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<div class="headertitle"><div class="title">bisection_method.cpp File Reference</div></div>
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<p>Solve the equation \(f(x)=0\) using <a href="https://en.wikipedia.org/wiki/Bisection_method" target="_blank">bisection method</a>
<a href="#details">More...</a></p>
<div class="textblock"><code>#include &lt;cmath&gt;</code><br />
<code>#include &lt;iostream&gt;</code><br />
<code>#include &lt;limits&gt;</code><br />
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Include dependency graph for bisection_method.cpp:</div>
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Macros</h2></td></tr>
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#define&#160;</td><td class="memItemRight" valign="bottom"><b>MAX_ITERATIONS</b>&#160;&#160;&#160;50000</td></tr>
<tr class="memdesc:a0a3abbca80bc98e7abcb3ae73abe0f14"><td class="mdescLeft">&#160;</td><td class="mdescRight">Maximum number of iterations to check. <br /></td></tr>
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Functions</h2></td></tr>
<tr class="memitem:a2003b5b2dcfff0769b957ab5c968b03d" id="r_a2003b5b2dcfff0769b957ab5c968b03d"><td class="memItemLeft" align="right" valign="top">static double&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="#a2003b5b2dcfff0769b957ab5c968b03d">eq</a> (double i)</td></tr>
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<tr class="memitem:a1ab31b90bc584c635ec159468ceed9b2" id="r_a1ab31b90bc584c635ec159468ceed9b2"><td class="memTemplParams" colspan="2">template&lt;typename T &gt; </td></tr>
<tr class="memitem:a1ab31b90bc584c635ec159468ceed9b2"><td class="memTemplItemLeft" align="right" valign="top">int&#160;</td><td class="memTemplItemRight" valign="bottom"><a class="el" href="#a1ab31b90bc584c635ec159468ceed9b2">sgn</a> (T val)</td></tr>
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<a name="details" id="details"></a><h2 class="groupheader">Detailed Description</h2>
<div class="textblock"><p>Solve the equation \(f(x)=0\) using <a href="https://en.wikipedia.org/wiki/Bisection_method" target="_blank">bisection method</a> </p>
<p>Given two points \(a\) and \(b\) such that \(f(a)&lt;0\) and \(f(b)&gt;0\), then the \((i+1)^\text{th}\) approximation is given by: </p><p class="formulaDsp">
\[
x_{i+1} = \frac{a_i+b_i}{2}
\]
</p>
<p> For the next iteration, the interval is selected as: \([a,x]\) if \(x&gt;0\) or \([x,b]\) if \(x&lt;0\). The Process is continued till a close enough approximation is achieved.</p>
<dl class="section see"><dt>See also</dt><dd><a class="el" href="../../de/dd3/newton__raphson__method_8cpp.html" title="Solve the equation using Newton-Raphson method for both real and complex solutions.">newton_raphson_method.cpp</a>, <a class="el" href="../../dd/d29/false__position_8cpp.html" title="Solve the equation using false position method, also known as the Secant method.">false_position.cpp</a>, secant_method.cpp </dd></dl>
</div><h2 class="groupheader">Macro Definition Documentation</h2>
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<h2 class="memtitle"><span class="permalink"><a href="#a002b2f4894492820fe708b1b7e7c5e70">&#9670;&#160;</a></span>EPSILON</h2>
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<td class="memname">#define EPSILON&#160;&#160;&#160; 1e-6</td>
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<div class="fragment"><div class="line"><span class="lineno"> 20</span><span class="preprocessor">#define EPSILON \</span></div>
<div class="line"><span class="lineno"> 21</span><span class="preprocessor"> 1e-6 </span><span class="comment">// std::numeric_limits&lt;double&gt;::epsilon() ///&lt; system accuracy limit</span></div>
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<h2 class="groupheader">Function Documentation</h2>
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<h2 class="memtitle"><span class="permalink"><a href="#a2003b5b2dcfff0769b957ab5c968b03d">&#9670;&#160;</a></span>eq()</h2>
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<td class="memname">static double eq </td>
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<td class="paramtype">double</td> <td class="paramname"><span class="paramname"><em>i</em></span></td><td>)</td>
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<p>define \(f(x)\) to find root for </p>
<div class="fragment"><div class="line"><span class="lineno"> 26</span> {</div>
<div class="line"><span class="lineno"> 27</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>(i, 3) - (4 * i) - 9); <span class="comment">// original equation</span></div>
<div class="line"><span class="lineno"> 28</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="#ae66f6b31b5ad750f1fe042a706a4e3d4">&#9670;&#160;</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"><span class="lineno"> 37</span> {</div>
<div class="line"><span class="lineno"> 38</span> <span class="keywordtype">double</span> a = -1, b = 1, x, z;</div>
<div class="line"><span class="lineno"> 39</span> <span class="keywordtype">int</span> i;</div>
<div class="line"><span class="lineno"> 40</span> </div>
<div class="line"><span class="lineno"> 41</span> <span class="comment">// loop to find initial intervals a, b</span></div>
<div class="line"><span class="lineno"> 42</span> <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i &lt; <a class="code hl_define" href="#a0a3abbca80bc98e7abcb3ae73abe0f14">MAX_ITERATIONS</a>; i++) {</div>
<div class="line"><span class="lineno"> 43</span> z = <a class="code hl_function" href="#a2003b5b2dcfff0769b957ab5c968b03d">eq</a>(a);</div>
<div class="line"><span class="lineno"> 44</span> x = <a class="code hl_function" href="#a2003b5b2dcfff0769b957ab5c968b03d">eq</a>(b);</div>
<div class="line"><span class="lineno"> 45</span> <span class="keywordflow">if</span> (<a class="code hl_function" href="#a1ab31b90bc584c635ec159468ceed9b2">sgn</a>(z) == <a class="code hl_function" href="#a1ab31b90bc584c635ec159468ceed9b2">sgn</a>(x)) { <span class="comment">// same signs, increase interval</span></div>
<div class="line"><span class="lineno"> 46</span> b++;</div>
<div class="line"><span class="lineno"> 47</span> a--;</div>
<div class="line"><span class="lineno"> 48</span> } <span class="keywordflow">else</span> { <span class="comment">// if opposite signs, we got our interval</span></div>
<div class="line"><span class="lineno"> 49</span> <span class="keywordflow">break</span>;</div>
<div class="line"><span class="lineno"> 50</span> }</div>
<div class="line"><span class="lineno"> 51</span> }</div>
<div class="line"><span class="lineno"> 52</span> </div>
<div class="line"><span class="lineno"> 53</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;\nFirst initial: &quot;</span> &lt;&lt; a;</div>
<div class="line"><span class="lineno"> 54</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;\nSecond initial: &quot;</span> &lt;&lt; b;</div>
<div class="line"><span class="lineno"> 55</span> </div>
<div class="line"><span class="lineno"> 56</span> <span class="comment">// start iterations</span></div>
<div class="line"><span class="lineno"> 57</span> <span class="keywordflow">for</span> (i = 0; i &lt; <a class="code hl_define" href="#a0a3abbca80bc98e7abcb3ae73abe0f14">MAX_ITERATIONS</a>; i++) {</div>
<div class="line"><span class="lineno"> 58</span> x = (a + b) / 2;</div>
<div class="line"><span class="lineno"> 59</span> z = <a class="code hl_function" href="#a2003b5b2dcfff0769b957ab5c968b03d">eq</a>(x);</div>
<div class="line"><span class="lineno"> 60</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;\n\nz: &quot;</span> &lt;&lt; z &lt;&lt; <span class="stringliteral">&quot;\t[&quot;</span> &lt;&lt; a &lt;&lt; <span class="stringliteral">&quot; , &quot;</span> &lt;&lt; b</div>
<div class="line"><span class="lineno"> 61</span> &lt;&lt; <span class="stringliteral">&quot; | Bisect: &quot;</span> &lt;&lt; x &lt;&lt; <span class="stringliteral">&quot;]&quot;</span>;</div>
<div class="line"><span class="lineno"> 62</span> </div>
<div class="line"><span class="lineno"> 63</span> <span class="keywordflow">if</span> (z &lt; 0) {</div>
<div class="line"><span class="lineno"> 64</span> a = x;</div>
<div class="line"><span class="lineno"> 65</span> } <span class="keywordflow">else</span> {</div>
<div class="line"><span class="lineno"> 66</span> b = x;</div>
<div class="line"><span class="lineno"> 67</span> }</div>
<div class="line"><span class="lineno"> 68</span> </div>
<div class="line"><span class="lineno"> 69</span> <span class="keywordflow">if</span> (std::abs(z) &lt; EPSILON) <span class="comment">// stoping criteria</span></div>
<div class="line"><span class="lineno"> 70</span> <span class="keywordflow">break</span>;</div>
<div class="line"><span class="lineno"> 71</span> }</div>
<div class="line"><span class="lineno"> 72</span> </div>
<div class="line"><span class="lineno"> 73</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;\n\nRoot: &quot;</span> &lt;&lt; x &lt;&lt; <span class="stringliteral">&quot;\t\tSteps: &quot;</span> &lt;&lt; i &lt;&lt; <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/endl.html">std::endl</a>;</div>
<div class="line"><span class="lineno"> 74</span> <span class="keywordflow">return</span> 0;</div>
<div class="line"><span class="lineno"> 75</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::cout</a></div></div>
<div class="ttc" id="abisection__method_8cpp_html_a0a3abbca80bc98e7abcb3ae73abe0f14"><div class="ttname"><a href="#a0a3abbca80bc98e7abcb3ae73abe0f14">MAX_ITERATIONS</a></div><div class="ttdeci">#define MAX_ITERATIONS</div><div class="ttdoc">Maximum number of iterations to check.</div><div class="ttdef"><b>Definition</b> bisection_method.cpp:22</div></div>
<div class="ttc" id="abisection__method_8cpp_html_a1ab31b90bc584c635ec159468ceed9b2"><div class="ttname"><a href="#a1ab31b90bc584c635ec159468ceed9b2">sgn</a></div><div class="ttdeci">int sgn(T val)</div><div class="ttdef"><b>Definition</b> bisection_method.cpp:32</div></div>
<div class="ttc" id="abisection__method_8cpp_html_a2003b5b2dcfff0769b957ab5c968b03d"><div class="ttname"><a href="#a2003b5b2dcfff0769b957ab5c968b03d">eq</a></div><div class="ttdeci">static double eq(double i)</div><div class="ttdef"><b>Definition</b> bisection_method.cpp:26</div></div>
<div class="ttc" id="aendl_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/io/manip/endl.html">std::endl</a></div><div class="ttdeci">T endl(T... args)</div></div>
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<h2 class="memtitle"><span class="permalink"><a href="#a1ab31b90bc584c635ec159468ceed9b2">&#9670;&#160;</a></span>sgn()</h2>
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<p>get the sign of any given number </p>
<div class="fragment"><div class="line"><span class="lineno"> 32</span> {</div>
<div class="line"><span class="lineno"> 33</span> <span class="keywordflow">return</span> (T(0) &lt; val) - (val &lt; T(0));</div>
<div class="line"><span class="lineno"> 34</span>}</div>
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