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<p>Faster computation for \(a^b\).
<a href="#details">More...</a></p>
<div class="textblock"><code>#include &lt;cassert&gt;</code><br />
<code>#include &lt;cmath&gt;</code><br />
<code>#include &lt;cstdint&gt;</code><br />
<code>#include &lt;cstdlib&gt;</code><br />
<code>#include &lt;ctime&gt;</code><br />
<code>#include &lt;iostream&gt;</code><br />
</div><div class="textblock"><div class="dynheader">
Include dependency graph for fast_power.cpp:</div>
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Functions</h2></td></tr>
<tr class="memitem:a8399b847626e5bba983cab7c7918b445"><td class="memTemplParams" colspan="2">template&lt;typename T &gt; </td></tr>
<tr class="memitem:a8399b847626e5bba983cab7c7918b445"><td class="memTemplItemLeft" align="right" valign="top">double&#160;</td><td class="memTemplItemRight" valign="bottom"><a class="el" href="../../d2/d0b/fast__power_8cpp.html#a8399b847626e5bba983cab7c7918b445">fast_power_recursive</a> (T <a class="el" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a>, T b)</td></tr>
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<tr class="memitem:a0d4e3f0099a6975b74a314a35bd0f94d"><td class="memTemplParams" colspan="2">template&lt;typename T &gt; </td></tr>
<tr class="memitem:a0d4e3f0099a6975b74a314a35bd0f94d"><td class="memTemplItemLeft" align="right" valign="top">double&#160;</td><td class="memTemplItemRight" valign="bottom"><a class="el" href="../../d2/d0b/fast__power_8cpp.html#a0d4e3f0099a6975b74a314a35bd0f94d">fast_power_linear</a> (T <a class="el" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a>, T b)</td></tr>
<tr class="separator:a0d4e3f0099a6975b74a314a35bd0f94d"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:ae66f6b31b5ad750f1fe042a706a4e3d4"><td class="memItemLeft" align="right" valign="top">int&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../d2/d0b/fast__power_8cpp.html#ae66f6b31b5ad750f1fe042a706a4e3d4">main</a> ()</td></tr>
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<a name="details" id="details"></a><h2 class="groupheader">Detailed Description</h2>
<div class="textblock"><p>Faster computation for \(a^b\). </p>
<p>Program that computes \(a^b\) in \(O(logN)\) time. It is based on formula that:</p><ol type="1">
<li>if \(b\) is even: \(a^b = a^\frac{b}{2} \cdot a^\frac{b}{2} = {a^\frac{b}{2}}^2\)</li>
<li>if \(b\) is odd: \(a^b = a^\frac{b-1}{2} \cdot a^\frac{b-1}{2} \cdot a = {a^\frac{b-1}{2}}^2 \cdot a\)</li>
</ol>
<p>We can compute \(a^b\) recursively using above algorithm. </p>
</div><h2 class="groupheader">Function Documentation</h2>
<a id="a0d4e3f0099a6975b74a314a35bd0f94d"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a0d4e3f0099a6975b74a314a35bd0f94d">&#9670;&nbsp;</a></span>fast_power_linear()</h2>
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template&lt;typename T &gt; </div>
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<td class="memname">double fast_power_linear </td>
<td>(</td>
<td class="paramtype">T&#160;</td>
<td class="paramname"><em>a</em>, </td>
</tr>
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<td class="paramkey"></td>
<td></td>
<td class="paramtype">T&#160;</td>
<td class="paramname"><em>b</em>&#160;</td>
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<p>Same algorithm with little different formula. It still calculates in \(O(\log N)\) </p>
<div class="fragment"><div class="line"><a name="l00050"></a><span class="lineno"> 50</span>&#160; {</div>
<div class="line"><a name="l00051"></a><span class="lineno"> 51</span>&#160; <span class="comment">// negative power. a^b = 1 / (a^-b)</span></div>
<div class="line"><a name="l00052"></a><span class="lineno"> 52</span>&#160; <span class="keywordflow">if</span> (b &lt; 0)</div>
<div class="line"><a name="l00053"></a><span class="lineno"> 53</span>&#160; <span class="keywordflow">return</span> 1.0 / <a class="code" href="../../d2/d0b/fast__power_8cpp.html#a0d4e3f0099a6975b74a314a35bd0f94d">fast_power_linear</a>(<a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a>, -b);</div>
<div class="line"><a name="l00054"></a><span class="lineno"> 54</span>&#160; </div>
<div class="line"><a name="l00055"></a><span class="lineno"> 55</span>&#160; <span class="keywordtype">double</span> result = 1;</div>
<div class="line"><a name="l00056"></a><span class="lineno"> 56</span>&#160; <span class="keywordflow">while</span> (b) {</div>
<div class="line"><a name="l00057"></a><span class="lineno"> 57</span>&#160; <span class="keywordflow">if</span> (b &amp; 1)</div>
<div class="line"><a name="l00058"></a><span class="lineno"> 58</span>&#160; result = result * <a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a>;</div>
<div class="line"><a name="l00059"></a><span class="lineno"> 59</span>&#160; <a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a> = <a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a> * <a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a>;</div>
<div class="line"><a name="l00060"></a><span class="lineno"> 60</span>&#160; b = b &gt;&gt; 1;</div>
<div class="line"><a name="l00061"></a><span class="lineno"> 61</span>&#160; }</div>
<div class="line"><a name="l00062"></a><span class="lineno"> 62</span>&#160; <span class="keywordflow">return</span> result;</div>
<div class="line"><a name="l00063"></a><span class="lineno"> 63</span>&#160;}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#a8399b847626e5bba983cab7c7918b445">&#9670;&nbsp;</a></span>fast_power_recursive()</h2>
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template&lt;typename T &gt; </div>
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<td class="memname">double fast_power_recursive </td>
<td>(</td>
<td class="paramtype">T&#160;</td>
<td class="paramname"><em>a</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">T&#160;</td>
<td class="paramname"><em>b</em>&#160;</td>
</tr>
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<td></td>
<td>)</td>
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<p>algorithm implementation for \(a^b\) </p>
<div class="fragment"><div class="line"><a name="l00026"></a><span class="lineno"> 26</span>&#160; {</div>
<div class="line"><a name="l00027"></a><span class="lineno"> 27</span>&#160; <span class="comment">// negative power. a^b = 1 / (a^-b)</span></div>
<div class="line"><a name="l00028"></a><span class="lineno"> 28</span>&#160; <span class="keywordflow">if</span> (b &lt; 0)</div>
<div class="line"><a name="l00029"></a><span class="lineno"> 29</span>&#160; <span class="keywordflow">return</span> 1.0 / <a class="code" href="../../d2/d0b/fast__power_8cpp.html#a8399b847626e5bba983cab7c7918b445">fast_power_recursive</a>(<a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a>, -b);</div>
<div class="line"><a name="l00030"></a><span class="lineno"> 30</span>&#160; </div>
<div class="line"><a name="l00031"></a><span class="lineno"> 31</span>&#160; <span class="keywordflow">if</span> (b == 0)</div>
<div class="line"><a name="l00032"></a><span class="lineno"> 32</span>&#160; <span class="keywordflow">return</span> 1;</div>
<div class="line"><a name="l00033"></a><span class="lineno"> 33</span>&#160; T bottom = <a class="code" href="../../d2/d0b/fast__power_8cpp.html#a8399b847626e5bba983cab7c7918b445">fast_power_recursive</a>(<a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a>, b &gt;&gt; 1);</div>
<div class="line"><a name="l00034"></a><span class="lineno"> 34</span>&#160; <span class="comment">// Since it is integer division b/2 = (b-1)/2 where b is odd.</span></div>
<div class="line"><a name="l00035"></a><span class="lineno"> 35</span>&#160; <span class="comment">// Therefore, case2 is easily solved by integer division.</span></div>
<div class="line"><a name="l00036"></a><span class="lineno"> 36</span>&#160; </div>
<div class="line"><a name="l00037"></a><span class="lineno"> 37</span>&#160; <span class="keywordtype">double</span> result;</div>
<div class="line"><a name="l00038"></a><span class="lineno"> 38</span>&#160; <span class="keywordflow">if</span> ((b &amp; 1) == 0) <span class="comment">// case1</span></div>
<div class="line"><a name="l00039"></a><span class="lineno"> 39</span>&#160; result = bottom * bottom;</div>
<div class="line"><a name="l00040"></a><span class="lineno"> 40</span>&#160; <span class="keywordflow">else</span> <span class="comment">// case2</span></div>
<div class="line"><a name="l00041"></a><span class="lineno"> 41</span>&#160; result = bottom * bottom * <a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a>;</div>
<div class="line"><a name="l00042"></a><span class="lineno"> 42</span>&#160; <span class="keywordflow">return</span> result;</div>
<div class="line"><a name="l00043"></a><span class="lineno"> 43</span>&#160;}</div>
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<a id="ae66f6b31b5ad750f1fe042a706a4e3d4"></a>
<h2 class="memtitle"><span class="permalink"><a href="#ae66f6b31b5ad750f1fe042a706a4e3d4">&#9670;&nbsp;</a></span>main()</h2>
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<td class="memname">int main </td>
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<p>Main function </p>
<div class="fragment"><div class="line"><a name="l00068"></a><span class="lineno"> 68</span>&#160; {</div>
<div class="line"><a name="l00069"></a><span class="lineno"> 69</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/random/srand.html">std::srand</a>(<a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/chrono/c/time.html">std::time</a>(<span class="keyword">nullptr</span>));</div>
<div class="line"><a name="l00070"></a><span class="lineno"> 70</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/ios_base/sync_with_stdio.html">std::ios_base::sync_with_stdio</a>(<span class="keyword">false</span>);</div>
<div class="line"><a name="l00071"></a><span class="lineno"> 71</span>&#160; </div>
<div class="line"><a name="l00072"></a><span class="lineno"> 72</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; <span class="stringliteral">&quot;Testing...&quot;</span> &lt;&lt; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/endl.html">std::endl</a>;</div>
<div class="line"><a name="l00073"></a><span class="lineno"> 73</span>&#160; <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i &lt; 20; i++) {</div>
<div class="line"><a name="l00074"></a><span class="lineno"> 74</span>&#160; <span class="keywordtype">int</span> <a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a> = <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/random/rand.html">std::rand</a>() % 20 - 10;</div>
<div class="line"><a name="l00075"></a><span class="lineno"> 75</span>&#160; <span class="keywordtype">int</span> b = <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/random/rand.html">std::rand</a>() % 20 - 10;</div>
<div class="line"><a name="l00076"></a><span class="lineno"> 76</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/endl.html">std::endl</a> &lt;&lt; <span class="stringliteral">&quot;Calculating &quot;</span> &lt;&lt; <a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a> &lt;&lt; <span class="stringliteral">&quot;^&quot;</span> &lt;&lt; b &lt;&lt; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/endl.html">std::endl</a>;</div>
<div class="line"><a name="l00077"></a><span class="lineno"> 77</span>&#160; assert(<a class="code" href="../../d2/d0b/fast__power_8cpp.html#a8399b847626e5bba983cab7c7918b445">fast_power_recursive</a>(<a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a>, b) == <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/math/pow.html">std::pow</a>(<a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a>, b));</div>
<div class="line"><a name="l00078"></a><span class="lineno"> 78</span>&#160; assert(<a class="code" href="../../d2/d0b/fast__power_8cpp.html#a0d4e3f0099a6975b74a314a35bd0f94d">fast_power_linear</a>(<a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a>, b) == <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/math/pow.html">std::pow</a>(<a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a>, b));</div>
<div class="line"><a name="l00079"></a><span class="lineno"> 79</span>&#160; </div>
<div class="line"><a name="l00080"></a><span class="lineno"> 80</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; <span class="stringliteral">&quot;------ &quot;</span> &lt;&lt; <a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a> &lt;&lt; <span class="stringliteral">&quot;^&quot;</span> &lt;&lt; b &lt;&lt; <span class="stringliteral">&quot; = &quot;</span></div>
<div class="line"><a name="l00081"></a><span class="lineno"> 81</span>&#160; &lt;&lt; <a class="code" href="../../d2/d0b/fast__power_8cpp.html#a8399b847626e5bba983cab7c7918b445">fast_power_recursive</a>(<a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a>, b) &lt;&lt; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/endl.html">std::endl</a>;</div>
<div class="line"><a name="l00082"></a><span class="lineno"> 82</span>&#160; }</div>
<div class="line"><a name="l00083"></a><span class="lineno"> 83</span>&#160; </div>
<div class="line"><a name="l00084"></a><span class="lineno"> 84</span>&#160; int64_t <a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a>, b;</div>
<div class="line"><a name="l00085"></a><span class="lineno"> 85</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_istream.html">std::cin</a> &gt;&gt; <a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a> &gt;&gt; b;</div>
<div class="line"><a name="l00086"></a><span class="lineno"> 86</span>&#160; </div>
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