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<div id="projectbrief">Set of algorithms implemented in C++.</div>
<div id="projectbrief">All the algorithms implemented in C++</div>
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@@ -122,22 +139,24 @@ Include dependency graph for ode_forward_euler.cpp:</div>
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<p><a href="../../db/dd3/ode__forward__euler_8cpp_source.html">Go to the source code of this file.</a></p>
<table class="memberdecls">
<tr class="heading"><td colspan="2"><h2 class="groupheader"><a id="func-members" name="func-members"></a>
Functions</h2></td></tr>
<tr class="memitem:abaeae8f62a018d197f0187a1c80a90fe" id="r_abaeae8f62a018d197f0187a1c80a90fe"><td class="memItemLeft" align="right" valign="top">void&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="#abaeae8f62a018d197f0187a1c80a90fe">problem</a> (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="memitem:abaeae8f62a018d197f0187a1c80a90fe" id="r_abaeae8f62a018d197f0187a1c80a90fe"><td class="memItemLeft" align="right" valign="top">void&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="#abaeae8f62a018d197f0187a1c80a90fe">problem</a> (const double &amp;x, std::valarray&lt; double &gt; *y, std::valarray&lt; double &gt; *dy)</td></tr>
<tr class="memdesc:abaeae8f62a018d197f0187a1c80a90fe"><td class="mdescLeft">&#160;</td><td class="mdescRight">Problem statement for a system with first-order differential equations. Updates the system differential variables. <br /></td></tr>
<tr class="separator:abaeae8f62a018d197f0187a1c80a90fe"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:af3adf7b092a87868917ee5fb4255192b" id="r_af3adf7b092a87868917ee5fb4255192b"><td class="memItemLeft" align="right" valign="top">void&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="#af3adf7b092a87868917ee5fb4255192b">exact_solution</a> (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)</td></tr>
<tr class="memitem:af3adf7b092a87868917ee5fb4255192b" id="r_af3adf7b092a87868917ee5fb4255192b"><td class="memItemLeft" align="right" valign="top">void&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="#af3adf7b092a87868917ee5fb4255192b">exact_solution</a> (const double &amp;x, std::valarray&lt; double &gt; *y)</td></tr>
<tr class="memdesc:af3adf7b092a87868917ee5fb4255192b"><td class="mdescLeft">&#160;</td><td class="mdescRight">Exact solution of the problem. Used for solution comparison. <br /></td></tr>
<tr class="separator:af3adf7b092a87868917ee5fb4255192b"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:ga195d23bbdfcb80e83c9cda45c9ad5723" id="r_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="memitem:ga195d23bbdfcb80e83c9cda45c9ad5723" id="r_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, std::valarray&lt; double &gt; *y, std::valarray&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>
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<tr class="memitem:gae0509f8843e2bc42de2abbd00a14b7b9" id="r_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, std::valarray&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:aa13517b8e5de1b75592052db7f7e237f" id="r_aa13517b8e5de1b75592052db7f7e237f"><td class="memItemLeft" align="right" valign="top">void&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="#aa13517b8e5de1b75592052db7f7e237f">save_exact_solution</a> (const double &amp;X0, const double &amp;X_MAX, const double &amp;step_size, const <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; double &gt; &amp;Y0)</td></tr>
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<div class="line">dy[0] = y[1];</div>
<div class="line">dy[1] = mu * (1.f - y[0] * y[0]) * y[1] - y[0];</div>
</div><!-- fragment --> <dl class="section see"><dt>See also</dt><dd><a class="el" href="../../d6/dd3/ode__midpoint__euler_8cpp.html" title="Solve a multivariable first order ordinary differential equation (ODEs) using midpoint Euler method">ode_midpoint_euler.cpp</a>, <a class="el" href="../../d3/d06/ode__semi__implicit__euler_8cpp.html" title="Solve a multivariable first order ordinary differential equation (ODEs) using semi implicit Euler met...">ode_semi_implicit_euler.cpp</a> </dd></dl>
<p class="definition">Definition in file <a class="el" href="../../db/dd3/ode__forward__euler_8cpp_source.html">ode_forward_euler.cpp</a>.</p>
</div><h2 class="groupheader">Function Documentation</h2>
<a id="af3adf7b092a87868917ee5fb4255192b" name="af3adf7b092a87868917ee5fb4255192b"></a>
<h2 class="memtitle"><span class="permalink"><a href="#af3adf7b092a87868917ee5fb4255192b">&#9670;&#160;</a></span>exact_solution()</h2>
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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; *</td> <td class="paramname"><span class="paramname"><em>y</em></span>&#160;)</td>
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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"> 69</span> y[0][1] = -<a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/math/sin.html">std::sin</a>(x);</div>
<div class="line"><span class="lineno"> 70</span>}</div>
<div class="ttc" id="acos_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/numeric/math/cos.html">std::cos</a></div><div class="ttdeci">T cos(T... args)</div></div>
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<p class="definition">Definition at line <a class="el" href="../../db/dd3/ode__forward__euler_8cpp_source.html#l00067">67</a> of file <a class="el" href="../../db/dd3/ode__forward__euler_8cpp_source.html">ode_forward_euler.cpp</a>.</p>
<div class="fragment"><div class="line"><span class="lineno"> 67</span> {</div>
<div class="line"><span class="lineno"> 68</span> y[0][0] = std::cos(x);</div>
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<p>Main Function </p>
<p class="definition">Definition at line <a class="el" href="../../db/dd3/ode__forward__euler_8cpp_source.html#l00189">189</a> of file <a class="el" href="../../db/dd3/ode__forward__euler_8cpp_source.html">ode_forward_euler.cpp</a>.</p>
<div class="fragment"><div class="line"><span class="lineno"> 189</span> {</div>
<div class="line"><span class="lineno"> 190</span> <span class="keywordtype">double</span> X0 = 0.f; <span class="comment">/* initial value of x0 */</span></div>
<div class="line"><span class="lineno"> 191</span> <span class="keywordtype">double</span> X_MAX = 10.F; <span class="comment">/* upper limit of integration */</span></div>
<div class="line"><span class="lineno"> 192</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray&lt;double&gt;</a> Y0{1.f, 0.f}; <span class="comment">/* initial value Y = y(x = x_0) */</span></div>
<div class="line"><span class="lineno"> 192</span> std::valarray&lt;double&gt; Y0{1.f, 0.f}; <span class="comment">/* initial value Y = y(x = x_0) */</span></div>
<div class="line"><span class="lineno"> 193</span> <span class="keywordtype">double</span> step_size = NAN;</div>
<div class="line"><span class="lineno"> 194</span> </div>
<div class="line"><span class="lineno"> 195</span> <span class="keywordflow">if</span> (argc == 1) {</div>
<div class="line"><span class="lineno"> 196</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;\nEnter the step size: &quot;</span>;</div>
<div class="line"><span class="lineno"> 197</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_istream.html">std::cin</a> &gt;&gt; step_size;</div>
<div class="line"><span class="lineno"> 196</span> std::cout &lt;&lt; <span class="stringliteral">&quot;\nEnter the step size: &quot;</span>;</div>
<div class="line"><span class="lineno"> 197</span> std::cin &gt;&gt; step_size;</div>
<div class="line"><span class="lineno"> 198</span> } <span class="keywordflow">else</span> {</div>
<div class="line"><span class="lineno"> 199</span> <span class="comment">// use commandline argument as independent variable step size</span></div>
<div class="line"><span class="lineno"> 200</span> step_size = <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/string/byte/atof.html">std::atof</a>(argv[1]);</div>
<div class="line"><span class="lineno"> 200</span> step_size = std::atof(argv[1]);</div>
<div class="line"><span class="lineno"> 201</span> }</div>
<div class="line"><span class="lineno"> 202</span> </div>
<div class="line"><span class="lineno"> 203</span> <span class="comment">// get approximate solution</span></div>
<div class="line"><span class="lineno"> 204</span> <span class="keywordtype">double</span> total_time = <a class="code hl_function" href="../../d6/d60/group__ode.html#gae0509f8843e2bc42de2abbd00a14b7b9">forward_euler</a>(step_size, X0, X_MAX, &amp;Y0, <span class="keyword">true</span>);</div>
<div class="line"><span class="lineno"> 205</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;\tTime = &quot;</span> &lt;&lt; total_time &lt;&lt; <span class="stringliteral">&quot; ms\n&quot;</span>;</div>
<div class="line"><span class="lineno"> 205</span> std::cout &lt;&lt; <span class="stringliteral">&quot;\tTime = &quot;</span> &lt;&lt; total_time &lt;&lt; <span class="stringliteral">&quot; ms\n&quot;</span>;</div>
<div class="line"><span class="lineno"> 206</span> </div>
<div class="line"><span class="lineno"> 207</span> <span class="comment">/* compute exact solution for comparion */</span></div>
<div class="line"><span class="lineno"> 208</span> <a class="code hl_function" href="#aa13517b8e5de1b75592052db7f7e237f">save_exact_solution</a>(X0, X_MAX, step_size, Y0);</div>
<div class="line"><span class="lineno"> 209</span> </div>
<div class="line"><span class="lineno"> 210</span> <span class="keywordflow">return</span> 0;</div>
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<div class="ttc" id="aatof_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/string/byte/atof.html">std::atof</a></div><div class="ttdeci">T atof(T... args)</div></div>
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<div class="ttc" id="agroup__ode_html_gae0509f8843e2bc42de2abbd00a14b7b9"><div class="ttname"><a href="../../d6/d60/group__ode.html#gae0509f8843e2bc42de2abbd00a14b7b9">forward_euler</a></div><div class="ttdeci">double forward_euler(double dx, double x0, double x_max, std::valarray&lt; double &gt; *y, bool save_to_file=false)</div><div class="ttdoc">Compute approximation using the forward-Euler method in the given limits.</div><div class="ttdef"><b>Definition</b> ode_forward_euler.cpp:102</div></div>
<div class="ttc" id="aode__forward__euler_8cpp_html_aa13517b8e5de1b75592052db7f7e237f"><div class="ttname"><a href="#aa13517b8e5de1b75592052db7f7e237f">save_exact_solution</a></div><div class="ttdeci">void save_exact_solution(const double &amp;X0, const double &amp;X_MAX, const double &amp;step_size, const std::valarray&lt; double &gt; &amp;Y0)</div><div class="ttdef"><b>Definition</b> ode_forward_euler.cpp:153</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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<p class="definition">Definition at line <a class="el" href="../../db/dd3/ode__forward__euler_8cpp_source.html#l00054">54</a> of file <a class="el" href="../../db/dd3/ode__forward__euler_8cpp_source.html">ode_forward_euler.cpp</a>.</p>
<div class="fragment"><div class="line"><span class="lineno"> 55</span> {</div>
<div class="line"><span class="lineno"> 56</span> <span class="keyword">const</span> <span class="keywordtype">double</span> omega = 1.F; <span class="comment">// some const for the problem</span></div>
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<p class="definition">Definition at line <a class="el" href="../../db/dd3/ode__forward__euler_8cpp_source.html#l00153">153</a> of file <a class="el" href="../../db/dd3/ode__forward__euler_8cpp_source.html">ode_forward_euler.cpp</a>.</p>
<div class="fragment"><div class="line"><span class="lineno"> 155</span> {</div>
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<a href="../../db/dd3/ode__forward__euler_8cpp.html">Go to the documentation of this file.</a><div class="fragment"><div class="line"><a id="l00001" name="l00001"></a><span class="lineno"> 1</span> </div>
<div class="line"><a id="l00039" name="l00039"></a><span class="lineno"> 39</span><span class="preprocessor">#include &lt;cmath&gt;</span></div>
<div class="line"><a id="l00040" name="l00040"></a><span class="lineno"> 40</span><span class="preprocessor">#include &lt;ctime&gt;</span></div>
<div class="line"><a id="l00041" name="l00041"></a><span class="lineno"> 41</span><span class="preprocessor">#include &lt;fstream&gt;</span></div>
<div class="line"><a id="l00042" name="l00042"></a><span class="lineno"> 42</span><span class="preprocessor">#include &lt;iostream&gt;</span></div>
<div class="line"><a id="l00043" name="l00043"></a><span class="lineno"> 43</span><span class="preprocessor">#include &lt;valarray&gt;</span></div>
<div class="line"><a id="l00044" name="l00044"></a><span class="lineno"> 44</span> </div>
<div class="foldopen" id="foldopen00054" data-start="{" data-end="}">
<div class="line"><a id="l00054" name="l00054"></a><span class="lineno"><a class="line" href="../../db/dd3/ode__forward__euler_8cpp.html#abaeae8f62a018d197f0187a1c80a90fe"> 54</a></span><span class="keywordtype">void</span> <a class="code hl_function" href="../../db/dd3/ode__forward__euler_8cpp.html#abaeae8f62a018d197f0187a1c80a90fe">problem</a>(<span class="keyword">const</span> <span class="keywordtype">double</span> &amp;x, std::valarray&lt;double&gt; *y,</div>
<div class="line"><a id="l00055" name="l00055"></a><span class="lineno"> 55</span> std::valarray&lt;double&gt; *dy) {</div>
<div class="line"><a id="l00056" name="l00056"></a><span class="lineno"> 56</span> <span class="keyword">const</span> <span class="keywordtype">double</span> omega = 1.F; <span class="comment">// some const for the problem</span></div>
<div class="line"><a id="l00057" name="l00057"></a><span class="lineno"> 57</span> (*dy)[0] = (*y)[1]; <span class="comment">// x dot // NOLINT</span></div>
<div class="line"><a id="l00058" name="l00058"></a><span class="lineno"> 58</span> (*dy)[1] = -omega * omega * (*y)[0]; <span class="comment">// y dot // NOLINT</span></div>
<div class="line"><a id="l00059" name="l00059"></a><span class="lineno"> 59</span>}</div>
</div>
<div class="line"><a id="l00060" name="l00060"></a><span class="lineno"> 60</span> </div>
<div class="foldopen" id="foldopen00067" data-start="{" data-end="}">
<div class="line"><a id="l00067" name="l00067"></a><span class="lineno"><a class="line" href="../../db/dd3/ode__forward__euler_8cpp.html#af3adf7b092a87868917ee5fb4255192b"> 67</a></span><span class="keywordtype">void</span> <a class="code hl_function" href="../../db/dd3/ode__forward__euler_8cpp.html#af3adf7b092a87868917ee5fb4255192b">exact_solution</a>(<span class="keyword">const</span> <span class="keywordtype">double</span> &amp;x, std::valarray&lt;double&gt; *y) {</div>
<div class="line"><a id="l00068" name="l00068"></a><span class="lineno"> 68</span> y[0][0] = std::cos(x);</div>
<div class="line"><a id="l00069" name="l00069"></a><span class="lineno"> 69</span> y[0][1] = -std::sin(x);</div>
<div class="line"><a id="l00070" name="l00070"></a><span class="lineno"> 70</span>}</div>
</div>
<div class="line"><a id="l00071" name="l00071"></a><span class="lineno"> 71</span> </div>
<div class="foldopen" id="foldopen00086" data-start="{" data-end="}">
<div class="line"><a id="l00086" name="l00086"></a><span class="lineno"><a class="line" href="../../d6/d60/group__ode.html#ga195d23bbdfcb80e83c9cda45c9ad5723"> 86</a></span><span class="keywordtype">void</span> <a class="code hl_function" href="../../d6/d60/group__ode.html#ga195d23bbdfcb80e83c9cda45c9ad5723">forward_euler_step</a>(<span class="keyword">const</span> <span class="keywordtype">double</span> dx, <span class="keyword">const</span> <span class="keywordtype">double</span> x,</div>
<div class="line"><a id="l00087" name="l00087"></a><span class="lineno"> 87</span> std::valarray&lt;double&gt; *y, std::valarray&lt;double&gt; *dy) {</div>
<div class="line"><a id="l00088" name="l00088"></a><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"><a id="l00089" name="l00089"></a><span class="lineno"> 89</span> *y += *dy * dx;</div>
<div class="line"><a id="l00090" name="l00090"></a><span class="lineno"> 90</span>}</div>
</div>
<div class="line"><a id="l00091" name="l00091"></a><span class="lineno"> 91</span> </div>
<div class="foldopen" id="foldopen00102" data-start="{" data-end="}">
<div class="line"><a id="l00102" name="l00102"></a><span class="lineno"><a class="line" href="../../d6/d60/group__ode.html#gae0509f8843e2bc42de2abbd00a14b7b9"> 102</a></span><span class="keywordtype">double</span> <a class="code hl_function" href="../../d6/d60/group__ode.html#gae0509f8843e2bc42de2abbd00a14b7b9">forward_euler</a>(<span class="keywordtype">double</span> dx, <span class="keywordtype">double</span> x0, <span class="keywordtype">double</span> x_max,</div>
<div class="line"><a id="l00103" name="l00103"></a><span class="lineno"> 103</span> std::valarray&lt;double&gt; *y, <span class="keywordtype">bool</span> save_to_file = <span class="keyword">false</span>) {</div>
<div class="line"><a id="l00104" name="l00104"></a><span class="lineno"> 104</span> std::valarray&lt;double&gt; dy = *y;</div>
<div class="line"><a id="l00105" name="l00105"></a><span class="lineno"> 105</span> </div>
<div class="line"><a id="l00106" name="l00106"></a><span class="lineno"> 106</span> std::ofstream fp;</div>
<div class="line"><a id="l00107" name="l00107"></a><span class="lineno"> 107</span> <span class="keywordflow">if</span> (save_to_file) {</div>
<div class="line"><a id="l00108" name="l00108"></a><span class="lineno"> 108</span> fp.open(<span class="stringliteral">&quot;forward_euler.csv&quot;</span>, std::ofstream::out);</div>
<div class="line"><a id="l00109" name="l00109"></a><span class="lineno"> 109</span> <span class="keywordflow">if</span> (!fp.is_open()) {</div>
<div class="line"><a id="l00110" name="l00110"></a><span class="lineno"> 110</span> std::perror(<span class="stringliteral">&quot;Error! &quot;</span>);</div>
<div class="line"><a id="l00111" name="l00111"></a><span class="lineno"> 111</span> }</div>
<div class="line"><a id="l00112" name="l00112"></a><span class="lineno"> 112</span> }</div>
<div class="line"><a id="l00113" name="l00113"></a><span class="lineno"> 113</span> </div>
<div class="line"><a id="l00114" name="l00114"></a><span class="lineno"> 114</span> std::size_t L = y-&gt;size();</div>
<div class="line"><a id="l00115" name="l00115"></a><span class="lineno"> 115</span> </div>
<div class="line"><a id="l00116" name="l00116"></a><span class="lineno"> 116</span> <span class="comment">/* start integration */</span></div>
<div class="line"><a id="l00117" name="l00117"></a><span class="lineno"> 117</span> std::clock_t t1 = std::clock();</div>
<div class="line"><a id="l00118" name="l00118"></a><span class="lineno"> 118</span> <span class="keywordtype">double</span> x = x0;</div>
<div class="line"><a id="l00119" name="l00119"></a><span class="lineno"> 119</span> </div>
<div class="line"><a id="l00120" name="l00120"></a><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"><a id="l00121" name="l00121"></a><span class="lineno"> 121</span> <span class="keywordflow">if</span> (save_to_file &amp;&amp; fp.is_open()) {</div>
<div class="line"><a id="l00122" name="l00122"></a><span class="lineno"> 122</span> <span class="comment">// write to file</span></div>
<div class="line"><a id="l00123" name="l00123"></a><span class="lineno"> 123</span> fp &lt;&lt; x &lt;&lt; <span class="stringliteral">&quot;,&quot;</span>;</div>
<div class="line"><a id="l00124" name="l00124"></a><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"><a id="l00125" name="l00125"></a><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"><a id="l00126" name="l00126"></a><span class="lineno"> 126</span> }</div>
<div class="line"><a id="l00127" name="l00127"></a><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"><a id="l00128" name="l00128"></a><span class="lineno"> 128</span> }</div>
<div class="line"><a id="l00129" name="l00129"></a><span class="lineno"> 129</span> </div>
<div class="line"><a id="l00130" name="l00130"></a><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"><a id="l00131" name="l00131"></a><span class="lineno"> 131</span> x += dx; <span class="comment">// update step</span></div>
<div class="line"><a id="l00132" name="l00132"></a><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"><a id="l00133" name="l00133"></a><span class="lineno"> 133</span> <span class="comment">/* end of integration */</span></div>
<div class="line"><a id="l00134" name="l00134"></a><span class="lineno"> 134</span> std::clock_t t2 = std::clock();</div>
<div class="line"><a id="l00135" name="l00135"></a><span class="lineno"> 135</span> </div>
<div class="line"><a id="l00136" name="l00136"></a><span class="lineno"> 136</span> <span class="keywordflow">if</span> (fp.is_open()) {</div>
<div class="line"><a id="l00137" name="l00137"></a><span class="lineno"> 137</span> fp.close();</div>
<div class="line"><a id="l00138" name="l00138"></a><span class="lineno"> 138</span> }</div>
<div class="line"><a id="l00139" name="l00139"></a><span class="lineno"> 139</span> </div>
<div class="line"><a id="l00140" name="l00140"></a><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"><a id="l00141" name="l00141"></a><span class="lineno"> 141</span>}</div>
</div>
<div class="line"><a id="l00142" name="l00142"></a><span class="lineno"> 142</span> </div>
<div class="foldopen" id="foldopen00153" data-start="{" data-end="}">
<div class="line"><a id="l00153" name="l00153"></a><span class="lineno"><a class="line" href="../../db/dd3/ode__forward__euler_8cpp.html#aa13517b8e5de1b75592052db7f7e237f"> 153</a></span><span class="keywordtype">void</span> <a class="code hl_function" href="../../db/dd3/ode__forward__euler_8cpp.html#aa13517b8e5de1b75592052db7f7e237f">save_exact_solution</a>(<span class="keyword">const</span> <span class="keywordtype">double</span> &amp;X0, <span class="keyword">const</span> <span class="keywordtype">double</span> &amp;X_MAX,</div>
<div class="line"><a id="l00154" name="l00154"></a><span class="lineno"> 154</span> <span class="keyword">const</span> <span class="keywordtype">double</span> &amp;step_size,</div>
<div class="line"><a id="l00155" name="l00155"></a><span class="lineno"> 155</span> <span class="keyword">const</span> std::valarray&lt;double&gt; &amp;Y0) {</div>
<div class="line"><a id="l00156" name="l00156"></a><span class="lineno"> 156</span> <span class="keywordtype">double</span> x = X0;</div>
<div class="line"><a id="l00157" name="l00157"></a><span class="lineno"> 157</span> std::valarray&lt;double&gt; y(Y0);</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> std::ofstream fp(<span class="stringliteral">&quot;exact.csv&quot;</span>, std::ostream::out);</div>
<div class="line"><a id="l00160" name="l00160"></a><span class="lineno"> 160</span> <span class="keywordflow">if</span> (!fp.is_open()) {</div>
<div class="line"><a id="l00161" name="l00161"></a><span class="lineno"> 161</span> std::perror(<span class="stringliteral">&quot;Error! &quot;</span>);</div>
<div class="line"><a id="l00162" name="l00162"></a><span class="lineno"> 162</span> <span class="keywordflow">return</span>;</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> std::cout &lt;&lt; <span class="stringliteral">&quot;Finding exact solution\n&quot;</span>;</div>
<div class="line"><a id="l00165" name="l00165"></a><span class="lineno"> 165</span> </div>
<div class="line"><a id="l00166" name="l00166"></a><span class="lineno"> 166</span> std::clock_t t1 = std::clock();</div>
<div class="line"><a id="l00167" name="l00167"></a><span class="lineno"> 167</span> <span class="keywordflow">do</span> {</div>
<div class="line"><a id="l00168" name="l00168"></a><span class="lineno"> 168</span> fp &lt;&lt; x &lt;&lt; <span class="stringliteral">&quot;,&quot;</span>;</div>
<div class="line"><a id="l00169" name="l00169"></a><span class="lineno"> 169</span> <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i &lt; y.size() - 1; i++) {</div>
<div class="line"><a id="l00170" name="l00170"></a><span class="lineno"> 170</span> fp &lt;&lt; y[i] &lt;&lt; <span class="stringliteral">&quot;,&quot;</span>; <span class="comment">// NOLINT</span></div>
<div class="line"><a id="l00171" name="l00171"></a><span class="lineno"> 171</span> }</div>
<div class="line"><a id="l00172" name="l00172"></a><span class="lineno"> 172</span> fp &lt;&lt; y[y.size() - 1] &lt;&lt; <span class="stringliteral">&quot;\n&quot;</span>; <span class="comment">// NOLINT</span></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_function" href="../../db/dd3/ode__forward__euler_8cpp.html#af3adf7b092a87868917ee5fb4255192b">exact_solution</a>(x, &amp;y);</div>
<div class="line"><a id="l00175" name="l00175"></a><span class="lineno"> 175</span> </div>
<div class="line"><a id="l00176" name="l00176"></a><span class="lineno"> 176</span> x += step_size;</div>
<div class="line"><a id="l00177" name="l00177"></a><span class="lineno"> 177</span> } <span class="keywordflow">while</span> (x &lt;= X_MAX);</div>
<div class="line"><a id="l00178" name="l00178"></a><span class="lineno"> 178</span> </div>
<div class="line"><a id="l00179" name="l00179"></a><span class="lineno"> 179</span> std::clock_t t2 = std::clock();</div>
<div class="line"><a id="l00180" name="l00180"></a><span class="lineno"> 180</span> <span class="keywordtype">double</span> total_time = <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"><a id="l00181" name="l00181"></a><span class="lineno"> 181</span> std::cout &lt;&lt; <span class="stringliteral">&quot;\tTime = &quot;</span> &lt;&lt; total_time &lt;&lt; <span class="stringliteral">&quot; ms\n&quot;</span>;</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> fp.close();</div>
<div class="line"><a id="l00184" name="l00184"></a><span class="lineno"> 184</span>}</div>
</div>
<div class="line"><a id="l00185" name="l00185"></a><span class="lineno"> 185</span> </div>
<div class="foldopen" id="foldopen00189" data-start="{" data-end="}">
<div class="line"><a id="l00189" name="l00189"></a><span class="lineno"><a class="line" href="../../db/dd3/ode__forward__euler_8cpp.html#a0ddf1224851353fc92bfbff6f499fa97"> 189</a></span><span class="keywordtype">int</span> <a class="code hl_function" href="../../dd/d1e/generate__parentheses_8cpp.html#gae66f6b31b5ad750f1fe042a706a4e3d4">main</a>(<span class="keywordtype">int</span> argc, <span class="keywordtype">char</span> *argv[]) {</div>
<div class="line"><a id="l00190" name="l00190"></a><span class="lineno"> 190</span> <span class="keywordtype">double</span> X0 = 0.f; <span class="comment">/* initial value of x0 */</span></div>
<div class="line"><a id="l00191" name="l00191"></a><span class="lineno"> 191</span> <span class="keywordtype">double</span> X_MAX = 10.F; <span class="comment">/* upper limit of integration */</span></div>
<div class="line"><a id="l00192" name="l00192"></a><span class="lineno"> 192</span> std::valarray&lt;double&gt; Y0{1.f, 0.f}; <span class="comment">/* initial value Y = y(x = x_0) */</span></div>
<div class="line"><a id="l00193" name="l00193"></a><span class="lineno"> 193</span> <span class="keywordtype">double</span> step_size = NAN;</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="keywordflow">if</span> (argc == 1) {</div>
<div class="line"><a id="l00196" name="l00196"></a><span class="lineno"> 196</span> std::cout &lt;&lt; <span class="stringliteral">&quot;\nEnter the step size: &quot;</span>;</div>
<div class="line"><a id="l00197" name="l00197"></a><span class="lineno"> 197</span> std::cin &gt;&gt; step_size;</div>
<div class="line"><a id="l00198" name="l00198"></a><span class="lineno"> 198</span> } <span class="keywordflow">else</span> {</div>
<div class="line"><a id="l00199" name="l00199"></a><span class="lineno"> 199</span> <span class="comment">// use commandline argument as independent variable step size</span></div>
<div class="line"><a id="l00200" name="l00200"></a><span class="lineno"> 200</span> step_size = std::atof(argv[1]);</div>
<div class="line"><a id="l00201" name="l00201"></a><span class="lineno"> 201</span> }</div>
<div class="line"><a id="l00202" name="l00202"></a><span class="lineno"> 202</span> </div>
<div class="line"><a id="l00203" name="l00203"></a><span class="lineno"> 203</span> <span class="comment">// get approximate solution</span></div>
<div class="line"><a id="l00204" name="l00204"></a><span class="lineno"> 204</span> <span class="keywordtype">double</span> total_time = <a class="code hl_function" href="../../d6/d60/group__ode.html#gae0509f8843e2bc42de2abbd00a14b7b9">forward_euler</a>(step_size, X0, X_MAX, &amp;Y0, <span class="keyword">true</span>);</div>
<div class="line"><a id="l00205" name="l00205"></a><span class="lineno"> 205</span> std::cout &lt;&lt; <span class="stringliteral">&quot;\tTime = &quot;</span> &lt;&lt; total_time &lt;&lt; <span class="stringliteral">&quot; ms\n&quot;</span>;</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> <span class="comment">/* compute exact solution for comparion */</span></div>
<div class="line"><a id="l00208" name="l00208"></a><span class="lineno"> 208</span> <a class="code hl_function" href="../../db/dd3/ode__forward__euler_8cpp.html#aa13517b8e5de1b75592052db7f7e237f">save_exact_solution</a>(X0, X_MAX, step_size, Y0);</div>
<div class="line"><a id="l00209" name="l00209"></a><span class="lineno"> 209</span> </div>
<div class="line"><a id="l00210" name="l00210"></a><span class="lineno"> 210</span> <span class="keywordflow">return</span> 0;</div>
<div class="line"><a id="l00211" name="l00211"></a><span class="lineno"> 211</span>}</div>
</div>
<div class="ttc" id="agenerate__parentheses_8cpp_html_gae66f6b31b5ad750f1fe042a706a4e3d4"><div class="ttname"><a href="../../dd/d1e/generate__parentheses_8cpp.html#gae66f6b31b5ad750f1fe042a706a4e3d4">main</a></div><div class="ttdeci">int main()</div><div class="ttdoc">Main function.</div><div class="ttdef"><b>Definition</b> <a href="../../dd/d1e/generate__parentheses_8cpp_source.html#l00110">generate_parentheses.cpp:110</a></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> <a href="#l00086">ode_forward_euler.cpp:86</a></div></div>
<div class="ttc" id="agroup__ode_html_gae0509f8843e2bc42de2abbd00a14b7b9"><div class="ttname"><a href="../../d6/d60/group__ode.html#gae0509f8843e2bc42de2abbd00a14b7b9">forward_euler</a></div><div class="ttdeci">double forward_euler(double dx, double x0, double x_max, std::valarray&lt; double &gt; *y, bool save_to_file=false)</div><div class="ttdoc">Compute approximation using the forward-Euler method in the given limits.</div><div class="ttdef"><b>Definition</b> <a href="#l00102">ode_forward_euler.cpp:102</a></div></div>
<div class="ttc" id="aode__forward__euler_8cpp_html_aa13517b8e5de1b75592052db7f7e237f"><div class="ttname"><a href="../../db/dd3/ode__forward__euler_8cpp.html#aa13517b8e5de1b75592052db7f7e237f">save_exact_solution</a></div><div class="ttdeci">void save_exact_solution(const double &amp;X0, const double &amp;X_MAX, const double &amp;step_size, const std::valarray&lt; double &gt; &amp;Y0)</div><div class="ttdef"><b>Definition</b> <a href="#l00153">ode_forward_euler.cpp:153</a></div></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> <a href="#l00054">ode_forward_euler.cpp:54</a></div></div>
<div class="ttc" id="aode__forward__euler_8cpp_html_af3adf7b092a87868917ee5fb4255192b"><div class="ttname"><a href="../../db/dd3/ode__forward__euler_8cpp.html#af3adf7b092a87868917ee5fb4255192b">exact_solution</a></div><div class="ttdeci">void exact_solution(const double &amp;x, std::valarray&lt; double &gt; *y)</div><div class="ttdoc">Exact solution of the problem. Used for solution comparison.</div><div class="ttdef"><b>Definition</b> <a href="#l00067">ode_forward_euler.cpp:67</a></div></div>
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