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de/dcf/binary__exponent_8cpp.html
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<title>Algorithms_in_C++: math/binary_exponent.cpp File Reference</title>
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@@ -41,7 +41,7 @@ MathJax.Hub.Config({
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@@ -120,7 +120,13 @@ Functions</h2></td></tr>
<div class="textblock"><p >C++ Program to find Binary Exponent Iteratively and Recursively. </p>
<p >Calculate \(a^b\) in \(O(\log(b))\) by converting \(b\) to a binary number. Binary exponentiation is also known as exponentiation by squaring. </p><dl class="section note"><dt>Note</dt><dd>This is a far better approach compared to naive method which provide \(O(b)\) operations.</dd></dl>
<p>Example: 10 in base 2 is 1010. </p><p class="formulaDsp">
\begin{eqnarray*} 2^{10_d} &amp;=&amp; 2^{1010_b} = 2^8 * 2^2\\ 2^1 &amp;=&amp; 2\\ 2^2 &amp;=&amp; (2^1)^2 = 2^2 = 4\\ 2^4 &amp;=&amp; (2^2)^2 = 4^2 = 16\\ 2^8 &amp;=&amp; (2^4)^2 = 16^2 = 256\\ \end{eqnarray*}
\begin{eqnarray*}
2^{10_d} &amp;=&amp; 2^{1010_b} = 2^8 * 2^2\\
2^1 &amp;=&amp; 2\\
2^2 &amp;=&amp; (2^1)^2 = 2^2 = 4\\
2^4 &amp;=&amp; (2^2)^2 = 4^2 = 16\\
2^8 &amp;=&amp; (2^4)^2 = 16^2 = 256\\
\end{eqnarray*}
</p>
<p> Hence to calculate 2^10 we only need to multiply \(2^8\) and \(2^2\) skipping \(2^1\) and \(2^4\). </p>
</div><h2 class="groupheader">Function Documentation</h2>
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