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<p><a href="https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods">Runge Kutta fourth order</a> method implementation
<a href="#details">More...</a></p>
<div class="textblock"><code>#include &lt;cassert&gt;</code><br />
<code>#include &lt;iostream&gt;</code><br />
<code>#include &lt;vector&gt;</code><br />
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Include dependency graph for rungekutta.cpp:</div>
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Namespaces</h2></td></tr>
<tr class="memitem:d1/de0/namespacenumerical__methods"><td class="memItemLeft" align="right" valign="top"> &#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../d1/de0/namespacenumerical__methods.html">numerical_methods</a></td></tr>
<tr class="memdesc:d1/de0/namespacenumerical__methods"><td class="mdescLeft">&#160;</td><td class="mdescRight">for io operations <br /></td></tr>
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<tr class="memitem:d2/de7/namespacerunge__kutta"><td class="memItemLeft" align="right" valign="top"> &#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../d2/de7/namespacerunge__kutta.html">runge_kutta</a></td></tr>
<tr class="memdesc:d2/de7/namespacerunge__kutta"><td class="mdescLeft">&#160;</td><td class="mdescRight">Functions for <a href="https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods">Runge Kutta fourth order</a> method. <br /></td></tr>
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Functions</h2></td></tr>
<tr class="memitem:a7b9f40c7b5e9749cc550f19be3dc8856"><td class="memItemLeft" align="right" valign="top">double&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../d1/da6/rungekutta_8cpp.html#a7b9f40c7b5e9749cc550f19be3dc8856">numerical_methods::runge_kutta::rungeKutta</a> (double init_x, const double &amp;init_y, const double &amp;x, const double &amp;<a class="el" href="../../d1/df3/hash__search_8cpp.html#a566eaf0ffafd50bc61e644561fd27001">h</a>)</td></tr>
<tr class="memdesc:a7b9f40c7b5e9749cc550f19be3dc8856"><td class="mdescLeft">&#160;</td><td class="mdescRight">the Runge Kutta method finds the value of integration of a function in the given limits. the lower limit of integration as the initial value and the upper limit is the given x <a href="../../d1/da6/rungekutta_8cpp.html#a7b9f40c7b5e9749cc550f19be3dc8856">More...</a><br /></td></tr>
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<tr><td colspan="2"><div class="groupHeader"></div></td></tr>
<tr class="memitem:a450497475f5607333f9aca8f88901579"><td class="memItemLeft" align="right" valign="top">static double&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../d1/da6/rungekutta_8cpp.html#a450497475f5607333f9aca8f88901579">change</a> (double x, double y)</td></tr>
<tr class="memdesc:a450497475f5607333f9aca8f88901579"><td class="mdescLeft">&#160;</td><td class="mdescRight">for using the vector container <a href="../../d1/da6/rungekutta_8cpp.html#a450497475f5607333f9aca8f88901579">More...</a><br /></td></tr>
<tr class="separator:a450497475f5607333f9aca8f88901579"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:aa8dca7b867074164d5f45b0f3851269d"><td class="memItemLeft" align="right" valign="top">static void&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../d1/da6/rungekutta_8cpp.html#aa8dca7b867074164d5f45b0f3851269d">test</a> ()</td></tr>
<tr class="memdesc:aa8dca7b867074164d5f45b0f3851269d"><td class="mdescLeft">&#160;</td><td class="mdescRight">Tests to check algorithm implementation. <a href="../../d1/da6/rungekutta_8cpp.html#aa8dca7b867074164d5f45b0f3851269d">More...</a><br /></td></tr>
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<tr class="memitem:ae66f6b31b5ad750f1fe042a706a4e3d4"><td class="memItemLeft" align="right" valign="top">int&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../d1/da6/rungekutta_8cpp.html#ae66f6b31b5ad750f1fe042a706a4e3d4">main</a> ()</td></tr>
<tr class="memdesc:ae66f6b31b5ad750f1fe042a706a4e3d4"><td class="mdescLeft">&#160;</td><td class="mdescRight">Main function. <a href="../../d1/da6/rungekutta_8cpp.html#ae66f6b31b5ad750f1fe042a706a4e3d4">More...</a><br /></td></tr>
<tr class="separator:ae66f6b31b5ad750f1fe042a706a4e3d4"><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><a href="https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods">Runge Kutta fourth order</a> method implementation </p>
<dl class="section author"><dt>Author</dt><dd><a href="http://github.com/rudra697">Rudra Prasad Das</a></dd></dl>
<p>It solves the unknown value of y for a given value of x only first order differential equations can be solved </p>
</div><h2 class="groupheader">Function Documentation</h2>
<a id="a450497475f5607333f9aca8f88901579"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a450497475f5607333f9aca8f88901579">&#9670;&nbsp;</a></span>change()</h2>
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<td class="memname">static double change </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>&#160;</td>
</tr>
<tr>
<td></td>
<td>)</td>
<td></td><td></td>
</tr>
</table>
</td>
<td class="mlabels-right">
<span class="mlabels"><span class="mlabel">static</span></span> </td>
</tr>
</table>
</div><div class="memdoc">
<p>for using the vector container </p>
<p>asserting the test functions for io operations</p>
<p>The <a class="el" href="../../d1/da6/rungekutta_8cpp.html#a450497475f5607333f9aca8f88901579" title="for using the vector container">change()</a> function is used to return the updated iterative value corresponding to the given function </p><dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">x</td><td>is the value corresponding to the x coordinate </td></tr>
<tr><td class="paramname">y</td><td>is the value corresponding to the y coordinate </td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>the computed function value at that call </dd></dl>
<dl class="section examples"><dt>Examples</dt><dd><a class="el" href="../../dc/dc4/_2_users_2runner_2work_2_c-_plus-_plus_2_c-_plus-_plus_2numerical_methods_2rungekutta_8cpp-example.html#a0">/Users/runner/work/C-Plus-Plus/C-Plus-Plus/numerical_methods/rungekutta.cpp</a>.</dd>
</dl>
<div class="fragment"><div class="line"><a name="l00033"></a><span class="lineno"> 33</span>&#160;{ <span class="keywordflow">return</span> ((x - y) / 2.0); }</div>
</div><!-- fragment -->
</div>
</div>
<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>
<td>(</td>
<td class="paramtype">void&#160;</td>
<td class="paramname"></td><td>)</td>
<td></td>
</tr>
</table>
</div><div class="memdoc">
<p>Main function. </p>
<dl class="section return"><dt>Returns</dt><dd>0 on exit </dd></dl>
<div class="fragment"><div class="line"><a name="l00130"></a><span class="lineno"> 130</span>&#160; {</div>
<div class="line"><a name="l00131"></a><span class="lineno"> 131</span>&#160; <a class="code" href="../../d1/da6/rungekutta_8cpp.html#aa8dca7b867074164d5f45b0f3851269d">test</a>(); <span class="comment">// Execute the tests</span></div>
<div class="line"><a name="l00132"></a><span class="lineno"> 132</span>&#160; <span class="keywordflow">return</span> 0;</div>
<div class="line"><a name="l00133"></a><span class="lineno"> 133</span>&#160;}</div>
<div class="ttc" id="arungekutta_8cpp_html_aa8dca7b867074164d5f45b0f3851269d"><div class="ttname"><a href="../../d1/da6/rungekutta_8cpp.html#aa8dca7b867074164d5f45b0f3851269d">test</a></div><div class="ttdeci">static void test()</div><div class="ttdoc">Tests to check algorithm implementation.</div><div class="ttdef"><b>Definition:</b> rungekutta.cpp:100</div></div>
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Here is the call graph for this function:</div>
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<h2 class="memtitle"><span class="permalink"><a href="#a7b9f40c7b5e9749cc550f19be3dc8856">&#9670;&nbsp;</a></span>rungeKutta()</h2>
<div class="memitem">
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<td class="memname">double numerical_methods::runge_kutta::rungeKutta </td>
<td>(</td>
<td class="paramtype">double&#160;</td>
<td class="paramname"><em>init_x</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">const double &amp;&#160;</td>
<td class="paramname"><em>init_y</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">const double &amp;&#160;</td>
<td class="paramname"><em>x</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">const double &amp;&#160;</td>
<td class="paramname"><em>h</em>&#160;</td>
</tr>
<tr>
<td></td>
<td>)</td>
<td></td><td></td>
</tr>
</table>
</div><div class="memdoc">
<p>the Runge Kutta method finds the value of integration of a function in the given limits. the lower limit of integration as the initial value and the upper limit is the given x </p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">init_x</td><td>is the value of initial x and is updated after each call </td></tr>
<tr><td class="paramname">init_y</td><td>is the value of initial x and is updated after each call </td></tr>
<tr><td class="paramname">x</td><td>is current iteration at which the function needs to be evaluated </td></tr>
<tr><td class="paramname">h</td><td>is the step value </td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>the value of y at thr required value of x from the initial conditions </dd></dl>
<dl class="section examples"><dt>Examples</dt><dd><a class="el" href="../../dc/dc4/_2_users_2runner_2work_2_c-_plus-_plus_2_c-_plus-_plus_2numerical_methods_2rungekutta_8cpp-example.html#a1">/Users/runner/work/C-Plus-Plus/C-Plus-Plus/numerical_methods/rungekutta.cpp</a>.</dd>
</dl>
<div class="fragment"><div class="line"><a name="l00058"></a><span class="lineno"> 58</span>&#160; {</div>
<div class="line"><a name="l00059"></a><span class="lineno"> 59</span>&#160; <span class="comment">// Count number of iterations</span></div>
<div class="line"><a name="l00060"></a><span class="lineno"> 60</span>&#160; <span class="comment">// using step size or</span></div>
<div class="line"><a name="l00061"></a><span class="lineno"> 61</span>&#160; <span class="comment">// step height h</span></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; <span class="comment">// n calucates the number of iterations</span></div>
<div class="line"><a name="l00064"></a><span class="lineno"> 64</span>&#160; <span class="comment">// k1, k2, k3, k4 are the Runge Kutta variables</span></div>
<div class="line"><a name="l00065"></a><span class="lineno"> 65</span>&#160; <span class="comment">// used for calculation of y at each iteration</span></div>
<div class="line"><a name="l00066"></a><span class="lineno"> 66</span>&#160; </div>
<div class="line"><a name="l00067"></a><span class="lineno"> 67</span>&#160; <span class="keyword">auto</span> n = <span class="keyword">static_cast&lt;</span>uint64_t<span class="keyword">&gt;</span>((x - init_x) / <a class="code" href="../../d1/df3/hash__search_8cpp.html#a566eaf0ffafd50bc61e644561fd27001">h</a>);</div>
<div class="line"><a name="l00068"></a><span class="lineno"> 68</span>&#160; <span class="comment">// used a vector container for the variables</span></div>
<div class="line"><a name="l00069"></a><span class="lineno"> 69</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/vector.html">std::vector&lt;double&gt;</a> k(4, 0.0);</div>
<div class="line"><a name="l00070"></a><span class="lineno"> 70</span>&#160; </div>
<div class="line"><a name="l00071"></a><span class="lineno"> 71</span>&#160; <span class="comment">// Iterate for number of iterations</span></div>
<div class="line"><a name="l00072"></a><span class="lineno"> 72</span>&#160; </div>
<div class="line"><a name="l00073"></a><span class="lineno"> 73</span>&#160; <span class="keywordtype">double</span> y = init_y;</div>
<div class="line"><a name="l00074"></a><span class="lineno"> 74</span>&#160; <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 1; i &lt;= n; ++i) {</div>
<div class="line"><a name="l00075"></a><span class="lineno"> 75</span>&#160; <span class="comment">// Apply Runge Kutta Formulas</span></div>
<div class="line"><a name="l00076"></a><span class="lineno"> 76</span>&#160; <span class="comment">// to find next value of y</span></div>
<div class="line"><a name="l00077"></a><span class="lineno"> 77</span>&#160; k[0] = <a class="code" href="../../d1/df3/hash__search_8cpp.html#a566eaf0ffafd50bc61e644561fd27001">h</a> * <a class="code" href="../../d1/da6/rungekutta_8cpp.html#a450497475f5607333f9aca8f88901579">change</a>(init_x, y);</div>
<div class="line"><a name="l00078"></a><span class="lineno"> 78</span>&#160; k[1] = <a class="code" href="../../d1/df3/hash__search_8cpp.html#a566eaf0ffafd50bc61e644561fd27001">h</a> * <a class="code" href="../../d1/da6/rungekutta_8cpp.html#a450497475f5607333f9aca8f88901579">change</a>(init_x + 0.5 * <a class="code" href="../../d1/df3/hash__search_8cpp.html#a566eaf0ffafd50bc61e644561fd27001">h</a>, y + 0.5 * k[0]);</div>
<div class="line"><a name="l00079"></a><span class="lineno"> 79</span>&#160; k[2] = <a class="code" href="../../d1/df3/hash__search_8cpp.html#a566eaf0ffafd50bc61e644561fd27001">h</a> * <a class="code" href="../../d1/da6/rungekutta_8cpp.html#a450497475f5607333f9aca8f88901579">change</a>(init_x + 0.5 * <a class="code" href="../../d1/df3/hash__search_8cpp.html#a566eaf0ffafd50bc61e644561fd27001">h</a>, y + 0.5 * k[1]);</div>
<div class="line"><a name="l00080"></a><span class="lineno"> 80</span>&#160; k[3] = <a class="code" href="../../d1/df3/hash__search_8cpp.html#a566eaf0ffafd50bc61e644561fd27001">h</a> * <a class="code" href="../../d1/da6/rungekutta_8cpp.html#a450497475f5607333f9aca8f88901579">change</a>(init_x + <a class="code" href="../../d1/df3/hash__search_8cpp.html#a566eaf0ffafd50bc61e644561fd27001">h</a>, y + k[2]);</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; <span class="comment">// Update next value of y</span></div>
<div class="line"><a name="l00083"></a><span class="lineno"> 83</span>&#160; </div>
<div class="line"><a name="l00084"></a><span class="lineno"> 84</span>&#160; y += (1.0 / 6.0) * (k[0] + 2 * k[1] + 2 * k[2] + k[3]);</div>
<div class="line"><a name="l00085"></a><span class="lineno"> 85</span>&#160; </div>
<div class="line"><a name="l00086"></a><span class="lineno"> 86</span>&#160; <span class="comment">// Update next value of x</span></div>
<div class="line"><a name="l00087"></a><span class="lineno"> 87</span>&#160; </div>
<div class="line"><a name="l00088"></a><span class="lineno"> 88</span>&#160; init_x += <a class="code" href="../../d1/df3/hash__search_8cpp.html#a566eaf0ffafd50bc61e644561fd27001">h</a>;</div>
<div class="line"><a name="l00089"></a><span class="lineno"> 89</span>&#160; }</div>
<div class="line"><a name="l00090"></a><span class="lineno"> 90</span>&#160; </div>
<div class="line"><a name="l00091"></a><span class="lineno"> 91</span>&#160; <span class="keywordflow">return</span> y;</div>
<div class="line"><a name="l00092"></a><span class="lineno"> 92</span>&#160;}</div>
<div class="ttc" id="ahash__search_8cpp_html_a566eaf0ffafd50bc61e644561fd27001"><div class="ttname"><a href="../../d1/df3/hash__search_8cpp.html#a566eaf0ffafd50bc61e644561fd27001">h</a></div><div class="ttdeci">int h(int key)</div><div class="ttdef"><b>Definition:</b> hash_search.cpp:45</div></div>
<div class="ttc" id="arungekutta_8cpp_html_a450497475f5607333f9aca8f88901579"><div class="ttname"><a href="../../d1/da6/rungekutta_8cpp.html#a450497475f5607333f9aca8f88901579">change</a></div><div class="ttdeci">static double change(double x, double y)</div><div class="ttdoc">for using the vector container</div><div class="ttdef"><b>Definition:</b> rungekutta.cpp:33</div></div>
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<h2 class="memtitle"><span class="permalink"><a href="#aa8dca7b867074164d5f45b0f3851269d">&#9670;&nbsp;</a></span>test()</h2>
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<td class="memname">static void test </td>
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<p>Tests to check algorithm implementation. </p>
<dl class="section return"><dt>Returns</dt><dd>void </dd></dl>
<div class="fragment"><div class="line"><a name="l00100"></a><span class="lineno"> 100</span>&#160; {</div>
<div class="line"><a name="l00101"></a><span class="lineno"> 101</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; <span class="stringliteral">&quot;The Runge Kutta function will be tested on the basis of &quot;</span></div>
<div class="line"><a name="l00102"></a><span class="lineno"> 102</span>&#160; <span class="stringliteral">&quot;precomputed values\n&quot;</span>;</div>
<div class="line"><a name="l00103"></a><span class="lineno"> 103</span>&#160; </div>
<div class="line"><a name="l00104"></a><span class="lineno"> 104</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; <span class="stringliteral">&quot;Test 1....&quot;</span></div>
<div class="line"><a name="l00105"></a><span class="lineno"> 105</span>&#160; &lt;&lt; <span class="stringliteral">&quot;\n&quot;</span>;</div>
<div class="line"><a name="l00106"></a><span class="lineno"> 106</span>&#160; <span class="keywordtype">double</span> valfirst = <a class="code" href="../../d1/da6/rungekutta_8cpp.html#a7b9f40c7b5e9749cc550f19be3dc8856">numerical_methods::runge_kutta::rungeKutta</a>(</div>
<div class="line"><a name="l00107"></a><span class="lineno"> 107</span>&#160; 2, 3, 4, 0.2); <span class="comment">// Tests the function with pre calculated values</span></div>
<div class="line"><a name="l00108"></a><span class="lineno"> 108</span>&#160; assert(valfirst == 3.10363932323749570);</div>
<div class="line"><a name="l00109"></a><span class="lineno"> 109</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; <span class="stringliteral">&quot;Passed Test 1\n&quot;</span>;</div>
<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; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; <span class="stringliteral">&quot;Test 2....&quot;</span></div>
<div class="line"><a name="l00112"></a><span class="lineno"> 112</span>&#160; &lt;&lt; <span class="stringliteral">&quot;\n&quot;</span>;</div>
<div class="line"><a name="l00113"></a><span class="lineno"> 113</span>&#160; <span class="keywordtype">double</span> valsec = <a class="code" href="../../d1/da6/rungekutta_8cpp.html#a7b9f40c7b5e9749cc550f19be3dc8856">numerical_methods::runge_kutta::rungeKutta</a>(</div>
<div class="line"><a name="l00114"></a><span class="lineno"> 114</span>&#160; 1, 2, 5, 0.1); <span class="comment">// The value of step changed</span></div>
<div class="line"><a name="l00115"></a><span class="lineno"> 115</span>&#160; assert(valsec == 3.40600589380261409);</div>
<div class="line"><a name="l00116"></a><span class="lineno"> 116</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; <span class="stringliteral">&quot;Passed Test 2\n&quot;</span>;</div>
<div class="line"><a name="l00117"></a><span class="lineno"> 117</span>&#160; </div>
<div class="line"><a name="l00118"></a><span class="lineno"> 118</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; <span class="stringliteral">&quot;Test 3....&quot;</span></div>
<div class="line"><a name="l00119"></a><span class="lineno"> 119</span>&#160; &lt;&lt; <span class="stringliteral">&quot;\n&quot;</span>;</div>
<div class="line"><a name="l00120"></a><span class="lineno"> 120</span>&#160; <span class="keywordtype">double</span> valthird = <a class="code" href="../../d1/da6/rungekutta_8cpp.html#a7b9f40c7b5e9749cc550f19be3dc8856">numerical_methods::runge_kutta::rungeKutta</a>(</div>
<div class="line"><a name="l00121"></a><span class="lineno"> 121</span>&#160; -1, 3, 4, 0.1); <span class="comment">// Tested with negative value</span></div>
<div class="line"><a name="l00122"></a><span class="lineno"> 122</span>&#160; assert(valthird == 2.49251005860244268);</div>
<div class="line"><a name="l00123"></a><span class="lineno"> 123</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; <span class="stringliteral">&quot;Passed Test 3\n&quot;</span>;</div>
<div class="line"><a name="l00124"></a><span class="lineno"> 124</span>&#160;}</div>
<div class="ttc" id="abasic_ostream_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a></div></div>
<div class="ttc" id="arungekutta_8cpp_html_a7b9f40c7b5e9749cc550f19be3dc8856"><div class="ttname"><a href="../../d1/da6/rungekutta_8cpp.html#a7b9f40c7b5e9749cc550f19be3dc8856">numerical_methods::runge_kutta::rungeKutta</a></div><div class="ttdeci">double rungeKutta(double init_x, const double &amp;init_y, const double &amp;x, const double &amp;h)</div><div class="ttdoc">the Runge Kutta method finds the value of integration of a function in the given limits....</div><div class="ttdef"><b>Definition:</b> rungekutta.cpp:57</div></div>
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