backup/C-Plus-Plus
1
0
Fork 0
mirror of https://github.com/TheAlgorithms/C-Plus-Plus.git synced 2026-03-21 20:31:43 +08:00
Code Issues Projects Releases Wiki Activity
Files
fa852a68013de92db82a6920a622e0539c42e508
C-Plus-Plus/d1/da6/rungekutta_8cpp_source.html
realstealthninja b41207008f Documentation for f1eddf4d65
2025-07-09 10:53:16 +00:00

222 lines
22 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 1.14.0"/>
<meta name="viewport" content="width=device-width, initial-scale=1"/>
<title>TheAlgorithms/C++: numerical_methods/rungekutta.cpp Source File</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="../../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 1.14.0 -->
<script type="text/javascript">
var searchBox = new SearchBox("searchBox", "../../search/",'.html');
</script>
<script type="text/javascript">
$(function() { codefold.init(); });
</script>
<script type="text/javascript" src="../../menudata.js"></script>
<script type="text/javascript" src="../../menu.js"></script>
<script type="text/javascript">
$(function() {
initMenu('../../',true,false,'search.php','Search',true);
$(function() { init_search(); });
});
</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">
$(function(){initNavTree('d1/da6/rungekutta_8cpp_source.html','../../',''); });
</script>
<div id="container">
<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="headertitle"><div class="title">rungekutta.cpp</div></div>
</div><!--header-->
<div class="contents">
<a href="../../d1/da6/rungekutta_8cpp.html">Go to the documentation of this file.</a><div class="fragment"><div class="line"><a id="l00001" name="l00001"></a><span class="lineno"> 1</span></div>
<div class="line"><a id="l00021" name="l00021"></a><span class="lineno"> 21</span><span class="preprocessor">#include &lt;cassert&gt;</span> </div>
<div class="line"><a id="l00022" name="l00022"></a><span class="lineno"> 22</span><span class="preprocessor">#include &lt;cstdint&gt;</span></div>
<div class="line"><a id="l00023" name="l00023"></a><span class="lineno"> 23</span><span class="preprocessor">#include &lt;iostream&gt;</span> </div>
<div class="line"><a id="l00024" name="l00024"></a><span class="lineno"> 24</span><span class="preprocessor">#include &lt;vector&gt;</span> </div>
<div class="line"><a id="l00033" name="l00033"></a><span class="lineno"><a class="line" href="../../d1/da6/rungekutta_8cpp.html#a450497475f5607333f9aca8f88901579"> 33</a></span><span class="keyword">static</span> <span class="keywordtype">double</span> <a class="code hl_function" href="../../d1/da6/rungekutta_8cpp.html#a450497475f5607333f9aca8f88901579">change</a>(<span class="keywordtype">double</span> x, <span class="keywordtype">double</span> y) { <span class="keywordflow">return</span> ((x - y) / 2.0); }</div>
<div class="line"><a id="l00034" name="l00034"></a><span class="lineno"> 34</span></div>
<div class="line"><a id="l00039" name="l00039"></a><span class="lineno"> 39</span><span class="keyword">namespace </span><a class="code hl_namespace" href="../../d1/de0/namespacenumerical__methods.html">numerical_methods</a> {</div>
<div class="line"><a id="l00045" name="l00045"></a><span class="lineno"> 45</span><span class="keyword">namespace </span><a class="code hl_namespace" href="../../d2/de7/namespacerunge__kutta.html">runge_kutta</a> {</div>
<div class="foldopen" id="foldopen00057" data-start="{" data-end="}">
<div class="line"><a id="l00057" name="l00057"></a><span class="lineno"><a class="line" href="../../d1/da6/rungekutta_8cpp.html#a7b9f40c7b5e9749cc550f19be3dc8856"> 57</a></span><span class="keywordtype">double</span> <a class="code hl_function" href="../../d1/da6/rungekutta_8cpp.html#a7b9f40c7b5e9749cc550f19be3dc8856">rungeKutta</a>(<span class="keywordtype">double</span> init_x, <span class="keyword">const</span> <span class="keywordtype">double</span> &amp;init_y, <span class="keyword">const</span> <span class="keywordtype">double</span> &amp;x,</div>
<div class="line"><a id="l00058" name="l00058"></a><span class="lineno"> 58</span> <span class="keyword">const</span> <span class="keywordtype">double</span> &amp;<a class="code hl_function" href="../../d1/df3/hash__search_8cpp.html#a566eaf0ffafd50bc61e644561fd27001">h</a>) {</div>
<div class="line"><a id="l00059" name="l00059"></a><span class="lineno"> 59</span> <span class="comment">// Count number of iterations</span></div>
<div class="line"><a id="l00060" name="l00060"></a><span class="lineno"> 60</span> <span class="comment">// using step size or</span></div>
<div class="line"><a id="l00061" name="l00061"></a><span class="lineno"> 61</span> <span class="comment">// step height h</span></div>
<div class="line"><a id="l00062" name="l00062"></a><span class="lineno"> 62</span> </div>
<div class="line"><a id="l00063" name="l00063"></a><span class="lineno"> 63</span> <span class="comment">// n calucates the number of iterations</span></div>
<div class="line"><a id="l00064" name="l00064"></a><span class="lineno"> 64</span> <span class="comment">// k1, k2, k3, k4 are the Runge Kutta variables</span></div>
<div class="line"><a id="l00065" name="l00065"></a><span class="lineno"> 65</span> <span class="comment">// used for calculation of y at each iteration</span></div>
<div class="line"><a id="l00066" name="l00066"></a><span class="lineno"> 66</span> </div>
<div class="line"><a id="l00067" name="l00067"></a><span class="lineno"> 67</span> <span class="keyword">auto</span> n = <span class="keyword">static_cast&lt;</span>uint64_t<span class="keyword">&gt;</span>((x - init_x) / <a class="code hl_function" href="../../d1/df3/hash__search_8cpp.html#a566eaf0ffafd50bc61e644561fd27001">h</a>);</div>
<div class="line"><a id="l00068" name="l00068"></a><span class="lineno"> 68</span> <span class="comment">// used a vector container for the variables</span></div>
<div class="line"><a id="l00069" name="l00069"></a><span class="lineno"> 69</span> std::vector&lt;double&gt; k(4, 0.0);</div>
<div class="line"><a id="l00070" name="l00070"></a><span class="lineno"> 70</span> </div>
<div class="line"><a id="l00071" name="l00071"></a><span class="lineno"> 71</span> <span class="comment">// Iterate for number of iterations</span></div>
<div class="line"><a id="l00072" name="l00072"></a><span class="lineno"> 72</span> </div>
<div class="line"><a id="l00073" name="l00073"></a><span class="lineno"> 73</span> <span class="keywordtype">double</span> y = init_y;</div>
<div class="line"><a id="l00074" name="l00074"></a><span class="lineno"> 74</span> <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 1; i &lt;= n; ++i) {</div>
<div class="line"><a id="l00075" name="l00075"></a><span class="lineno"> 75</span> <span class="comment">// Apply Runge Kutta Formulas</span></div>
<div class="line"><a id="l00076" name="l00076"></a><span class="lineno"> 76</span> <span class="comment">// to find next value of y</span></div>
<div class="line"><a id="l00077" name="l00077"></a><span class="lineno"> 77</span> k[0] = <a class="code hl_function" href="../../d1/df3/hash__search_8cpp.html#a566eaf0ffafd50bc61e644561fd27001">h</a> * <a class="code hl_function" href="../../d1/da6/rungekutta_8cpp.html#a450497475f5607333f9aca8f88901579">change</a>(init_x, y);</div>
<div class="line"><a id="l00078" name="l00078"></a><span class="lineno"> 78</span> k[1] = <a class="code hl_function" href="../../d1/df3/hash__search_8cpp.html#a566eaf0ffafd50bc61e644561fd27001">h</a> * <a class="code hl_function" href="../../d1/da6/rungekutta_8cpp.html#a450497475f5607333f9aca8f88901579">change</a>(init_x + 0.5 * <a class="code hl_function" href="../../d1/df3/hash__search_8cpp.html#a566eaf0ffafd50bc61e644561fd27001">h</a>, y + 0.5 * k[0]);</div>
<div class="line"><a id="l00079" name="l00079"></a><span class="lineno"> 79</span> k[2] = <a class="code hl_function" href="../../d1/df3/hash__search_8cpp.html#a566eaf0ffafd50bc61e644561fd27001">h</a> * <a class="code hl_function" href="../../d1/da6/rungekutta_8cpp.html#a450497475f5607333f9aca8f88901579">change</a>(init_x + 0.5 * <a class="code hl_function" href="../../d1/df3/hash__search_8cpp.html#a566eaf0ffafd50bc61e644561fd27001">h</a>, y + 0.5 * k[1]);</div>
<div class="line"><a id="l00080" name="l00080"></a><span class="lineno"> 80</span> k[3] = <a class="code hl_function" href="../../d1/df3/hash__search_8cpp.html#a566eaf0ffafd50bc61e644561fd27001">h</a> * <a class="code hl_function" href="../../d1/da6/rungekutta_8cpp.html#a450497475f5607333f9aca8f88901579">change</a>(init_x + <a class="code hl_function" href="../../d1/df3/hash__search_8cpp.html#a566eaf0ffafd50bc61e644561fd27001">h</a>, y + k[2]);</div>
<div class="line"><a id="l00081" name="l00081"></a><span class="lineno"> 81</span> </div>
<div class="line"><a id="l00082" name="l00082"></a><span class="lineno"> 82</span> <span class="comment">// Update next value of y</span></div>
<div class="line"><a id="l00083" name="l00083"></a><span class="lineno"> 83</span> </div>
<div class="line"><a id="l00084" name="l00084"></a><span class="lineno"> 84</span> y += (1.0 / 6.0) * (k[0] + 2 * k[1] + 2 * k[2] + k[3]);</div>
<div class="line"><a id="l00085" name="l00085"></a><span class="lineno"> 85</span> </div>
<div class="line"><a id="l00086" name="l00086"></a><span class="lineno"> 86</span> <span class="comment">// Update next value of x</span></div>
<div class="line"><a id="l00087" name="l00087"></a><span class="lineno"> 87</span> </div>
<div class="line"><a id="l00088" name="l00088"></a><span class="lineno"> 88</span> init_x += <a class="code hl_function" href="../../d1/df3/hash__search_8cpp.html#a566eaf0ffafd50bc61e644561fd27001">h</a>;</div>
<div class="line"><a id="l00089" name="l00089"></a><span class="lineno"> 89</span> }</div>
<div class="line"><a id="l00090" name="l00090"></a><span class="lineno"> 90</span> </div>
<div class="line"><a id="l00091" name="l00091"></a><span class="lineno"> 91</span> <span class="keywordflow">return</span> y;</div>
<div class="line"><a id="l00092" name="l00092"></a><span class="lineno"> 92</span>}</div>
</div>
<div class="line"><a id="l00093" name="l00093"></a><span class="lineno"> 93</span>} <span class="comment">// namespace runge_kutta</span></div>
<div class="line"><a id="l00094" name="l00094"></a><span class="lineno"> 94</span>} <span class="comment">// namespace numerical_methods</span></div>
<div class="line"><a id="l00095" name="l00095"></a><span class="lineno"> 95</span></div>
<div class="foldopen" id="foldopen00100" data-start="{" data-end="}">
<div class="line"><a id="l00100" name="l00100"></a><span class="lineno"><a class="line" href="../../d1/da6/rungekutta_8cpp.html#aa8dca7b867074164d5f45b0f3851269d"> 100</a></span><span class="keyword">static</span> <span class="keywordtype">void</span> <a class="code hl_function" href="../../d1/da6/rungekutta_8cpp.html#aa8dca7b867074164d5f45b0f3851269d">test</a>() {</div>
<div class="line"><a id="l00101" name="l00101"></a><span class="lineno"> 101</span> std::cout &lt;&lt; <span class="stringliteral">&quot;The Runge Kutta function will be tested on the basis of &quot;</span></div>
<div class="line"><a id="l00102" name="l00102"></a><span class="lineno"> 102</span> <span class="stringliteral">&quot;precomputed values\n&quot;</span>;</div>
<div class="line"><a id="l00103" name="l00103"></a><span class="lineno"> 103</span> </div>
<div class="line"><a id="l00104" name="l00104"></a><span class="lineno"> 104</span> std::cout &lt;&lt; <span class="stringliteral">&quot;Test 1....&quot;</span></div>
<div class="line"><a id="l00105" name="l00105"></a><span class="lineno"> 105</span> &lt;&lt; <span class="stringliteral">&quot;\n&quot;</span>;</div>
<div class="line"><a id="l00106" name="l00106"></a><span class="lineno"> 106</span> <span class="keywordtype">double</span> valfirst = <a class="code hl_function" href="../../d1/da6/rungekutta_8cpp.html#a7b9f40c7b5e9749cc550f19be3dc8856">numerical_methods::runge_kutta::rungeKutta</a>(</div>
<div class="line"><a id="l00107" name="l00107"></a><span class="lineno"> 107</span> 2, 3, 4, 0.2); <span class="comment">// Tests the function with pre calculated values</span></div>
<div class="line"><a id="l00108" name="l00108"></a><span class="lineno"> 108</span> assert(valfirst == 3.10363932323749570);</div>
<div class="line"><a id="l00109" name="l00109"></a><span class="lineno"> 109</span> std::cout &lt;&lt; <span class="stringliteral">&quot;Passed Test 1\n&quot;</span>;</div>
<div class="line"><a id="l00110" name="l00110"></a><span class="lineno"> 110</span> </div>
<div class="line"><a id="l00111" name="l00111"></a><span class="lineno"> 111</span> std::cout &lt;&lt; <span class="stringliteral">&quot;Test 2....&quot;</span></div>
<div class="line"><a id="l00112" name="l00112"></a><span class="lineno"> 112</span> &lt;&lt; <span class="stringliteral">&quot;\n&quot;</span>;</div>
<div class="line"><a id="l00113" name="l00113"></a><span class="lineno"> 113</span> <span class="keywordtype">double</span> valsec = <a class="code hl_function" href="../../d1/da6/rungekutta_8cpp.html#a7b9f40c7b5e9749cc550f19be3dc8856">numerical_methods::runge_kutta::rungeKutta</a>(</div>
<div class="line"><a id="l00114" name="l00114"></a><span class="lineno"> 114</span> 1, 2, 5, 0.1); <span class="comment">// The value of step changed</span></div>
<div class="line"><a id="l00115" name="l00115"></a><span class="lineno"> 115</span> assert(valsec == 3.40600589380261409);</div>
<div class="line"><a id="l00116" name="l00116"></a><span class="lineno"> 116</span> std::cout &lt;&lt; <span class="stringliteral">&quot;Passed Test 2\n&quot;</span>;</div>
<div class="line"><a id="l00117" name="l00117"></a><span class="lineno"> 117</span> </div>
<div class="line"><a id="l00118" name="l00118"></a><span class="lineno"> 118</span> std::cout &lt;&lt; <span class="stringliteral">&quot;Test 3....&quot;</span></div>
<div class="line"><a id="l00119" name="l00119"></a><span class="lineno"> 119</span> &lt;&lt; <span class="stringliteral">&quot;\n&quot;</span>;</div>
<div class="line"><a id="l00120" name="l00120"></a><span class="lineno"> 120</span> <span class="keywordtype">double</span> valthird = <a class="code hl_function" href="../../d1/da6/rungekutta_8cpp.html#a7b9f40c7b5e9749cc550f19be3dc8856">numerical_methods::runge_kutta::rungeKutta</a>(</div>
<div class="line"><a id="l00121" name="l00121"></a><span class="lineno"> 121</span> -1, 3, 4, 0.1); <span class="comment">// Tested with negative value</span></div>
<div class="line"><a id="l00122" name="l00122"></a><span class="lineno"> 122</span> assert(valthird == 2.49251005860244268);</div>
<div class="line"><a id="l00123" name="l00123"></a><span class="lineno"> 123</span> std::cout &lt;&lt; <span class="stringliteral">&quot;Passed Test 3\n&quot;</span>;</div>
<div class="line"><a id="l00124" name="l00124"></a><span class="lineno"> 124</span>}</div>
</div>
<div class="line"><a id="l00125" name="l00125"></a><span class="lineno"> 125</span></div>
<div class="foldopen" id="foldopen00130" data-start="{" data-end="}">
<div class="line"><a id="l00130" name="l00130"></a><span class="lineno"><a class="line" href="../../d1/da6/rungekutta_8cpp.html#ae66f6b31b5ad750f1fe042a706a4e3d4"> 130</a></span><span class="keywordtype">int</span> <a class="code hl_function" href="../../d1/da6/rungekutta_8cpp.html#ae66f6b31b5ad750f1fe042a706a4e3d4">main</a>() {</div>
<div class="line"><a id="l00131" name="l00131"></a><span class="lineno"> 131</span> <a class="code hl_function" href="../../d1/da6/rungekutta_8cpp.html#aa8dca7b867074164d5f45b0f3851269d">test</a>(); <span class="comment">// Execute the tests</span></div>
<div class="line"><a id="l00132" name="l00132"></a><span class="lineno"> 132</span> <span class="keywordflow">return</span> 0;</div>
<div class="line"><a id="l00133" name="l00133"></a><span class="lineno"> 133</span>}</div>
</div>
<div class="ttc" id="ahash__search_8cpp_html_a566eaf0ffafd50bc61e644561fd27001"><div class="ttname"><a href="../../d1/df3/hash__search_8cpp.html#a566eaf0ffafd50bc61e644561fd27001">h</a></div><div class="ttdeci">int h(int key)</div><div class="ttdef"><b>Definition</b> <a href="../../d1/df3/hash__search_8cpp_source.html#l00045">hash_search.cpp:45</a></div></div>
<div class="ttc" id="anamespacenumerical__methods_html"><div class="ttname"><a href="../../d1/de0/namespacenumerical__methods.html">numerical_methods</a></div><div class="ttdoc">for assert</div></div>
<div class="ttc" id="anamespacerunge__kutta_html"><div class="ttname"><a href="../../d2/de7/namespacerunge__kutta.html">runge_kutta</a></div><div class="ttdoc">Functions for Runge Kutta fourth order method.</div></div>
<div class="ttc" id="arungekutta_8cpp_html_a450497475f5607333f9aca8f88901579"><div class="ttname"><a href="../../d1/da6/rungekutta_8cpp.html#a450497475f5607333f9aca8f88901579">change</a></div><div class="ttdeci">static double change(double x, double y)</div><div class="ttdoc">asserting the test functions</div><div class="ttdef"><b>Definition</b> <a href="#l00033">rungekutta.cpp:33</a></div></div>
<div class="ttc" id="arungekutta_8cpp_html_a7b9f40c7b5e9749cc550f19be3dc8856"><div class="ttname"><a href="../../d1/da6/rungekutta_8cpp.html#a7b9f40c7b5e9749cc550f19be3dc8856">numerical_methods::runge_kutta::rungeKutta</a></div><div class="ttdeci">double rungeKutta(double init_x, const double &amp;init_y, const double &amp;x, const double &amp;h)</div><div class="ttdoc">the Runge Kutta method finds the value of integration of a function in the given limits....</div><div class="ttdef"><b>Definition</b> <a href="#l00057">rungekutta.cpp:57</a></div></div>
<div class="ttc" id="arungekutta_8cpp_html_aa8dca7b867074164d5f45b0f3851269d"><div class="ttname"><a href="../../d1/da6/rungekutta_8cpp.html#aa8dca7b867074164d5f45b0f3851269d">test</a></div><div class="ttdeci">static void test()</div><div class="ttdoc">Tests to check algorithm implementation.</div><div class="ttdef"><b>Definition</b> <a href="#l00100">rungekutta.cpp:100</a></div></div>
<div class="ttc" id="arungekutta_8cpp_html_ae66f6b31b5ad750f1fe042a706a4e3d4"><div class="ttname"><a href="../../d1/da6/rungekutta_8cpp.html#ae66f6b31b5ad750f1fe042a706a4e3d4">main</a></div><div class="ttdeci">int main()</div><div class="ttdoc">Main function.</div><div class="ttdef"><b>Definition</b> <a href="#l00130">rungekutta.cpp:130</a></div></div>
</div><!-- fragment --></div><!-- contents -->
</div><!-- doc-content -->
</div><!-- container -->
<!-- start footer part -->
<div id="nav-path" class="navpath"><!-- id is needed for treeview function! -->
<ul>
<li class="navelem"><a href="../../dir_9c6faab82c22511b50177aa2e38e2780.html">numerical_methods</a></li><li class="navelem"><a href="../../d1/da6/rungekutta_8cpp.html">rungekutta.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> 1.14.0 </li>
</ul>
</div>
</body>
</html>
Reference in New Issue View Git Blame Copy Permalink