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<div class="title">Ordinary Differential Equations</div> </div>
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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:gaa197c6b7085ac9ea71abc4c063e6f52f"><td class="memItemLeft" align="right" valign="top">void </td><td class="memItemRight" valign="bottom"><a class="el" href="../../d6/d60/group__ode.html#gaa197c6b7085ac9ea71abc4c063e6f52f">forward_euler_step</a> (const double dx, const double &x, <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>< double > *y, <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>< double > *dy)</td></tr>
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<tr class="memdesc:gaa197c6b7085ac9ea71abc4c063e6f52f"><td class="mdescLeft"> </td><td class="mdescRight">Compute next step approximation using the forward-Euler method. <a href="../../d6/d60/group__ode.html#gaa197c6b7085ac9ea71abc4c063e6f52f">More...</a><br /></td></tr>
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<tr class="separator:gaa197c6b7085ac9ea71abc4c063e6f52f"><td class="memSeparator" colspan="2"> </td></tr>
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<tr class="memitem:gae0509f8843e2bc42de2abbd00a14b7b9"><td class="memItemLeft" align="right" valign="top">double </td><td class="memItemRight" valign="bottom"><a class="el" href="../../d6/d60/group__ode.html#gae0509f8843e2bc42de2abbd00a14b7b9">forward_euler</a> (double dx, double x0, double x_max, <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>< double > *y, bool save_to_file=false)</td></tr>
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<tr class="memdesc:gae0509f8843e2bc42de2abbd00a14b7b9"><td class="mdescLeft"> </td><td class="mdescRight">Compute approximation using the forward-Euler method in the given limits. <a href="../../d6/d60/group__ode.html#gae0509f8843e2bc42de2abbd00a14b7b9">More...</a><br /></td></tr>
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<tr class="separator:gae0509f8843e2bc42de2abbd00a14b7b9"><td class="memSeparator" colspan="2"> </td></tr>
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<tr class="memitem:ga8c319db420c3d97a83e9dcca803b6812"><td class="memItemLeft" align="right" valign="top">void </td><td class="memItemRight" valign="bottom"><a class="el" href="../../d6/d60/group__ode.html#ga8c319db420c3d97a83e9dcca803b6812">midpoint_euler_step</a> (const double dx, const double &x, <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>< double > *y, <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>< double > *dy)</td></tr>
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<tr class="memdesc:ga8c319db420c3d97a83e9dcca803b6812"><td class="mdescLeft"> </td><td class="mdescRight">Compute next step approximation using the midpoint-Euler method. <a href="../../d6/d60/group__ode.html#ga8c319db420c3d97a83e9dcca803b6812">More...</a><br /></td></tr>
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<tr class="separator:ga8c319db420c3d97a83e9dcca803b6812"><td class="memSeparator" colspan="2"> </td></tr>
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<tr class="memitem:ga36ff0710861ca39d957046c0b09b2985"><td class="memItemLeft" align="right" valign="top">double </td><td class="memItemRight" valign="bottom"><a class="el" href="../../d6/d60/group__ode.html#ga36ff0710861ca39d957046c0b09b2985">midpoint_euler</a> (double dx, double x0, double x_max, <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>< double > *y, bool save_to_file=false)</td></tr>
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<tr class="memdesc:ga36ff0710861ca39d957046c0b09b2985"><td class="mdescLeft"> </td><td class="mdescRight">Compute approximation using the midpoint-Euler method in the given limits. <a href="../../d6/d60/group__ode.html#ga36ff0710861ca39d957046c0b09b2985">More...</a><br /></td></tr>
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<tr class="separator:ga36ff0710861ca39d957046c0b09b2985"><td class="memSeparator" colspan="2"> </td></tr>
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<tr class="memitem:ga827bf009831ddc477c5fa8891d5cb35f"><td class="memItemLeft" align="right" valign="top">void </td><td class="memItemRight" valign="bottom"><a class="el" href="../../d6/d60/group__ode.html#ga827bf009831ddc477c5fa8891d5cb35f">semi_implicit_euler_step</a> (const double dx, const double &x, <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>< double > *y, <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>< double > *dy)</td></tr>
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<tr class="memdesc:ga827bf009831ddc477c5fa8891d5cb35f"><td class="mdescLeft"> </td><td class="mdescRight">Compute next step approximation using the semi-implicit-Euler method. <a href="../../d6/d60/group__ode.html#ga827bf009831ddc477c5fa8891d5cb35f">More...</a><br /></td></tr>
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<tr class="separator:ga827bf009831ddc477c5fa8891d5cb35f"><td class="memSeparator" colspan="2"> </td></tr>
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<tr class="memitem:ga3874d294ab00fe9ce8731c5b7991a089"><td class="memItemLeft" align="right" valign="top">double </td><td class="memItemRight" valign="bottom"><a class="el" href="../../d6/d60/group__ode.html#ga3874d294ab00fe9ce8731c5b7991a089">semi_implicit_euler</a> (double dx, double x0, double x_max, <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>< double > *y, bool save_to_file=false)</td></tr>
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<tr class="memdesc:ga3874d294ab00fe9ce8731c5b7991a089"><td class="mdescLeft"> </td><td class="mdescRight">Compute approximation using the semi-implicit-Euler method in the given limits. <a href="../../d6/d60/group__ode.html#ga3874d294ab00fe9ce8731c5b7991a089">More...</a><br /></td></tr>
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<tr class="separator:ga3874d294ab00fe9ce8731c5b7991a089"><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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<p>Integration functions for implementations with solving <a href="https://en.wikipedia.org/wiki/Ordinary_differential_equation">ordinary differential equations</a> (ODEs) of any order and and any number of independent variables. </p>
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<h2 class="groupheader">Function Documentation</h2>
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<a id="gae0509f8843e2bc42de2abbd00a14b7b9"></a>
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<h2 class="memtitle"><span class="permalink"><a href="#gae0509f8843e2bc42de2abbd00a14b7b9">◆ </a></span>forward_euler()</h2>
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<td class="memname">double forward_euler </td>
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<td>(</td>
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<td class="paramtype">double </td>
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<td class="paramname"><em>dx</em>, </td>
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<tr>
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<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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<tr>
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<td class="paramkey"></td>
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<td></td>
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<td class="paramtype"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>< double > * </td>
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<td class="paramname"><em>y</em>, </td>
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<tr>
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<td class="paramkey"></td>
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<td></td>
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<td class="paramtype">bool </td>
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<td class="paramname"><em>save_to_file</em> = <code>false</code> </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>Compute approximation using the forward-Euler method in the given limits. </p>
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<dl class="params"><dt>Parameters</dt><dd>
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<table class="params">
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<tr><td class="paramdir">[in]</td><td class="paramname">dx</td><td>step size </td></tr>
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<tr><td class="paramdir">[in]</td><td class="paramname">x0</td><td>initial value of independent variable </td></tr>
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<tr><td class="paramdir">[in]</td><td class="paramname">x_max</td><td>final value of independent variable </td></tr>
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<tr><td class="paramdir">[in,out]</td><td class="paramname">y</td><td>take \(y_n\) and compute \(y_{n+1}\) </td></tr>
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<tr><td class="paramdir">[in]</td><td class="paramname">save_to_file</td><td>flag to save results to a CSV file (1) or not (0) </td></tr>
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</table>
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</dd>
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</dl>
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<dl class="section return"><dt>Returns</dt><dd>time taken for computation in seconds </dd></dl>
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<div class="fragment"><div class="line"><a name="l00103"></a><span class="lineno"> 103</span>  {</div>
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<div class="line"><a name="l00104"></a><span class="lineno"> 104</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray<double></a> dy = y[0];</div>
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<div class="line"><a name="l00105"></a><span class="lineno"> 105</span>  </div>
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<div class="line"><a name="l00106"></a><span class="lineno"> 106</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ofstream.html">std::ofstream</a> fp;</div>
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<div class="line"><a name="l00107"></a><span class="lineno"> 107</span>  <span class="keywordflow">if</span> (save_to_file) {</div>
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<div class="line"><a name="l00108"></a><span class="lineno"> 108</span>  fp.<a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ofstream/open.html">open</a>(<span class="stringliteral">"forward_euler.csv"</span>, std::ofstream::out);</div>
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<div class="line"><a name="l00109"></a><span class="lineno"> 109</span>  <span class="keywordflow">if</span> (!fp.<a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ofstream/is_open.html">is_open</a>()) {</div>
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<div class="line"><a name="l00110"></a><span class="lineno"> 110</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/c/perror.html">std::perror</a>(<span class="stringliteral">"Error! "</span>);</div>
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<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>  }</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>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/types/size_t.html">std::size_t</a> L = y->size();</div>
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<div class="line"><a name="l00115"></a><span class="lineno"> 115</span>  </div>
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<div class="line"><a name="l00116"></a><span class="lineno"> 116</span>  <span class="comment">/* start integration */</span></div>
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<div class="line"><a name="l00117"></a><span class="lineno"> 117</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/chrono/c/clock_t.html">std::clock_t</a> t1 = <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/chrono/c/clock.html">std::clock</a>();</div>
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<div class="line"><a name="l00118"></a><span class="lineno"> 118</span>  <span class="keywordtype">double</span> x = x0;</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>  <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="l00121"></a><span class="lineno"> 121</span>  <span class="keywordflow">if</span> (save_to_file && fp.<a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ofstream/is_open.html">is_open</a>()) {</div>
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<div class="line"><a name="l00122"></a><span class="lineno"> 122</span>  <span class="comment">// write to file</span></div>
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<div class="line"><a name="l00123"></a><span class="lineno"> 123</span>  fp << x << <span class="stringliteral">","</span>;</div>
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<div class="line"><a name="l00124"></a><span class="lineno"> 124</span>  <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i < L - 1; i++) {</div>
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<div class="line"><a name="l00125"></a><span class="lineno"> 125</span>  fp << y[0][i] << <span class="stringliteral">","</span>;</div>
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<div class="line"><a name="l00126"></a><span class="lineno"> 126</span>  }</div>
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<div class="line"><a name="l00127"></a><span class="lineno"> 127</span>  fp << y[0][L - 1] << <span class="stringliteral">"\n"</span>;</div>
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<div class="line"><a name="l00128"></a><span class="lineno"> 128</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>  <a class="code" href="../../d6/d60/group__ode.html#gaa197c6b7085ac9ea71abc4c063e6f52f">forward_euler_step</a>(dx, x, y, &dy); <span class="comment">// perform integration</span></div>
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<div class="line"><a name="l00131"></a><span class="lineno"> 131</span>  x += dx; <span class="comment">// update step</span></div>
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<div class="line"><a name="l00132"></a><span class="lineno"> 132</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="l00133"></a><span class="lineno"> 133</span>  <span class="comment">/* end of integration */</span></div>
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<div class="line"><a name="l00134"></a><span class="lineno"> 134</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/chrono/c/clock_t.html">std::clock_t</a> t2 = <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/chrono/c/clock.html">std::clock</a>();</div>
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<div class="line"><a name="l00135"></a><span class="lineno"> 135</span>  </div>
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<div class="line"><a name="l00136"></a><span class="lineno"> 136</span>  <span class="keywordflow">if</span> (fp.<a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ofstream/is_open.html">is_open</a>())</div>
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<div class="line"><a name="l00137"></a><span class="lineno"> 137</span>  fp.<a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ofstream/close.html">close</a>();</div>
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<div class="line"><a name="l00138"></a><span class="lineno"> 138</span>  </div>
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<div class="line"><a name="l00139"></a><span class="lineno"> 139</span>  <span class="keywordflow">return</span> <span class="keyword">static_cast<</span><span class="keywordtype">double</span><span class="keyword">></span>(t2 - t1) / CLOCKS_PER_SEC;</div>
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<div class="line"><a name="l00140"></a><span class="lineno"> 140</span> }</div>
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<h2 class="memtitle"><span class="permalink"><a href="#gaa197c6b7085ac9ea71abc4c063e6f52f">◆ </a></span>forward_euler_step()</h2>
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<td class="memname">void forward_euler_step </td>
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<td>(</td>
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<td class="paramtype">const double </td>
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<td class="paramname"><em>dx</em>, </td>
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<td class="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"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>< double > * </td>
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<td class="paramname"><em>y</em>, </td>
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<td class="paramkey"></td>
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<td></td>
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<td class="paramtype"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>< double > * </td>
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<td class="paramname"><em>dy</em> </td>
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<p>Compute next step approximation using the forward-Euler method. </p>
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<p class="formulaDsp">
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\[y_{n+1}=y_n + dx\cdot f\left(x_n,y_n\right)\]
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</p>
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<dl class="params"><dt>Parameters</dt><dd>
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<table class="params">
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<tr><td class="paramdir">[in]</td><td class="paramname">dx</td><td>step size </td></tr>
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<tr><td class="paramdir">[in]</td><td class="paramname">x</td><td>take \(x_n\) and compute \(x_{n+1}\) </td></tr>
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<tr><td class="paramdir">[in,out]</td><td class="paramname">y</td><td>take \(y_n\) and compute \(y_{n+1}\) </td></tr>
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<tr><td class="paramdir">[in,out]</td><td class="paramname">dy</td><td>compute \(f\left(x_n,y_n\right)\) </td></tr>
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<div class="fragment"><div class="line"><a name="l00087"></a><span class="lineno"> 87</span>  {</div>
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<div class="line"><a name="l00088"></a><span class="lineno"> 88</span>  <a class="code" href="../../db/dd3/ode__forward__euler_8cpp.html#abaeae8f62a018d197f0187a1c80a90fe">problem</a>(x, y, dy);</div>
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<div class="line"><a name="l00089"></a><span class="lineno"> 89</span>  y[0] += dy[0] * dx;</div>
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<div class="line"><a name="l00090"></a><span class="lineno"> 90</span> }</div>
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<h2 class="memtitle"><span class="permalink"><a href="#ga36ff0710861ca39d957046c0b09b2985">◆ </a></span>midpoint_euler()</h2>
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<td class="memname">double midpoint_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 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"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>< double > * </td>
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<td class="paramname"><em>y</em>, </td>
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<td class="paramkey"></td>
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<td></td>
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<td class="paramtype">bool </td>
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<td class="paramname"><em>save_to_file</em> = <code>false</code> </td>
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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 approximation using the midpoint-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="l00108"></a><span class="lineno"> 108</span>  {</div>
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<div class="line"><a name="l00109"></a><span class="lineno"> 109</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray<double></a> dy = y[0];</div>
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<div class="line"><a name="l00110"></a><span class="lineno"> 110</span>  </div>
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<div class="line"><a name="l00111"></a><span class="lineno"> 111</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ofstream.html">std::ofstream</a> fp;</div>
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<div class="line"><a name="l00112"></a><span class="lineno"> 112</span>  <span class="keywordflow">if</span> (save_to_file) {</div>
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<div class="line"><a name="l00113"></a><span class="lineno"> 113</span>  fp.<a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ofstream/open.html">open</a>(<span class="stringliteral">"midpoint_euler.csv"</span>, std::ofstream::out);</div>
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<div class="line"><a name="l00114"></a><span class="lineno"> 114</span>  <span class="keywordflow">if</span> (!fp.<a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ofstream/is_open.html">is_open</a>()) {</div>
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<div class="line"><a name="l00115"></a><span class="lineno"> 115</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/c/perror.html">std::perror</a>(<span class="stringliteral">"Error! "</span>);</div>
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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>  }</div>
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<div class="line"><a name="l00118"></a><span class="lineno"> 118</span>  </div>
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<div class="line"><a name="l00119"></a><span class="lineno"> 119</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/types/size_t.html">std::size_t</a> L = y->size();</div>
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<div class="line"><a name="l00120"></a><span class="lineno"> 120</span>  </div>
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<div class="line"><a name="l00121"></a><span class="lineno"> 121</span>  <span class="comment">/* start integration */</span></div>
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<div class="line"><a name="l00122"></a><span class="lineno"> 122</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/chrono/c/clock_t.html">std::clock_t</a> t1 = <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/chrono/c/clock.html">std::clock</a>();</div>
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<div class="line"><a name="l00123"></a><span class="lineno"> 123</span>  <span class="keywordtype">double</span> x = x0;</div>
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<div class="line"><a name="l00124"></a><span class="lineno"> 124</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="l00125"></a><span class="lineno"> 125</span>  <span class="keywordflow">if</span> (save_to_file && fp.<a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ofstream/is_open.html">is_open</a>()) {</div>
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<div class="line"><a name="l00126"></a><span class="lineno"> 126</span>  <span class="comment">// write to file</span></div>
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<div class="line"><a name="l00127"></a><span class="lineno"> 127</span>  fp << x << <span class="stringliteral">","</span>;</div>
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<div class="line"><a name="l00128"></a><span class="lineno"> 128</span>  <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i < L - 1; i++) {</div>
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<div class="line"><a name="l00129"></a><span class="lineno"> 129</span>  fp << y[0][i] << <span class="stringliteral">","</span>;</div>
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<div class="line"><a name="l00130"></a><span class="lineno"> 130</span>  }</div>
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<div class="line"><a name="l00131"></a><span class="lineno"> 131</span>  fp << y[0][L - 1] << <span class="stringliteral">"\n"</span>;</div>
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<div class="line"><a name="l00132"></a><span class="lineno"> 132</span>  }</div>
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<div class="line"><a name="l00133"></a><span class="lineno"> 133</span>  </div>
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<div class="line"><a name="l00134"></a><span class="lineno"> 134</span>  <a class="code" href="../../d6/d60/group__ode.html#ga8c319db420c3d97a83e9dcca803b6812">midpoint_euler_step</a>(dx, x, y, &dy); <span class="comment">// perform integration</span></div>
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<div class="line"><a name="l00135"></a><span class="lineno"> 135</span>  x += dx; <span class="comment">// update step</span></div>
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<div class="line"><a name="l00136"></a><span class="lineno"> 136</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="l00137"></a><span class="lineno"> 137</span>  <span class="comment">/* end of integration */</span></div>
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<div class="line"><a name="l00138"></a><span class="lineno"> 138</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/chrono/c/clock_t.html">std::clock_t</a> t2 = <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/chrono/c/clock.html">std::clock</a>();</div>
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<div class="line"><a name="l00139"></a><span class="lineno"> 139</span>  </div>
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<div class="line"><a name="l00140"></a><span class="lineno"> 140</span>  <span class="keywordflow">if</span> (fp.<a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ofstream/is_open.html">is_open</a>())</div>
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<div class="line"><a name="l00141"></a><span class="lineno"> 141</span>  fp.<a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ofstream/close.html">close</a>();</div>
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<div class="line"><a name="l00142"></a><span class="lineno"> 142</span>  </div>
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<div class="line"><a name="l00143"></a><span class="lineno"> 143</span>  <span class="keywordflow">return</span> <span class="keyword">static_cast<</span><span class="keywordtype">double</span><span class="keyword">></span>(t2 - t1) / CLOCKS_PER_SEC;</div>
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<div class="line"><a name="l00144"></a><span class="lineno"> 144</span> }</div>
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<h2 class="memtitle"><span class="permalink"><a href="#ga8c319db420c3d97a83e9dcca803b6812">◆ </a></span>midpoint_euler_step()</h2>
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<td class="memname">void midpoint_euler_step </td>
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<td>(</td>
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<td class="paramtype">const double </td>
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<td class="paramname"><em>dx</em>, </td>
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<td class="paramkey"></td>
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<td></td>
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<td class="paramtype">const double & </td>
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<td class="paramname"><em>x</em>, </td>
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<td></td>
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<td class="paramtype"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>< double > * </td>
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<td class="paramname"><em>y</em>, </td>
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<td></td>
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<td class="paramtype"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>< double > * </td>
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<td class="paramname"><em>dy</em> </td>
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<p>Compute next step approximation using the midpoint-Euler method. </p>
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<p class="formulaDsp">
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\[y_{n+1} = y_n + dx\, f\left(x_n+\frac{1}{2}dx, y_n + \frac{1}{2}dx\,f\left(x_n,y_n\right)\right)\]
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</p>
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<dl class="params"><dt>Parameters</dt><dd>
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<table class="params">
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<tr><td class="paramdir">[in]</td><td class="paramname">dx</td><td>step size </td></tr>
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<tr><td class="paramdir">[in]</td><td class="paramname">x</td><td>take \(x_n\) and compute \(x_{n+1}\) </td></tr>
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<tr><td class="paramdir">[in,out]</td><td class="paramname">y</td><td>take \(y_n\) and compute \(y_{n+1}\) </td></tr>
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<tr><td class="paramdir">[in,out]</td><td class="paramname">dy</td><td>compute \(f\left(x_n,y_n\right)\) </td></tr>
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</table>
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</dd>
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</dl>
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<div class="fragment"><div class="line"><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>  <a class="code" href="../../d6/dd3/ode__midpoint__euler_8cpp.html#abaeae8f62a018d197f0187a1c80a90fe">problem</a>(x, y, dy);</div>
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<div class="line"><a name="l00088"></a><span class="lineno"> 88</span>  <span class="keywordtype">double</span> tmp_x = x + 0.5 * dx;</div>
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<div class="line"><a name="l00089"></a><span class="lineno"> 89</span>  </div>
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<div class="line"><a name="l00090"></a><span class="lineno"> 90</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray<double></a> tmp_y = y[0] + dy[0] * (0.5 * dx);</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="../../d6/dd3/ode__midpoint__euler_8cpp.html#abaeae8f62a018d197f0187a1c80a90fe">problem</a>(tmp_x, &tmp_y, dy);</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>  y[0] += dy[0] * dx;</div>
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<div class="line"><a name="l00095"></a><span class="lineno"> 95</span> }</div>
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<h2 class="memtitle"><span class="permalink"><a href="#ga3874d294ab00fe9ce8731c5b7991a089">◆ </a></span>semi_implicit_euler()</h2>
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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 class="paramtype">double </td>
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<td class="paramname"><em>x0</em>, </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"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>< double > * </td>
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<td class="paramname"><em>y</em>, </td>
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<td class="paramkey"></td>
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<td class="paramtype">bool </td>
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<td class="paramname"><em>save_to_file</em> = <code>false</code> </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="l00105"></a><span class="lineno"> 105</span>  {</div>
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<div class="line"><a name="l00106"></a><span class="lineno"> 106</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray<double></a> dy = y[0];</div>
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<div class="line"><a name="l00107"></a><span class="lineno"> 107</span>  </div>
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<div class="line"><a name="l00108"></a><span class="lineno"> 108</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ofstream.html">std::ofstream</a> fp;</div>
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<div class="line"><a name="l00109"></a><span class="lineno"> 109</span>  <span class="keywordflow">if</span> (save_to_file) {</div>
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<div class="line"><a name="l00110"></a><span class="lineno"> 110</span>  fp.<a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ofstream/open.html">open</a>(<span class="stringliteral">"semi_implicit_euler.csv"</span>, std::ofstream::out);</div>
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<div class="line"><a name="l00111"></a><span class="lineno"> 111</span>  <span class="keywordflow">if</span> (!fp.<a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ofstream/is_open.html">is_open</a>()) {</div>
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<div class="line"><a name="l00112"></a><span class="lineno"> 112</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/c/perror.html">std::perror</a>(<span class="stringliteral">"Error! "</span>);</div>
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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>  }</div>
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<div class="line"><a name="l00115"></a><span class="lineno"> 115</span>  </div>
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<div class="line"><a name="l00116"></a><span class="lineno"> 116</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/types/size_t.html">std::size_t</a> L = y->size();</div>
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<div class="line"><a name="l00117"></a><span class="lineno"> 117</span>  </div>
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<div class="line"><a name="l00118"></a><span class="lineno"> 118</span>  <span class="comment">/* start integration */</span></div>
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<div class="line"><a name="l00119"></a><span class="lineno"> 119</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/chrono/c/clock_t.html">std::clock_t</a> t1 = <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/chrono/c/clock.html">std::clock</a>();</div>
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<div class="line"><a name="l00120"></a><span class="lineno"> 120</span>  <span class="keywordtype">double</span> x = x0;</div>
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<div class="line"><a name="l00121"></a><span class="lineno"> 121</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="l00122"></a><span class="lineno"> 122</span>  <span class="keywordflow">if</span> (save_to_file && fp.<a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ofstream/is_open.html">is_open</a>()) {</div>
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<div class="line"><a name="l00123"></a><span class="lineno"> 123</span>  <span class="comment">// write to file</span></div>
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<div class="line"><a name="l00124"></a><span class="lineno"> 124</span>  fp << x << <span class="stringliteral">","</span>;</div>
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<div class="line"><a name="l00125"></a><span class="lineno"> 125</span>  <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i < L - 1; i++) {</div>
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<div class="line"><a name="l00126"></a><span class="lineno"> 126</span>  fp << y[0][i] << <span class="stringliteral">","</span>;</div>
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<div class="line"><a name="l00127"></a><span class="lineno"> 127</span>  }</div>
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<div class="line"><a name="l00128"></a><span class="lineno"> 128</span>  fp << y[0][L - 1] << <span class="stringliteral">"\n"</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>  </div>
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<div class="line"><a name="l00131"></a><span class="lineno"> 131</span>  <a class="code" href="../../d6/d60/group__ode.html#ga827bf009831ddc477c5fa8891d5cb35f">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="l00132"></a><span class="lineno"> 132</span>  x += dx; <span class="comment">// update step</span></div>
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<div class="line"><a name="l00133"></a><span class="lineno"> 133</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="l00134"></a><span class="lineno"> 134</span>  <span class="comment">/* end of integration */</span></div>
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<div class="line"><a name="l00135"></a><span class="lineno"> 135</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/chrono/c/clock_t.html">std::clock_t</a> t2 = <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/chrono/c/clock.html">std::clock</a>();</div>
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<div class="line"><a name="l00136"></a><span class="lineno"> 136</span>  </div>
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<div class="line"><a name="l00137"></a><span class="lineno"> 137</span>  <span class="keywordflow">if</span> (fp.<a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ofstream/is_open.html">is_open</a>())</div>
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<div class="line"><a name="l00138"></a><span class="lineno"> 138</span>  fp.<a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ofstream/close.html">close</a>();</div>
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<div class="line"><a name="l00139"></a><span class="lineno"> 139</span>  </div>
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<div class="line"><a name="l00140"></a><span class="lineno"> 140</span>  <span class="keywordflow">return</span> <span class="keyword">static_cast<</span><span class="keywordtype">double</span><span class="keyword">></span>(t2 - t1) / CLOCKS_PER_SEC;</div>
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<div class="line"><a name="l00141"></a><span class="lineno"> 141</span> }</div>
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<h2 class="memtitle"><span class="permalink"><a href="#ga827bf009831ddc477c5fa8891d5cb35f">◆ </a></span>semi_implicit_euler_step()</h2>
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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">const double </td>
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<td class="paramname"><em>dx</em>, </td>
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<td class="paramkey"></td>
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<td></td>
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<td class="paramtype">const double & </td>
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<td class="paramname"><em>x</em>, </td>
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<td class="paramkey"></td>
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<td></td>
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<td class="paramtype"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>< double > * </td>
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<td class="paramname"><em>y</em>, </td>
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<td class="paramkey"></td>
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<td></td>
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<td class="paramtype"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>< double > * </td>
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<td class="paramname"><em>dy</em> </td>
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<td>)</td>
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</div><div class="memdoc">
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<p>Compute next step approximation using the semi-implicit-Euler method. </p>
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<p class="formulaDsp">
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\[y_{n+1}=y_n + dx\cdot f\left(x_n,y_n\right)\]
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</p>
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<dl class="params"><dt>Parameters</dt><dd>
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<table class="params">
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<tr><td class="paramdir">[in]</td><td class="paramname">dx</td><td>step size </td></tr>
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<tr><td class="paramdir">[in]</td><td class="paramname">x</td><td>take \(x_n\) and compute \(x_{n+1}\) </td></tr>
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<tr><td class="paramdir">[in,out]</td><td class="paramname">y</td><td>take \(y_n\) and compute \(y_{n+1}\) </td></tr>
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<tr><td class="paramdir">[in,out]</td><td class="paramname">dy</td><td>compute \(f\left(x_n,y_n\right)\) </td></tr>
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</table>
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</dd>
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</dl>
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<div class="fragment"><div class="line"><a name="l00084"></a><span class="lineno"> 84</span>  {</div>
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<div class="line"><a name="l00085"></a><span class="lineno"> 85</span>  <a class="code" href="../../d3/d06/ode__semi__implicit__euler_8cpp.html#abaeae8f62a018d197f0187a1c80a90fe">problem</a>(x, y, dy); <span class="comment">// update dy once</span></div>
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<div class="line"><a name="l00086"></a><span class="lineno"> 86</span>  y[0][0] += dx * dy[0][0]; <span class="comment">// update y0</span></div>
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<div class="line"><a name="l00087"></a><span class="lineno"> 87</span>  <a class="code" href="../../d3/d06/ode__semi__implicit__euler_8cpp.html#abaeae8f62a018d197f0187a1c80a90fe">problem</a>(x, y, dy); <span class="comment">// update dy once more</span></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>  dy[0][0] = 0.f; <span class="comment">// ignore y0</span></div>
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<div class="line"><a name="l00090"></a><span class="lineno"> 90</span>  y[0] += dy[0] * dx; <span class="comment">// update remaining using new dy</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="ttc" id="agroup__ode_html_ga827bf009831ddc477c5fa8891d5cb35f"><div class="ttname"><a href="../../d6/d60/group__ode.html#ga827bf009831ddc477c5fa8891d5cb35f">semi_implicit_euler_step</a></div><div class="ttdeci">void semi_implicit_euler_step(const double dx, const double &x, std::valarray< double > *y, std::valarray< 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.cpp:82</div></div>
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<div class="ttc" id="aclock_t_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/chrono/c/clock_t.html">std::clock_t</a></div></div>
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<div class="ttc" id="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 &x, std::valarray< double > *y, std::valarray< 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.cpp:85</div></div>
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<div class="ttc" id="agroup__ode_html_gaa197c6b7085ac9ea71abc4c063e6f52f"><div class="ttname"><a href="../../d6/d60/group__ode.html#gaa197c6b7085ac9ea71abc4c063e6f52f">forward_euler_step</a></div><div class="ttdeci">void forward_euler_step(const double dx, const double &x, std::valarray< double > *y, std::valarray< double > *dy)</div><div class="ttdoc">Compute next step approximation using the forward-Euler method.</div><div class="ttdef"><b>Definition:</b> ode_forward_euler.cpp:86</div></div>
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<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 &x, std::valarray< double > *y, std::valarray< 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_forward_euler.cpp:54</div></div>
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<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 &x, std::valarray< double > *y, std::valarray< 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.cpp:53</div></div>
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