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@@ -118,10 +120,10 @@ Include dependency graph for ode_semi_implicit_euler.cpp:</div>
</div><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="../../d3/d06/ode__semi__implicit__euler_8cpp.html#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, <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: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="../../d3/d06/ode__semi__implicit__euler_8cpp.html#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, <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="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:ga827bf009831ddc477c5fa8891d5cb35f" id="r_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>
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<tr class="memitem:ga3874d294ab00fe9ce8731c5b7991a089" id="r_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>
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<a name="details" id="details"></a><h2 class="groupheader">Detailed Description</h2>
@@ -153,7 +155,7 @@ v(x) &amp;=&amp; -\sin(x)\\
\end{eqnarray*}
</p>
<p> The computation results are stored to a text file <code>semi_implicit_euler.csv</code> and the exact soltuion results in <code>exact.csv</code> 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>
<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="../../d3/d06/ode__semi__implicit__euler_8cpp.html#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>
<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>
<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="../../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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<tr>
<td class="memname">void exact_solution </td>
<td>(</td>
<td class="paramtype">const double &amp;&#160;</td>
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<td></td>
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<td class="memname">int main </td>
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<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"> 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="../../d3/d06/ode__semi__implicit__euler_8cpp.html#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>
<div class="line"><span class="lineno"> 210</span> <span class="keywordflow">return</span> 0;</div>
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<div class="ttc" id="abasic_istream_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/io/basic_istream.html">std::cin</a></div></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&lt; double &gt; *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> ode_semi_implicit_euler.cpp:103</div></div>
<div class="ttc" id="aode__semi__implicit__euler_8cpp_html_aa13517b8e5de1b75592052db7f7e237f"><div class="ttname"><a href="../../d3/d06/ode__semi__implicit__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> ode_semi_implicit_euler.cpp:153</div></div>
<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 &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_semi_implicit_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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Here is the call graph for this function:</div>
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<tr>
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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>
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<td></td>
<td>)</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; *</td> <td class="paramname"><span class="paramname"><em>dy</em></span>&#160;)</td>
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<tr>
<td class="memname">void save_exact_solution </td>
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<td class="paramtype">const double &amp;&#160;</td>
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<td class="paramtype">const double &amp;&#160;</td>
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<td class="paramtype">const double &amp;&#160;</td>
<td class="paramname"><em>step_size</em>, </td>
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</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">const <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; double &gt; &amp;&#160;</td>
<td class="paramname"><em>Y0</em>&#160;</td>
</tr>
<tr>
<td></td>
<td>)</td>
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<td class="paramtype">const <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; double &gt; &amp;</td> <td class="paramname"><span class="paramname"><em>Y0</em></span>&#160;)</td>
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<div class="line"><span class="lineno"> 171</span> }</div>
<div class="line"><span class="lineno"> 172</span> fp &lt;&lt; y[y.size() - 1] &lt;&lt; <span class="stringliteral">&quot;\n&quot;</span>;</div>
<div class="line"><span class="lineno"> 173</span> </div>
<div class="line"><span class="lineno"> 174</span> <a class="code hl_function" href="../../d3/d06/ode__semi__implicit__euler_8cpp.html#af3adf7b092a87868917ee5fb4255192b">exact_solution</a>(x, &amp;y);</div>
<div class="line"><span class="lineno"> 174</span> <a class="code hl_function" href="#af3adf7b092a87868917ee5fb4255192b">exact_solution</a>(x, &amp;y);</div>
<div class="line"><span class="lineno"> 175</span> </div>
<div class="line"><span class="lineno"> 176</span> x += step_size;</div>
<div class="line"><span class="lineno"> 177</span> } <span class="keywordflow">while</span> (x &lt;= X_MAX);</div>
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<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="aode__semi__implicit__euler_8cpp_html_af3adf7b092a87868917ee5fb4255192b"><div class="ttname"><a href="../../d3/d06/ode__semi__implicit__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> ode_semi_implicit_euler.cpp:66</div></div>
<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 &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> ode_semi_implicit_euler.cpp:66</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>
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