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<p>Solve the equation \(f(x)=0\) using <a href="https://en.wikipedia.org/wiki/Bisection_method">bisection method</a>
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<a href="#details">More...</a></p>
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<div class="textblock"><code>#include <cmath></code><br />
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<code>#include <iostream></code><br />
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<code>#include <limits></code><br />
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</div><div class="textblock"><div class="dynheader">
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Include dependency graph for bisection_method.cpp:</div>
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<div class="dyncontent">
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Macros</h2></td></tr>
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#define </td><td class="memItemRight" valign="bottom"><b>EPSILON</b>    1e-6</td></tr>
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<tr class="memitem:a0a3abbca80bc98e7abcb3ae73abe0f14"><td class="memItemLeft" align="right" valign="top"><a id="a0a3abbca80bc98e7abcb3ae73abe0f14"></a>
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#define </td><td class="memItemRight" valign="bottom"><a class="el" href="../../d7/d6a/bisection__method_8cpp.html#a0a3abbca80bc98e7abcb3ae73abe0f14">MAX_ITERATIONS</a>   50000</td></tr>
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<tr class="memdesc:a0a3abbca80bc98e7abcb3ae73abe0f14"><td class="mdescLeft"> </td><td class="mdescRight">Maximum number of iterations to check. <br /></td></tr>
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Functions</h2></td></tr>
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<tr class="memitem:a2003b5b2dcfff0769b957ab5c968b03d"><td class="memItemLeft" align="right" valign="top">static double </td><td class="memItemRight" valign="bottom"><a class="el" href="../../d7/d6a/bisection__method_8cpp.html#a2003b5b2dcfff0769b957ab5c968b03d">eq</a> (double i)</td></tr>
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<tr class="separator:a2003b5b2dcfff0769b957ab5c968b03d"><td class="memSeparator" colspan="2"> </td></tr>
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<tr class="memitem:a1ab31b90bc584c635ec159468ceed9b2"><td class="memTemplParams" colspan="2">template<typename T > </td></tr>
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<tr class="memitem:a1ab31b90bc584c635ec159468ceed9b2"><td class="memTemplItemLeft" align="right" valign="top">int </td><td class="memTemplItemRight" valign="bottom"><a class="el" href="../../d7/d6a/bisection__method_8cpp.html#a1ab31b90bc584c635ec159468ceed9b2">sgn</a> (T val)</td></tr>
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<tr class="separator:a1ab31b90bc584c635ec159468ceed9b2"><td class="memSeparator" colspan="2"> </td></tr>
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<tr class="memitem:ae66f6b31b5ad750f1fe042a706a4e3d4"><td class="memItemLeft" align="right" valign="top">int </td><td class="memItemRight" valign="bottom"><a class="el" href="../../d7/d6a/bisection__method_8cpp.html#ae66f6b31b5ad750f1fe042a706a4e3d4">main</a> ()</td></tr>
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</table>
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<a name="details" id="details"></a><h2 class="groupheader">Detailed Description</h2>
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<div class="textblock"><p>Solve the equation \(f(x)=0\) using <a href="https://en.wikipedia.org/wiki/Bisection_method">bisection method</a> </p>
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<p>Given two points \(a\) and \(b\) such that \(f(a)<0\) and \(f(b)>0\), then the \((i+1)^\text{th}\) approximation is given by: </p><p class="formulaDsp">
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\[ x_{i+1} = \frac{a_i+b_i}{2} \]
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</p>
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<p> For the next iteration, the interval is selected as: \([a,x]\) if \(x>0\) or \([x,b]\) if \(x<0\). The Process is continued till a close enough approximation is achieved.</p>
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<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>
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</div><h2 class="groupheader">Function Documentation</h2>
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<a id="a2003b5b2dcfff0769b957ab5c968b03d"></a>
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<h2 class="memtitle"><span class="permalink"><a href="#a2003b5b2dcfff0769b957ab5c968b03d">◆ </a></span>eq()</h2>
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<td class="memname">static double eq </td>
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<td>(</td>
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<td class="paramtype">double </td>
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<td class="paramname"><em>i</em></td><td>)</td>
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<td></td>
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<span class="mlabels"><span class="mlabel">static</span></span> </td>
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<p>define \(f(x)\) to find root for </p>
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<div class="fragment"><div class="line"><a name="l00026"></a><span class="lineno"> 26</span>  {</div>
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<div class="line"><a name="l00027"></a><span class="lineno"> 27</span>  <span class="keywordflow">return</span> (<a class="codeRef" 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>
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<div class="line"><a name="l00028"></a><span class="lineno"> 28</span> }</div>
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<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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</div><!-- fragment --><div class="dynheader">
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Here is the call graph for this function:</div>
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<div class="dyncontent">
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<div class="center"><iframe scrolling="no" frameborder="0" src="../../d7/d6a/bisection__method_8cpp_a2003b5b2dcfff0769b957ab5c968b03d_cgraph.svg" width="170" height="38"><p><b>This browser is not able to show SVG: try Firefox, Chrome, Safari, or Opera instead.</b></p></iframe>
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<a id="ae66f6b31b5ad750f1fe042a706a4e3d4"></a>
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<h2 class="memtitle"><span class="permalink"><a href="#ae66f6b31b5ad750f1fe042a706a4e3d4">◆ </a></span>main()</h2>
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<td class="memname">int main </td>
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<td>(</td>
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<td class="paramtype">void </td>
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<td class="paramname"></td><td>)</td>
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<td></td>
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<p>main function </p>
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<div class="fragment"><div class="line"><a name="l00037"></a><span class="lineno"> 37</span>  {</div>
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<div class="line"><a name="l00038"></a><span class="lineno"> 38</span>  <span class="keywordtype">double</span> a = -1, b = 1, x, z;</div>
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<div class="line"><a name="l00039"></a><span class="lineno"> 39</span>  <span class="keywordtype">int</span> i;</div>
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<div class="line"><a name="l00040"></a><span class="lineno"> 40</span>  </div>
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<div class="line"><a name="l00041"></a><span class="lineno"> 41</span>  <span class="comment">// loop to find initial intervals a, b</span></div>
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<div class="line"><a name="l00042"></a><span class="lineno"> 42</span>  <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i < <a class="code" href="../../d7/d6a/bisection__method_8cpp.html#a0a3abbca80bc98e7abcb3ae73abe0f14">MAX_ITERATIONS</a>; i++) {</div>
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<div class="line"><a name="l00043"></a><span class="lineno"> 43</span>  z = <a class="code" href="../../d7/d6a/bisection__method_8cpp.html#a2003b5b2dcfff0769b957ab5c968b03d">eq</a>(a);</div>
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<div class="line"><a name="l00044"></a><span class="lineno"> 44</span>  x = <a class="code" href="../../d7/d6a/bisection__method_8cpp.html#a2003b5b2dcfff0769b957ab5c968b03d">eq</a>(b);</div>
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<div class="line"><a name="l00045"></a><span class="lineno"> 45</span>  <span class="keywordflow">if</span> (<a class="code" href="../../d7/d6a/bisection__method_8cpp.html#a1ab31b90bc584c635ec159468ceed9b2">sgn</a>(z) == <a class="code" href="../../d7/d6a/bisection__method_8cpp.html#a1ab31b90bc584c635ec159468ceed9b2">sgn</a>(x)) { <span class="comment">// same signs, increase interval</span></div>
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<div class="line"><a name="l00046"></a><span class="lineno"> 46</span>  b++;</div>
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<div class="line"><a name="l00047"></a><span class="lineno"> 47</span>  a--;</div>
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<div class="line"><a name="l00048"></a><span class="lineno"> 48</span>  } <span class="keywordflow">else</span> { <span class="comment">// if opposite signs, we got our interval</span></div>
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<div class="line"><a name="l00049"></a><span class="lineno"> 49</span>  <span class="keywordflow">break</span>;</div>
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<div class="line"><a name="l00050"></a><span class="lineno"> 50</span>  }</div>
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<div class="line"><a name="l00051"></a><span class="lineno"> 51</span>  }</div>
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<div class="line"><a name="l00052"></a><span class="lineno"> 52</span>  </div>
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<div class="line"><a name="l00053"></a><span class="lineno"> 53</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> << <span class="stringliteral">"\nFirst initial: "</span> << a;</div>
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<div class="line"><a name="l00054"></a><span class="lineno"> 54</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> << <span class="stringliteral">"\nSecond initial: "</span> << b;</div>
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<div class="line"><a name="l00055"></a><span class="lineno"> 55</span>  </div>
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<div class="line"><a name="l00056"></a><span class="lineno"> 56</span>  <span class="comment">// start iterations</span></div>
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<div class="line"><a name="l00057"></a><span class="lineno"> 57</span>  <span class="keywordflow">for</span> (i = 0; i < <a class="code" href="../../d7/d6a/bisection__method_8cpp.html#a0a3abbca80bc98e7abcb3ae73abe0f14">MAX_ITERATIONS</a>; i++) {</div>
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<div class="line"><a name="l00058"></a><span class="lineno"> 58</span>  x = (a + b) / 2;</div>
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<div class="line"><a name="l00059"></a><span class="lineno"> 59</span>  z = <a class="code" href="../../d7/d6a/bisection__method_8cpp.html#a2003b5b2dcfff0769b957ab5c968b03d">eq</a>(x);</div>
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<div class="line"><a name="l00060"></a><span class="lineno"> 60</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> << <span class="stringliteral">"\n\nz: "</span> << z << <span class="stringliteral">"\t["</span> << a << <span class="stringliteral">" , "</span> << b</div>
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<div class="line"><a name="l00061"></a><span class="lineno"> 61</span>  << <span class="stringliteral">" | Bisect: "</span> << x << <span class="stringliteral">"]"</span>;</div>
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<div class="line"><a name="l00062"></a><span class="lineno"> 62</span>  </div>
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<div class="line"><a name="l00063"></a><span class="lineno"> 63</span>  <span class="keywordflow">if</span> (z < 0) {</div>
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<div class="line"><a name="l00064"></a><span class="lineno"> 64</span>  a = x;</div>
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<div class="line"><a name="l00065"></a><span class="lineno"> 65</span>  } <span class="keywordflow">else</span> {</div>
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<div class="line"><a name="l00066"></a><span class="lineno"> 66</span>  b = x;</div>
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<div class="line"><a name="l00067"></a><span class="lineno"> 67</span>  }</div>
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<div class="line"><a name="l00068"></a><span class="lineno"> 68</span>  </div>
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<div class="line"><a name="l00069"></a><span class="lineno"> 69</span>  <span class="keywordflow">if</span> (std::abs(z) < <a class="code" href="../../de/dd3/newton__raphson__method_8cpp.html#a75f355966e0a97665eaf5a4c79baa7af">EPSILON</a>) <span class="comment">// stoping criteria</span></div>
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<div class="line"><a name="l00070"></a><span class="lineno"> 70</span>  <span class="keywordflow">break</span>;</div>
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<div class="line"><a name="l00071"></a><span class="lineno"> 71</span>  }</div>
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<div class="line"><a name="l00072"></a><span class="lineno"> 72</span>  </div>
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<div class="line"><a name="l00073"></a><span class="lineno"> 73</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> << <span class="stringliteral">"\n\nRoot: "</span> << x << <span class="stringliteral">"\t\tSteps: "</span> << i << <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/endl.html">std::endl</a>;</div>
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<div class="line"><a name="l00074"></a><span class="lineno"> 74</span>  <span class="keywordflow">return</span> 0;</div>
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<div class="line"><a name="l00075"></a><span class="lineno"> 75</span> }</div>
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<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>
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<div class="ttc" id="abisection__method_8cpp_html_a0a3abbca80bc98e7abcb3ae73abe0f14"><div class="ttname"><a href="../../d7/d6a/bisection__method_8cpp.html#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>
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<div class="ttc" id="abisection__method_8cpp_html_a1ab31b90bc584c635ec159468ceed9b2"><div class="ttname"><a href="../../d7/d6a/bisection__method_8cpp.html#a1ab31b90bc584c635ec159468ceed9b2">sgn</a></div><div class="ttdeci">int sgn(T val)</div><div class="ttdef"><b>Definition:</b> bisection_method.cpp:32</div></div>
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<div class="ttc" id="abisection__method_8cpp_html_a2003b5b2dcfff0769b957ab5c968b03d"><div class="ttname"><a href="../../d7/d6a/bisection__method_8cpp.html#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>
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<div class="ttc" id="anewton__raphson__method_8cpp_html_a75f355966e0a97665eaf5a4c79baa7af"><div class="ttname"><a href="../../de/dd3/newton__raphson__method_8cpp.html#a75f355966e0a97665eaf5a4c79baa7af">EPSILON</a></div><div class="ttdeci">constexpr double EPSILON</div><div class="ttdoc">system accuracy limit</div><div class="ttdef"><b>Definition:</b> newton_raphson_method.cpp:20</div></div>
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<h2 class="memtitle"><span class="permalink"><a href="#a1ab31b90bc584c635ec159468ceed9b2">◆ </a></span>sgn()</h2>
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template<typename T > </div>
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<td class="memname">int sgn </td>
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<td>(</td>
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<td class="paramtype">T </td>
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<td class="paramname"><em>val</em></td><td>)</td>
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<p>get the sign of any given number </p>
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<div class="fragment"><div class="line"><a name="l00032"></a><span class="lineno"> 32</span>  {</div>
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<div class="line"><a name="l00033"></a><span class="lineno"> 33</span>  <span class="keywordflow">return</span> (T(0) < val) - (val < T(0));</div>
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<div class="line"><a name="l00034"></a><span class="lineno"> 34</span> }</div>
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