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 <span id="projectnumber">1.0.0</span>
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<div class="title">ode_semi_implicit_euler.c File Reference</div> </div>
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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/Semi-implicit_Euler_method">semi implicit 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_semi_implicit_euler.c:</div>
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<div class="dyncontent">
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<div class="center"><iframe scrolling="no" frameborder="0" src="../../d9/d85/ode__semi__implicit__euler_8c__incl.svg" width="331" height="127"><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><table class="memberdecls">
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<tr class="heading"><td colspan="2"><h2 class="groupheader"><a 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"></a>
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#define </td><td class="memItemRight" valign="bottom"><a class="el" href="../../d4/d99/ode__semi__implicit__euler_8c.html#a9ceb646336224ee890a269d0b4600d09">order</a>   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/d99/ode__semi__implicit__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 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/d99/ode__semi__implicit__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/d99/ode__semi__implicit__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/d99/ode__semi__implicit__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/d99/ode__semi__implicit__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:a720b7e995d2bbc615f94a2c7dbcf84eb"><td class="memItemLeft" align="right" valign="top">void </td><td class="memItemRight" valign="bottom"><a class="el" href="../../d4/d99/ode__semi__implicit__euler_8c.html#a720b7e995d2bbc615f94a2c7dbcf84eb">semi_implicit_euler_step</a> (double dx, double *x, double *y, double *dy)</td></tr>
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<tr class="memdesc:a720b7e995d2bbc615f94a2c7dbcf84eb"><td class="mdescLeft"> </td><td class="mdescRight">Compute next step approximation using the semi-implicit-Euler method. <a href="../../d4/d99/ode__semi__implicit__euler_8c.html#a720b7e995d2bbc615f94a2c7dbcf84eb">More...</a><br /></td></tr>
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<tr class="separator:a720b7e995d2bbc615f94a2c7dbcf84eb"><td class="memSeparator" colspan="2"> </td></tr>
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<tr class="memitem:ad80059877222f885b549f2d0a3dc6b55"><td class="memItemLeft" align="right" valign="top">double </td><td class="memItemRight" valign="bottom"><a class="el" href="../../d4/d99/ode__semi__implicit__euler_8c.html#ad80059877222f885b549f2d0a3dc6b55">semi_implicit_euler</a> (double dx, double x0, double x_max, double *y, char save_to_file)</td></tr>
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<tr class="memdesc:ad80059877222f885b549f2d0a3dc6b55"><td class="mdescLeft"> </td><td class="mdescRight">Compute approximation using the semi-implicit-Euler method in the given limits. <a href="../../d4/d99/ode__semi__implicit__euler_8c.html#ad80059877222f885b549f2d0a3dc6b55">More...</a><br /></td></tr>
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<tr class="separator:ad80059877222f885b549f2d0a3dc6b55"><td class="memSeparator" colspan="2"> </td></tr>
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<tr class="memitem:a0ddf1224851353fc92bfbff6f499fa97"><td class="memItemLeft" align="right" valign="top"><a id="a0ddf1224851353fc92bfbff6f499fa97"></a>
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int </td><td class="memItemRight" valign="bottom"><a class="el" href="../../d4/d99/ode__semi__implicit__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. <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">ordinary differential equation (ODEs)</a> using <a href="https://en.wikipedia.org/wiki/Semi-implicit_Euler_method">semi implicit 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>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/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">Van der Pol oscillator</a>, change the <a class="el" href="../../d4/d99/ode__semi__implicit__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 --><p><a href="https://en.wikipedia.org/wiki/Van_der_Pol_oscillator"><img src="https://raw.githubusercontent.com/TheAlgorithms/C/docs/images/numerical_methods/van_der_pol_implicit_euler.svg" alt="Van der Pol Oscillator solution" style="pointer-events: none;" class="inline"/></a></p>
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<dl class="section see"><dt>See also</dt><dd><a class="el" href="../../d4/d07/ode__forward__euler_8c.html" title="Solve a multivariable first order ordinary differential equation (ODEs) using forward Euler method">ode_forward_euler.c</a>, <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> </dd></dl>
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</div><h2 class="groupheader">Function Documentation</h2>
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<a id="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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<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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<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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<td></td>
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<td>)</td>
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<td></td><td></td>
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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"><a name="l00072"></a><span class="lineno"> 72</span> {</div>
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<div class="line"><a name="l00073"></a><span class="lineno"> 73</span>  y[0] = cos(x[0]);</div>
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<div class="line"><a name="l00074"></a><span class="lineno"> 74</span>  y[1] = -sin(x[0]);</div>
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<div class="line"><a name="l00075"></a><span class="lineno"> 75</span> }</div>
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</div><!-- fragment -->
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</div>
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</div>
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<a id="a97075291390a68c262ed66e157a57eb4"></a>
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<h2 class="memtitle"><span class="permalink"><a href="#a97075291390a68c262ed66e157a57eb4">◆ </a></span>problem()</h2>
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<div class="memitem">
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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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</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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<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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<td></td><td></td>
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</div><div class="memdoc">
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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"><a name="l00059"></a><span class="lineno"> 59</span> {</div>
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<div class="line"><a name="l00060"></a><span class="lineno"> 60</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"><a name="l00061"></a><span class="lineno"> 61</span>  dy[0] = y[1]; <span class="comment">// x dot</span></div>
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<div class="line"><a name="l00062"></a><span class="lineno"> 62</span>  dy[1] = -omega * omega * y[0]; <span class="comment">// y dot</span></div>
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<div class="line"><a name="l00063"></a><span class="lineno"> 63</span> }</div>
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</div><!-- fragment -->
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<a id="ad80059877222f885b549f2d0a3dc6b55"></a>
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<h2 class="memtitle"><span class="permalink"><a href="#ad80059877222f885b549f2d0a3dc6b55">◆ </a></span>semi_implicit_euler()</h2>
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<div class="memitem">
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<td class="memname">double semi_implicit_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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<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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<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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<td class="paramkey"></td>
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<td></td>
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<td class="paramtype">double * </td>
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<p>Compute approximation using the semi-implicit-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"><a name="l00111"></a><span class="lineno"> 111</span> {</div>
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<div class="line"><a name="l00112"></a><span class="lineno"> 112</span>  <span class="keywordtype">double</span> dy[<a class="code" href="../../d4/d99/ode__semi__implicit__euler_8c.html#a9ceb646336224ee890a269d0b4600d09">order</a>];</div>
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<div class="line"><a name="l00113"></a><span class="lineno"> 113</span>  </div>
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<div class="line"><a name="l00114"></a><span class="lineno"> 114</span>  FILE *fp = NULL;</div>
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<div class="line"><a name="l00115"></a><span class="lineno"> 115</span>  <span class="keywordflow">if</span> (save_to_file)</div>
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<div class="line"><a name="l00116"></a><span class="lineno"> 116</span>  {</div>
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<div class="line"><a name="l00117"></a><span class="lineno"> 117</span>  fp = fopen(<span class="stringliteral">"semi_implicit_euler.csv"</span>, <span class="stringliteral">"w+"</span>);</div>
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<div class="line"><a name="l00118"></a><span class="lineno"> 118</span>  <span class="keywordflow">if</span> (fp == NULL)</div>
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<div class="line"><a name="l00119"></a><span class="lineno"> 119</span>  {</div>
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<div class="line"><a name="l00120"></a><span class="lineno"> 120</span>  perror(<span class="stringliteral">"Error! "</span>);</div>
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<div class="line"><a name="l00121"></a><span class="lineno"> 121</span>  <span class="keywordflow">return</span> -1;</div>
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<div class="line"><a name="l00122"></a><span class="lineno"> 122</span>  }</div>
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<div class="line"><a name="l00123"></a><span class="lineno"> 123</span>  }</div>
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<div class="line"><a name="l00124"></a><span class="lineno"> 124</span>  </div>
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<div class="line"><a name="l00125"></a><span class="lineno"> 125</span>  <span class="comment">/* start integration */</span></div>
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<div class="line"><a name="l00126"></a><span class="lineno"> 126</span>  clock_t t1 = clock();</div>
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<div class="line"><a name="l00127"></a><span class="lineno"> 127</span>  <span class="keywordtype">double</span> x = x0;</div>
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<div class="line"><a name="l00128"></a><span class="lineno"> 128</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"><a name="l00129"></a><span class="lineno"> 129</span>  {</div>
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<div class="line"><a name="l00130"></a><span class="lineno"> 130</span>  <span class="keywordflow">if</span> (save_to_file && fp)</div>
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<div class="line"><a name="l00131"></a><span class="lineno"> 131</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"><a name="l00132"></a><span class="lineno"> 132</span>  <a class="code" href="../../d4/d99/ode__semi__implicit__euler_8c.html#a720b7e995d2bbc615f94a2c7dbcf84eb">semi_implicit_euler_step</a>(dx, &x, y, dy); <span class="comment">// perform integration</span></div>
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<div class="line"><a name="l00133"></a><span class="lineno"> 133</span>  x += dx; <span class="comment">// update step</span></div>
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<div class="line"><a name="l00134"></a><span class="lineno"> 134</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"><a name="l00135"></a><span class="lineno"> 135</span>  <span class="comment">/* end of integration */</span></div>
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<div class="line"><a name="l00136"></a><span class="lineno"> 136</span>  clock_t t2 = clock();</div>
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<div class="line"><a name="l00137"></a><span class="lineno"> 137</span>  </div>
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<div class="line"><a name="l00138"></a><span class="lineno"> 138</span>  <span class="keywordflow">if</span> (save_to_file && fp)</div>
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<div class="line"><a name="l00139"></a><span class="lineno"> 139</span>  fclose(fp);</div>
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<div class="line"><a name="l00140"></a><span class="lineno"> 140</span>  </div>
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<div class="line"><a name="l00141"></a><span class="lineno"> 141</span>  <span class="keywordflow">return</span> (<span class="keywordtype">double</span>)(t2 - t1) / CLOCKS_PER_SEC;</div>
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<div class="line"><a name="l00142"></a><span class="lineno"> 142</span> }</div>
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<h2 class="memtitle"><span class="permalink"><a href="#a720b7e995d2bbc615f94a2c7dbcf84eb">◆ </a></span>semi_implicit_euler_step()</h2>
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<div class="memitem">
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<td class="memname">void semi_implicit_euler_step </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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<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</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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<td></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 semi-implicit-Euler method. </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 \(y_n+\frac{1}{2}dx\,f\left(x_n,y_n\right)\) </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"><a name="l00086"></a><span class="lineno"> 86</span> {</div>
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<div class="line"><a name="l00087"></a><span class="lineno"> 87</span>  <span class="keywordtype">int</span> o;</div>
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<div class="line"><a name="l00088"></a><span class="lineno"> 88</span>  </div>
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<div class="line"><a name="l00089"></a><span class="lineno"> 89</span>  <a class="code" href="../../d4/d99/ode__semi__implicit__euler_8c.html#a97075291390a68c262ed66e157a57eb4">problem</a>(x, y, dy); <span class="comment">// update dy once</span></div>
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<div class="line"><a name="l00090"></a><span class="lineno"> 90</span>  y[0] += dx * dy[0]; <span class="comment">// update y0</span></div>
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<div class="line"><a name="l00091"></a><span class="lineno"> 91</span>  </div>
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<div class="line"><a name="l00092"></a><span class="lineno"> 92</span>  <a class="code" href="../../d4/d99/ode__semi__implicit__euler_8c.html#a97075291390a68c262ed66e157a57eb4">problem</a>(x, y, dy); <span class="comment">// update dy once more</span></div>
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<div class="line"><a name="l00093"></a><span class="lineno"> 93</span>  </div>
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<div class="line"><a name="l00094"></a><span class="lineno"> 94</span>  <span class="keywordflow">for</span> (o = 1; o < <a class="code" href="../../d4/d99/ode__semi__implicit__euler_8c.html#a9ceb646336224ee890a269d0b4600d09">order</a>; o++)</div>
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<div class="line"><a name="l00095"></a><span class="lineno"> 95</span>  y[o] += dx * dy[o]; <span class="comment">// update remaining using new dy</span></div>
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<div class="line"><a name="l00096"></a><span class="lineno"> 96</span>  *x += dx;</div>
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<div class="line"><a name="l00097"></a><span class="lineno"> 97</span> }</div>
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</div><!-- fragment --><div class="dynheader">
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<div class="center"><iframe scrolling="no" frameborder="0" src="../../d4/d99/ode__semi__implicit__euler_8c_a720b7e995d2bbc615f94a2c7dbcf84eb_cgraph.svg" width="292" height="38"><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><!-- contents -->
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<div class="ttc" id="aode__semi__implicit__euler_8c_html_a97075291390a68c262ed66e157a57eb4"><div class="ttname"><a href="../../d4/d99/ode__semi__implicit__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_semi_implicit_euler.c:58</div></div>
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<div class="ttc" id="aode__semi__implicit__euler_8c_html_a9ceb646336224ee890a269d0b4600d09"><div class="ttname"><a href="../../d4/d99/ode__semi__implicit__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_semi_implicit_euler.c:47</div></div>
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<div class="ttc" id="aode__semi__implicit__euler_8c_html_a720b7e995d2bbc615f94a2c7dbcf84eb"><div class="ttname"><a href="../../d4/d99/ode__semi__implicit__euler_8c.html#a720b7e995d2bbc615f94a2c7dbcf84eb">semi_implicit_euler_step</a></div><div class="ttdeci">void semi_implicit_euler_step(double dx, double *x, double *y, double *dy)</div><div class="ttdoc">Compute next step approximation using the semi-implicit-Euler method.</div><div class="ttdef"><b>Definition:</b> ode_semi_implicit_euler.c:85</div></div>
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