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<div class="title">golden_search_extrema.cpp File Reference</div> </div>
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<p>Find extrema of a univariate real function in a given interval using <a href="https://en.wikipedia.org/wiki/Golden-section_search">golden section search algorithm</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 golden_search_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="separator:a525335710b53cb064ca56b936120431e"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:a002b2f4894492820fe708b1b7e7c5e70"><td class="memItemLeft" align="right" valign="top"><a id="a002b2f4894492820fe708b1b7e7c5e70"></a>
#define&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../d6/d7a/golden__search__extrema_8cpp.html#a002b2f4894492820fe708b1b7e7c5e70">EPSILON</a>&#160;&#160;&#160;1e-7</td></tr>
<tr class="memdesc:a002b2f4894492820fe708b1b7e7c5e70"><td class="mdescLeft">&#160;</td><td class="mdescRight">solution accuracy limit <br /></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="../../d6/d7a/golden__search__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;<a class="el" href="../../d4/d32/fibonacci__fast_8cpp.html#a3ba232425d45f9e9c0b87a8cf7ab69d9">f</a>, double lim_a, double lim_b)</td></tr>
<tr class="memdesc:a1aa76a6d5fd4d333f9072beff1dc486b"><td class="mdescLeft">&#160;</td><td class="mdescRight">Get the minima of a function in the given interval. To get the maxima, simply negate the function. The golden ratio used here is: <a href="../../d6/d7a/golden__search__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"><a id="a1440a7779ac56f47a3f355ce4a8c7da0"></a>
void&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../d6/d7a/golden__search__extrema_8cpp.html#a1440a7779ac56f47a3f355ce4a8c7da0">test1</a> ()</td></tr>
<tr class="memdesc:a1440a7779ac56f47a3f355ce4a8c7da0"><td class="mdescLeft">&#160;</td><td class="mdescRight">Test function to find minima for the function \(f(x)= (x-2)^2\) in the interval \([1,5]\) <br />
Expected result = 2. <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"><a id="a0283886819c7c140a023582b7269e2d0"></a>
void&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../d6/d7a/golden__search__extrema_8cpp.html#a0283886819c7c140a023582b7269e2d0">test2</a> ()</td></tr>
<tr class="memdesc:a0283886819c7c140a023582b7269e2d0"><td class="mdescLeft">&#160;</td><td class="mdescRight">Test function to find <em>maxima</em> 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="separator:a0283886819c7c140a023582b7269e2d0"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:a6d0455dd5c30adda100e95f0423c786e"><td class="memItemLeft" align="right" valign="top"><a id="a6d0455dd5c30adda100e95f0423c786e"></a>
void&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../d6/d7a/golden__search__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: \(\pi\approx 3.14159265358979312\). <br /></td></tr>
<tr class="separator:a6d0455dd5c30adda100e95f0423c786e"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:ae66f6b31b5ad750f1fe042a706a4e3d4"><td class="memItemLeft" align="right" valign="top">int&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../d6/d7a/golden__search__extrema_8cpp.html#ae66f6b31b5ad750f1fe042a706a4e3d4">main</a> ()</td></tr>
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<a name="details" id="details"></a><h2 class="groupheader">Detailed Description</h2>
<div class="textblock"><p>Find extrema of a univariate real function in a given interval using <a href="https://en.wikipedia.org/wiki/Golden-section_search">golden section search algorithm</a>. </p>
<dl class="section see"><dt>See also</dt><dd><a class="el" href="../../db/d01/brent__method__extrema_8cpp.html" title="Find real extrema of a univariate real function in a given interval using Brent&#39;s method.">brent_method_extrema.cpp</a> </dd></dl>
<dl class="section author"><dt>Author</dt><dd><a href="https://github.com/kvedala">Krishna Vedala</a> </dd></dl>
</div><h2 class="groupheader">Function Documentation</h2>
<a id="a1aa76a6d5fd4d333f9072beff1dc486b"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a1aa76a6d5fd4d333f9072beff1dc486b">&#9670;&nbsp;</a></span>get_minima()</h2>
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<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>
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</div><div class="memdoc">
<p>Get the minima of a function in the given interval. To get the maxima, simply negate the function. The golden ratio used here is: </p>
<p class="formulaDsp">
\[ k=\frac{3-\sqrt{5}}{2} \approx 0.381966\ldots\]
</p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">f</td><td>function to get minima 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>local minima found in the interval </dd></dl>
<div class="fragment"><div class="line"><a name="l00030"></a><span class="lineno"> 30</span>&#160; {</div>
<div class="line"><a name="l00031"></a><span class="lineno"> 31</span>&#160; uint32_t iters = 0;</div>
<div class="line"><a name="l00032"></a><span class="lineno"> 32</span>&#160; <span class="keywordtype">double</span> c, d;</div>
<div class="line"><a name="l00033"></a><span class="lineno"> 33</span>&#160; <span class="keywordtype">double</span> prev_mean, mean = <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/types/numeric_limits/infinity.html">std::numeric_limits&lt;double&gt;::infinity</a>();</div>
<div class="line"><a name="l00034"></a><span class="lineno"> 34</span>&#160; </div>
<div class="line"><a name="l00035"></a><span class="lineno"> 35</span>&#160; <span class="comment">// golden ratio value</span></div>
<div class="line"><a name="l00036"></a><span class="lineno"> 36</span>&#160; <span class="keyword">const</span> <span class="keywordtype">double</span> M_GOLDEN_RATIO = (1.f + <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/math/sqrt.html">std::sqrt</a>(5.<a class="code" href="../../d4/d32/fibonacci__fast_8cpp.html#a3ba232425d45f9e9c0b87a8cf7ab69d9">f</a>)) / 2.<a class="code" href="../../d4/d32/fibonacci__fast_8cpp.html#a3ba232425d45f9e9c0b87a8cf7ab69d9">f</a>;</div>
<div class="line"><a name="l00037"></a><span class="lineno"> 37</span>&#160; </div>
<div class="line"><a name="l00038"></a><span class="lineno"> 38</span>&#160; <span class="comment">// ensure that lim_a &lt; lim_b</span></div>
<div class="line"><a name="l00039"></a><span class="lineno"> 39</span>&#160; <span class="keywordflow">if</span> (lim_a &gt; lim_b) {</div>
<div class="line"><a name="l00040"></a><span class="lineno"> 40</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/algorithm/swap.html">std::swap</a>(lim_a, lim_b);</div>
<div class="line"><a name="l00041"></a><span class="lineno"> 41</span>&#160; } <span class="keywordflow">else</span> <span class="keywordflow">if</span> (std::abs(lim_a - lim_b) &lt;= <a class="code" href="../../d6/d7a/golden__search__extrema_8cpp.html#a002b2f4894492820fe708b1b7e7c5e70">EPSILON</a>) {</div>
<div class="line"><a name="l00042"></a><span class="lineno"> 42</span>&#160; <a class="codeRef" 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" href="../../d6/d7a/golden__search__extrema_8cpp.html#a002b2f4894492820fe708b1b7e7c5e70">EPSILON</a> &lt;&lt; <span class="stringliteral">&quot;\n&quot;</span>;</div>
<div class="line"><a name="l00043"></a><span class="lineno"> 43</span>&#160; <span class="keywordflow">return</span> lim_a;</div>
<div class="line"><a name="l00044"></a><span class="lineno"> 44</span>&#160; }</div>
<div class="line"><a name="l00045"></a><span class="lineno"> 45</span>&#160; </div>
<div class="line"><a name="l00046"></a><span class="lineno"> 46</span>&#160; <span class="keywordflow">do</span> {</div>
<div class="line"><a name="l00047"></a><span class="lineno"> 47</span>&#160; prev_mean = mean;</div>
<div class="line"><a name="l00048"></a><span class="lineno"> 48</span>&#160; </div>
<div class="line"><a name="l00049"></a><span class="lineno"> 49</span>&#160; <span class="comment">// compute the section ratio width</span></div>
<div class="line"><a name="l00050"></a><span class="lineno"> 50</span>&#160; <span class="keywordtype">double</span> ratio = (lim_b - lim_a) / M_GOLDEN_RATIO;</div>
<div class="line"><a name="l00051"></a><span class="lineno"> 51</span>&#160; c = lim_b - ratio; <span class="comment">// right-side section start</span></div>
<div class="line"><a name="l00052"></a><span class="lineno"> 52</span>&#160; d = lim_a + ratio; <span class="comment">// left-side section end</span></div>
<div class="line"><a name="l00053"></a><span class="lineno"> 53</span>&#160; </div>
<div class="line"><a name="l00054"></a><span class="lineno"> 54</span>&#160; <span class="keywordflow">if</span> (<a class="code" href="../../d4/d32/fibonacci__fast_8cpp.html#a3ba232425d45f9e9c0b87a8cf7ab69d9">f</a>(c) &lt; <a class="code" href="../../d4/d32/fibonacci__fast_8cpp.html#a3ba232425d45f9e9c0b87a8cf7ab69d9">f</a>(d)) {</div>
<div class="line"><a name="l00055"></a><span class="lineno"> 55</span>&#160; <span class="comment">// select left section</span></div>
<div class="line"><a name="l00056"></a><span class="lineno"> 56</span>&#160; lim_b = d;</div>
<div class="line"><a name="l00057"></a><span class="lineno"> 57</span>&#160; } <span class="keywordflow">else</span> {</div>
<div class="line"><a name="l00058"></a><span class="lineno"> 58</span>&#160; <span class="comment">// selct right section</span></div>
<div class="line"><a name="l00059"></a><span class="lineno"> 59</span>&#160; lim_a = c;</div>
<div class="line"><a name="l00060"></a><span class="lineno"> 60</span>&#160; }</div>
<div class="line"><a name="l00061"></a><span class="lineno"> 61</span>&#160; </div>
<div class="line"><a name="l00062"></a><span class="lineno"> 62</span>&#160; mean = (lim_a + lim_b) / 2.<a class="code" href="../../d4/d32/fibonacci__fast_8cpp.html#a3ba232425d45f9e9c0b87a8cf7ab69d9">f</a>;</div>
<div class="line"><a name="l00063"></a><span class="lineno"> 63</span>&#160; iters++;</div>
<div class="line"><a name="l00064"></a><span class="lineno"> 64</span>&#160; </div>
<div class="line"><a name="l00065"></a><span class="lineno"> 65</span>&#160; <span class="comment">// continue till the interval width is greater than sqrt(system epsilon)</span></div>
<div class="line"><a name="l00066"></a><span class="lineno"> 66</span>&#160; } <span class="keywordflow">while</span> (std::abs(lim_a - lim_b) &gt; <a class="code" href="../../d6/d7a/golden__search__extrema_8cpp.html#a002b2f4894492820fe708b1b7e7c5e70">EPSILON</a>);</div>
<div class="line"><a name="l00067"></a><span class="lineno"> 67</span>&#160; </div>
<div class="line"><a name="l00068"></a><span class="lineno"> 68</span>&#160; <a class="codeRef" 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 name="l00069"></a><span class="lineno"> 69</span>&#160; <span class="keywordflow">return</span> prev_mean;</div>
<div class="line"><a name="l00070"></a><span class="lineno"> 70</span>&#160;}</div>
</div><!-- fragment -->
</div>
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<a id="ae66f6b31b5ad750f1fe042a706a4e3d4"></a>
<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>
<td>(</td>
<td class="paramname"></td><td>)</td>
<td></td>
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</div><div class="memdoc">
<p>Main function </p>
<div class="fragment"><div class="line"><a name="l00139"></a><span class="lineno"> 139</span>&#160; {</div>
<div class="line"><a name="l00140"></a><span class="lineno"> 140</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a>.precision(9);</div>
<div class="line"><a name="l00141"></a><span class="lineno"> 141</span>&#160; </div>
<div class="line"><a name="l00142"></a><span class="lineno"> 142</span>&#160; <a class="codeRef" 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" href="../../d6/d7a/golden__search__extrema_8cpp.html#a002b2f4894492820fe708b1b7e7c5e70">EPSILON</a></div>
<div class="line"><a name="l00143"></a><span class="lineno"> 143</span>&#160; &lt;&lt; <span class="stringliteral">&quot;\n&quot;</span>;</div>
<div class="line"><a name="l00144"></a><span class="lineno"> 144</span>&#160; </div>
<div class="line"><a name="l00145"></a><span class="lineno"> 145</span>&#160; <a class="code" href="../../d6/d7a/golden__search__extrema_8cpp.html#a1440a7779ac56f47a3f355ce4a8c7da0">test1</a>();</div>
<div class="line"><a name="l00146"></a><span class="lineno"> 146</span>&#160; <a class="code" href="../../d6/d7a/golden__search__extrema_8cpp.html#a0283886819c7c140a023582b7269e2d0">test2</a>();</div>
<div class="line"><a name="l00147"></a><span class="lineno"> 147</span>&#160; <a class="code" href="../../d6/d7a/golden__search__extrema_8cpp.html#a6d0455dd5c30adda100e95f0423c786e">test3</a>();</div>
<div class="line"><a name="l00148"></a><span class="lineno"> 148</span>&#160; </div>
<div class="line"><a name="l00149"></a><span class="lineno"> 149</span>&#160; <span class="keywordflow">return</span> 0;</div>
<div class="line"><a name="l00150"></a><span class="lineno"> 150</span>&#160;}</div>
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<div class="ttc" id="agolden__search__extrema_8cpp_html_a002b2f4894492820fe708b1b7e7c5e70"><div class="ttname"><a href="../../d6/d7a/golden__search__extrema_8cpp.html#a002b2f4894492820fe708b1b7e7c5e70">EPSILON</a></div><div class="ttdeci">#define EPSILON</div><div class="ttdoc">solution accuracy limit</div><div class="ttdef"><b>Definition:</b> golden_search_extrema.cpp:17</div></div>
<div class="ttc" id="afibonacci__fast_8cpp_html_a3ba232425d45f9e9c0b87a8cf7ab69d9"><div class="ttname"><a href="../../d4/d32/fibonacci__fast_8cpp.html#a3ba232425d45f9e9c0b87a8cf7ab69d9">f</a></div><div class="ttdeci">uint64_t f[MAX]</div><div class="ttdef"><b>Definition:</b> fibonacci_fast.cpp:27</div></div>
<div class="ttc" id="agolden__search__extrema_8cpp_html_a6d0455dd5c30adda100e95f0423c786e"><div class="ttname"><a href="../../d6/d7a/golden__search__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> golden_search_extrema.cpp:123</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>
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<div class="ttc" id="ainfinity_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/types/numeric_limits/infinity.html">std::numeric_limits::infinity</a></div><div class="ttdeci">T infinity(T... args)</div></div>
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