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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" target="_blank">bisection method</a> of root-finding.
<a href="#details">More...</a></p>
<div class="textblock"><code>#include &lt;cassert&gt;</code><br />
<code>#include &lt;iostream&gt;</code><br />
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Include dependency graph for sqrt_double.cpp:</div>
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<p><a href="../../da/d24/sqrt__double_8cpp_source.html">Go to the source code of this file.</a></p>
<table class="memberdecls">
<tr class="heading"><td colspan="2"><h2 id="header-func-members" class="groupheader"><a id="func-members" name="func-members"></a>
Functions</h2></td></tr>
<tr class="memitem:ae662282ad0740d2063ac404ca3bd74fc" id="r_ae662282ad0740d2063ac404ca3bd74fc"><td class="memItemLeft" align="right" valign="top">double&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="#ae662282ad0740d2063ac404ca3bd74fc">Sqrt</a> (double a)</td></tr>
<tr class="memitem:ae66f6b31b5ad750f1fe042a706a4e3d4" id="r_ae66f6b31b5ad750f1fe042a706a4e3d4"><td class="memItemLeft" align="right" valign="top">int&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="#ae66f6b31b5ad750f1fe042a706a4e3d4">main</a> ()</td></tr>
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<a name="details" id="details"></a><h2 id="header-details" class="groupheader">Detailed Description</h2>
<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" target="_blank">bisection method</a> of root-finding. </p>
<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>
<p class="definition">Definition in file <a class="el" href="../../da/d24/sqrt__double_8cpp_source.html">sqrt_double.cpp</a>.</p>
</div><a name="doc-func-members" id="doc-func-members"></a><h2 id="header-doc-func-members" class="groupheader">Function Documentation</h2>
<a id="ae66f6b31b5ad750f1fe042a706a4e3d4" name="ae66f6b31b5ad750f1fe042a706a4e3d4"></a>
<h2 class="memtitle"><span class="permalink"><a href="#ae66f6b31b5ad750f1fe042a706a4e3d4">&#9670;&#160;</a></span>main()</h2>
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<td class="memname">int main </td>
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<td class="paramtype">void</td> <td class="paramname"><span class="paramname"><em></em></span></td><td>)</td>
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<p>main function </p>
<p class="definition">Definition at line <a class="el" href="../../da/d24/sqrt__double_8cpp_source.html#l00042">42</a> of file <a class="el" href="../../da/d24/sqrt__double_8cpp_source.html">sqrt_double.cpp</a>.</p>
<div class="fragment"><div class="line"><span class="lineno"> 42</span> {</div>
<div class="line"><span class="lineno"> 43</span> <span class="keywordtype">double</span> n{};</div>
<div class="line"><span class="lineno"> 44</span> std::cin &gt;&gt; n;</div>
<div class="line"><span class="lineno"> 45</span> assert(n &gt;= 0);</div>
<div class="line"><span class="lineno"> 46</span> <span class="comment">// Change this line for a better precision</span></div>
<div class="line"><span class="lineno"> 47</span> std::cout.precision(12);</div>
<div class="line"><span class="lineno"> 48</span> std::cout &lt;&lt; std::fixed &lt;&lt; <a class="code hl_function" href="#ae662282ad0740d2063ac404ca3bd74fc">Sqrt</a>(n);</div>
<div class="line"><span class="lineno"> 49</span>}</div>
<div class="ttc" id="asqrt__double_8cpp_html_ae662282ad0740d2063ac404ca3bd74fc"><div class="ttname"><a href="#ae662282ad0740d2063ac404ca3bd74fc">Sqrt</a></div><div class="ttdeci">double Sqrt(double a)</div><div class="ttdef"><b>Definition</b> <a href="../../da/d24/sqrt__double_8cpp_source.html#l00016">sqrt_double.cpp:16</a></div></div>
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<a id="ae662282ad0740d2063ac404ca3bd74fc" name="ae662282ad0740d2063ac404ca3bd74fc"></a>
<h2 class="memtitle"><span class="permalink"><a href="#ae662282ad0740d2063ac404ca3bd74fc">&#9670;&#160;</a></span>Sqrt()</h2>
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<td class="memname">double Sqrt </td>
<td>(</td>
<td class="paramtype">double</td> <td class="paramname"><span class="paramname"><em>a</em></span></td><td>)</td>
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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>
<p class="definition">Definition at line <a class="el" href="../../da/d24/sqrt__double_8cpp_source.html#l00016">16</a> of file <a class="el" href="../../da/d24/sqrt__double_8cpp_source.html">sqrt_double.cpp</a>.</p>
<div class="fragment"><div class="line"><span class="lineno"> 16</span> {</div>
<div class="line"><span class="lineno"> 17</span> <span class="keywordflow">if</span> (a &gt; 0 &amp;&amp; a &lt; 1) {</div>
<div class="line"><span class="lineno"> 18</span> <span class="keywordflow">return</span> 1 / <a class="code hl_function" href="#ae662282ad0740d2063ac404ca3bd74fc">Sqrt</a>(1 / a);</div>
<div class="line"><span class="lineno"> 19</span> }</div>
<div class="line"><span class="lineno"> 20</span> <span class="keywordtype">double</span> <a class="code hl_function" href="../../d4/d18/composite__simpson__rule_8cpp.html#a6d8df83a6f26ce24a75d3b358b7f5b8a">l</a> = 0, r = a;</div>
<div class="line"><span class="lineno"> 21</span> <span class="comment">/* Epsilon is the precision.</span></div>
<div class="line"><span class="lineno"> 22</span><span class="comment"> A great precision is</span></div>
<div class="line"><span class="lineno"> 23</span><span class="comment"> between 1e-7 and 1e-12.</span></div>
<div class="line"><span class="lineno"> 24</span><span class="comment"> double epsilon = 1e-12;</span></div>
<div class="line"><span class="lineno"> 25</span><span class="comment"> */</span></div>
<div class="line"><span class="lineno"> 26</span> <span class="keywordtype">double</span> epsilon = 1e-12;</div>
<div class="line"><span class="lineno"> 27</span> <span class="keywordflow">while</span> (l &lt;= r) {</div>
<div class="line"><span class="lineno"> 28</span> <span class="keywordtype">double</span> mid = (<a class="code hl_function" href="../../d4/d18/composite__simpson__rule_8cpp.html#a6d8df83a6f26ce24a75d3b358b7f5b8a">l</a> + r) / 2;</div>
<div class="line"><span class="lineno"> 29</span> <span class="keywordflow">if</span> (mid * mid &gt; a) {</div>
<div class="line"><span class="lineno"> 30</span> r = mid;</div>
<div class="line"><span class="lineno"> 31</span> } <span class="keywordflow">else</span> {</div>
<div class="line"><span class="lineno"> 32</span> <span class="keywordflow">if</span> (a - mid * mid &lt; epsilon) {</div>
<div class="line"><span class="lineno"> 33</span> <span class="keywordflow">return</span> mid;</div>
<div class="line"><span class="lineno"> 34</span> }</div>
<div class="line"><span class="lineno"> 35</span> <a class="code hl_function" href="../../d4/d18/composite__simpson__rule_8cpp.html#a6d8df83a6f26ce24a75d3b358b7f5b8a">l</a> = mid;</div>
<div class="line"><span class="lineno"> 36</span> }</div>
<div class="line"><span class="lineno"> 37</span> }</div>
<div class="line"><span class="lineno"> 38</span> <span class="keywordflow">return</span> -1;</div>
<div class="line"><span class="lineno"> 39</span>}</div>
<div class="ttc" id="acomposite__simpson__rule_8cpp_html_a6d8df83a6f26ce24a75d3b358b7f5b8a"><div class="ttname"><a href="../../d4/d18/composite__simpson__rule_8cpp.html#a6d8df83a6f26ce24a75d3b358b7f5b8a">numerical_methods::simpson_method::l</a></div><div class="ttdeci">double l(double x)</div><div class="ttdoc">Another test function.</div><div class="ttdef"><b>Definition</b> <a href="../../d4/d18/composite__simpson__rule_8cpp_source.html#l00119">composite_simpson_rule.cpp:119</a></div></div>
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