mirror of
https://github.com/TheAlgorithms/C-Plus-Plus.git
synced 2026-03-21 04:18:34 +08:00
516 lines
38 KiB
HTML
516 lines
38 KiB
HTML
<!-- 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/brent_method_extrema.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="../../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"> 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('db/d01/brent__method__extrema_8cpp.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">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 <cstdint></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" loading="lazy" frameborder="0" src="../../d9/d0e/brent__method__extrema_8cpp__incl.svg" width="515" height="126"><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="../../db/d01/brent__method__extrema_8cpp_source.html">Go to the source code of this file.</a></p>
|
|
<table class="memberdecls">
|
|
<tr class="heading"><td colspan="2"><h2 id="header-define-members" class="groupheader"><a id="define-members" name="define-members"></a>
|
|
Macros</h2></td></tr>
|
|
<tr class="memitem:a525335710b53cb064ca56b936120431e" id="r_a525335710b53cb064ca56b936120431e"><td class="memItemLeft" align="right" valign="top">#define </td><td class="memItemRight" valign="bottom"><a class="el" href="#a525335710b53cb064ca56b936120431e">_USE_MATH_DEFINES</a></td></tr>
|
|
<tr class="memdesc:a525335710b53cb064ca56b936120431e"><td class="mdescLeft"> </td><td class="mdescRight">required for MS Visual C++ <br /></td></tr>
|
|
<tr class="memitem:a002b2f4894492820fe708b1b7e7c5e70" id="r_a002b2f4894492820fe708b1b7e7c5e70"><td class="memItemLeft" align="right" valign="top">#define </td><td class="memItemRight" valign="bottom"><a class="el" href="#a002b2f4894492820fe708b1b7e7c5e70">EPSILON</a></td></tr>
|
|
<tr class="memdesc:a002b2f4894492820fe708b1b7e7c5e70"><td class="mdescLeft"> </td><td class="mdescRight">system accuracy limit <br /></td></tr>
|
|
</table><table class="memberdecls">
|
|
<tr class="heading"><td colspan="2"><h2 id="header-func-members" class="groupheader"><a id="func-members" name="func-members"></a>
|
|
Functions</h2></td></tr>
|
|
<tr class="memitem:a1aa76a6d5fd4d333f9072beff1dc486b" id="r_a1aa76a6d5fd4d333f9072beff1dc486b"><td class="memItemLeft" align="right" valign="top">double </td><td class="memItemRight" valign="bottom"><a class="el" href="#a1aa76a6d5fd4d333f9072beff1dc486b">get_minima</a> (const std::function< 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. <br /></td></tr>
|
|
<tr class="memitem:a1440a7779ac56f47a3f355ce4a8c7da0" id="r_a1440a7779ac56f47a3f355ce4a8c7da0"><td class="memItemLeft" align="right" valign="top">void </td><td class="memItemRight" valign="bottom"><a class="el" href="#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. <br /></td></tr>
|
|
<tr class="memitem:a0283886819c7c140a023582b7269e2d0" id="r_a0283886819c7c140a023582b7269e2d0"><td class="memItemLeft" align="right" valign="top">void </td><td class="memItemRight" valign="bottom"><a class="el" href="#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: \(e\approx 2.71828182845904509\). <br /></td></tr>
|
|
<tr class="memitem:a6d0455dd5c30adda100e95f0423c786e" id="r_a6d0455dd5c30adda100e95f0423c786e"><td class="memItemLeft" align="right" valign="top">void </td><td class="memItemRight" valign="bottom"><a class="el" href="#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: \(\pi\approx 3.14159265358979312\). <br /></td></tr>
|
|
<tr class="memitem:ae66f6b31b5ad750f1fe042a706a4e3d4" id="r_ae66f6b31b5ad750f1fe042a706a4e3d4"><td class="memItemLeft" align="right" valign="top">int </td><td class="memItemRight" valign="bottom"><a class="el" href="#ae66f6b31b5ad750f1fe042a706a4e3d4">main</a> ()</td></tr>
|
|
</table>
|
|
<a name="details" id="details"></a><h2 id="header-details" 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>
|
|
|
|
<p class="definition">Definition in file <a class="el" href="../../db/d01/brent__method__extrema_8cpp_source.html">brent_method_extrema.cpp</a>.</p>
|
|
</div><a name="doc-define-members" id="doc-define-members"></a><h2 id="header-doc-define-members" class="groupheader">Macro Definition Documentation</h2>
|
|
<a id="a525335710b53cb064ca56b936120431e" name="a525335710b53cb064ca56b936120431e"></a>
|
|
<h2 class="memtitle"><span class="permalink"><a href="#a525335710b53cb064ca56b936120431e">◆ </a></span>_USE_MATH_DEFINES</h2>
|
|
|
|
<div class="memitem">
|
|
<div class="memproto">
|
|
<table class="memname">
|
|
<tr>
|
|
<td class="memname">#define _USE_MATH_DEFINES</td>
|
|
</tr>
|
|
</table>
|
|
</div><div class="memdoc">
|
|
|
|
<p>required for MS Visual C++ </p>
|
|
|
|
<p class="definition">Definition at line <a class="el" href="../../db/d01/brent__method__extrema_8cpp_source.html#l00016">16</a> of file <a class="el" href="../../db/d01/brent__method__extrema_8cpp_source.html">brent_method_extrema.cpp</a>.</p>
|
|
|
|
</div>
|
|
</div>
|
|
<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"> std::sqrt( \</div>
|
|
<div class="line"> std::numeric_limits<double>::epsilon())</div>
|
|
</div><!-- fragment -->
|
|
<p>system accuracy limit </p>
|
|
|
|
<p class="definition">Definition at line <a class="el" href="../../db/d01/brent__method__extrema_8cpp_source.html#l00024">24</a> of file <a class="el" href="../../db/d01/brent__method__extrema_8cpp_source.html">brent_method_extrema.cpp</a>.</p>
|
|
<div class="fragment"><div class="line"><span class="lineno"> 24</span><span class="preprocessor">#define EPSILON \</span></div>
|
|
<div class="line"><span class="lineno"> 25</span><span class="preprocessor"> std::sqrt( \</span></div>
|
|
<div class="line"><span class="lineno"> 26</span><span class="preprocessor"> std::numeric_limits<double>::epsilon()) </span></div>
|
|
</div><!-- fragment -->
|
|
</div>
|
|
</div>
|
|
<a name="doc-func-members" id="doc-func-members"></a><h2 id="header-doc-func-members" 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 std::function< double(double)> &</td> <td class="paramname"><span class="paramname"><em>f</em></span>, </td>
|
|
</tr>
|
|
<tr>
|
|
<td class="paramkey"></td>
|
|
<td></td>
|
|
<td class="paramtype">double</td> <td class="paramname"><span class="paramname"><em>lim_a</em></span>, </td>
|
|
</tr>
|
|
<tr>
|
|
<td class="paramkey"></td>
|
|
<td></td>
|
|
<td class="paramtype">double</td> <td class="paramname"><span class="paramname"><em>lim_b</em></span> )</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>
|
|
|
|
<p class="definition">Definition at line <a class="el" href="../../db/d01/brent__method__extrema_8cpp_source.html#l00036">36</a> of file <a class="el" href="../../db/d01/brent__method__extrema_8cpp_source.html">brent_method_extrema.cpp</a>.</p>
|
|
<div class="fragment"><div class="line"><span class="lineno"> 37</span> {</div>
|
|
<div class="line"><span class="lineno"> 38</span> uint32_t iters = 0;</div>
|
|
<div class="line"><span class="lineno"> 39</span> </div>
|
|
<div class="line"><span class="lineno"> 40</span> <span class="keywordflow">if</span> (lim_a > lim_b) {</div>
|
|
<div class="line"><span class="lineno"> 41</span> std::swap(lim_a, lim_b);</div>
|
|
<div class="line"><span class="lineno"> 42</span> } <span class="keywordflow">else</span> <span class="keywordflow">if</span> (std::abs(lim_a - lim_b) <= EPSILON) {</div>
|
|
<div class="line"><span class="lineno"> 43</span> std::cerr << <span class="stringliteral">"Search range must be greater than "</span> << EPSILON << <span class="stringliteral">"\n"</span>;</div>
|
|
<div class="line"><span class="lineno"> 44</span> <span class="keywordflow">return</span> lim_a;</div>
|
|
<div class="line"><span class="lineno"> 45</span> }</div>
|
|
<div class="line"><span class="lineno"> 46</span> </div>
|
|
<div class="line"><span class="lineno"> 47</span> <span class="comment">// golden ratio value</span></div>
|
|
<div class="line"><span class="lineno"> 48</span> <span class="keyword">const</span> <span class="keywordtype">double</span> M_GOLDEN_RATIO = (3.f - std::sqrt(5.f)) / 2.f;</div>
|
|
<div class="line"><span class="lineno"> 49</span> </div>
|
|
<div class="line"><span class="lineno"> 50</span> <span class="keywordtype">double</span> v = lim_a + M_GOLDEN_RATIO * (lim_b - lim_a);</div>
|
|
<div class="line"><span class="lineno"> 51</span> <span class="keywordtype">double</span> u, w = v, x = v;</div>
|
|
<div class="line"><span class="lineno"> 52</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"> 53</span> <span class="keywordtype">double</span> fw = fv, fx = fv;</div>
|
|
<div class="line"><span class="lineno"> 54</span> </div>
|
|
<div class="line"><span class="lineno"> 55</span> <span class="keywordtype">double</span> mid_point = (lim_a + lim_b) / 2.f;</div>
|
|
<div class="line"><span class="lineno"> 56</span> <span class="keywordtype">double</span> p = 0, q = 0, r = 0;</div>
|
|
<div class="line"><span class="lineno"> 57</span> </div>
|
|
<div class="line"><span class="lineno"> 58</span> <span class="keywordtype">double</span> d, e = 0;</div>
|
|
<div class="line"><span class="lineno"> 59</span> <span class="keywordtype">double</span> tolerance, tolerance2;</div>
|
|
<div class="line"><span class="lineno"> 60</span> </div>
|
|
<div class="line"><span class="lineno"> 61</span> <span class="keywordflow">do</span> {</div>
|
|
<div class="line"><span class="lineno"> 62</span> mid_point = (lim_a + lim_b) / 2.f;</div>
|
|
<div class="line"><span class="lineno"> 63</span> tolerance = EPSILON * std::abs(x);</div>
|
|
<div class="line"><span class="lineno"> 64</span> tolerance2 = 2 * tolerance;</div>
|
|
<div class="line"><span class="lineno"> 65</span> </div>
|
|
<div class="line"><span class="lineno"> 66</span> <span class="keywordflow">if</span> (std::abs(e) > tolerance2) {</div>
|
|
<div class="line"><span class="lineno"> 67</span> <span class="comment">// fit parabola</span></div>
|
|
<div class="line"><span class="lineno"> 68</span> r = (x - w) * (fx - fv);</div>
|
|
<div class="line"><span class="lineno"> 69</span> q = (x - v) * (fx - fw);</div>
|
|
<div class="line"><span class="lineno"> 70</span> p = (x - v) * q - (x - w) * r;</div>
|
|
<div class="line"><span class="lineno"> 71</span> q = 2.f * (q - r);</div>
|
|
<div class="line"><span class="lineno"> 72</span> <span class="keywordflow">if</span> (q > 0)</div>
|
|
<div class="line"><span class="lineno"> 73</span> p = -p;</div>
|
|
<div class="line"><span class="lineno"> 74</span> <span class="keywordflow">else</span></div>
|
|
<div class="line"><span class="lineno"> 75</span> q = -q;</div>
|
|
<div class="line"><span class="lineno"> 76</span> r = e;</div>
|
|
<div class="line"><span class="lineno"> 77</span> e = d;</div>
|
|
<div class="line"><span class="lineno"> 78</span> }</div>
|
|
<div class="line"><span class="lineno"> 79</span> </div>
|
|
<div class="line"><span class="lineno"> 80</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"> 81</span> <span class="comment">// parabolic interpolation step</span></div>
|
|
<div class="line"><span class="lineno"> 82</span> d = p / q;</div>
|
|
<div class="line"><span class="lineno"> 83</span> u = x + d;</div>
|
|
<div class="line"><span class="lineno"> 84</span> <span class="keywordflow">if</span> (u - lim_a < tolerance2 || lim_b - u < tolerance2)</div>
|
|
<div class="line"><span class="lineno"> 85</span> d = x < mid_point ? tolerance : -tolerance;</div>
|
|
<div class="line"><span class="lineno"> 86</span> } <span class="keywordflow">else</span> {</div>
|
|
<div class="line"><span class="lineno"> 87</span> <span class="comment">// golden section interpolation step</span></div>
|
|
<div class="line"><span class="lineno"> 88</span> e = (x < mid_point ? lim_b : lim_a) - x;</div>
|
|
<div class="line"><span class="lineno"> 89</span> d = M_GOLDEN_RATIO * e;</div>
|
|
<div class="line"><span class="lineno"> 90</span> }</div>
|
|
<div class="line"><span class="lineno"> 91</span> </div>
|
|
<div class="line"><span class="lineno"> 92</span> <span class="comment">// evaluate not too close to x</span></div>
|
|
<div class="line"><span class="lineno"> 93</span> <span class="keywordflow">if</span> (std::abs(d) >= tolerance)</div>
|
|
<div class="line"><span class="lineno"> 94</span> u = d;</div>
|
|
<div class="line"><span class="lineno"> 95</span> <span class="keywordflow">else</span> <span class="keywordflow">if</span> (d > 0)</div>
|
|
<div class="line"><span class="lineno"> 96</span> u = tolerance;</div>
|
|
<div class="line"><span class="lineno"> 97</span> <span class="keywordflow">else</span></div>
|
|
<div class="line"><span class="lineno"> 98</span> u = -tolerance;</div>
|
|
<div class="line"><span class="lineno"> 99</span> u += x;</div>
|
|
<div class="line"><span class="lineno"> 100</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"> 101</span> </div>
|
|
<div class="line"><span class="lineno"> 102</span> <span class="comment">// update variables</span></div>
|
|
<div class="line"><span class="lineno"> 103</span> <span class="keywordflow">if</span> (fu <= fx) {</div>
|
|
<div class="line"><span class="lineno"> 104</span> <span class="keywordflow">if</span> (u < x)</div>
|
|
<div class="line"><span class="lineno"> 105</span> lim_b = x;</div>
|
|
<div class="line"><span class="lineno"> 106</span> <span class="keywordflow">else</span></div>
|
|
<div class="line"><span class="lineno"> 107</span> lim_a = x;</div>
|
|
<div class="line"><span class="lineno"> 108</span> v = w;</div>
|
|
<div class="line"><span class="lineno"> 109</span> fv = fw;</div>
|
|
<div class="line"><span class="lineno"> 110</span> w = x;</div>
|
|
<div class="line"><span class="lineno"> 111</span> fw = fx;</div>
|
|
<div class="line"><span class="lineno"> 112</span> x = u;</div>
|
|
<div class="line"><span class="lineno"> 113</span> fx = fu;</div>
|
|
<div class="line"><span class="lineno"> 114</span> } <span class="keywordflow">else</span> {</div>
|
|
<div class="line"><span class="lineno"> 115</span> <span class="keywordflow">if</span> (u < x)</div>
|
|
<div class="line"><span class="lineno"> 116</span> lim_a = u;</div>
|
|
<div class="line"><span class="lineno"> 117</span> <span class="keywordflow">else</span></div>
|
|
<div class="line"><span class="lineno"> 118</span> lim_b = u;</div>
|
|
<div class="line"><span class="lineno"> 119</span> <span class="keywordflow">if</span> (fu <= fw || x == w) {</div>
|
|
<div class="line"><span class="lineno"> 120</span> v = w;</div>
|
|
<div class="line"><span class="lineno"> 121</span> fv = fw;</div>
|
|
<div class="line"><span class="lineno"> 122</span> w = u;</div>
|
|
<div class="line"><span class="lineno"> 123</span> fw = fu;</div>
|
|
<div class="line"><span class="lineno"> 124</span> } <span class="keywordflow">else</span> <span class="keywordflow">if</span> (fu <= fv || v == x || v == w) {</div>
|
|
<div class="line"><span class="lineno"> 125</span> v = u;</div>
|
|
<div class="line"><span class="lineno"> 126</span> fv = fu;</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> </div>
|
|
<div class="line"><span class="lineno"> 130</span> iters++;</div>
|
|
<div class="line"><span class="lineno"> 131</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"> 132</span> </div>
|
|
<div class="line"><span class="lineno"> 133</span> std::cout << <span class="stringliteral">" (iters: "</span> << iters << <span class="stringliteral">") "</span>;</div>
|
|
<div class="line"><span class="lineno"> 134</span> </div>
|
|
<div class="line"><span class="lineno"> 135</span> <span class="keywordflow">return</span> x;</div>
|
|
<div class="line"><span class="lineno"> 136</span>}</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> <a href="../../d4/d18/composite__simpson__rule_8cpp_source.html#l00113">composite_simpson_rule.cpp:113</a></div></div>
|
|
</div><!-- fragment -->
|
|
</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"><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="../../db/d01/brent__method__extrema_8cpp_source.html#l00205">205</a> of file <a class="el" href="../../db/d01/brent__method__extrema_8cpp_source.html">brent_method_extrema.cpp</a>.</p>
|
|
<div class="fragment"><div class="line"><span class="lineno"> 205</span> {</div>
|
|
<div class="line"><span class="lineno"> 206</span> std::cout.precision(18);</div>
|
|
<div class="line"><span class="lineno"> 207</span> </div>
|
|
<div class="line"><span class="lineno"> 208</span> std::cout << <span class="stringliteral">"Computations performed with machine epsilon: "</span> << EPSILON</div>
|
|
<div class="line"><span class="lineno"> 209</span> << <span class="stringliteral">"\n"</span>;</div>
|
|
<div class="line"><span class="lineno"> 210</span> </div>
|
|
<div class="line"><span class="lineno"> 211</span> <a class="code hl_function" href="#a1440a7779ac56f47a3f355ce4a8c7da0">test1</a>();</div>
|
|
<div class="line"><span class="lineno"> 212</span> <a class="code hl_function" href="#a0283886819c7c140a023582b7269e2d0">test2</a>();</div>
|
|
<div class="line"><span class="lineno"> 213</span> <a class="code hl_function" href="#a6d0455dd5c30adda100e95f0423c786e">test3</a>();</div>
|
|
<div class="line"><span class="lineno"> 214</span> </div>
|
|
<div class="line"><span class="lineno"> 215</span> <span class="keywordflow">return</span> 0;</div>
|
|
<div class="line"><span class="lineno"> 216</span>}</div>
|
|
<div class="ttc" id="abrent__method__extrema_8cpp_html_a0283886819c7c140a023582b7269e2d0"><div class="ttname"><a href="#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> <a href="../../db/d01/brent__method__extrema_8cpp_source.html#l00166">brent_method_extrema.cpp:166</a></div></div>
|
|
<div class="ttc" id="abrent__method__extrema_8cpp_html_a1440a7779ac56f47a3f355ce4a8c7da0"><div class="ttname"><a href="#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> <a href="../../db/d01/brent__method__extrema_8cpp_source.html#l00144">brent_method_extrema.cpp:144</a></div></div>
|
|
<div class="ttc" id="abrent__method__extrema_8cpp_html_a6d0455dd5c30adda100e95f0423c786e"><div class="ttname"><a href="#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> <a href="../../db/d01/brent__method__extrema_8cpp_source.html#l00189">brent_method_extrema.cpp:189</a></div></div>
|
|
</div><!-- fragment -->
|
|
</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"><span class="paramname"><em></em></span></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>
|
|
|
|
<p class="definition">Definition at line <a class="el" href="../../db/d01/brent__method__extrema_8cpp_source.html#l00144">144</a> of file <a class="el" href="../../db/d01/brent__method__extrema_8cpp_source.html">brent_method_extrema.cpp</a>.</p>
|
|
<div class="fragment"><div class="line"><span class="lineno"> 144</span> {</div>
|
|
<div class="line"><span class="lineno"> 145</span> <span class="comment">// define the function to minimize as a lambda function</span></div>
|
|
<div class="line"><span class="lineno"> 146</span> std::function<double(<span class="keywordtype">double</span>)> f1 = [](<span class="keywordtype">double</span> x) {</div>
|
|
<div class="line"><span class="lineno"> 147</span> <span class="keywordflow">return</span> (x - 2) * (x - 2);</div>
|
|
<div class="line"><span class="lineno"> 148</span> };</div>
|
|
<div class="line"><span class="lineno"> 149</span> </div>
|
|
<div class="line"><span class="lineno"> 150</span> std::cout << <span class="stringliteral">"Test 1.... "</span>;</div>
|
|
<div class="line"><span class="lineno"> 151</span> </div>
|
|
<div class="line"><span class="lineno"> 152</span> <span class="keywordtype">double</span> minima = <a class="code hl_function" href="#a1aa76a6d5fd4d333f9072beff1dc486b">get_minima</a>(f1, -1, 5);</div>
|
|
<div class="line"><span class="lineno"> 153</span> </div>
|
|
<div class="line"><span class="lineno"> 154</span> std::cout << minima << <span class="stringliteral">"..."</span>;</div>
|
|
<div class="line"><span class="lineno"> 155</span> </div>
|
|
<div class="line"><span class="lineno"> 156</span> assert(std::abs(minima - 2) < EPSILON);</div>
|
|
<div class="line"><span class="lineno"> 157</span> std::cout << <span class="stringliteral">"passed\n"</span>;</div>
|
|
<div class="line"><span class="lineno"> 158</span>}</div>
|
|
<div class="ttc" id="abrent__method__extrema_8cpp_html_a1aa76a6d5fd4d333f9072beff1dc486b"><div class="ttname"><a href="#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> <a href="../../db/d01/brent__method__extrema_8cpp_source.html#l00036">brent_method_extrema.cpp:36</a></div></div>
|
|
</div><!-- fragment -->
|
|
</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"><span class="paramname"><em></em></span></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>
|
|
|
|
<p class="definition">Definition at line <a class="el" href="../../db/d01/brent__method__extrema_8cpp_source.html#l00166">166</a> of file <a class="el" href="../../db/d01/brent__method__extrema_8cpp_source.html">brent_method_extrema.cpp</a>.</p>
|
|
<div class="fragment"><div class="line"><span class="lineno"> 166</span> {</div>
|
|
<div class="line"><span class="lineno"> 167</span> <span class="comment">// define the function to maximize as a lambda function</span></div>
|
|
<div class="line"><span class="lineno"> 168</span> <span class="comment">// since we are maximixing, we negated the function return value</span></div>
|
|
<div class="line"><span class="lineno"> 169</span> std::function<double(<span class="keywordtype">double</span>)> func = [](<span class="keywordtype">double</span> x) {</div>
|
|
<div class="line"><span class="lineno"> 170</span> <span class="keywordflow">return</span> -std::pow(x, 1.f / x);</div>
|
|
<div class="line"><span class="lineno"> 171</span> };</div>
|
|
<div class="line"><span class="lineno"> 172</span> </div>
|
|
<div class="line"><span class="lineno"> 173</span> std::cout << <span class="stringliteral">"Test 2.... "</span>;</div>
|
|
<div class="line"><span class="lineno"> 174</span> </div>
|
|
<div class="line"><span class="lineno"> 175</span> <span class="keywordtype">double</span> minima = <a class="code hl_function" href="#a1aa76a6d5fd4d333f9072beff1dc486b">get_minima</a>(func, -2, 5);</div>
|
|
<div class="line"><span class="lineno"> 176</span> </div>
|
|
<div class="line"><span class="lineno"> 177</span> std::cout << minima << <span class="stringliteral">" ("</span> << M_E << <span class="stringliteral">")..."</span>;</div>
|
|
<div class="line"><span class="lineno"> 178</span> </div>
|
|
<div class="line"><span class="lineno"> 179</span> assert(std::abs(minima - M_E) < EPSILON);</div>
|
|
<div class="line"><span class="lineno"> 180</span> std::cout << <span class="stringliteral">"passed\n"</span>;</div>
|
|
<div class="line"><span class="lineno"> 181</span>}</div>
|
|
</div><!-- fragment -->
|
|
</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"><span class="paramname"><em></em></span></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>
|
|
|
|
<p class="definition">Definition at line <a class="el" href="../../db/d01/brent__method__extrema_8cpp_source.html#l00189">189</a> of file <a class="el" href="../../db/d01/brent__method__extrema_8cpp_source.html">brent_method_extrema.cpp</a>.</p>
|
|
<div class="fragment"><div class="line"><span class="lineno"> 189</span> {</div>
|
|
<div class="line"><span class="lineno"> 190</span> <span class="comment">// define the function to maximize as a lambda function</span></div>
|
|
<div class="line"><span class="lineno"> 191</span> <span class="comment">// since we are maximixing, we negated the function return value</span></div>
|
|
<div class="line"><span class="lineno"> 192</span> std::function<double(<span class="keywordtype">double</span>)> func = [](<span class="keywordtype">double</span> x) { <span class="keywordflow">return</span> std::cos(x); };</div>
|
|
<div class="line"><span class="lineno"> 193</span> </div>
|
|
<div class="line"><span class="lineno"> 194</span> std::cout << <span class="stringliteral">"Test 3.... "</span>;</div>
|
|
<div class="line"><span class="lineno"> 195</span> </div>
|
|
<div class="line"><span class="lineno"> 196</span> <span class="keywordtype">double</span> minima = <a class="code hl_function" href="#a1aa76a6d5fd4d333f9072beff1dc486b">get_minima</a>(func, -4, 12);</div>
|
|
<div class="line"><span class="lineno"> 197</span> </div>
|
|
<div class="line"><span class="lineno"> 198</span> std::cout << minima << <span class="stringliteral">" ("</span> << M_PI << <span class="stringliteral">")..."</span>;</div>
|
|
<div class="line"><span class="lineno"> 199</span> </div>
|
|
<div class="line"><span class="lineno"> 200</span> assert(std::abs(minima - M_PI) < EPSILON);</div>
|
|
<div class="line"><span class="lineno"> 201</span> std::cout << <span class="stringliteral">"passed\n"</span>;</div>
|
|
<div class="line"><span class="lineno"> 202</span>}</div>
|
|
</div><!-- fragment -->
|
|
</div>
|
|
</div>
|
|
</div><!-- contents -->
|
|
</div><!-- doc-content -->
|
|
<div id="page-nav" class="page-nav-panel">
|
|
<div id="page-nav-resize-handle"></div>
|
|
<div id="page-nav-tree">
|
|
<div id="page-nav-contents">
|
|
</div><!-- page-nav-contents -->
|
|
</div><!-- page-nav-tree -->
|
|
</div><!-- page-nav -->
|
|
</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="../../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> 1.14.0 </li>
|
|
</ul>
|
|
</div>
|
|
</body>
|
|
</html>
|