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<div class="headertitle"><div class="title">durand_kerner_roots.cpp File Reference</div></div>
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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" target="_blank">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 <cstdint></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="394"><p><b>This browser is not able to show SVG: try Firefox, Chrome, Safari, or Opera instead.</b></p></iframe></div></div>
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<p><a href="../../da/df2/durand__kerner__roots_8cpp_source.html">Go to the source code of this file.</a></p>
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<table class="memberdecls">
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<tr class="heading"><td colspan="2"><h2 class="groupheader"><a id="define-members" name="define-members"></a>
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Macros</h2></td></tr>
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<tr class="memitem:af270a96662132d0385cb6b4637c5a689" id="r_af270a96662132d0385cb6b4637c5a689"><td class="memItemLeft" align="right" valign="top">#define </td><td class="memItemRight" valign="bottom"><a class="el" href="#af270a96662132d0385cb6b4637c5a689">ACCURACY</a>   1e-10</td></tr>
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<tr class="separator:af270a96662132d0385cb6b4637c5a689"><td class="memSeparator" colspan="2"> </td></tr>
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<tr class="memitem:a212979f5fc9dcf0a420e90b9b21d780a" id="r_a212979f5fc9dcf0a420e90b9b21d780a"><td class="memItemLeft" align="right" valign="top"><a id="a212979f5fc9dcf0a420e90b9b21d780a" name="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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Functions</h2></td></tr>
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<tr class="memitem:ae2ab49ba1969d7a5988f9bc2c8924569" id="r_ae2ab49ba1969d7a5988f9bc2c8924569"><td class="memItemLeft" align="right" valign="top">std::complex< double > </td><td class="memItemRight" valign="bottom"><a class="el" href="#ae2ab49ba1969d7a5988f9bc2c8924569">poly_function</a> (const std::valarray< double > &coeffs, std::complex< double > x)</td></tr>
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<tr class="separator:ae2ab49ba1969d7a5988f9bc2c8924569"><td class="memSeparator" colspan="2"> </td></tr>
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<tr class="memitem:a90219e35062007d1f1b68e9af071ab5c" id="r_a90219e35062007d1f1b68e9af071ab5c"><td class="memItemLeft" align="right" valign="top">const char * </td><td class="memItemRight" valign="bottom"><a class="el" href="#a90219e35062007d1f1b68e9af071ab5c">complex_str</a> (const std::complex< double > &x)</td></tr>
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<tr class="separator:a90219e35062007d1f1b68e9af071ab5c"><td class="memSeparator" colspan="2"> </td></tr>
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<tr class="memitem:a024b8bc4755863315456d573a6732377" id="r_a024b8bc4755863315456d573a6732377"><td class="memItemLeft" align="right" valign="top">bool </td><td class="memItemRight" valign="bottom"><a class="el" href="#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:a2c35b320ace8677f9b331faf94f8b2fd" id="r_a2c35b320ace8677f9b331faf94f8b2fd"><td class="memItemLeft" align="right" valign="top">std::pair< uint32_t, double > </td><td class="memItemRight" valign="bottom"><a class="el" href="#a2c35b320ace8677f9b331faf94f8b2fd">durand_kerner_algo</a> (const std::valarray< double > &coeffs, std::valarray< std::complex< double > > *roots, bool write_log=false)</td></tr>
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<tr class="separator:a2c35b320ace8677f9b331faf94f8b2fd"><td class="memSeparator" colspan="2"> </td></tr>
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<tr class="memitem:a1440a7779ac56f47a3f355ce4a8c7da0" id="r_a1440a7779ac56f47a3f355ce4a8c7da0"><td class="memItemLeft" align="right" valign="top">void </td><td class="memItemRight" valign="bottom"><a class="el" href="#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" id="r_a0283886819c7c140a023582b7269e2d0"><td class="memItemLeft" align="right" valign="top">void </td><td class="memItemRight" valign="bottom"><a class="el" href="#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" id="r_a3c04138a5bfe5d72780bb7e82a18e627"><td class="memItemLeft" align="right" valign="top">int </td><td class="memItemRight" valign="bottom"><a class="el" href="#a3c04138a5bfe5d72780bb7e82a18e627">main</a> (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" target="_blank">Durand Kerner algorithm</a> </p>
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<dl class="section author"><dt>Author</dt><dd><a href="https://github.com/kvedala" target="_blank">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 hl_namespace" 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 class="ttc" id="anamespacenumerical__methods_html"><div class="ttname"><a href="../../d1/de0/namespacenumerical__methods.html">numerical_methods</a></div><div class="ttdoc">for assert</div></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." width="400" style="pointer-events: none;" 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" width="400" style="pointer-events: none;" class="inline"/> </p>
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<p class="definition">Definition in file <a class="el" href="../../da/df2/durand__kerner__roots_8cpp_source.html">durand_kerner_roots.cpp</a>.</p>
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</div><h2 class="groupheader">Macro Definition Documentation</h2>
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<a id="af270a96662132d0385cb6b4637c5a689" name="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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<p class="definition">Definition at line <a class="el" href="../../da/df2/durand__kerner__roots_8cpp_source.html#l00046">46</a> of file <a class="el" href="../../da/df2/durand__kerner__roots_8cpp_source.html">durand_kerner_roots.cpp</a>.</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" name="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> <td class="paramname"><span class="paramname"><em>delta</em></span></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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<p class="definition">Definition at line <a class="el" href="../../da/df2/durand__kerner__roots_8cpp_source.html#l00092">92</a> of file <a class="el" href="../../da/df2/durand__kerner__roots_8cpp_source.html">durand_kerner_roots.cpp</a>.</p>
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<div class="fragment"><div class="line"><span class="lineno"> 92</span> {</div>
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<div class="line"><span class="lineno"> 93</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"><span class="lineno"> 94</span> <span class="keywordflow">if</span> (std::abs(past_delta - delta) <= <a class="code hl_define" href="#af270a96662132d0385cb6b4637c5a689">ACCURACY</a> || delta < <a class="code hl_define" href="#af270a96662132d0385cb6b4637c5a689">ACCURACY</a>)</div>
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<div class="line"><span class="lineno"> 95</span> <span class="keywordflow">return</span> <span class="keyword">true</span>;</div>
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<div class="line"><span class="lineno"> 96</span> past_delta = delta;</div>
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<div class="line"><span class="lineno"> 97</span> <span class="keywordflow">return</span> <span class="keyword">false</span>;</div>
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<div class="line"><span class="lineno"> 98</span>}</div>
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<div class="ttc" id="adurand__kerner__roots_8cpp_html_af270a96662132d0385cb6b4637c5a689"><div class="ttname"><a href="#af270a96662132d0385cb6b4637c5a689">ACCURACY</a></div><div class="ttdeci">#define ACCURACY</div><div class="ttdef"><b>Definition</b> <a href="../../da/df2/durand__kerner__roots_8cpp_source.html#l00046">durand_kerner_roots.cpp:46</a></div></div>
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</div><!-- fragment -->
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</div>
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</div>
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<a id="a90219e35062007d1f1b68e9af071ab5c" name="a90219e35062007d1f1b68e9af071ab5c"></a>
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<h2 class="memtitle"><span class="permalink"><a href="#a90219e35062007d1f1b68e9af071ab5c">◆ </a></span>complex_str()</h2>
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<div class="memitem">
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<tr>
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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 std::complex< double > &</td> <td class="paramname"><span class="paramname"><em>x</em></span></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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<dl class="section return"><dt>Returns</dt><dd>pointer to converted string </dd></dl>
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<p class="definition">Definition at line <a class="el" href="../../da/df2/durand__kerner__roots_8cpp_source.html#l00077">77</a> of file <a class="el" href="../../da/df2/durand__kerner__roots_8cpp_source.html">durand_kerner_roots.cpp</a>.</p>
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<div class="fragment"><div class="line"><span class="lineno"> 77</span> {</div>
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<div class="line"><span class="lineno"> 78</span><span class="preprocessor">#define MAX_BUFF_SIZE 50</span></div>
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<div class="line"><span class="lineno"> 79</span> <span class="keyword">static</span> <span class="keywordtype">char</span> msg[MAX_BUFF_SIZE];</div>
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<div class="line"><span class="lineno"> 80</span> </div>
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<div class="line"><span class="lineno"> 81</span> std::snprintf(msg, MAX_BUFF_SIZE, <span class="stringliteral">"% 7.04g%+7.04gj"</span>, x.real(), x.imag());</div>
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<div class="line"><span class="lineno"> 82</span> </div>
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<div class="line"><span class="lineno"> 83</span> <span class="keywordflow">return</span> msg;</div>
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<div class="line"><span class="lineno"> 84</span>}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#a2c35b320ace8677f9b331faf94f8b2fd">◆ </a></span>durand_kerner_algo()</h2>
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<td class="memname">std::pair< uint32_t, double > durand_kerner_algo </td>
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<td>(</td>
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<td class="paramtype">const std::valarray< double > &</td> <td class="paramname"><span class="paramname"><em>coeffs</em></span>, </td>
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<td class="paramtype">std::valarray< std::complex< double > > *</td> <td class="paramname"><span class="paramname"><em>roots</em></span>, </td>
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<td class="paramtype">bool</td> <td class="paramname"><span class="paramname"><em>write_log</em></span><span class="paramdefsep"> = </span><span class="paramdefval">false</span> )</td>
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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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<p class="definition">Definition at line <a class="el" href="../../da/df2/durand__kerner__roots_8cpp_source.html#l00110">110</a> of file <a class="el" href="../../da/df2/durand__kerner__roots_8cpp_source.html">durand_kerner_roots.cpp</a>.</p>
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<div class="fragment"><div class="line"><span class="lineno"> 112</span> {</div>
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<div class="line"><span class="lineno"> 113</span> <span class="keywordtype">long</span> <span class="keywordtype">double</span> tol_condition = 1;</div>
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<div class="line"><span class="lineno"> 114</span> uint32_t iter = 0;</div>
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<div class="line"><span class="lineno"> 115</span> <span class="keywordtype">int</span> n;</div>
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<div class="line"><span class="lineno"> 116</span> std::ofstream log_file;</div>
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<div class="line"><span class="lineno"> 117</span> </div>
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<div class="line"><span class="lineno"> 118</span> <span class="keywordflow">if</span> (write_log) {</div>
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<div class="line"><span class="lineno"> 119</span> <span class="comment">/*</span></div>
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<div class="line"><span class="lineno"> 120</span><span class="comment"> * store intermediate values to a CSV file</span></div>
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<div class="line"><span class="lineno"> 121</span><span class="comment"> */</span></div>
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<div class="line"><span class="lineno"> 122</span> log_file.open(<span class="stringliteral">"durand_kerner.log.csv"</span>);</div>
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<div class="line"><span class="lineno"> 123</span> <span class="keywordflow">if</span> (!log_file.is_open()) {</div>
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<div class="line"><span class="lineno"> 124</span> perror(<span class="stringliteral">"Unable to create a storage log file!"</span>);</div>
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<div class="line"><span class="lineno"> 125</span> std::exit(EXIT_FAILURE);</div>
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<div class="line"><span class="lineno"> 126</span> }</div>
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<div class="line"><span class="lineno"> 127</span> log_file << <span class="stringliteral">"iter#,"</span>;</div>
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<div class="line"><span class="lineno"> 128</span> </div>
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<div class="line"><span class="lineno"> 129</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"><span class="lineno"> 130</span> </div>
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<div class="line"><span class="lineno"> 131</span> log_file << <span class="stringliteral">"avg. correction"</span>;</div>
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<div class="line"><span class="lineno"> 132</span> log_file << <span class="stringliteral">"\n0,"</span>;</div>
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<div class="line"><span class="lineno"> 133</span> <span class="keywordflow">for</span> (n = 0; n < roots->size(); n++)</div>
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<div class="line"><span class="lineno"> 134</span> log_file << <a class="code hl_function" href="#a90219e35062007d1f1b68e9af071ab5c">complex_str</a>((*roots)[n]) << <span class="stringliteral">","</span>;</div>
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<div class="line"><span class="lineno"> 135</span> }</div>
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<div class="line"><span class="lineno"> 136</span> </div>
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<div class="line"><span class="lineno"> 137</span> <span class="keywordtype">bool</span> break_loop = <span class="keyword">false</span>;</div>
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<div class="line"><span class="lineno"> 138</span> <span class="keywordflow">while</span> (!<a class="code hl_function" href="#a024b8bc4755863315456d573a6732377">check_termination</a>(tol_condition) && iter < INT16_MAX &&</div>
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<div class="line"><span class="lineno"> 139</span> !break_loop) {</div>
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<div class="line"><span class="lineno"> 140</span> tol_condition = 0;</div>
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<div class="line"><span class="lineno"> 141</span> iter++;</div>
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<div class="line"><span class="lineno"> 142</span> break_loop = <span class="keyword">false</span>;</div>
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<div class="line"><span class="lineno"> 143</span> </div>
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<div class="line"><span class="lineno"> 144</span> <span class="keywordflow">if</span> (log_file.is_open())</div>
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<div class="line"><span class="lineno"> 145</span> log_file << <span class="stringliteral">"\n"</span> << iter << <span class="stringliteral">","</span>;</div>
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<div class="line"><span class="lineno"> 146</span> </div>
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<div class="line"><span class="lineno"> 147</span><span class="preprocessor">#ifdef _OPENMP</span></div>
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<div class="line"><span class="lineno"> 148</span><span class="preprocessor">#pragma omp parallel for shared(break_loop, tol_condition)</span></div>
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<div class="line"><span class="lineno"> 149</span><span class="preprocessor">#endif</span></div>
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<div class="line"><span class="lineno"> 150</span> <span class="keywordflow">for</span> (n = 0; n < roots->size(); n++) {</div>
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<div class="line"><span class="lineno"> 151</span> <span class="keywordflow">if</span> (break_loop)</div>
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<div class="line"><span class="lineno"> 152</span> <span class="keywordflow">continue</span>;</div>
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<div class="line"><span class="lineno"> 153</span> </div>
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<div class="line"><span class="lineno"> 154</span> std::complex<double> numerator, denominator;</div>
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<div class="line"><span class="lineno"> 155</span> numerator = <a class="code hl_function" href="#ae2ab49ba1969d7a5988f9bc2c8924569">poly_function</a>(coeffs, (*roots)[n]);</div>
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<div class="line"><span class="lineno"> 156</span> denominator = 1.0;</div>
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<div class="line"><span class="lineno"> 157</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"><span class="lineno"> 158</span> <span class="keywordflow">if</span> (i != n)</div>
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<div class="line"><span class="lineno"> 159</span> denominator *= (*roots)[n] - (*roots)[i];</div>
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<div class="line"><span class="lineno"> 160</span> </div>
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<div class="line"><span class="lineno"> 161</span> std::complex<long double> delta = numerator / denominator;</div>
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<div class="line"><span class="lineno"> 162</span> </div>
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<div class="line"><span class="lineno"> 163</span> <span class="keywordflow">if</span> (std::isnan(std::abs(delta)) || std::isinf(std::abs(delta))) {</div>
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<div class="line"><span class="lineno"> 164</span> std::cerr << <span class="stringliteral">"\n\nOverflow/underrun error - got value = "</span></div>
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<div class="line"><span class="lineno"> 165</span> << std::abs(delta) << <span class="stringliteral">"\n"</span>;</div>
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<div class="line"><span class="lineno"> 166</span> <span class="comment">// return std::pair<uint32_t, double>(iter, tol_condition);</span></div>
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<div class="line"><span class="lineno"> 167</span> break_loop = <span class="keyword">true</span>;</div>
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<div class="line"><span class="lineno"> 168</span> }</div>
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<div class="line"><span class="lineno"> 169</span> </div>
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<div class="line"><span class="lineno"> 170</span> (*roots)[n] -= delta;</div>
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<div class="line"><span class="lineno"> 171</span> </div>
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<div class="line"><span class="lineno"> 172</span><span class="preprocessor">#ifdef _OPENMP</span></div>
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<div class="line"><span class="lineno"> 173</span><span class="preprocessor">#pragma omp critical</span></div>
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<div class="line"><span class="lineno"> 174</span><span class="preprocessor">#endif</span></div>
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<div class="line"><span class="lineno"> 175</span> tol_condition = std::max(tol_condition, std::abs(std::abs(delta)));</div>
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<div class="line"><span class="lineno"> 176</span> }</div>
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<div class="line"><span class="lineno"> 177</span> <span class="comment">// tol_condition /= (degree - 1);</span></div>
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<div class="line"><span class="lineno"> 178</span> </div>
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<div class="line"><span class="lineno"> 179</span> <span class="keywordflow">if</span> (break_loop)</div>
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<div class="line"><span class="lineno"> 180</span> <span class="keywordflow">break</span>;</div>
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<div class="line"><span class="lineno"> 181</span> </div>
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<div class="line"><span class="lineno"> 182</span> <span class="keywordflow">if</span> (log_file.is_open()) {</div>
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<div class="line"><span class="lineno"> 183</span> <span class="keywordflow">for</span> (n = 0; n < roots->size(); n++)</div>
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<div class="line"><span class="lineno"> 184</span> log_file << <a class="code hl_function" href="#a90219e35062007d1f1b68e9af071ab5c">complex_str</a>((*roots)[n]) << <span class="stringliteral">","</span>;</div>
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<div class="line"><span class="lineno"> 185</span> }</div>
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<div class="line"><span class="lineno"> 186</span> </div>
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<div class="line"><span class="lineno"> 187</span><span class="preprocessor">#if defined(DEBUG) || !defined(NDEBUG)</span></div>
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<div class="line"><span class="lineno"> 188</span> <span class="keywordflow">if</span> (iter % 500 == 0) {</div>
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<div class="line"><span class="lineno"> 189</span> std::cout << <span class="stringliteral">"Iter: "</span> << iter << <span class="stringliteral">"\t"</span>;</div>
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<div class="line"><span class="lineno"> 190</span> <span class="keywordflow">for</span> (n = 0; n < roots->size(); n++)</div>
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<div class="line"><span class="lineno"> 191</span> std::cout << <span class="stringliteral">"\t"</span> << <a class="code hl_function" href="#a90219e35062007d1f1b68e9af071ab5c">complex_str</a>((*roots)[n]);</div>
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<div class="line"><span class="lineno"> 192</span> std::cout << <span class="stringliteral">"\t\tabsolute average change: "</span> << tol_condition</div>
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<div class="line"><span class="lineno"> 193</span> << <span class="stringliteral">"\n"</span>;</div>
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<div class="line"><span class="lineno"> 194</span> }</div>
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<div class="line"><span class="lineno"> 195</span><span class="preprocessor">#endif</span></div>
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<div class="line"><span class="lineno"> 196</span> </div>
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<div class="line"><span class="lineno"> 197</span> <span class="keywordflow">if</span> (log_file.is_open())</div>
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<div class="line"><span class="lineno"> 198</span> log_file << tol_condition;</div>
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<div class="line"><span class="lineno"> 199</span> }</div>
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<div class="line"><span class="lineno"> 200</span> </div>
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<div class="line"><span class="lineno"> 201</span> <span class="keywordflow">return</span> std::pair<uint32_t, long double>(iter, tol_condition);</div>
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<div class="line"><span class="lineno"> 202</span>}</div>
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<div class="ttc" id="adurand__kerner__roots_8cpp_html_a024b8bc4755863315456d573a6732377"><div class="ttname"><a href="#a024b8bc4755863315456d573a6732377">check_termination</a></div><div class="ttdeci">bool check_termination(long double delta)</div><div class="ttdef"><b>Definition</b> <a href="../../da/df2/durand__kerner__roots_8cpp_source.html#l00092">durand_kerner_roots.cpp:92</a></div></div>
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<div class="ttc" id="adurand__kerner__roots_8cpp_html_a90219e35062007d1f1b68e9af071ab5c"><div class="ttname"><a href="#a90219e35062007d1f1b68e9af071ab5c">complex_str</a></div><div class="ttdeci">const char * complex_str(const std::complex< double > &x)</div><div class="ttdef"><b>Definition</b> <a href="../../da/df2/durand__kerner__roots_8cpp_source.html#l00077">durand_kerner_roots.cpp:77</a></div></div>
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<div class="ttc" id="adurand__kerner__roots_8cpp_html_ae2ab49ba1969d7a5988f9bc2c8924569"><div class="ttname"><a href="#ae2ab49ba1969d7a5988f9bc2c8924569">poly_function</a></div><div class="ttdeci">std::complex< double > poly_function(const std::valarray< double > &coeffs, std::complex< double > x)</div><div class="ttdef"><b>Definition</b> <a href="../../da/df2/durand__kerner__roots_8cpp_source.html#l00054">durand_kerner_roots.cpp:54</a></div></div>
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<h2 class="memtitle"><span class="permalink"><a href="#a3c04138a5bfe5d72780bb7e82a18e627">◆ </a></span>main()</h2>
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<p class="definition">Definition at line <a class="el" href="../../da/df2/durand__kerner__roots_8cpp_source.html#l00284">284</a> of file <a class="el" href="../../da/df2/durand__kerner__roots_8cpp_source.html">durand_kerner_roots.cpp</a>.</p>
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<div class="fragment"><div class="line"><span class="lineno"> 284</span> {</div>
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<div class="line"><span class="lineno"> 285</span> <span class="comment">/* initialize random seed: */</span></div>
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<div class="line"><span class="lineno"> 286</span> std::srand(std::time(<span class="keyword">nullptr</span>));</div>
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<div class="line"><span class="lineno"> 287</span> </div>
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<div class="line"><span class="lineno"> 288</span> <span class="keywordflow">if</span> (argc < 2) {</div>
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<div class="line"><span class="lineno"> 289</span> <a class="code hl_function" href="#a1440a7779ac56f47a3f355ce4a8c7da0">test1</a>(); <span class="comment">// run tests when no input is provided</span></div>
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<div class="line"><span class="lineno"> 290</span> <a class="code hl_function" href="#a0283886819c7c140a023582b7269e2d0">test2</a>(); <span class="comment">// and skip tests when input polynomial is provided</span></div>
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<div class="line"><span class="lineno"> 291</span> std::cout << <span class="stringliteral">"Please pass the coefficients of the polynomial as "</span></div>
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<div class="line"><span class="lineno"> 292</span> <span class="stringliteral">"commandline "</span></div>
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<div class="line"><span class="lineno"> 293</span> <span class="stringliteral">"arguments.\n"</span>;</div>
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<div class="line"><span class="lineno"> 294</span> <span class="keywordflow">return</span> 0;</div>
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<div class="line"><span class="lineno"> 295</span> }</div>
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<div class="line"><span class="lineno"> 296</span> </div>
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<div class="line"><span class="lineno"> 297</span> <span class="keywordtype">int</span> n, degree = argc - 1; <span class="comment">// detected polynomial degree</span></div>
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<div class="line"><span class="lineno"> 298</span> std::valarray<double> coeffs(degree); <span class="comment">// create coefficiencts array</span></div>
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<div class="line"><span class="lineno"> 299</span> </div>
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<div class="line"><span class="lineno"> 300</span> <span class="comment">// number of roots = degree - 1</span></div>
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<div class="line"><span class="lineno"> 301</span> std::valarray<std::complex<double>> s0(degree - 1);</div>
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<div class="line"><span class="lineno"> 302</span> </div>
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<div class="line"><span class="lineno"> 303</span> std::cout << <span class="stringliteral">"Computing the roots for:\n\t"</span>;</div>
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<div class="line"><span class="lineno"> 304</span> <span class="keywordflow">for</span> (n = 0; n < degree; n++) {</div>
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<div class="line"><span class="lineno"> 305</span> coeffs[n] = strtod(argv[n + 1], <span class="keyword">nullptr</span>);</div>
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<div class="line"><span class="lineno"> 306</span> <span class="keywordflow">if</span> (n < degree - 1 && coeffs[n] != 0)</div>
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<div class="line"><span class="lineno"> 307</span> std::cout << <span class="stringliteral">"("</span> << coeffs[n] << <span class="stringliteral">") x^"</span> << degree - n - 1 << <span class="stringliteral">" + "</span>;</div>
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<div class="line"><span class="lineno"> 308</span> <span class="keywordflow">else</span> <span class="keywordflow">if</span> (coeffs[n] != 0)</div>
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<div class="line"><span class="lineno"> 309</span> std::cout << <span class="stringliteral">"("</span> << coeffs[n] << <span class="stringliteral">") x^"</span> << degree - n - 1</div>
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<div class="line"><span class="lineno"> 310</span> << <span class="stringliteral">" = 0\n"</span>;</div>
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<div class="line"><span class="lineno"> 311</span> </div>
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<div class="line"><span class="lineno"> 312</span> <span class="comment">/* initialize root approximations with random values */</span></div>
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<div class="line"><span class="lineno"> 313</span> <span class="keywordflow">if</span> (n < degree - 1) {</div>
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<div class="line"><span class="lineno"> 314</span> s0[n] = std::complex<double>(std::rand() % 100, std::rand() % 100);</div>
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<div class="line"><span class="lineno"> 315</span> s0[n] -= 50.f;</div>
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<div class="line"><span class="lineno"> 316</span> s0[n] /= 50.f;</div>
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<div class="line"><span class="lineno"> 317</span> }</div>
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<div class="line"><span class="lineno"> 318</span> }</div>
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<div class="line"><span class="lineno"> 319</span> </div>
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<div class="line"><span class="lineno"> 320</span> <span class="comment">// numerical errors less when the first coefficient is "1"</span></div>
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<div class="line"><span class="lineno"> 321</span> <span class="comment">// hence, we normalize the first coefficient</span></div>
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<div class="line"><span class="lineno"> 322</span> {</div>
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<div class="line"><span class="lineno"> 323</span> <span class="keywordtype">double</span> tmp = coeffs[0];</div>
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<div class="line"><span class="lineno"> 324</span> coeffs /= tmp;</div>
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<div class="line"><span class="lineno"> 325</span> }</div>
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<div class="line"><span class="lineno"> 326</span> </div>
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<div class="line"><span class="lineno"> 327</span> clock_t end_time, start_time = clock();</div>
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<div class="line"><span class="lineno"> 328</span> <span class="keyword">auto</span> <a class="code hl_function" href="../../de/dc3/fibonacci__sum_8cpp.html#aadb40ac4c74a7efc0680b83eeee138aa">result</a> = <a class="code hl_function" href="#a2c35b320ace8677f9b331faf94f8b2fd">durand_kerner_algo</a>(coeffs, &s0, <span class="keyword">true</span>);</div>
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<div class="line"><span class="lineno"> 329</span> end_time = clock();</div>
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<div class="line"><span class="lineno"> 330</span> </div>
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<div class="line"><span class="lineno"> 331</span> std::cout << <span class="stringliteral">"\nIterations: "</span> << <a class="code hl_function" href="../../de/dc3/fibonacci__sum_8cpp.html#aadb40ac4c74a7efc0680b83eeee138aa">result</a>.first << <span class="stringliteral">"\n"</span>;</div>
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<div class="line"><span class="lineno"> 332</span> <span class="keywordflow">for</span> (n = 0; n < degree - 1; n++)</div>
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<div class="line"><span class="lineno"> 333</span> std::cout << <span class="stringliteral">"\t"</span> << <a class="code hl_function" href="#a90219e35062007d1f1b68e9af071ab5c">complex_str</a>(s0[n]) << <span class="stringliteral">"\n"</span>;</div>
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<div class="line"><span class="lineno"> 334</span> std::cout << <span class="stringliteral">"absolute average change: "</span> << <a class="code hl_function" href="../../de/dc3/fibonacci__sum_8cpp.html#aadb40ac4c74a7efc0680b83eeee138aa">result</a>.second << <span class="stringliteral">"\n"</span>;</div>
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<div class="line"><span class="lineno"> 335</span> std::cout << <span class="stringliteral">"Time taken: "</span></div>
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<div class="line"><span class="lineno"> 336</span> << <span class="keyword">static_cast<</span><span class="keywordtype">double</span><span class="keyword">></span>(end_time - start_time) / CLOCKS_PER_SEC</div>
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<div class="line"><span class="lineno"> 337</span> << <span class="stringliteral">" sec\n"</span>;</div>
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<div class="line"><span class="lineno"> 338</span> </div>
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<div class="line"><span class="lineno"> 339</span> <span class="keywordflow">return</span> 0;</div>
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<div class="line"><span class="lineno"> 340</span>}</div>
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<div class="ttc" id="adurand__kerner__roots_8cpp_html_a0283886819c7c140a023582b7269e2d0"><div class="ttname"><a href="#a0283886819c7c140a023582b7269e2d0">test2</a></div><div class="ttdeci">void test2()</div><div class="ttdef"><b>Definition</b> <a href="../../da/df2/durand__kerner__roots_8cpp_source.html#l00243">durand_kerner_roots.cpp:243</a></div></div>
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<div class="ttc" id="adurand__kerner__roots_8cpp_html_a1440a7779ac56f47a3f355ce4a8c7da0"><div class="ttname"><a href="#a1440a7779ac56f47a3f355ce4a8c7da0">test1</a></div><div class="ttdeci">void test1()</div><div class="ttdef"><b>Definition</b> <a href="../../da/df2/durand__kerner__roots_8cpp_source.html#l00208">durand_kerner_roots.cpp:208</a></div></div>
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<div class="ttc" id="adurand__kerner__roots_8cpp_html_a2c35b320ace8677f9b331faf94f8b2fd"><div class="ttname"><a href="#a2c35b320ace8677f9b331faf94f8b2fd">durand_kerner_algo</a></div><div class="ttdeci">std::pair< uint32_t, double > durand_kerner_algo(const std::valarray< double > &coeffs, std::valarray< std::complex< double > > *roots, bool write_log=false)</div><div class="ttdef"><b>Definition</b> <a href="../../da/df2/durand__kerner__roots_8cpp_source.html#l00110">durand_kerner_roots.cpp:110</a></div></div>
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<div class="ttc" id="afibonacci__sum_8cpp_html_aadb40ac4c74a7efc0680b83eeee138aa"><div class="ttname"><a href="../../de/dc3/fibonacci__sum_8cpp.html#aadb40ac4c74a7efc0680b83eeee138aa">math::fibonacci_sum::result</a></div><div class="ttdeci">uint64_t result(uint64_t n)</div><div class="ttdef"><b>Definition</b> <a href="../../de/dc3/fibonacci__sum_8cpp_source.html#l00077">fibonacci_sum.cpp:77</a></div></div>
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<h2 class="memtitle"><span class="permalink"><a href="#ae2ab49ba1969d7a5988f9bc2c8924569">◆ </a></span>poly_function()</h2>
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<td class="memname">std::complex< double > poly_function </td>
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<td>(</td>
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<td class="paramtype">const std::valarray< double > &</td> <td class="paramname"><span class="paramname"><em>coeffs</em></span>, </td>
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<td class="paramtype">std::complex< double ></td> <td class="paramname"><span class="paramname"><em>x</em></span> )</td>
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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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</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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<p class="definition">Definition at line <a class="el" href="../../da/df2/durand__kerner__roots_8cpp_source.html#l00054">54</a> of file <a class="el" href="../../da/df2/durand__kerner__roots_8cpp_source.html">durand_kerner_roots.cpp</a>.</p>
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<div class="fragment"><div class="line"><span class="lineno"> 55</span> {</div>
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<div class="line"><span class="lineno"> 56</span> <span class="keywordtype">double</span> real = 0.f, imag = 0.f;</div>
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<div class="line"><span class="lineno"> 57</span> <span class="keywordtype">int</span> n;</div>
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<div class="line"><span class="lineno"> 58</span> </div>
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<div class="line"><span class="lineno"> 59</span> <span class="comment">// #ifdef _OPENMP</span></div>
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<div class="line"><span class="lineno"> 60</span> <span class="comment">// #pragma omp target teams distribute reduction(+ : real, imag)</span></div>
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<div class="line"><span class="lineno"> 61</span> <span class="comment">// #endif</span></div>
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<div class="line"><span class="lineno"> 62</span> <span class="keywordflow">for</span> (n = 0; n < coeffs.size(); n++) {</div>
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<div class="line"><span class="lineno"> 63</span> std::complex<double> tmp =</div>
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<div class="line"><span class="lineno"> 64</span> coeffs[n] * std::pow(x, coeffs.size() - n - 1);</div>
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<div class="line"><span class="lineno"> 65</span> real += tmp.real();</div>
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<div class="line"><span class="lineno"> 66</span> imag += tmp.imag();</div>
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<div class="line"><span class="lineno"> 67</span> }</div>
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<div class="line"><span class="lineno"> 68</span> </div>
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<div class="line"><span class="lineno"> 69</span> <span class="keywordflow">return</span> std::complex<double>(real, imag);</div>
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<div class="line"><span class="lineno"> 70</span>}</div>
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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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<td>(</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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<p class="definition">Definition at line <a class="el" href="../../da/df2/durand__kerner__roots_8cpp_source.html#l00208">208</a> of file <a class="el" href="../../da/df2/durand__kerner__roots_8cpp_source.html">durand_kerner_roots.cpp</a>.</p>
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<div class="fragment"><div class="line"><span class="lineno"> 208</span> {</div>
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<div class="line"><span class="lineno"> 209</span> <span class="keyword">const</span> std::valarray<double> coeffs = {1, 0, 4}; <span class="comment">// x^2 - 2 = 0</span></div>
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<div class="line"><span class="lineno"> 210</span> std::valarray<std::complex<double>> roots(2);</div>
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<div class="line"><span class="lineno"> 211</span> std::valarray<std::complex<double>> expected = {</div>
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<div class="line"><span class="lineno"> 212</span> std::complex<double>(0., 2.),</div>
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<div class="line"><span class="lineno"> 213</span> std::complex<double>(0., -2.) <span class="comment">// known expected roots</span></div>
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<div class="line"><span class="lineno"> 214</span> };</div>
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<div class="line"><span class="lineno"> 215</span> </div>
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<div class="line"><span class="lineno"> 216</span> <span class="comment">/* initialize root approximations with random values */</span></div>
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<div class="line"><span class="lineno"> 217</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"><span class="lineno"> 218</span> roots[n] = std::complex<double>(std::rand() % 100, std::rand() % 100);</div>
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<div class="line"><span class="lineno"> 219</span> roots[n] -= 50.f;</div>
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<div class="line"><span class="lineno"> 220</span> roots[n] /= 25.f;</div>
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<div class="line"><span class="lineno"> 221</span> }</div>
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<div class="line"><span class="lineno"> 222</span> </div>
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<div class="line"><span class="lineno"> 223</span> <span class="keyword">auto</span> <a class="code hl_function" href="../../de/dc3/fibonacci__sum_8cpp.html#aadb40ac4c74a7efc0680b83eeee138aa">result</a> = <a class="code hl_function" href="#a2c35b320ace8677f9b331faf94f8b2fd">durand_kerner_algo</a>(coeffs, &roots, <span class="keyword">false</span>);</div>
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<div class="line"><span class="lineno"> 224</span> </div>
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<div class="line"><span class="lineno"> 225</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"><span class="lineno"> 226</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"><span class="lineno"> 227</span> <span class="comment">// expected roots</span></div>
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<div class="line"><span class="lineno"> 228</span> <span class="keywordtype">bool</span> err1 = <span class="keyword">false</span>;</div>
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<div class="line"><span class="lineno"> 229</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"><span class="lineno"> 230</span> err1 |= std::abs(std::abs(roots[i] - expected[j])) < 1e-3;</div>
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<div class="line"><span class="lineno"> 231</span> assert(err1);</div>
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<div class="line"><span class="lineno"> 232</span> }</div>
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<div class="line"><span class="lineno"> 233</span> </div>
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<div class="line"><span class="lineno"> 234</span> std::cout << <span class="stringliteral">"Test 1 passed! - "</span> << <a class="code hl_function" href="../../de/dc3/fibonacci__sum_8cpp.html#aadb40ac4c74a7efc0680b83eeee138aa">result</a>.first << <span class="stringliteral">" iterations, "</span></div>
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<div class="line"><span class="lineno"> 235</span> << <a class="code hl_function" href="../../de/dc3/fibonacci__sum_8cpp.html#aadb40ac4c74a7efc0680b83eeee138aa">result</a>.second << <span class="stringliteral">" accuracy"</span></div>
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<div class="line"><span class="lineno"> 236</span> << <span class="stringliteral">"\n"</span>;</div>
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<div class="line"><span class="lineno"> 237</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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<p class="definition">Definition at line <a class="el" href="../../da/df2/durand__kerner__roots_8cpp_source.html#l00243">243</a> of file <a class="el" href="../../da/df2/durand__kerner__roots_8cpp_source.html">durand_kerner_roots.cpp</a>.</p>
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<div class="fragment"><div class="line"><span class="lineno"> 243</span> {</div>
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<div class="line"><span class="lineno"> 244</span> <span class="keyword">const</span> std::valarray<double> coeffs = {<span class="comment">// 0.015625 x^3 - 1 = 0</span></div>
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<div class="line"><span class="lineno"> 245</span> 1. / 64., 0., 0., -1.};</div>
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<div class="line"><span class="lineno"> 246</span> std::valarray<std::complex<double>> roots(3);</div>
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<div class="line"><span class="lineno"> 247</span> <span class="keyword">const</span> std::valarray<std::complex<double>> expected = {</div>
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<div class="line"><span class="lineno"> 248</span> std::complex<double>(4., 0.), std::complex<double>(-2., 3.46410162),</div>
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<div class="line"><span class="lineno"> 249</span> std::complex<double>(-2., -3.46410162) <span class="comment">// known expected roots</span></div>
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<div class="line"><span class="lineno"> 250</span> };</div>
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<div class="line"><span class="lineno"> 251</span> </div>
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<div class="line"><span class="lineno"> 252</span> <span class="comment">/* initialize root approximations with random values */</span></div>
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<div class="line"><span class="lineno"> 253</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"><span class="lineno"> 254</span> roots[n] = std::complex<double>(std::rand() % 100, std::rand() % 100);</div>
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<div class="line"><span class="lineno"> 255</span> roots[n] -= 50.f;</div>
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<div class="line"><span class="lineno"> 256</span> roots[n] /= 25.f;</div>
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<div class="line"><span class="lineno"> 257</span> }</div>
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<div class="line"><span class="lineno"> 258</span> </div>
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<div class="line"><span class="lineno"> 259</span> <span class="keyword">auto</span> <a class="code hl_function" href="../../de/dc3/fibonacci__sum_8cpp.html#aadb40ac4c74a7efc0680b83eeee138aa">result</a> = <a class="code hl_function" href="#a2c35b320ace8677f9b331faf94f8b2fd">durand_kerner_algo</a>(coeffs, &roots, <span class="keyword">false</span>);</div>
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<div class="line"><span class="lineno"> 260</span> </div>
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<div class="line"><span class="lineno"> 261</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"><span class="lineno"> 262</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"><span class="lineno"> 263</span> <span class="comment">// expected roots</span></div>
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<div class="line"><span class="lineno"> 264</span> <span class="keywordtype">bool</span> err1 = <span class="keyword">false</span>;</div>
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<div class="line"><span class="lineno"> 265</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"><span class="lineno"> 266</span> err1 |= std::abs(std::abs(roots[i] - expected[j])) < 1e-3;</div>
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<div class="line"><span class="lineno"> 267</span> assert(err1);</div>
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<div class="line"><span class="lineno"> 268</span> }</div>
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<div class="line"><span class="lineno"> 269</span> </div>
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<div class="line"><span class="lineno"> 270</span> std::cout << <span class="stringliteral">"Test 2 passed! - "</span> << <a class="code hl_function" href="../../de/dc3/fibonacci__sum_8cpp.html#aadb40ac4c74a7efc0680b83eeee138aa">result</a>.first << <span class="stringliteral">" iterations, "</span></div>
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<div class="line"><span class="lineno"> 271</span> << <a class="code hl_function" href="../../de/dc3/fibonacci__sum_8cpp.html#aadb40ac4c74a7efc0680b83eeee138aa">result</a>.second << <span class="stringliteral">" accuracy"</span></div>
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<div class="line"><span class="lineno"> 272</span> << <span class="stringliteral">"\n"</span>;</div>
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<div class="line"><span class="lineno"> 273</span>}</div>
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