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<div class="title">durand_kerner_roots.cpp File Reference</div> </div>
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<div class="contents">
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<p>Compute all possible approximate roots of any given polynomial using <a href="https://en.wikipedia.org/wiki/Durand%E2%80%93Kerner_method">Durand Kerner algorithm</a>
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<a href="#details">More...</a></p>
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<div class="textblock"><code>#include <algorithm></code><br />
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<code>#include <cassert></code><br />
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<code>#include <cmath></code><br />
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<code>#include <complex></code><br />
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<code>#include <cstdlib></code><br />
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<code>#include <ctime></code><br />
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<code>#include <fstream></code><br />
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<code>#include <iostream></code><br />
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<code>#include <valarray></code><br />
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</div><div class="textblock"><div class="dynheader">
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Include dependency graph for durand_kerner_roots.cpp:</div>
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<div class="dyncontent">
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<div class="center"><div class="zoom"><iframe scrolling="no" frameborder="0" src="../../d4/d26/durand__kerner__roots_8cpp__incl.svg" width="100%" height="395"><p><b>This browser is not able to show SVG: try Firefox, Chrome, Safari, or Opera instead.</b></p></iframe></div>
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Macros</h2></td></tr>
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<tr class="memitem:af270a96662132d0385cb6b4637c5a689"><td class="memItemLeft" align="right" valign="top">#define </td><td class="memItemRight" valign="bottom"><a class="el" href="../../da/df2/durand__kerner__roots_8cpp.html#af270a96662132d0385cb6b4637c5a689">ACCURACY</a>   1e-10</td></tr>
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<tr class="memitem:a212979f5fc9dcf0a420e90b9b21d780a"><td class="memItemLeft" align="right" valign="top"><a id="a212979f5fc9dcf0a420e90b9b21d780a"></a>
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#define </td><td class="memItemRight" valign="bottom"><b>MAX_BUFF_SIZE</b>   50</td></tr>
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<tr class="separator:a212979f5fc9dcf0a420e90b9b21d780a"><td class="memSeparator" colspan="2"> </td></tr>
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</table><table class="memberdecls">
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Functions</h2></td></tr>
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<tr class="memitem:a51637ff89e55ca749993d3e84b95d153"><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>< double > </td><td class="memItemRight" valign="bottom"><a class="el" href="../../da/df2/durand__kerner__roots_8cpp.html#a51637ff89e55ca749993d3e84b95d153">poly_function</a> (const <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>< double > &coeffs, <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/complex.html">std::complex</a>< double > x)</td></tr>
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<tr class="separator:a51637ff89e55ca749993d3e84b95d153"><td class="memSeparator" colspan="2"> </td></tr>
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<tr class="memitem:a3bada3e58ad8d1b00918db2ffe83a740"><td class="memItemLeft" align="right" valign="top">const char * </td><td class="memItemRight" valign="bottom"><a class="el" href="../../da/df2/durand__kerner__roots_8cpp.html#a3bada3e58ad8d1b00918db2ffe83a740">complex_str</a> (const <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/complex.html">std::complex</a>< double > &x)</td></tr>
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<tr class="separator:a3bada3e58ad8d1b00918db2ffe83a740"><td class="memSeparator" colspan="2"> </td></tr>
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<tr class="memitem:a024b8bc4755863315456d573a6732377"><td class="memItemLeft" align="right" valign="top">bool </td><td class="memItemRight" valign="bottom"><a class="el" href="../../da/df2/durand__kerner__roots_8cpp.html#a024b8bc4755863315456d573a6732377">check_termination</a> (long double delta)</td></tr>
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<tr class="separator:a024b8bc4755863315456d573a6732377"><td class="memSeparator" colspan="2"> </td></tr>
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<tr class="memitem:ac38b9cfd2a39d856e0feba3b5d241ded"><td class="memItemLeft" align="right" valign="top"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/utility/pair.html">std::pair</a>< uint32_t, double > </td><td class="memItemRight" valign="bottom"><a class="el" href="../../da/df2/durand__kerner__roots_8cpp.html#ac38b9cfd2a39d856e0feba3b5d241ded">durand_kerner_algo</a> (const <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>< double > &coeffs, <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>< <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/complex.html">std::complex</a>< double >> *roots, bool write_log=false)</td></tr>
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<tr class="separator:ac38b9cfd2a39d856e0feba3b5d241ded"><td class="memSeparator" colspan="2"> </td></tr>
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<tr class="memitem:a1440a7779ac56f47a3f355ce4a8c7da0"><td class="memItemLeft" align="right" valign="top">void </td><td class="memItemRight" valign="bottom"><a class="el" href="../../da/df2/durand__kerner__roots_8cpp.html#a1440a7779ac56f47a3f355ce4a8c7da0">test1</a> ()</td></tr>
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<tr class="separator:a1440a7779ac56f47a3f355ce4a8c7da0"><td class="memSeparator" colspan="2"> </td></tr>
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<tr class="memitem:a0283886819c7c140a023582b7269e2d0"><td class="memItemLeft" align="right" valign="top">void </td><td class="memItemRight" valign="bottom"><a class="el" href="../../da/df2/durand__kerner__roots_8cpp.html#a0283886819c7c140a023582b7269e2d0">test2</a> ()</td></tr>
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<tr class="separator:a0283886819c7c140a023582b7269e2d0"><td class="memSeparator" colspan="2"> </td></tr>
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<tr class="memitem:a3c04138a5bfe5d72780bb7e82a18e627"><td class="memItemLeft" align="right" valign="top"><a id="a3c04138a5bfe5d72780bb7e82a18e627"></a>
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int </td><td class="memItemRight" valign="bottom"><b>main</b> (int argc, char **argv)</td></tr>
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<tr class="separator:a3c04138a5bfe5d72780bb7e82a18e627"><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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<div class="textblock"><p>Compute all possible approximate roots of any given polynomial using <a href="https://en.wikipedia.org/wiki/Durand%E2%80%93Kerner_method">Durand Kerner algorithm</a> </p>
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<dl class="section author"><dt>Author</dt><dd><a href="https://github.com/kvedala">Krishna Vedala</a></dd></dl>
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<p>Test the algorithm online: <a href="https://gist.github.com/kvedala/27f1b0b6502af935f6917673ec43bcd7">https://gist.github.com/kvedala/27f1b0b6502af935f6917673ec43bcd7</a></p>
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<p>Try the highly unstable Wilkinson's polynomial: </p><div class="fragment"><div class="line">./<a class="code" href="../../d1/de0/namespacenumerical__methods.html">numerical_methods</a>/durand_kerner_roots 1 -210 20615 -1256850 53327946</div>
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<div class="line">-1672280820 40171771630 -756111184500 11310276995381 -135585182899530</div>
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<div class="line">1307535010540395 -10142299865511450 63030812099294896 -311333643161390640</div>
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<div class="line">1206647803780373360 -3599979517947607200 8037811822645051776</div>
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<div class="line">-12870931245150988800 13803759753640704000 -8752948036761600000</div>
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<div class="line">2432902008176640000</div>
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</div><!-- fragment --><p>Sample implementation results to compute approximate roots of the equation \(x^4-1=0\):<br />
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<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
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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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roots and their convergence to a final approximation along with the iterative
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approximations" style="pointer-events: none;" width="400" class="inline"/> </p>
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</div><h2 class="groupheader">Macro Definition Documentation</h2>
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<a id="af270a96662132d0385cb6b4637c5a689"></a>
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<h2 class="memtitle"><span class="permalink"><a href="#af270a96662132d0385cb6b4637c5a689">◆ </a></span>ACCURACY</h2>
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<div class="memitem">
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<div class="memproto">
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<table class="memname">
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<tr>
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<td class="memname">#define ACCURACY   1e-10</td>
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</tr>
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</table>
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</div><div class="memdoc">
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<p>maximum accuracy limit </p>
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</div>
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</div>
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<h2 class="groupheader">Function Documentation</h2>
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<a id="a024b8bc4755863315456d573a6732377"></a>
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<h2 class="memtitle"><span class="permalink"><a href="#a024b8bc4755863315456d573a6732377">◆ </a></span>check_termination()</h2>
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<div class="memitem">
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<div class="memproto">
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<table class="memname">
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<td class="memname">bool check_termination </td>
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<td>(</td>
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<td class="paramtype">long double </td>
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<td class="paramname"><em>delta</em></td><td>)</td>
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<td></td>
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</tr>
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</table>
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</div><div class="memdoc">
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<p>check for termination condition </p><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">delta</td><td>point at which to evaluate the polynomial </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><code>false</code> if termination not reached </dd>
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<dd>
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<code>true</code> if termination reached </dd></dl>
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<div class="fragment"><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>  <span class="keyword">static</span> <span class="keywordtype">long</span> <span class="keywordtype">double</span> past_delta = INFINITY;</div>
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<div class="line"><a name="l00093"></a><span class="lineno"> 93</span>  <span class="keywordflow">if</span> (std::abs(past_delta - delta) <= <a class="code" href="../../da/df2/durand__kerner__roots_8cpp.html#af270a96662132d0385cb6b4637c5a689">ACCURACY</a> || delta < <a class="code" href="../../da/df2/durand__kerner__roots_8cpp.html#af270a96662132d0385cb6b4637c5a689">ACCURACY</a>)</div>
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<div class="line"><a name="l00094"></a><span class="lineno"> 94</span>  <span class="keywordflow">return</span> <span class="keyword">true</span>;</div>
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<div class="line"><a name="l00095"></a><span class="lineno"> 95</span>  past_delta = delta;</div>
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<div class="line"><a name="l00096"></a><span class="lineno"> 96</span>  <span class="keywordflow">return</span> <span class="keyword">false</span>;</div>
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<div class="line"><a name="l00097"></a><span class="lineno"> 97</span> }</div>
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</div><!-- fragment -->
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</div>
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</div>
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<a id="a3bada3e58ad8d1b00918db2ffe83a740"></a>
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<h2 class="memtitle"><span class="permalink"><a href="#a3bada3e58ad8d1b00918db2ffe83a740">◆ </a></span>complex_str()</h2>
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<td class="memname">const char* complex_str </td>
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<td>(</td>
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<td class="paramtype">const <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/complex.html">std::complex</a>< double > & </td>
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<td class="paramname"><em>x</em></td><td>)</td>
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<td></td>
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</div><div class="memdoc">
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<p>create a textual form of complex number </p><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">x</td><td>point at which to evaluate the polynomial </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>pointer to converted string </dd></dl>
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<div class="fragment"><div class="line"><a name="l00076"></a><span class="lineno"> 76</span>  {</div>
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<div class="line"><a name="l00077"></a><span class="lineno"> 77</span> <span class="preprocessor">#define MAX_BUFF_SIZE 50</span></div>
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<div class="line"><a name="l00078"></a><span class="lineno"> 78</span>  <span class="keyword">static</span> <span class="keywordtype">char</span> msg[MAX_BUFF_SIZE];</div>
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<div class="line"><a name="l00079"></a><span class="lineno"> 79</span>  </div>
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<div class="line"><a name="l00080"></a><span class="lineno"> 80</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/c/fprintf.html">std::snprintf</a>(msg, MAX_BUFF_SIZE, <span class="stringliteral">"% 7.04g%+7.04gj"</span>, x.<a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/complex/real.html">real</a>(), x.<a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/complex/imag.html">imag</a>());</div>
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<div class="line"><a name="l00081"></a><span class="lineno"> 81</span>  </div>
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<div class="line"><a name="l00082"></a><span class="lineno"> 82</span>  <span class="keywordflow">return</span> msg;</div>
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<div class="line"><a name="l00083"></a><span class="lineno"> 83</span> }</div>
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</div><!-- fragment -->
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<a id="ac38b9cfd2a39d856e0feba3b5d241ded"></a>
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<h2 class="memtitle"><span class="permalink"><a href="#ac38b9cfd2a39d856e0feba3b5d241ded">◆ </a></span>durand_kerner_algo()</h2>
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<td class="memname"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/utility/pair.html">std::pair</a><uint32_t, double> durand_kerner_algo </td>
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<td>(</td>
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<td class="paramtype">const <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>coeffs</em>, </td>
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</tr>
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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>< <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/complex.html">std::complex</a>< double >> * </td>
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<td class="paramname"><em>roots</em>, </td>
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</tr>
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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>write_log</em> = <code>false</code> </td>
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</tr>
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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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</tr>
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</table>
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</div><div class="memdoc">
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<p>Implements Durand Kerner iterative algorithm to compute all roots of a polynomial.</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">coeffs</td><td>coefficients of the polynomial </td></tr>
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<tr><td class="paramdir">[out]</td><td class="paramname">roots</td><td>the computed roots of the polynomial </td></tr>
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<tr><td class="paramdir">[in]</td><td class="paramname">write_log</td><td>flag whether to save the log file (default = <code>false</code>) </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>pair of values - number of iterations taken and final accuracy achieved </dd></dl>
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<div class="fragment"><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>  <span class="keywordtype">long</span> <span class="keywordtype">double</span> tol_condition = 1;</div>
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<div class="line"><a name="l00113"></a><span class="lineno"> 113</span>  uint32_t iter = 0;</div>
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<div class="line"><a name="l00114"></a><span class="lineno"> 114</span>  <span class="keywordtype">int</span> n;</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/basic_ofstream.html">std::ofstream</a> log_file;</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>  <span class="keywordflow">if</span> (write_log) {</div>
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<div class="line"><a name="l00118"></a><span class="lineno"> 118</span>  <span class="comment">/*</span></div>
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<div class="line"><a name="l00119"></a><span class="lineno"> 119</span> <span class="comment"> * store intermediate values to a CSV file</span></div>
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<div class="line"><a name="l00120"></a><span class="lineno"> 120</span> <span class="comment"> */</span></div>
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<div class="line"><a name="l00121"></a><span class="lineno"> 121</span>  log_file.<a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ofstream/open.html">open</a>(<span class="stringliteral">"durand_kerner.log.csv"</span>);</div>
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<div class="line"><a name="l00122"></a><span class="lineno"> 122</span>  <span class="keywordflow">if</span> (!log_file.<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>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/c/perror.html">perror</a>(<span class="stringliteral">"Unable to create a storage log file!"</span>);</div>
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<div class="line"><a name="l00124"></a><span class="lineno"> 124</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/utility/program/exit.html">std::exit</a>(EXIT_FAILURE);</div>
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<div class="line"><a name="l00125"></a><span class="lineno"> 125</span>  }</div>
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<div class="line"><a name="l00126"></a><span class="lineno"> 126</span>  log_file << <span class="stringliteral">"iter#,"</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>  <span class="keywordflow">for</span> (n = 0; n < roots->size(); n++) log_file << <span class="stringliteral">"root_"</span> << n << <span class="stringliteral">","</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>  log_file << <span class="stringliteral">"avg. correction"</span>;</div>
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<div class="line"><a name="l00131"></a><span class="lineno"> 131</span>  log_file << <span class="stringliteral">"\n0,"</span>;</div>
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<div class="line"><a name="l00132"></a><span class="lineno"> 132</span>  <span class="keywordflow">for</span> (n = 0; n < roots->size(); n++)</div>
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<div class="line"><a name="l00133"></a><span class="lineno"> 133</span>  log_file << <a class="code" href="../../da/df2/durand__kerner__roots_8cpp.html#a3bada3e58ad8d1b00918db2ffe83a740">complex_str</a>((*roots)[n]) << <span class="stringliteral">","</span>;</div>
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<div class="line"><a name="l00134"></a><span class="lineno"> 134</span>  }</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="keywordtype">bool</span> break_loop = <span class="keyword">false</span>;</div>
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<div class="line"><a name="l00137"></a><span class="lineno"> 137</span>  <span class="keywordflow">while</span> (!<a class="code" href="../../da/df2/durand__kerner__roots_8cpp.html#a024b8bc4755863315456d573a6732377">check_termination</a>(tol_condition) && iter < INT16_MAX &&</div>
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<div class="line"><a name="l00138"></a><span class="lineno"> 138</span>  !break_loop) {</div>
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<div class="line"><a name="l00139"></a><span class="lineno"> 139</span>  tol_condition = 0;</div>
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<div class="line"><a name="l00140"></a><span class="lineno"> 140</span>  iter++;</div>
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<div class="line"><a name="l00141"></a><span class="lineno"> 141</span>  break_loop = <span class="keyword">false</span>;</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">if</span> (log_file.<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="l00144"></a><span class="lineno"> 144</span>  log_file << <span class="stringliteral">"\n"</span> << iter << <span class="stringliteral">","</span>;</div>
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<div class="line"><a name="l00145"></a><span class="lineno"> 145</span>  </div>
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<div class="line"><a name="l00146"></a><span class="lineno"> 146</span> <span class="preprocessor">#ifdef _OPENMP</span></div>
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<div class="line"><a name="l00147"></a><span class="lineno"> 147</span> <span class="preprocessor">#pragma omp parallel for shared(break_loop, tol_condition)</span></div>
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<div class="line"><a name="l00148"></a><span class="lineno"> 148</span> <span class="preprocessor">#endif</span></div>
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<div class="line"><a name="l00149"></a><span class="lineno"> 149</span>  <span class="keywordflow">for</span> (n = 0; n < roots->size(); n++) {</div>
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<div class="line"><a name="l00150"></a><span class="lineno"> 150</span>  <span class="keywordflow">if</span> (break_loop)</div>
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<div class="line"><a name="l00151"></a><span class="lineno"> 151</span>  <span class="keywordflow">continue</span>;</div>
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<div class="line"><a name="l00152"></a><span class="lineno"> 152</span>  </div>
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<div class="line"><a name="l00153"></a><span class="lineno"> 153</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/complex.html">std::complex<double></a> numerator, denominator;</div>
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<div class="line"><a name="l00154"></a><span class="lineno"> 154</span>  numerator = <a class="code" href="../../da/df2/durand__kerner__roots_8cpp.html#a51637ff89e55ca749993d3e84b95d153">poly_function</a>(coeffs, (*roots)[n]);</div>
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<div class="line"><a name="l00155"></a><span class="lineno"> 155</span>  denominator = 1.0;</div>
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<div class="line"><a name="l00156"></a><span class="lineno"> 156</span>  <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i < roots->size(); i++)</div>
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<div class="line"><a name="l00157"></a><span class="lineno"> 157</span>  <span class="keywordflow">if</span> (i != n)</div>
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<div class="line"><a name="l00158"></a><span class="lineno"> 158</span>  denominator *= (*roots)[n] - (*roots)[i];</div>
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<div class="line"><a name="l00159"></a><span class="lineno"> 159</span>  </div>
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<div class="line"><a name="l00160"></a><span class="lineno"> 160</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/complex.html">std::complex<long double></a> delta = numerator / denominator;</div>
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<div class="line"><a name="l00161"></a><span class="lineno"> 161</span>  </div>
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<div class="line"><a name="l00162"></a><span class="lineno"> 162</span>  <span class="keywordflow">if</span> (<a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/math/isnan.html">std::isnan</a>(std::abs(delta)) || <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/math/isinf.html">std::isinf</a>(std::abs(delta))) {</div>
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<div class="line"><a name="l00163"></a><span class="lineno"> 163</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cerr</a> << <span class="stringliteral">"\n\nOverflow/underrun error - got value = "</span></div>
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<div class="line"><a name="l00164"></a><span class="lineno"> 164</span>  << std::abs(delta) << <span class="stringliteral">"\n"</span>;</div>
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<div class="line"><a name="l00165"></a><span class="lineno"> 165</span>  <span class="comment">// return std::pair<uint32_t, double>(iter, tol_condition);</span></div>
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<div class="line"><a name="l00166"></a><span class="lineno"> 166</span>  break_loop = <span class="keyword">true</span>;</div>
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<div class="line"><a name="l00167"></a><span class="lineno"> 167</span>  }</div>
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<div class="line"><a name="l00168"></a><span class="lineno"> 168</span>  </div>
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<div class="line"><a name="l00169"></a><span class="lineno"> 169</span>  (*roots)[n] -= delta;</div>
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<div class="line"><a name="l00170"></a><span class="lineno"> 170</span>  </div>
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<div class="line"><a name="l00171"></a><span class="lineno"> 171</span> <span class="preprocessor">#ifdef _OPENMP</span></div>
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<div class="line"><a name="l00172"></a><span class="lineno"> 172</span> <span class="preprocessor">#pragma omp critical</span></div>
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<div class="line"><a name="l00173"></a><span class="lineno"> 173</span> <span class="preprocessor">#endif</span></div>
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<div class="line"><a name="l00174"></a><span class="lineno"> 174</span>  tol_condition = <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/algorithm/max.html">std::max</a>(tol_condition, std::abs(std::abs(delta)));</div>
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<div class="line"><a name="l00175"></a><span class="lineno"> 175</span>  }</div>
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<div class="line"><a name="l00176"></a><span class="lineno"> 176</span>  <span class="comment">// tol_condition /= (degree - 1);</span></div>
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<div class="line"><a name="l00177"></a><span class="lineno"> 177</span>  </div>
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<div class="line"><a name="l00178"></a><span class="lineno"> 178</span>  <span class="keywordflow">if</span> (break_loop)</div>
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<div class="line"><a name="l00179"></a><span class="lineno"> 179</span>  <span class="keywordflow">break</span>;</div>
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<div class="line"><a name="l00180"></a><span class="lineno"> 180</span>  </div>
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<div class="line"><a name="l00181"></a><span class="lineno"> 181</span>  <span class="keywordflow">if</span> (log_file.<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="l00182"></a><span class="lineno"> 182</span>  <span class="keywordflow">for</span> (n = 0; n < roots->size(); n++)</div>
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<div class="line"><a name="l00183"></a><span class="lineno"> 183</span>  log_file << <a class="code" href="../../da/df2/durand__kerner__roots_8cpp.html#a3bada3e58ad8d1b00918db2ffe83a740">complex_str</a>((*roots)[n]) << <span class="stringliteral">","</span>;</div>
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<div class="line"><a name="l00184"></a><span class="lineno"> 184</span>  }</div>
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<div class="line"><a name="l00185"></a><span class="lineno"> 185</span>  </div>
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<div class="line"><a name="l00186"></a><span class="lineno"> 186</span> <span class="preprocessor">#if defined(DEBUG) || !defined(NDEBUG)</span></div>
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<div class="line"><a name="l00187"></a><span class="lineno"> 187</span>  <span class="keywordflow">if</span> (iter % 500 == 0) {</div>
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<div class="line"><a name="l00188"></a><span class="lineno"> 188</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> << <span class="stringliteral">"Iter: "</span> << iter << <span class="stringliteral">"\t"</span>;</div>
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<div class="line"><a name="l00189"></a><span class="lineno"> 189</span>  <span class="keywordflow">for</span> (n = 0; n < roots->size(); n++)</div>
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<div class="line"><a name="l00190"></a><span class="lineno"> 190</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> << <span class="stringliteral">"\t"</span> << <a class="code" href="../../da/df2/durand__kerner__roots_8cpp.html#a3bada3e58ad8d1b00918db2ffe83a740">complex_str</a>((*roots)[n]);</div>
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<div class="line"><a name="l00191"></a><span class="lineno"> 191</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> << <span class="stringliteral">"\t\tabsolute average change: "</span> << tol_condition</div>
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<div class="line"><a name="l00192"></a><span class="lineno"> 192</span>  << <span class="stringliteral">"\n"</span>;</div>
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<div class="line"><a name="l00193"></a><span class="lineno"> 193</span>  }</div>
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<div class="line"><a name="l00194"></a><span class="lineno"> 194</span> <span class="preprocessor">#endif</span></div>
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<div class="line"><a name="l00195"></a><span class="lineno"> 195</span>  </div>
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<div class="line"><a name="l00196"></a><span class="lineno"> 196</span>  <span class="keywordflow">if</span> (log_file.<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="l00197"></a><span class="lineno"> 197</span>  log_file << tol_condition;</div>
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<div class="line"><a name="l00198"></a><span class="lineno"> 198</span>  }</div>
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<div class="line"><a name="l00199"></a><span class="lineno"> 199</span>  </div>
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<div class="line"><a name="l00200"></a><span class="lineno"> 200</span>  <span class="keywordflow">return</span> <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/utility/pair.html">std::pair<uint32_t, long double></a>(iter, tol_condition);</div>
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<div class="line"><a name="l00201"></a><span class="lineno"> 201</span> }</div>
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</div><!-- fragment --><div class="dynheader">
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Here is the call graph for this function:</div>
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<div class="dyncontent">
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<div class="center"><iframe scrolling="no" frameborder="0" src="../../da/df2/durand__kerner__roots_8cpp_ac38b9cfd2a39d856e0feba3b5d241ded_cgraph.svg" width="527" height="494"><p><b>This browser is not able to show SVG: try Firefox, Chrome, Safari, or Opera instead.</b></p></iframe>
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<a id="a51637ff89e55ca749993d3e84b95d153"></a>
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<h2 class="memtitle"><span class="permalink"><a href="#a51637ff89e55ca749993d3e84b95d153">◆ </a></span>poly_function()</h2>
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<div class="memitem">
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<div class="memproto">
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<table class="memname">
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<tr>
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<td class="memname"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/complex.html">std::complex</a><double> poly_function </td>
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<td>(</td>
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<td class="paramtype">const <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>coeffs</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/complex.html">std::complex</a>< double > </td>
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<td class="paramname"><em>x</em> </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>Evaluate the value of a polynomial with given coefficients </p><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">coeffs</td><td>coefficients of the polynomial </td></tr>
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<tr><td class="paramdir">[in]</td><td class="paramname">x</td><td>point at which to evaluate the polynomial </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>\(f(x)\) </dd></dl>
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<div class="fragment"><div class="line"><a name="l00054"></a><span class="lineno"> 54</span>  {</div>
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<div class="line"><a name="l00055"></a><span class="lineno"> 55</span>  <span class="keywordtype">double</span> real = 0.f, imag = 0.f;</div>
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<div class="line"><a name="l00056"></a><span class="lineno"> 56</span>  <span class="keywordtype">int</span> n;</div>
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<div class="line"><a name="l00057"></a><span class="lineno"> 57</span>  </div>
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<div class="line"><a name="l00058"></a><span class="lineno"> 58</span>  <span class="comment">// #ifdef _OPENMP</span></div>
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<div class="line"><a name="l00059"></a><span class="lineno"> 59</span>  <span class="comment">// #pragma omp target teams distribute reduction(+ : real, imag)</span></div>
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<div class="line"><a name="l00060"></a><span class="lineno"> 60</span>  <span class="comment">// #endif</span></div>
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<div class="line"><a name="l00061"></a><span class="lineno"> 61</span>  <span class="keywordflow">for</span> (n = 0; n < coeffs.size(); n++) {</div>
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<div class="line"><a name="l00062"></a><span class="lineno"> 62</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/complex.html">std::complex<double></a> tmp =</div>
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<div class="line"><a name="l00063"></a><span class="lineno"> 63</span>  coeffs[n] * <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/math/pow.html">std::pow</a>(x, coeffs.size() - n - 1);</div>
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<div class="line"><a name="l00064"></a><span class="lineno"> 64</span>  real += tmp.<a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/complex/real.html">real</a>();</div>
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<div class="line"><a name="l00065"></a><span class="lineno"> 65</span>  imag += tmp.<a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/complex/imag.html">imag</a>();</div>
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<div class="line"><a name="l00066"></a><span class="lineno"> 66</span>  }</div>
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<div class="line"><a name="l00067"></a><span class="lineno"> 67</span>  </div>
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<div class="line"><a name="l00068"></a><span class="lineno"> 68</span>  <span class="keywordflow">return</span> <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/complex.html">std::complex<double></a>(real, imag);</div>
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<div class="line"><a name="l00069"></a><span class="lineno"> 69</span> }</div>
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</div><!-- fragment --><div class="dynheader">
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<h2 class="memtitle"><span class="permalink"><a href="#a1440a7779ac56f47a3f355ce4a8c7da0">◆ </a></span>test1()</h2>
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<td class="memname">void test1 </td>
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<p>Self test the algorithm by checking the roots for \(x^2+4=0\) to which the roots are \(0 \pm 2i\) </p>
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<div class="fragment"><div class="line"><a name="l00207"></a><span class="lineno"> 207</span>  {</div>
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<div class="line"><a name="l00208"></a><span class="lineno"> 208</span>  <span class="keyword">const</span> <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray<double></a> coeffs = {1, 0, 4}; <span class="comment">// x^2 - 2 = 0</span></div>
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<div class="line"><a name="l00209"></a><span class="lineno"> 209</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray<std::complex<double></a>> roots(2);</div>
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<div class="line"><a name="l00210"></a><span class="lineno"> 210</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray<std::complex<double></a>> expected = {</div>
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<div class="line"><a name="l00211"></a><span class="lineno"> 211</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/complex.html">std::complex<double></a>(0., 2.),</div>
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<div class="line"><a name="l00212"></a><span class="lineno"> 212</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/complex.html">std::complex<double></a>(0., -2.) <span class="comment">// known expected roots</span></div>
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<div class="line"><a name="l00213"></a><span class="lineno"> 213</span>  };</div>
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<div class="line"><a name="l00214"></a><span class="lineno"> 214</span>  </div>
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<div class="line"><a name="l00215"></a><span class="lineno"> 215</span>  <span class="comment">/* initialize root approximations with random values */</span></div>
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<div class="line"><a name="l00216"></a><span class="lineno"> 216</span>  <span class="keywordflow">for</span> (<span class="keywordtype">int</span> n = 0; n < roots.size(); n++) {</div>
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<div class="line"><a name="l00217"></a><span class="lineno"> 217</span>  roots[n] = <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/complex.html">std::complex<double></a>(<a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/random/rand.html">std::rand</a>() % 100, <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/random/rand.html">std::rand</a>() % 100);</div>
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<div class="line"><a name="l00218"></a><span class="lineno"> 218</span>  roots[n] -= 50.f;</div>
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<div class="line"><a name="l00219"></a><span class="lineno"> 219</span>  roots[n] /= 25.f;</div>
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<div class="line"><a name="l00220"></a><span class="lineno"> 220</span>  }</div>
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<div class="line"><a name="l00221"></a><span class="lineno"> 221</span>  </div>
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<div class="line"><a name="l00222"></a><span class="lineno"> 222</span>  <span class="keyword">auto</span> <a class="code" href="../../de/dc3/fibonacci__sum_8cpp.html#aadb40ac4c74a7efc0680b83eeee138aa">result</a> = <a class="code" href="../../da/df2/durand__kerner__roots_8cpp.html#ac38b9cfd2a39d856e0feba3b5d241ded">durand_kerner_algo</a>(coeffs, &roots, <span class="keyword">false</span>);</div>
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<div class="line"><a name="l00223"></a><span class="lineno"> 223</span>  </div>
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<div class="line"><a name="l00224"></a><span class="lineno"> 224</span>  <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i < roots.size(); i++) {</div>
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<div class="line"><a name="l00225"></a><span class="lineno"> 225</span>  <span class="comment">// check if approximations are have < 0.1% error with one of the</span></div>
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<div class="line"><a name="l00226"></a><span class="lineno"> 226</span>  <span class="comment">// expected roots</span></div>
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<div class="line"><a name="l00227"></a><span class="lineno"> 227</span>  <span class="keywordtype">bool</span> err1 = <span class="keyword">false</span>;</div>
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<div class="line"><a name="l00228"></a><span class="lineno"> 228</span>  <span class="keywordflow">for</span> (<span class="keywordtype">int</span> j = 0; j < roots.size(); j++)</div>
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<div class="line"><a name="l00229"></a><span class="lineno"> 229</span>  err1 |= std::abs(std::abs(roots[i] - expected[j])) < 1e-3;</div>
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<div class="line"><a name="l00230"></a><span class="lineno"> 230</span>  assert(err1);</div>
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<div class="line"><a name="l00231"></a><span class="lineno"> 231</span>  }</div>
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<div class="line"><a name="l00232"></a><span class="lineno"> 232</span>  </div>
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<div class="line"><a name="l00233"></a><span class="lineno"> 233</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> << <span class="stringliteral">"Test 1 passed! - "</span> << <a class="code" href="../../de/dc3/fibonacci__sum_8cpp.html#aadb40ac4c74a7efc0680b83eeee138aa">result</a>.first << <span class="stringliteral">" iterations, "</span></div>
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<div class="line"><a name="l00234"></a><span class="lineno"> 234</span>  << <a class="code" href="../../de/dc3/fibonacci__sum_8cpp.html#aadb40ac4c74a7efc0680b83eeee138aa">result</a>.second << <span class="stringliteral">" accuracy"</span></div>
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<div class="line"><a name="l00235"></a><span class="lineno"> 235</span>  << <span class="stringliteral">"\n"</span>;</div>
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<div class="line"><a name="l00236"></a><span class="lineno"> 236</span> }</div>
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<h2 class="memtitle"><span class="permalink"><a href="#a0283886819c7c140a023582b7269e2d0">◆ </a></span>test2()</h2>
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<td class="memname">void test2 </td>
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<p>Self test the algorithm by checking the roots for \(0.015625x^3-1=0\) to which the roots are \((4+0i),\,(-2\pm3.464i)\) </p>
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<div class="fragment"><div class="line"><a name="l00242"></a><span class="lineno"> 242</span>  {</div>
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<div class="line"><a name="l00243"></a><span class="lineno"> 243</span>  <span class="keyword">const</span> <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray<double></a> coeffs = {<span class="comment">// 0.015625 x^3 - 1 = 0</span></div>
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<div class="line"><a name="l00244"></a><span class="lineno"> 244</span>  1. / 64., 0., 0., -1.};</div>
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<div class="line"><a name="l00245"></a><span class="lineno"> 245</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray<std::complex<double></a>> roots(3);</div>
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<div class="line"><a name="l00246"></a><span class="lineno"> 246</span>  <span class="keyword">const</span> <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray<std::complex<double></a>> expected = {</div>
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<div class="line"><a name="l00247"></a><span class="lineno"> 247</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/complex.html">std::complex<double></a>(4., 0.), <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/complex.html">std::complex<double></a>(-2., 3.46410162),</div>
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<div class="line"><a name="l00248"></a><span class="lineno"> 248</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/complex.html">std::complex<double></a>(-2., -3.46410162) <span class="comment">// known expected roots</span></div>
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<div class="line"><a name="l00249"></a><span class="lineno"> 249</span>  };</div>
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<div class="line"><a name="l00250"></a><span class="lineno"> 250</span>  </div>
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<div class="line"><a name="l00251"></a><span class="lineno"> 251</span>  <span class="comment">/* initialize root approximations with random values */</span></div>
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<div class="line"><a name="l00252"></a><span class="lineno"> 252</span>  <span class="keywordflow">for</span> (<span class="keywordtype">int</span> n = 0; n < roots.size(); n++) {</div>
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<div class="line"><a name="l00253"></a><span class="lineno"> 253</span>  roots[n] = <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/complex.html">std::complex<double></a>(<a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/random/rand.html">std::rand</a>() % 100, <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/random/rand.html">std::rand</a>() % 100);</div>
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<div class="line"><a name="l00254"></a><span class="lineno"> 254</span>  roots[n] -= 50.f;</div>
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<div class="line"><a name="l00255"></a><span class="lineno"> 255</span>  roots[n] /= 25.f;</div>
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<div class="line"><a name="l00256"></a><span class="lineno"> 256</span>  }</div>
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<div class="line"><a name="l00257"></a><span class="lineno"> 257</span>  </div>
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<div class="line"><a name="l00258"></a><span class="lineno"> 258</span>  <span class="keyword">auto</span> <a class="code" href="../../de/dc3/fibonacci__sum_8cpp.html#aadb40ac4c74a7efc0680b83eeee138aa">result</a> = <a class="code" href="../../da/df2/durand__kerner__roots_8cpp.html#ac38b9cfd2a39d856e0feba3b5d241ded">durand_kerner_algo</a>(coeffs, &roots, <span class="keyword">false</span>);</div>
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<div class="line"><a name="l00259"></a><span class="lineno"> 259</span>  </div>
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<div class="line"><a name="l00260"></a><span class="lineno"> 260</span>  <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i < roots.size(); i++) {</div>
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<div class="line"><a name="l00261"></a><span class="lineno"> 261</span>  <span class="comment">// check if approximations are have < 0.1% error with one of the</span></div>
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<div class="line"><a name="l00262"></a><span class="lineno"> 262</span>  <span class="comment">// expected roots</span></div>
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<div class="line"><a name="l00263"></a><span class="lineno"> 263</span>  <span class="keywordtype">bool</span> err1 = <span class="keyword">false</span>;</div>
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<div class="line"><a name="l00264"></a><span class="lineno"> 264</span>  <span class="keywordflow">for</span> (<span class="keywordtype">int</span> j = 0; j < roots.size(); j++)</div>
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<div class="line"><a name="l00265"></a><span class="lineno"> 265</span>  err1 |= std::abs(std::abs(roots[i] - expected[j])) < 1e-3;</div>
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<div class="line"><a name="l00266"></a><span class="lineno"> 266</span>  assert(err1);</div>
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<div class="line"><a name="l00267"></a><span class="lineno"> 267</span>  }</div>
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<div class="line"><a name="l00268"></a><span class="lineno"> 268</span>  </div>
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<div class="line"><a name="l00269"></a><span class="lineno"> 269</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> << <span class="stringliteral">"Test 2 passed! - "</span> << <a class="code" href="../../de/dc3/fibonacci__sum_8cpp.html#aadb40ac4c74a7efc0680b83eeee138aa">result</a>.first << <span class="stringliteral">" iterations, "</span></div>
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<div class="line"><a name="l00270"></a><span class="lineno"> 270</span>  << <a class="code" href="../../de/dc3/fibonacci__sum_8cpp.html#aadb40ac4c74a7efc0680b83eeee138aa">result</a>.second << <span class="stringliteral">" accuracy"</span></div>
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<div class="line"><a name="l00271"></a><span class="lineno"> 271</span>  << <span class="stringliteral">"\n"</span>;</div>
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<div class="line"><a name="l00272"></a><span class="lineno"> 272</span> }</div>
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