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<a href="#define-members">Macros</a> |
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<a href="#func-members">Functions</a> </div>
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<div class="headertitle"><div class="title">ode_forward_euler.c File Reference</div></div>
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</div><!--header-->
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<div class="contents">
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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/Numerical_methods_for_ordinary_differential_equations#Euler_method" target="_blank">forward Euler method</a>
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<a href="#details">More...</a></p>
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<div class="textblock"><code>#include <math.h></code><br />
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<code>#include <stdio.h></code><br />
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<code>#include <stdlib.h></code><br />
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<code>#include <time.h></code><br />
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</div><div class="textblock"><div class="dynheader">
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Include dependency graph for ode_forward_euler.c:</div>
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<div class="dyncontent">
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<div class="center"><iframe scrolling="no" frameborder="0" src="../../d1/d6c/ode__forward__euler_8c__incl.svg" width="331" height="124"><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><table class="memberdecls">
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<tr class="heading"><td colspan="2"><h2 class="groupheader"><a id="define-members" name="define-members"></a>
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Macros</h2></td></tr>
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<tr class="memitem:a9ceb646336224ee890a269d0b4600d09"><td class="memItemLeft" align="right" valign="top"><a id="a9ceb646336224ee890a269d0b4600d09" name="a9ceb646336224ee890a269d0b4600d09"></a>
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#define </td><td class="memItemRight" valign="bottom"><b>order</b>   2</td></tr>
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<tr class="memdesc:a9ceb646336224ee890a269d0b4600d09"><td class="mdescLeft"> </td><td class="mdescRight">number of dependent variables in <a class="el" href="../../d4/d07/ode__forward__euler_8c.html#a97075291390a68c262ed66e157a57eb4" title="Problem statement for a system with first-order differential equations.">problem</a> <br /></td></tr>
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<tr class="separator:a9ceb646336224ee890a269d0b4600d09"><td class="memSeparator" colspan="2"> </td></tr>
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</table><table class="memberdecls">
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<tr class="heading"><td colspan="2"><h2 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:a97075291390a68c262ed66e157a57eb4"><td class="memItemLeft" align="right" valign="top">void </td><td class="memItemRight" valign="bottom"><a class="el" href="../../d4/d07/ode__forward__euler_8c.html#a97075291390a68c262ed66e157a57eb4">problem</a> (const double *x, double *y, double *dy)</td></tr>
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<tr class="memdesc:a97075291390a68c262ed66e157a57eb4"><td class="mdescLeft"> </td><td class="mdescRight">Problem statement for a system with first-order differential equations. <a href="../../d4/d07/ode__forward__euler_8c.html#a97075291390a68c262ed66e157a57eb4">More...</a><br /></td></tr>
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<tr class="separator:a97075291390a68c262ed66e157a57eb4"><td class="memSeparator" colspan="2"> </td></tr>
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<tr class="memitem:a8caee977b26888d34040b122e0e28e3a"><td class="memItemLeft" align="right" valign="top">void </td><td class="memItemRight" valign="bottom"><a class="el" href="../../d4/d07/ode__forward__euler_8c.html#a8caee977b26888d34040b122e0e28e3a">exact_solution</a> (const double *x, double *y)</td></tr>
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<tr class="memdesc:a8caee977b26888d34040b122e0e28e3a"><td class="mdescLeft"> </td><td class="mdescRight">Exact solution of the problem. <a href="../../d4/d07/ode__forward__euler_8c.html#a8caee977b26888d34040b122e0e28e3a">More...</a><br /></td></tr>
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<tr class="separator:a8caee977b26888d34040b122e0e28e3a"><td class="memSeparator" colspan="2"> </td></tr>
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<tr class="memitem:ae6c9413953c8d9d4bc9e374b29586350"><td class="memItemLeft" align="right" valign="top">void </td><td class="memItemRight" valign="bottom"><a class="el" href="../../d4/d07/ode__forward__euler_8c.html#ae6c9413953c8d9d4bc9e374b29586350">forward_euler_step</a> (const double dx, const double *x, double *y, double *dy)</td></tr>
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<tr class="memdesc:ae6c9413953c8d9d4bc9e374b29586350"><td class="mdescLeft"> </td><td class="mdescRight">Compute next step approximation using the forward-Euler method. <a href="../../d4/d07/ode__forward__euler_8c.html#ae6c9413953c8d9d4bc9e374b29586350">More...</a><br /></td></tr>
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<tr class="separator:ae6c9413953c8d9d4bc9e374b29586350"><td class="memSeparator" colspan="2"> </td></tr>
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<tr class="memitem:aaf88ad8f9f7c39fc38f3f03d6fea9df9"><td class="memItemLeft" align="right" valign="top">double </td><td class="memItemRight" valign="bottom"><a class="el" href="../../d4/d07/ode__forward__euler_8c.html#aaf88ad8f9f7c39fc38f3f03d6fea9df9">forward_euler</a> (double dx, double x0, double x_max, double *y, char save_to_file)</td></tr>
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<tr class="memdesc:aaf88ad8f9f7c39fc38f3f03d6fea9df9"><td class="mdescLeft"> </td><td class="mdescRight">Compute approximation using the forward-Euler method in the given limits. <a href="../../d4/d07/ode__forward__euler_8c.html#aaf88ad8f9f7c39fc38f3f03d6fea9df9">More...</a><br /></td></tr>
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<tr class="separator:aaf88ad8f9f7c39fc38f3f03d6fea9df9"><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="../../d4/d07/ode__forward__euler_8c.html#a0ddf1224851353fc92bfbff6f499fa97">main</a> (int argc, char *argv[])</td></tr>
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<tr class="memdesc:a0ddf1224851353fc92bfbff6f499fa97"><td class="mdescLeft"> </td><td class="mdescRight">Main Function. <a href="../../d4/d07/ode__forward__euler_8c.html#a0ddf1224851353fc92bfbff6f499fa97">More...</a><br /></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>
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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/Numerical_methods_for_ordinary_differential_equations#Euler_method" target="_blank">forward 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*}
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\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*}
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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 <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/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" target="_blank">Van der Pol oscillator</a>, change the <a class="el" href="../../d4/d07/ode__forward__euler_8c.html#a97075291390a68c262ed66e157a57eb4" title="Problem statement for a system with first-order differential equations.">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="../../d1/dc2/ode__midpoint__euler_8c.html" title="Solve a multivariable first order ordinary differential equation (ODEs) using midpoint Euler method">ode_midpoint_euler.c</a>, <a class="el" href="../../d4/d99/ode__semi__implicit__euler_8c.html" title="Solve a multivariable first order ordinary differential equation (ODEs) using semi implicit Euler met...">ode_semi_implicit_euler.c</a> </dd></dl>
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</div><h2 class="groupheader">Function Documentation</h2>
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<a id="a8caee977b26888d34040b122e0e28e3a" name="a8caee977b26888d34040b122e0e28e3a"></a>
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<h2 class="memtitle"><span class="permalink"><a href="#a8caee977b26888d34040b122e0e28e3a">◆ </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>
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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">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. </p>
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<p >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"><span class="lineno"> 69</span>{</div>
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<div class="line"><span class="lineno"> 70</span> y[0] = cos(x[0]);</div>
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<div class="line"><span class="lineno"> 71</span> y[1] = -sin(x[0]);</div>
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<div class="line"><span class="lineno"> 72</span>}</div>
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</div><!-- fragment -->
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</div>
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</div>
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<a id="aaf88ad8f9f7c39fc38f3f03d6fea9df9" name="aaf88ad8f9f7c39fc38f3f03d6fea9df9"></a>
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<h2 class="memtitle"><span class="permalink"><a href="#aaf88ad8f9f7c39fc38f3f03d6fea9df9">◆ </a></span>forward_euler()</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">double forward_euler </td>
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<td>(</td>
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<td class="paramtype">double </td>
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<td class="paramname"><em>dx</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">double </td>
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<td class="paramname"><em>x0</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">double </td>
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<td class="paramname"><em>x_max</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">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 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>save_to_file</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>Compute approximation using the forward-Euler method in the given limits. </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">dx</td><td>step size </td></tr>
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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,out]</td><td class="paramname">y</td><td>take \(y_n\) and compute \(y_{n+1}\) </td></tr>
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<tr><td class="paramdir">[in]</td><td class="paramname">save_to_file</td><td>flag to save results to a CSV file (1) or not (0) </td></tr>
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</table>
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</dd>
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</dl>
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<dl class="section return"><dt>Returns</dt><dd>time taken for computation in seconds </dd></dl>
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<div class="fragment"><div class="line"><span class="lineno"> 101</span>{</div>
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<div class="line"><span class="lineno"> 102</span> <span class="keywordtype">double</span> dy[<a class="code hl_define" href="../../d4/d07/ode__forward__euler_8c.html#a9ceb646336224ee890a269d0b4600d09">order</a>];</div>
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<div class="line"><span class="lineno"> 103</span> </div>
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<div class="line"><span class="lineno"> 104</span> FILE *fp = NULL;</div>
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<div class="line"><span class="lineno"> 105</span> <span class="keywordflow">if</span> (save_to_file)</div>
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<div class="line"><span class="lineno"> 106</span> {</div>
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<div class="line"><span class="lineno"> 107</span> fp = fopen(<span class="stringliteral">"forward_euler.csv"</span>, <span class="stringliteral">"w+"</span>);</div>
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<div class="line"><span class="lineno"> 108</span> <span class="keywordflow">if</span> (fp == NULL)</div>
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<div class="line"><span class="lineno"> 109</span> {</div>
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<div class="line"><span class="lineno"> 110</span> perror(<span class="stringliteral">"Error! "</span>);</div>
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<div class="line"><span class="lineno"> 111</span> <span class="keywordflow">return</span> -1;</div>
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<div class="line"><span class="lineno"> 112</span> }</div>
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<div class="line"><span class="lineno"> 113</span> }</div>
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<div class="line"><span class="lineno"> 114</span> </div>
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<div class="line"><span class="lineno"> 115</span> <span class="comment">/* start integration */</span></div>
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<div class="line"><span class="lineno"> 116</span> clock_t t1 = clock();</div>
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<div class="line"><span class="lineno"> 117</span> <span class="keywordtype">double</span> x = x0;</div>
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<div class="line"><span class="lineno"> 118</span> <span class="keywordflow">do</span> <span class="comment">// iterate for each step of independent variable</span></div>
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<div class="line"><span class="lineno"> 119</span> {</div>
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<div class="line"><span class="lineno"> 120</span> <span class="keywordflow">if</span> (save_to_file && fp)</div>
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<div class="line"><span class="lineno"> 121</span> fprintf(fp, <span class="stringliteral">"%.4g,%.4g,%.4g\n"</span>, x, y[0], y[1]); <span class="comment">// write to file</span></div>
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<div class="line"><span class="lineno"> 122</span> <a class="code hl_function" href="../../d4/d07/ode__forward__euler_8c.html#ae6c9413953c8d9d4bc9e374b29586350">forward_euler_step</a>(dx, &x, y, dy); <span class="comment">// perform integration</span></div>
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<div class="line"><span class="lineno"> 123</span> x += dx; <span class="comment">// update step</span></div>
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<div class="line"><span class="lineno"> 124</span> } <span class="keywordflow">while</span> (x <= x_max); <span class="comment">// till upper limit of independent variable</span></div>
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<div class="line"><span class="lineno"> 125</span> <span class="comment">/* end of integration */</span></div>
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<div class="line"><span class="lineno"> 126</span> clock_t t2 = clock();</div>
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<div class="line"><span class="lineno"> 127</span> </div>
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<div class="line"><span class="lineno"> 128</span> <span class="keywordflow">if</span> (save_to_file && fp)</div>
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<div class="line"><span class="lineno"> 129</span> fclose(fp);</div>
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<div class="line"><span class="lineno"> 130</span> </div>
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<div class="line"><span class="lineno"> 131</span> <span class="keywordflow">return</span> (<span class="keywordtype">double</span>)(t2 - t1) / CLOCKS_PER_SEC;</div>
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<div class="line"><span class="lineno"> 132</span>}</div>
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<div class="ttc" id="aode__forward__euler_8c_html_a9ceb646336224ee890a269d0b4600d09"><div class="ttname"><a href="../../d4/d07/ode__forward__euler_8c.html#a9ceb646336224ee890a269d0b4600d09">order</a></div><div class="ttdeci">#define order</div><div class="ttdoc">number of dependent variables in problem</div><div class="ttdef"><b>Definition:</b> ode_forward_euler.c:44</div></div>
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<div class="ttc" id="aode__forward__euler_8c_html_ae6c9413953c8d9d4bc9e374b29586350"><div class="ttname"><a href="../../d4/d07/ode__forward__euler_8c.html#ae6c9413953c8d9d4bc9e374b29586350">forward_euler_step</a></div><div class="ttdeci">void forward_euler_step(const double dx, const double *x, double *y, double *dy)</div><div class="ttdoc">Compute next step approximation using the forward-Euler method.</div><div class="ttdef"><b>Definition:</b> ode_forward_euler.c:82</div></div>
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<h2 class="memtitle"><span class="permalink"><a href="#ae6c9413953c8d9d4bc9e374b29586350">◆ </a></span>forward_euler_step()</h2>
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<td class="memname">void forward_euler_step </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>dx</em>, </td>
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<td class="paramkey"></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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<td class="paramkey"></td>
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<td></td>
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<td class="paramtype">double * </td>
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<td class="paramname"><em>y</em>, </td>
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<td class="paramkey"></td>
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<td></td>
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<td class="paramtype">double * </td>
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<td class="paramname"><em>dy</em> </td>
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<td>)</td>
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</div><div class="memdoc">
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<p>Compute next step approximation using the forward-Euler method. </p>
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<p class="formulaDsp">
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\[y_{n+1}=y_n + dx\cdot f\left(x_n,y_n\right)\]
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</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">dx</td><td>step size </td></tr>
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<tr><td class="paramdir">[in,out]</td><td class="paramname">x</td><td>take \(x_n\) and compute \(x_{n+1}\) </td></tr>
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<tr><td class="paramdir">[in,out]</td><td class="paramname">y</td><td>take \(y_n\) and compute \(y_{n+1}\) </td></tr>
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<tr><td class="paramdir">[in,out]</td><td class="paramname">dy</td><td>compute \(f\left(x_n,y_n\right)\) </td></tr>
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</table>
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</dd>
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</dl>
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<div class="fragment"><div class="line"><span class="lineno"> 83</span>{</div>
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<div class="line"><span class="lineno"> 84</span> <span class="keywordtype">int</span> o;</div>
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<div class="line"><span class="lineno"> 85</span> <a class="code hl_function" href="../../d4/d07/ode__forward__euler_8c.html#a97075291390a68c262ed66e157a57eb4">problem</a>(x, y, dy);</div>
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<div class="line"><span class="lineno"> 86</span> <span class="keywordflow">for</span> (o = 0; o < <a class="code hl_define" href="../../d4/d07/ode__forward__euler_8c.html#a9ceb646336224ee890a269d0b4600d09">order</a>; o++) y[o] += dx * dy[o];</div>
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<div class="line"><span class="lineno"> 87</span>}</div>
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<div class="ttc" id="aode__forward__euler_8c_html_a97075291390a68c262ed66e157a57eb4"><div class="ttname"><a href="../../d4/d07/ode__forward__euler_8c.html#a97075291390a68c262ed66e157a57eb4">problem</a></div><div class="ttdeci">void problem(const double *x, double *y, double *dy)</div><div class="ttdoc">Problem statement for a system with first-order differential equations.</div><div class="ttdef"><b>Definition:</b> ode_forward_euler.c:55</div></div>
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<h2 class="memtitle"><span class="permalink"><a href="#a0ddf1224851353fc92bfbff6f499fa97">◆ </a></span>main()</h2>
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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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<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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<td></td>
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<td>)</td>
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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"><span class="lineno"> 138</span>{</div>
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<div class="line"><span class="lineno"> 139</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"> 140</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"> 141</span> <span class="keywordtype">double</span> 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"> 142</span> <span class="keywordtype">double</span> step_size;</div>
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<div class="line"><span class="lineno"> 143</span> </div>
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<div class="line"><span class="lineno"> 144</span> <span class="keywordflow">if</span> (argc == 1)</div>
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<div class="line"><span class="lineno"> 145</span> {</div>
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<div class="line"><span class="lineno"> 146</span> printf(<span class="stringliteral">"\nEnter the step size: "</span>);</div>
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<div class="line"><span class="lineno"> 147</span> scanf(<span class="stringliteral">"%lg"</span>, &step_size);</div>
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<div class="line"><span class="lineno"> 148</span> }</div>
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<div class="line"><span class="lineno"> 149</span> <span class="keywordflow">else</span></div>
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<div class="line"><span class="lineno"> 150</span> <span class="comment">// use commandline argument as independent variable step size</span></div>
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<div class="line"><span class="lineno"> 151</span> step_size = atof(argv[1]);</div>
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<div class="line"><span class="lineno"> 152</span> </div>
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<div class="line"><span class="lineno"> 153</span> <span class="comment">// get approximate solution</span></div>
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<div class="line"><span class="lineno"> 154</span> <span class="keywordtype">double</span> total_time = <a class="code hl_function" href="../../d4/d07/ode__forward__euler_8c.html#aaf88ad8f9f7c39fc38f3f03d6fea9df9">forward_euler</a>(step_size, X0, X_MAX, Y0, 1);</div>
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<div class="line"><span class="lineno"> 155</span> printf(<span class="stringliteral">"\tTime = %.6g ms\n"</span>, total_time);</div>
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<div class="line"><span class="lineno"> 156</span> </div>
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<div class="line"><span class="lineno"> 157</span> <span class="comment">/* compute exact solution for comparion */</span></div>
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<div class="line"><span class="lineno"> 158</span> FILE *fp = fopen(<span class="stringliteral">"exact.csv"</span>, <span class="stringliteral">"w+"</span>);</div>
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<div class="line"><span class="lineno"> 159</span> <span class="keywordflow">if</span> (fp == NULL)</div>
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<div class="line"><span class="lineno"> 160</span> {</div>
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<div class="line"><span class="lineno"> 161</span> perror(<span class="stringliteral">"Error! "</span>);</div>
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<div class="line"><span class="lineno"> 162</span> <span class="keywordflow">return</span> -1;</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> <span class="keywordtype">double</span> x = X0;</div>
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<div class="line"><span class="lineno"> 165</span> <span class="keywordtype">double</span> *y = &(Y0[0]);</div>
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<div class="line"><span class="lineno"> 166</span> printf(<span class="stringliteral">"Finding exact solution\n"</span>);</div>
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<div class="line"><span class="lineno"> 167</span> clock_t t1 = clock();</div>
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<div class="line"><span class="lineno"> 168</span> </div>
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<div class="line"><span class="lineno"> 169</span> <span class="keywordflow">do</span></div>
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<div class="line"><span class="lineno"> 170</span> {</div>
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<div class="line"><span class="lineno"> 171</span> fprintf(fp, <span class="stringliteral">"%.4g,%.4g,%.4g\n"</span>, x, y[0], y[1]); <span class="comment">// write to file</span></div>
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<div class="line"><span class="lineno"> 172</span> <a class="code hl_function" href="../../d4/d07/ode__forward__euler_8c.html#a8caee977b26888d34040b122e0e28e3a">exact_solution</a>(&x, y);</div>
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<div class="line"><span class="lineno"> 173</span> x += step_size;</div>
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<div class="line"><span class="lineno"> 174</span> } <span class="keywordflow">while</span> (x <= X_MAX);</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> clock_t t2 = clock();</div>
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<div class="line"><span class="lineno"> 177</span> total_time = (t2 - t1) / CLOCKS_PER_SEC;</div>
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<div class="line"><span class="lineno"> 178</span> printf(<span class="stringliteral">"\tTime = %.6g ms\n"</span>, total_time);</div>
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<div class="line"><span class="lineno"> 179</span> fclose(fp);</div>
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<div class="line"><span class="lineno"> 180</span> </div>
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<div class="line"><span class="lineno"> 181</span> <span class="keywordflow">return</span> 0;</div>
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<div class="line"><span class="lineno"> 182</span>}</div>
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<div class="ttc" id="aode__forward__euler_8c_html_a8caee977b26888d34040b122e0e28e3a"><div class="ttname"><a href="../../d4/d07/ode__forward__euler_8c.html#a8caee977b26888d34040b122e0e28e3a">exact_solution</a></div><div class="ttdeci">void exact_solution(const double *x, double *y)</div><div class="ttdoc">Exact solution of the problem.</div><div class="ttdef"><b>Definition:</b> ode_forward_euler.c:68</div></div>
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<div class="ttc" id="aode__forward__euler_8c_html_aaf88ad8f9f7c39fc38f3f03d6fea9df9"><div class="ttname"><a href="../../d4/d07/ode__forward__euler_8c.html#aaf88ad8f9f7c39fc38f3f03d6fea9df9">forward_euler</a></div><div class="ttdeci">double forward_euler(double dx, double x0, double x_max, double *y, char save_to_file)</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.c:99</div></div>
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<a id="a97075291390a68c262ed66e157a57eb4" name="a97075291390a68c262ed66e157a57eb4"></a>
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<h2 class="memtitle"><span class="permalink"><a href="#a97075291390a68c262ed66e157a57eb4">◆ </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>
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<td class="paramname"><em>x</em>, </td>
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<td class="paramkey"></td>
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<td class="paramtype">double * </td>
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<td class="paramname"><em>y</em>, </td>
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<td class="paramkey"></td>
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<td></td>
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<td class="paramtype">double * </td>
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<td class="paramname"><em>dy</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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<p>Problem statement for a system with first-order differential equations. </p>
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<p >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>
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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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</table>
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</dd>
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</dl>
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<div class="fragment"><div class="line"><span class="lineno"> 56</span>{</div>
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<div class="line"><span class="lineno"> 57</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"> 58</span> dy[0] = y[1]; <span class="comment">// x dot</span></div>
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<div class="line"><span class="lineno"> 59</span> dy[1] = -omega * omega * y[0]; <span class="comment">// y dot</span></div>
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<div class="line"><span class="lineno"> 60</span>}</div>
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