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<div id="projectbrief">Set of algorithms implemented in C++.</div>
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<div class="title">sqrt_double.cpp File Reference</div> </div>
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<p>Calculate the square root of any positive real number in \(O(\log N)\) time, with precision fixed using <a href="https://en.wikipedia.org/wiki/Bisection_method">bisection method</a> of root-finding.
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<a href="#details">More...</a></p>
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<div class="textblock"><code>#include <cassert></code><br />
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<code>#include <iostream></code><br />
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</div><div class="textblock"><div class="dynheader">
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Include dependency graph for sqrt_double.cpp:</div>
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<div class="dyncontent">
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Functions</h2></td></tr>
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<tr class="memitem:ae662282ad0740d2063ac404ca3bd74fc"><td class="memItemLeft" align="right" valign="top">double </td><td class="memItemRight" valign="bottom"><a class="el" href="../../da/d24/sqrt__double_8cpp.html#ae662282ad0740d2063ac404ca3bd74fc">Sqrt</a> (double <a class="el" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a>)</td></tr>
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<tr class="separator:ae662282ad0740d2063ac404ca3bd74fc"><td class="memSeparator" colspan="2"> </td></tr>
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<tr class="memitem:ae66f6b31b5ad750f1fe042a706a4e3d4"><td class="memItemLeft" align="right" valign="top">int </td><td class="memItemRight" valign="bottom"><a class="el" href="../../da/d24/sqrt__double_8cpp.html#ae66f6b31b5ad750f1fe042a706a4e3d4">main</a> ()</td></tr>
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<tr class="separator:ae66f6b31b5ad750f1fe042a706a4e3d4"><td class="memSeparator" colspan="2"> </td></tr>
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<a name="details" id="details"></a><h2 class="groupheader">Detailed Description</h2>
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<div class="textblock"><p>Calculate the square root of any positive real number in \(O(\log N)\) time, with precision fixed using <a href="https://en.wikipedia.org/wiki/Bisection_method">bisection method</a> of root-finding. </p>
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<dl class="section see"><dt>See also</dt><dd>Can be implemented using faster and better algorithms like <a class="el" href="../../de/dd3/newton__raphson__method_8cpp.html" title="Solve the equation using Newton-Raphson method for both real and complex solutions.">newton_raphson_method.cpp</a> and <a class="el" href="../../dd/d29/false__position_8cpp.html" title="Solve the equation using false position method, also known as the Secant method.">false_position.cpp</a> </dd></dl>
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</div><h2 class="groupheader">Function Documentation</h2>
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<a id="ae66f6b31b5ad750f1fe042a706a4e3d4"></a>
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<h2 class="memtitle"><span class="permalink"><a href="#ae66f6b31b5ad750f1fe042a706a4e3d4">◆ </a></span>main()</h2>
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<div class="memitem">
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<td class="memname">int main </td>
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<td>(</td>
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<td class="paramname"></td><td>)</td>
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<td></td>
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</tr>
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</div><div class="memdoc">
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<p>main function </p>
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<div class="fragment"><div class="line"><a name="l00042"></a><span class="lineno"> 42</span>  {</div>
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<div class="line"><a name="l00043"></a><span class="lineno"> 43</span>  <span class="keywordtype">double</span> n{};</div>
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<div class="line"><a name="l00044"></a><span class="lineno"> 44</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_istream.html">std::cin</a> >> n;</div>
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<div class="line"><a name="l00045"></a><span class="lineno"> 45</span>  assert(n >= 0);</div>
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<div class="line"><a name="l00046"></a><span class="lineno"> 46</span>  <span class="comment">// Change this line for a better precision</span></div>
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<div class="line"><a name="l00047"></a><span class="lineno"> 47</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a>.precision(12);</div>
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<div class="line"><a name="l00048"></a><span class="lineno"> 48</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> << <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/fixed.html">std::fixed</a> << <a class="code" href="../../da/d24/sqrt__double_8cpp.html#ae662282ad0740d2063ac404ca3bd74fc">Sqrt</a>(n);</div>
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<div class="line"><a name="l00049"></a><span class="lineno"> 49</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="center"><iframe scrolling="no" frameborder="0" src="../../da/d24/sqrt__double_8cpp_ae66f6b31b5ad750f1fe042a706a4e3d4_cgraph.svg" width="186" height="88"><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="ae662282ad0740d2063ac404ca3bd74fc"></a>
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<h2 class="memtitle"><span class="permalink"><a href="#ae662282ad0740d2063ac404ca3bd74fc">◆ </a></span>Sqrt()</h2>
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<td class="memname">double Sqrt </td>
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<td>(</td>
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<td class="paramtype">double </td>
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<td class="paramname"><em>a</em></td><td>)</td>
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<td></td>
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</div><div class="memdoc">
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<p>Bisection method implemented for the function \(x^2-a=0\) whose roots are \(\pm\sqrt{a}\) and only the positive root is returned. </p>
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<div class="fragment"><div class="line"><a name="l00016"></a><span class="lineno"> 16</span>  {</div>
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<div class="line"><a name="l00017"></a><span class="lineno"> 17</span>  <span class="keywordflow">if</span> (<a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a> > 0 && <a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a> < 1) {</div>
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<div class="line"><a name="l00018"></a><span class="lineno"> 18</span>  <span class="keywordflow">return</span> 1 / <a class="code" href="../../da/d24/sqrt__double_8cpp.html#ae662282ad0740d2063ac404ca3bd74fc">Sqrt</a>(1 / <a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a>);</div>
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<div class="line"><a name="l00019"></a><span class="lineno"> 19</span>  }</div>
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<div class="line"><a name="l00020"></a><span class="lineno"> 20</span>  <span class="keywordtype">double</span> l = 0, r = <a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a>;</div>
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<div class="line"><a name="l00021"></a><span class="lineno"> 21</span>  <span class="comment">/* Epsilon is the precision.</span></div>
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<div class="line"><a name="l00022"></a><span class="lineno"> 22</span> <span class="comment"> A great precision is</span></div>
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<div class="line"><a name="l00023"></a><span class="lineno"> 23</span> <span class="comment"> between 1e-7 and 1e-12.</span></div>
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<div class="line"><a name="l00024"></a><span class="lineno"> 24</span> <span class="comment"> double epsilon = 1e-12;</span></div>
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<div class="line"><a name="l00025"></a><span class="lineno"> 25</span> <span class="comment"> */</span></div>
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<div class="line"><a name="l00026"></a><span class="lineno"> 26</span>  <span class="keywordtype">double</span> epsilon = 1e-12;</div>
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<div class="line"><a name="l00027"></a><span class="lineno"> 27</span>  <span class="keywordflow">while</span> (l <= r) {</div>
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<div class="line"><a name="l00028"></a><span class="lineno"> 28</span>  <span class="keywordtype">double</span> mid = (l + r) / 2;</div>
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<div class="line"><a name="l00029"></a><span class="lineno"> 29</span>  <span class="keywordflow">if</span> (mid * mid > <a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a>) {</div>
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<div class="line"><a name="l00030"></a><span class="lineno"> 30</span>  r = mid;</div>
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<div class="line"><a name="l00031"></a><span class="lineno"> 31</span>  } <span class="keywordflow">else</span> {</div>
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<div class="line"><a name="l00032"></a><span class="lineno"> 32</span>  <span class="keywordflow">if</span> (<a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a> - mid * mid < epsilon) {</div>
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<div class="line"><a name="l00033"></a><span class="lineno"> 33</span>  <span class="keywordflow">return</span> mid;</div>
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<div class="line"><a name="l00034"></a><span class="lineno"> 34</span>  }</div>
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<div class="line"><a name="l00035"></a><span class="lineno"> 35</span>  l = mid;</div>
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<div class="line"><a name="l00036"></a><span class="lineno"> 36</span>  }</div>
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<div class="line"><a name="l00037"></a><span class="lineno"> 37</span>  }</div>
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<div class="line"><a name="l00038"></a><span class="lineno"> 38</span>  <span class="keywordflow">return</span> -1;</div>
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<div class="line"><a name="l00039"></a><span class="lineno"> 39</span> }</div>
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<div class="ttc" id="asqrt__double_8cpp_html_ae662282ad0740d2063ac404ca3bd74fc"><div class="ttname"><a href="../../da/d24/sqrt__double_8cpp.html#ae662282ad0740d2063ac404ca3bd74fc">Sqrt</a></div><div class="ttdeci">double Sqrt(double a)</div><div class="ttdef"><b>Definition:</b> sqrt_double.cpp:16</div></div>
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<div class="ttc" id="abasic_ostream_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a></div></div>
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<div class="ttc" id="amatrix__exponentiation_8cpp_html_a9ecd98ce0064e2f7f97bb6a92f58ee72"><div class="ttname"><a href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a></div><div class="ttdeci">vector< ll > a</div><div class="ttdef"><b>Definition:</b> matrix_exponentiation.cpp:53</div></div>
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<div class="ttc" id="afixed_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/io/manip/fixed.html">std::fixed</a></div><div class="ttdeci">T fixed(T... args)</div></div>
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<div class="ttc" id="abasic_istream_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/io/basic_istream.html">std::cin</a></div></div>
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