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<p>Solve the equation \(f(x)=0\) using <a href="https://en.wikipedia.org/wiki/Regula_falsi">false position method</a>, also known as the Secant method.
<a href="#details">More...</a></p>
<div class="textblock"><code>#include &lt;cmath&gt;</code><br />
<code>#include &lt;cstdlib&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 false_position.cpp:</div>
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#define&#160;</td><td class="memItemRight" valign="bottom"><b>EPSILON</b>&#160;&#160;&#160;1e-6</td></tr>
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#define&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../dd/d29/false__position_8cpp.html#a0a3abbca80bc98e7abcb3ae73abe0f14">MAX_ITERATIONS</a>&#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"><td class="memItemLeft" align="right" valign="top">static double&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../dd/d29/false__position_8cpp.html#a2003b5b2dcfff0769b957ab5c968b03d">eq</a> (double i)</td></tr>
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<tr class="memitem: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="../../dd/d29/false__position_8cpp.html#a1ab31b90bc584c635ec159468ceed9b2">sgn</a> (T val)</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="../../dd/d29/false__position_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>Solve the equation \(f(x)=0\) using <a href="https://en.wikipedia.org/wiki/Regula_falsi">false position method</a>, also known as the Secant method. </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">
\[ x_{i+1} = \frac{a_i\cdot f(b_i) - b_i\cdot f(a_i)}{f(b_i) - f(a_i)} \]
</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="../../d7/d6a/bisection__method_8cpp.html" title="Solve the equation using bisection method">bisection_method.cpp</a> </dd></dl>
</div><h2 class="groupheader">Function Documentation</h2>
<a id="a2003b5b2dcfff0769b957ab5c968b03d"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a2003b5b2dcfff0769b957ab5c968b03d">&#9670;&nbsp;</a></span>eq()</h2>
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<td class="memname">static double eq </td>
<td>(</td>
<td class="paramtype">double&#160;</td>
<td class="paramname"><em>i</em></td><td>)</td>
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<p>define \(f(x)\) to find root for </p>
<div class="fragment"><div class="line"><a name="l00028"></a><span class="lineno"> 28</span>&#160; {</div>
<div class="line"><a name="l00029"></a><span class="lineno"> 29</span>&#160; <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">// origial equation</span></div>
<div class="line"><a name="l00030"></a><span class="lineno"> 30</span>&#160;}</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 name="l00039"></a><span class="lineno"> 39</span>&#160; {</div>
<div class="line"><a name="l00040"></a><span class="lineno"> 40</span>&#160; <span class="keywordtype">double</span> <a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a> = -1, b = 1, x, z, m, n, c;</div>
<div class="line"><a name="l00041"></a><span class="lineno"> 41</span>&#160; <span class="keywordtype">int</span> i;</div>
<div class="line"><a name="l00042"></a><span class="lineno"> 42</span>&#160; </div>
<div class="line"><a name="l00043"></a><span class="lineno"> 43</span>&#160; <span class="comment">// loop to find initial intervals a, b</span></div>
<div class="line"><a name="l00044"></a><span class="lineno"> 44</span>&#160; <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i &lt; <a class="code" href="../../dd/d29/false__position_8cpp.html#a0a3abbca80bc98e7abcb3ae73abe0f14">MAX_ITERATIONS</a>; i++) {</div>
<div class="line"><a name="l00045"></a><span class="lineno"> 45</span>&#160; z = <a class="code" href="../../dd/d29/false__position_8cpp.html#a2003b5b2dcfff0769b957ab5c968b03d">eq</a>(<a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a>);</div>
<div class="line"><a name="l00046"></a><span class="lineno"> 46</span>&#160; x = <a class="code" href="../../dd/d29/false__position_8cpp.html#a2003b5b2dcfff0769b957ab5c968b03d">eq</a>(b);</div>
<div class="line"><a name="l00047"></a><span class="lineno"> 47</span>&#160; <span class="keywordflow">if</span> (<a class="code" href="../../dd/d29/false__position_8cpp.html#a1ab31b90bc584c635ec159468ceed9b2">sgn</a>(z) == <a class="code" href="../../dd/d29/false__position_8cpp.html#a1ab31b90bc584c635ec159468ceed9b2">sgn</a>(x)) { <span class="comment">// same signs, increase interval</span></div>
<div class="line"><a name="l00048"></a><span class="lineno"> 48</span>&#160; b++;</div>
<div class="line"><a name="l00049"></a><span class="lineno"> 49</span>&#160; <a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a>--;</div>
<div class="line"><a name="l00050"></a><span class="lineno"> 50</span>&#160; } <span class="keywordflow">else</span> { <span class="comment">// if opposite signs, we got our interval</span></div>
<div class="line"><a name="l00051"></a><span class="lineno"> 51</span>&#160; <span class="keywordflow">break</span>;</div>
<div class="line"><a name="l00052"></a><span class="lineno"> 52</span>&#160; }</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; </div>
<div class="line"><a name="l00055"></a><span class="lineno"> 55</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;\nFirst initial: &quot;</span> &lt;&lt; <a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a>;</div>
<div class="line"><a name="l00056"></a><span class="lineno"> 56</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;\nSecond initial: &quot;</span> &lt;&lt; b;</div>
<div class="line"><a name="l00057"></a><span class="lineno"> 57</span>&#160; </div>
<div class="line"><a name="l00058"></a><span class="lineno"> 58</span>&#160; <span class="keywordflow">for</span> (i = 0; i &lt; <a class="code" href="../../dd/d29/false__position_8cpp.html#a0a3abbca80bc98e7abcb3ae73abe0f14">MAX_ITERATIONS</a>; i++) {</div>
<div class="line"><a name="l00059"></a><span class="lineno"> 59</span>&#160; m = <a class="code" href="../../dd/d29/false__position_8cpp.html#a2003b5b2dcfff0769b957ab5c968b03d">eq</a>(<a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a>);</div>
<div class="line"><a name="l00060"></a><span class="lineno"> 60</span>&#160; n = <a class="code" href="../../dd/d29/false__position_8cpp.html#a2003b5b2dcfff0769b957ab5c968b03d">eq</a>(b);</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; c = ((<a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a> * n) - (b * m)) / (n - m);</div>
<div class="line"><a name="l00063"></a><span class="lineno"> 63</span>&#160; </div>
<div class="line"><a name="l00064"></a><span class="lineno"> 64</span>&#160; <a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a> = c;</div>
<div class="line"><a name="l00065"></a><span class="lineno"> 65</span>&#160; z = <a class="code" href="../../dd/d29/false__position_8cpp.html#a2003b5b2dcfff0769b957ab5c968b03d">eq</a>(c);</div>
<div class="line"><a name="l00066"></a><span class="lineno"> 66</span>&#160; </div>
<div class="line"><a name="l00067"></a><span class="lineno"> 67</span>&#160; <span class="keywordflow">if</span> (std::abs(z) &lt; EPSILON) { <span class="comment">// stoping criteria</span></div>
<div class="line"><a name="l00068"></a><span class="lineno"> 68</span>&#160; <span class="keywordflow">break</span>;</div>
<div class="line"><a name="l00069"></a><span class="lineno"> 69</span>&#160; }</div>
<div class="line"><a name="l00070"></a><span class="lineno"> 70</span>&#160; }</div>
<div class="line"><a name="l00071"></a><span class="lineno"> 71</span>&#160; </div>
<div class="line"><a name="l00072"></a><span class="lineno"> 72</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;\n\nRoot: &quot;</span> &lt;&lt; c &lt;&lt; <span class="stringliteral">&quot;\t\tSteps: &quot;</span> &lt;&lt; i &lt;&lt; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/endl.html">std::endl</a>;</div>
<div class="line"><a name="l00073"></a><span class="lineno"> 73</span>&#160; <span class="keywordflow">return</span> 0;</div>
<div class="line"><a name="l00074"></a><span class="lineno"> 74</span>&#160;}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#a1ab31b90bc584c635ec159468ceed9b2">&#9670;&nbsp;</a></span>sgn()</h2>
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template&lt;typename T &gt; </div>
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<td class="memname">int sgn </td>
<td>(</td>
<td class="paramtype">T&#160;</td>
<td class="paramname"><em>val</em></td><td>)</td>
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<p>get the sign of any given number </p>
<div class="fragment"><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="keywordflow">return</span> (T(0) &lt; val) - (val &lt; T(0));</div>
<div class="line"><a name="l00036"></a><span class="lineno"> 36</span>&#160;}</div>
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<div class="ttc" id="afalse__position_8cpp_html_a0a3abbca80bc98e7abcb3ae73abe0f14"><div class="ttname"><a href="../../dd/d29/false__position_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> false_position.cpp:24</div></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>
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<div class="ttc" id="amatrix__exponentiation_8cpp_html_a9ecd98ce0064e2f7f97bb6a92f58ee72"><div class="ttname"><a href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a></div><div class="ttdeci">vector&lt; ll &gt; a</div><div class="ttdef"><b>Definition:</b> matrix_exponentiation.cpp:53</div></div>
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