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<div class="headertitle"><div class="title">ode_semi_implicit_euler.cpp File Reference</div></div>
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<p>Solve a multivariable first order <a href="https://en.wikipedia.org/wiki/Ordinary_differential_equation" target="_blank">ordinary differential equation (ODEs)</a> using <a href="https://en.wikipedia.org/wiki/Semi-implicit_Euler_method" target="_blank">semi implicit Euler method</a>
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<a href="#details">More...</a></p>
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<div class="textblock"><code>#include <cmath></code><br />
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<code>#include <ctime></code><br />
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<code>#include <fstream></code><br />
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<code>#include <iostream></code><br />
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<code>#include <valarray></code><br />
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</div><div class="textblock"><div class="dynheader">
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Include dependency graph for ode_semi_implicit_euler.cpp:</div>
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<div class="dyncontent">
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<div class="center"><iframe scrolling="no" loading="lazy" frameborder="0" src="../../d6/db4/ode__semi__implicit__euler_8cpp__incl.svg" width="423" height="126"><p><b>This browser is not able to show SVG: try Firefox, Chrome, Safari, or Opera instead.</b></p></iframe></div>
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<p><a href="../../d3/d06/ode__semi__implicit__euler_8cpp_source.html">Go to the source code of this file.</a></p>
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<table class="memberdecls">
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<tr class="heading"><td colspan="2"><h2 id="header-func-members" class="groupheader"><a id="func-members" name="func-members"></a>
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Functions</h2></td></tr>
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<tr class="memitem:abaeae8f62a018d197f0187a1c80a90fe" id="r_abaeae8f62a018d197f0187a1c80a90fe"><td class="memItemLeft" align="right" valign="top">void </td><td class="memItemRight" valign="bottom"><a class="el" href="#abaeae8f62a018d197f0187a1c80a90fe">problem</a> (const double &x, std::valarray< double > *y, std::valarray< double > *dy)</td></tr>
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<tr class="memdesc:abaeae8f62a018d197f0187a1c80a90fe"><td class="mdescLeft"> </td><td class="mdescRight">Problem statement for a system with first-order differential equations. Updates the system differential variables. <br /></td></tr>
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<tr class="memitem:af3adf7b092a87868917ee5fb4255192b" id="r_af3adf7b092a87868917ee5fb4255192b"><td class="memItemLeft" align="right" valign="top">void </td><td class="memItemRight" valign="bottom"><a class="el" href="#af3adf7b092a87868917ee5fb4255192b">exact_solution</a> (const double &x, std::valarray< double > *y)</td></tr>
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<tr class="memdesc:af3adf7b092a87868917ee5fb4255192b"><td class="mdescLeft"> </td><td class="mdescRight">Exact solution of the problem. Used for solution comparison. <br /></td></tr>
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<tr class="memitem:ga827bf009831ddc477c5fa8891d5cb35f" id="r_ga827bf009831ddc477c5fa8891d5cb35f"><td class="memItemLeft" align="right" valign="top">void </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 &x, std::valarray< double > *y, std::valarray< double > *dy)</td></tr>
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<tr class="memdesc:ga827bf009831ddc477c5fa8891d5cb35f"><td class="mdescLeft"> </td><td class="mdescRight">Compute next step approximation using the semi-implicit-Euler method. <br /></td></tr>
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<tr class="memitem:ga3874d294ab00fe9ce8731c5b7991a089" id="r_ga3874d294ab00fe9ce8731c5b7991a089"><td class="memItemLeft" align="right" valign="top">double </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, std::valarray< double > *y, bool save_to_file=false)</td></tr>
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<tr class="memdesc:ga3874d294ab00fe9ce8731c5b7991a089"><td class="mdescLeft"> </td><td class="mdescRight">Compute approximation using the semi-implicit-Euler method in the given limits. <br /></td></tr>
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<tr class="memitem:aa13517b8e5de1b75592052db7f7e237f" id="r_aa13517b8e5de1b75592052db7f7e237f"><td class="memItemLeft" align="right" valign="top">void </td><td class="memItemRight" valign="bottom"><a class="el" href="#aa13517b8e5de1b75592052db7f7e237f">save_exact_solution</a> (const double &X0, const double &X_MAX, const double &step_size, const std::valarray< double > &Y0)</td></tr>
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<tr class="memitem:a0ddf1224851353fc92bfbff6f499fa97" id="r_a0ddf1224851353fc92bfbff6f499fa97"><td class="memItemLeft" align="right" valign="top">int </td><td class="memItemRight" valign="bottom"><a class="el" href="#a0ddf1224851353fc92bfbff6f499fa97">main</a> (int argc, char *argv[])</td></tr>
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</table>
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<a name="details" id="details"></a><h2 id="header-details" class="groupheader">Detailed Description</h2>
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<div class="textblock"><p>Solve a multivariable first order <a href="https://en.wikipedia.org/wiki/Ordinary_differential_equation" target="_blank">ordinary differential equation (ODEs)</a> using <a href="https://en.wikipedia.org/wiki/Semi-implicit_Euler_method" target="_blank">semi implicit Euler method</a> </p>
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<dl class="section author"><dt>Authors</dt><dd><a href="https://github.com/kvedala" target="_blank">Krishna Vedala</a> </dd></dl>
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<p>The ODE being solved is: </p><p class="formulaDsp">
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\begin{eqnarray*} \dot{u} &=& v\\
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\dot{v} &=& -\omega^2 u\\
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\omega &=& 1\\
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[x_0, u_0, v_0] &=& [0,1,0]\qquad\ldots\text{(initial values)}
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\end{eqnarray*}
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</p>
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<p> The exact solution for the above problem is: </p><p class="formulaDsp">
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\begin{eqnarray*} u(x) &=& \cos(x)\\
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v(x) &=& -\sin(x)\\
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\end{eqnarray*}
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</p>
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<p> The computation results are stored to a text file <span class="tt">semi_implicit_euler.csv</span> and the exact soltuion results in <span class="tt">exact.csv</span> for comparison. <img src="https://raw.githubusercontent.com/TheAlgorithms/C-Plus-Plus/docs/images/numerical_methods/ode_semi_implicit_euler.svg" alt="Implementation solution" style="pointer-events: none;" class="inline"/></p>
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<p>To implement <a href="https://en.wikipedia.org/wiki/Van_der_Pol_oscillator" target="_blank">Van der Pol oscillator</a>, change the <a class="el" href="#abaeae8f62a018d197f0187a1c80a90fe" title="Problem statement for a system with first-order differential equations. Updates the system differenti...">problem</a> function to: </p><div class="fragment"><div class="line"><span class="keyword">const</span> <span class="keywordtype">double</span> mu = 2.0;</div>
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<div class="line">dy[0] = y[1];</div>
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<div class="line">dy[1] = mu * (1.f - y[0] * y[0]) * y[1] - y[0];</div>
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</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="../../db/dd3/ode__forward__euler_8cpp.html" title="Solve a multivariable first order ordinary differential equation (ODEs) using forward Euler method">ode_forward_euler.cpp</a> </dd></dl>
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<p class="definition">Definition in file <a class="el" href="../../d3/d06/ode__semi__implicit__euler_8cpp_source.html">ode_semi_implicit_euler.cpp</a>.</p>
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</div><a name="doc-func-members" id="doc-func-members"></a><h2 id="header-doc-func-members" class="groupheader">Function Documentation</h2>
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<a id="af3adf7b092a87868917ee5fb4255192b" name="af3adf7b092a87868917ee5fb4255192b"></a>
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<h2 class="memtitle"><span class="permalink"><a href="#af3adf7b092a87868917ee5fb4255192b">◆ </a></span>exact_solution()</h2>
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<div class="memitem">
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<div class="memproto">
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<table class="memname">
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<tr>
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<td class="memname">void exact_solution </td>
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<td>(</td>
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<td class="paramtype">const double &</td> <td class="paramname"><span class="paramname"><em>x</em></span>, </td>
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</tr>
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<tr>
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<td class="paramkey"></td>
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<td></td>
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<td class="paramtype">std::valarray< double > *</td> <td class="paramname"><span class="paramname"><em>y</em></span> )</td>
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</tr>
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</table>
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</div><div class="memdoc">
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<p>Exact solution of the problem. Used for solution comparison. </p>
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<dl class="params"><dt>Parameters</dt><dd>
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<table class="params">
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<tr><td class="paramdir">[in]</td><td class="paramname">x</td><td>independent variable </td></tr>
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<tr><td class="paramdir">[in,out]</td><td class="paramname">y</td><td>dependent variable </td></tr>
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</table>
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</dd>
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</dl>
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<p class="definition">Definition at line <a class="el" href="../../d3/d06/ode__semi__implicit__euler_8cpp_source.html#l00066">66</a> of file <a class="el" href="../../d3/d06/ode__semi__implicit__euler_8cpp_source.html">ode_semi_implicit_euler.cpp</a>.</p>
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<div class="fragment"><div class="line"><span class="lineno"> 66</span> {</div>
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<div class="line"><span class="lineno"> 67</span> y[0][0] = std::cos(x);</div>
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<div class="line"><span class="lineno"> 68</span> y[0][1] = -std::sin(x);</div>
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<div class="line"><span class="lineno"> 69</span>}</div>
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</div><!-- fragment -->
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</div>
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</div>
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<a id="a0ddf1224851353fc92bfbff6f499fa97" name="a0ddf1224851353fc92bfbff6f499fa97"></a>
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<h2 class="memtitle"><span class="permalink"><a href="#a0ddf1224851353fc92bfbff6f499fa97">◆ </a></span>main()</h2>
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<div class="memitem">
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<div class="memproto">
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<table class="memname">
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<tr>
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<td class="memname">int main </td>
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<td>(</td>
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<td class="paramtype">int</td> <td class="paramname"><span class="paramname"><em>argc</em></span>, </td>
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</tr>
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<tr>
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<td class="paramkey"></td>
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<td></td>
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<td class="paramtype">char *</td> <td class="paramname"><span class="paramname"><em>argv</em></span>[] )</td>
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</tr>
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</table>
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</div><div class="memdoc">
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<p>Main Function </p>
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<p class="definition">Definition at line <a class="el" href="../../d3/d06/ode__semi__implicit__euler_8cpp_source.html#l00189">189</a> of file <a class="el" href="../../d3/d06/ode__semi__implicit__euler_8cpp_source.html">ode_semi_implicit_euler.cpp</a>.</p>
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<div class="fragment"><div class="line"><span class="lineno"> 189</span> {</div>
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<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>
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<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>
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<div class="line"><span class="lineno"> 192</span> std::valarray<double> Y0 = {1.f, 0.f}; <span class="comment">/* initial value Y = y(x = x_0) */</span></div>
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<div class="line"><span class="lineno"> 193</span> <span class="keywordtype">double</span> step_size;</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="keywordflow">if</span> (argc == 1) {</div>
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<div class="line"><span class="lineno"> 196</span> std::cout << <span class="stringliteral">"\nEnter the step size: "</span>;</div>
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<div class="line"><span class="lineno"> 197</span> std::cin >> step_size;</div>
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<div class="line"><span class="lineno"> 198</span> } <span class="keywordflow">else</span> {</div>
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<div class="line"><span class="lineno"> 199</span> <span class="comment">// use commandline argument as independent variable step size</span></div>
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<div class="line"><span class="lineno"> 200</span> step_size = std::atof(argv[1]);</div>
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<div class="line"><span class="lineno"> 201</span> }</div>
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<div class="line"><span class="lineno"> 202</span> </div>
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<div class="line"><span class="lineno"> 203</span> <span class="comment">// get approximate solution</span></div>
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<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#ga3874d294ab00fe9ce8731c5b7991a089">semi_implicit_euler</a>(step_size, X0, X_MAX, &Y0, <span class="keyword">true</span>);</div>
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<div class="line"><span class="lineno"> 205</span> std::cout << <span class="stringliteral">"\tTime = "</span> << total_time << <span class="stringliteral">" ms\n"</span>;</div>
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<div class="line"><span class="lineno"> 206</span> </div>
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<div class="line"><span class="lineno"> 207</span> <span class="comment">/* compute exact solution for comparion */</span></div>
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<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>
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<div class="line"><span class="lineno"> 209</span> </div>
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<div class="line"><span class="lineno"> 210</span> <span class="keywordflow">return</span> 0;</div>
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<div class="line"><span class="lineno"> 211</span>}</div>
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<div class="ttc" id="agroup__ode_html_ga3874d294ab00fe9ce8731c5b7991a089"><div class="ttname"><a href="../../d6/d60/group__ode.html#ga3874d294ab00fe9ce8731c5b7991a089">semi_implicit_euler</a></div><div class="ttdeci">double semi_implicit_euler(double dx, double x0, double x_max, std::valarray< double > *y, bool save_to_file=false)</div><div class="ttdoc">Compute approximation using the semi-implicit-Euler method in the given limits.</div><div class="ttdef"><b>Definition</b> <a href="../../d3/d06/ode__semi__implicit__euler_8cpp_source.html#l00103">ode_semi_implicit_euler.cpp:103</a></div></div>
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<div class="ttc" id="aode__semi__implicit__euler_8cpp_html_aa13517b8e5de1b75592052db7f7e237f"><div class="ttname"><a href="#aa13517b8e5de1b75592052db7f7e237f">save_exact_solution</a></div><div class="ttdeci">void save_exact_solution(const double &X0, const double &X_MAX, const double &step_size, const std::valarray< double > &Y0)</div><div class="ttdef"><b>Definition</b> <a href="../../d3/d06/ode__semi__implicit__euler_8cpp_source.html#l00153">ode_semi_implicit_euler.cpp:153</a></div></div>
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</div><!-- fragment -->
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</div>
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<h2 class="memtitle"><span class="permalink"><a href="#abaeae8f62a018d197f0187a1c80a90fe">◆ </a></span>problem()</h2>
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<td class="memname">void problem </td>
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<td>(</td>
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<td class="paramtype">const double &</td> <td class="paramname"><span class="paramname"><em>x</em></span>, </td>
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<td class="paramkey"></td>
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<td class="paramtype">std::valarray< double > *</td> <td class="paramname"><span class="paramname"><em>y</em></span>, </td>
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<td class="paramtype">std::valarray< double > *</td> <td class="paramname"><span class="paramname"><em>dy</em></span> )</td>
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<p>Problem statement for a system with first-order differential equations. Updates the system differential variables. </p>
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<dl class="section note"><dt>Note</dt><dd>This function can be updated to and ode of any order.</dd></dl>
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<dl class="params"><dt>Parameters</dt><dd>
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<table class="params">
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<tr><td class="paramdir">[in]</td><td class="paramname">x</td><td>independent variable(s) </td></tr>
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<tr><td class="paramdir">[in,out]</td><td class="paramname">y</td><td>dependent variable(s) </td></tr>
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<tr><td class="paramdir">[in,out]</td><td class="paramname">dy</td><td>first-derivative of dependent variable(s) </td></tr>
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</dd>
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</dl>
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<p class="definition">Definition at line <a class="el" href="../../d3/d06/ode__semi__implicit__euler_8cpp_source.html#l00053">53</a> of file <a class="el" href="../../d3/d06/ode__semi__implicit__euler_8cpp_source.html">ode_semi_implicit_euler.cpp</a>.</p>
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<div class="fragment"><div class="line"><span class="lineno"> 54</span> {</div>
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<div class="line"><span class="lineno"> 55</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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<div class="line"><span class="lineno"> 56</span> dy[0][0] = y[0][1]; <span class="comment">// x dot</span></div>
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<div class="line"><span class="lineno"> 57</span> dy[0][1] = -omega * omega * y[0][0]; <span class="comment">// y dot</span></div>
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<div class="line"><span class="lineno"> 58</span>}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#aa13517b8e5de1b75592052db7f7e237f">◆ </a></span>save_exact_solution()</h2>
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<td class="memname">void save_exact_solution </td>
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<td>(</td>
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<td class="paramtype">const double &</td> <td class="paramname"><span class="paramname"><em>X0</em></span>, </td>
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<td class="paramtype">const double &</td> <td class="paramname"><span class="paramname"><em>X_MAX</em></span>, </td>
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<td class="paramtype">const double &</td> <td class="paramname"><span class="paramname"><em>step_size</em></span>, </td>
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<td class="paramtype">const std::valarray< double > &</td> <td class="paramname"><span class="paramname"><em>Y0</em></span> )</td>
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<p>Function to compute and save exact solution for comparison</p>
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<dl class="params"><dt>Parameters</dt><dd>
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<table class="params">
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<tr><td class="paramdir">[in]</td><td class="paramname">X0</td><td>initial value of independent variable </td></tr>
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<tr><td class="paramdir">[in]</td><td class="paramname">X_MAX</td><td>final value of independent variable </td></tr>
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<tr><td class="paramdir">[in]</td><td class="paramname">step_size</td><td>independent variable step size </td></tr>
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<tr><td class="paramdir">[in]</td><td class="paramname">Y0</td><td>initial values of dependent variables </td></tr>
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</table>
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<p class="definition">Definition at line <a class="el" href="../../d3/d06/ode__semi__implicit__euler_8cpp_source.html#l00153">153</a> of file <a class="el" href="../../d3/d06/ode__semi__implicit__euler_8cpp_source.html">ode_semi_implicit_euler.cpp</a>.</p>
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<div class="fragment"><div class="line"><span class="lineno"> 155</span> {</div>
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<div class="line"><span class="lineno"> 156</span> <span class="keywordtype">double</span> x = X0;</div>
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<div class="line"><span class="lineno"> 157</span> std::valarray<double> y = Y0;</div>
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<div class="line"><span class="lineno"> 158</span> </div>
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<div class="line"><span class="lineno"> 159</span> std::ofstream fp(<span class="stringliteral">"exact.csv"</span>, std::ostream::out);</div>
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<div class="line"><span class="lineno"> 160</span> <span class="keywordflow">if</span> (!fp.is_open()) {</div>
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<div class="line"><span class="lineno"> 161</span> std::perror(<span class="stringliteral">"Error! "</span>);</div>
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<div class="line"><span class="lineno"> 162</span> <span class="keywordflow">return</span>;</div>
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<div class="line"><span class="lineno"> 163</span> }</div>
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<div class="line"><span class="lineno"> 164</span> std::cout << <span class="stringliteral">"Finding exact solution\n"</span>;</div>
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<div class="line"><span class="lineno"> 165</span> </div>
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<div class="line"><span class="lineno"> 166</span> std::clock_t t1 = std::clock();</div>
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<div class="line"><span class="lineno"> 167</span> <span class="keywordflow">do</span> {</div>
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<div class="line"><span class="lineno"> 168</span> fp << x << <span class="stringliteral">","</span>;</div>
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<div class="line"><span class="lineno"> 169</span> <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i < y.size() - 1; i++) {</div>
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<div class="line"><span class="lineno"> 170</span> fp << y[i] << <span class="stringliteral">","</span>;</div>
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<div class="line"><span class="lineno"> 171</span> }</div>
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<div class="line"><span class="lineno"> 172</span> fp << y[y.size() - 1] << <span class="stringliteral">"\n"</span>;</div>
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<div class="line"><span class="lineno"> 173</span> </div>
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<div class="line"><span class="lineno"> 174</span> <a class="code hl_function" href="#af3adf7b092a87868917ee5fb4255192b">exact_solution</a>(x, &y);</div>
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<div class="line"><span class="lineno"> 175</span> </div>
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<div class="line"><span class="lineno"> 176</span> x += step_size;</div>
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<div class="line"><span class="lineno"> 177</span> } <span class="keywordflow">while</span> (x <= X_MAX);</div>
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<div class="line"><span class="lineno"> 178</span> </div>
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<div class="line"><span class="lineno"> 179</span> std::clock_t t2 = std::clock();</div>
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<div class="line"><span class="lineno"> 180</span> <span class="keywordtype">double</span> total_time = <span class="keyword">static_cast<</span><span class="keywordtype">double</span><span class="keyword">></span>(t2 - t1) / CLOCKS_PER_SEC;</div>
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<div class="line"><span class="lineno"> 181</span> std::cout << <span class="stringliteral">"\tTime = "</span> << total_time << <span class="stringliteral">" ms\n"</span>;</div>
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<div class="line"><span class="lineno"> 182</span> </div>
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<div class="line"><span class="lineno"> 183</span> fp.close();</div>
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<div class="line"><span class="lineno"> 184</span>}</div>
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<div class="ttc" id="aode__semi__implicit__euler_8cpp_html_af3adf7b092a87868917ee5fb4255192b"><div class="ttname"><a href="#af3adf7b092a87868917ee5fb4255192b">exact_solution</a></div><div class="ttdeci">void exact_solution(const double &x, std::valarray< double > *y)</div><div class="ttdoc">Exact solution of the problem. Used for solution comparison.</div><div class="ttdef"><b>Definition</b> <a href="../../d3/d06/ode__semi__implicit__euler_8cpp_source.html#l00066">ode_semi_implicit_euler.cpp:66</a></div></div>
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