backup/C-Plus-Plus
1
0
Fork 0
mirror of https://github.com/TheAlgorithms/C-Plus-Plus.git synced 2026-03-21 04:18:34 +08:00
Code Issues Projects Releases Wiki Activity
Files
282f05f7f2934750a8e5386dd22f85268e7b367d
C-Plus-Plus/d2/d0b/fast__power_8cpp.html
realstealthninja a7bccf7d01 Documentation for 8b1eab204b
2024-11-04 12:43:05 +00:00

312 lines
21 KiB
HTML
Raw Blame History

<!-- HTML header for doxygen 1.12.0-->
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "https://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" lang="en-US">
<head>
<meta http-equiv="Content-Type" content="text/xhtml;charset=UTF-8"/>
<meta http-equiv="X-UA-Compatible" content="IE=11"/>
<meta name="generator" content="Doxygen 1.12.0"/>
<meta name="viewport" content="width=device-width, initial-scale=1"/>
<title>TheAlgorithms/C++: math/fast_power.cpp File Reference</title>
<link rel="icon" href="../../favicon.svg" type="image/x-icon" />
<link href="../../tabs.css" rel="stylesheet" type="text/css"/>
<script type="text/javascript" src="../../jquery.js"></script>
<script type="text/javascript" src="../../dynsections.js"></script>
<script type="text/javascript" src="https://cdn.jsdelivr.net/npm/@xpack-3rd-party/doxygen-awesome-css@2.2.0-1/doxygen-awesome-darkmode-toggle.js"></script>
<script type="text/javascript">
DoxygenAwesomeDarkModeToggle.init()
</script>
<script type="text/javascript" src="../../clipboard.js"></script>
<link href="../../navtree.css" rel="stylesheet" type="text/css"/>
<script type="text/javascript" src="../../navtreedata.js"></script>
<script type="text/javascript" src="../../navtree.js"></script>
<script type="text/javascript" src="../../resize.js"></script>
<script type="text/javascript" src="../../cookie.js"></script>
<link href="../../search/search.css" rel="stylesheet" type="text/css"/>
<script type="text/javascript" src="../../search/searchdata.js"></script>
<script type="text/javascript" src="../../search/search.js"></script>
<script type="text/javascript">
window.MathJax = {
options: {
ignoreHtmlClass: 'tex2jax_ignore',
processHtmlClass: 'tex2jax_process'
},
loader: {
load: ['[tex]/ams']
},
tex: {
macros: {},
packages: ['base','configmacros','ams']
}
};
</script>
<script type="text/javascript" id="MathJax-script" async="async" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js"></script>
<link href="../../doxygen.css" rel="stylesheet" type="text/css" />
<link href="../../doxygen-awesome.css" rel="stylesheet" type="text/css"/>
</head>
<body>
<div id="top"><!-- do not remove this div, it is closed by doxygen! -->
<div id="titlearea">
<table cellspacing="0" cellpadding="0">
<tbody>
<tr id="projectrow">
<td id="projectlogo"><img alt="Logo" src="../../project_logo.png"/></td>
<td id="projectalign">
<div id="projectname">TheAlgorithms/C++<span id="projectnumber">&#160;1.0.0</span>
</div>
<div id="projectbrief">All the algorithms implemented in C++</div>
</td>
</tr>
</tbody>
</table>
</div>
<!-- end header part -->
<!-- Generated by Doxygen 1.12.0 -->
<script type="text/javascript">
/* @license magnet:?xt=urn:btih:d3d9a9a6595521f9666a5e94cc830dab83b65699&amp;dn=expat.txt MIT */
var searchBox = new SearchBox("searchBox", "../../search/",'.html');
/* @license-end */
</script>
<script type="text/javascript">
/* @license magnet:?xt=urn:btih:d3d9a9a6595521f9666a5e94cc830dab83b65699&amp;dn=expat.txt MIT */
$(function() { codefold.init(1); });
/* @license-end */
</script>
<script type="text/javascript" src="../../menudata.js"></script>
<script type="text/javascript" src="../../menu.js"></script>
<script type="text/javascript">
/* @license magnet:?xt=urn:btih:d3d9a9a6595521f9666a5e94cc830dab83b65699&amp;dn=expat.txt MIT */
$(function() {
initMenu('../../',true,false,'search.php','Search',true);
$(function() { init_search(); });
});
/* @license-end */
</script>
<div id="main-nav"></div>
</div><!-- top -->
<div id="side-nav" class="ui-resizable side-nav-resizable">
<div id="nav-tree">
<div id="nav-tree-contents">
<div id="nav-sync" class="sync"></div>
</div>
</div>
<div id="splitbar" style="-moz-user-select:none;"
class="ui-resizable-handle">
</div>
</div>
<script type="text/javascript">
/* @license magnet:?xt=urn:btih:d3d9a9a6595521f9666a5e94cc830dab83b65699&amp;dn=expat.txt MIT */
$(function(){initNavTree('d2/d0b/fast__power_8cpp.html','../../'); initResizable(true); });
/* @license-end */
</script>
<div id="doc-content">
<!-- window showing the filter options -->
<div id="MSearchSelectWindow"
onmouseover="return searchBox.OnSearchSelectShow()"
onmouseout="return searchBox.OnSearchSelectHide()"
onkeydown="return searchBox.OnSearchSelectKey(event)">
</div>
<!-- iframe showing the search results (closed by default) -->
<div id="MSearchResultsWindow">
<div id="MSearchResults">
<div class="SRPage">
<div id="SRIndex">
<div id="SRResults"></div>
<div class="SRStatus" id="Loading">Loading...</div>
<div class="SRStatus" id="Searching">Searching...</div>
<div class="SRStatus" id="NoMatches">No Matches</div>
</div>
</div>
</div>
</div>
<div class="header">
<div class="summary">
<a href="#func-members">Functions</a> </div>
<div class="headertitle"><div class="title">fast_power.cpp File Reference</div></div>
</div><!--header-->
<div class="contents">
<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>
<div class="dyncontent">
<div class="center"><iframe scrolling="no" frameborder="0" src="../../db/d30/fast__power_8cpp__incl.svg" width="498" height="111"><p><b>This browser is not able to show SVG: try Firefox, Chrome, Safari, or Opera instead.</b></p></iframe></div>
</div>
</div>
<p><a href="../../d2/d0b/fast__power_8cpp_source.html">Go to the source code of this file.</a></p>
<table class="memberdecls">
<tr class="heading"><td colspan="2"><h2 class="groupheader"><a id="func-members" name="func-members"></a>
Functions</h2></td></tr>
<tr class="memitem:a8399b847626e5bba983cab7c7918b445" id="r_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="#a8399b847626e5bba983cab7c7918b445">fast_power_recursive</a> (T a, T b)</td></tr>
<tr class="separator:a8399b847626e5bba983cab7c7918b445"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:a0d4e3f0099a6975b74a314a35bd0f94d" id="r_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="#a0d4e3f0099a6975b74a314a35bd0f94d">fast_power_linear</a> (T a, T b)</td></tr>
<tr class="separator:a0d4e3f0099a6975b74a314a35bd0f94d"><td class="memSeparator" colspan="2">&#160;</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>
<tr class="separator:ae66f6b31b5ad750f1fe042a706a4e3d4"><td class="memSeparator" colspan="2">&#160;</td></tr>
</table>
<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>
<p class="definition">Definition in file <a class="el" href="../../d2/d0b/fast__power_8cpp_source.html">fast_power.cpp</a>.</p>
</div><h2 class="groupheader">Function Documentation</h2>
<a id="a0d4e3f0099a6975b74a314a35bd0f94d" name="a0d4e3f0099a6975b74a314a35bd0f94d"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a0d4e3f0099a6975b74a314a35bd0f94d">&#9670;&#160;</a></span>fast_power_linear()</h2>
<div class="memitem">
<div class="memproto">
<div class="memtemplate">
template&lt;typename T &gt; </div>
<table class="memname">
<tr>
<td class="memname">double fast_power_linear </td>
<td>(</td>
<td class="paramtype">T</td> <td class="paramname"><span class="paramname"><em>a</em></span>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">T</td> <td class="paramname"><span class="paramname"><em>b</em></span>&#160;)</td>
</tr>
</table>
</div><div class="memdoc">
<p>Same algorithm with little different formula. It still calculates in \(O(\log N)\) </p>
<p class="definition">Definition at line <a class="el" href="../../d2/d0b/fast__power_8cpp_source.html#l00050">50</a> of file <a class="el" href="../../d2/d0b/fast__power_8cpp_source.html">fast_power.cpp</a>.</p>
<div class="fragment"><div class="line"><span class="lineno"> 50</span> {</div>
<div class="line"><span class="lineno"> 51</span> <span class="comment">// negative power. a^b = 1 / (a^-b)</span></div>
<div class="line"><span class="lineno"> 52</span> <span class="keywordflow">if</span> (b &lt; 0)</div>
<div class="line"><span class="lineno"> 53</span> <span class="keywordflow">return</span> 1.0 / <a class="code hl_function" href="#a0d4e3f0099a6975b74a314a35bd0f94d">fast_power_linear</a>(a, -b);</div>
<div class="line"><span class="lineno"> 54</span> </div>
<div class="line"><span class="lineno"> 55</span> <span class="keywordtype">double</span> <a class="code hl_function" href="../../de/dc3/fibonacci__sum_8cpp.html#aadb40ac4c74a7efc0680b83eeee138aa">result</a> = 1;</div>
<div class="line"><span class="lineno"> 56</span> <span class="keywordflow">while</span> (b) {</div>
<div class="line"><span class="lineno"> 57</span> <span class="keywordflow">if</span> (b &amp; 1)</div>
<div class="line"><span class="lineno"> 58</span> <a class="code hl_function" href="../../de/dc3/fibonacci__sum_8cpp.html#aadb40ac4c74a7efc0680b83eeee138aa">result</a> = <a class="code hl_function" href="../../de/dc3/fibonacci__sum_8cpp.html#aadb40ac4c74a7efc0680b83eeee138aa">result</a> * a;</div>
<div class="line"><span class="lineno"> 59</span> a = a * a;</div>
<div class="line"><span class="lineno"> 60</span> b = b &gt;&gt; 1;</div>
<div class="line"><span class="lineno"> 61</span> }</div>
<div class="line"><span class="lineno"> 62</span> <span class="keywordflow">return</span> <a class="code hl_function" href="../../de/dc3/fibonacci__sum_8cpp.html#aadb40ac4c74a7efc0680b83eeee138aa">result</a>;</div>
<div class="line"><span class="lineno"> 63</span>}</div>
<div class="ttc" id="afast__power_8cpp_html_a0d4e3f0099a6975b74a314a35bd0f94d"><div class="ttname"><a href="#a0d4e3f0099a6975b74a314a35bd0f94d">fast_power_linear</a></div><div class="ttdeci">double fast_power_linear(T a, T b)</div><div class="ttdef"><b>Definition</b> <a href="../../d2/d0b/fast__power_8cpp_source.html#l00050">fast_power.cpp:50</a></div></div>
<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>
</div><!-- fragment -->
</div>
</div>
<a id="a8399b847626e5bba983cab7c7918b445" name="a8399b847626e5bba983cab7c7918b445"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a8399b847626e5bba983cab7c7918b445">&#9670;&#160;</a></span>fast_power_recursive()</h2>
<div class="memitem">
<div class="memproto">
<div class="memtemplate">
template&lt;typename T &gt; </div>
<table class="memname">
<tr>
<td class="memname">double fast_power_recursive </td>
<td>(</td>
<td class="paramtype">T</td> <td class="paramname"><span class="paramname"><em>a</em></span>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">T</td> <td class="paramname"><span class="paramname"><em>b</em></span>&#160;)</td>
</tr>
</table>
</div><div class="memdoc">
<p>algorithm implementation for \(a^b\) </p>
<p class="definition">Definition at line <a class="el" href="../../d2/d0b/fast__power_8cpp_source.html#l00026">26</a> of file <a class="el" href="../../d2/d0b/fast__power_8cpp_source.html">fast_power.cpp</a>.</p>
<div class="fragment"><div class="line"><span class="lineno"> 26</span> {</div>
<div class="line"><span class="lineno"> 27</span> <span class="comment">// negative power. a^b = 1 / (a^-b)</span></div>
<div class="line"><span class="lineno"> 28</span> <span class="keywordflow">if</span> (b &lt; 0)</div>
<div class="line"><span class="lineno"> 29</span> <span class="keywordflow">return</span> 1.0 / <a class="code hl_function" href="#a8399b847626e5bba983cab7c7918b445">fast_power_recursive</a>(a, -b);</div>
<div class="line"><span class="lineno"> 30</span> </div>
<div class="line"><span class="lineno"> 31</span> <span class="keywordflow">if</span> (b == 0)</div>
<div class="line"><span class="lineno"> 32</span> <span class="keywordflow">return</span> 1;</div>
<div class="line"><span class="lineno"> 33</span> T bottom = <a class="code hl_function" href="#a8399b847626e5bba983cab7c7918b445">fast_power_recursive</a>(a, b &gt;&gt; 1);</div>
<div class="line"><span class="lineno"> 34</span> <span class="comment">// Since it is integer division b/2 = (b-1)/2 where b is odd.</span></div>
<div class="line"><span class="lineno"> 35</span> <span class="comment">// Therefore, case2 is easily solved by integer division.</span></div>
<div class="line"><span class="lineno"> 36</span> </div>
<div class="line"><span class="lineno"> 37</span> <span class="keywordtype">double</span> <a class="code hl_function" href="../../de/dc3/fibonacci__sum_8cpp.html#aadb40ac4c74a7efc0680b83eeee138aa">result</a>;</div>
<div class="line"><span class="lineno"> 38</span> <span class="keywordflow">if</span> ((b &amp; 1) == 0) <span class="comment">// case1</span></div>
<div class="line"><span class="lineno"> 39</span> <a class="code hl_function" href="../../de/dc3/fibonacci__sum_8cpp.html#aadb40ac4c74a7efc0680b83eeee138aa">result</a> = bottom * bottom;</div>
<div class="line"><span class="lineno"> 40</span> <span class="keywordflow">else</span> <span class="comment">// case2</span></div>
<div class="line"><span class="lineno"> 41</span> <a class="code hl_function" href="../../de/dc3/fibonacci__sum_8cpp.html#aadb40ac4c74a7efc0680b83eeee138aa">result</a> = bottom * bottom * a;</div>
<div class="line"><span class="lineno"> 42</span> <span class="keywordflow">return</span> <a class="code hl_function" href="../../de/dc3/fibonacci__sum_8cpp.html#aadb40ac4c74a7efc0680b83eeee138aa">result</a>;</div>
<div class="line"><span class="lineno"> 43</span>}</div>
<div class="ttc" id="afast__power_8cpp_html_a8399b847626e5bba983cab7c7918b445"><div class="ttname"><a href="#a8399b847626e5bba983cab7c7918b445">fast_power_recursive</a></div><div class="ttdeci">double fast_power_recursive(T a, T b)</div><div class="ttdef"><b>Definition</b> <a href="../../d2/d0b/fast__power_8cpp_source.html#l00026">fast_power.cpp:26</a></div></div>
</div><!-- fragment -->
</div>
</div>
<a id="ae66f6b31b5ad750f1fe042a706a4e3d4" name="ae66f6b31b5ad750f1fe042a706a4e3d4"></a>
<h2 class="memtitle"><span class="permalink"><a href="#ae66f6b31b5ad750f1fe042a706a4e3d4">&#9670;&#160;</a></span>main()</h2>
<div class="memitem">
<div class="memproto">
<table class="memname">
<tr>
<td class="memname">int main </td>
<td>(</td>
<td class="paramtype">void</td> <td class="paramname"><span class="paramname"><em></em></span></td><td>)</td>
<td></td>
</tr>
</table>
</div><div class="memdoc">
<p>Main function </p>
<p class="definition">Definition at line <a class="el" href="../../d2/d0b/fast__power_8cpp_source.html#l00068">68</a> of file <a class="el" href="../../d2/d0b/fast__power_8cpp_source.html">fast_power.cpp</a>.</p>
<div class="fragment"><div class="line"><span class="lineno"> 68</span> {</div>
<div class="line"><span class="lineno"> 69</span> std::srand(std::time(<span class="keyword">nullptr</span>));</div>
<div class="line"><span class="lineno"> 70</span> std::ios_base::sync_with_stdio(<span class="keyword">false</span>);</div>
<div class="line"><span class="lineno"> 71</span> </div>
<div class="line"><span class="lineno"> 72</span> std::cout &lt;&lt; <span class="stringliteral">&quot;Testing...&quot;</span> &lt;&lt; std::endl;</div>
<div class="line"><span class="lineno"> 73</span> <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i &lt; 20; i++) {</div>
<div class="line"><span class="lineno"> 74</span> <span class="keywordtype">int</span> a = std::rand() % 20 - 10;</div>
<div class="line"><span class="lineno"> 75</span> <span class="keywordtype">int</span> b = std::rand() % 20 - 10;</div>
<div class="line"><span class="lineno"> 76</span> std::cout &lt;&lt; std::endl &lt;&lt; <span class="stringliteral">&quot;Calculating &quot;</span> &lt;&lt; a &lt;&lt; <span class="stringliteral">&quot;^&quot;</span> &lt;&lt; b &lt;&lt; std::endl;</div>
<div class="line"><span class="lineno"> 77</span> assert(<a class="code hl_function" href="#a8399b847626e5bba983cab7c7918b445">fast_power_recursive</a>(a, b) == std::pow(a, b));</div>
<div class="line"><span class="lineno"> 78</span> assert(<a class="code hl_function" href="#a0d4e3f0099a6975b74a314a35bd0f94d">fast_power_linear</a>(a, b) == std::pow(a, b));</div>
<div class="line"><span class="lineno"> 79</span> </div>
<div class="line"><span class="lineno"> 80</span> std::cout &lt;&lt; <span class="stringliteral">&quot;------ &quot;</span> &lt;&lt; a &lt;&lt; <span class="stringliteral">&quot;^&quot;</span> &lt;&lt; b &lt;&lt; <span class="stringliteral">&quot; = &quot;</span></div>
<div class="line"><span class="lineno"> 81</span> &lt;&lt; <a class="code hl_function" href="#a8399b847626e5bba983cab7c7918b445">fast_power_recursive</a>(a, b) &lt;&lt; std::endl;</div>
<div class="line"><span class="lineno"> 82</span> }</div>
<div class="line"><span class="lineno"> 83</span> </div>
<div class="line"><span class="lineno"> 84</span> int64_t a, b;</div>
<div class="line"><span class="lineno"> 85</span> std::cin &gt;&gt; a &gt;&gt; b;</div>
<div class="line"><span class="lineno"> 86</span> </div>
<div class="line"><span class="lineno"> 87</span> std::cout &lt;&lt; a &lt;&lt; <span class="stringliteral">&quot;^&quot;</span> &lt;&lt; b &lt;&lt; <span class="stringliteral">&quot; = &quot;</span> &lt;&lt; <a class="code hl_function" href="#a8399b847626e5bba983cab7c7918b445">fast_power_recursive</a>(a, b)</div>
<div class="line"><span class="lineno"> 88</span> &lt;&lt; std::endl;</div>
<div class="line"><span class="lineno"> 89</span> </div>
<div class="line"><span class="lineno"> 90</span> std::cout &lt;&lt; a &lt;&lt; <span class="stringliteral">&quot;^&quot;</span> &lt;&lt; b &lt;&lt; <span class="stringliteral">&quot; = &quot;</span> &lt;&lt; <a class="code hl_function" href="#a0d4e3f0099a6975b74a314a35bd0f94d">fast_power_linear</a>(a, b) &lt;&lt; std::endl;</div>
<div class="line"><span class="lineno"> 91</span> </div>
<div class="line"><span class="lineno"> 92</span> <span class="keywordflow">return</span> 0;</div>
<div class="line"><span class="lineno"> 93</span>}</div>
</div><!-- fragment -->
</div>
</div>
</div><!-- contents -->
</div><!-- doc-content -->
<!-- start footer part -->
<div id="nav-path" class="navpath"><!-- id is needed for treeview function! -->
<ul>
<li class="navelem"><a class="el" href="../../dir_296d53ceaeaa7e099814a6def439fe8a.html">math</a></li><li class="navelem"><a class="el" href="../../d2/d0b/fast__power_8cpp.html">fast_power.cpp</a></li>
<li class="footer">Generated by <a href="https://www.doxygen.org/index.html"><img class="footer" src="../../doxygen.svg" width="104" height="31" alt="doxygen"/></a> 1.12.0 </li>
</ul>
</div>
</body>
</html>
Reference in New Issue View Git Blame Copy Permalink