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<div class="headertitle"><div class="title">brent_method_extrema.cpp File Reference</div></div>
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<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 &lt;cassert&gt;</code><br />
<code>#include &lt;cmath&gt;</code><br />
<code>#include &lt;functional&gt;</code><br />
<code>#include &lt;iostream&gt;</code><br />
<code>#include &lt;limits&gt;</code><br />
</div><div class="textblock"><div class="dynheader">
Include dependency graph for brent_method_extrema.cpp:</div>
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Macros</h2></td></tr>
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#define&#160;</td><td class="memItemRight" valign="bottom"><b>_USE_MATH_DEFINES</b></td></tr>
<tr class="memdesc:a525335710b53cb064ca56b936120431e"><td class="mdescLeft">&#160;</td><td class="mdescRight">required for MS Visual C++ <br /></td></tr>
<tr class="separator:a525335710b53cb064ca56b936120431e"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:a002b2f4894492820fe708b1b7e7c5e70"><td class="memItemLeft" align="right" valign="top">#define&#160;</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">&#160;</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">&#160;</td></tr>
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Functions</h2></td></tr>
<tr class="memitem:a1aa76a6d5fd4d333f9072beff1dc486b"><td class="memItemLeft" align="right" valign="top">double&#160;</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>&lt; double(double)&gt; &amp;f, double lim_a, double lim_b)</td></tr>
<tr class="memdesc:a1aa76a6d5fd4d333f9072beff1dc486b"><td class="mdescLeft">&#160;</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">&#160;</td></tr>
<tr class="memitem:a1440a7779ac56f47a3f355ce4a8c7da0"><td class="memItemLeft" align="right" valign="top">void&#160;</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">&#160;</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">&#160;</td></tr>
<tr class="memitem:a0283886819c7c140a023582b7269e2d0"><td class="memItemLeft" align="right" valign="top">void&#160;</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">&#160;</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">&#160;</td></tr>
<tr class="memitem:a6d0455dd5c30adda100e95f0423c786e"><td class="memItemLeft" align="right" valign="top">void&#160;</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">&#160;</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>
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<tr class="memitem:ae66f6b31b5ad750f1fe042a706a4e3d4"><td class="memItemLeft" align="right" valign="top">int&#160;</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">&#160;</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">&#9670;&nbsp;</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&lt;double&gt;::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">&#9670;&nbsp;</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>&lt; double(double)&gt; &amp;&#160;</td>
<td class="paramname"><em>f</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">double&#160;</td>
<td class="paramname"><em>lim_a</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">double&#160;</td>
<td class="paramname"><em>lim_b</em>&#160;</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"><a id="l00036" name="l00036"></a><span class="lineno"> 36</span> {</div>
<div class="line"><a id="l00037" name="l00037"></a><span class="lineno"> 37</span> uint32_t iters = 0;</div>
<div class="line"><a id="l00038" name="l00038"></a><span class="lineno"> 38</span> </div>
<div class="line"><a id="l00039" name="l00039"></a><span class="lineno"> 39</span> <span class="keywordflow">if</span> (lim_a &gt; lim_b) {</div>
<div class="line"><a id="l00040" name="l00040"></a><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"><a id="l00041" name="l00041"></a><span class="lineno"> 41</span> } <span class="keywordflow">else</span> <span class="keywordflow">if</span> (std::abs(lim_a - lim_b) &lt;= <a class="code hl_define" href="../../db/d01/brent__method__extrema_8cpp.html#a002b2f4894492820fe708b1b7e7c5e70">EPSILON</a>) {</div>
<div class="line"><a id="l00042" name="l00042"></a><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> &lt;&lt; <span class="stringliteral">&quot;Search range must be greater than &quot;</span> &lt;&lt; <a class="code hl_define" href="../../db/d01/brent__method__extrema_8cpp.html#a002b2f4894492820fe708b1b7e7c5e70">EPSILON</a> &lt;&lt; <span class="stringliteral">&quot;\n&quot;</span>;</div>
<div class="line"><a id="l00043" name="l00043"></a><span class="lineno"> 43</span> <span class="keywordflow">return</span> lim_a;</div>
<div class="line"><a id="l00044" name="l00044"></a><span class="lineno"> 44</span> }</div>
<div class="line"><a id="l00045" name="l00045"></a><span class="lineno"> 45</span> </div>
<div class="line"><a id="l00046" name="l00046"></a><span class="lineno"> 46</span> <span class="comment">// golden ratio value</span></div>
<div class="line"><a id="l00047" name="l00047"></a><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"><a id="l00048" name="l00048"></a><span class="lineno"> 48</span> </div>
<div class="line"><a id="l00049" name="l00049"></a><span class="lineno"> 49</span> <span class="keywordtype">double</span> v = lim_a + M_GOLDEN_RATIO * (lim_b - lim_a);</div>
<div class="line"><a id="l00050" name="l00050"></a><span class="lineno"> 50</span> <span class="keywordtype">double</span> u, w = v, x = v;</div>
<div class="line"><a id="l00051" name="l00051"></a><span class="lineno"> 51</span> <span class="keywordtype">double</span> fu, fv = f(v);</div>
<div class="line"><a id="l00052" name="l00052"></a><span class="lineno"> 52</span> <span class="keywordtype">double</span> fw = fv, fx = fv;</div>
<div class="line"><a id="l00053" name="l00053"></a><span class="lineno"> 53</span> </div>
<div class="line"><a id="l00054" name="l00054"></a><span class="lineno"> 54</span> <span class="keywordtype">double</span> mid_point = (lim_a + lim_b) / 2.f;</div>
<div class="line"><a id="l00055" name="l00055"></a><span class="lineno"> 55</span> <span class="keywordtype">double</span> p = 0, q = 0, r = 0;</div>
<div class="line"><a id="l00056" name="l00056"></a><span class="lineno"> 56</span> </div>
<div class="line"><a id="l00057" name="l00057"></a><span class="lineno"> 57</span> <span class="keywordtype">double</span> d, e = 0;</div>
<div class="line"><a id="l00058" name="l00058"></a><span class="lineno"> 58</span> <span class="keywordtype">double</span> tolerance, tolerance2;</div>
<div class="line"><a id="l00059" name="l00059"></a><span class="lineno"> 59</span> </div>
<div class="line"><a id="l00060" name="l00060"></a><span class="lineno"> 60</span> <span class="keywordflow">do</span> {</div>
<div class="line"><a id="l00061" name="l00061"></a><span class="lineno"> 61</span> mid_point = (lim_a + lim_b) / 2.f;</div>
<div class="line"><a id="l00062" name="l00062"></a><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"><a id="l00063" name="l00063"></a><span class="lineno"> 63</span> tolerance2 = 2 * tolerance;</div>
<div class="line"><a id="l00064" name="l00064"></a><span class="lineno"> 64</span> </div>
<div class="line"><a id="l00065" name="l00065"></a><span class="lineno"> 65</span> <span class="keywordflow">if</span> (std::abs(e) &gt; tolerance2) {</div>
<div class="line"><a id="l00066" name="l00066"></a><span class="lineno"> 66</span> <span class="comment">// fit parabola</span></div>
<div class="line"><a id="l00067" name="l00067"></a><span class="lineno"> 67</span> r = (x - w) * (fx - fv);</div>
<div class="line"><a id="l00068" name="l00068"></a><span class="lineno"> 68</span> q = (x - v) * (fx - fw);</div>
<div class="line"><a id="l00069" name="l00069"></a><span class="lineno"> 69</span> p = (x - v) * q - (x - w) * r;</div>
<div class="line"><a id="l00070" name="l00070"></a><span class="lineno"> 70</span> q = 2.f * (q - r);</div>
<div class="line"><a id="l00071" name="l00071"></a><span class="lineno"> 71</span> <span class="keywordflow">if</span> (q &gt; 0)</div>
<div class="line"><a id="l00072" name="l00072"></a><span class="lineno"> 72</span> p = -p;</div>
<div class="line"><a id="l00073" name="l00073"></a><span class="lineno"> 73</span> <span class="keywordflow">else</span></div>
<div class="line"><a id="l00074" name="l00074"></a><span class="lineno"> 74</span> q = -q;</div>
<div class="line"><a id="l00075" name="l00075"></a><span class="lineno"> 75</span> r = e;</div>
<div class="line"><a id="l00076" name="l00076"></a><span class="lineno"> 76</span> e = d;</div>
<div class="line"><a id="l00077" name="l00077"></a><span class="lineno"> 77</span> }</div>
<div class="line"><a id="l00078" name="l00078"></a><span class="lineno"> 78</span> </div>
<div class="line"><a id="l00079" name="l00079"></a><span class="lineno"> 79</span> <span class="keywordflow">if</span> (std::abs(p) &lt; std::abs(0.5 * q * r) &amp;&amp; p &lt; q * (lim_b - x)) {</div>
<div class="line"><a id="l00080" name="l00080"></a><span class="lineno"> 80</span> <span class="comment">// parabolic interpolation step</span></div>
<div class="line"><a id="l00081" name="l00081"></a><span class="lineno"> 81</span> d = p / q;</div>
<div class="line"><a id="l00082" name="l00082"></a><span class="lineno"> 82</span> u = x + d;</div>
<div class="line"><a id="l00083" name="l00083"></a><span class="lineno"> 83</span> <span class="keywordflow">if</span> (u - lim_a &lt; tolerance2 || lim_b - u &lt; tolerance2)</div>
<div class="line"><a id="l00084" name="l00084"></a><span class="lineno"> 84</span> d = x &lt; mid_point ? tolerance : -tolerance;</div>
<div class="line"><a id="l00085" name="l00085"></a><span class="lineno"> 85</span> } <span class="keywordflow">else</span> {</div>
<div class="line"><a id="l00086" name="l00086"></a><span class="lineno"> 86</span> <span class="comment">// golden section interpolation step</span></div>
<div class="line"><a id="l00087" name="l00087"></a><span class="lineno"> 87</span> e = (x &lt; mid_point ? lim_b : lim_a) - x;</div>
<div class="line"><a id="l00088" name="l00088"></a><span class="lineno"> 88</span> d = M_GOLDEN_RATIO * e;</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="comment">// evaluate not too close to x</span></div>
<div class="line"><a id="l00092" name="l00092"></a><span class="lineno"> 92</span> <span class="keywordflow">if</span> (std::abs(d) &gt;= tolerance)</div>
<div class="line"><a id="l00093" name="l00093"></a><span class="lineno"> 93</span> u = d;</div>
<div class="line"><a id="l00094" name="l00094"></a><span class="lineno"> 94</span> <span class="keywordflow">else</span> <span class="keywordflow">if</span> (d &gt; 0)</div>
<div class="line"><a id="l00095" name="l00095"></a><span class="lineno"> 95</span> u = tolerance;</div>
<div class="line"><a id="l00096" name="l00096"></a><span class="lineno"> 96</span> <span class="keywordflow">else</span></div>
<div class="line"><a id="l00097" name="l00097"></a><span class="lineno"> 97</span> u = -tolerance;</div>
<div class="line"><a id="l00098" name="l00098"></a><span class="lineno"> 98</span> u += x;</div>
<div class="line"><a id="l00099" name="l00099"></a><span class="lineno"> 99</span> fu = f(u);</div>
<div class="line"><a id="l00100" name="l00100"></a><span class="lineno"> 100</span> </div>
<div class="line"><a id="l00101" name="l00101"></a><span class="lineno"> 101</span> <span class="comment">// update variables</span></div>
<div class="line"><a id="l00102" name="l00102"></a><span class="lineno"> 102</span> <span class="keywordflow">if</span> (fu &lt;= fx) {</div>
<div class="line"><a id="l00103" name="l00103"></a><span class="lineno"> 103</span> <span class="keywordflow">if</span> (u &lt; x)</div>
<div class="line"><a id="l00104" name="l00104"></a><span class="lineno"> 104</span> lim_b = x;</div>
<div class="line"><a id="l00105" name="l00105"></a><span class="lineno"> 105</span> <span class="keywordflow">else</span></div>
<div class="line"><a id="l00106" name="l00106"></a><span class="lineno"> 106</span> lim_a = x;</div>
<div class="line"><a id="l00107" name="l00107"></a><span class="lineno"> 107</span> v = w;</div>
<div class="line"><a id="l00108" name="l00108"></a><span class="lineno"> 108</span> fv = fw;</div>
<div class="line"><a id="l00109" name="l00109"></a><span class="lineno"> 109</span> w = x;</div>
<div class="line"><a id="l00110" name="l00110"></a><span class="lineno"> 110</span> fw = fx;</div>
<div class="line"><a id="l00111" name="l00111"></a><span class="lineno"> 111</span> x = u;</div>
<div class="line"><a id="l00112" name="l00112"></a><span class="lineno"> 112</span> fx = fu;</div>
<div class="line"><a id="l00113" name="l00113"></a><span class="lineno"> 113</span> } <span class="keywordflow">else</span> {</div>
<div class="line"><a id="l00114" name="l00114"></a><span class="lineno"> 114</span> <span class="keywordflow">if</span> (u &lt; x)</div>
<div class="line"><a id="l00115" name="l00115"></a><span class="lineno"> 115</span> lim_a = u;</div>
<div class="line"><a id="l00116" name="l00116"></a><span class="lineno"> 116</span> <span class="keywordflow">else</span></div>
<div class="line"><a id="l00117" name="l00117"></a><span class="lineno"> 117</span> lim_b = u;</div>
<div class="line"><a id="l00118" name="l00118"></a><span class="lineno"> 118</span> <span class="keywordflow">if</span> (fu &lt;= fw || x == w) {</div>
<div class="line"><a id="l00119" name="l00119"></a><span class="lineno"> 119</span> v = w;</div>
<div class="line"><a id="l00120" name="l00120"></a><span class="lineno"> 120</span> fv = fw;</div>
<div class="line"><a id="l00121" name="l00121"></a><span class="lineno"> 121</span> w = u;</div>
<div class="line"><a id="l00122" name="l00122"></a><span class="lineno"> 122</span> fw = fu;</div>
<div class="line"><a id="l00123" name="l00123"></a><span class="lineno"> 123</span> } <span class="keywordflow">else</span> <span class="keywordflow">if</span> (fu &lt;= fv || v == x || v == w) {</div>
<div class="line"><a id="l00124" name="l00124"></a><span class="lineno"> 124</span> v = u;</div>
<div class="line"><a id="l00125" name="l00125"></a><span class="lineno"> 125</span> fv = fu;</div>
<div class="line"><a id="l00126" name="l00126"></a><span class="lineno"> 126</span> }</div>
<div class="line"><a id="l00127" name="l00127"></a><span class="lineno"> 127</span> }</div>
<div class="line"><a id="l00128" name="l00128"></a><span class="lineno"> 128</span> </div>
<div class="line"><a id="l00129" name="l00129"></a><span class="lineno"> 129</span> iters++;</div>
<div class="line"><a id="l00130" name="l00130"></a><span class="lineno"> 130</span> } <span class="keywordflow">while</span> (std::abs(x - mid_point) &gt; (tolerance - (lim_b - lim_a) / 2.f));</div>
<div class="line"><a id="l00131" name="l00131"></a><span class="lineno"> 131</span> </div>
<div class="line"><a id="l00132" name="l00132"></a><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> &lt;&lt; <span class="stringliteral">&quot; (iters: &quot;</span> &lt;&lt; iters &lt;&lt; <span class="stringliteral">&quot;) &quot;</span>;</div>
<div class="line"><a id="l00133" name="l00133"></a><span class="lineno"> 133</span> </div>
<div class="line"><a id="l00134" name="l00134"></a><span class="lineno"> 134</span> <span class="keywordflow">return</span> x;</div>
<div class="line"><a id="l00135" name="l00135"></a><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="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>
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<h2 class="memtitle"><span class="permalink"><a href="#ae66f6b31b5ad750f1fe042a706a4e3d4">&#9670;&nbsp;</a></span>main()</h2>
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<td class="memname">int main </td>
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<p >Main function </p>
<div class="fragment"><div class="line"><a id="l00204" name="l00204"></a><span class="lineno"> 204</span> {</div>
<div class="line"><a id="l00205" name="l00205"></a><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"><a id="l00206" name="l00206"></a><span class="lineno"> 206</span> </div>
<div class="line"><a id="l00207" name="l00207"></a><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> &lt;&lt; <span class="stringliteral">&quot;Computations performed with machine epsilon: &quot;</span> &lt;&lt; <a class="code hl_define" href="../../db/d01/brent__method__extrema_8cpp.html#a002b2f4894492820fe708b1b7e7c5e70">EPSILON</a></div>
<div class="line"><a id="l00208" name="l00208"></a><span class="lineno"> 208</span> &lt;&lt; <span class="stringliteral">&quot;\n&quot;</span>;</div>
<div class="line"><a id="l00209" name="l00209"></a><span class="lineno"> 209</span> </div>
<div class="line"><a id="l00210" name="l00210"></a><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"><a id="l00211" name="l00211"></a><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"><a id="l00212" name="l00212"></a><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"><a id="l00213" name="l00213"></a><span class="lineno"> 213</span> </div>
<div class="line"><a id="l00214" name="l00214"></a><span class="lineno"> 214</span> <span class="keywordflow">return</span> 0;</div>
<div class="line"><a id="l00215" name="l00215"></a><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>
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<h2 class="memtitle"><span class="permalink"><a href="#a1440a7779ac56f47a3f355ce4a8c7da0">&#9670;&nbsp;</a></span>test1()</h2>
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<td class="memname">void test1 </td>
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<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"><a id="l00143" name="l00143"></a><span class="lineno"> 143</span> {</div>
<div class="line"><a id="l00144" name="l00144"></a><span class="lineno"> 144</span> <span class="comment">// define the function to minimize as a lambda function</span></div>
<div class="line"><a id="l00145" name="l00145"></a><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>&lt;double(<span class="keywordtype">double</span>)&gt; f1 = [](<span class="keywordtype">double</span> x) {</div>
<div class="line"><a id="l00146" name="l00146"></a><span class="lineno"> 146</span> <span class="keywordflow">return</span> (x - 2) * (x - 2);</div>
<div class="line"><a id="l00147" name="l00147"></a><span class="lineno"> 147</span> };</div>
<div class="line"><a id="l00148" name="l00148"></a><span class="lineno"> 148</span> </div>
<div class="line"><a id="l00149" name="l00149"></a><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> &lt;&lt; <span class="stringliteral">&quot;Test 1.... &quot;</span>;</div>
<div class="line"><a id="l00150" name="l00150"></a><span class="lineno"> 150</span> </div>
<div class="line"><a id="l00151" name="l00151"></a><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"><a id="l00152" name="l00152"></a><span class="lineno"> 152</span> </div>
<div class="line"><a id="l00153" name="l00153"></a><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> &lt;&lt; minima &lt;&lt; <span class="stringliteral">&quot;...&quot;</span>;</div>
<div class="line"><a id="l00154" name="l00154"></a><span class="lineno"> 154</span> </div>
<div class="line"><a id="l00155" name="l00155"></a><span class="lineno"> 155</span> assert(std::abs(minima - 2) &lt; <a class="code hl_define" href="../../db/d01/brent__method__extrema_8cpp.html#a002b2f4894492820fe708b1b7e7c5e70">EPSILON</a>);</div>
<div class="line"><a id="l00156" name="l00156"></a><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> &lt;&lt; <span class="stringliteral">&quot;passed\n&quot;</span>;</div>
<div class="line"><a id="l00157" name="l00157"></a><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&lt; double(double)&gt; &amp;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>
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<h2 class="memtitle"><span class="permalink"><a href="#a0283886819c7c140a023582b7269e2d0">&#9670;&nbsp;</a></span>test2()</h2>
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<td class="memname">void test2 </td>
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<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"><a id="l00165" name="l00165"></a><span class="lineno"> 165</span> {</div>
<div class="line"><a id="l00166" name="l00166"></a><span class="lineno"> 166</span> <span class="comment">// define the function to maximize as a lambda function</span></div>
<div class="line"><a id="l00167" name="l00167"></a><span class="lineno"> 167</span> <span class="comment">// since we are maximixing, we negated the function return value</span></div>
<div class="line"><a id="l00168" name="l00168"></a><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>&lt;double(<span class="keywordtype">double</span>)&gt; func = [](<span class="keywordtype">double</span> x) {</div>
<div class="line"><a id="l00169" name="l00169"></a><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"><a id="l00170" name="l00170"></a><span class="lineno"> 170</span> };</div>
<div class="line"><a id="l00171" name="l00171"></a><span class="lineno"> 171</span> </div>
<div class="line"><a id="l00172" name="l00172"></a><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> &lt;&lt; <span class="stringliteral">&quot;Test 2.... &quot;</span>;</div>
<div class="line"><a id="l00173" name="l00173"></a><span class="lineno"> 173</span> </div>
<div class="line"><a id="l00174" name="l00174"></a><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"><a id="l00175" name="l00175"></a><span class="lineno"> 175</span> </div>
<div class="line"><a id="l00176" name="l00176"></a><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> &lt;&lt; minima &lt;&lt; <span class="stringliteral">&quot; (&quot;</span> &lt;&lt; M_E &lt;&lt; <span class="stringliteral">&quot;)...&quot;</span>;</div>
<div class="line"><a id="l00177" name="l00177"></a><span class="lineno"> 177</span> </div>
<div class="line"><a id="l00178" name="l00178"></a><span class="lineno"> 178</span> assert(std::abs(minima - M_E) &lt; <a class="code hl_define" href="../../db/d01/brent__method__extrema_8cpp.html#a002b2f4894492820fe708b1b7e7c5e70">EPSILON</a>);</div>
<div class="line"><a id="l00179" name="l00179"></a><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> &lt;&lt; <span class="stringliteral">&quot;passed\n&quot;</span>;</div>
<div class="line"><a id="l00180" name="l00180"></a><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>
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<h2 class="memtitle"><span class="permalink"><a href="#a6d0455dd5c30adda100e95f0423c786e">&#9670;&nbsp;</a></span>test3()</h2>
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<td class="memname">void test3 </td>
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<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"><a id="l00188" name="l00188"></a><span class="lineno"> 188</span> {</div>
<div class="line"><a id="l00189" name="l00189"></a><span class="lineno"> 189</span> <span class="comment">// define the function to maximize as a lambda function</span></div>
<div class="line"><a id="l00190" name="l00190"></a><span class="lineno"> 190</span> <span class="comment">// since we are maximixing, we negated the function return value</span></div>
<div class="line"><a id="l00191" name="l00191"></a><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>&lt;double(<span class="keywordtype">double</span>)&gt; 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"><a id="l00192" name="l00192"></a><span class="lineno"> 192</span> </div>
<div class="line"><a id="l00193" name="l00193"></a><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> &lt;&lt; <span class="stringliteral">&quot;Test 3.... &quot;</span>;</div>
<div class="line"><a id="l00194" name="l00194"></a><span class="lineno"> 194</span> </div>
<div class="line"><a id="l00195" name="l00195"></a><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"><a id="l00196" name="l00196"></a><span class="lineno"> 196</span> </div>
<div class="line"><a id="l00197" name="l00197"></a><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> &lt;&lt; minima &lt;&lt; <span class="stringliteral">&quot; (&quot;</span> &lt;&lt; M_PI &lt;&lt; <span class="stringliteral">&quot;)...&quot;</span>;</div>
<div class="line"><a id="l00198" name="l00198"></a><span class="lineno"> 198</span> </div>
<div class="line"><a id="l00199" name="l00199"></a><span class="lineno"> 199</span> assert(std::abs(minima - M_PI) &lt; <a class="code hl_define" href="../../db/d01/brent__method__extrema_8cpp.html#a002b2f4894492820fe708b1b7e7c5e70">EPSILON</a>);</div>
<div class="line"><a id="l00200" name="l00200"></a><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> &lt;&lt; <span class="stringliteral">&quot;passed\n&quot;</span>;</div>
<div class="line"><a id="l00201" name="l00201"></a><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>
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