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<p>Solve the equation \(f(x)=0\) using <a href="https://en.wikipedia.org/wiki/Newton%27s_method" target="_blank">Newton-Raphson method</a> for both real and complex solutions.
<a href="#details">More...</a></p>
<div class="textblock"><code>#include &lt;cmath&gt;</code><br />
<code>#include &lt;ctime&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 newton_raphson_method.cpp:</div>
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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="../../de/dd3/newton__raphson__method_8cpp.html#a2003b5b2dcfff0769b957ab5c968b03d">eq</a> (double i)</td></tr>
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<tr class="memitem:a3d3f7f41977394680af6ebbed96f3386" id="r_a3d3f7f41977394680af6ebbed96f3386"><td class="memItemLeft" align="right" valign="top">static double&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../de/dd3/newton__raphson__method_8cpp.html#a3d3f7f41977394680af6ebbed96f3386">eq_der</a> (double i)</td></tr>
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<tr class="memitem:ae66f6b31b5ad750f1fe042a706a4e3d4" id="r_ae66f6b31b5ad750f1fe042a706a4e3d4"><td class="memItemLeft" align="right" valign="top">int&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../de/dd3/newton__raphson__method_8cpp.html#ae66f6b31b5ad750f1fe042a706a4e3d4">main</a> ()</td></tr>
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Variables</h2></td></tr>
<tr class="memitem:a75f355966e0a97665eaf5a4c79baa7af" id="r_a75f355966e0a97665eaf5a4c79baa7af"><td class="memItemLeft" align="right" valign="top"><a id="a75f355966e0a97665eaf5a4c79baa7af" name="a75f355966e0a97665eaf5a4c79baa7af"></a>
constexpr double&#160;</td><td class="memItemRight" valign="bottom"><b>EPSILON</b> = 1e-10</td></tr>
<tr class="memdesc:a75f355966e0a97665eaf5a4c79baa7af"><td class="mdescLeft">&#160;</td><td class="mdescRight">system accuracy limit <br /></td></tr>
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<tr class="memitem:a5c5ff05b9f37ae59dad67e1d4f6cd51d" id="r_a5c5ff05b9f37ae59dad67e1d4f6cd51d"><td class="memItemLeft" align="right" valign="top"><a id="a5c5ff05b9f37ae59dad67e1d4f6cd51d" name="a5c5ff05b9f37ae59dad67e1d4f6cd51d"></a>
constexpr int16_t&#160;</td><td class="memItemRight" valign="bottom"><b>MAX_ITERATIONS</b> = INT16_MAX</td></tr>
<tr class="memdesc:a5c5ff05b9f37ae59dad67e1d4f6cd51d"><td class="mdescLeft">&#160;</td><td class="mdescRight">Maximum number of iterations. <br /></td></tr>
<tr class="separator:a5c5ff05b9f37ae59dad67e1d4f6cd51d"><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>Solve the equation \(f(x)=0\) using <a href="https://en.wikipedia.org/wiki/Newton%27s_method" target="_blank">Newton-Raphson method</a> for both real and complex solutions. </p>
<p>The \((i+1)^\text{th}\) approximation is given by: </p><p class="formulaDsp">
\[
x_{i+1} = x_i - \frac{f(x_i)}{f&#39;(x_i)}
\]
</p>
<dl class="section author"><dt>Author</dt><dd><a href="https://github.com/kvedala" target="_blank">Krishna Vedala</a> </dd></dl>
<dl class="section see"><dt>See also</dt><dd><a class="el" href="../../d7/d6a/bisection__method_8cpp.html" title="Solve the equation using bisection method">bisection_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> </dd></dl>
</div><h2 class="groupheader">Function Documentation</h2>
<a id="a2003b5b2dcfff0769b957ab5c968b03d" name="a2003b5b2dcfff0769b957ab5c968b03d"></a>
<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>
<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. Currently defined as: </p><p class="formulaDsp">
\[
f(x) = x^3 - 4x - 9
\]
</p>
<div class="fragment"><div class="line"><span class="lineno"> 29</span> {</div>
<div class="line"><span class="lineno"> 30</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"> 31</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="#a3d3f7f41977394680af6ebbed96f3386">&#9670;&#160;</a></span>eq_der()</h2>
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<td class="memname">static double eq_der </td>
<td>(</td>
<td class="paramtype">double&#160;</td>
<td class="paramname"><em>i</em></td><td>)</td>
<td></td>
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<p>define the derivative function \(f&#39;(x)\) For the current problem, it is: </p><p class="formulaDsp">
\[
f&#39;(x) = 3x^2 - 4
\]
</p>
<div class="fragment"><div class="line"><span class="lineno"> 39</span> {</div>
<div class="line"><span class="lineno"> 40</span> <span class="keywordflow">return</span> ((3 * <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/math/pow.html">std::pow</a>(i, 2)) - 4); <span class="comment">// derivative of equation</span></div>
<div class="line"><span class="lineno"> 41</span>}</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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<td class="paramtype">void&#160;</td>
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<p>Main function </p>
<div class="fragment"><div class="line"><span class="lineno"> 44</span> {</div>
<div class="line"><span class="lineno"> 45</span> <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/random/srand.html">std::srand</a>(<a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/chrono/c/time.html">std::time</a>(<span class="keyword">nullptr</span>)); <span class="comment">// initialize randomizer</span></div>
<div class="line"><span class="lineno"> 46</span> </div>
<div class="line"><span class="lineno"> 47</span> <span class="keywordtype">double</span> z = NAN, c = <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/random/rand.html">std::rand</a>() % 100, m = NAN, n = NAN;</div>
<div class="line"><span class="lineno"> 48</span> <span class="keywordtype">int</span> i = 0;</div>
<div class="line"><span class="lineno"> 49</span> </div>
<div class="line"><span class="lineno"> 50</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;\nInitial approximation: &quot;</span> &lt;&lt; c;</div>
<div class="line"><span class="lineno"> 51</span> </div>
<div class="line"><span class="lineno"> 52</span> <span class="comment">// start iterations</span></div>
<div class="line"><span class="lineno"> 53</span> <span class="keywordflow">for</span> (i = 0; i &lt; <a class="code hl_variable" href="../../de/dd3/newton__raphson__method_8cpp.html#a5c5ff05b9f37ae59dad67e1d4f6cd51d">MAX_ITERATIONS</a>; i++) {</div>
<div class="line"><span class="lineno"> 54</span> m = <a class="code hl_function" href="../../de/dd3/newton__raphson__method_8cpp.html#a2003b5b2dcfff0769b957ab5c968b03d">eq</a>(c);</div>
<div class="line"><span class="lineno"> 55</span> n = <a class="code hl_function" href="../../de/dd3/newton__raphson__method_8cpp.html#a3d3f7f41977394680af6ebbed96f3386">eq_der</a>(c);</div>
<div class="line"><span class="lineno"> 56</span> </div>
<div class="line"><span class="lineno"> 57</span> z = c - (m / n);</div>
<div class="line"><span class="lineno"> 58</span> c = z;</div>
<div class="line"><span class="lineno"> 59</span> </div>
<div class="line"><span class="lineno"> 60</span> <span class="keywordflow">if</span> (std::abs(m) &lt; <a class="code hl_variable" href="../../de/dd3/newton__raphson__method_8cpp.html#a75f355966e0a97665eaf5a4c79baa7af">EPSILON</a>) { <span class="comment">// stoping criteria</span></div>
<div class="line"><span class="lineno"> 61</span> <span class="keywordflow">break</span>;</div>
<div class="line"><span class="lineno"> 62</span> }</div>
<div class="line"><span class="lineno"> 63</span> }</div>
<div class="line"><span class="lineno"> 64</span> </div>
<div class="line"><span class="lineno"> 65</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; z &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"> 66</span> <span class="keywordflow">return</span> 0;</div>
<div class="line"><span class="lineno"> 67</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="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>
<div class="ttc" id="anewton__raphson__method_8cpp_html_a2003b5b2dcfff0769b957ab5c968b03d"><div class="ttname"><a href="../../de/dd3/newton__raphson__method_8cpp.html#a2003b5b2dcfff0769b957ab5c968b03d">eq</a></div><div class="ttdeci">static double eq(double i)</div><div class="ttdef"><b>Definition</b> newton_raphson_method.cpp:29</div></div>
<div class="ttc" id="anewton__raphson__method_8cpp_html_a3d3f7f41977394680af6ebbed96f3386"><div class="ttname"><a href="../../de/dd3/newton__raphson__method_8cpp.html#a3d3f7f41977394680af6ebbed96f3386">eq_der</a></div><div class="ttdeci">static double eq_der(double i)</div><div class="ttdef"><b>Definition</b> newton_raphson_method.cpp:39</div></div>
<div class="ttc" id="anewton__raphson__method_8cpp_html_a5c5ff05b9f37ae59dad67e1d4f6cd51d"><div class="ttname"><a href="../../de/dd3/newton__raphson__method_8cpp.html#a5c5ff05b9f37ae59dad67e1d4f6cd51d">MAX_ITERATIONS</a></div><div class="ttdeci">constexpr int16_t MAX_ITERATIONS</div><div class="ttdoc">Maximum number of iterations.</div><div class="ttdef"><b>Definition</b> newton_raphson_method.cpp:21</div></div>
<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>
<div class="ttc" id="arand_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/numeric/random/rand.html">std::rand</a></div><div class="ttdeci">T rand(T... args)</div></div>
<div class="ttc" id="asrand_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/numeric/random/srand.html">std::srand</a></div><div class="ttdeci">T srand(T... args)</div></div>
<div class="ttc" id="atime_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/chrono/c/time.html">std::time</a></div><div class="ttdeci">T time(T... args)</div></div>
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