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<div id="projectbrief">Set of algorithms implemented in C++.</div>
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<a href="#func-members">Functions</a> </div>
<div class="headertitle"><div class="title">Ordinary Differential Equations</div></div>
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Functions</h2></td></tr>
<tr class="memitem:ga195d23bbdfcb80e83c9cda45c9ad5723"><td class="memItemLeft" align="right" valign="top">void&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../d6/d60/group__ode.html#ga195d23bbdfcb80e83c9cda45c9ad5723">forward_euler_step</a> (const double dx, const double x, <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; double &gt; *y, <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; double &gt; *dy)</td></tr>
<tr class="memdesc:ga195d23bbdfcb80e83c9cda45c9ad5723"><td class="mdescLeft">&#160;</td><td class="mdescRight">Compute next step approximation using the forward-Euler method. <br /></td></tr>
<tr class="separator:ga195d23bbdfcb80e83c9cda45c9ad5723"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:gae0509f8843e2bc42de2abbd00a14b7b9"><td class="memItemLeft" align="right" valign="top">double&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../d6/d60/group__ode.html#gae0509f8843e2bc42de2abbd00a14b7b9">forward_euler</a> (double dx, double x0, double x_max, <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; double &gt; *y, bool save_to_file=false)</td></tr>
<tr class="memdesc:gae0509f8843e2bc42de2abbd00a14b7b9"><td class="mdescLeft">&#160;</td><td class="mdescRight">Compute approximation using the forward-Euler method in the given limits. <br /></td></tr>
<tr class="separator:gae0509f8843e2bc42de2abbd00a14b7b9"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:ga8c319db420c3d97a83e9dcca803b6812"><td class="memItemLeft" align="right" valign="top">void&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../d6/d60/group__ode.html#ga8c319db420c3d97a83e9dcca803b6812">midpoint_euler_step</a> (const double dx, const double &amp;x, <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; double &gt; *y, <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; double &gt; *dy)</td></tr>
<tr class="memdesc:ga8c319db420c3d97a83e9dcca803b6812"><td class="mdescLeft">&#160;</td><td class="mdescRight">Compute next step approximation using the midpoint-Euler method. <br /></td></tr>
<tr class="separator:ga8c319db420c3d97a83e9dcca803b6812"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:ga36ff0710861ca39d957046c0b09b2985"><td class="memItemLeft" align="right" valign="top">double&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../d6/d60/group__ode.html#ga36ff0710861ca39d957046c0b09b2985">midpoint_euler</a> (double dx, double x0, double x_max, <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; double &gt; *y, bool save_to_file=false)</td></tr>
<tr class="memdesc:ga36ff0710861ca39d957046c0b09b2985"><td class="mdescLeft">&#160;</td><td class="mdescRight">Compute approximation using the midpoint-Euler method in the given limits. <br /></td></tr>
<tr class="separator:ga36ff0710861ca39d957046c0b09b2985"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:ga827bf009831ddc477c5fa8891d5cb35f"><td class="memItemLeft" align="right" valign="top">void&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../d6/d60/group__ode.html#ga827bf009831ddc477c5fa8891d5cb35f">semi_implicit_euler_step</a> (const double dx, const double &amp;x, <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; double &gt; *y, <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; double &gt; *dy)</td></tr>
<tr class="memdesc:ga827bf009831ddc477c5fa8891d5cb35f"><td class="mdescLeft">&#160;</td><td class="mdescRight">Compute next step approximation using the semi-implicit-Euler method. <br /></td></tr>
<tr class="separator:ga827bf009831ddc477c5fa8891d5cb35f"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:ga3874d294ab00fe9ce8731c5b7991a089"><td class="memItemLeft" align="right" valign="top">double&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../d6/d60/group__ode.html#ga3874d294ab00fe9ce8731c5b7991a089">semi_implicit_euler</a> (double dx, double x0, double x_max, <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; double &gt; *y, bool save_to_file=false)</td></tr>
<tr class="memdesc:ga3874d294ab00fe9ce8731c5b7991a089"><td class="mdescLeft">&#160;</td><td class="mdescRight">Compute approximation using the semi-implicit-Euler method in the given limits. <br /></td></tr>
<tr class="separator:ga3874d294ab00fe9ce8731c5b7991a089"><td class="memSeparator" colspan="2">&#160;</td></tr>
</table>
<a name="details" id="details"></a><h2 class="groupheader">Detailed Description</h2>
<p>Integration functions for implementations with solving <a href="https://en.wikipedia.org/wiki/Ordinary_differential_equation" target="_blank">ordinary differential equations</a> (ODEs) of any order and and any number of independent variables. </p>
<h2 class="groupheader">Function Documentation</h2>
<a id="gae0509f8843e2bc42de2abbd00a14b7b9" name="gae0509f8843e2bc42de2abbd00a14b7b9"></a>
<h2 class="memtitle"><span class="permalink"><a href="#gae0509f8843e2bc42de2abbd00a14b7b9">&#9670;&#160;</a></span>forward_euler()</h2>
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<td class="memname">double forward_euler </td>
<td>(</td>
<td class="paramtype">double&#160;</td>
<td class="paramname"><em>dx</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">double&#160;</td>
<td class="paramname"><em>x0</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">double&#160;</td>
<td class="paramname"><em>x_max</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; double &gt; *&#160;</td>
<td class="paramname"><em>y</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">bool&#160;</td>
<td class="paramname"><em>save_to_file</em> = <code>false</code>&#160;</td>
</tr>
<tr>
<td></td>
<td>)</td>
<td></td><td></td>
</tr>
</table>
</div><div class="memdoc">
<p>Compute approximation using the forward-Euler method in the given limits. </p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramdir">[in]</td><td class="paramname">dx</td><td>step size </td></tr>
<tr><td class="paramdir">[in]</td><td class="paramname">x0</td><td>initial value of independent variable </td></tr>
<tr><td class="paramdir">[in]</td><td class="paramname">x_max</td><td>final value of independent variable </td></tr>
<tr><td class="paramdir">[in,out]</td><td class="paramname">y</td><td>take \(y_n\) and compute \(y_{n+1}\) </td></tr>
<tr><td class="paramdir">[in]</td><td class="paramname">save_to_file</td><td>flag to save results to a CSV file (1) or not (0) </td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>time taken for computation in seconds </dd></dl>
<div class="fragment"><div class="line"><span class="lineno"> 103</span> {</div>
<div class="line"><span class="lineno"> 104</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray&lt;double&gt;</a> dy = *y;</div>
<div class="line"><span class="lineno"> 105</span> </div>
<div class="line"><span class="lineno"> 106</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ofstream.html">std::ofstream</a> fp;</div>
<div class="line"><span class="lineno"> 107</span> <span class="keywordflow">if</span> (save_to_file) {</div>
<div class="line"><span class="lineno"> 108</span> fp.<a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ofstream/open.html">open</a>(<span class="stringliteral">&quot;forward_euler.csv&quot;</span>, std::ofstream::out);</div>
<div class="line"><span class="lineno"> 109</span> <span class="keywordflow">if</span> (!fp.<a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ofstream/is_open.html">is_open</a>()) {</div>
<div class="line"><span class="lineno"> 110</span> <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/c/perror.html">std::perror</a>(<span class="stringliteral">&quot;Error! &quot;</span>);</div>
<div class="line"><span class="lineno"> 111</span> }</div>
<div class="line"><span class="lineno"> 112</span> }</div>
<div class="line"><span class="lineno"> 113</span> </div>
<div class="line"><span class="lineno"> 114</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/types/size_t.html">std::size_t</a> L = y-&gt;size();</div>
<div class="line"><span class="lineno"> 115</span> </div>
<div class="line"><span class="lineno"> 116</span> <span class="comment">/* start integration */</span></div>
<div class="line"><span class="lineno"> 117</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/chrono/c/clock_t.html">std::clock_t</a> t1 = <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/chrono/c/clock.html">std::clock</a>();</div>
<div class="line"><span class="lineno"> 118</span> <span class="keywordtype">double</span> x = x0;</div>
<div class="line"><span class="lineno"> 119</span> </div>
<div class="line"><span class="lineno"> 120</span> <span class="keywordflow">do</span> { <span class="comment">// iterate for each step of independent variable</span></div>
<div class="line"><span class="lineno"> 121</span> <span class="keywordflow">if</span> (save_to_file &amp;&amp; fp.<a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ofstream/is_open.html">is_open</a>()) {</div>
<div class="line"><span class="lineno"> 122</span> <span class="comment">// write to file</span></div>
<div class="line"><span class="lineno"> 123</span> fp &lt;&lt; x &lt;&lt; <span class="stringliteral">&quot;,&quot;</span>;</div>
<div class="line"><span class="lineno"> 124</span> <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i &lt; L - 1; i++) {</div>
<div class="line"><span class="lineno"> 125</span> fp &lt;&lt; y[0][i] &lt;&lt; <span class="stringliteral">&quot;,&quot;</span>; <span class="comment">// NOLINT</span></div>
<div class="line"><span class="lineno"> 126</span> }</div>
<div class="line"><span class="lineno"> 127</span> fp &lt;&lt; y[0][L - 1] &lt;&lt; <span class="stringliteral">&quot;\n&quot;</span>; <span class="comment">// NOLINT</span></div>
<div class="line"><span class="lineno"> 128</span> }</div>
<div class="line"><span class="lineno"> 129</span> </div>
<div class="line"><span class="lineno"> 130</span> <a class="code hl_function" href="../../d6/d60/group__ode.html#ga195d23bbdfcb80e83c9cda45c9ad5723">forward_euler_step</a>(dx, x, y, &amp;dy); <span class="comment">// perform integration</span></div>
<div class="line"><span class="lineno"> 131</span> x += dx; <span class="comment">// update step</span></div>
<div class="line"><span class="lineno"> 132</span> } <span class="keywordflow">while</span> (x &lt;= x_max); <span class="comment">// till upper limit of independent variable</span></div>
<div class="line"><span class="lineno"> 133</span> <span class="comment">/* end of integration */</span></div>
<div class="line"><span class="lineno"> 134</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/chrono/c/clock_t.html">std::clock_t</a> t2 = <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/chrono/c/clock.html">std::clock</a>();</div>
<div class="line"><span class="lineno"> 135</span> </div>
<div class="line"><span class="lineno"> 136</span> <span class="keywordflow">if</span> (fp.<a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ofstream/is_open.html">is_open</a>()) {</div>
<div class="line"><span class="lineno"> 137</span> fp.<a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ofstream/close.html">close</a>();</div>
<div class="line"><span class="lineno"> 138</span> }</div>
<div class="line"><span class="lineno"> 139</span> </div>
<div class="line"><span class="lineno"> 140</span> <span class="keywordflow">return</span> <span class="keyword">static_cast&lt;</span><span class="keywordtype">double</span><span class="keyword">&gt;</span>(t2 - t1) / CLOCKS_PER_SEC;</div>
<div class="line"><span class="lineno"> 141</span>}</div>
<div class="ttc" id="abasic_ofstream_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/io/basic_ofstream.html">std::ofstream</a></div></div>
<div class="ttc" id="aclock_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/chrono/c/clock.html">std::clock</a></div><div class="ttdeci">T clock(T... args)</div></div>
<div class="ttc" id="aclock_t_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/chrono/c/clock_t.html">std::clock_t</a></div></div>
<div class="ttc" id="aclose_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/io/basic_ofstream/close.html">std::ofstream::close</a></div><div class="ttdeci">T close(T... args)</div></div>
<div class="ttc" id="agroup__ode_html_ga195d23bbdfcb80e83c9cda45c9ad5723"><div class="ttname"><a href="../../d6/d60/group__ode.html#ga195d23bbdfcb80e83c9cda45c9ad5723">forward_euler_step</a></div><div class="ttdeci">void forward_euler_step(const double dx, const double x, std::valarray&lt; double &gt; *y, std::valarray&lt; double &gt; *dy)</div><div class="ttdoc">Compute next step approximation using the forward-Euler method.</div><div class="ttdef"><b>Definition</b> ode_forward_euler.cpp:86</div></div>
<div class="ttc" id="ais_open_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/io/basic_ofstream/is_open.html">std::ofstream::is_open</a></div><div class="ttdeci">T is_open(T... args)</div></div>
<div class="ttc" id="aopen_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/io/basic_ofstream/open.html">std::ofstream::open</a></div><div class="ttdeci">T open(T... args)</div></div>
<div class="ttc" id="aperror_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/io/c/perror.html">std::perror</a></div><div class="ttdeci">T perror(T... args)</div></div>
<div class="ttc" id="asize_t_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/types/size_t.html">std::size_t</a></div></div>
<div class="ttc" id="avalarray_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a></div></div>
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<h2 class="memtitle"><span class="permalink"><a href="#ga195d23bbdfcb80e83c9cda45c9ad5723">&#9670;&#160;</a></span>forward_euler_step()</h2>
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<td class="memname">void forward_euler_step </td>
<td>(</td>
<td class="paramtype">const double&#160;</td>
<td class="paramname"><em>dx</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">const double&#160;</td>
<td class="paramname"><em>x</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; double &gt; *&#160;</td>
<td class="paramname"><em>y</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; double &gt; *&#160;</td>
<td class="paramname"><em>dy</em>&#160;</td>
</tr>
<tr>
<td></td>
<td>)</td>
<td></td><td></td>
</tr>
</table>
</div><div class="memdoc">
<p>Compute next step approximation using the forward-Euler method. </p>
<p class="formulaDsp">
\[y_{n+1}=y_n + dx\cdot f\left(x_n,y_n\right)\]
</p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramdir">[in]</td><td class="paramname">dx</td><td>step size </td></tr>
<tr><td class="paramdir">[in]</td><td class="paramname">x</td><td>take \(x_n\) and compute \(x_{n+1}\) </td></tr>
<tr><td class="paramdir">[in,out]</td><td class="paramname">y</td><td>take \(y_n\) and compute \(y_{n+1}\) </td></tr>
<tr><td class="paramdir">[in,out]</td><td class="paramname">dy</td><td>compute \(f\left(x_n,y_n\right)\) </td></tr>
</table>
</dd>
</dl>
<div class="fragment"><div class="line"><span class="lineno"> 87</span> {</div>
<div class="line"><span class="lineno"> 88</span> <a class="code hl_function" href="../../db/dd3/ode__forward__euler_8cpp.html#abaeae8f62a018d197f0187a1c80a90fe">problem</a>(x, y, dy);</div>
<div class="line"><span class="lineno"> 89</span> *y += *dy * dx;</div>
<div class="line"><span class="lineno"> 90</span>}</div>
<div class="ttc" id="aode__forward__euler_8cpp_html_abaeae8f62a018d197f0187a1c80a90fe"><div class="ttname"><a href="../../db/dd3/ode__forward__euler_8cpp.html#abaeae8f62a018d197f0187a1c80a90fe">problem</a></div><div class="ttdeci">void problem(const double &amp;x, std::valarray&lt; double &gt; *y, std::valarray&lt; double &gt; *dy)</div><div class="ttdoc">Problem statement for a system with first-order differential equations. Updates the system differenti...</div><div class="ttdef"><b>Definition</b> ode_forward_euler.cpp:54</div></div>
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<h2 class="memtitle"><span class="permalink"><a href="#ga36ff0710861ca39d957046c0b09b2985">&#9670;&#160;</a></span>midpoint_euler()</h2>
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<td class="memname">double midpoint_euler </td>
<td>(</td>
<td class="paramtype">double&#160;</td>
<td class="paramname"><em>dx</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">double&#160;</td>
<td class="paramname"><em>x0</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">double&#160;</td>
<td class="paramname"><em>x_max</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; double &gt; *&#160;</td>
<td class="paramname"><em>y</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">bool&#160;</td>
<td class="paramname"><em>save_to_file</em> = <code>false</code>&#160;</td>
</tr>
<tr>
<td></td>
<td>)</td>
<td></td><td></td>
</tr>
</table>
</div><div class="memdoc">
<p>Compute approximation using the midpoint-Euler method in the given limits. </p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramdir">[in]</td><td class="paramname">dx</td><td>step size </td></tr>
<tr><td class="paramdir">[in]</td><td class="paramname">x0</td><td>initial value of independent variable </td></tr>
<tr><td class="paramdir">[in]</td><td class="paramname">x_max</td><td>final value of independent variable </td></tr>
<tr><td class="paramdir">[in,out]</td><td class="paramname">y</td><td>take \(y_n\) and compute \(y_{n+1}\) </td></tr>
<tr><td class="paramdir">[in]</td><td class="paramname">save_to_file</td><td>flag to save results to a CSV file (1) or not (0) </td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>time taken for computation in seconds </dd></dl>
<div class="fragment"><div class="line"><span class="lineno"> 108</span> {</div>
<div class="line"><span class="lineno"> 109</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray&lt;double&gt;</a> dy = y[0];</div>
<div class="line"><span class="lineno"> 110</span> </div>
<div class="line"><span class="lineno"> 111</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ofstream.html">std::ofstream</a> fp;</div>
<div class="line"><span class="lineno"> 112</span> <span class="keywordflow">if</span> (save_to_file) {</div>
<div class="line"><span class="lineno"> 113</span> fp.<a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ofstream/open.html">open</a>(<span class="stringliteral">&quot;midpoint_euler.csv&quot;</span>, std::ofstream::out);</div>
<div class="line"><span class="lineno"> 114</span> <span class="keywordflow">if</span> (!fp.<a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ofstream/is_open.html">is_open</a>()) {</div>
<div class="line"><span class="lineno"> 115</span> <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/c/perror.html">std::perror</a>(<span class="stringliteral">&quot;Error! &quot;</span>);</div>
<div class="line"><span class="lineno"> 116</span> }</div>
<div class="line"><span class="lineno"> 117</span> }</div>
<div class="line"><span class="lineno"> 118</span> </div>
<div class="line"><span class="lineno"> 119</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/types/size_t.html">std::size_t</a> L = y-&gt;size();</div>
<div class="line"><span class="lineno"> 120</span> </div>
<div class="line"><span class="lineno"> 121</span> <span class="comment">/* start integration */</span></div>
<div class="line"><span class="lineno"> 122</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/chrono/c/clock_t.html">std::clock_t</a> t1 = <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/chrono/c/clock.html">std::clock</a>();</div>
<div class="line"><span class="lineno"> 123</span> <span class="keywordtype">double</span> x = x0;</div>
<div class="line"><span class="lineno"> 124</span> <span class="keywordflow">do</span> { <span class="comment">// iterate for each step of independent variable</span></div>
<div class="line"><span class="lineno"> 125</span> <span class="keywordflow">if</span> (save_to_file &amp;&amp; fp.<a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ofstream/is_open.html">is_open</a>()) {</div>
<div class="line"><span class="lineno"> 126</span> <span class="comment">// write to file</span></div>
<div class="line"><span class="lineno"> 127</span> fp &lt;&lt; x &lt;&lt; <span class="stringliteral">&quot;,&quot;</span>;</div>
<div class="line"><span class="lineno"> 128</span> <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i &lt; L - 1; i++) {</div>
<div class="line"><span class="lineno"> 129</span> fp &lt;&lt; y[0][i] &lt;&lt; <span class="stringliteral">&quot;,&quot;</span>;</div>
<div class="line"><span class="lineno"> 130</span> }</div>
<div class="line"><span class="lineno"> 131</span> fp &lt;&lt; y[0][L - 1] &lt;&lt; <span class="stringliteral">&quot;\n&quot;</span>;</div>
<div class="line"><span class="lineno"> 132</span> }</div>
<div class="line"><span class="lineno"> 133</span> </div>
<div class="line"><span class="lineno"> 134</span> <a class="code hl_function" href="../../d6/d60/group__ode.html#ga8c319db420c3d97a83e9dcca803b6812">midpoint_euler_step</a>(dx, x, y, &amp;dy); <span class="comment">// perform integration</span></div>
<div class="line"><span class="lineno"> 135</span> x += dx; <span class="comment">// update step</span></div>
<div class="line"><span class="lineno"> 136</span> } <span class="keywordflow">while</span> (x &lt;= x_max); <span class="comment">// till upper limit of independent variable</span></div>
<div class="line"><span class="lineno"> 137</span> <span class="comment">/* end of integration */</span></div>
<div class="line"><span class="lineno"> 138</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/chrono/c/clock_t.html">std::clock_t</a> t2 = <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/chrono/c/clock.html">std::clock</a>();</div>
<div class="line"><span class="lineno"> 139</span> </div>
<div class="line"><span class="lineno"> 140</span> <span class="keywordflow">if</span> (fp.<a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ofstream/is_open.html">is_open</a>())</div>
<div class="line"><span class="lineno"> 141</span> fp.<a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ofstream/close.html">close</a>();</div>
<div class="line"><span class="lineno"> 142</span> </div>
<div class="line"><span class="lineno"> 143</span> <span class="keywordflow">return</span> <span class="keyword">static_cast&lt;</span><span class="keywordtype">double</span><span class="keyword">&gt;</span>(t2 - t1) / CLOCKS_PER_SEC;</div>
<div class="line"><span class="lineno"> 144</span>}</div>
<div class="ttc" id="agroup__ode_html_ga8c319db420c3d97a83e9dcca803b6812"><div class="ttname"><a href="../../d6/d60/group__ode.html#ga8c319db420c3d97a83e9dcca803b6812">midpoint_euler_step</a></div><div class="ttdeci">void midpoint_euler_step(const double dx, const double &amp;x, std::valarray&lt; double &gt; *y, std::valarray&lt; double &gt; *dy)</div><div class="ttdoc">Compute next step approximation using the midpoint-Euler method.</div><div class="ttdef"><b>Definition</b> ode_midpoint_euler.cpp:85</div></div>
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<h2 class="memtitle"><span class="permalink"><a href="#ga8c319db420c3d97a83e9dcca803b6812">&#9670;&#160;</a></span>midpoint_euler_step()</h2>
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<td class="memname">void midpoint_euler_step </td>
<td>(</td>
<td class="paramtype">const double&#160;</td>
<td class="paramname"><em>dx</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">const double &amp;&#160;</td>
<td class="paramname"><em>x</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; double &gt; *&#160;</td>
<td class="paramname"><em>y</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; double &gt; *&#160;</td>
<td class="paramname"><em>dy</em>&#160;</td>
</tr>
<tr>
<td></td>
<td>)</td>
<td></td><td></td>
</tr>
</table>
</div><div class="memdoc">
<p>Compute next step approximation using the midpoint-Euler method. </p>
<p class="formulaDsp">
\[y_{n+1} = y_n + dx\, f\left(x_n+\frac{1}{2}dx,
y_n + \frac{1}{2}dx\,f\left(x_n,y_n\right)\right)\]
</p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramdir">[in]</td><td class="paramname">dx</td><td>step size </td></tr>
<tr><td class="paramdir">[in]</td><td class="paramname">x</td><td>take \(x_n\) and compute \(x_{n+1}\) </td></tr>
<tr><td class="paramdir">[in,out]</td><td class="paramname">y</td><td>take \(y_n\) and compute \(y_{n+1}\) </td></tr>
<tr><td class="paramdir">[in,out]</td><td class="paramname">dy</td><td>compute \(f\left(x_n,y_n\right)\) </td></tr>
</table>
</dd>
</dl>
<div class="fragment"><div class="line"><span class="lineno"> 86</span> {</div>
<div class="line"><span class="lineno"> 87</span> <a class="code hl_function" href="../../d6/dd3/ode__midpoint__euler_8cpp.html#abaeae8f62a018d197f0187a1c80a90fe">problem</a>(x, y, dy);</div>
<div class="line"><span class="lineno"> 88</span> <span class="keywordtype">double</span> tmp_x = x + 0.5 * dx;</div>
<div class="line"><span class="lineno"> 89</span> </div>
<div class="line"><span class="lineno"> 90</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray&lt;double&gt;</a> tmp_y = y[0] + dy[0] * (0.5 * dx);</div>
<div class="line"><span class="lineno"> 91</span> </div>
<div class="line"><span class="lineno"> 92</span> <a class="code hl_function" href="../../d6/dd3/ode__midpoint__euler_8cpp.html#abaeae8f62a018d197f0187a1c80a90fe">problem</a>(tmp_x, &amp;tmp_y, dy);</div>
<div class="line"><span class="lineno"> 93</span> </div>
<div class="line"><span class="lineno"> 94</span> y[0] += dy[0] * dx;</div>
<div class="line"><span class="lineno"> 95</span>}</div>
<div class="ttc" id="aode__midpoint__euler_8cpp_html_abaeae8f62a018d197f0187a1c80a90fe"><div class="ttname"><a href="../../d6/dd3/ode__midpoint__euler_8cpp.html#abaeae8f62a018d197f0187a1c80a90fe">problem</a></div><div class="ttdeci">void problem(const double &amp;x, std::valarray&lt; double &gt; *y, std::valarray&lt; double &gt; *dy)</div><div class="ttdoc">Problem statement for a system with first-order differential equations. Updates the system differenti...</div><div class="ttdef"><b>Definition</b> ode_midpoint_euler.cpp:53</div></div>
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<h2 class="memtitle"><span class="permalink"><a href="#ga3874d294ab00fe9ce8731c5b7991a089">&#9670;&#160;</a></span>semi_implicit_euler()</h2>
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<td class="memname">double semi_implicit_euler </td>
<td>(</td>
<td class="paramtype">double&#160;</td>
<td class="paramname"><em>dx</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">double&#160;</td>
<td class="paramname"><em>x0</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">double&#160;</td>
<td class="paramname"><em>x_max</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; double &gt; *&#160;</td>
<td class="paramname"><em>y</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">bool&#160;</td>
<td class="paramname"><em>save_to_file</em> = <code>false</code>&#160;</td>
</tr>
<tr>
<td></td>
<td>)</td>
<td></td><td></td>
</tr>
</table>
</div><div class="memdoc">
<p>Compute approximation using the semi-implicit-Euler method in the given limits. </p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramdir">[in]</td><td class="paramname">dx</td><td>step size </td></tr>
<tr><td class="paramdir">[in]</td><td class="paramname">x0</td><td>initial value of independent variable </td></tr>
<tr><td class="paramdir">[in]</td><td class="paramname">x_max</td><td>final value of independent variable </td></tr>
<tr><td class="paramdir">[in,out]</td><td class="paramname">y</td><td>take \(y_n\) and compute \(y_{n+1}\) </td></tr>
<tr><td class="paramdir">[in]</td><td class="paramname">save_to_file</td><td>flag to save results to a CSV file (1) or not (0) </td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>time taken for computation in seconds </dd></dl>
<div class="fragment"><div class="line"><span class="lineno"> 105</span> {</div>
<div class="line"><span class="lineno"> 106</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray&lt;double&gt;</a> dy = y[0];</div>
<div class="line"><span class="lineno"> 107</span> </div>
<div class="line"><span class="lineno"> 108</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ofstream.html">std::ofstream</a> fp;</div>
<div class="line"><span class="lineno"> 109</span> <span class="keywordflow">if</span> (save_to_file) {</div>
<div class="line"><span class="lineno"> 110</span> fp.<a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ofstream/open.html">open</a>(<span class="stringliteral">&quot;semi_implicit_euler.csv&quot;</span>, std::ofstream::out);</div>
<div class="line"><span class="lineno"> 111</span> <span class="keywordflow">if</span> (!fp.<a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ofstream/is_open.html">is_open</a>()) {</div>
<div class="line"><span class="lineno"> 112</span> <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/c/perror.html">std::perror</a>(<span class="stringliteral">&quot;Error! &quot;</span>);</div>
<div class="line"><span class="lineno"> 113</span> }</div>
<div class="line"><span class="lineno"> 114</span> }</div>
<div class="line"><span class="lineno"> 115</span> </div>
<div class="line"><span class="lineno"> 116</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/types/size_t.html">std::size_t</a> L = y-&gt;size();</div>
<div class="line"><span class="lineno"> 117</span> </div>
<div class="line"><span class="lineno"> 118</span> <span class="comment">/* start integration */</span></div>
<div class="line"><span class="lineno"> 119</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/chrono/c/clock_t.html">std::clock_t</a> t1 = <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/chrono/c/clock.html">std::clock</a>();</div>
<div class="line"><span class="lineno"> 120</span> <span class="keywordtype">double</span> x = x0;</div>
<div class="line"><span class="lineno"> 121</span> <span class="keywordflow">do</span> { <span class="comment">// iterate for each step of independent variable</span></div>
<div class="line"><span class="lineno"> 122</span> <span class="keywordflow">if</span> (save_to_file &amp;&amp; fp.<a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ofstream/is_open.html">is_open</a>()) {</div>
<div class="line"><span class="lineno"> 123</span> <span class="comment">// write to file</span></div>
<div class="line"><span class="lineno"> 124</span> fp &lt;&lt; x &lt;&lt; <span class="stringliteral">&quot;,&quot;</span>;</div>
<div class="line"><span class="lineno"> 125</span> <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i &lt; L - 1; i++) {</div>
<div class="line"><span class="lineno"> 126</span> fp &lt;&lt; y[0][i] &lt;&lt; <span class="stringliteral">&quot;,&quot;</span>;</div>
<div class="line"><span class="lineno"> 127</span> }</div>
<div class="line"><span class="lineno"> 128</span> fp &lt;&lt; y[0][L - 1] &lt;&lt; <span class="stringliteral">&quot;\n&quot;</span>;</div>
<div class="line"><span class="lineno"> 129</span> }</div>
<div class="line"><span class="lineno"> 130</span> </div>
<div class="line"><span class="lineno"> 131</span> <a class="code hl_function" href="../../d6/d60/group__ode.html#ga827bf009831ddc477c5fa8891d5cb35f">semi_implicit_euler_step</a>(dx, x, y, &amp;dy); <span class="comment">// perform integration</span></div>
<div class="line"><span class="lineno"> 132</span> x += dx; <span class="comment">// update step</span></div>
<div class="line"><span class="lineno"> 133</span> } <span class="keywordflow">while</span> (x &lt;= x_max); <span class="comment">// till upper limit of independent variable</span></div>
<div class="line"><span class="lineno"> 134</span> <span class="comment">/* end of integration */</span></div>
<div class="line"><span class="lineno"> 135</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/chrono/c/clock_t.html">std::clock_t</a> t2 = <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/chrono/c/clock.html">std::clock</a>();</div>
<div class="line"><span class="lineno"> 136</span> </div>
<div class="line"><span class="lineno"> 137</span> <span class="keywordflow">if</span> (fp.<a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ofstream/is_open.html">is_open</a>())</div>
<div class="line"><span class="lineno"> 138</span> fp.<a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ofstream/close.html">close</a>();</div>
<div class="line"><span class="lineno"> 139</span> </div>
<div class="line"><span class="lineno"> 140</span> <span class="keywordflow">return</span> <span class="keyword">static_cast&lt;</span><span class="keywordtype">double</span><span class="keyword">&gt;</span>(t2 - t1) / CLOCKS_PER_SEC;</div>
<div class="line"><span class="lineno"> 141</span>}</div>
<div class="ttc" id="agroup__ode_html_ga827bf009831ddc477c5fa8891d5cb35f"><div class="ttname"><a href="../../d6/d60/group__ode.html#ga827bf009831ddc477c5fa8891d5cb35f">semi_implicit_euler_step</a></div><div class="ttdeci">void semi_implicit_euler_step(const double dx, const double &amp;x, std::valarray&lt; double &gt; *y, std::valarray&lt; double &gt; *dy)</div><div class="ttdoc">Compute next step approximation using the semi-implicit-Euler method.</div><div class="ttdef"><b>Definition</b> ode_semi_implicit_euler.cpp:82</div></div>
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<h2 class="memtitle"><span class="permalink"><a href="#ga827bf009831ddc477c5fa8891d5cb35f">&#9670;&#160;</a></span>semi_implicit_euler_step()</h2>
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<td class="memname">void semi_implicit_euler_step </td>
<td>(</td>
<td class="paramtype">const double&#160;</td>
<td class="paramname"><em>dx</em>, </td>
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<td class="paramtype">const double &amp;&#160;</td>
<td class="paramname"><em>x</em>, </td>
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<td class="paramtype"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; double &gt; *&#160;</td>
<td class="paramname"><em>y</em>, </td>
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<td class="paramkey"></td>
<td></td>
<td class="paramtype"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; double &gt; *&#160;</td>
<td class="paramname"><em>dy</em>&#160;</td>
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<p>Compute next step approximation using the semi-implicit-Euler method. </p>
<p class="formulaDsp">
\[y_{n+1}=y_n + dx\cdot f\left(x_n,y_n\right)\]
</p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramdir">[in]</td><td class="paramname">dx</td><td>step size </td></tr>
<tr><td class="paramdir">[in]</td><td class="paramname">x</td><td>take \(x_n\) and compute \(x_{n+1}\) </td></tr>
<tr><td class="paramdir">[in,out]</td><td class="paramname">y</td><td>take \(y_n\) and compute \(y_{n+1}\) </td></tr>
<tr><td class="paramdir">[in,out]</td><td class="paramname">dy</td><td>compute \(f\left(x_n,y_n\right)\) </td></tr>
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</dd>
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<div class="fragment"><div class="line"><span class="lineno"> 84</span> {</div>
<div class="line"><span class="lineno"> 85</span> <a class="code hl_function" href="../../d3/d06/ode__semi__implicit__euler_8cpp.html#abaeae8f62a018d197f0187a1c80a90fe">problem</a>(x, y, dy); <span class="comment">// update dy once</span></div>
<div class="line"><span class="lineno"> 86</span> y[0][0] += dx * dy[0][0]; <span class="comment">// update y0</span></div>
<div class="line"><span class="lineno"> 87</span> <a class="code hl_function" href="../../d3/d06/ode__semi__implicit__euler_8cpp.html#abaeae8f62a018d197f0187a1c80a90fe">problem</a>(x, y, dy); <span class="comment">// update dy once more</span></div>
<div class="line"><span class="lineno"> 88</span> </div>
<div class="line"><span class="lineno"> 89</span> dy[0][0] = 0.f; <span class="comment">// ignore y0</span></div>
<div class="line"><span class="lineno"> 90</span> y[0] += dy[0] * dx; <span class="comment">// update remaining using new dy</span></div>
<div class="line"><span class="lineno"> 91</span>}</div>
<div class="ttc" id="aode__semi__implicit__euler_8cpp_html_abaeae8f62a018d197f0187a1c80a90fe"><div class="ttname"><a href="../../d3/d06/ode__semi__implicit__euler_8cpp.html#abaeae8f62a018d197f0187a1c80a90fe">problem</a></div><div class="ttdeci">void problem(const double &amp;x, std::valarray&lt; double &gt; *y, std::valarray&lt; double &gt; *dy)</div><div class="ttdoc">Problem statement for a system with first-order differential equations. Updates the system differenti...</div><div class="ttdef"><b>Definition</b> ode_semi_implicit_euler.cpp:53</div></div>
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