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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;cstdint&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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<p><a href="../../de/dd3/newton__raphson__method_8cpp_source.html">Go to the source code of this file.</a></p>
<table class="memberdecls">
<tr class="heading"><td colspan="2"><h2 id="header-func-members" class="groupheader"><a id="func-members" name="func-members"></a>
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>
<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="#a3d3f7f41977394680af6ebbed96f3386">eq_der</a> (double i)</td></tr>
<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="#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">constexpr double&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="#a75f355966e0a97665eaf5a4c79baa7af">EPSILON</a> = 1e-10</td></tr>
<tr class="memdesc:a75f355966e0a97665eaf5a4c79baa7af"><td class="mdescLeft">&#160;</td><td class="mdescRight">system accuracy limit <br /></td></tr>
<tr class="memitem:a5c5ff05b9f37ae59dad67e1d4f6cd51d" id="r_a5c5ff05b9f37ae59dad67e1d4f6cd51d"><td class="memItemLeft" align="right" valign="top">constexpr int16_t&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="#a5c5ff05b9f37ae59dad67e1d4f6cd51d">MAX_ITERATIONS</a> = INT16_MAX</td></tr>
<tr class="memdesc:a5c5ff05b9f37ae59dad67e1d4f6cd51d"><td class="mdescLeft">&#160;</td><td class="mdescRight">Maximum number of iterations. <br /></td></tr>
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<a name="details" id="details"></a><h2 id="header-details" 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>
<p class="definition">Definition in file <a class="el" href="../../de/dd3/newton__raphson__method_8cpp_source.html">newton_raphson_method.cpp</a>.</p>
</div><a name="doc-func-members" id="doc-func-members"></a><h2 id="header-doc-func-members" 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">double eq </td>
<td>(</td>
<td class="paramtype">double</td> <td class="paramname"><span class="paramname"><em>i</em></span></td><td>)</td>
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<span class="mlabels"><span class="mlabel static">static</span></span> </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>
<p class="definition">Definition at line <a class="el" href="../../de/dd3/newton__raphson__method_8cpp_source.html#l00030">30</a> of file <a class="el" href="../../de/dd3/newton__raphson__method_8cpp_source.html">newton_raphson_method.cpp</a>.</p>
<div class="fragment"><div class="line"><span class="lineno"> 30</span> {</div>
<div class="line"><span class="lineno"> 31</span> <span class="keywordflow">return</span> (std::pow(i, 3) - (4 * i) - 9); <span class="comment">// original equation</span></div>
<div class="line"><span class="lineno"> 32</span>}</div>
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<a id="a3d3f7f41977394680af6ebbed96f3386" name="a3d3f7f41977394680af6ebbed96f3386"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a3d3f7f41977394680af6ebbed96f3386">&#9670;&#160;</a></span>eq_der()</h2>
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<td class="memname">double eq_der </td>
<td>(</td>
<td class="paramtype">double</td> <td class="paramname"><span class="paramname"><em>i</em></span></td><td>)</td>
<td></td>
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<span class="mlabels"><span class="mlabel static">static</span></span> </td>
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</div><div class="memdoc">
<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>
<p class="definition">Definition at line <a class="el" href="../../de/dd3/newton__raphson__method_8cpp_source.html#l00040">40</a> of file <a class="el" href="../../de/dd3/newton__raphson__method_8cpp_source.html">newton_raphson_method.cpp</a>.</p>
<div class="fragment"><div class="line"><span class="lineno"> 40</span> {</div>
<div class="line"><span class="lineno"> 41</span> <span class="keywordflow">return</span> ((3 * std::pow(i, 2)) - 4); <span class="comment">// derivative of equation</span></div>
<div class="line"><span class="lineno"> 42</span>}</div>
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<a id="ae66f6b31b5ad750f1fe042a706a4e3d4" name="ae66f6b31b5ad750f1fe042a706a4e3d4"></a>
<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</td> <td class="paramname"><span class="paramname"><em></em></span></td><td>)</td>
<td></td>
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<p>Main function </p>
<p class="definition">Definition at line <a class="el" href="../../de/dd3/newton__raphson__method_8cpp_source.html#l00045">45</a> of file <a class="el" href="../../de/dd3/newton__raphson__method_8cpp_source.html">newton_raphson_method.cpp</a>.</p>
<div class="fragment"><div class="line"><span class="lineno"> 45</span> {</div>
<div class="line"><span class="lineno"> 46</span> std::srand(std::time(<span class="keyword">nullptr</span>)); <span class="comment">// initialize randomizer</span></div>
<div class="line"><span class="lineno"> 47</span> </div>
<div class="line"><span class="lineno"> 48</span> <span class="keywordtype">double</span> z = NAN, c = std::rand() % 100, m = NAN, n = NAN;</div>
<div class="line"><span class="lineno"> 49</span> <span class="keywordtype">int</span> i = 0;</div>
<div class="line"><span class="lineno"> 50</span> </div>
<div class="line"><span class="lineno"> 51</span> std::cout &lt;&lt; <span class="stringliteral">&quot;\nInitial approximation: &quot;</span> &lt;&lt; c;</div>
<div class="line"><span class="lineno"> 52</span> </div>
<div class="line"><span class="lineno"> 53</span> <span class="comment">// start iterations</span></div>
<div class="line"><span class="lineno"> 54</span> <span class="keywordflow">for</span> (i = 0; i &lt; <a class="code hl_define" href="../../d7/d6a/bisection__method_8cpp.html#a0a3abbca80bc98e7abcb3ae73abe0f14">MAX_ITERATIONS</a>; i++) {</div>
<div class="line"><span class="lineno"> 55</span> m = <a class="code hl_function" href="#a2003b5b2dcfff0769b957ab5c968b03d">eq</a>(c);</div>
<div class="line"><span class="lineno"> 56</span> n = <a class="code hl_function" href="#a3d3f7f41977394680af6ebbed96f3386">eq_der</a>(c);</div>
<div class="line"><span class="lineno"> 57</span> </div>
<div class="line"><span class="lineno"> 58</span> z = c - (m / n);</div>
<div class="line"><span class="lineno"> 59</span> c = z;</div>
<div class="line"><span class="lineno"> 60</span> </div>
<div class="line"><span class="lineno"> 61</span> <span class="keywordflow">if</span> (std::abs(m) &lt; EPSILON) { <span class="comment">// stoping criteria</span></div>
<div class="line"><span class="lineno"> 62</span> <span class="keywordflow">break</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> </div>
<div class="line"><span class="lineno"> 66</span> std::cout &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; std::endl;</div>
<div class="line"><span class="lineno"> 67</span> <span class="keywordflow">return</span> 0;</div>
<div class="line"><span class="lineno"> 68</span>}</div>
<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> <a href="../../d7/d6a/bisection__method_8cpp_source.html#l00022">bisection_method.cpp:22</a></div></div>
<div class="ttc" id="anewton__raphson__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> <a href="../../de/dd3/newton__raphson__method_8cpp_source.html#l00030">newton_raphson_method.cpp:30</a></div></div>
<div class="ttc" id="anewton__raphson__method_8cpp_html_a3d3f7f41977394680af6ebbed96f3386"><div class="ttname"><a href="#a3d3f7f41977394680af6ebbed96f3386">eq_der</a></div><div class="ttdeci">static double eq_der(double i)</div><div class="ttdef"><b>Definition</b> <a href="../../de/dd3/newton__raphson__method_8cpp_source.html#l00040">newton_raphson_method.cpp:40</a></div></div>
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<a name="doc-var-members" id="doc-var-members"></a><h2 id="header-doc-var-members" class="groupheader">Variable Documentation</h2>
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<h2 class="memtitle"><span class="permalink"><a href="#a75f355966e0a97665eaf5a4c79baa7af">&#9670;&#160;</a></span>EPSILON</h2>
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<td class="memname">double EPSILON = 1e-10</td>
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<span class="mlabels"><span class="mlabel constexpr">constexpr</span></span> </td>
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<p>system accuracy limit </p>
<p class="definition">Definition at line <a class="el" href="../../de/dd3/newton__raphson__method_8cpp_source.html#l00021">21</a> of file <a class="el" href="../../de/dd3/newton__raphson__method_8cpp_source.html">newton_raphson_method.cpp</a>.</p>
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<h2 class="memtitle"><span class="permalink"><a href="#a5c5ff05b9f37ae59dad67e1d4f6cd51d">&#9670;&#160;</a></span>MAX_ITERATIONS</h2>
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<td class="memname">int16_t MAX_ITERATIONS = INT16_MAX</td>
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<span class="mlabels"><span class="mlabel constexpr">constexpr</span></span> </td>
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<p>Maximum number of iterations. </p>
<p class="definition">Definition at line <a class="el" href="../../de/dd3/newton__raphson__method_8cpp_source.html#l00022">22</a> of file <a class="el" href="../../de/dd3/newton__raphson__method_8cpp_source.html">newton_raphson_method.cpp</a>.</p>
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