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<a href="#pub-attribs">Data Fields</a> </div>
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<div class="title">pid Struct Reference</div> </div>
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Data Fields</h2></td></tr>
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<a name="details" id="details"></a><h2 class="groupheader">Detailed Description</h2>
<div class="textblock"><p>PID Controller</p>
<p>The PID controller is a linear control algorithm that has three terms:</p><ul>
<li>Proportional: A simple scaling of the error value by a gain kP</li>
<li>Integral: Integration of the error value over time, then multipled by gain kI</li>
<li>Derivative: Rate of change of the error value over time, multiplied by gain kD</li>
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<p>Terms of the controller can be removed by setting their gain to 0, creating a PI (kD = 0) or PD (kI = 0) controller. Depending on the control problem at hand, some terms may not increase the performance of the system, or may have a negative effect.</p>
<p>For a more mathematical expanation of the PID Controller, see <a href="https://en.wikipedia.org/wiki/PID_controller">https://en.wikipedia.org/wiki/PID_controller</a></p>
<p>Limitations of this implementation:</p><ul>
<li>Since this implementation is just for demonstration, the pid_step function takes the dt as a parameter, and it can be provided by the user in <a class="el" href="../../d7/dd8/c__atoi__str__to__integer_8c.html#a3c04138a5bfe5d72780bb7e82a18e627">main()</a>. This allows deterministic experimentation with the algorithm, rather than using time(NULL) which would make the function non-deterministic.</li>
</ul>
<p>Inputs: e(t) - Current error at time t. For example, how far a servo is off the desired angle Output: u(t) - Controller output at time t. </p>
</div><hr/>The documentation for this struct was generated from the following file:<ul>
<li>misc/pid.c</li>
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<p><a href="https://projecteuler.net/problem=4">Problem 4</a> solution
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Functions</h2></td></tr>
<tr class="memitem:adf9bea8d35848959bde5b3f277edf0c4"><td class="memItemLeft" align="right" valign="top">int&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../d0/d6c/problem__4_2sol_8c.html#adf9bea8d35848959bde5b3f277edf0c4">is_palindromic</a> (unsigned int n)</td></tr>
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<a name="details" id="details"></a><h2 class="groupheader">Detailed Description</h2>
<div class="textblock"><p><a href="https://projecteuler.net/problem=4">Problem 4</a> solution </p>
</div><h2 class="groupheader">Function Documentation</h2>
<a id="adf9bea8d35848959bde5b3f277edf0c4"></a>
<h2 class="memtitle"><span class="permalink"><a href="#adf9bea8d35848959bde5b3f277edf0c4">&#9670;&nbsp;</a></span>is_palindromic()</h2>
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<td class="memname">int is_palindromic </td>
<td>(</td>
<td class="paramtype">unsigned int&#160;</td>
<td class="paramname"><em>n</em></td><td>)</td>
<td></td>
</tr>
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<p>Check if number is palindromic </p><dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramdir">[in]</td><td class="paramname">n</td><td>number to check </td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>1 if palindromic </dd>
<dd>
0 if not palindromic </dd></dl>
<div class="fragment"><div class="line"><a name="l00013"></a><span class="lineno"> 13</span>&#160;{</div>
<div class="line"><a name="l00014"></a><span class="lineno"> 14</span>&#160; <span class="keywordtype">unsigned</span> <span class="keywordtype">int</span> reversed = 0, t = n;</div>
<div class="line"><a name="l00015"></a><span class="lineno"> 15</span>&#160; </div>
<div class="line"><a name="l00016"></a><span class="lineno"> 16</span>&#160; <span class="keywordflow">while</span> (t &gt; 0)</div>
<div class="line"><a name="l00017"></a><span class="lineno"> 17</span>&#160; {</div>
<div class="line"><a name="l00018"></a><span class="lineno"> 18</span>&#160; reversed = 10 * reversed + (t % 10);</div>
<div class="line"><a name="l00019"></a><span class="lineno"> 19</span>&#160; t /= 10;</div>
<div class="line"><a name="l00020"></a><span class="lineno"> 20</span>&#160; }</div>
<div class="line"><a name="l00021"></a><span class="lineno"> 21</span>&#160; <span class="keywordflow">return</span> reversed == n;</div>
<div class="line"><a name="l00022"></a><span class="lineno"> 22</span>&#160;}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#a840291bc02cba5474a4cb46a9b9566fe">&#9670;&nbsp;</a></span>main()</h2>
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<td class="memname">int main </td>
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<td class="paramtype">void&#160;</td>
<td class="paramname"></td><td>)</td>
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<p>Main function </p>
<div class="fragment"><div class="line"><a name="l00026"></a><span class="lineno"> 26</span>&#160;{</div>
<div class="line"><a name="l00027"></a><span class="lineno"> 27</span>&#160; <span class="keywordtype">unsigned</span> <span class="keywordtype">int</span> i, j, <a class="code" href="../../d2/df6/kohonen__som__topology_8c.html#affe776513b24d84b39af8ab0930fef7f">max</a> = 0;</div>
<div class="line"><a name="l00028"></a><span class="lineno"> 28</span>&#160; <span class="keywordflow">for</span> (i = 100; i &lt;= 999; i++)</div>
<div class="line"><a name="l00029"></a><span class="lineno"> 29</span>&#160; {</div>
<div class="line"><a name="l00030"></a><span class="lineno"> 30</span>&#160; <span class="keywordflow">for</span> (j = 100; j &lt;= 999; j++)</div>
<div class="line"><a name="l00031"></a><span class="lineno"> 31</span>&#160; {</div>
<div class="line"><a name="l00032"></a><span class="lineno"> 32</span>&#160; <span class="keywordtype">unsigned</span> <span class="keywordtype">int</span> p = i * j;</div>
<div class="line"><a name="l00033"></a><span class="lineno"> 33</span>&#160; <span class="keywordflow">if</span> (<a class="code" href="../../d0/d6c/problem__4_2sol_8c.html#adf9bea8d35848959bde5b3f277edf0c4">is_palindromic</a>(p) &amp;&amp; p &gt; <a class="code" href="../../d2/df6/kohonen__som__topology_8c.html#affe776513b24d84b39af8ab0930fef7f">max</a>)</div>
<div class="line"><a name="l00034"></a><span class="lineno"> 34</span>&#160; {</div>
<div class="line"><a name="l00035"></a><span class="lineno"> 35</span>&#160; <a class="code" href="../../d2/df6/kohonen__som__topology_8c.html#affe776513b24d84b39af8ab0930fef7f">max</a> = p;</div>
<div class="line"><a name="l00036"></a><span class="lineno"> 36</span>&#160; }</div>
<div class="line"><a name="l00037"></a><span class="lineno"> 37</span>&#160; }</div>
<div class="line"><a name="l00038"></a><span class="lineno"> 38</span>&#160; }</div>
<div class="line"><a name="l00039"></a><span class="lineno"> 39</span>&#160; printf(<span class="stringliteral">&quot;%u\n&quot;</span>, <a class="code" href="../../d2/df6/kohonen__som__topology_8c.html#affe776513b24d84b39af8ab0930fef7f">max</a>);</div>
<div class="line"><a name="l00040"></a><span class="lineno"> 40</span>&#160; <span class="keywordflow">return</span> 0;</div>
<div class="line"><a name="l00041"></a><span class="lineno"> 41</span>&#160;}</div>
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<p><a href="https://projecteuler.net/problem=10">Problem 10</a> solution
<a href="#details">More...</a></p>
<div class="textblock"><code>#include &lt;math.h&gt;</code><br />
<code>#include &lt;stdio.h&gt;</code><br />
<code>#include &lt;stdlib.h&gt;</code><br />
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Functions</h2></td></tr>
<tr class="memitem:acc871ab6bfead702e983a7f9c412915f"><td class="memItemLeft" align="right" valign="top">char&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../d0/d6d/problem__10_2sol1_8c.html#acc871ab6bfead702e983a7f9c412915f">is_prime</a> (unsigned long n)</td></tr>
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<tr class="memitem:ae3d987cb2ad0ddb0c3caa4c2506a20e5"><td class="memItemLeft" align="right" valign="top">unsigned long long&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../d0/d6d/problem__10_2sol1_8c.html#ae3d987cb2ad0ddb0c3caa4c2506a20e5">sum_of_primes</a> (unsigned long <a class="el" href="../../db/d01/problem__13_2sol1_8c.html#a0240ac851181b84ac374872dc5434ee4">N</a>)</td></tr>
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<tr class="memitem:a0ddf1224851353fc92bfbff6f499fa97"><td class="memItemLeft" align="right" valign="top">int&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../d0/d6d/problem__10_2sol1_8c.html#a0ddf1224851353fc92bfbff6f499fa97">main</a> (int argc, char *argv[])</td></tr>
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<a name="details" id="details"></a><h2 class="groupheader">Detailed Description</h2>
<div class="textblock"><p><a href="https://projecteuler.net/problem=10">Problem 10</a> solution </p>
<dl class="section author"><dt>Author</dt><dd><a href="https://github.com/kvedala">Krishna Vedala</a> </dd></dl>
</div><h2 class="groupheader">Function Documentation</h2>
<a id="acc871ab6bfead702e983a7f9c412915f"></a>
<h2 class="memtitle"><span class="permalink"><a href="#acc871ab6bfead702e983a7f9c412915f">&#9670;&nbsp;</a></span>is_prime()</h2>
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<td class="memname">char is_prime </td>
<td>(</td>
<td class="paramtype">unsigned long&#160;</td>
<td class="paramname"><em>n</em></td><td>)</td>
<td></td>
</tr>
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<p>Function to check if a number is prime </p>
<div class="fragment"><div class="line"><a name="l00012"></a><span class="lineno"> 12</span>&#160;{</div>
<div class="line"><a name="l00013"></a><span class="lineno"> 13</span>&#160; <span class="keywordflow">for</span> (<span class="keywordtype">unsigned</span> <span class="keywordtype">long</span> i = 2; i &lt; sqrtl(n) + 1; i++)</div>
<div class="line"><a name="l00014"></a><span class="lineno"> 14</span>&#160; <span class="keywordflow">if</span> (n % i == 0)</div>
<div class="line"><a name="l00015"></a><span class="lineno"> 15</span>&#160; <span class="keywordflow">return</span> 0;</div>
<div class="line"><a name="l00016"></a><span class="lineno"> 16</span>&#160; </div>
<div class="line"><a name="l00017"></a><span class="lineno"> 17</span>&#160; <span class="keywordflow">return</span> 1;</div>
<div class="line"><a name="l00018"></a><span class="lineno"> 18</span>&#160;}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#a0ddf1224851353fc92bfbff6f499fa97">&#9670;&nbsp;</a></span>main()</h2>
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<td class="memname">int main </td>
<td>(</td>
<td class="paramtype">int&#160;</td>
<td class="paramname"><em>argc</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">char *&#160;</td>
<td class="paramname"><em>argv</em>[]&#160;</td>
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<p>Main function </p>
<div class="fragment"><div class="line"><a name="l00034"></a><span class="lineno"> 34</span>&#160;{</div>
<div class="line"><a name="l00035"></a><span class="lineno"> 35</span>&#160; <span class="keywordtype">unsigned</span> <span class="keywordtype">long</span> n = 100;</div>
<div class="line"><a name="l00036"></a><span class="lineno"> 36</span>&#160; </div>
<div class="line"><a name="l00037"></a><span class="lineno"> 37</span>&#160; <span class="keywordflow">if</span> (argc == 2) <span class="comment">/* if command line argument is provided */</span></div>
<div class="line"><a name="l00038"></a><span class="lineno"> 38</span>&#160; n = atol(argv[1]); <span class="comment">/* use that as the upper limit */</span></div>
<div class="line"><a name="l00039"></a><span class="lineno"> 39</span>&#160; </div>
<div class="line"><a name="l00040"></a><span class="lineno"> 40</span>&#160; printf(<span class="stringliteral">&quot;%ld: %llu\n&quot;</span>, n, <a class="code" href="../../d0/d6d/problem__10_2sol1_8c.html#ae3d987cb2ad0ddb0c3caa4c2506a20e5">sum_of_primes</a>(n));</div>
<div class="line"><a name="l00041"></a><span class="lineno"> 41</span>&#160; </div>
<div class="line"><a name="l00042"></a><span class="lineno"> 42</span>&#160; <span class="keywordflow">return</span> 0;</div>
<div class="line"><a name="l00043"></a><span class="lineno"> 43</span>&#160;}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#ae3d987cb2ad0ddb0c3caa4c2506a20e5">&#9670;&nbsp;</a></span>sum_of_primes()</h2>
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<td class="memname">unsigned long long sum_of_primes </td>
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<td class="paramname"><em>N</em></td><td>)</td>
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<p>Computes sum of prime numbers less than N </p>
<div class="fragment"><div class="line"><a name="l00022"></a><span class="lineno"> 22</span>&#160;{</div>
<div class="line"><a name="l00023"></a><span class="lineno"> 23</span>&#160; <span class="keywordtype">unsigned</span> <span class="keywordtype">long</span> <span class="keywordtype">long</span> sum = 2;</div>
<div class="line"><a name="l00024"></a><span class="lineno"> 24</span>&#160; </div>
<div class="line"><a name="l00025"></a><span class="lineno"> 25</span>&#160; <span class="keywordflow">for</span> (<span class="keywordtype">long</span> i = 3; i &lt; <a class="code" href="../../db/d01/problem__13_2sol1_8c.html#a0240ac851181b84ac374872dc5434ee4">N</a>; i += 2) <span class="comment">/* skip even numbers */</span></div>
<div class="line"><a name="l00026"></a><span class="lineno"> 26</span>&#160; <span class="keywordflow">if</span> (<a class="code" href="../../d0/d6d/problem__10_2sol1_8c.html#acc871ab6bfead702e983a7f9c412915f">is_prime</a>(i))</div>
<div class="line"><a name="l00027"></a><span class="lineno"> 27</span>&#160; sum += i;</div>
<div class="line"><a name="l00028"></a><span class="lineno"> 28</span>&#160; </div>
<div class="line"><a name="l00029"></a><span class="lineno"> 29</span>&#160; <span class="keywordflow">return</span> sum;</div>
<div class="line"><a name="l00030"></a><span class="lineno"> 30</span>&#160;}</div>
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<div class="ttc" id="aproblem__10_2sol1_8c_html_acc871ab6bfead702e983a7f9c412915f"><div class="ttname"><a href="../../d0/d6d/problem__10_2sol1_8c.html#acc871ab6bfead702e983a7f9c412915f">is_prime</a></div><div class="ttdeci">char is_prime(unsigned long n)</div><div class="ttdef"><b>Definition:</b> sol1.c:11</div></div>
<div class="ttc" id="aproblem__13_2sol1_8c_html_a0240ac851181b84ac374872dc5434ee4"><div class="ttname"><a href="../../db/d01/problem__13_2sol1_8c.html#a0240ac851181b84ac374872dc5434ee4">N</a></div><div class="ttdeci">#define N</div><div class="ttdef"><b>Definition:</b> sol1.c:111</div></div>
<div class="ttc" id="aproblem__10_2sol1_8c_html_ae3d987cb2ad0ddb0c3caa4c2506a20e5"><div class="ttname"><a href="../../d0/d6d/problem__10_2sol1_8c.html#ae3d987cb2ad0ddb0c3caa4c2506a20e5">sum_of_primes</a></div><div class="ttdeci">unsigned long long sum_of_primes(unsigned long N)</div><div class="ttdef"><b>Definition:</b> sol1.c:21</div></div>
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<p><a href="https://projecteuler.net/problem=2">Problem 2</a> solution
<a href="#details">More...</a></p>
<div class="textblock"><code>#include &lt;stdio.h&gt;</code><br />
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Include dependency graph for so1.c:</div>
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Functions</h2></td></tr>
<tr class="memitem:ae66f6b31b5ad750f1fe042a706a4e3d4"><td class="memItemLeft" align="right" valign="top">int&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../d0/d7f/so1_8c.html#ae66f6b31b5ad750f1fe042a706a4e3d4">main</a> ()</td></tr>
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<a name="details" id="details"></a><h2 class="groupheader">Detailed Description</h2>
<div class="textblock"><p><a href="https://projecteuler.net/problem=2">Problem 2</a> solution </p>
<p>Problem:</p>
<p>Each new term in the Fibonacci sequence is generated by adding the previous two terms. By starting with 1 and 2, the first 10 terms will be: <code>1,2,3,5,8,13,21,34,55,89,..</code> By considering the terms in the Fibonacci sequence whose values do not exceed n, find the sum of the even-valued terms. e.g. for n=10, we have {2,8}, sum is 10. </p>
</div><h2 class="groupheader">Function Documentation</h2>
<a id="ae66f6b31b5ad750f1fe042a706a4e3d4"></a>
<h2 class="memtitle"><span class="permalink"><a href="#ae66f6b31b5ad750f1fe042a706a4e3d4">&#9670;&nbsp;</a></span>main()</h2>
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<td class="memname">int main </td>
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<td class="paramname"></td><td>)</td>
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<p>Main function </p>
<div class="fragment"><div class="line"><a name="l00018"></a><span class="lineno"> 18</span>&#160;{</div>
<div class="line"><a name="l00019"></a><span class="lineno"> 19</span>&#160; <span class="keywordtype">int</span> n = 0;</div>
<div class="line"><a name="l00020"></a><span class="lineno"> 20</span>&#160; <span class="keywordtype">int</span> sum = 0;</div>
<div class="line"><a name="l00021"></a><span class="lineno"> 21</span>&#160; <span class="keywordtype">int</span> i = 1;</div>
<div class="line"><a name="l00022"></a><span class="lineno"> 22</span>&#160; <span class="keywordtype">int</span> j = 2;</div>
<div class="line"><a name="l00023"></a><span class="lineno"> 23</span>&#160; <span class="keywordtype">int</span> temp;</div>
<div class="line"><a name="l00024"></a><span class="lineno"> 24</span>&#160; scanf(<span class="stringliteral">&quot;%d&quot;</span>, &amp;n);</div>
<div class="line"><a name="l00025"></a><span class="lineno"> 25</span>&#160; </div>
<div class="line"><a name="l00026"></a><span class="lineno"> 26</span>&#160; <span class="keywordflow">while</span> (j &lt;= n)</div>
<div class="line"><a name="l00027"></a><span class="lineno"> 27</span>&#160; {</div>
<div class="line"><a name="l00028"></a><span class="lineno"> 28</span>&#160; <span class="keywordflow">if</span> ((j &amp; 1) == 0) <span class="comment">// can also use(j%2 == 0)</span></div>
<div class="line"><a name="l00029"></a><span class="lineno"> 29</span>&#160; sum += j;</div>
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Content</h2>
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<li>stack-Header file for import.</li>
<li>stack.c implementation of the stack</li>
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<p><a href="https://projecteuler.net/problem=7">Problem 7</a> solution
<a href="#details">More...</a></p>
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Functions</h2></td></tr>
<tr class="memitem:a840291bc02cba5474a4cb46a9b9566fe"><td class="memItemLeft" align="right" valign="top">int&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../d1/d2f/problem__7_2sol_8c.html#a840291bc02cba5474a4cb46a9b9566fe">main</a> (void)</td></tr>
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<div class="textblock"><p><a href="https://projecteuler.net/problem=7">Problem 7</a> solution </p>
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<h2 class="memtitle"><span class="permalink"><a href="#a840291bc02cba5474a4cb46a9b9566fe">&#9670;&nbsp;</a></span>main()</h2>
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<td class="memname">int main </td>
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<td class="paramtype">void&#160;</td>
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<p>Main function </p>
<div class="fragment"><div class="line"><a name="l00010"></a><span class="lineno"> 10</span>&#160;{</div>
<div class="line"><a name="l00011"></a><span class="lineno"> 11</span>&#160; <span class="keywordtype">char</span> *sieve;</div>
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<div class="line"><a name="l00013"></a><span class="lineno"> 13</span>&#160; <span class="keywordtype">unsigned</span> count = 0;</div>
<div class="line"><a name="l00014"></a><span class="lineno"> 14</span>&#160; <span class="keywordtype">size_t</span> n = 1000000;</div>
<div class="line"><a name="l00015"></a><span class="lineno"> 15</span>&#160; <span class="keyword">const</span> <span class="keywordtype">unsigned</span> target = 10001;</div>
<div class="line"><a name="l00016"></a><span class="lineno"> 16</span>&#160; </div>
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<div class="line"><a name="l00019"></a><span class="lineno"> 19</span>&#160; {</div>
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<div class="line"><a name="l00021"></a><span class="lineno"> 21</span>&#160; {</div>
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<div class="line"><a name="l00025"></a><span class="lineno"> 25</span>&#160; {</div>
<div class="line"><a name="l00026"></a><span class="lineno"> 26</span>&#160; printf(<span class="stringliteral">&quot;%lu\n&quot;</span>, i);</div>
<div class="line"><a name="l00027"></a><span class="lineno"> 27</span>&#160; <span class="keywordflow">break</span>;</div>
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<div class="line"><a name="l00030"></a><span class="lineno"> 30</span>&#160; {</div>
<div class="line"><a name="l00031"></a><span class="lineno"> 31</span>&#160; sieve[j] = 1;</div>
<div class="line"><a name="l00032"></a><span class="lineno"> 32</span>&#160; }</div>
<div class="line"><a name="l00033"></a><span class="lineno"> 33</span>&#160; }</div>
<div class="line"><a name="l00034"></a><span class="lineno"> 34</span>&#160; }</div>
<div class="line"><a name="l00035"></a><span class="lineno"> 35</span>&#160; free(sieve);</div>
<div class="line"><a name="l00036"></a><span class="lineno"> 36</span>&#160; <span class="keywordflow">return</span> 0;</div>
<div class="line"><a name="l00037"></a><span class="lineno"> 37</span>&#160;}</div>
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<div class="title">ode_midpoint_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/Midpoint_method">midpoint Euler method</a>
<a href="#details">More...</a></p>
<div class="textblock"><code>#include &lt;math.h&gt;</code><br />
<code>#include &lt;stdio.h&gt;</code><br />
<code>#include &lt;stdlib.h&gt;</code><br />
<code>#include &lt;time.h&gt;</code><br />
</div><div class="textblock"><div class="dynheader">
Include dependency graph for ode_midpoint_euler.c:</div>
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Macros</h2></td></tr>
<tr class="memitem:a9ceb646336224ee890a269d0b4600d09"><td class="memItemLeft" align="right" valign="top">#define&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../d1/dc2/ode__midpoint__euler_8c.html#a9ceb646336224ee890a269d0b4600d09">order</a>&#160;&#160;&#160;2</td></tr>
<tr class="separator:a9ceb646336224ee890a269d0b4600d09"><td class="memSeparator" colspan="2">&#160;</td></tr>
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Functions</h2></td></tr>
<tr class="memitem:a97075291390a68c262ed66e157a57eb4"><td class="memItemLeft" align="right" valign="top">void&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../d1/dc2/ode__midpoint__euler_8c.html#a97075291390a68c262ed66e157a57eb4">problem</a> (const double *x, double *y, double *dy)</td></tr>
<tr class="memdesc:a97075291390a68c262ed66e157a57eb4"><td class="mdescLeft">&#160;</td><td class="mdescRight">Problem statement for a system with first-order differential equations. Updates the system differential variables. <a href="../../d1/dc2/ode__midpoint__euler_8c.html#a97075291390a68c262ed66e157a57eb4">More...</a><br /></td></tr>
<tr class="separator:a97075291390a68c262ed66e157a57eb4"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:a8caee977b26888d34040b122e0e28e3a"><td class="memItemLeft" align="right" valign="top">void&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../d1/dc2/ode__midpoint__euler_8c.html#a8caee977b26888d34040b122e0e28e3a">exact_solution</a> (const double *x, double *y)</td></tr>
<tr class="memdesc:a8caee977b26888d34040b122e0e28e3a"><td class="mdescLeft">&#160;</td><td class="mdescRight">Exact solution of the problem. Used for solution comparison. <a href="../../d1/dc2/ode__midpoint__euler_8c.html#a8caee977b26888d34040b122e0e28e3a">More...</a><br /></td></tr>
<tr class="separator:a8caee977b26888d34040b122e0e28e3a"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:affe6cc2ab040b94a29e6c41782f72d51"><td class="memItemLeft" align="right" valign="top">void&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../d1/dc2/ode__midpoint__euler_8c.html#affe6cc2ab040b94a29e6c41782f72d51">midpoint_euler_step</a> (double dx, double *x, double *y, double *dy)</td></tr>
<tr class="memdesc:affe6cc2ab040b94a29e6c41782f72d51"><td class="mdescLeft">&#160;</td><td class="mdescRight">Compute next step approximation using the midpoint-Euler method. <a href="../../d1/dc2/ode__midpoint__euler_8c.html#affe6cc2ab040b94a29e6c41782f72d51">More...</a><br /></td></tr>
<tr class="separator:affe6cc2ab040b94a29e6c41782f72d51"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:a148003d8b261d040c1c41e73b40af1dd"><td class="memItemLeft" align="right" valign="top">double&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../d1/dc2/ode__midpoint__euler_8c.html#a148003d8b261d040c1c41e73b40af1dd">midpoint_euler</a> (double dx, double x0, double x_max, double *y, char save_to_file)</td></tr>
<tr class="memdesc:a148003d8b261d040c1c41e73b40af1dd"><td class="mdescLeft">&#160;</td><td class="mdescRight">Compute approximation using the midpoint-Euler method in the given limits. <a href="../../d1/dc2/ode__midpoint__euler_8c.html#a148003d8b261d040c1c41e73b40af1dd">More...</a><br /></td></tr>
<tr class="separator:a148003d8b261d040c1c41e73b40af1dd"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:a0ddf1224851353fc92bfbff6f499fa97"><td class="memItemLeft" align="right" valign="top">int&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../d1/dc2/ode__midpoint__euler_8c.html#a0ddf1224851353fc92bfbff6f499fa97">main</a> (int argc, char *argv[])</td></tr>
<tr class="separator:a0ddf1224851353fc92bfbff6f499fa97"><td class="memSeparator" colspan="2">&#160;</td></tr>
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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/Midpoint_method">midpoint Euler method</a> </p>
<dl class="section author"><dt>Authors</dt><dd><a href="https://github.com/kvedala">Krishna Vedala</a></dd></dl>
<p>The ODE being solved is: </p><p class="formulaDsp">
\begin{eqnarray*} \dot{u} &amp;=&amp; v\\ \dot{v} &amp;=&amp; -\omega^2 u\\ \omega &amp;=&amp; 1\\ [x_0, u_0, v_0] &amp;=&amp; [0,1,0]\qquad\ldots\text{(initial values)} \end{eqnarray*}
</p>
<p> The exact solution for the above problem is: </p><p class="formulaDsp">
\begin{eqnarray*} u(x) &amp;=&amp; \cos(x)\\ v(x) &amp;=&amp; -\sin(x)\\ \end{eqnarray*}
</p>
<p> The computation results are stored to a text file <code>midpoint_euler.csv</code> and the exact soltuion results in <code>exact.csv</code> for comparison. <img src="https://raw.githubusercontent.com/kvedala/C/docs/images/numerical_methods/ode_midpoint_euler.svg" alt="Implementation solution" style="pointer-events: none;" class="inline"/></p>
<p>To implement <a href="https://en.wikipedia.org/wiki/Van_der_Pol_oscillator">Van der Pol oscillator</a>, change the <a class="el" href="../../d1/dc2/ode__midpoint__euler_8c.html#a97075291390a68c262ed66e157a57eb4" title="Problem statement for a system with first-order differential equations. Updates the system differenti...">problem</a> function to: </p><div class="fragment"><div class="line"><span class="keyword">const</span> <span class="keywordtype">double</span> mu = 2.0;</div>
<div class="line">dy[0] = y[1];</div>
<div class="line">dy[1] = mu * (1.f - y[0] * y[0]) * y[1] - y[0];</div>
</div><!-- fragment --><dl class="section see"><dt>See also</dt><dd><a class="el" href="../../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="../../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>
</div><h2 class="groupheader">Macro Definition Documentation</h2>
<a id="a9ceb646336224ee890a269d0b4600d09"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a9ceb646336224ee890a269d0b4600d09">&#9670;&nbsp;</a></span>order</h2>
<div class="memitem">
<div class="memproto">
<table class="memname">
<tr>
<td class="memname">#define order&#160;&#160;&#160;2</td>
</tr>
</table>
</div><div class="memdoc">
<p>number of dependent variables in <a class="el" href="../../d1/dc2/ode__midpoint__euler_8c.html#a97075291390a68c262ed66e157a57eb4" title="Problem statement for a system with first-order differential equations. Updates the system differenti...">problem</a> </p>
</div>
</div>
<h2 class="groupheader">Function Documentation</h2>
<a id="a8caee977b26888d34040b122e0e28e3a"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a8caee977b26888d34040b122e0e28e3a">&#9670;&nbsp;</a></span>exact_solution()</h2>
<div class="memitem">
<div class="memproto">
<table class="memname">
<tr>
<td class="memname">void exact_solution </td>
<td>(</td>
<td class="paramtype">const double *&#160;</td>
<td class="paramname"><em>x</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">double *&#160;</td>
<td class="paramname"><em>y</em>&#160;</td>
</tr>
<tr>
<td></td>
<td>)</td>
<td></td><td></td>
</tr>
</table>
</div><div class="memdoc">
<p>Exact solution of the problem. Used for solution comparison. </p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramdir">[in]</td><td class="paramname">x</td><td>independent variable </td></tr>
<tr><td class="paramdir">[in,out]</td><td class="paramname">y</td><td>dependent variable </td></tr>
</table>
</dd>
</dl>
<div class="fragment"><div class="line"><a name="l00068"></a><span class="lineno"> 68</span>&#160;{</div>
<div class="line"><a name="l00069"></a><span class="lineno"> 69</span>&#160; y[0] = cos(x[0]);</div>
<div class="line"><a name="l00070"></a><span class="lineno"> 70</span>&#160; y[1] = -sin(x[0]);</div>
<div class="line"><a name="l00071"></a><span class="lineno"> 71</span>&#160;}</div>
</div><!-- fragment -->
</div>
</div>
<a id="a0ddf1224851353fc92bfbff6f499fa97"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a0ddf1224851353fc92bfbff6f499fa97">&#9670;&nbsp;</a></span>main()</h2>
<div class="memitem">
<div class="memproto">
<table class="memname">
<tr>
<td class="memname">int main </td>
<td>(</td>
<td class="paramtype">int&#160;</td>
<td class="paramname"><em>argc</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">char *&#160;</td>
<td class="paramname"><em>argv</em>[]&#160;</td>
</tr>
<tr>
<td></td>
<td>)</td>
<td></td><td></td>
</tr>
</table>
</div><div class="memdoc">
<p>Main Function </p>
<div class="fragment"><div class="line"><a name="l00147"></a><span class="lineno"> 147</span>&#160;{</div>
<div class="line"><a name="l00148"></a><span class="lineno"> 148</span>&#160; <span class="keywordtype">double</span> X0 = 0.f; <span class="comment">/* initial value of x0 */</span></div>
<div class="line"><a name="l00149"></a><span class="lineno"> 149</span>&#160; <span class="keywordtype">double</span> X_MAX = 10.F; <span class="comment">/* upper limit of integration */</span></div>
<div class="line"><a name="l00150"></a><span class="lineno"> 150</span>&#160; <span class="keywordtype">double</span> Y0[] = {1.f, 0.f}; <span class="comment">/* initial value Y = y(x = x_0) */</span></div>
<div class="line"><a name="l00151"></a><span class="lineno"> 151</span>&#160; <span class="keywordtype">double</span> step_size;</div>
<div class="line"><a name="l00152"></a><span class="lineno"> 152</span>&#160; </div>
<div class="line"><a name="l00153"></a><span class="lineno"> 153</span>&#160; <span class="keywordflow">if</span> (argc == 1)</div>
<div class="line"><a name="l00154"></a><span class="lineno"> 154</span>&#160; {</div>
<div class="line"><a name="l00155"></a><span class="lineno"> 155</span>&#160; printf(<span class="stringliteral">&quot;\nEnter the step size: &quot;</span>);</div>
<div class="line"><a name="l00156"></a><span class="lineno"> 156</span>&#160; scanf(<span class="stringliteral">&quot;%lg&quot;</span>, &amp;step_size);</div>
<div class="line"><a name="l00157"></a><span class="lineno"> 157</span>&#160; }</div>
<div class="line"><a name="l00158"></a><span class="lineno"> 158</span>&#160; <span class="keywordflow">else</span></div>
<div class="line"><a name="l00159"></a><span class="lineno"> 159</span>&#160; <span class="comment">// use commandline argument as independent variable step size</span></div>
<div class="line"><a name="l00160"></a><span class="lineno"> 160</span>&#160; step_size = atof(argv[1]);</div>
<div class="line"><a name="l00161"></a><span class="lineno"> 161</span>&#160; </div>
<div class="line"><a name="l00162"></a><span class="lineno"> 162</span>&#160; <span class="comment">// get approximate solution</span></div>
<div class="line"><a name="l00163"></a><span class="lineno"> 163</span>&#160; <span class="keywordtype">double</span> total_time = <a class="code" href="../../d1/dc2/ode__midpoint__euler_8c.html#a148003d8b261d040c1c41e73b40af1dd">midpoint_euler</a>(step_size, X0, X_MAX, Y0, 1);</div>
<div class="line"><a name="l00164"></a><span class="lineno"> 164</span>&#160; printf(<span class="stringliteral">&quot;\tTime = %.6g ms\n&quot;</span>, total_time);</div>
<div class="line"><a name="l00165"></a><span class="lineno"> 165</span>&#160; </div>
<div class="line"><a name="l00166"></a><span class="lineno"> 166</span>&#160; <span class="comment">/* compute exact solution for comparion */</span></div>
<div class="line"><a name="l00167"></a><span class="lineno"> 167</span>&#160; FILE *fp = fopen(<span class="stringliteral">&quot;exact.csv&quot;</span>, <span class="stringliteral">&quot;w+&quot;</span>);</div>
<div class="line"><a name="l00168"></a><span class="lineno"> 168</span>&#160; <span class="keywordflow">if</span> (fp == NULL)</div>
<div class="line"><a name="l00169"></a><span class="lineno"> 169</span>&#160; {</div>
<div class="line"><a name="l00170"></a><span class="lineno"> 170</span>&#160; perror(<span class="stringliteral">&quot;Error! &quot;</span>);</div>
<div class="line"><a name="l00171"></a><span class="lineno"> 171</span>&#160; <span class="keywordflow">return</span> -1;</div>
<div class="line"><a name="l00172"></a><span class="lineno"> 172</span>&#160; }</div>
<div class="line"><a name="l00173"></a><span class="lineno"> 173</span>&#160; <span class="keywordtype">double</span> x = X0;</div>
<div class="line"><a name="l00174"></a><span class="lineno"> 174</span>&#160; <span class="keywordtype">double</span> *y = &amp;(Y0[0]);</div>
<div class="line"><a name="l00175"></a><span class="lineno"> 175</span>&#160; printf(<span class="stringliteral">&quot;Finding exact solution\n&quot;</span>);</div>
<div class="line"><a name="l00176"></a><span class="lineno"> 176</span>&#160; clock_t t1 = clock();</div>
<div class="line"><a name="l00177"></a><span class="lineno"> 177</span>&#160; </div>
<div class="line"><a name="l00178"></a><span class="lineno"> 178</span>&#160; <span class="keywordflow">do</span></div>
<div class="line"><a name="l00179"></a><span class="lineno"> 179</span>&#160; {</div>
<div class="line"><a name="l00180"></a><span class="lineno"> 180</span>&#160; fprintf(fp, <span class="stringliteral">&quot;%.4g,%.4g,%.4g\n&quot;</span>, x, y[0], y[1]); <span class="comment">// write to file</span></div>
<div class="line"><a name="l00181"></a><span class="lineno"> 181</span>&#160; <a class="code" href="../../d1/dc2/ode__midpoint__euler_8c.html#a8caee977b26888d34040b122e0e28e3a">exact_solution</a>(&amp;x, y);</div>
<div class="line"><a name="l00182"></a><span class="lineno"> 182</span>&#160; x += step_size;</div>
<div class="line"><a name="l00183"></a><span class="lineno"> 183</span>&#160; } <span class="keywordflow">while</span> (x &lt;= X_MAX);</div>
<div class="line"><a name="l00184"></a><span class="lineno"> 184</span>&#160; </div>
<div class="line"><a name="l00185"></a><span class="lineno"> 185</span>&#160; clock_t t2 = clock();</div>
<div class="line"><a name="l00186"></a><span class="lineno"> 186</span>&#160; total_time = (t2 - t1) / CLOCKS_PER_SEC;</div>
<div class="line"><a name="l00187"></a><span class="lineno"> 187</span>&#160; printf(<span class="stringliteral">&quot;\tTime = %.6g ms\n&quot;</span>, total_time);</div>
<div class="line"><a name="l00188"></a><span class="lineno"> 188</span>&#160; fclose(fp);</div>
<div class="line"><a name="l00189"></a><span class="lineno"> 189</span>&#160; </div>
<div class="line"><a name="l00190"></a><span class="lineno"> 190</span>&#160; <span class="keywordflow">return</span> 0;</div>
<div class="line"><a name="l00191"></a><span class="lineno"> 191</span>&#160;}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#a148003d8b261d040c1c41e73b40af1dd">&#9670;&nbsp;</a></span>midpoint_euler()</h2>
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<td class="memname">double midpoint_euler </td>
<td>(</td>
<td class="paramtype">double&#160;</td>
<td class="paramname"><em>dx</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">double&#160;</td>
<td class="paramname"><em>x0</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">double&#160;</td>
<td class="paramname"><em>x_max</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">double *&#160;</td>
<td class="paramname"><em>y</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">char&#160;</td>
<td class="paramname"><em>save_to_file</em>&#160;</td>
</tr>
<tr>
<td></td>
<td>)</td>
<td></td><td></td>
</tr>
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</div><div class="memdoc">
<p>Compute approximation using the midpoint-Euler method in the given limits. </p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramdir">[in]</td><td class="paramname">dx</td><td>step size </td></tr>
<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,out]</td><td class="paramname">y</td><td>take \(y_n\) and compute \(y_{n+1}\) </td></tr>
<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>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>time taken for computation in seconds </dd></dl>
<div class="fragment"><div class="line"><a name="l00110"></a><span class="lineno"> 110</span>&#160;{</div>
<div class="line"><a name="l00111"></a><span class="lineno"> 111</span>&#160; <span class="keywordtype">double</span> dy[<a class="code" href="../../d1/dc2/ode__midpoint__euler_8c.html#a9ceb646336224ee890a269d0b4600d09">order</a>];</div>
<div class="line"><a name="l00112"></a><span class="lineno"> 112</span>&#160; </div>
<div class="line"><a name="l00113"></a><span class="lineno"> 113</span>&#160; FILE *fp = NULL;</div>
<div class="line"><a name="l00114"></a><span class="lineno"> 114</span>&#160; <span class="keywordflow">if</span> (save_to_file)</div>
<div class="line"><a name="l00115"></a><span class="lineno"> 115</span>&#160; {</div>
<div class="line"><a name="l00116"></a><span class="lineno"> 116</span>&#160; fp = fopen(<span class="stringliteral">&quot;midpoint_euler.csv&quot;</span>, <span class="stringliteral">&quot;w+&quot;</span>);</div>
<div class="line"><a name="l00117"></a><span class="lineno"> 117</span>&#160; <span class="keywordflow">if</span> (fp == NULL)</div>
<div class="line"><a name="l00118"></a><span class="lineno"> 118</span>&#160; {</div>
<div class="line"><a name="l00119"></a><span class="lineno"> 119</span>&#160; perror(<span class="stringliteral">&quot;Error! &quot;</span>);</div>
<div class="line"><a name="l00120"></a><span class="lineno"> 120</span>&#160; <span class="keywordflow">return</span> -1;</div>
<div class="line"><a name="l00121"></a><span class="lineno"> 121</span>&#160; }</div>
<div class="line"><a name="l00122"></a><span class="lineno"> 122</span>&#160; }</div>
<div class="line"><a name="l00123"></a><span class="lineno"> 123</span>&#160; </div>
<div class="line"><a name="l00124"></a><span class="lineno"> 124</span>&#160; <span class="comment">/* start integration */</span></div>
<div class="line"><a name="l00125"></a><span class="lineno"> 125</span>&#160; clock_t t1 = clock();</div>
<div class="line"><a name="l00126"></a><span class="lineno"> 126</span>&#160; <span class="keywordtype">double</span> x = x0;</div>
<div class="line"><a name="l00127"></a><span class="lineno"> 127</span>&#160; <span class="keywordflow">do</span> <span class="comment">// iterate for each step of independent variable</span></div>
<div class="line"><a name="l00128"></a><span class="lineno"> 128</span>&#160; {</div>
<div class="line"><a name="l00129"></a><span class="lineno"> 129</span>&#160; <span class="keywordflow">if</span> (save_to_file &amp;&amp; fp)</div>
<div class="line"><a name="l00130"></a><span class="lineno"> 130</span>&#160; fprintf(fp, <span class="stringliteral">&quot;%.4g,%.4g,%.4g\n&quot;</span>, x, y[0], y[1]); <span class="comment">// write to file</span></div>
<div class="line"><a name="l00131"></a><span class="lineno"> 131</span>&#160; <a class="code" href="../../d1/dc2/ode__midpoint__euler_8c.html#affe6cc2ab040b94a29e6c41782f72d51">midpoint_euler_step</a>(dx, &amp;x, y, dy); <span class="comment">// perform integration</span></div>
<div class="line"><a name="l00132"></a><span class="lineno"> 132</span>&#160; x += dx; <span class="comment">// update step</span></div>
<div class="line"><a name="l00133"></a><span class="lineno"> 133</span>&#160; } <span class="keywordflow">while</span> (x &lt;= x_max); <span class="comment">// till upper limit of independent variable</span></div>
<div class="line"><a name="l00134"></a><span class="lineno"> 134</span>&#160; <span class="comment">/* end of integration */</span></div>
<div class="line"><a name="l00135"></a><span class="lineno"> 135</span>&#160; clock_t t2 = clock();</div>
<div class="line"><a name="l00136"></a><span class="lineno"> 136</span>&#160; </div>
<div class="line"><a name="l00137"></a><span class="lineno"> 137</span>&#160; <span class="keywordflow">if</span> (save_to_file &amp;&amp; fp)</div>
<div class="line"><a name="l00138"></a><span class="lineno"> 138</span>&#160; fclose(fp);</div>
<div class="line"><a name="l00139"></a><span class="lineno"> 139</span>&#160; </div>
<div class="line"><a name="l00140"></a><span class="lineno"> 140</span>&#160; <span class="keywordflow">return</span> (<span class="keywordtype">double</span>)(t2 - t1) / CLOCKS_PER_SEC;</div>
<div class="line"><a name="l00141"></a><span class="lineno"> 141</span>&#160;}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#affe6cc2ab040b94a29e6c41782f72d51">&#9670;&nbsp;</a></span>midpoint_euler_step()</h2>
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<td class="memname">void midpoint_euler_step </td>
<td>(</td>
<td class="paramtype">double&#160;</td>
<td class="paramname"><em>dx</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">double *&#160;</td>
<td class="paramname"><em>x</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">double *&#160;</td>
<td class="paramname"><em>y</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">double *&#160;</td>
<td class="paramname"><em>dy</em>&#160;</td>
</tr>
<tr>
<td></td>
<td>)</td>
<td></td><td></td>
</tr>
</table>
</div><div class="memdoc">
<p>Compute next step approximation using the midpoint-Euler method. </p>
<p class="formulaDsp">
\[y_{n+1} = y_n + dx\, f\left(x_n+\frac{1}{2}dx, y_n + \frac{1}{2}dx\,f\left(x_n,y_n\right)\right)\]
</p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramdir">[in]</td><td class="paramname">dx</td><td>step size </td></tr>
<tr><td class="paramdir">[in,out]</td><td class="paramname">x</td><td>take \(x_n\) and compute \(x_{n+1}\) </td></tr>
<tr><td class="paramdir">[in,out]</td><td class="paramname">y</td><td>take \(y_n\) and compute \(y_{n+1}\) </td></tr>
<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>
</table>
</dd>
</dl>
<div class="fragment"><div class="line"><a name="l00084"></a><span class="lineno"> 84</span>&#160;{</div>
<div class="line"><a name="l00085"></a><span class="lineno"> 85</span>&#160; <a class="code" href="../../d1/dc2/ode__midpoint__euler_8c.html#a97075291390a68c262ed66e157a57eb4">problem</a>(x, y, dy);</div>
<div class="line"><a name="l00086"></a><span class="lineno"> 86</span>&#160; <span class="keywordtype">double</span> tmp_x = (*x) + 0.5 * dx;</div>
<div class="line"><a name="l00087"></a><span class="lineno"> 87</span>&#160; <span class="keywordtype">double</span> tmp_y[<a class="code" href="../../d1/dc2/ode__midpoint__euler_8c.html#a9ceb646336224ee890a269d0b4600d09">order</a>];</div>
<div class="line"><a name="l00088"></a><span class="lineno"> 88</span>&#160; <span class="keywordtype">int</span> o;</div>
<div class="line"><a name="l00089"></a><span class="lineno"> 89</span>&#160; <span class="keywordflow">for</span> (o = 0; o &lt; <a class="code" href="../../d1/dc2/ode__midpoint__euler_8c.html#a9ceb646336224ee890a269d0b4600d09">order</a>; o++)</div>
<div class="line"><a name="l00090"></a><span class="lineno"> 90</span>&#160; tmp_y[o] = y[o] + 0.5 * dx * dy[o];</div>
<div class="line"><a name="l00091"></a><span class="lineno"> 91</span>&#160; </div>
<div class="line"><a name="l00092"></a><span class="lineno"> 92</span>&#160; <a class="code" href="../../d1/dc2/ode__midpoint__euler_8c.html#a97075291390a68c262ed66e157a57eb4">problem</a>(&amp;tmp_x, tmp_y, dy);</div>
<div class="line"><a name="l00093"></a><span class="lineno"> 93</span>&#160; </div>
<div class="line"><a name="l00094"></a><span class="lineno"> 94</span>&#160; <span class="keywordflow">for</span> (o = 0; o &lt; <a class="code" href="../../d1/dc2/ode__midpoint__euler_8c.html#a9ceb646336224ee890a269d0b4600d09">order</a>; o++)</div>
<div class="line"><a name="l00095"></a><span class="lineno"> 95</span>&#160; y[o] += dx * dy[o];</div>
<div class="line"><a name="l00096"></a><span class="lineno"> 96</span>&#160;}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#a97075291390a68c262ed66e157a57eb4">&#9670;&nbsp;</a></span>problem()</h2>
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<td class="memname">void problem </td>
<td>(</td>
<td class="paramtype">const double *&#160;</td>
<td class="paramname"><em>x</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">double *&#160;</td>
<td class="paramname"><em>y</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">double *&#160;</td>
<td class="paramname"><em>dy</em>&#160;</td>
</tr>
<tr>
<td></td>
<td>)</td>
<td></td><td></td>
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<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>&#160;{</div>
<div class="line"><a name="l00056"></a><span class="lineno"> 56</span>&#160; <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>&#160; dy[0] = y[1]; <span class="comment">// x dot</span></div>
<div class="line"><a name="l00058"></a><span class="lineno"> 58</span>&#160; dy[1] = -omega * omega * y[0]; <span class="comment">// y dot</span></div>
<div class="line"><a name="l00059"></a><span class="lineno"> 59</span>&#160;}</div>
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<div class="ttc" id="aode__midpoint__euler_8c_html_a8caee977b26888d34040b122e0e28e3a"><div class="ttname"><a href="../../d1/dc2/ode__midpoint__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. Used for solution comparison.</div><div class="ttdef"><b>Definition:</b> ode_midpoint_euler.c:67</div></div>
<div class="ttc" id="aode__midpoint__euler_8c_html_affe6cc2ab040b94a29e6c41782f72d51"><div class="ttname"><a href="../../d1/dc2/ode__midpoint__euler_8c.html#affe6cc2ab040b94a29e6c41782f72d51">midpoint_euler_step</a></div><div class="ttdeci">void midpoint_euler_step(double dx, double *x, double *y, double *dy)</div><div class="ttdoc">Compute next step approximation using the midpoint-Euler method.</div><div class="ttdef"><b>Definition:</b> ode_midpoint_euler.c:83</div></div>
<div class="ttc" id="aode__midpoint__euler_8c_html_a9ceb646336224ee890a269d0b4600d09"><div class="ttname"><a href="../../d1/dc2/ode__midpoint__euler_8c.html#a9ceb646336224ee890a269d0b4600d09">order</a></div><div class="ttdeci">#define order</div><div class="ttdef"><b>Definition:</b> ode_midpoint_euler.c:43</div></div>
<div class="ttc" id="aode__midpoint__euler_8c_html_a148003d8b261d040c1c41e73b40af1dd"><div class="ttname"><a href="../../d1/dc2/ode__midpoint__euler_8c.html#a148003d8b261d040c1c41e73b40af1dd">midpoint_euler</a></div><div class="ttdeci">double midpoint_euler(double dx, double x0, double x_max, double *y, char save_to_file)</div><div class="ttdoc">Compute approximation using the midpoint-Euler method in the given limits.</div><div class="ttdef"><b>Definition:</b> ode_midpoint_euler.c:108</div></div>
<div class="ttc" id="aode__midpoint__euler_8c_html_a97075291390a68c262ed66e157a57eb4"><div class="ttname"><a href="../../d1/dc2/ode__midpoint__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. Updates the system differenti...</div><div class="ttdef"><b>Definition:</b> ode_midpoint_euler.c:54</div></div>
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<a href="#define-members">Macros</a> &#124;
<a href="#func-members">Functions</a> </div>
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<div class="title">sol1.c File Reference</div> </div>
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<p><a href="https://projecteuler.net/problem=26">Problem 26</a> solution
<a href="#details">More...</a></p>
<div class="textblock"><code>#include &lt;stdio.h&gt;</code><br />
<code>#include &lt;stdlib.h&gt;</code><br />
<code>#include &lt;string.h&gt;</code><br />
<code>#include &lt;time.h&gt;</code><br />
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Include dependency graph for sol1.c:</div>
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Macros</h2></td></tr>
<tr class="memitem:a619eec3220cebd7c5e455edbb14e9b12"><td class="memItemLeft" align="right" valign="top">#define&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../d1/df9/problem__26_2sol1_8c.html#a619eec3220cebd7c5e455edbb14e9b12">MAX_DENO</a>&#160;&#160;&#160;2000</td></tr>
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<tr class="memitem:aabf4f709c8199e41cf279c77112345fe"><td class="memItemLeft" align="right" valign="top">#define&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../d1/df9/problem__26_2sol1_8c.html#aabf4f709c8199e41cf279c77112345fe">MAX_LEN</a>&#160;&#160;&#160;(<a class="el" href="../../d1/df9/problem__26_2sol1_8c.html#a619eec3220cebd7c5e455edbb14e9b12">MAX_DENO</a> + 10)</td></tr>
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Functions</h2></td></tr>
<tr class="memitem:ac70138609ef6aa6fabca57aca8681e83"><td class="memItemLeft" align="right" valign="top">int&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../d1/df9/problem__26_2sol1_8c.html#ac70138609ef6aa6fabca57aca8681e83">compare</a> (const void *a, const void *b)</td></tr>
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<tr class="memitem:a0ddf1224851353fc92bfbff6f499fa97"><td class="memItemLeft" align="right" valign="top">int&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../d1/df9/problem__26_2sol1_8c.html#a0ddf1224851353fc92bfbff6f499fa97">main</a> (int argc, char *argv[])</td></tr>
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<a name="details" id="details"></a><h2 class="groupheader">Detailed Description</h2>
<div class="textblock"><p><a href="https://projecteuler.net/problem=26">Problem 26</a> solution </p>
<dl class="section author"><dt>Author</dt><dd><a href="https://github.com/kvedala">Krishna Vedala</a> </dd></dl>
</div><h2 class="groupheader">Macro Definition Documentation</h2>
<a id="a619eec3220cebd7c5e455edbb14e9b12"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a619eec3220cebd7c5e455edbb14e9b12">&#9670;&nbsp;</a></span>MAX_DENO</h2>
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<td class="memname">#define MAX_DENO&#160;&#160;&#160;2000</td>
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<p>limit of unit fractions </p>
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<a id="aabf4f709c8199e41cf279c77112345fe"></a>
<h2 class="memtitle"><span class="permalink"><a href="#aabf4f709c8199e41cf279c77112345fe">&#9670;&nbsp;</a></span>MAX_LEN</h2>
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<td class="memname">#define MAX_LEN&#160;&#160;&#160;(<a class="el" href="../../d1/df9/problem__26_2sol1_8c.html#a619eec3220cebd7c5e455edbb14e9b12">MAX_DENO</a> + 10)</td>
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<p>length of resulting recurring fraction number </p>
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<h2 class="groupheader">Function Documentation</h2>
<a id="ac70138609ef6aa6fabca57aca8681e83"></a>
<h2 class="memtitle"><span class="permalink"><a href="#ac70138609ef6aa6fabca57aca8681e83">&#9670;&nbsp;</a></span>compare()</h2>
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<td class="memname">int compare </td>
<td>(</td>
<td class="paramtype">const void *&#160;</td>
<td class="paramname"><em>a</em>, </td>
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<td class="paramkey"></td>
<td></td>
<td class="paramtype">const void *&#160;</td>
<td class="paramname"><em>b</em>&#160;</td>
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<td>)</td>
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<p>comparison function for use with internal <code>qsort</code> algorithm </p>
<div class="fragment"><div class="line"><a name="l00020"></a><span class="lineno"> 20</span>&#160;{</div>
<div class="line"><a name="l00021"></a><span class="lineno"> 21</span>&#160; <span class="keywordflow">return</span> (*(<span class="keywordtype">unsigned</span> <span class="keywordtype">short</span> *)a - *(<span class="keywordtype">unsigned</span> <span class="keywordtype">short</span> *)b);</div>
<div class="line"><a name="l00022"></a><span class="lineno"> 22</span>&#160;}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#a0ddf1224851353fc92bfbff6f499fa97">&#9670;&nbsp;</a></span>main()</h2>
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<td class="memname">int main </td>
<td>(</td>
<td class="paramtype">int&#160;</td>
<td class="paramname"><em>argc</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">char *&#160;</td>
<td class="paramname"><em>argv</em>[]&#160;</td>
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<td>)</td>
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<p>Main function </p>
<div class="fragment"><div class="line"><a name="l00026"></a><span class="lineno"> 26</span>&#160;{</div>
<div class="line"><a name="l00027"></a><span class="lineno"> 27</span>&#160; <span class="keywordtype">unsigned</span> <span class="keywordtype">short</span> max_digits = 0, max_idx_number = 0;</div>
<div class="line"><a name="l00028"></a><span class="lineno"> 28</span>&#160; </div>
<div class="line"><a name="l00029"></a><span class="lineno"> 29</span>&#160; clock_t start_time = clock();</div>
<div class="line"><a name="l00030"></a><span class="lineno"> 30</span>&#160; <span class="keywordtype">short</span> deno;</div>
<div class="line"><a name="l00031"></a><span class="lineno"> 31</span>&#160;<span class="preprocessor">#ifdef _OPENMP</span></div>
<div class="line"><a name="l00032"></a><span class="lineno"> 32</span>&#160;<span class="preprocessor">#pragma omp for</span></div>
<div class="line"><a name="l00033"></a><span class="lineno"> 33</span>&#160;<span class="preprocessor">#endif</span></div>
<div class="line"><a name="l00034"></a><span class="lineno"> 34</span>&#160; <span class="keywordflow">for</span> (deno = 2; deno &lt; <a class="code" href="../../d1/df9/problem__26_2sol1_8c.html#a619eec3220cebd7c5e455edbb14e9b12">MAX_DENO</a>; deno++)</div>
<div class="line"><a name="l00035"></a><span class="lineno"> 35</span>&#160; {</div>
<div class="line"><a name="l00036"></a><span class="lineno"> 36</span>&#160; <span class="keywordtype">unsigned</span> <span class="keywordtype">short</span> remainders[<a class="code" href="../../d1/df9/problem__26_2sol1_8c.html#aabf4f709c8199e41cf279c77112345fe">MAX_LEN</a>];</div>
<div class="line"><a name="l00037"></a><span class="lineno"> 37</span>&#160; <span class="keywordtype">unsigned</span> <span class="keywordtype">short</span> rem = 1, *rem_ptr = remainders;</div>
<div class="line"><a name="l00038"></a><span class="lineno"> 38</span>&#160; memset(remainders, (<span class="keywordtype">unsigned</span> <span class="keywordtype">short</span>)-1,</div>
<div class="line"><a name="l00039"></a><span class="lineno"> 39</span>&#160; <a class="code" href="../../d1/df9/problem__26_2sol1_8c.html#aabf4f709c8199e41cf279c77112345fe">MAX_LEN</a> * <span class="keyword">sizeof</span>(<span class="keywordtype">unsigned</span> <span class="keywordtype">short</span>));</div>
<div class="line"><a name="l00040"></a><span class="lineno"> 40</span>&#160; <span class="comment">// remainders[0] = 1;</span></div>
<div class="line"><a name="l00041"></a><span class="lineno"> 41</span>&#160; <span class="comment">// printf(&quot;1/%-4u\t &quot;, deno);</span></div>
<div class="line"><a name="l00042"></a><span class="lineno"> 42</span>&#160; <span class="keywordtype">unsigned</span> <span class="keywordtype">short</span> index = 0, num_digits;</div>
<div class="line"><a name="l00043"></a><span class="lineno"> 43</span>&#160; </div>
<div class="line"><a name="l00044"></a><span class="lineno"> 44</span>&#160; <span class="keywordflow">while</span> (rem != 0)</div>
<div class="line"><a name="l00045"></a><span class="lineno"> 45</span>&#160; {</div>
<div class="line"><a name="l00046"></a><span class="lineno"> 46</span>&#160; rem = (rem * 10) % deno;</div>
<div class="line"><a name="l00047"></a><span class="lineno"> 47</span>&#160; <span class="keywordflow">if</span> (rem == 0)</div>
<div class="line"><a name="l00048"></a><span class="lineno"> 48</span>&#160; {</div>
<div class="line"><a name="l00049"></a><span class="lineno"> 49</span>&#160; index = 0;</div>
<div class="line"><a name="l00050"></a><span class="lineno"> 50</span>&#160; <span class="keywordflow">break</span>;</div>
<div class="line"><a name="l00051"></a><span class="lineno"> 51</span>&#160; }</div>
<div class="line"><a name="l00052"></a><span class="lineno"> 52</span>&#160; rem_ptr = (<span class="keywordtype">unsigned</span> <span class="keywordtype">short</span> *)bsearch(</div>
<div class="line"><a name="l00053"></a><span class="lineno"> 53</span>&#160; &amp;rem, remainders, <a class="code" href="../../d1/df9/problem__26_2sol1_8c.html#aabf4f709c8199e41cf279c77112345fe">MAX_LEN</a>, <span class="keyword">sizeof</span>(<span class="keywordtype">unsigned</span> <span class="keywordtype">short</span>), <a class="code" href="../../d1/df9/problem__26_2sol1_8c.html#ac70138609ef6aa6fabca57aca8681e83">compare</a>);</div>
<div class="line"><a name="l00054"></a><span class="lineno"> 54</span>&#160; <span class="comment">// printf(&quot;%2d, &quot;, rem);</span></div>
<div class="line"><a name="l00055"></a><span class="lineno"> 55</span>&#160; <span class="comment">// printf(&quot;(%14p), &quot;, rem_ptr);</span></div>
<div class="line"><a name="l00056"></a><span class="lineno"> 56</span>&#160; <span class="keywordflow">if</span> (rem_ptr != NULL)</div>
<div class="line"><a name="l00057"></a><span class="lineno"> 57</span>&#160; <span class="keywordflow">break</span>;</div>
<div class="line"><a name="l00058"></a><span class="lineno"> 58</span>&#160; remainders[index] = rem;</div>
<div class="line"><a name="l00059"></a><span class="lineno"> 59</span>&#160; rem_ptr = remainders;</div>
<div class="line"><a name="l00060"></a><span class="lineno"> 60</span>&#160; index++;</div>
<div class="line"><a name="l00061"></a><span class="lineno"> 61</span>&#160; }</div>
<div class="line"><a name="l00062"></a><span class="lineno"> 62</span>&#160; </div>
<div class="line"><a name="l00063"></a><span class="lineno"> 63</span>&#160; num_digits = index - (rem_ptr - remainders);</div>
<div class="line"><a name="l00064"></a><span class="lineno"> 64</span>&#160; <span class="comment">// printf(&quot;\n\t(%14p, %14p, %4u, %4u)\n&quot;, rem_ptr, remainders, index,</span></div>
<div class="line"><a name="l00065"></a><span class="lineno"> 65</span>&#160; <span class="comment">// num_digits);</span></div>
<div class="line"><a name="l00066"></a><span class="lineno"> 66</span>&#160;<span class="preprocessor">#ifdef _OPENMP</span></div>
<div class="line"><a name="l00067"></a><span class="lineno"> 67</span>&#160;<span class="preprocessor">#pragma omp critical</span></div>
<div class="line"><a name="l00068"></a><span class="lineno"> 68</span>&#160; {</div>
<div class="line"><a name="l00069"></a><span class="lineno"> 69</span>&#160;<span class="preprocessor">#endif</span></div>
<div class="line"><a name="l00070"></a><span class="lineno"> 70</span>&#160; <span class="keywordflow">if</span> (num_digits &gt; max_digits)</div>
<div class="line"><a name="l00071"></a><span class="lineno"> 71</span>&#160; {</div>
<div class="line"><a name="l00072"></a><span class="lineno"> 72</span>&#160; max_digits = num_digits;</div>
<div class="line"><a name="l00073"></a><span class="lineno"> 73</span>&#160; max_idx_number = deno;</div>
<div class="line"><a name="l00074"></a><span class="lineno"> 74</span>&#160; <span class="comment">// printf(&quot;\t (%u, %u)\n &quot;, max_digits, max_idx_number);</span></div>
<div class="line"><a name="l00075"></a><span class="lineno"> 75</span>&#160; }</div>
<div class="line"><a name="l00076"></a><span class="lineno"> 76</span>&#160;<span class="preprocessor">#ifdef _OPENMP</span></div>
<div class="line"><a name="l00077"></a><span class="lineno"> 77</span>&#160; }</div>
<div class="line"><a name="l00078"></a><span class="lineno"> 78</span>&#160;<span class="preprocessor">#endif</span></div>
<div class="line"><a name="l00079"></a><span class="lineno"> 79</span>&#160; }</div>
<div class="line"><a name="l00080"></a><span class="lineno"> 80</span>&#160; clock_t end_time = clock();</div>
<div class="line"><a name="l00081"></a><span class="lineno"> 81</span>&#160; </div>
<div class="line"><a name="l00082"></a><span class="lineno"> 82</span>&#160; printf(<span class="stringliteral">&quot;Time taken: %.4g ms\n&quot;</span>,</div>
<div class="line"><a name="l00083"></a><span class="lineno"> 83</span>&#160; 1e3 * (<span class="keywordtype">double</span>)(end_time - start_time) / CLOCKS_PER_SEC);</div>
<div class="line"><a name="l00084"></a><span class="lineno"> 84</span>&#160; printf(<span class="stringliteral">&quot;Maximum digits: %hu\t Denominator: %hu\n&quot;</span>, max_digits,</div>
<div class="line"><a name="l00085"></a><span class="lineno"> 85</span>&#160; max_idx_number);</div>
<div class="line"><a name="l00086"></a><span class="lineno"> 86</span>&#160; </div>
<div class="line"><a name="l00087"></a><span class="lineno"> 87</span>&#160; <span class="keywordflow">return</span> 0;</div>
<div class="line"><a name="l00088"></a><span class="lineno"> 88</span>&#160;}</div>
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<div class="ttc" id="aproblem__26_2sol1_8c_html_ac70138609ef6aa6fabca57aca8681e83"><div class="ttname"><a href="../../d1/df9/problem__26_2sol1_8c.html#ac70138609ef6aa6fabca57aca8681e83">compare</a></div><div class="ttdeci">int compare(const void *a, const void *b)</div><div class="ttdef"><b>Definition:</b> sol1.c:19</div></div>
<div class="ttc" id="aproblem__26_2sol1_8c_html_aabf4f709c8199e41cf279c77112345fe"><div class="ttname"><a href="../../d1/df9/problem__26_2sol1_8c.html#aabf4f709c8199e41cf279c77112345fe">MAX_LEN</a></div><div class="ttdeci">#define MAX_LEN</div><div class="ttdef"><b>Definition:</b> sol1.c:15</div></div>
<div class="ttc" id="aproblem__26_2sol1_8c_html_a619eec3220cebd7c5e455edbb14e9b12"><div class="ttname"><a href="../../d1/df9/problem__26_2sol1_8c.html#a619eec3220cebd7c5e455edbb14e9b12">MAX_DENO</a></div><div class="ttdeci">#define MAX_DENO</div><div class="ttdef"><b>Definition:</b> sol1.c:14</div></div>
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