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d1/da6/rungekutta_8cpp.html
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@@ -112,7 +112,7 @@ Include dependency graph for rungekutta.cpp:</div>
<tr class="heading"><td colspan="2"><h2 class="groupheader"><a id="namespaces" name="namespaces"></a>
Namespaces</h2></td></tr>
<tr class="memitem:d1/de0/namespacenumerical__methods"><td class="memItemLeft" align="right" valign="top">namespace &#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 <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/map.html">std::map</a> container <br /></td></tr>
<tr class="memdesc:d1/de0/namespacenumerical__methods"><td class="mdescLeft">&#160;</td><td class="mdescRight">for IO operations <br /></td></tr>
<tr class="separator:"><td class="memSeparator" colspan="2">&#160;</td></tr>
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<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" target="_blank">Runge Kutta fourth order</a> method. <br /></td></tr>

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<p>for <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/map.html">std::map</a> container
<p>for IO operations
<a href="../../d1/de0/namespacenumerical__methods.html#details">More...</a></p>
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Functions</h2></td></tr>
<tr class="memitem:a28e67885f8606564cc8335f483f63309"><td class="memItemLeft" align="right" valign="top">double&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../d1/de0/namespacenumerical__methods.html#a28e67885f8606564cc8335f483f63309">babylonian_method</a> (double radicand)</td></tr>
<tr class="memdesc:a28e67885f8606564cc8335f483f63309"><td class="mdescLeft">&#160;</td><td class="mdescRight">Babylonian methods is an iterative function which returns square root of radicand. <a href="../../d1/de0/namespacenumerical__methods.html#a28e67885f8606564cc8335f483f63309">More...</a><br /></td></tr>
<tr class="separator:a28e67885f8606564cc8335f483f63309"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:a158fd271b9a53e8f3f60b08b18857150"><td class="memItemLeft" align="right" valign="top"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/complex.html">std::complex</a>&lt; double &gt; *&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../d1/de0/namespacenumerical__methods.html#a158fd271b9a53e8f3f60b08b18857150">FastFourierTransform</a> (<a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/complex.html">std::complex</a>&lt; double &gt; *p, uint8_t n)</td></tr>
<tr class="memdesc:a158fd271b9a53e8f3f60b08b18857150"><td class="mdescLeft">&#160;</td><td class="mdescRight">FastFourierTransform is a recursive function which returns list of complex numbers. <a href="../../d1/de0/namespacenumerical__methods.html#a158fd271b9a53e8f3f60b08b18857150">More...</a><br /></td></tr>
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<a name="details" id="details"></a><h2 class="groupheader">Detailed Description</h2>
<div class="textblock"><p >for <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/map.html">std::map</a> container </p>
<div class="textblock"><p >for IO operations </p>
<p >Numerical Methods.</p>
<p >for <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/vector.html">std::vector</a></p>
<p >for io operations</p>
<p >for <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/map.html">std::map</a> container</p>
<p >for assert</p>
<p >Numerical algorithms/methods</p>
<p >for assert for math functions for integer allocation for <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/string/byte/atof.html">std::atof</a> for <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/utility/functional/function.html">std::function</a> for IO operations</p>
<p >Numerical algorithms/methods</p>
<p >for math operations</p>
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<p >for assert for mathematical-related functions for storing points and coefficents for IO operations</p>
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<a id="a28e67885f8606564cc8335f483f63309" name="a28e67885f8606564cc8335f483f63309"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a28e67885f8606564cc8335f483f63309">&#9670;&nbsp;</a></span>babylonian_method()</h2>
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<td class="memname">double numerical_methods::babylonian_method </td>
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<p>Babylonian methods is an iterative function which returns square root of radicand. </p>
<dl class="params"><dt>Parameters</dt><dd>
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<p >To find initial root or rough approximation</p>
<p >Real Initial value will be i-1 as loop stops on +1 value</p>
<p >Storing previous value for comparison</p>
<p >Storing calculated value for comparison</p>
<p >Temp variable to x0 and x1</p>
<p >Newly calculated root</p>
<p >Returning final root</p>
<div class="fragment"><div class="line"><a id="l00031" name="l00031"></a><span class="lineno"> 31</span> {</div>
<div class="line"><a id="l00032" name="l00032"></a><span class="lineno"> 32</span> <span class="keywordtype">int</span> i = 1; <span class="comment">/// To find initial root or rough approximation</span></div>
<div class="line"><a id="l00033" name="l00033"></a><span class="lineno"> 33</span> </div>
<div class="line"><a id="l00034" name="l00034"></a><span class="lineno"> 34</span> <span class="keywordflow">while</span> (i * i &lt;= radicand) {</div>
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<div class="line"><a id="l00037" name="l00037"></a><span class="lineno"> 37</span> </div>
<div class="line"><a id="l00038" name="l00038"></a><span class="lineno"> 38</span> i--; <span class="comment">/// Real Initial value will be i-1 as loop stops on +1 value</span></div>
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<div class="line"><a id="l00042" name="l00042"></a><span class="lineno"> 42</span> (radicand / x0 + x0) / 2; <span class="comment">/// Storing calculated value for comparison</span></div>
<div class="line"><a id="l00043" name="l00043"></a><span class="lineno"> 43</span> <span class="keywordtype">double</span> temp = NAN; <span class="comment">/// Temp variable to x0 and x1</span></div>
<div class="line"><a id="l00044" name="l00044"></a><span class="lineno"> 44</span> </div>
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<div class="line"><a id="l00046" name="l00046"></a><span class="lineno"> 46</span> temp = (radicand / x1 + x1) / 2; <span class="comment">/// Newly calculated root</span></div>
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<div class="line"><a id="l00048" name="l00048"></a><span class="lineno"> 48</span> x1 = temp;</div>
<div class="line"><a id="l00049" name="l00049"></a><span class="lineno"> 49</span> }</div>
<div class="line"><a id="l00050" name="l00050"></a><span class="lineno"> 50</span> </div>
<div class="line"><a id="l00051" name="l00051"></a><span class="lineno"> 51</span> <span class="keywordflow">return</span> x1; <span class="comment">/// Returning final root</span></div>
<div class="line"><a id="l00052" name="l00052"></a><span class="lineno"> 52</span>}</div>
<div class="ttc" id="amax_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/algorithm/max.html">std::max</a></div><div class="ttdeci">T max(T... args)</div></div>
<div class="ttc" id="amin_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/algorithm/min.html">std::min</a></div><div class="ttdeci">T min(T... args)</div></div>
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<tr class="memitem:d1/de0/namespacenumerical__methods"><td class="memItemLeft" align="right" valign="top">namespace &#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../d1/de0/namespacenumerical__methods.html">numerical_methods</a></td></tr>
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<h1><a class="anchor" id="autotoc_md58"></a>
Numerical Methods</h1>
<ul>
<li><a href="https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/numerical_methods/babylonian_method.cpp" target="_blank">Babylonian Method</a></li>
<li><a href="https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/numerical_methods/bisection_method.cpp" target="_blank">Bisection Method</a></li>
<li><a href="https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/numerical_methods/brent_method_extrema.cpp" target="_blank">Brent Method Extrema</a></li>
<li><a href="https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/numerical_methods/composite_simpson_rule.cpp" target="_blank">Composite Simpson Rule</a></li>

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</div><!-- fragment --><p> Sample implementation results to compute approximate roots of the equation \(x^4-1=0\):<br />
<img src="https://raw.githubusercontent.com/TheAlgorithms/C-Plus-Plus/docs/images/numerical_methods/durand_kerner_error.svg" alt="Error evolution during root approximations computed every
iteration." style="pointer-events: none;" width="400" class="inline"/> <img src="https://raw.githubusercontent.com/TheAlgorithms/C-Plus-Plus/docs/images/numerical_methods/durand_kerner_roots.svg" alt="Roots evolution - shows the initial approximation of the

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<p><a href="https://en.wikipedia.org/wiki/Methods_of_computing_square_roots#Babylonian_method" target="_blank">A babylonian method (BM)</a> is an algorithm that computes the square root.
<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 &quot;math.h&quot;</code><br />
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Include dependency graph for babylonian_method.cpp:</div>
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Functions</h2></td></tr>
<tr class="memitem:a28e67885f8606564cc8335f483f63309"><td class="memItemLeft" align="right" valign="top">double&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../d1/de0/namespacenumerical__methods.html#a28e67885f8606564cc8335f483f63309">numerical_methods::babylonian_method</a> (double radicand)</td></tr>
<tr class="memdesc:a28e67885f8606564cc8335f483f63309"><td class="mdescLeft">&#160;</td><td class="mdescRight">Babylonian methods is an iterative function which returns square root of radicand. <a href="../../d1/de0/namespacenumerical__methods.html#a28e67885f8606564cc8335f483f63309">More...</a><br /></td></tr>
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<tr class="memitem:aa8dca7b867074164d5f45b0f3851269d"><td class="memItemLeft" align="right" valign="top">static void&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../dc/d9c/babylonian__method_8cpp.html#aa8dca7b867074164d5f45b0f3851269d">test</a> ()</td></tr>
<tr class="memdesc:aa8dca7b867074164d5f45b0f3851269d"><td class="mdescLeft">&#160;</td><td class="mdescRight">Self-test implementations. <a href="../../dc/d9c/babylonian__method_8cpp.html#aa8dca7b867074164d5f45b0f3851269d">More...</a><br /></td></tr>
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<tr class="memitem:abf9e6b7e6f15df4b525a2e7705ba3089"><td class="memItemLeft" align="right" valign="top">int&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../dc/d9c/babylonian__method_8cpp.html#abf9e6b7e6f15df4b525a2e7705ba3089">main</a> (int argc, char const *argv[])</td></tr>
<tr class="memdesc:abf9e6b7e6f15df4b525a2e7705ba3089"><td class="mdescLeft">&#160;</td><td class="mdescRight">Main function. <a href="../../dc/d9c/babylonian__method_8cpp.html#abf9e6b7e6f15df4b525a2e7705ba3089">More...</a><br /></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/Methods_of_computing_square_roots#Babylonian_method" target="_blank">A babylonian method (BM)</a> is an algorithm that computes the square root. </p>
<p >This algorithm has an application in use case scenario where a user wants find accurate square roots of large numbers </p><dl class="section author"><dt>Author</dt><dd><a href="https://github.com/ameyachawlaggsipu" target="_blank">Ameya Chawla</a> </dd></dl>
</div><h2 class="groupheader">Function Documentation</h2>
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<h2 class="memtitle"><span class="permalink"><a href="#abf9e6b7e6f15df4b525a2e7705ba3089">&#9670;&nbsp;</a></span>main()</h2>
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<td class="memname">int main </td>
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<td class="paramtype">int&#160;</td>
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<p>Main function. </p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">argc</td><td>commandline argument count (ignored) </td></tr>
<tr><td class="paramname">argv</td><td>commandline array of arguments (ignored) calls automated test function to test the working of fast fourier transform. </td></tr>
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</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>0 on exit </dd></dl>
<div class="fragment"><div class="line"><a id="l00097" name="l00097"></a><span class="lineno"> 97</span> {</div>
<div class="line"><a id="l00098" name="l00098"></a><span class="lineno"> 98</span> <a class="code hl_function" href="../../dc/d9c/babylonian__method_8cpp.html#aa8dca7b867074164d5f45b0f3851269d">test</a>(); <span class="comment">// run self-test implementations</span></div>
<div class="line"><a id="l00099" name="l00099"></a><span class="lineno"> 99</span> <span class="comment">// with 2 defined test cases</span></div>
<div class="line"><a id="l00100" name="l00100"></a><span class="lineno"> 100</span> <span class="keywordflow">return</span> 0;</div>
<div class="line"><a id="l00101" name="l00101"></a><span class="lineno"> 101</span>}</div>
<div class="ttc" id="ababylonian__method_8cpp_html_aa8dca7b867074164d5f45b0f3851269d"><div class="ttname"><a href="../../dc/d9c/babylonian__method_8cpp.html#aa8dca7b867074164d5f45b0f3851269d">test</a></div><div class="ttdeci">static void test()</div><div class="ttdoc">Self-test implementations.</div><div class="ttdef"><b>Definition:</b> babylonian_method.cpp:63</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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<p>Self-test implementations. </p>
<p >Declaring two test cases and checking for the error in predicted and true value is less than 0.0001. </p><dl class="section return"><dt>Returns</dt><dd>void </dd></dl>
<p >Testcase 1</p>
<p >Testcase 2</p>
<p >Real Output 1</p>
<p >Real Output 2</p>
<p >Test result for testcase 1</p>
<p >Test result for testcase 2</p>
<p >Testing for test Case 1</p>
<p >Testing for test Case 2</p>
<div class="fragment"><div class="line"><a id="l00063" name="l00063"></a><span class="lineno"> 63</span> {</div>
<div class="line"><a id="l00064" name="l00064"></a><span class="lineno"> 64</span> <span class="comment">/* descriptions of the following test */</span></div>
<div class="line"><a id="l00065" name="l00065"></a><span class="lineno"> 65</span> </div>
<div class="line"><a id="l00066" name="l00066"></a><span class="lineno"> 66</span> <span class="keyword">auto</span> testcase1 = 125348; <span class="comment">/// Testcase 1</span></div>
<div class="line"><a id="l00067" name="l00067"></a><span class="lineno"> 67</span> <span class="keyword">auto</span> testcase2 = 752080; <span class="comment">/// Testcase 2</span></div>
<div class="line"><a id="l00068" name="l00068"></a><span class="lineno"> 68</span> </div>
<div class="line"><a id="l00069" name="l00069"></a><span class="lineno"> 69</span> <span class="keyword">auto</span> real_output1 = 354.045194855; <span class="comment">/// Real Output 1</span></div>
<div class="line"><a id="l00070" name="l00070"></a><span class="lineno"> 70</span> <span class="keyword">auto</span> real_output2 = 867.225460881; <span class="comment">/// Real Output 2</span></div>
<div class="line"><a id="l00071" name="l00071"></a><span class="lineno"> 71</span> </div>
<div class="line"><a id="l00072" name="l00072"></a><span class="lineno"> 72</span> <span class="keyword">auto</span> test_result1 = <a class="code hl_function" href="../../d1/de0/namespacenumerical__methods.html#a28e67885f8606564cc8335f483f63309">numerical_methods::babylonian_method</a>(testcase1);<span class="comment"></span></div>
<div class="line"><a id="l00073" name="l00073"></a><span class="lineno"> 73</span><span class="comment"> /// Test result for testcase 1</span></div>
<div class="line"><a id="l00074" name="l00074"></a><span class="lineno"> 74</span><span class="comment"></span> <span class="keyword">auto</span> test_result2 = <a class="code hl_function" href="../../d1/de0/namespacenumerical__methods.html#a28e67885f8606564cc8335f483f63309">numerical_methods::babylonian_method</a>(testcase2);<span class="comment"></span></div>
<div class="line"><a id="l00075" name="l00075"></a><span class="lineno"> 75</span><span class="comment"> /// Test result for testcase 2</span></div>
<div class="line"><a id="l00076" name="l00076"></a><span class="lineno"> 76</span><span class="comment"></span> </div>
<div class="line"><a id="l00077" name="l00077"></a><span class="lineno"> 77</span> assert(<a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/algorithm/max.html">std::max</a>(test_result1, real_output1) -</div>
<div class="line"><a id="l00078" name="l00078"></a><span class="lineno"> 78</span> <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/algorithm/min.html">std::min</a>(test_result1, real_output1) &lt;</div>
<div class="line"><a id="l00079" name="l00079"></a><span class="lineno"> 79</span> 0.0001);<span class="comment"></span></div>
<div class="line"><a id="l00080" name="l00080"></a><span class="lineno"> 80</span><span class="comment"> /// Testing for test Case 1</span></div>
<div class="line"><a id="l00081" name="l00081"></a><span class="lineno"> 81</span><span class="comment"></span> assert(<a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/algorithm/max.html">std::max</a>(test_result2, real_output2) -</div>
<div class="line"><a id="l00082" name="l00082"></a><span class="lineno"> 82</span> <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/algorithm/min.html">std::min</a>(test_result2, real_output2) &lt;</div>
<div class="line"><a id="l00083" name="l00083"></a><span class="lineno"> 83</span> 0.0001);<span class="comment"></span></div>
<div class="line"><a id="l00084" name="l00084"></a><span class="lineno"> 84</span><span class="comment"> /// Testing for test Case 2</span></div>
<div class="line"><a id="l00085" name="l00085"></a><span class="lineno"> 85</span><span class="comment"></span> </div>
<div class="line"><a id="l00086" name="l00086"></a><span class="lineno"> 86</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; <span class="stringliteral">&quot;All tests have successfully passed!\n&quot;</span>;</div>
<div class="line"><a id="l00087" name="l00087"></a><span class="lineno"> 87</span>}</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="amax_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/algorithm/max.html">std::max</a></div><div class="ttdeci">T max(T... args)</div></div>
<div class="ttc" id="amin_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/algorithm/min.html">std::min</a></div><div class="ttdeci">T min(T... args)</div></div>
<div class="ttc" id="anamespacenumerical__methods_html_a28e67885f8606564cc8335f483f63309"><div class="ttname"><a href="../../d1/de0/namespacenumerical__methods.html#a28e67885f8606564cc8335f483f63309">numerical_methods::babylonian_method</a></div><div class="ttdeci">double babylonian_method(double radicand)</div><div class="ttdoc">Babylonian methods is an iterative function which returns square root of radicand.</div><div class="ttdef"><b>Definition:</b> babylonian_method.cpp:31</div></div>
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<tr id="row_15_19_" class="even" style="display:none;"><td class="entry"><span style="width:32px;display:inline-block;">&#160;</span><span class="icondoc"></span><a class="el" href="d1/da6/rungekutta_8cpp.html" target="_self">rungekutta.cpp</a></td><td class="desc"><a href="https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods" target="_blank">Runge Kutta fourth order</a> method implementation </td></tr>
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<tr id="row_15_6_" class="even" style="display:none;"><td class="entry"><span style="width:32px;display:inline-block;">&#160;</span><span class="icondoc"></span><a class="el" href="d8/d9a/fast__fourier__transform_8cpp.html" target="_self">fast_fourier_transform.cpp</a></td><td class="desc"><a href="https://medium.com/@aiswaryamathur/understanding-fast-fouriertransform-from-scratch-to-solve-polynomial-multiplication-8018d511162f" target="_blank">A fast Fourier transform (FFT)</a> is an algorithm that computes the discrete Fourier transform (DFT) of a sequence, or its inverse (IDFT) </td></tr>
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