mirror of
https://github.com/TheAlgorithms/C-Plus-Plus.git
synced 2026-03-22 04:42:10 +08:00
506 lines
42 KiB
HTML
506 lines
42 KiB
HTML
<!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">
|
|
<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 1.9.4"/>
|
|
<meta name="viewport" content="width=device-width, initial-scale=1"/>
|
|
<title>Algorithms_in_C++: numerical_methods/brent_method_extrema.cpp File Reference</title>
|
|
<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>
|
|
<link href="../../navtree.css" rel="stylesheet" type="text/css"/>
|
|
<script type="text/javascript" src="../../resize.js"></script>
|
|
<script type="text/javascript" src="../../navtreedata.js"></script>
|
|
<script type="text/javascript" src="../../navtree.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/x-mathjax-config">
|
|
MathJax.Hub.Config({
|
|
extensions: ["tex2jax.js", "TeX/AMSmath.js", "TeX/AMSsymbols.js"],
|
|
jax: ["input/TeX","output/HTML-CSS"],
|
|
});
|
|
</script>
|
|
<script type="text/javascript" async="async" src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.7/MathJax.js?config=TeX-MML-AM_CHTML/MathJax.js"></script>
|
|
<link href="../../doxygen.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="projectalign">
|
|
<div id="projectname">Algorithms_in_C++<span id="projectnumber"> 1.0.0</span>
|
|
</div>
|
|
<div id="projectbrief">Set of algorithms implemented in C++.</div>
|
|
</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</div>
|
|
<!-- end header part -->
|
|
<!-- Generated by Doxygen 1.9.4 -->
|
|
<script type="text/javascript">
|
|
/* @license magnet:?xt=urn:btih:d3d9a9a6595521f9666a5e94cc830dab83b65699&dn=expat.txt MIT */
|
|
var searchBox = new SearchBox("searchBox", "../../search",'Search','.html');
|
|
/* @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&dn=expat.txt MIT */
|
|
$(function() {
|
|
initMenu('../../',true,false,'search.php','Search');
|
|
$(document).ready(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&dn=expat.txt MIT */
|
|
$(document).ready(function(){initNavTree('db/d01/brent__method__extrema_8cpp.html','../../'); initResizable(); });
|
|
/* @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">
|
|
<iframe src="javascript:void(0)" frameborder="0"
|
|
name="MSearchResults" id="MSearchResults">
|
|
</iframe>
|
|
</div>
|
|
|
|
<div class="header">
|
|
<div class="summary">
|
|
<a href="#define-members">Macros</a> |
|
|
<a href="#func-members">Functions</a> </div>
|
|
<div class="headertitle"><div class="title">brent_method_extrema.cpp File Reference</div></div>
|
|
</div><!--header-->
|
|
<div class="contents">
|
|
|
|
<p>Find real extrema of a univariate real function in a given interval using <a href="https://en.wikipedia.org/wiki/Brent%27s_method" target="_blank">Brent's method</a>.
|
|
<a href="#details">More...</a></p>
|
|
<div class="textblock"><code>#include <cassert></code><br />
|
|
<code>#include <cmath></code><br />
|
|
<code>#include <functional></code><br />
|
|
<code>#include <iostream></code><br />
|
|
<code>#include <limits></code><br />
|
|
</div><div class="textblock"><div class="dynheader">
|
|
Include dependency graph for brent_method_extrema.cpp:</div>
|
|
<div class="dyncontent">
|
|
<div class="center"><iframe scrolling="no" frameborder="0" src="../../d9/d0e/brent__method__extrema_8cpp__incl.svg" width="438" height="127"><p><b>This browser is not able to show SVG: try Firefox, Chrome, Safari, or Opera instead.</b></p></iframe>
|
|
</div>
|
|
</div>
|
|
</div><table class="memberdecls">
|
|
<tr class="heading"><td colspan="2"><h2 class="groupheader"><a id="define-members" name="define-members"></a>
|
|
Macros</h2></td></tr>
|
|
<tr class="memitem:a525335710b53cb064ca56b936120431e"><td class="memItemLeft" align="right" valign="top"><a id="a525335710b53cb064ca56b936120431e" name="a525335710b53cb064ca56b936120431e"></a>
|
|
#define </td><td class="memItemRight" valign="bottom"><b>_USE_MATH_DEFINES</b></td></tr>
|
|
<tr class="memdesc:a525335710b53cb064ca56b936120431e"><td class="mdescLeft"> </td><td class="mdescRight">required for MS Visual C++ <br /></td></tr>
|
|
<tr class="separator:a525335710b53cb064ca56b936120431e"><td class="memSeparator" colspan="2"> </td></tr>
|
|
<tr class="memitem:a002b2f4894492820fe708b1b7e7c5e70"><td class="memItemLeft" align="right" valign="top">#define </td><td class="memItemRight" valign="bottom"><a class="el" href="../../db/d01/brent__method__extrema_8cpp.html#a002b2f4894492820fe708b1b7e7c5e70">EPSILON</a></td></tr>
|
|
<tr class="memdesc:a002b2f4894492820fe708b1b7e7c5e70"><td class="mdescLeft"> </td><td class="mdescRight">system accuracy limit <a href="../../db/d01/brent__method__extrema_8cpp.html#a002b2f4894492820fe708b1b7e7c5e70">More...</a><br /></td></tr>
|
|
<tr class="separator:a002b2f4894492820fe708b1b7e7c5e70"><td class="memSeparator" colspan="2"> </td></tr>
|
|
</table><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:a1aa76a6d5fd4d333f9072beff1dc486b"><td class="memItemLeft" align="right" valign="top">double </td><td class="memItemRight" valign="bottom"><a class="el" href="../../db/d01/brent__method__extrema_8cpp.html#a1aa76a6d5fd4d333f9072beff1dc486b">get_minima</a> (const <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/utility/functional/function.html">std::function</a>< double(double)> &f, double lim_a, double lim_b)</td></tr>
|
|
<tr class="memdesc:a1aa76a6d5fd4d333f9072beff1dc486b"><td class="mdescLeft"> </td><td class="mdescRight">Get the real root of a function in the given interval. <a href="../../db/d01/brent__method__extrema_8cpp.html#a1aa76a6d5fd4d333f9072beff1dc486b">More...</a><br /></td></tr>
|
|
<tr class="separator:a1aa76a6d5fd4d333f9072beff1dc486b"><td class="memSeparator" colspan="2"> </td></tr>
|
|
<tr class="memitem:a1440a7779ac56f47a3f355ce4a8c7da0"><td class="memItemLeft" align="right" valign="top">void </td><td class="memItemRight" valign="bottom"><a class="el" href="../../db/d01/brent__method__extrema_8cpp.html#a1440a7779ac56f47a3f355ce4a8c7da0">test1</a> ()</td></tr>
|
|
<tr class="memdesc:a1440a7779ac56f47a3f355ce4a8c7da0"><td class="mdescLeft"> </td><td class="mdescRight">Test function to find root for the function \(f(x)= (x-2)^2\) in the interval \([1,5]\) <br />
|
|
Expected result = 2. <a href="../../db/d01/brent__method__extrema_8cpp.html#a1440a7779ac56f47a3f355ce4a8c7da0">More...</a><br /></td></tr>
|
|
<tr class="separator:a1440a7779ac56f47a3f355ce4a8c7da0"><td class="memSeparator" colspan="2"> </td></tr>
|
|
<tr class="memitem:a0283886819c7c140a023582b7269e2d0"><td class="memItemLeft" align="right" valign="top">void </td><td class="memItemRight" valign="bottom"><a class="el" href="../../db/d01/brent__method__extrema_8cpp.html#a0283886819c7c140a023582b7269e2d0">test2</a> ()</td></tr>
|
|
<tr class="memdesc:a0283886819c7c140a023582b7269e2d0"><td class="mdescLeft"> </td><td class="mdescRight">Test function to find root for the function \(f(x)= x^{\frac{1}{x}}\) in the interval \([-2,10]\) <br />
|
|
Expected result: \(e\approx 2.71828182845904509\). <a href="../../db/d01/brent__method__extrema_8cpp.html#a0283886819c7c140a023582b7269e2d0">More...</a><br /></td></tr>
|
|
<tr class="separator:a0283886819c7c140a023582b7269e2d0"><td class="memSeparator" colspan="2"> </td></tr>
|
|
<tr class="memitem:a6d0455dd5c30adda100e95f0423c786e"><td class="memItemLeft" align="right" valign="top">void </td><td class="memItemRight" valign="bottom"><a class="el" href="../../db/d01/brent__method__extrema_8cpp.html#a6d0455dd5c30adda100e95f0423c786e">test3</a> ()</td></tr>
|
|
<tr class="memdesc:a6d0455dd5c30adda100e95f0423c786e"><td class="mdescLeft"> </td><td class="mdescRight">Test function to find <em>maxima</em> for the function \(f(x)= \cos x\) in the interval \([0,12]\) <br />
|
|
Expected result: \(\pi\approx 3.14159265358979312\). <a href="../../db/d01/brent__method__extrema_8cpp.html#a6d0455dd5c30adda100e95f0423c786e">More...</a><br /></td></tr>
|
|
<tr class="separator:a6d0455dd5c30adda100e95f0423c786e"><td class="memSeparator" colspan="2"> </td></tr>
|
|
<tr class="memitem:ae66f6b31b5ad750f1fe042a706a4e3d4"><td class="memItemLeft" align="right" valign="top">int </td><td class="memItemRight" valign="bottom"><a class="el" href="../../db/d01/brent__method__extrema_8cpp.html#ae66f6b31b5ad750f1fe042a706a4e3d4">main</a> ()</td></tr>
|
|
<tr class="separator:ae66f6b31b5ad750f1fe042a706a4e3d4"><td class="memSeparator" colspan="2"> </td></tr>
|
|
</table>
|
|
<a name="details" id="details"></a><h2 class="groupheader">Detailed Description</h2>
|
|
<div class="textblock"><p >Find real extrema of a univariate real function in a given interval using <a href="https://en.wikipedia.org/wiki/Brent%27s_method" target="_blank">Brent's method</a>. </p>
|
|
<p >Refer the algorithm discoverer's publication <a href="https://maths-people.anu.edu.au/~brent/pd/rpb011i.pdf" target="_blank">online</a> and also associated book: </p><blockquote class="doxtable">
|
|
<p >‍R. P. Brent, Algorithms for Minimization without Derivatives, Prentice-Hall, Englewood Cliffs, New Jersey, 1973 </p>
|
|
</blockquote>
|
|
<dl class="section see"><dt>See also</dt><dd><a class="el" href="../../d6/d7a/golden__search__extrema_8cpp.html" title="Find extrema of a univariate real function in a given interval using golden section search algorithm.">golden_search_extrema.cpp</a></dd></dl>
|
|
<dl class="section author"><dt>Author</dt><dd><a href="https://github.com/kvedala" target="_blank">Krishna Vedala</a> </dd></dl>
|
|
</div><h2 class="groupheader">Macro Definition Documentation</h2>
|
|
<a id="a002b2f4894492820fe708b1b7e7c5e70" name="a002b2f4894492820fe708b1b7e7c5e70"></a>
|
|
<h2 class="memtitle"><span class="permalink"><a href="#a002b2f4894492820fe708b1b7e7c5e70">◆ </a></span>EPSILON</h2>
|
|
|
|
<div class="memitem">
|
|
<div class="memproto">
|
|
<table class="memname">
|
|
<tr>
|
|
<td class="memname">#define EPSILON</td>
|
|
</tr>
|
|
</table>
|
|
</div><div class="memdoc">
|
|
<b>Value:</b><div class="fragment"><div class="line"> <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/math/sqrt.html">std::sqrt</a>( \</div>
|
|
<div class="line"> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/types/numeric_limits.html">std::numeric_limits<double>::epsilon</a>())</div>
|
|
<div class="ttc" id="anumeric_limits_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/types/numeric_limits.html">std::numeric_limits</a></div></div>
|
|
<div class="ttc" id="asqrt_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/numeric/math/sqrt.html">std::sqrt</a></div><div class="ttdeci">T sqrt(T... args)</div></div>
|
|
</div><!-- fragment -->
|
|
<p>system accuracy limit </p>
|
|
|
|
</div>
|
|
</div>
|
|
<h2 class="groupheader">Function Documentation</h2>
|
|
<a id="a1aa76a6d5fd4d333f9072beff1dc486b" name="a1aa76a6d5fd4d333f9072beff1dc486b"></a>
|
|
<h2 class="memtitle"><span class="permalink"><a href="#a1aa76a6d5fd4d333f9072beff1dc486b">◆ </a></span>get_minima()</h2>
|
|
|
|
<div class="memitem">
|
|
<div class="memproto">
|
|
<table class="memname">
|
|
<tr>
|
|
<td class="memname">double get_minima </td>
|
|
<td>(</td>
|
|
<td class="paramtype">const <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/utility/functional/function.html">std::function</a>< double(double)> & </td>
|
|
<td class="paramname"><em>f</em>, </td>
|
|
</tr>
|
|
<tr>
|
|
<td class="paramkey"></td>
|
|
<td></td>
|
|
<td class="paramtype">double </td>
|
|
<td class="paramname"><em>lim_a</em>, </td>
|
|
</tr>
|
|
<tr>
|
|
<td class="paramkey"></td>
|
|
<td></td>
|
|
<td class="paramtype">double </td>
|
|
<td class="paramname"><em>lim_b</em> </td>
|
|
</tr>
|
|
<tr>
|
|
<td></td>
|
|
<td>)</td>
|
|
<td></td><td></td>
|
|
</tr>
|
|
</table>
|
|
</div><div class="memdoc">
|
|
|
|
<p>Get the real root of a function in the given interval. </p>
|
|
<dl class="params"><dt>Parameters</dt><dd>
|
|
<table class="params">
|
|
<tr><td class="paramname">f</td><td>function to get root for </td></tr>
|
|
<tr><td class="paramname">lim_a</td><td>lower limit of search window </td></tr>
|
|
<tr><td class="paramname">lim_b</td><td>upper limit of search window </td></tr>
|
|
</table>
|
|
</dd>
|
|
</dl>
|
|
<dl class="section return"><dt>Returns</dt><dd>root found in the interval </dd></dl>
|
|
<div class="fragment"><div class="line"><span class="lineno"> 36</span> {</div>
|
|
<div class="line"><span class="lineno"> 37</span> uint32_t iters = 0;</div>
|
|
<div class="line"><span class="lineno"> 38</span> </div>
|
|
<div class="line"><span class="lineno"> 39</span> <span class="keywordflow">if</span> (lim_a > lim_b) {</div>
|
|
<div class="line"><span class="lineno"> 40</span> <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/algorithm/swap.html">std::swap</a>(lim_a, lim_b);</div>
|
|
<div class="line"><span class="lineno"> 41</span> } <span class="keywordflow">else</span> <span class="keywordflow">if</span> (std::abs(lim_a - lim_b) <= <a class="code hl_define" href="../../db/d01/brent__method__extrema_8cpp.html#a002b2f4894492820fe708b1b7e7c5e70">EPSILON</a>) {</div>
|
|
<div class="line"><span class="lineno"> 42</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cerr</a> << <span class="stringliteral">"Search range must be greater than "</span> << <a class="code hl_define" href="../../db/d01/brent__method__extrema_8cpp.html#a002b2f4894492820fe708b1b7e7c5e70">EPSILON</a> << <span class="stringliteral">"\n"</span>;</div>
|
|
<div class="line"><span class="lineno"> 43</span> <span class="keywordflow">return</span> lim_a;</div>
|
|
<div class="line"><span class="lineno"> 44</span> }</div>
|
|
<div class="line"><span class="lineno"> 45</span> </div>
|
|
<div class="line"><span class="lineno"> 46</span> <span class="comment">// golden ratio value</span></div>
|
|
<div class="line"><span class="lineno"> 47</span> <span class="keyword">const</span> <span class="keywordtype">double</span> M_GOLDEN_RATIO = (3.f - <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/math/sqrt.html">std::sqrt</a>(5.f)) / 2.f;</div>
|
|
<div class="line"><span class="lineno"> 48</span> </div>
|
|
<div class="line"><span class="lineno"> 49</span> <span class="keywordtype">double</span> v = lim_a + M_GOLDEN_RATIO * (lim_b - lim_a);</div>
|
|
<div class="line"><span class="lineno"> 50</span> <span class="keywordtype">double</span> u, w = v, x = v;</div>
|
|
<div class="line"><span class="lineno"> 51</span> <span class="keywordtype">double</span> fu, fv = <a class="code hl_function" href="../../d4/d18/composite__simpson__rule_8cpp.html#a4251b4df4748a0b9c43a48f61bdd2397">f</a>(v);</div>
|
|
<div class="line"><span class="lineno"> 52</span> <span class="keywordtype">double</span> fw = fv, fx = fv;</div>
|
|
<div class="line"><span class="lineno"> 53</span> </div>
|
|
<div class="line"><span class="lineno"> 54</span> <span class="keywordtype">double</span> mid_point = (lim_a + lim_b) / 2.f;</div>
|
|
<div class="line"><span class="lineno"> 55</span> <span class="keywordtype">double</span> p = 0, q = 0, r = 0;</div>
|
|
<div class="line"><span class="lineno"> 56</span> </div>
|
|
<div class="line"><span class="lineno"> 57</span> <span class="keywordtype">double</span> d, e = 0;</div>
|
|
<div class="line"><span class="lineno"> 58</span> <span class="keywordtype">double</span> tolerance, tolerance2;</div>
|
|
<div class="line"><span class="lineno"> 59</span> </div>
|
|
<div class="line"><span class="lineno"> 60</span> <span class="keywordflow">do</span> {</div>
|
|
<div class="line"><span class="lineno"> 61</span> mid_point = (lim_a + lim_b) / 2.f;</div>
|
|
<div class="line"><span class="lineno"> 62</span> tolerance = <a class="code hl_define" href="../../db/d01/brent__method__extrema_8cpp.html#a002b2f4894492820fe708b1b7e7c5e70">EPSILON</a> * std::abs(x);</div>
|
|
<div class="line"><span class="lineno"> 63</span> tolerance2 = 2 * tolerance;</div>
|
|
<div class="line"><span class="lineno"> 64</span> </div>
|
|
<div class="line"><span class="lineno"> 65</span> <span class="keywordflow">if</span> (std::abs(e) > tolerance2) {</div>
|
|
<div class="line"><span class="lineno"> 66</span> <span class="comment">// fit parabola</span></div>
|
|
<div class="line"><span class="lineno"> 67</span> r = (x - w) * (fx - fv);</div>
|
|
<div class="line"><span class="lineno"> 68</span> q = (x - v) * (fx - fw);</div>
|
|
<div class="line"><span class="lineno"> 69</span> p = (x - v) * q - (x - w) * r;</div>
|
|
<div class="line"><span class="lineno"> 70</span> q = 2.f * (q - r);</div>
|
|
<div class="line"><span class="lineno"> 71</span> <span class="keywordflow">if</span> (q > 0)</div>
|
|
<div class="line"><span class="lineno"> 72</span> p = -p;</div>
|
|
<div class="line"><span class="lineno"> 73</span> <span class="keywordflow">else</span></div>
|
|
<div class="line"><span class="lineno"> 74</span> q = -q;</div>
|
|
<div class="line"><span class="lineno"> 75</span> r = e;</div>
|
|
<div class="line"><span class="lineno"> 76</span> e = d;</div>
|
|
<div class="line"><span class="lineno"> 77</span> }</div>
|
|
<div class="line"><span class="lineno"> 78</span> </div>
|
|
<div class="line"><span class="lineno"> 79</span> <span class="keywordflow">if</span> (std::abs(p) < std::abs(0.5 * q * r) && p < q * (lim_b - x)) {</div>
|
|
<div class="line"><span class="lineno"> 80</span> <span class="comment">// parabolic interpolation step</span></div>
|
|
<div class="line"><span class="lineno"> 81</span> d = p / q;</div>
|
|
<div class="line"><span class="lineno"> 82</span> u = x + d;</div>
|
|
<div class="line"><span class="lineno"> 83</span> <span class="keywordflow">if</span> (u - lim_a < tolerance2 || lim_b - u < tolerance2)</div>
|
|
<div class="line"><span class="lineno"> 84</span> d = x < mid_point ? tolerance : -tolerance;</div>
|
|
<div class="line"><span class="lineno"> 85</span> } <span class="keywordflow">else</span> {</div>
|
|
<div class="line"><span class="lineno"> 86</span> <span class="comment">// golden section interpolation step</span></div>
|
|
<div class="line"><span class="lineno"> 87</span> e = (x < mid_point ? lim_b : lim_a) - x;</div>
|
|
<div class="line"><span class="lineno"> 88</span> d = M_GOLDEN_RATIO * e;</div>
|
|
<div class="line"><span class="lineno"> 89</span> }</div>
|
|
<div class="line"><span class="lineno"> 90</span> </div>
|
|
<div class="line"><span class="lineno"> 91</span> <span class="comment">// evaluate not too close to x</span></div>
|
|
<div class="line"><span class="lineno"> 92</span> <span class="keywordflow">if</span> (std::abs(d) >= tolerance)</div>
|
|
<div class="line"><span class="lineno"> 93</span> u = d;</div>
|
|
<div class="line"><span class="lineno"> 94</span> <span class="keywordflow">else</span> <span class="keywordflow">if</span> (d > 0)</div>
|
|
<div class="line"><span class="lineno"> 95</span> u = tolerance;</div>
|
|
<div class="line"><span class="lineno"> 96</span> <span class="keywordflow">else</span></div>
|
|
<div class="line"><span class="lineno"> 97</span> u = -tolerance;</div>
|
|
<div class="line"><span class="lineno"> 98</span> u += x;</div>
|
|
<div class="line"><span class="lineno"> 99</span> fu = <a class="code hl_function" href="../../d4/d18/composite__simpson__rule_8cpp.html#a4251b4df4748a0b9c43a48f61bdd2397">f</a>(u);</div>
|
|
<div class="line"><span class="lineno"> 100</span> </div>
|
|
<div class="line"><span class="lineno"> 101</span> <span class="comment">// update variables</span></div>
|
|
<div class="line"><span class="lineno"> 102</span> <span class="keywordflow">if</span> (fu <= fx) {</div>
|
|
<div class="line"><span class="lineno"> 103</span> <span class="keywordflow">if</span> (u < x)</div>
|
|
<div class="line"><span class="lineno"> 104</span> lim_b = x;</div>
|
|
<div class="line"><span class="lineno"> 105</span> <span class="keywordflow">else</span></div>
|
|
<div class="line"><span class="lineno"> 106</span> lim_a = x;</div>
|
|
<div class="line"><span class="lineno"> 107</span> v = w;</div>
|
|
<div class="line"><span class="lineno"> 108</span> fv = fw;</div>
|
|
<div class="line"><span class="lineno"> 109</span> w = x;</div>
|
|
<div class="line"><span class="lineno"> 110</span> fw = fx;</div>
|
|
<div class="line"><span class="lineno"> 111</span> x = u;</div>
|
|
<div class="line"><span class="lineno"> 112</span> fx = fu;</div>
|
|
<div class="line"><span class="lineno"> 113</span> } <span class="keywordflow">else</span> {</div>
|
|
<div class="line"><span class="lineno"> 114</span> <span class="keywordflow">if</span> (u < x)</div>
|
|
<div class="line"><span class="lineno"> 115</span> lim_a = u;</div>
|
|
<div class="line"><span class="lineno"> 116</span> <span class="keywordflow">else</span></div>
|
|
<div class="line"><span class="lineno"> 117</span> lim_b = u;</div>
|
|
<div class="line"><span class="lineno"> 118</span> <span class="keywordflow">if</span> (fu <= fw || x == w) {</div>
|
|
<div class="line"><span class="lineno"> 119</span> v = w;</div>
|
|
<div class="line"><span class="lineno"> 120</span> fv = fw;</div>
|
|
<div class="line"><span class="lineno"> 121</span> w = u;</div>
|
|
<div class="line"><span class="lineno"> 122</span> fw = fu;</div>
|
|
<div class="line"><span class="lineno"> 123</span> } <span class="keywordflow">else</span> <span class="keywordflow">if</span> (fu <= fv || v == x || v == w) {</div>
|
|
<div class="line"><span class="lineno"> 124</span> v = u;</div>
|
|
<div class="line"><span class="lineno"> 125</span> fv = fu;</div>
|
|
<div class="line"><span class="lineno"> 126</span> }</div>
|
|
<div class="line"><span class="lineno"> 127</span> }</div>
|
|
<div class="line"><span class="lineno"> 128</span> </div>
|
|
<div class="line"><span class="lineno"> 129</span> iters++;</div>
|
|
<div class="line"><span class="lineno"> 130</span> } <span class="keywordflow">while</span> (std::abs(x - mid_point) > (tolerance - (lim_b - lim_a) / 2.f));</div>
|
|
<div class="line"><span class="lineno"> 131</span> </div>
|
|
<div class="line"><span class="lineno"> 132</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> << <span class="stringliteral">" (iters: "</span> << iters << <span class="stringliteral">") "</span>;</div>
|
|
<div class="line"><span class="lineno"> 133</span> </div>
|
|
<div class="line"><span class="lineno"> 134</span> <span class="keywordflow">return</span> x;</div>
|
|
<div class="line"><span class="lineno"> 135</span>}</div>
|
|
<div class="ttc" id="abasic_ostream_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cerr</a></div></div>
|
|
<div class="ttc" id="abrent__method__extrema_8cpp_html_a002b2f4894492820fe708b1b7e7c5e70"><div class="ttname"><a href="../../db/d01/brent__method__extrema_8cpp.html#a002b2f4894492820fe708b1b7e7c5e70">EPSILON</a></div><div class="ttdeci">#define EPSILON</div><div class="ttdoc">system accuracy limit</div><div class="ttdef"><b>Definition:</b> brent_method_extrema.cpp:23</div></div>
|
|
<div class="ttc" id="acomposite__simpson__rule_8cpp_html_a4251b4df4748a0b9c43a48f61bdd2397"><div class="ttname"><a href="../../d4/d18/composite__simpson__rule_8cpp.html#a4251b4df4748a0b9c43a48f61bdd2397">numerical_methods::simpson_method::f</a></div><div class="ttdeci">double f(double x)</div><div class="ttdoc">A function f(x) that will be used to test the method.</div><div class="ttdef"><b>Definition:</b> composite_simpson_rule.cpp:113</div></div>
|
|
<div class="ttc" id="aswap_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/algorithm/swap.html">std::swap</a></div><div class="ttdeci">T swap(T... args)</div></div>
|
|
</div><!-- fragment --><div class="dynheader">
|
|
Here is the call graph for this function:</div>
|
|
<div class="dyncontent">
|
|
<div class="center"><iframe scrolling="no" frameborder="0" src="../../db/d01/brent__method__extrema_8cpp_a1aa76a6d5fd4d333f9072beff1dc486b_cgraph.svg" width="227" height="88"><p><b>This browser is not able to show SVG: try Firefox, Chrome, Safari, or Opera instead.</b></p></iframe>
|
|
</div>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<a id="ae66f6b31b5ad750f1fe042a706a4e3d4" name="ae66f6b31b5ad750f1fe042a706a4e3d4"></a>
|
|
<h2 class="memtitle"><span class="permalink"><a href="#ae66f6b31b5ad750f1fe042a706a4e3d4">◆ </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"></td><td>)</td>
|
|
<td></td>
|
|
</tr>
|
|
</table>
|
|
</div><div class="memdoc">
|
|
<p >Main function </p>
|
|
<div class="fragment"><div class="line"><span class="lineno"> 204</span> {</div>
|
|
<div class="line"><span class="lineno"> 205</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a>.precision(18);</div>
|
|
<div class="line"><span class="lineno"> 206</span> </div>
|
|
<div class="line"><span class="lineno"> 207</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> << <span class="stringliteral">"Computations performed with machine epsilon: "</span> << <a class="code hl_define" href="../../db/d01/brent__method__extrema_8cpp.html#a002b2f4894492820fe708b1b7e7c5e70">EPSILON</a></div>
|
|
<div class="line"><span class="lineno"> 208</span> << <span class="stringliteral">"\n"</span>;</div>
|
|
<div class="line"><span class="lineno"> 209</span> </div>
|
|
<div class="line"><span class="lineno"> 210</span> <a class="code hl_function" href="../../db/d01/brent__method__extrema_8cpp.html#a1440a7779ac56f47a3f355ce4a8c7da0">test1</a>();</div>
|
|
<div class="line"><span class="lineno"> 211</span> <a class="code hl_function" href="../../db/d01/brent__method__extrema_8cpp.html#a0283886819c7c140a023582b7269e2d0">test2</a>();</div>
|
|
<div class="line"><span class="lineno"> 212</span> <a class="code hl_function" href="../../db/d01/brent__method__extrema_8cpp.html#a6d0455dd5c30adda100e95f0423c786e">test3</a>();</div>
|
|
<div class="line"><span class="lineno"> 213</span> </div>
|
|
<div class="line"><span class="lineno"> 214</span> <span class="keywordflow">return</span> 0;</div>
|
|
<div class="line"><span class="lineno"> 215</span>}</div>
|
|
<div class="ttc" id="abrent__method__extrema_8cpp_html_a0283886819c7c140a023582b7269e2d0"><div class="ttname"><a href="../../db/d01/brent__method__extrema_8cpp.html#a0283886819c7c140a023582b7269e2d0">test2</a></div><div class="ttdeci">void test2()</div><div class="ttdoc">Test function to find root for the function in the interval Expected result: .</div><div class="ttdef"><b>Definition:</b> brent_method_extrema.cpp:165</div></div>
|
|
<div class="ttc" id="abrent__method__extrema_8cpp_html_a1440a7779ac56f47a3f355ce4a8c7da0"><div class="ttname"><a href="../../db/d01/brent__method__extrema_8cpp.html#a1440a7779ac56f47a3f355ce4a8c7da0">test1</a></div><div class="ttdeci">void test1()</div><div class="ttdoc">Test function to find root for the function in the interval Expected result = 2.</div><div class="ttdef"><b>Definition:</b> brent_method_extrema.cpp:143</div></div>
|
|
<div class="ttc" id="abrent__method__extrema_8cpp_html_a6d0455dd5c30adda100e95f0423c786e"><div class="ttname"><a href="../../db/d01/brent__method__extrema_8cpp.html#a6d0455dd5c30adda100e95f0423c786e">test3</a></div><div class="ttdeci">void test3()</div><div class="ttdoc">Test function to find maxima for the function in the interval Expected result: .</div><div class="ttdef"><b>Definition:</b> brent_method_extrema.cpp:188</div></div>
|
|
</div><!-- fragment --><div class="dynheader">
|
|
Here is the call graph for this function:</div>
|
|
<div class="dyncontent">
|
|
<div class="center"><iframe scrolling="no" frameborder="0" src="../../db/d01/brent__method__extrema_8cpp_ae66f6b31b5ad750f1fe042a706a4e3d4_cgraph.svg" width="424" height="139"><p><b>This browser is not able to show SVG: try Firefox, Chrome, Safari, or Opera instead.</b></p></iframe>
|
|
</div>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<a id="a1440a7779ac56f47a3f355ce4a8c7da0" name="a1440a7779ac56f47a3f355ce4a8c7da0"></a>
|
|
<h2 class="memtitle"><span class="permalink"><a href="#a1440a7779ac56f47a3f355ce4a8c7da0">◆ </a></span>test1()</h2>
|
|
|
|
<div class="memitem">
|
|
<div class="memproto">
|
|
<table class="memname">
|
|
<tr>
|
|
<td class="memname">void test1 </td>
|
|
<td>(</td>
|
|
<td class="paramname"></td><td>)</td>
|
|
<td></td>
|
|
</tr>
|
|
</table>
|
|
</div><div class="memdoc">
|
|
|
|
<p>Test function to find root for the function \(f(x)= (x-2)^2\) in the interval \([1,5]\) <br />
|
|
Expected result = 2. </p>
|
|
<div class="fragment"><div class="line"><span class="lineno"> 143</span> {</div>
|
|
<div class="line"><span class="lineno"> 144</span> <span class="comment">// define the function to minimize as a lambda function</span></div>
|
|
<div class="line"><span class="lineno"> 145</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/utility/functional/function.html">std::function</a><double(<span class="keywordtype">double</span>)> f1 = [](<span class="keywordtype">double</span> x) {</div>
|
|
<div class="line"><span class="lineno"> 146</span> <span class="keywordflow">return</span> (x - 2) * (x - 2);</div>
|
|
<div class="line"><span class="lineno"> 147</span> };</div>
|
|
<div class="line"><span class="lineno"> 148</span> </div>
|
|
<div class="line"><span class="lineno"> 149</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> << <span class="stringliteral">"Test 1.... "</span>;</div>
|
|
<div class="line"><span class="lineno"> 150</span> </div>
|
|
<div class="line"><span class="lineno"> 151</span> <span class="keywordtype">double</span> minima = <a class="code hl_function" href="../../db/d01/brent__method__extrema_8cpp.html#a1aa76a6d5fd4d333f9072beff1dc486b">get_minima</a>(f1, -1, 5);</div>
|
|
<div class="line"><span class="lineno"> 152</span> </div>
|
|
<div class="line"><span class="lineno"> 153</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> << minima << <span class="stringliteral">"..."</span>;</div>
|
|
<div class="line"><span class="lineno"> 154</span> </div>
|
|
<div class="line"><span class="lineno"> 155</span> assert(std::abs(minima - 2) < <a class="code hl_define" href="../../db/d01/brent__method__extrema_8cpp.html#a002b2f4894492820fe708b1b7e7c5e70">EPSILON</a>);</div>
|
|
<div class="line"><span class="lineno"> 156</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> << <span class="stringliteral">"passed\n"</span>;</div>
|
|
<div class="line"><span class="lineno"> 157</span>}</div>
|
|
<div class="ttc" id="abrent__method__extrema_8cpp_html_a1aa76a6d5fd4d333f9072beff1dc486b"><div class="ttname"><a href="../../db/d01/brent__method__extrema_8cpp.html#a1aa76a6d5fd4d333f9072beff1dc486b">get_minima</a></div><div class="ttdeci">double get_minima(const std::function< double(double)> &f, double lim_a, double lim_b)</div><div class="ttdoc">Get the real root of a function in the given interval.</div><div class="ttdef"><b>Definition:</b> brent_method_extrema.cpp:35</div></div>
|
|
<div class="ttc" id="afunction_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/utility/functional/function.html">std::function</a></div></div>
|
|
</div><!-- fragment --><div class="dynheader">
|
|
Here is the call graph for this function:</div>
|
|
<div class="dyncontent">
|
|
<div class="center"><iframe scrolling="no" frameborder="0" src="../../db/d01/brent__method__extrema_8cpp_a1440a7779ac56f47a3f355ce4a8c7da0_cgraph.svg" width="326" height="88"><p><b>This browser is not able to show SVG: try Firefox, Chrome, Safari, or Opera instead.</b></p></iframe>
|
|
</div>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<a id="a0283886819c7c140a023582b7269e2d0" name="a0283886819c7c140a023582b7269e2d0"></a>
|
|
<h2 class="memtitle"><span class="permalink"><a href="#a0283886819c7c140a023582b7269e2d0">◆ </a></span>test2()</h2>
|
|
|
|
<div class="memitem">
|
|
<div class="memproto">
|
|
<table class="memname">
|
|
<tr>
|
|
<td class="memname">void test2 </td>
|
|
<td>(</td>
|
|
<td class="paramname"></td><td>)</td>
|
|
<td></td>
|
|
</tr>
|
|
</table>
|
|
</div><div class="memdoc">
|
|
|
|
<p>Test function to find root for the function \(f(x)= x^{\frac{1}{x}}\) in the interval \([-2,10]\) <br />
|
|
Expected result: \(e\approx 2.71828182845904509\). </p>
|
|
<div class="fragment"><div class="line"><span class="lineno"> 165</span> {</div>
|
|
<div class="line"><span class="lineno"> 166</span> <span class="comment">// define the function to maximize as a lambda function</span></div>
|
|
<div class="line"><span class="lineno"> 167</span> <span class="comment">// since we are maximixing, we negated the function return value</span></div>
|
|
<div class="line"><span class="lineno"> 168</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/utility/functional/function.html">std::function</a><double(<span class="keywordtype">double</span>)> func = [](<span class="keywordtype">double</span> x) {</div>
|
|
<div class="line"><span class="lineno"> 169</span> <span class="keywordflow">return</span> -<a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/math/pow.html">std::pow</a>(x, 1.f / x);</div>
|
|
<div class="line"><span class="lineno"> 170</span> };</div>
|
|
<div class="line"><span class="lineno"> 171</span> </div>
|
|
<div class="line"><span class="lineno"> 172</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> << <span class="stringliteral">"Test 2.... "</span>;</div>
|
|
<div class="line"><span class="lineno"> 173</span> </div>
|
|
<div class="line"><span class="lineno"> 174</span> <span class="keywordtype">double</span> minima = <a class="code hl_function" href="../../db/d01/brent__method__extrema_8cpp.html#a1aa76a6d5fd4d333f9072beff1dc486b">get_minima</a>(func, -2, 5);</div>
|
|
<div class="line"><span class="lineno"> 175</span> </div>
|
|
<div class="line"><span class="lineno"> 176</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> << minima << <span class="stringliteral">" ("</span> << M_E << <span class="stringliteral">")..."</span>;</div>
|
|
<div class="line"><span class="lineno"> 177</span> </div>
|
|
<div class="line"><span class="lineno"> 178</span> assert(std::abs(minima - M_E) < <a class="code hl_define" href="../../db/d01/brent__method__extrema_8cpp.html#a002b2f4894492820fe708b1b7e7c5e70">EPSILON</a>);</div>
|
|
<div class="line"><span class="lineno"> 179</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> << <span class="stringliteral">"passed\n"</span>;</div>
|
|
<div class="line"><span class="lineno"> 180</span>}</div>
|
|
<div class="ttc" id="apow_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/numeric/math/pow.html">std::pow</a></div><div class="ttdeci">T pow(T... args)</div></div>
|
|
</div><!-- fragment --><div class="dynheader">
|
|
Here is the call graph for this function:</div>
|
|
<div class="dyncontent">
|
|
<div class="center"><iframe scrolling="no" frameborder="0" src="../../db/d01/brent__method__extrema_8cpp_a0283886819c7c140a023582b7269e2d0_cgraph.svg" width="326" height="114"><p><b>This browser is not able to show SVG: try Firefox, Chrome, Safari, or Opera instead.</b></p></iframe>
|
|
</div>
|
|
</div>
|
|
|
|
</div>
|
|
</div>
|
|
<a id="a6d0455dd5c30adda100e95f0423c786e" name="a6d0455dd5c30adda100e95f0423c786e"></a>
|
|
<h2 class="memtitle"><span class="permalink"><a href="#a6d0455dd5c30adda100e95f0423c786e">◆ </a></span>test3()</h2>
|
|
|
|
<div class="memitem">
|
|
<div class="memproto">
|
|
<table class="memname">
|
|
<tr>
|
|
<td class="memname">void test3 </td>
|
|
<td>(</td>
|
|
<td class="paramname"></td><td>)</td>
|
|
<td></td>
|
|
</tr>
|
|
</table>
|
|
</div><div class="memdoc">
|
|
|
|
<p>Test function to find <em>maxima</em> for the function \(f(x)= \cos x\) in the interval \([0,12]\) <br />
|
|
Expected result: \(\pi\approx 3.14159265358979312\). </p>
|
|
<div class="fragment"><div class="line"><span class="lineno"> 188</span> {</div>
|
|
<div class="line"><span class="lineno"> 189</span> <span class="comment">// define the function to maximize as a lambda function</span></div>
|
|
<div class="line"><span class="lineno"> 190</span> <span class="comment">// since we are maximixing, we negated the function return value</span></div>
|
|
<div class="line"><span class="lineno"> 191</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/utility/functional/function.html">std::function</a><double(<span class="keywordtype">double</span>)> func = [](<span class="keywordtype">double</span> x) { <span class="keywordflow">return</span> <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/math/cos.html">std::cos</a>(x); };</div>
|
|
<div class="line"><span class="lineno"> 192</span> </div>
|
|
<div class="line"><span class="lineno"> 193</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> << <span class="stringliteral">"Test 3.... "</span>;</div>
|
|
<div class="line"><span class="lineno"> 194</span> </div>
|
|
<div class="line"><span class="lineno"> 195</span> <span class="keywordtype">double</span> minima = <a class="code hl_function" href="../../db/d01/brent__method__extrema_8cpp.html#a1aa76a6d5fd4d333f9072beff1dc486b">get_minima</a>(func, -4, 12);</div>
|
|
<div class="line"><span class="lineno"> 196</span> </div>
|
|
<div class="line"><span class="lineno"> 197</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> << minima << <span class="stringliteral">" ("</span> << M_PI << <span class="stringliteral">")..."</span>;</div>
|
|
<div class="line"><span class="lineno"> 198</span> </div>
|
|
<div class="line"><span class="lineno"> 199</span> assert(std::abs(minima - M_PI) < <a class="code hl_define" href="../../db/d01/brent__method__extrema_8cpp.html#a002b2f4894492820fe708b1b7e7c5e70">EPSILON</a>);</div>
|
|
<div class="line"><span class="lineno"> 200</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> << <span class="stringliteral">"passed\n"</span>;</div>
|
|
<div class="line"><span class="lineno"> 201</span>}</div>
|
|
<div class="ttc" id="acos_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/numeric/math/cos.html">std::cos</a></div><div class="ttdeci">T cos(T... args)</div></div>
|
|
</div><!-- fragment --><div class="dynheader">
|
|
Here is the call graph for this function:</div>
|
|
<div class="dyncontent">
|
|
<div class="center"><iframe scrolling="no" frameborder="0" src="../../db/d01/brent__method__extrema_8cpp_a6d0455dd5c30adda100e95f0423c786e_cgraph.svg" width="326" height="114"><p><b>This browser is not able to show SVG: try Firefox, Chrome, Safari, or Opera instead.</b></p></iframe>
|
|
</div>
|
|
</div>
|
|
|
|
</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_9c6faab82c22511b50177aa2e38e2780.html">numerical_methods</a></li><li class="navelem"><a class="el" href="../../db/d01/brent__method__extrema_8cpp.html">brent_method_extrema.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> 1.9.4 </li>
|
|
</ul>
|
|
</div>
|
|
</body>
|
|
</html>
|