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<p>Solve a multivariable first order <a href="https://en.wikipedia.org/wiki/Ordinary_differential_equation">ordinary differential equation (ODEs)</a> using <a href="https://en.wikipedia.org/wiki/Numerical_methods_for_ordinary_differential_equations#Euler_method">forward 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_forward_euler.cpp:</div>
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<div class="dyncontent">
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<tr class="heading"><td colspan="2"><h2 class="groupheader"><a name="func-members"></a>
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Functions</h2></td></tr>
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||||
<tr class="memitem:abaeae8f62a018d197f0187a1c80a90fe"><td class="memItemLeft" align="right" valign="top">void </td><td class="memItemRight" valign="bottom"><a class="el" href="../../db/dd3/ode__forward__euler_8cpp.html#abaeae8f62a018d197f0187a1c80a90fe">problem</a> (const double &x, <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>< double > *y, <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>< 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. <a href="../../db/dd3/ode__forward__euler_8cpp.html#abaeae8f62a018d197f0187a1c80a90fe">More...</a><br /></td></tr>
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<tr class="separator:abaeae8f62a018d197f0187a1c80a90fe"><td class="memSeparator" colspan="2"> </td></tr>
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<tr class="memitem:af3adf7b092a87868917ee5fb4255192b"><td class="memItemLeft" align="right" valign="top">void </td><td class="memItemRight" valign="bottom"><a class="el" href="../../db/dd3/ode__forward__euler_8cpp.html#af3adf7b092a87868917ee5fb4255192b">exact_solution</a> (const double &x, <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>< 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. <a href="../../db/dd3/ode__forward__euler_8cpp.html#af3adf7b092a87868917ee5fb4255192b">More...</a><br /></td></tr>
|
||||
<tr class="separator:af3adf7b092a87868917ee5fb4255192b"><td class="memSeparator" colspan="2"> </td></tr>
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||||
<tr class="memitem:gaa197c6b7085ac9ea71abc4c063e6f52f"><td class="memItemLeft" align="right" valign="top">void </td><td class="memItemRight" valign="bottom"><a class="el" href="../../d6/d60/group__ode.html#gaa197c6b7085ac9ea71abc4c063e6f52f">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>< double > *y, <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>< double > *dy)</td></tr>
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||||
<tr class="memdesc:gaa197c6b7085ac9ea71abc4c063e6f52f"><td class="mdescLeft"> </td><td class="mdescRight">Compute next step approximation using the forward-Euler method. <a href="../../d6/d60/group__ode.html#gaa197c6b7085ac9ea71abc4c063e6f52f">More...</a><br /></td></tr>
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||||
<tr class="separator:gaa197c6b7085ac9ea71abc4c063e6f52f"><td class="memSeparator" colspan="2"> </td></tr>
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<tr class="memitem:gae0509f8843e2bc42de2abbd00a14b7b9"><td class="memItemLeft" align="right" valign="top">double </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>< double > *y, bool save_to_file=false)</td></tr>
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||||
<tr class="memdesc:gae0509f8843e2bc42de2abbd00a14b7b9"><td class="mdescLeft"> </td><td class="mdescRight">Compute approximation using the forward-Euler method in the given limits. <a href="../../d6/d60/group__ode.html#gae0509f8843e2bc42de2abbd00a14b7b9">More...</a><br /></td></tr>
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||||
<tr class="separator:gae0509f8843e2bc42de2abbd00a14b7b9"><td class="memSeparator" colspan="2"> </td></tr>
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<tr class="memitem:aa13517b8e5de1b75592052db7f7e237f"><td class="memItemLeft" align="right" valign="top">void </td><td class="memItemRight" valign="bottom"><a class="el" href="../../db/dd3/ode__forward__euler_8cpp.html#aa13517b8e5de1b75592052db7f7e237f">save_exact_solution</a> (const double &X0, const double &X_MAX, const double &step_size, const <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>< double > &Y0)</td></tr>
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<tr class="separator:aa13517b8e5de1b75592052db7f7e237f"><td class="memSeparator" colspan="2"> </td></tr>
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<tr class="memitem:a0ddf1224851353fc92bfbff6f499fa97"><td class="memItemLeft" align="right" valign="top">int </td><td class="memItemRight" valign="bottom"><a class="el" href="../../db/dd3/ode__forward__euler_8cpp.html#a0ddf1224851353fc92bfbff6f499fa97">main</a> (int argc, char *argv[])</td></tr>
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<tr class="separator:a0ddf1224851353fc92bfbff6f499fa97"><td class="memSeparator" colspan="2"> </td></tr>
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</table>
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<a name="details" id="details"></a><h2 class="groupheader">Detailed Description</h2>
|
||||
<div class="textblock"><p>Solve a multivariable first order <a href="https://en.wikipedia.org/wiki/Ordinary_differential_equation">ordinary differential equation (ODEs)</a> using <a href="https://en.wikipedia.org/wiki/Numerical_methods_for_ordinary_differential_equations#Euler_method">forward Euler method</a> </p>
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<dl class="section author"><dt>Authors</dt><dd><a href="https://github.com/kvedala">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\\ \dot{v} &=& -\omega^2 u\\ \omega &=& 1\\ [x_0, u_0, v_0] &=& [0,1,0]\qquad\ldots\text{(initial values)} \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)\\ v(x) &=& -\sin(x)\\ \end{eqnarray*}
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</p>
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<p> The computation results are stored to a text file <code>forward_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_forward_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">Van der Pol oscillator</a>, change the <a class="el" href="../../db/dd3/ode__forward__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>
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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="../../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>
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</div><h2 class="groupheader">Function Documentation</h2>
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||||
<a id="af3adf7b092a87868917ee5fb4255192b"></a>
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||||
<h2 class="memtitle"><span class="permalink"><a href="#af3adf7b092a87868917ee5fb4255192b">◆ </a></span>exact_solution()</h2>
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||||
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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>
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<td class="paramname"><em>x</em>, </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"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>< double > * </td>
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<td class="paramname"><em>y</em> </td>
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</tr>
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<tr>
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<td></td>
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<td>)</td>
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<td></td><td></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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<div class="fragment"><div class="line"><a name="l00067"></a><span class="lineno"> 67</span>  {</div>
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<div class="line"><a name="l00068"></a><span class="lineno"> 68</span>  y[0][0] = <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/math/cos.html">std::cos</a>(x);</div>
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<div class="line"><a name="l00069"></a><span class="lineno"> 69</span>  y[0][1] = -<a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/math/sin.html">std::sin</a>(x);</div>
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<div class="line"><a name="l00070"></a><span class="lineno"> 70</span> }</div>
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</div><!-- fragment --><div class="dynheader">
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||||
Here is the call graph for this function:</div>
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<div class="dyncontent">
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||||
<div class="center"><iframe scrolling="no" frameborder="0" src="../../db/dd3/ode__forward__euler_8cpp_af3adf7b092a87868917ee5fb4255192b_cgraph.svg" width="235" height="88"><p><b>This browser is not able to show SVG: try Firefox, Chrome, Safari, or Opera instead.</b></p></iframe>
|
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</div>
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</div>
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</div>
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</div>
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<a id="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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<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>
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<td class="paramname"><em>argc</em>, </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>
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<td class="paramname"><em>argv</em>[] </td>
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</tr>
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<tr>
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<td></td>
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<td>)</td>
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<td></td><td></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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<div class="fragment"><div class="line"><a name="l00188"></a><span class="lineno"> 188</span>  {</div>
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<div class="line"><a name="l00189"></a><span class="lineno"> 189</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"><a name="l00190"></a><span class="lineno"> 190</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"><a name="l00191"></a><span class="lineno"> 191</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray<double></a> Y0 = {1.f, 0.f}; <span class="comment">/* initial value Y = y(x = x_0) */</span></div>
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<div class="line"><a name="l00192"></a><span class="lineno"> 192</span>  <span class="keywordtype">double</span> step_size;</div>
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<div class="line"><a name="l00193"></a><span class="lineno"> 193</span>  </div>
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<div class="line"><a name="l00194"></a><span class="lineno"> 194</span>  <span class="keywordflow">if</span> (argc == 1) {</div>
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<div class="line"><a name="l00195"></a><span class="lineno"> 195</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> << <span class="stringliteral">"\nEnter the step size: "</span>;</div>
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<div class="line"><a name="l00196"></a><span class="lineno"> 196</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_istream.html">std::cin</a> >> step_size;</div>
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<div class="line"><a name="l00197"></a><span class="lineno"> 197</span>  } <span class="keywordflow">else</span> {</div>
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<div class="line"><a name="l00198"></a><span class="lineno"> 198</span>  <span class="comment">// use commandline argument as independent variable step size</span></div>
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<div class="line"><a name="l00199"></a><span class="lineno"> 199</span>  step_size = <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/string/byte/atof.html">std::atof</a>(argv[1]);</div>
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<div class="line"><a name="l00200"></a><span class="lineno"> 200</span>  }</div>
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<div class="line"><a name="l00201"></a><span class="lineno"> 201</span>  </div>
|
||||
<div class="line"><a name="l00202"></a><span class="lineno"> 202</span>  <span class="comment">// get approximate solution</span></div>
|
||||
<div class="line"><a name="l00203"></a><span class="lineno"> 203</span>  <span class="keywordtype">double</span> total_time = <a class="code" href="../../d6/d60/group__ode.html#gae0509f8843e2bc42de2abbd00a14b7b9">forward_euler</a>(step_size, X0, X_MAX, &Y0, <span class="keyword">true</span>);</div>
|
||||
<div class="line"><a name="l00204"></a><span class="lineno"> 204</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> << <span class="stringliteral">"\tTime = "</span> << total_time << <span class="stringliteral">" ms\n"</span>;</div>
|
||||
<div class="line"><a name="l00205"></a><span class="lineno"> 205</span>  </div>
|
||||
<div class="line"><a name="l00206"></a><span class="lineno"> 206</span>  <span class="comment">/* compute exact solution for comparion */</span></div>
|
||||
<div class="line"><a name="l00207"></a><span class="lineno"> 207</span>  <a class="code" href="../../db/dd3/ode__forward__euler_8cpp.html#aa13517b8e5de1b75592052db7f7e237f">save_exact_solution</a>(X0, X_MAX, step_size, Y0);</div>
|
||||
<div class="line"><a name="l00208"></a><span class="lineno"> 208</span>  </div>
|
||||
<div class="line"><a name="l00209"></a><span class="lineno"> 209</span>  <span class="keywordflow">return</span> 0;</div>
|
||||
<div class="line"><a name="l00210"></a><span class="lineno"> 210</span> }</div>
|
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</div><!-- fragment --><div class="dynheader">
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Here is the call graph for this function:</div>
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||||
<div class="dyncontent">
|
||||
<div class="center"><iframe scrolling="no" frameborder="0" src="../../db/dd3/ode__forward__euler_8cpp_a0ddf1224851353fc92bfbff6f499fa97_cgraph.svg" width="568" height="367"><p><b>This browser is not able to show SVG: try Firefox, Chrome, Safari, or Opera instead.</b></p></iframe>
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<a id="abaeae8f62a018d197f0187a1c80a90fe"></a>
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<h2 class="memtitle"><span class="permalink"><a href="#abaeae8f62a018d197f0187a1c80a90fe">◆ </a></span>problem()</h2>
|
||||
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||||
<div class="memitem">
|
||||
<div class="memproto">
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<table class="memname">
|
||||
<tr>
|
||||
<td class="memname">void problem </td>
|
||||
<td>(</td>
|
||||
<td class="paramtype">const double & </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>< double > * </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>< double > * </td>
|
||||
<td class="paramname"><em>dy</em> </td>
|
||||
</tr>
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||||
<tr>
|
||||
<td></td>
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||||
<td>)</td>
|
||||
<td></td><td></td>
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||||
</tr>
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||||
</table>
|
||||
</div><div class="memdoc">
|
||||
|
||||
<p>Problem statement for a system with first-order differential equations. Updates the system differential variables. </p>
|
||||
<dl class="section note"><dt>Note</dt><dd>This function can be updated to and ode of any order.</dd></dl>
|
||||
<dl class="params"><dt>Parameters</dt><dd>
|
||||
<table class="params">
|
||||
<tr><td class="paramdir">[in]</td><td class="paramname">x</td><td>independent variable(s) </td></tr>
|
||||
<tr><td class="paramdir">[in,out]</td><td class="paramname">y</td><td>dependent variable(s) </td></tr>
|
||||
<tr><td class="paramdir">[in,out]</td><td class="paramname">dy</td><td>first-derivative of dependent variable(s) </td></tr>
|
||||
</table>
|
||||
</dd>
|
||||
</dl>
|
||||
<div class="fragment"><div class="line"><a name="l00055"></a><span class="lineno"> 55</span>  {</div>
|
||||
<div class="line"><a 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 name="l00057"></a><span class="lineno"> 57</span>  dy[0][0] = y[0][1]; <span class="comment">// x dot</span></div>
|
||||
<div class="line"><a name="l00058"></a><span class="lineno"> 58</span>  dy[0][1] = -omega * omega * y[0][0]; <span class="comment">// y dot</span></div>
|
||||
<div class="line"><a name="l00059"></a><span class="lineno"> 59</span> }</div>
|
||||
</div><!-- fragment -->
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||||
</div>
|
||||
</div>
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<a id="aa13517b8e5de1b75592052db7f7e237f"></a>
|
||||
<h2 class="memtitle"><span class="permalink"><a href="#aa13517b8e5de1b75592052db7f7e237f">◆ </a></span>save_exact_solution()</h2>
|
||||
|
||||
<div class="memitem">
|
||||
<div class="memproto">
|
||||
<table class="memname">
|
||||
<tr>
|
||||
<td class="memname">void save_exact_solution </td>
|
||||
<td>(</td>
|
||||
<td class="paramtype">const double & </td>
|
||||
<td class="paramname"><em>X0</em>, </td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="paramkey"></td>
|
||||
<td></td>
|
||||
<td class="paramtype">const double & </td>
|
||||
<td class="paramname"><em>X_MAX</em>, </td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td class="paramkey"></td>
|
||||
<td></td>
|
||||
<td class="paramtype">const double & </td>
|
||||
<td class="paramname"><em>step_size</em>, </td>
|
||||
</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>< double > & </td>
|
||||
<td class="paramname"><em>Y0</em> </td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td></td>
|
||||
<td>)</td>
|
||||
<td></td><td></td>
|
||||
</tr>
|
||||
</table>
|
||||
</div><div class="memdoc">
|
||||
<p>Function to compute and save exact solution for comparison</p>
|
||||
<dl class="params"><dt>Parameters</dt><dd>
|
||||
<table class="params">
|
||||
<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]</td><td class="paramname">step_size</td><td>independent variable step size </td></tr>
|
||||
<tr><td class="paramdir">[in]</td><td class="paramname">Y0</td><td>initial values of dependent variables </td></tr>
|
||||
</table>
|
||||
</dd>
|
||||
</dl>
|
||||
<div class="fragment"><div class="line"><a name="l00154"></a><span class="lineno"> 154</span>  {</div>
|
||||
<div class="line"><a name="l00155"></a><span class="lineno"> 155</span>  <span class="keywordtype">double</span> x = X0;</div>
|
||||
<div class="line"><a name="l00156"></a><span class="lineno"> 156</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray<double></a> y = Y0;</div>
|
||||
<div class="line"><a name="l00157"></a><span class="lineno"> 157</span>  </div>
|
||||
<div class="line"><a name="l00158"></a><span class="lineno"> 158</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ofstream.html">std::ofstream</a> fp(<span class="stringliteral">"exact.csv"</span>, std::ostream::out);</div>
|
||||
<div class="line"><a name="l00159"></a><span class="lineno"> 159</span>  <span class="keywordflow">if</span> (!fp.is_open()) {</div>
|
||||
<div class="line"><a name="l00160"></a><span class="lineno"> 160</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/c/perror.html">std::perror</a>(<span class="stringliteral">"Error! "</span>);</div>
|
||||
<div class="line"><a name="l00161"></a><span class="lineno"> 161</span>  <span class="keywordflow">return</span>;</div>
|
||||
<div class="line"><a name="l00162"></a><span class="lineno"> 162</span>  }</div>
|
||||
<div class="line"><a name="l00163"></a><span class="lineno"> 163</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> << <span class="stringliteral">"Finding exact solution\n"</span>;</div>
|
||||
<div class="line"><a name="l00164"></a><span class="lineno"> 164</span>  </div>
|
||||
<div class="line"><a name="l00165"></a><span class="lineno"> 165</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/chrono/c/clock_t.html">std::clock_t</a> t1 = <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/chrono/c/clock.html">std::clock</a>();</div>
|
||||
<div class="line"><a name="l00166"></a><span class="lineno"> 166</span>  <span class="keywordflow">do</span> {</div>
|
||||
<div class="line"><a name="l00167"></a><span class="lineno"> 167</span>  fp << x << <span class="stringliteral">","</span>;</div>
|
||||
<div class="line"><a name="l00168"></a><span class="lineno"> 168</span>  <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i < y.size() - 1; i++) {</div>
|
||||
<div class="line"><a name="l00169"></a><span class="lineno"> 169</span>  fp << y[i] << <span class="stringliteral">","</span>;</div>
|
||||
<div class="line"><a name="l00170"></a><span class="lineno"> 170</span>  }</div>
|
||||
<div class="line"><a name="l00171"></a><span class="lineno"> 171</span>  fp << y[y.size() - 1] << <span class="stringliteral">"\n"</span>;</div>
|
||||
<div class="line"><a name="l00172"></a><span class="lineno"> 172</span>  </div>
|
||||
<div class="line"><a name="l00173"></a><span class="lineno"> 173</span>  <a class="code" href="../../db/dd3/ode__forward__euler_8cpp.html#af3adf7b092a87868917ee5fb4255192b">exact_solution</a>(x, &y);</div>
|
||||
<div class="line"><a name="l00174"></a><span class="lineno"> 174</span>  </div>
|
||||
<div class="line"><a name="l00175"></a><span class="lineno"> 175</span>  x += step_size;</div>
|
||||
<div class="line"><a name="l00176"></a><span class="lineno"> 176</span>  } <span class="keywordflow">while</span> (x <= X_MAX);</div>
|
||||
<div class="line"><a name="l00177"></a><span class="lineno"> 177</span>  </div>
|
||||
<div class="line"><a name="l00178"></a><span class="lineno"> 178</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/chrono/c/clock_t.html">std::clock_t</a> t2 = <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/chrono/c/clock.html">std::clock</a>();</div>
|
||||
<div class="line"><a name="l00179"></a><span class="lineno"> 179</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>
|
||||
<div class="line"><a name="l00180"></a><span class="lineno"> 180</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> << <span class="stringliteral">"\tTime = "</span> << total_time << <span class="stringliteral">" ms\n"</span>;</div>
|
||||
<div class="line"><a name="l00181"></a><span class="lineno"> 181</span>  </div>
|
||||
<div class="line"><a name="l00182"></a><span class="lineno"> 182</span>  fp.close();</div>
|
||||
<div class="line"><a name="l00183"></a><span class="lineno"> 183</span> }</div>
|
||||
</div><!-- fragment --><div class="dynheader">
|
||||
Here is the call graph for this function:</div>
|
||||
<div class="dyncontent">
|
||||
<div class="center"><iframe scrolling="no" frameborder="0" src="../../db/dd3/ode__forward__euler_8cpp_aa13517b8e5de1b75592052db7f7e237f_cgraph.svg" width="470" height="240"><p><b>This browser is not able to show SVG: try Firefox, Chrome, Safari, or Opera instead.</b></p></iframe>
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|
||||
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||||
</div>
|
||||
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|
||||
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|
||||
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|
||||
<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>
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<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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|
||||
<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>
|
||||
<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>
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|
||||
<div class="ttc" id="abasic_ostream_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a></div></div>
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|
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<div class="ttc" id="asin_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/numeric/math/sin.html">std::sin</a></div><div class="ttdeci">T sin(T... args)</div></div>
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