>
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<div class="line"><a id="l00078" name="l00078"></a><span class="lineno"> 78</span> <span class="keywordflow">for</span> (uint32_t t = 0; t < num_samples + discard; ++t) {</div>
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<div class="line"><a id="l00079" name="l00079"></a><span class="lineno"> 79</span> <span class="comment">// Generate a new proposal according to some mutation strategy.</span></div>
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<div class="line"><a id="l00080" name="l00080"></a><span class="lineno"> 80</span> <span class="comment">// This is arbitrary and can be swapped.</span></div>
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<div class="line"><a id="l00081" name="l00081"></a><span class="lineno"> 81</span> <span class="keywordtype">double</span> x_dash = normal(generator) + x_t;</div>
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<div class="line"><a id="l00082" name="l00082"></a><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>
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<div class="line"><a id="l00083" name="l00083"></a><span class="lineno"> 83</span> <span class="keywordtype">double</span> u = uniform(generator);</div>
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<div class="line"><a id="l00084" name="l00084"></a><span class="lineno"> 84</span> </div>
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<div class="line"><a id="l00085" name="l00085"></a><span class="lineno"> 85</span> <span class="comment">// Accept "new state" according to the acceptance_probability</span></div>
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<div class="line"><a id="l00086" name="l00086"></a><span class="lineno"> 86</span> <span class="keywordflow">if</span> (u <= acceptance_probability) {</div>
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<div class="line"><a id="l00087" name="l00087"></a><span class="lineno"> 87</span> x_t = x_dash;</div>
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<div class="line"><a id="l00088" name="l00088"></a><span class="lineno"> 88</span> }</div>
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<div class="line"><a id="l00089" name="l00089"></a><span class="lineno"> 89</span> </div>
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<div class="line"><a id="l00090" name="l00090"></a><span class="lineno"> 90</span> <span class="keywordflow">if</span> (t >= discard) {</div>
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<div class="line"><a id="l00091" name="l00091"></a><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>
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<div class="line"><a id="l00092" name="l00092"></a><span class="lineno"> 92</span> }</div>
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<div class="line"><a id="l00093" name="l00093"></a><span class="lineno"> 93</span> }</div>
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<div class="line"><a id="l00094" name="l00094"></a><span class="lineno"> 94</span> </div>
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<div class="line"><a id="l00095" name="l00095"></a><span class="lineno"> 95</span> <span class="keywordflow">return</span> samples;</div>
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<div class="line"><a id="l00096" name="l00096"></a><span class="lineno"> 96</span>}</div>
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<div class="fragment"><div class="line"><span class="lineno"> 67</span> {</div>
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<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<double></a> samples;</div>
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<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>
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<div class="line"><span class="lineno"> 70</span> </div>
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<div class="line"><span class="lineno"> 71</span> <span class="keywordtype">double</span> x_t = start_point;</div>
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<div class="line"><span class="lineno"> 72</span> </div>
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<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>
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<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<double></a> uniform(0.0, 1.0);</div>
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<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<double></a> normal(0.0, 1.0);</div>
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<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>
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<div class="line"><span class="lineno"> 77</span> </div>
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<div class="line"><span class="lineno"> 78</span> <span class="keywordflow">for</span> (uint32_t t = 0; t < num_samples + discard; ++t) {</div>
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<div class="line"><span class="lineno"> 79</span> <span class="comment">// Generate a new proposal according to some mutation strategy.</span></div>
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<div class="line"><span class="lineno"> 80</span> <span class="comment">// This is arbitrary and can be swapped.</span></div>
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<div class="line"><span class="lineno"> 81</span> <span class="keywordtype">double</span> x_dash = normal(generator) + x_t;</div>
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<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>
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<div class="line"><span class="lineno"> 83</span> <span class="keywordtype">double</span> u = uniform(generator);</div>
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<div class="line"><span class="lineno"> 84</span> </div>
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<div class="line"><span class="lineno"> 85</span> <span class="comment">// Accept "new state" according to the acceptance_probability</span></div>
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<div class="line"><span class="lineno"> 86</span> <span class="keywordflow">if</span> (u <= acceptance_probability) {</div>
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<div class="line"><span class="lineno"> 87</span> x_t = x_dash;</div>
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<div class="line"><span class="lineno"> 88</span> }</div>
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<div class="line"><span class="lineno"> 89</span> </div>
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<div class="line"><span class="lineno"> 90</span> <span class="keywordflow">if</span> (t >= discard) {</div>
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<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>
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<div class="line"><span class="lineno"> 92</span> }</div>
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<div class="line"><span class="lineno"> 93</span> }</div>
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<div class="line"><span class="lineno"> 94</span> </div>
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<div class="line"><span class="lineno"> 95</span> <span class="keywordflow">return</span> samples;</div>
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<div class="line"><span class="lineno"> 96</span>}</div>
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<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>
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<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="apush_back_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/container/vector/push_back.html">std::vector::push_back</a></div><div class="ttdeci">T push_back(T... args)</div></div>
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@@ -238,7 +238,7 @@ Functions</h2></td></tr>
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<div class="ttc" id="areserve_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/container/vector/reserve.html">std::vector::reserve</a></div><div class="ttdeci">T reserve(T... args)</div></div>
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<div class="ttc" id="atime_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/chrono/c/time.html">std::time</a></div><div class="ttdeci">T time(T... args)</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>
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<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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<div class="ttc" id="avector_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/container/vector.html">std::vector< double ></a></div></div>
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@@ -297,17 +297,17 @@ Here is the call graph for this function:</div>
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</dd>
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</dl>
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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>
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<div class="fragment"><div class="line"><a id="l00114" name="l00114"></a><span class="lineno"> 114</span> {</div>
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<div class="line"><a id="l00115" name="l00115"></a><span class="lineno"> 115</span> <span class="keywordtype">double</span> integral = 0.0;</div>
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<div class="line"><a id="l00116" name="l00116"></a><span class="lineno"> 116</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/vector.html">std::vector<double></a> samples =</div>
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<div class="line"><a id="l00117" name="l00117"></a><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>
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<div class="line"><a id="l00118" name="l00118"></a><span class="lineno"> 118</span> </div>
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<div class="line"><a id="l00119" name="l00119"></a><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"><a id="l00120" name="l00120"></a><span class="lineno"> 120</span> integral += <span class="keyword">function</span>(sample) / pdf(sample);</div>
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<div class="line"><a id="l00121" name="l00121"></a><span class="lineno"> 121</span> }</div>
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<div class="line"><a id="l00122" name="l00122"></a><span class="lineno"> 122</span> </div>
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<div class="line"><a id="l00123" name="l00123"></a><span class="lineno"> 123</span> <span class="keywordflow">return</span> integral / <span class="keyword">static_cast<</span><span class="keywordtype">double</span><span class="keyword">></span>(samples.size());</div>
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<div class="line"><a id="l00124" name="l00124"></a><span class="lineno"> 124</span>}</div>
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<div class="fragment"><div class="line"><span class="lineno"> 114</span> {</div>
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<div class="line"><span class="lineno"> 115</span> <span class="keywordtype">double</span> integral = 0.0;</div>
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<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<double></a> samples =</div>
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<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>
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<div class="line"><span class="lineno"> 118</span> </div>
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<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"> 120</span> integral += function(sample) / pdf(sample);</div>
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<div class="line"><span class="lineno"> 121</span> }</div>
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<div class="line"><span class="lineno"> 122</span> </div>
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<div class="line"><span class="lineno"> 123</span> <span class="keywordflow">return</span> integral / <span class="keyword">static_cast<</span><span class="keywordtype">double</span><span class="keyword">></span>(samples.size());</div>
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<div class="line"><span class="lineno"> 124</span>}</div>
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<div class="ttc" id="aintegral__approximation2_8cpp_html_a71249ee535f16f8ed2e9cc8f0199a2cf"><div class="ttname"><a href="../../db/d40/integral__approximation2_8cpp.html#a71249ee535f16f8ed2e9cc8f0199a2cf">math::monte_carlo::generate_samples</a></div><div class="ttdeci">std::vector< double > generate_samples(const double &start_point, const Function &pdf, const uint32_t &num_samples, const uint32_t &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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<p>Main function. </p>
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<dl class="section return"><dt>Returns</dt><dd>0 on exit </dd></dl>
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<div class="fragment"><div class="line"><a id="l00215" name="l00215"></a><span class="lineno"> 215</span> {</div>
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<div class="line"><a id="l00216" name="l00216"></a><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"><a id="l00217" name="l00217"></a><span class="lineno"> 217</span> <span class="keywordflow">return</span> 0;</div>
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<div class="line"><a id="l00218" name="l00218"></a><span class="lineno"> 218</span>}</div>
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<div class="fragment"><div class="line"><span class="lineno"> 215</span> {</div>
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<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"> 217</span> <span class="keywordflow">return</span> 0;</div>
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<div class="line"><span class="lineno"> 218</span>}</div>
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<div class="ttc" id="aintegral__approximation2_8cpp_html_aa8dca7b867074164d5f45b0f3851269d"><div class="ttname"><a href="../../db/d40/integral__approximation2_8cpp.html#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>
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<dl class="section return"><dt>Returns</dt><dd>void </dd></dl>
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<div class="fragment"><div class="line"><a id="l00133" name="l00133"></a><span class="lineno"> 133</span> {</div>
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<div class="line"><a id="l00134" name="l00134"></a><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> << <span class="stringliteral">"Disclaimer: Because this is a randomized algorithm,"</span></div>
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<div class="line"><a id="l00135" name="l00135"></a><span class="lineno"> 135</span> << <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/endl.html">std::endl</a>;</div>
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<div class="line"><a id="l00136" name="l00136"></a><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>
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<div class="line"><a id="l00137" name="l00137"></a><span class="lineno"> 137</span> << <span class="stringliteral">"it may happen that singular samples deviate from the true result."</span></div>
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<div class="line"><a id="l00138" name="l00138"></a><span class="lineno"> 138</span> << <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/endl.html">std::endl</a></div>
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<div class="line"><a id="l00139" name="l00139"></a><span class="lineno"> 139</span> << <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/endl.html">std::endl</a>;</div>
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<div class="line"><a id="l00140" name="l00140"></a><span class="lineno"> 140</span> ;</div>
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<div class="line"><a id="l00141" name="l00141"></a><span class="lineno"> 141</span> </div>
|
||||
<div class="line"><a id="l00142" name="l00142"></a><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"><a id="l00143" name="l00143"></a><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"><a id="l00144" name="l00144"></a><span class="lineno"> 144</span> <span class="keywordtype">double</span> integral = 0;</div>
|
||||
<div class="line"><a id="l00145" name="l00145"></a><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"><a id="l00146" name="l00146"></a><span class="lineno"> 146</span> </div>
|
||||
<div class="line"><a id="l00147" name="l00147"></a><span class="lineno"> 147</span> <span class="comment">/* \int_{-2}^{2} -x^2 + 4 dx */</span></div>
|
||||
<div class="line"><a id="l00148" name="l00148"></a><span class="lineno"> 148</span> <a class="code hl_function" href="../../d4/d18/composite__simpson__rule_8cpp.html#a4251b4df4748a0b9c43a48f61bdd2397">f</a> = [&](<span class="keywordtype">double</span>& x) { <span class="keywordflow">return</span> -x * x + 4.0; };</div>
|
||||
<div class="line"><a id="l00149" name="l00149"></a><span class="lineno"> 149</span> </div>
|
||||
<div class="line"><a id="l00150" name="l00150"></a><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"><a id="l00151" name="l00151"></a><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"><a id="l00152" name="l00152"></a><span class="lineno"> 152</span> pdf = [&](<span class="keywordtype">double</span>& x) {</div>
|
||||
<div class="line"><a id="l00153" name="l00153"></a><span class="lineno"> 153</span> <span class="keywordflow">if</span> (x >= lower_bound && x <= -1.0) {</div>
|
||||
<div class="line"><a id="l00154" name="l00154"></a><span class="lineno"> 154</span> <span class="keywordflow">return</span> 0.1;</div>
|
||||
<div class="line"><a id="l00155" name="l00155"></a><span class="lineno"> 155</span> }</div>
|
||||
<div class="line"><a id="l00156" name="l00156"></a><span class="lineno"> 156</span> <span class="keywordflow">if</span> (x <= upper_bound && x >= 1.0) {</div>
|
||||
<div class="line"><a id="l00157" name="l00157"></a><span class="lineno"> 157</span> <span class="keywordflow">return</span> 0.1;</div>
|
||||
<div class="line"><a id="l00158" name="l00158"></a><span class="lineno"> 158</span> }</div>
|
||||
<div class="line"><a id="l00159" name="l00159"></a><span class="lineno"> 159</span> <span class="keywordflow">if</span> (x > -1.0 && x < 1.0) {</div>
|
||||
<div class="line"><a id="l00160" name="l00160"></a><span class="lineno"> 160</span> <span class="keywordflow">return</span> 0.4;</div>
|
||||
<div class="line"><a id="l00161" name="l00161"></a><span class="lineno"> 161</span> }</div>
|
||||
<div class="line"><a id="l00162" name="l00162"></a><span class="lineno"> 162</span> <span class="keywordflow">return</span> 0.0;</div>
|
||||
<div class="line"><a id="l00163" name="l00163"></a><span class="lineno"> 163</span> };</div>
|
||||
<div class="line"><a id="l00164" name="l00164"></a><span class="lineno"> 164</span> </div>
|
||||
<div class="line"><a id="l00165" name="l00165"></a><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"><a id="l00166" name="l00166"></a><span class="lineno"> 166</span> (upper_bound - lower_bound) / 2.0, f, pdf);</div>
|
||||
<div class="line"><a id="l00167" name="l00167"></a><span class="lineno"> 167</span> </div>
|
||||
<div class="line"><a id="l00168" name="l00168"></a><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> << <span class="stringliteral">"This number should be close to 10.666666: "</span> << integral</div>
|
||||
<div class="line"><a id="l00169" name="l00169"></a><span class="lineno"> 169</span> << <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"><a id="l00170" name="l00170"></a><span class="lineno"> 170</span> </div>
|
||||
<div class="line"><a id="l00171" name="l00171"></a><span class="lineno"> 171</span> <span class="comment">/* \int_{0}^{1} e^x dx */</span></div>
|
||||
<div class="line"><a id="l00172" name="l00172"></a><span class="lineno"> 172</span> <a class="code hl_function" href="../../d4/d18/composite__simpson__rule_8cpp.html#a4251b4df4748a0b9c43a48f61bdd2397">f</a> = [&](<span class="keywordtype">double</span>& 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"><a id="l00173" name="l00173"></a><span class="lineno"> 173</span> </div>
|
||||
<div class="line"><a id="l00174" name="l00174"></a><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"><a id="l00175" name="l00175"></a><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"><a id="l00176" name="l00176"></a><span class="lineno"> 176</span> pdf = [&](<span class="keywordtype">double</span>& x) {</div>
|
||||
<div class="line"><a id="l00177" name="l00177"></a><span class="lineno"> 177</span> <span class="keywordflow">if</span> (x >= lower_bound && x <= 0.2) {</div>
|
||||
<div class="line"><a id="l00178" name="l00178"></a><span class="lineno"> 178</span> <span class="keywordflow">return</span> 0.1;</div>
|
||||
<div class="line"><a id="l00179" name="l00179"></a><span class="lineno"> 179</span> }</div>
|
||||
<div class="line"><a id="l00180" name="l00180"></a><span class="lineno"> 180</span> <span class="keywordflow">if</span> (x > 0.2 && x <= 0.4) {</div>
|
||||
<div class="line"><a id="l00181" name="l00181"></a><span class="lineno"> 181</span> <span class="keywordflow">return</span> 0.4;</div>
|
||||
<div class="line"><a id="l00182" name="l00182"></a><span class="lineno"> 182</span> }</div>
|
||||
<div class="line"><a id="l00183" name="l00183"></a><span class="lineno"> 183</span> <span class="keywordflow">if</span> (x > 0.4 && x < upper_bound) {</div>
|
||||
<div class="line"><a id="l00184" name="l00184"></a><span class="lineno"> 184</span> <span class="keywordflow">return</span> 1.5;</div>
|
||||
<div class="line"><a id="l00185" name="l00185"></a><span class="lineno"> 185</span> }</div>
|
||||
<div class="line"><a id="l00186" name="l00186"></a><span class="lineno"> 186</span> <span class="keywordflow">return</span> 0.0;</div>
|
||||
<div class="line"><a id="l00187" name="l00187"></a><span class="lineno"> 187</span> };</div>
|
||||
<div class="line"><a id="l00188" name="l00188"></a><span class="lineno"> 188</span> </div>
|
||||
<div class="line"><a id="l00189" name="l00189"></a><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"><a id="l00190" name="l00190"></a><span class="lineno"> 190</span> (upper_bound - lower_bound) / 2.0, f, pdf);</div>
|
||||
<div class="line"><a id="l00191" name="l00191"></a><span class="lineno"> 191</span> </div>
|
||||
<div class="line"><a id="l00192" name="l00192"></a><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> << <span class="stringliteral">"This number should be close to 1.7182818: "</span> << integral</div>
|
||||
<div class="line"><a id="l00193" name="l00193"></a><span class="lineno"> 193</span> << <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"><a id="l00194" name="l00194"></a><span class="lineno"> 194</span> </div>
|
||||
<div class="line"><a id="l00195" name="l00195"></a><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"><a id="l00196" name="l00196"></a><span class="lineno"> 196</span><span class="comment"> This is a difficult integral because of its infinite domain.</span></div>
|
||||
<div class="line"><a id="l00197" name="l00197"></a><span class="lineno"> 197</span><span class="comment"> Therefore, it may deviate largely from the expected result.</span></div>
|
||||
<div class="line"><a id="l00198" name="l00198"></a><span class="lineno"> 198</span><span class="comment"> */</span></div>
|
||||
<div class="line"><a id="l00199" name="l00199"></a><span class="lineno"> 199</span> <a class="code hl_function" href="../../d4/d18/composite__simpson__rule_8cpp.html#a4251b4df4748a0b9c43a48f61bdd2397">f</a> = [&](<span class="keywordtype">double</span>& 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"><a id="l00200" name="l00200"></a><span class="lineno"> 200</span> </div>
|
||||
<div class="line"><a id="l00201" name="l00201"></a><span class="lineno"> 201</span> pdf = [&](<span class="keywordtype">double</span>& x) {</div>
|
||||
<div class="line"><a id="l00202" name="l00202"></a><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"><a id="l00203" name="l00203"></a><span class="lineno"> 203</span> };</div>
|
||||
<div class="line"><a id="l00204" name="l00204"></a><span class="lineno"> 204</span> </div>
|
||||
<div class="line"><a id="l00205" name="l00205"></a><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"><a id="l00206" name="l00206"></a><span class="lineno"> 206</span> </div>
|
||||
<div class="line"><a id="l00207" name="l00207"></a><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> << <span class="stringliteral">"This number should be close to 1.0: "</span> << integral</div>
|
||||
<div class="line"><a id="l00208" name="l00208"></a><span class="lineno"> 208</span> << <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"><a id="l00209" name="l00209"></a><span class="lineno"> 209</span>}</div>
|
||||
<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> << <span class="stringliteral">"Disclaimer: Because this is a randomized algorithm,"</span></div>
|
||||
<div class="line"><span class="lineno"> 135</span> << <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"> 137</span> << <span class="stringliteral">"it may happen that singular samples deviate from the true result."</span></div>
|
||||
<div class="line"><span class="lineno"> 138</span> << <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> << <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"> 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"> 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"> 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> = [&](<span class="keywordtype">double</span>& 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"> 152</span> pdf = [&](<span class="keywordtype">double</span>& x) {</div>
|
||||
<div class="line"><span class="lineno"> 153</span> <span class="keywordflow">if</span> (x >= lower_bound && x <= -1.0) {</div>
|
||||
<div class="line"><span class="lineno"> 154</span> <span class="keywordflow">return</span> 0.1;</div>
|
||||
<div class="line"><span class="lineno"> 155</span> }</div>
|
||||
<div class="line"><span class="lineno"> 156</span> <span class="keywordflow">if</span> (x <= upper_bound && x >= 1.0) {</div>
|
||||
<div class="line"><span class="lineno"> 157</span> <span class="keywordflow">return</span> 0.1;</div>
|
||||
<div class="line"><span class="lineno"> 158</span> }</div>
|
||||
<div class="line"><span class="lineno"> 159</span> <span class="keywordflow">if</span> (x > -1.0 && x < 1.0) {</div>
|
||||
<div class="line"><span class="lineno"> 160</span> <span class="keywordflow">return</span> 0.4;</div>
|
||||
<div class="line"><span class="lineno"> 161</span> }</div>
|
||||
<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"> 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> << <span class="stringliteral">"This number should be close to 10.666666: "</span> << integral</div>
|
||||
<div class="line"><span class="lineno"> 169</span> << <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"> 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> = [&](<span class="keywordtype">double</span>& 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"> 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"> 176</span> pdf = [&](<span class="keywordtype">double</span>& x) {</div>
|
||||
<div class="line"><span class="lineno"> 177</span> <span class="keywordflow">if</span> (x >= lower_bound && x <= 0.2) {</div>
|
||||
<div class="line"><span class="lineno"> 178</span> <span class="keywordflow">return</span> 0.1;</div>
|
||||
<div class="line"><span class="lineno"> 179</span> }</div>
|
||||
<div class="line"><span class="lineno"> 180</span> <span class="keywordflow">if</span> (x > 0.2 && x <= 0.4) {</div>
|
||||
<div class="line"><span class="lineno"> 181</span> <span class="keywordflow">return</span> 0.4;</div>
|
||||
<div class="line"><span class="lineno"> 182</span> }</div>
|
||||
<div class="line"><span class="lineno"> 183</span> <span class="keywordflow">if</span> (x > 0.4 && x < upper_bound) {</div>
|
||||
<div class="line"><span class="lineno"> 184</span> <span class="keywordflow">return</span> 1.5;</div>
|
||||
<div class="line"><span class="lineno"> 185</span> }</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"> 190</span> (upper_bound - lower_bound) / 2.0, f, pdf);</div>
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<div class="line"><span class="lineno"> 191</span> </div>
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<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> << <span class="stringliteral">"This number should be close to 1.7182818: "</span> << integral</div>
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<div class="line"><span class="lineno"> 193</span> << <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/endl.html">std::endl</a>;</div>
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<div class="line"><span class="lineno"> 194</span> </div>
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<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>
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<div class="line"><span class="lineno"> 196</span><span class="comment"> This is a difficult integral because of its infinite domain.</span></div>
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<div class="line"><span class="lineno"> 197</span><span class="comment"> Therefore, it may deviate largely from the expected result.</span></div>
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<div class="line"><span class="lineno"> 198</span><span class="comment"> */</span></div>
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<div class="line"><span class="lineno"> 199</span> <a class="code hl_function" href="../../d4/d18/composite__simpson__rule_8cpp.html#a4251b4df4748a0b9c43a48f61bdd2397">f</a> = [&](<span class="keywordtype">double</span>& 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"> 200</span> </div>
|
||||
<div class="line"><span class="lineno"> 201</span> pdf = [&](<span class="keywordtype">double</span>& 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"> 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"> 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> << <span class="stringliteral">"This number should be close to 1.0: "</span> << integral</div>
|
||||
<div class="line"><span class="lineno"> 208</span> << <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"> 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>
|
||||
@@ -477,7 +477,7 @@ Here is the call graph for this function:</div>
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