Documentation for af6e2c8991

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realstealthninja
2026-09-21 04:13:18 +00:00
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@@ -4,7 +4,7 @@
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<title>TheAlgorithms/C++: numerical_methods/ode_midpoint_euler.cpp Source File</title>
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@@ -59,22 +43,47 @@ window.MathJax = {
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@@ -125,7 +130,7 @@ $(function(){initNavTree('d6/dd3/ode__midpoint__euler_8cpp_source.html','../../'
<div class="line"><a id="l00042" name="l00042"></a><span class="lineno"> 42</span><span class="preprocessor">#include &lt;valarray&gt;</span></div>
<div class="line"><a id="l00043" name="l00043"></a><span class="lineno"> 43</span></div>
<div class="foldopen" id="foldopen00053" data-start="{" data-end="}">
<div class="line"><a id="l00053" name="l00053"></a><span class="lineno"><a class="line" href="../../d6/dd3/ode__midpoint__euler_8cpp.html#abaeae8f62a018d197f0187a1c80a90fe"> 53</a></span><span class="keywordtype">void</span> <a class="code hl_function" href="../../d6/dd3/ode__midpoint__euler_8cpp.html#abaeae8f62a018d197f0187a1c80a90fe">problem</a>(<span class="keyword">const</span> <span class="keywordtype">double</span> &amp;x, std::valarray&lt;double&gt; *y,</div>
<div class="line"><a id="l00053" name="l00053"></a><span class="lineno"><a class="line" href="../../d6/dd3/ode__midpoint__euler_8cpp.html#abaeae8f62a018d197f0187a1c80a90fe"> 53</a></span><span class="keywordtype">void</span> <a class="code hl_function" href="../../db/dd3/ode__forward__euler_8cpp.html#abaeae8f62a018d197f0187a1c80a90fe">problem</a>(<span class="keyword">const</span> <span class="keywordtype">double</span> &amp;x, std::valarray&lt;double&gt; *y,</div>
<div class="line"><a id="l00054" name="l00054"></a><span class="lineno"> 54</span> std::valarray&lt;double&gt; *dy) {</div>
<div class="line"><a id="l00055" name="l00055"></a><span class="lineno"> 55</span> <span class="keyword">const</span> <span class="keywordtype">double</span> omega = 1.F; <span class="comment">// some const for the problem</span></div>
<div class="line"><a id="l00056" name="l00056"></a><span class="lineno"> 56</span> dy[0][0] = y[0][1]; <span class="comment">// x dot</span></div>
@@ -134,7 +139,7 @@ $(function(){initNavTree('d6/dd3/ode__midpoint__euler_8cpp_source.html','../../'
</div>
<div class="line"><a id="l00059" name="l00059"></a><span class="lineno"> 59</span></div>
<div class="foldopen" id="foldopen00066" data-start="{" data-end="}">
<div class="line"><a id="l00066" name="l00066"></a><span class="lineno"><a class="line" href="../../d6/dd3/ode__midpoint__euler_8cpp.html#af3adf7b092a87868917ee5fb4255192b"> 66</a></span><span class="keywordtype">void</span> <a class="code hl_function" href="../../d6/dd3/ode__midpoint__euler_8cpp.html#af3adf7b092a87868917ee5fb4255192b">exact_solution</a>(<span class="keyword">const</span> <span class="keywordtype">double</span> &amp;x, std::valarray&lt;double&gt; *y) {</div>
<div class="line"><a id="l00066" name="l00066"></a><span class="lineno"><a class="line" href="../../d6/dd3/ode__midpoint__euler_8cpp.html#af3adf7b092a87868917ee5fb4255192b"> 66</a></span><span class="keywordtype">void</span> <a class="code hl_function" href="../../db/dd3/ode__forward__euler_8cpp.html#af3adf7b092a87868917ee5fb4255192b">exact_solution</a>(<span class="keyword">const</span> <span class="keywordtype">double</span> &amp;x, std::valarray&lt;double&gt; *y) {</div>
<div class="line"><a id="l00067" name="l00067"></a><span class="lineno"> 67</span> y[0][0] = std::cos(x);</div>
<div class="line"><a id="l00068" name="l00068"></a><span class="lineno"> 68</span> y[0][1] = -std::sin(x);</div>
<div class="line"><a id="l00069" name="l00069"></a><span class="lineno"> 69</span>}</div>
@@ -143,12 +148,12 @@ $(function(){initNavTree('d6/dd3/ode__midpoint__euler_8cpp_source.html','../../'
<div class="foldopen" id="foldopen00085" data-start="{" data-end="}">
<div class="line"><a id="l00085" name="l00085"></a><span class="lineno"><a class="line" href="../../d6/d60/group__ode.html#ga8c319db420c3d97a83e9dcca803b6812"> 85</a></span><span class="keywordtype">void</span> <a class="code hl_function" href="../../d6/d60/group__ode.html#ga8c319db420c3d97a83e9dcca803b6812">midpoint_euler_step</a>(<span class="keyword">const</span> <span class="keywordtype">double</span> dx, <span class="keyword">const</span> <span class="keywordtype">double</span> &amp;x,</div>
<div class="line"><a id="l00086" name="l00086"></a><span class="lineno"> 86</span> std::valarray&lt;double&gt; *y, std::valarray&lt;double&gt; *dy) {</div>
<div class="line"><a id="l00087" name="l00087"></a><span class="lineno"> 87</span> <a class="code hl_function" href="../../d6/dd3/ode__midpoint__euler_8cpp.html#abaeae8f62a018d197f0187a1c80a90fe">problem</a>(x, y, dy);</div>
<div class="line"><a id="l00087" name="l00087"></a><span class="lineno"> 87</span> <a class="code hl_function" href="../../db/dd3/ode__forward__euler_8cpp.html#abaeae8f62a018d197f0187a1c80a90fe">problem</a>(x, y, dy);</div>
<div class="line"><a id="l00088" name="l00088"></a><span class="lineno"> 88</span> <span class="keywordtype">double</span> tmp_x = x + 0.5 * dx;</div>
<div class="line"><a id="l00089" name="l00089"></a><span class="lineno"> 89</span> </div>
<div class="line"><a id="l00090" name="l00090"></a><span class="lineno"> 90</span> std::valarray&lt;double&gt; tmp_y = y[0] + dy[0] * (0.5 * dx);</div>
<div class="line"><a id="l00091" name="l00091"></a><span class="lineno"> 91</span> </div>
<div class="line"><a id="l00092" name="l00092"></a><span class="lineno"> 92</span> <a class="code hl_function" href="../../d6/dd3/ode__midpoint__euler_8cpp.html#abaeae8f62a018d197f0187a1c80a90fe">problem</a>(tmp_x, &amp;tmp_y, dy);</div>
<div class="line"><a id="l00092" name="l00092"></a><span class="lineno"> 92</span> <a class="code hl_function" href="../../db/dd3/ode__forward__euler_8cpp.html#abaeae8f62a018d197f0187a1c80a90fe">problem</a>(tmp_x, &amp;tmp_y, dy);</div>
<div class="line"><a id="l00093" name="l00093"></a><span class="lineno"> 93</span> </div>
<div class="line"><a id="l00094" name="l00094"></a><span class="lineno"> 94</span> y[0] += dy[0] * dx;</div>
<div class="line"><a id="l00095" name="l00095"></a><span class="lineno"> 95</span>}</div>
@@ -197,7 +202,7 @@ $(function(){initNavTree('d6/dd3/ode__midpoint__euler_8cpp_source.html','../../'
<div class="line"><a id="l00145" name="l00145"></a><span class="lineno"> 145</span></div>
<div class="line"><a id="l00147" name="l00147"></a><span class="lineno"> 147</span></div>
<div class="foldopen" id="foldopen00156" data-start="{" data-end="}">
<div class="line"><a id="l00156" name="l00156"></a><span class="lineno"><a class="line" href="../../d6/dd3/ode__midpoint__euler_8cpp.html#aa13517b8e5de1b75592052db7f7e237f"> 156</a></span><span class="keywordtype">void</span> <a class="code hl_function" href="../../d6/dd3/ode__midpoint__euler_8cpp.html#aa13517b8e5de1b75592052db7f7e237f">save_exact_solution</a>(<span class="keyword">const</span> <span class="keywordtype">double</span> &amp;X0, <span class="keyword">const</span> <span class="keywordtype">double</span> &amp;X_MAX,</div>
<div class="line"><a id="l00156" name="l00156"></a><span class="lineno"><a class="line" href="../../d6/dd3/ode__midpoint__euler_8cpp.html#aa13517b8e5de1b75592052db7f7e237f"> 156</a></span><span class="keywordtype">void</span> <a class="code hl_function" href="../../db/dd3/ode__forward__euler_8cpp.html#aa13517b8e5de1b75592052db7f7e237f">save_exact_solution</a>(<span class="keyword">const</span> <span class="keywordtype">double</span> &amp;X0, <span class="keyword">const</span> <span class="keywordtype">double</span> &amp;X_MAX,</div>
<div class="line"><a id="l00157" name="l00157"></a><span class="lineno"> 157</span> <span class="keyword">const</span> <span class="keywordtype">double</span> &amp;step_size,</div>
<div class="line"><a id="l00158" name="l00158"></a><span class="lineno"> 158</span> <span class="keyword">const</span> std::valarray&lt;double&gt; &amp;Y0) {</div>
<div class="line"><a id="l00159" name="l00159"></a><span class="lineno"> 159</span> <span class="keywordtype">double</span> x = X0;</div>
@@ -218,7 +223,7 @@ $(function(){initNavTree('d6/dd3/ode__midpoint__euler_8cpp_source.html','../../'
<div class="line"><a id="l00174" name="l00174"></a><span class="lineno"> 174</span> }</div>
<div class="line"><a id="l00175" name="l00175"></a><span class="lineno"> 175</span> fp &lt;&lt; y[y.size() - 1] &lt;&lt; <span class="stringliteral">&quot;\n&quot;</span>;</div>
<div class="line"><a id="l00176" name="l00176"></a><span class="lineno"> 176</span> </div>
<div class="line"><a id="l00177" name="l00177"></a><span class="lineno"> 177</span> <a class="code hl_function" href="../../d6/dd3/ode__midpoint__euler_8cpp.html#af3adf7b092a87868917ee5fb4255192b">exact_solution</a>(x, &amp;y);</div>
<div class="line"><a id="l00177" name="l00177"></a><span class="lineno"> 177</span> <a class="code hl_function" href="../../db/dd3/ode__forward__euler_8cpp.html#af3adf7b092a87868917ee5fb4255192b">exact_solution</a>(x, &amp;y);</div>
<div class="line"><a id="l00178" name="l00178"></a><span class="lineno"> 178</span> </div>
<div class="line"><a id="l00179" name="l00179"></a><span class="lineno"> 179</span> x += step_size;</div>
<div class="line"><a id="l00180" name="l00180"></a><span class="lineno"> 180</span> } <span class="keywordflow">while</span> (x &lt;= X_MAX);</div>
@@ -251,7 +256,7 @@ $(function(){initNavTree('d6/dd3/ode__midpoint__euler_8cpp_source.html','../../'
<div class="line"><a id="l00208" name="l00208"></a><span class="lineno"> 208</span> std::cout &lt;&lt; <span class="stringliteral">&quot;\tTime = &quot;</span> &lt;&lt; total_time &lt;&lt; <span class="stringliteral">&quot; ms\n&quot;</span>;</div>
<div class="line"><a id="l00209" name="l00209"></a><span class="lineno"> 209</span> </div>
<div class="line"><a id="l00210" name="l00210"></a><span class="lineno"> 210</span> <span class="comment">/* compute exact solution for comparion */</span></div>
<div class="line"><a id="l00211" name="l00211"></a><span class="lineno"> 211</span> <a class="code hl_function" href="../../d6/dd3/ode__midpoint__euler_8cpp.html#aa13517b8e5de1b75592052db7f7e237f">save_exact_solution</a>(X0, X_MAX, step_size, Y0);</div>
<div class="line"><a id="l00211" name="l00211"></a><span class="lineno"> 211</span> <a class="code hl_function" href="../../db/dd3/ode__forward__euler_8cpp.html#aa13517b8e5de1b75592052db7f7e237f">save_exact_solution</a>(X0, X_MAX, step_size, Y0);</div>
<div class="line"><a id="l00212" name="l00212"></a><span class="lineno"> 212</span> </div>
<div class="line"><a id="l00213" name="l00213"></a><span class="lineno"> 213</span> <span class="keywordflow">return</span> 0;</div>
<div class="line"><a id="l00214" name="l00214"></a><span class="lineno"> 214</span>}</div>
@@ -259,17 +264,17 @@ $(function(){initNavTree('d6/dd3/ode__midpoint__euler_8cpp_source.html','../../'
<div class="ttc" id="agenerate__parentheses_8cpp_html_gae66f6b31b5ad750f1fe042a706a4e3d4"><div class="ttname"><a href="../../dd/d1e/generate__parentheses_8cpp.html#gae66f6b31b5ad750f1fe042a706a4e3d4">main</a></div><div class="ttdeci">int main()</div><div class="ttdoc">Main function.</div><div class="ttdef"><b>Definition</b> <a href="../../dd/d1e/generate__parentheses_8cpp_source.html#l00110">generate_parentheses.cpp:110</a></div></div>
<div class="ttc" id="agroup__ode_html_ga36ff0710861ca39d957046c0b09b2985"><div class="ttname"><a href="../../d6/d60/group__ode.html#ga36ff0710861ca39d957046c0b09b2985">midpoint_euler</a></div><div class="ttdeci">double midpoint_euler(double dx, double x0, double x_max, std::valarray&lt; double &gt; *y, bool save_to_file=false)</div><div class="ttdoc">Compute approximation using the midpoint-Euler method in the given limits.</div><div class="ttdef"><b>Definition</b> <a href="#l00107">ode_midpoint_euler.cpp:107</a></div></div>
<div class="ttc" id="agroup__ode_html_ga8c319db420c3d97a83e9dcca803b6812"><div class="ttname"><a href="../../d6/d60/group__ode.html#ga8c319db420c3d97a83e9dcca803b6812">midpoint_euler_step</a></div><div class="ttdeci">void midpoint_euler_step(const double dx, const double &amp;x, std::valarray&lt; double &gt; *y, std::valarray&lt; double &gt; *dy)</div><div class="ttdoc">Compute next step approximation using the midpoint-Euler method.</div><div class="ttdef"><b>Definition</b> <a href="#l00085">ode_midpoint_euler.cpp:85</a></div></div>
<div class="ttc" id="aode__midpoint__euler_8cpp_html_aa13517b8e5de1b75592052db7f7e237f"><div class="ttname"><a href="../../d6/dd3/ode__midpoint__euler_8cpp.html#aa13517b8e5de1b75592052db7f7e237f">save_exact_solution</a></div><div class="ttdeci">void save_exact_solution(const double &amp;X0, const double &amp;X_MAX, const double &amp;step_size, const std::valarray&lt; double &gt; &amp;Y0)</div><div class="ttdef"><b>Definition</b> <a href="#l00156">ode_midpoint_euler.cpp:156</a></div></div>
<div class="ttc" id="aode__midpoint__euler_8cpp_html_abaeae8f62a018d197f0187a1c80a90fe"><div class="ttname"><a href="../../d6/dd3/ode__midpoint__euler_8cpp.html#abaeae8f62a018d197f0187a1c80a90fe">problem</a></div><div class="ttdeci">void problem(const double &amp;x, std::valarray&lt; double &gt; *y, std::valarray&lt; double &gt; *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> <a href="#l00053">ode_midpoint_euler.cpp:53</a></div></div>
<div class="ttc" id="aode__midpoint__euler_8cpp_html_af3adf7b092a87868917ee5fb4255192b"><div class="ttname"><a href="../../d6/dd3/ode__midpoint__euler_8cpp.html#af3adf7b092a87868917ee5fb4255192b">exact_solution</a></div><div class="ttdeci">void exact_solution(const double &amp;x, std::valarray&lt; double &gt; *y)</div><div class="ttdoc">Exact solution of the problem. Used for solution comparison.</div><div class="ttdef"><b>Definition</b> <a href="#l00066">ode_midpoint_euler.cpp:66</a></div></div>
<div class="ttc" id="aode__forward__euler_8cpp_html_aa13517b8e5de1b75592052db7f7e237f"><div class="ttname"><a href="../../db/dd3/ode__forward__euler_8cpp.html#aa13517b8e5de1b75592052db7f7e237f">save_exact_solution</a></div><div class="ttdeci">void save_exact_solution(const double &amp;X0, const double &amp;X_MAX, const double &amp;step_size, const std::valarray&lt; double &gt; &amp;Y0)</div><div class="ttdef"><b>Definition</b> <a href="../../db/dd3/ode__forward__euler_8cpp_source.html#l00153">ode_forward_euler.cpp:153</a></div></div>
<div class="ttc" id="aode__forward__euler_8cpp_html_abaeae8f62a018d197f0187a1c80a90fe"><div class="ttname"><a href="../../db/dd3/ode__forward__euler_8cpp.html#abaeae8f62a018d197f0187a1c80a90fe">problem</a></div><div class="ttdeci">void problem(const double &amp;x, std::valarray&lt; double &gt; *y, std::valarray&lt; double &gt; *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> <a href="../../db/dd3/ode__forward__euler_8cpp_source.html#l00054">ode_forward_euler.cpp:54</a></div></div>
<div class="ttc" id="aode__forward__euler_8cpp_html_af3adf7b092a87868917ee5fb4255192b"><div class="ttname"><a href="../../db/dd3/ode__forward__euler_8cpp.html#af3adf7b092a87868917ee5fb4255192b">exact_solution</a></div><div class="ttdeci">void exact_solution(const double &amp;x, std::valarray&lt; double &gt; *y)</div><div class="ttdoc">Exact solution of the problem. Used for solution comparison.</div><div class="ttdef"><b>Definition</b> <a href="../../db/dd3/ode__forward__euler_8cpp_source.html#l00067">ode_forward_euler.cpp:67</a></div></div>
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