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<div class="title">string_fibonacci.cpp File Reference</div> </div>
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<p>This Programme returns the Nth fibonacci as a string.
<a href="#details">More...</a></p>
<div class="textblock"><code>#include &lt;iostream&gt;</code><br />
<code>#include &lt;cstring&gt;</code><br />
</div><div class="textblock"><div class="dynheader">
Include dependency graph for string_fibonacci.cpp:</div>
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Functions</h2></td></tr>
<tr class="memitem:a28052eee05d43c2ebc5147c52bd50c35"><td class="memItemLeft" align="right" valign="top"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/string/basic_string.html">std::string</a>&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../de/d47/string__fibonacci_8cpp.html#a28052eee05d43c2ebc5147c52bd50c35">add</a> (<a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/string/basic_string.html">std::string</a> <a class="el" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a>, <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/string/basic_string.html">std::string</a> b)</td></tr>
<tr class="separator:a28052eee05d43c2ebc5147c52bd50c35"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:ad8055ee368a5d5b24de01bdde6bf8fca"><td class="memItemLeft" align="right" valign="top">void&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../de/d47/string__fibonacci_8cpp.html#ad8055ee368a5d5b24de01bdde6bf8fca">fib_Accurate</a> (uint64_t n)</td></tr>
<tr class="separator:ad8055ee368a5d5b24de01bdde6bf8fca"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:ae66f6b31b5ad750f1fe042a706a4e3d4"><td class="memItemLeft" align="right" valign="top">int&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../de/d47/string__fibonacci_8cpp.html#ae66f6b31b5ad750f1fe042a706a4e3d4">main</a> ()</td></tr>
<tr class="separator:ae66f6b31b5ad750f1fe042a706a4e3d4"><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>This Programme returns the Nth fibonacci as a string. </p>
<p>The method used is manual addition with carry and placing it in a string which is called string addition This makes it have no bounds or limits</p>
<dl class="section see"><dt>See also</dt><dd><a class="el" href="../../de/de4/fibonacci__large_8cpp.html" title="Computes N^th Fibonacci number given as input argument. Uses custom build arbitrary integers library ...">fibonacci_large.cpp</a>, <a class="el" href="../../d4/d32/fibonacci__fast_8cpp.html" title="Faster computation of Fibonacci series.">fibonacci_fast.cpp</a>, <a class="el" href="../../d9/d89/fibonacci_8cpp.html" title="Generate fibonacci sequence.">fibonacci.cpp</a> </dd></dl>
</div><h2 class="groupheader">Function Documentation</h2>
<a id="a28052eee05d43c2ebc5147c52bd50c35"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a28052eee05d43c2ebc5147c52bd50c35">&#9670;&nbsp;</a></span>add()</h2>
<div class="memitem">
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<td class="memname"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/string/basic_string.html">std::string</a> add </td>
<td>(</td>
<td class="paramtype"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/string/basic_string.html">std::string</a>&#160;</td>
<td class="paramname"><em>a</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/string/basic_string.html">std::string</a>&#160;</td>
<td class="paramname"><em>b</em>&#160;</td>
</tr>
<tr>
<td></td>
<td>)</td>
<td></td><td></td>
</tr>
</table>
</div><div class="memdoc">
<p>function to add two string numbers </p><dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramdir">[in]</td><td class="paramname">a</td><td>first number in string to add </td></tr>
<tr><td class="paramdir">[in]</td><td class="paramname">b</td><td>second number in string to add </td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>sum as a <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/string/basic_string.html" title="STL class.">std::string</a> </dd></dl>
<div class="fragment"><div class="line"><a name="l00024"></a><span class="lineno"> 24</span>&#160; {</div>
<div class="line"><a name="l00025"></a><span class="lineno"> 25</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/string/basic_string.html">std::string</a> temp = <span class="stringliteral">&quot;&quot;</span>;</div>
<div class="line"><a name="l00026"></a><span class="lineno"> 26</span>&#160; </div>
<div class="line"><a name="l00027"></a><span class="lineno"> 27</span>&#160; <span class="comment">// carry flag</span></div>
<div class="line"><a name="l00028"></a><span class="lineno"> 28</span>&#160; <span class="keywordtype">int</span> carry = 0;</div>
<div class="line"><a name="l00029"></a><span class="lineno"> 29</span>&#160; </div>
<div class="line"><a name="l00030"></a><span class="lineno"> 30</span>&#160; <span class="comment">// fills up with zeros</span></div>
<div class="line"><a name="l00031"></a><span class="lineno"> 31</span>&#160; <span class="keywordflow">while</span> (<a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a>.length() &lt; b.length()) {</div>
<div class="line"><a name="l00032"></a><span class="lineno"> 32</span>&#160; <a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a> = <span class="stringliteral">&quot;0&quot;</span> + <a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a>;</div>
<div class="line"><a name="l00033"></a><span class="lineno"> 33</span>&#160; }</div>
<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="comment">// fills up with zeros</span></div>
<div class="line"><a name="l00036"></a><span class="lineno"> 36</span>&#160; <span class="keywordflow">while</span> (b.length() &lt; <a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a>.length()) {</div>
<div class="line"><a name="l00037"></a><span class="lineno"> 37</span>&#160; b = <span class="stringliteral">&quot;0&quot;</span> + b;</div>
<div class="line"><a name="l00038"></a><span class="lineno"> 38</span>&#160; }</div>
<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="comment">// adds the numbers a and b</span></div>
<div class="line"><a name="l00041"></a><span class="lineno"> 41</span>&#160; <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = <a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a>.length() - 1; i &gt;= 0; i--) {</div>
<div class="line"><a name="l00042"></a><span class="lineno"> 42</span>&#160; <span class="keywordtype">char</span> val = <span class="keyword">static_cast&lt;</span><span class="keywordtype">char</span><span class="keyword">&gt;</span>(((<a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a>[i] - 48) + (b[i] - 48)) + 48 + carry);</div>
<div class="line"><a name="l00043"></a><span class="lineno"> 43</span>&#160; <span class="keywordflow">if</span> (val &gt; 57) {</div>
<div class="line"><a name="l00044"></a><span class="lineno"> 44</span>&#160; carry = 1;</div>
<div class="line"><a name="l00045"></a><span class="lineno"> 45</span>&#160; val -= 10;</div>
<div class="line"><a name="l00046"></a><span class="lineno"> 46</span>&#160; } <span class="keywordflow">else</span> {</div>
<div class="line"><a name="l00047"></a><span class="lineno"> 47</span>&#160; carry = 0;</div>
<div class="line"><a name="l00048"></a><span class="lineno"> 48</span>&#160; }</div>
<div class="line"><a name="l00049"></a><span class="lineno"> 49</span>&#160; temp = val + temp;</div>
<div class="line"><a name="l00050"></a><span class="lineno"> 50</span>&#160; }</div>
<div class="line"><a name="l00051"></a><span class="lineno"> 51</span>&#160; </div>
<div class="line"><a name="l00052"></a><span class="lineno"> 52</span>&#160; <span class="comment">// processes the carry flag</span></div>
<div class="line"><a name="l00053"></a><span class="lineno"> 53</span>&#160; <span class="keywordflow">if</span> (carry == 1) {</div>
<div class="line"><a name="l00054"></a><span class="lineno"> 54</span>&#160; temp = <span class="stringliteral">&quot;1&quot;</span> + temp;</div>
<div class="line"><a name="l00055"></a><span class="lineno"> 55</span>&#160; }</div>
<div class="line"><a name="l00056"></a><span class="lineno"> 56</span>&#160; </div>
<div class="line"><a name="l00057"></a><span class="lineno"> 57</span>&#160; <span class="comment">// removes leading zeros.</span></div>
<div class="line"><a name="l00058"></a><span class="lineno"> 58</span>&#160; <span class="keywordflow">while</span> (temp[0] == <span class="charliteral">&#39;0&#39;</span> &amp;&amp; temp.<a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/string/basic_string/size.html">length</a>() &gt; 1) {</div>
<div class="line"><a name="l00059"></a><span class="lineno"> 59</span>&#160; temp = temp.<a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/string/basic_string/substr.html">substr</a>(1);</div>
<div class="line"><a name="l00060"></a><span class="lineno"> 60</span>&#160; }</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; <span class="keywordflow">return</span> temp;</div>
<div class="line"><a name="l00063"></a><span class="lineno"> 63</span>&#160;}</div>
</div><!-- fragment -->
</div>
</div>
<a id="ad8055ee368a5d5b24de01bdde6bf8fca"></a>
<h2 class="memtitle"><span class="permalink"><a href="#ad8055ee368a5d5b24de01bdde6bf8fca">&#9670;&nbsp;</a></span>fib_Accurate()</h2>
<div class="memitem">
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<td class="memname">void fib_Accurate </td>
<td>(</td>
<td class="paramtype">uint64_t&#160;</td>
<td class="paramname"><em>n</em></td><td>)</td>
<td></td>
</tr>
</table>
</div><div class="memdoc">
<p>Fibonacci iterator </p><dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramdir">[in]</td><td class="paramname">n</td><td>n^th Fibonacci number </td></tr>
</table>
</dd>
</dl>
<div class="fragment"><div class="line"><a name="l00068"></a><span class="lineno"> 68</span>&#160; {</div>
<div class="line"><a name="l00069"></a><span class="lineno"> 69</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/string/basic_string.html">std::string</a> tmp = <span class="stringliteral">&quot;&quot;</span>;</div>
<div class="line"><a name="l00070"></a><span class="lineno"> 70</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/string/basic_string.html">std::string</a> fibMinus1 = <span class="stringliteral">&quot;1&quot;</span>;</div>
<div class="line"><a name="l00071"></a><span class="lineno"> 71</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/string/basic_string.html">std::string</a> fibMinus2 = <span class="stringliteral">&quot;0&quot;</span>;</div>
<div class="line"><a name="l00072"></a><span class="lineno"> 72</span>&#160; <span class="keywordflow">for</span> (uint64_t i = 0; i &lt; n; i++) {</div>
<div class="line"><a name="l00073"></a><span class="lineno"> 73</span>&#160; tmp = <a class="code" href="../../de/d47/string__fibonacci_8cpp.html#a28052eee05d43c2ebc5147c52bd50c35">add</a>(fibMinus1, fibMinus2);</div>
<div class="line"><a name="l00074"></a><span class="lineno"> 74</span>&#160; fibMinus2 = fibMinus1;</div>
<div class="line"><a name="l00075"></a><span class="lineno"> 75</span>&#160; fibMinus1 = tmp;</div>
<div class="line"><a name="l00076"></a><span class="lineno"> 76</span>&#160; }</div>
<div class="line"><a name="l00077"></a><span class="lineno"> 77</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; fibMinus2;</div>
<div class="line"><a name="l00078"></a><span class="lineno"> 78</span>&#160;}</div>
</div><!-- fragment --><div class="dynheader">
Here is the call graph for this function:</div>
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<h2 class="memtitle"><span class="permalink"><a href="#ae66f6b31b5ad750f1fe042a706a4e3d4">&#9670;&nbsp;</a></span>main()</h2>
<div class="memitem">
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<td class="memname">int main </td>
<td>(</td>
<td class="paramname"></td><td>)</td>
<td></td>
</tr>
</table>
</div><div class="memdoc">
<p>main function </p>
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<div class="textblock"><p>Structure of trie node. This struct doesn't need a constructor as we are initializing using intializer list which is more efficient than if we had done so with constructor. </p>
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<p>The <a href="https://en.wikipedia.org/wiki/Knuth–Morris–Pratt_algorithm">Knuth-Morris-Pratt Algorithm</a> for finding a pattern within a piece of text with complexity O(n + m)
<a href="#details">More...</a></p>
<div class="textblock"><code>#include &lt;iostream&gt;</code><br />
<code>#include &lt;cstring&gt;</code><br />
<code>#include &lt;vector&gt;</code><br />
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<tr class="memdesc:d9/d03/namespacestring__search"><td class="mdescLeft">&#160;</td><td class="mdescRight">String search algorithms. <br /></td></tr>
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<tr class="memitem:a26a58225ce7d3fa9d4c2f5349a65ed93"><td class="memItemLeft" align="right" valign="top">bool&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../d9/d03/namespacestring__search.html#a26a58225ce7d3fa9d4c2f5349a65ed93">string_search::kmp</a> (const <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/string/basic_string.html">std::string</a> &amp;pattern, const <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/string/basic_string.html">std::string</a> &amp;text)</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="../../de/d6a/knuth__morris__pratt_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>The <a href="https://en.wikipedia.org/wiki/Knuth–Morris–Pratt_algorithm">Knuth-Morris-Pratt Algorithm</a> for finding a pattern within a piece of text with complexity O(n + m) </p>
<ol type="1">
<li>Preprocess pattern to identify any suffixes that are identical to prefixes. This tells us where to continue from if we get a mismatch between a character in our pattern and the text.</li>
<li>Step through the text one character at a time and compare it to a character in the pattern updating our location within the pattern if necessary </li>
</ol>
</div><h2 class="groupheader">Function Documentation</h2>
<a id="ae66f6b31b5ad750f1fe042a706a4e3d4"></a>
<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>
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<div class="line"><a name="l00078"></a><span class="lineno"> 78</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/string/basic_string.html">std::string</a> pattern = <span class="stringliteral">&quot;abc1abc12l&quot;</span>;</div>
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<div class="line"><a name="l00084"></a><span class="lineno"> 84</span>&#160; }</div>
<div class="line"><a name="l00085"></a><span class="lineno"> 85</span>&#160; </div>
<div class="line"><a name="l00086"></a><span class="lineno"> 86</span>&#160; text = <span class="stringliteral">&quot;abcabc&quot;</span>;</div>
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<div class="line"><a name="l00089"></a><span class="lineno"> 89</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;Found&quot;</span> &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="l00090"></a><span class="lineno"> 90</span>&#160; } <span class="keywordflow">else</span> {</div>
<div class="line"><a name="l00091"></a><span class="lineno"> 91</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;Not Found&quot;</span> &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="l00092"></a><span class="lineno"> 92</span>&#160; }</div>
<div class="line"><a name="l00093"></a><span class="lineno"> 93</span>&#160; </div>
<div class="line"><a name="l00094"></a><span class="lineno"> 94</span>&#160; <span class="keywordflow">return</span> 0;</div>
<div class="line"><a name="l00095"></a><span class="lineno"> 95</span>&#160;}</div>
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<p>Compute real eigen values and eigen vectors of a symmetric matrix using <a href="https://en.wikipedia.org/wiki/QR_decomposition">QR decomposition</a> method.
<a href="#details">More...</a></p>
<div class="textblock"><code>#include &lt;cassert&gt;</code><br />
<code>#include &lt;cmath&gt;</code><br />
<code>#include &lt;cstdlib&gt;</code><br />
<code>#include &lt;ctime&gt;</code><br />
<code>#include &lt;iostream&gt;</code><br />
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<a name="details" id="details"></a><h2 class="groupheader">Detailed Description</h2>
<div class="textblock"><p>Compute real eigen values and eigen vectors of a symmetric matrix using <a href="https://en.wikipedia.org/wiki/QR_decomposition">QR decomposition</a> method. </p>
<dl class="section author"><dt>Author</dt><dd><a href="https://github.com/kvedala">Krishna Vedala</a> </dd></dl>
</div><h2 class="groupheader">Macro Definition Documentation</h2>
<a id="aee57a411f07599034f5ceb8cc7d65b40"></a>
<h2 class="memtitle"><span class="permalink"><a href="#aee57a411f07599034f5ceb8cc7d65b40">&#9670;&nbsp;</a></span>LIMS</h2>
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<td class="memname">#define LIMS&#160;&#160;&#160;9</td>
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<p>limit of range of matrix values </p>
</div>
</div>
<h2 class="groupheader">Function Documentation</h2>
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<h2 class="memtitle"><span class="permalink"><a href="#a33cb0a68c36aa26fd599c7c66da86ed7">&#9670;&nbsp;</a></span>create_matrix()</h2>
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<td class="memname">void create_matrix </td>
<td>(</td>
<td class="paramtype"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; double &gt;&gt; *&#160;</td>
<td class="paramname"><em>A</em></td><td>)</td>
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<p>create a symmetric square matrix of given size with random elements. A symmetric square matrix will <em>always</em> have real eigen values.</p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramdir">[out]</td><td class="paramname">A</td><td>matrix to create (must be pre-allocated in memory) </td></tr>
</table>
</dd>
</dl>
<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="keywordtype">int</span> i, j, tmp, lim2 = <a class="code" href="../../de/d75/qr__eigen__values_8cpp.html#aee57a411f07599034f5ceb8cc7d65b40">LIMS</a> &gt;&gt; 1;</div>
<div class="line"><a name="l00030"></a><span class="lineno"> 30</span>&#160; <span class="keywordtype">int</span> N = A-&gt;size();</div>
<div class="line"><a name="l00031"></a><span class="lineno"> 31</span>&#160; </div>
<div class="line"><a name="l00032"></a><span class="lineno"> 32</span>&#160;<span class="preprocessor">#ifdef _OPENMP</span></div>
<div class="line"><a name="l00033"></a><span class="lineno"> 33</span>&#160;<span class="preprocessor">#pragma omp for</span></div>
<div class="line"><a name="l00034"></a><span class="lineno"> 34</span>&#160;<span class="preprocessor">#endif</span></div>
<div class="line"><a name="l00035"></a><span class="lineno"> 35</span>&#160; <span class="keywordflow">for</span> (i = 0; i &lt; N; i++) {</div>
<div class="line"><a name="l00036"></a><span class="lineno"> 36</span>&#160; A[0][i][i] = (<a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/random/rand.html">std::rand</a>() % <a class="code" href="../../de/d75/qr__eigen__values_8cpp.html#aee57a411f07599034f5ceb8cc7d65b40">LIMS</a>) - lim2;</div>
<div class="line"><a name="l00037"></a><span class="lineno"> 37</span>&#160; <span class="keywordflow">for</span> (j = i + 1; j &lt; N; j++) {</div>
<div class="line"><a name="l00038"></a><span class="lineno"> 38</span>&#160; tmp = (<a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/random/rand.html">std::rand</a>() % <a class="code" href="../../de/d75/qr__eigen__values_8cpp.html#aee57a411f07599034f5ceb8cc7d65b40">LIMS</a>) - lim2;</div>
<div class="line"><a name="l00039"></a><span class="lineno"> 39</span>&#160; A[0][i][j] = tmp; <span class="comment">// summetrically distribute random values</span></div>
<div class="line"><a name="l00040"></a><span class="lineno"> 40</span>&#160; A[0][j][i] = tmp;</div>
<div class="line"><a name="l00041"></a><span class="lineno"> 41</span>&#160; }</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;}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#a3c04138a5bfe5d72780bb7e82a18e627">&#9670;&nbsp;</a></span>main()</h2>
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<td class="memname">int main </td>
<td>(</td>
<td class="paramtype">int&#160;</td>
<td class="paramname"><em>argc</em>, </td>
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<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">char **&#160;</td>
<td class="paramname"><em>argv</em>&#160;</td>
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<td></td>
<td>)</td>
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<p>main function </p>
<div class="fragment"><div class="line"><a name="l00243"></a><span class="lineno"> 243</span>&#160; {</div>
<div class="line"><a name="l00244"></a><span class="lineno"> 244</span>&#160; <span class="keywordtype">int</span> mat_size = 5;</div>
<div class="line"><a name="l00245"></a><span class="lineno"> 245</span>&#160; <span class="keywordflow">if</span> (argc == 2) {</div>
<div class="line"><a name="l00246"></a><span class="lineno"> 246</span>&#160; mat_size = <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/string/byte/atoi.html">atoi</a>(argv[1]);</div>
<div class="line"><a name="l00247"></a><span class="lineno"> 247</span>&#160; } <span class="keywordflow">else</span> { <span class="comment">// if invalid input argument is given run tests</span></div>
<div class="line"><a name="l00248"></a><span class="lineno"> 248</span>&#160; <a class="code" href="../../de/d75/qr__eigen__values_8cpp.html#a1440a7779ac56f47a3f355ce4a8c7da0">test1</a>();</div>
<div class="line"><a name="l00249"></a><span class="lineno"> 249</span>&#160; <a class="code" href="../../de/d75/qr__eigen__values_8cpp.html#a0283886819c7c140a023582b7269e2d0">test2</a>();</div>
<div class="line"><a name="l00250"></a><span class="lineno"> 250</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;Usage: ./qr_eigen_values [mat_size]\n&quot;</span>;</div>
<div class="line"><a name="l00251"></a><span class="lineno"> 251</span>&#160; <span class="keywordflow">return</span> 0;</div>
<div class="line"><a name="l00252"></a><span class="lineno"> 252</span>&#160; }</div>
<div class="line"><a name="l00253"></a><span class="lineno"> 253</span>&#160; </div>
<div class="line"><a name="l00254"></a><span class="lineno"> 254</span>&#160; <span class="keywordflow">if</span> (mat_size &lt; 2) {</div>
<div class="line"><a name="l00255"></a><span class="lineno"> 255</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/c/fprintf.html">fprintf</a>(stderr, <span class="stringliteral">&quot;Matrix size should be &gt; 2\n&quot;</span>);</div>
<div class="line"><a name="l00256"></a><span class="lineno"> 256</span>&#160; <span class="keywordflow">return</span> -1;</div>
<div class="line"><a name="l00257"></a><span class="lineno"> 257</span>&#160; }</div>
<div class="line"><a name="l00258"></a><span class="lineno"> 258</span>&#160; </div>
<div class="line"><a name="l00259"></a><span class="lineno"> 259</span>&#160; <span class="comment">// initialize random number generator</span></div>
<div class="line"><a name="l00260"></a><span class="lineno"> 260</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/random/srand.html">std::srand</a>(<a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/chrono/c/time.html">std::time</a>(<span class="keyword">nullptr</span>));</div>
<div class="line"><a name="l00261"></a><span class="lineno"> 261</span>&#160; </div>
<div class="line"><a name="l00262"></a><span class="lineno"> 262</span>&#160; <span class="keywordtype">int</span> i, rows = mat_size, columns = mat_size;</div>
<div class="line"><a name="l00263"></a><span class="lineno"> 263</span>&#160; </div>
<div class="line"><a name="l00264"></a><span class="lineno"> 264</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray&lt;std::valarray&lt;double&gt;</a>&gt; A(rows);</div>
<div class="line"><a name="l00265"></a><span class="lineno"> 265</span>&#160; </div>
<div class="line"><a name="l00266"></a><span class="lineno"> 266</span>&#160; <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i &lt; rows; i++) {</div>
<div class="line"><a name="l00267"></a><span class="lineno"> 267</span>&#160; A[i] = <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray&lt;double&gt;</a>(columns);</div>
<div class="line"><a name="l00268"></a><span class="lineno"> 268</span>&#160; }</div>
<div class="line"><a name="l00269"></a><span class="lineno"> 269</span>&#160; </div>
<div class="line"><a name="l00270"></a><span class="lineno"> 270</span>&#160; <span class="comment">/* create a random matrix */</span></div>
<div class="line"><a name="l00271"></a><span class="lineno"> 271</span>&#160; <a class="code" href="../../de/d75/qr__eigen__values_8cpp.html#a33cb0a68c36aa26fd599c7c66da86ed7">create_matrix</a>(&amp;A);</div>
<div class="line"><a name="l00272"></a><span class="lineno"> 272</span>&#160; </div>
<div class="line"><a name="l00273"></a><span class="lineno"> 273</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; A &lt;&lt; <span class="stringliteral">&quot;\n&quot;</span>;</div>
<div class="line"><a name="l00274"></a><span class="lineno"> 274</span>&#160; </div>
<div class="line"><a name="l00275"></a><span class="lineno"> 275</span>&#160; clock_t t1 = <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/chrono/c/clock.html">clock</a>();</div>
<div class="line"><a name="l00276"></a><span class="lineno"> 276</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray&lt;double&gt;</a> eigen_vals = <a class="code" href="../../d2/d3b/namespaceqr__algorithm.html#a763896dbb4a7e95c4a1e614ac0819d66">qr_algorithm::eigen_values</a>(&amp;A);</div>
<div class="line"><a name="l00277"></a><span class="lineno"> 277</span>&#160; <span class="keywordtype">double</span> dtime = <span class="keyword">static_cast&lt;</span><span class="keywordtype">double</span><span class="keyword">&gt;</span>(<a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/chrono/c/clock.html">clock</a>() - t1) / CLOCKS_PER_SEC;</div>
<div class="line"><a name="l00278"></a><span class="lineno"> 278</span>&#160; </div>
<div class="line"><a name="l00279"></a><span class="lineno"> 279</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;Eigen vals: &quot;</span>;</div>
<div class="line"><a name="l00280"></a><span class="lineno"> 280</span>&#160; <span class="keywordflow">for</span> (i = 0; i &lt; mat_size; i++) <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; eigen_vals[i] &lt;&lt; <span class="stringliteral">&quot;\t&quot;</span>;</div>
<div class="line"><a name="l00281"></a><span class="lineno"> 281</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;\nTime taken to compute: &quot;</span> &lt;&lt; dtime &lt;&lt; <span class="stringliteral">&quot; sec\n&quot;</span>;</div>
<div class="line"><a name="l00282"></a><span class="lineno"> 282</span>&#160; </div>
<div class="line"><a name="l00283"></a><span class="lineno"> 283</span>&#160; <span class="keywordflow">return</span> 0;</div>
<div class="line"><a name="l00284"></a><span class="lineno"> 284</span>&#160;}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#a7558fd7779bf207157ced66f275951a0">&#9670;&nbsp;</a></span>mat_mul()</h2>
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<td class="memname">void mat_mul </td>
<td>(</td>
<td class="paramtype">const <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; double &gt;&gt; &amp;&#160;</td>
<td class="paramname"><em>A</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">const <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; double &gt;&gt; &amp;&#160;</td>
<td class="paramname"><em>B</em>, </td>
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<td class="paramkey"></td>
<td></td>
<td class="paramtype"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray</a>&lt; double &gt;&gt; *&#160;</td>
<td class="paramname"><em>OUT</em>&#160;</td>
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<p>Perform multiplication of two matrices.</p><ul>
<li>R2 must be equal to C1</li>
<li>Resultant matrix size should be R1xC2 <dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramdir">[in]</td><td class="paramname">A</td><td>first matrix to multiply </td></tr>
<tr><td class="paramdir">[in]</td><td class="paramname">B</td><td>second matrix to multiply </td></tr>
<tr><td class="paramdir">[out]</td><td class="paramname">OUT</td><td>output matrix (must be pre-allocated) </td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>pointer to resultant matrix </dd></dl>
</li>
</ul>
<div class="fragment"><div class="line"><a name="l00056"></a><span class="lineno"> 56</span>&#160; {</div>
<div class="line"><a name="l00057"></a><span class="lineno"> 57</span>&#160; <span class="keywordtype">int</span> R1 = A.size();</div>
<div class="line"><a name="l00058"></a><span class="lineno"> 58</span>&#160; <span class="keywordtype">int</span> C1 = A[0].size();</div>
<div class="line"><a name="l00059"></a><span class="lineno"> 59</span>&#160; <span class="keywordtype">int</span> R2 = B.size();</div>
<div class="line"><a name="l00060"></a><span class="lineno"> 60</span>&#160; <span class="keywordtype">int</span> C2 = B[0].size();</div>
<div class="line"><a name="l00061"></a><span class="lineno"> 61</span>&#160; <span class="keywordflow">if</span> (C1 != R2) {</div>
<div class="line"><a name="l00062"></a><span class="lineno"> 62</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/c/perror.html">perror</a>(<span class="stringliteral">&quot;Matrix dimensions mismatch!&quot;</span>);</div>
<div class="line"><a name="l00063"></a><span class="lineno"> 63</span>&#160; <span class="keywordflow">return</span>;</div>
<div class="line"><a name="l00064"></a><span class="lineno"> 64</span>&#160; }</div>
<div class="line"><a name="l00065"></a><span class="lineno"> 65</span>&#160; </div>
<div class="line"><a name="l00066"></a><span class="lineno"> 66</span>&#160; <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i &lt; R1; i++) {</div>
<div class="line"><a name="l00067"></a><span class="lineno"> 67</span>&#160; <span class="keywordflow">for</span> (<span class="keywordtype">int</span> j = 0; j &lt; C2; j++) {</div>
<div class="line"><a name="l00068"></a><span class="lineno"> 68</span>&#160; OUT[0][i][j] = 0.f;</div>
<div class="line"><a name="l00069"></a><span class="lineno"> 69</span>&#160; <span class="keywordflow">for</span> (<span class="keywordtype">int</span> <a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#afcf0d19607cd0edd8ae333d2b9c220c0">k</a> = 0; <a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#afcf0d19607cd0edd8ae333d2b9c220c0">k</a> &lt; C1; <a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#afcf0d19607cd0edd8ae333d2b9c220c0">k</a>++) {</div>
<div class="line"><a name="l00070"></a><span class="lineno"> 70</span>&#160; OUT[0][i][j] += A[i][<a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#afcf0d19607cd0edd8ae333d2b9c220c0">k</a>] * B[<a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#afcf0d19607cd0edd8ae333d2b9c220c0">k</a>][j];</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; }</div>
<div class="line"><a name="l00073"></a><span class="lineno"> 73</span>&#160; }</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="#a1440a7779ac56f47a3f355ce4a8c7da0">&#9670;&nbsp;</a></span>test1()</h2>
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<td class="memname">void test1 </td>
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<td class="paramname"></td><td>)</td>
<td></td>
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<p>test function to compute eigen values of a 2x2 matrix </p><p class="formulaDsp">
\[\begin{bmatrix} 5 &amp; 7\\ 7 &amp; 11 \end{bmatrix}\]
</p>
<p> which are approximately, {15.56158, 0.384227} </p>
<div class="fragment"><div class="line"><a name="l00177"></a><span class="lineno"> 177</span>&#160; {</div>
<div class="line"><a name="l00178"></a><span class="lineno"> 178</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray&lt;std::valarray&lt;double&gt;</a>&gt; X = {{5, 7}, {7, 11}};</div>
<div class="line"><a name="l00179"></a><span class="lineno"> 179</span>&#160; <span class="keywordtype">double</span> y[] = {15.56158, 0.384227}; <span class="comment">// corresponding y-values</span></div>
<div class="line"><a name="l00180"></a><span class="lineno"> 180</span>&#160; </div>
<div class="line"><a name="l00181"></a><span class="lineno"> 181</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;------- Test 1 -------&quot;</span> &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="l00182"></a><span class="lineno"> 182</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray&lt;double&gt;</a> eig_vals = <a class="code" href="../../d2/d3b/namespaceqr__algorithm.html#a763896dbb4a7e95c4a1e614ac0819d66">qr_algorithm::eigen_values</a>(&amp;X);</div>
<div class="line"><a name="l00183"></a><span class="lineno"> 183</span>&#160; </div>
<div class="line"><a name="l00184"></a><span class="lineno"> 184</span>&#160; <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i &lt; 2; i++) {</div>
<div class="line"><a name="l00185"></a><span class="lineno"> 185</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; i + 1 &lt;&lt; <span class="stringliteral">&quot;/2 Checking for &quot;</span> &lt;&lt; y[i] &lt;&lt; <span class="stringliteral">&quot; --&gt; &quot;</span>;</div>
<div class="line"><a name="l00186"></a><span class="lineno"> 186</span>&#160; <span class="keywordtype">bool</span> result = <span class="keyword">false</span>;</div>
<div class="line"><a name="l00187"></a><span class="lineno"> 187</span>&#160; <span class="keywordflow">for</span> (<span class="keywordtype">int</span> j = 0; j &lt; 2 &amp;&amp; !result; j++) {</div>
<div class="line"><a name="l00188"></a><span class="lineno"> 188</span>&#160; <span class="keywordflow">if</span> (std::abs(y[i] - eig_vals[j]) &lt; 0.1) {</div>
<div class="line"><a name="l00189"></a><span class="lineno"> 189</span>&#160; result = <span class="keyword">true</span>;</div>
<div class="line"><a name="l00190"></a><span class="lineno"> 190</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;(&quot;</span> &lt;&lt; eig_vals[j] &lt;&lt; <span class="stringliteral">&quot;) &quot;</span>;</div>
<div class="line"><a name="l00191"></a><span class="lineno"> 191</span>&#160; }</div>
<div class="line"><a name="l00192"></a><span class="lineno"> 192</span>&#160; }</div>
<div class="line"><a name="l00193"></a><span class="lineno"> 193</span>&#160; assert(result); <span class="comment">// ensure that i^th expected eigen value was computed</span></div>
<div class="line"><a name="l00194"></a><span class="lineno"> 194</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;found\n&quot;</span>;</div>
<div class="line"><a name="l00195"></a><span class="lineno"> 195</span>&#160; }</div>
<div class="line"><a name="l00196"></a><span class="lineno"> 196</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;Test 1 Passed\n\n&quot;</span>;</div>
<div class="line"><a name="l00197"></a><span class="lineno"> 197</span>&#160;}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#a0283886819c7c140a023582b7269e2d0">&#9670;&nbsp;</a></span>test2()</h2>
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<td class="memname">void test2 </td>
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<p>test function to compute eigen values of a 2x2 matrix </p><p class="formulaDsp">
\[\begin{bmatrix} -4&amp; 4&amp; 2&amp; 0&amp; -3\\ 4&amp; -4&amp; 4&amp; -3&amp; -1\\ 2&amp; 4&amp; 4&amp; 3&amp; -3\\ 0&amp; -3&amp; 3&amp; -1&amp;-1\\ -3&amp; -1&amp; -3&amp; -3&amp; 0 \end{bmatrix}\]
</p>
<p> which are approximately, {9.27648, -9.26948, 2.0181, -1.03516, -5.98994} </p>
<div class="fragment"><div class="line"><a name="l00210"></a><span class="lineno"> 210</span>&#160; {</div>
<div class="line"><a name="l00211"></a><span class="lineno"> 211</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray&lt;std::valarray&lt;double&gt;</a>&gt; X = {{-4, 4, 2, 0, -3},</div>
<div class="line"><a name="l00212"></a><span class="lineno"> 212</span>&#160; {4, -4, 4, -3, -1},</div>
<div class="line"><a name="l00213"></a><span class="lineno"> 213</span>&#160; {2, 4, 4, 3, -3},</div>
<div class="line"><a name="l00214"></a><span class="lineno"> 214</span>&#160; {0, -3, 3, -1, -3},</div>
<div class="line"><a name="l00215"></a><span class="lineno"> 215</span>&#160; {-3, -1, -3, -3, 0}};</div>
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<div class="line"><a name="l00217"></a><span class="lineno"> 217</span>&#160; -5.98994}; <span class="comment">// corresponding y-values</span></div>
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<div class="line"><a name="l00219"></a><span class="lineno"> 219</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;------- Test 2 -------&quot;</span> &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="l00220"></a><span class="lineno"> 220</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/valarray.html">std::valarray&lt;double&gt;</a> eig_vals = <a class="code" href="../../d2/d3b/namespaceqr__algorithm.html#a763896dbb4a7e95c4a1e614ac0819d66">qr_algorithm::eigen_values</a>(&amp;X);</div>
<div class="line"><a name="l00221"></a><span class="lineno"> 221</span>&#160; </div>
<div class="line"><a name="l00222"></a><span class="lineno"> 222</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; X &lt;&lt; <span class="stringliteral">&quot;\n&quot;</span></div>
<div class="line"><a name="l00223"></a><span class="lineno"> 223</span>&#160; &lt;&lt; <span class="stringliteral">&quot;Eigen values: &quot;</span> &lt;&lt; eig_vals &lt;&lt; <span class="stringliteral">&quot;\n&quot;</span>;</div>
<div class="line"><a name="l00224"></a><span class="lineno"> 224</span>&#160; </div>
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<div class="line"><a name="l00226"></a><span class="lineno"> 226</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; i + 1 &lt;&lt; <span class="stringliteral">&quot;/5 Checking for &quot;</span> &lt;&lt; y[i] &lt;&lt; <span class="stringliteral">&quot; --&gt; &quot;</span>;</div>
<div class="line"><a name="l00227"></a><span class="lineno"> 227</span>&#160; <span class="keywordtype">bool</span> result = <span class="keyword">false</span>;</div>
<div class="line"><a name="l00228"></a><span class="lineno"> 228</span>&#160; <span class="keywordflow">for</span> (<span class="keywordtype">int</span> j = 0; j &lt; 5 &amp;&amp; !result; j++) {</div>
<div class="line"><a name="l00229"></a><span class="lineno"> 229</span>&#160; <span class="keywordflow">if</span> (std::abs(y[i] - eig_vals[j]) &lt; 0.1) {</div>
<div class="line"><a name="l00230"></a><span class="lineno"> 230</span>&#160; result = <span class="keyword">true</span>;</div>
<div class="line"><a name="l00231"></a><span class="lineno"> 231</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;(&quot;</span> &lt;&lt; eig_vals[j] &lt;&lt; <span class="stringliteral">&quot;) &quot;</span>;</div>
<div class="line"><a name="l00232"></a><span class="lineno"> 232</span>&#160; }</div>
<div class="line"><a name="l00233"></a><span class="lineno"> 233</span>&#160; }</div>
<div class="line"><a name="l00234"></a><span class="lineno"> 234</span>&#160; assert(result); <span class="comment">// ensure that i^th expected eigen value was computed</span></div>
<div class="line"><a name="l00235"></a><span class="lineno"> 235</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;found\n&quot;</span>;</div>
<div class="line"><a name="l00236"></a><span class="lineno"> 236</span>&#160; }</div>
<div class="line"><a name="l00237"></a><span class="lineno"> 237</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;Test 2 Passed\n\n&quot;</span>;</div>
<div class="line"><a name="l00238"></a><span class="lineno"> 238</span>&#160;}</div>
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<p>This Programme Converts a given decimal number in the range [0,4000) to both Lower case and Upper case Roman Numeral.
<a href="#details">More...</a></p>
<div class="textblock"><code>#include &lt;cmath&gt;</code><br />
<code>#include &lt;cstdio&gt;</code><br />
<code>#include &lt;cstring&gt;</code><br />
<code>#include &lt;iostream&gt;</code><br />
</div><div class="textblock"><div class="dynheader">
Include dependency graph for decimal_to_roman_numeral.cpp:</div>
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Functions</h2></td></tr>
<tr class="memitem:a88203bd297e8405160c132faa1187780"><td class="memItemLeft" align="right" valign="top"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/string/basic_string.html">std::string</a>&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../de/d85/decimal__to__roman__numeral_8cpp.html#a88203bd297e8405160c132faa1187780">fill</a> (char c, int n)</td></tr>
<tr class="separator:a88203bd297e8405160c132faa1187780"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:a003fb4e1b08279fe4cd50fbbc2782c2d"><td class="memItemLeft" align="right" valign="top"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/string/basic_string.html">std::string</a>&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../de/d85/decimal__to__roman__numeral_8cpp.html#a003fb4e1b08279fe4cd50fbbc2782c2d">tolowerRoman</a> (int n)</td></tr>
<tr class="separator:a003fb4e1b08279fe4cd50fbbc2782c2d"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:a214743638eff1336f835310049aef979"><td class="memItemLeft" align="right" valign="top"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/string/basic_string.html">std::string</a>&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../de/d85/decimal__to__roman__numeral_8cpp.html#a214743638eff1336f835310049aef979">toupperRoman</a> (int n)</td></tr>
<tr class="separator:a214743638eff1336f835310049aef979"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:ae66f6b31b5ad750f1fe042a706a4e3d4"><td class="memItemLeft" align="right" valign="top">int&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../de/d85/decimal__to__roman__numeral_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>This Programme Converts a given decimal number in the range [0,4000) to both Lower case and Upper case Roman Numeral. </p>
</div><h2 class="groupheader">Function Documentation</h2>
<a id="a88203bd297e8405160c132faa1187780"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a88203bd297e8405160c132faa1187780">&#9670;&nbsp;</a></span>fill()</h2>
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<div class="memproto">
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<td class="memname"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/string/basic_string.html">std::string</a> fill </td>
<td>(</td>
<td class="paramtype">char&#160;</td>
<td class="paramname"><em>c</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">int&#160;</td>
<td class="paramname"><em>n</em>&#160;</td>
</tr>
<tr>
<td></td>
<td>)</td>
<td></td><td></td>
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</div><div class="memdoc">
<p>This functions fills a string with character c, n times and returns it </p><dl class="section note"><dt>Note</dt><dd>This can probably be replace by <code>memcpy</code> function. </dd></dl>
<div class="fragment"><div class="line"><a name="l00015"></a><span class="lineno"> 15</span>&#160; {</div>
<div class="line"><a name="l00016"></a><span class="lineno"> 16</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/string/basic_string.html">std::string</a> s = <span class="stringliteral">&quot;&quot;</span>;</div>
<div class="line"><a name="l00017"></a><span class="lineno"> 17</span>&#160; <span class="keywordflow">while</span> (n--) s += c;</div>
<div class="line"><a name="l00018"></a><span class="lineno"> 18</span>&#160; <span class="keywordflow">return</span> s;</div>
<div class="line"><a name="l00019"></a><span class="lineno"> 19</span>&#160;}</div>
</div><!-- fragment -->
</div>
</div>
<a id="ae66f6b31b5ad750f1fe042a706a4e3d4"></a>
<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>
<td>(</td>
<td class="paramname"></td><td>)</td>
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<p>main function </p>
<div class="fragment"><div class="line"><a name="l00090"></a><span class="lineno"> 90</span>&#160; {</div>
<div class="line"><a name="l00091"></a><span class="lineno"> 91</span>&#160; <span class="keywordtype">int</span> n;</div>
<div class="line"><a name="l00092"></a><span class="lineno"> 92</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;\t\tRoman numbers converter\n\n&quot;</span>;</div>
<div class="line"><a name="l00093"></a><span class="lineno"> 93</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;Type in decimal number between 0 up to 4000 (exclusive): &quot;</span>;</div>
<div class="line"><a name="l00094"></a><span class="lineno"> 94</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_istream.html">std::cin</a> &gt;&gt; n;</div>
<div class="line"><a name="l00095"></a><span class="lineno"> 95</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; n &lt;&lt; <span class="stringliteral">&quot; in Upper Roman Numerals is &quot;</span> &lt;&lt; <a class="code" href="../../de/d85/decimal__to__roman__numeral_8cpp.html#a214743638eff1336f835310049aef979">toupperRoman</a>(n) &lt;&lt; <span class="stringliteral">&quot;\n&quot;</span>;</div>
<div class="line"><a name="l00096"></a><span class="lineno"> 96</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; n &lt;&lt; <span class="stringliteral">&quot; in Lower Roman Numerals is &quot;</span> &lt;&lt; <a class="code" href="../../de/d85/decimal__to__roman__numeral_8cpp.html#a003fb4e1b08279fe4cd50fbbc2782c2d">tolowerRoman</a>(n) &lt;&lt; <span class="stringliteral">&quot;\n&quot;</span>;</div>
<div class="line"><a name="l00097"></a><span class="lineno"> 97</span>&#160; <span class="keywordflow">return</span> 0;</div>
<div class="line"><a name="l00098"></a><span class="lineno"> 98</span>&#160;}</div>
</div><!-- fragment --><div class="dynheader">
Here is the call graph for this function:</div>
<div class="dyncontent">
<div class="center"><iframe scrolling="no" frameborder="0" src="../../de/d85/decimal__to__roman__numeral_8cpp_ae66f6b31b5ad750f1fe042a706a4e3d4_cgraph.svg" width="303" height="88"><p><b>This browser is not able to show SVG: try Firefox, Chrome, Safari, or Opera instead.</b></p></iframe>
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<h2 class="memtitle"><span class="permalink"><a href="#a003fb4e1b08279fe4cd50fbbc2782c2d">&#9670;&nbsp;</a></span>tolowerRoman()</h2>
<div class="memitem">
<div class="memproto">
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<td class="memname"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/string/basic_string.html">std::string</a> tolowerRoman </td>
<td>(</td>
<td class="paramtype">int&#160;</td>
<td class="paramname"><em>n</em></td><td>)</td>
<td></td>
</tr>
</table>
</div><div class="memdoc">
<p>to convert to lowercase Roman Numeral the function works recursively </p>
<div class="fragment"><div class="line"><a name="l00024"></a><span class="lineno"> 24</span>&#160; {</div>
<div class="line"><a name="l00025"></a><span class="lineno"> 25</span>&#160; <span class="keywordflow">if</span> (n &lt; 4)</div>
<div class="line"><a name="l00026"></a><span class="lineno"> 26</span>&#160; <span class="keywordflow">return</span> <a class="code" href="../../de/d85/decimal__to__roman__numeral_8cpp.html#a88203bd297e8405160c132faa1187780">fill</a>(<span class="charliteral">&#39;i&#39;</span>, n);</div>
<div class="line"><a name="l00027"></a><span class="lineno"> 27</span>&#160; <span class="keywordflow">if</span> (n &lt; 6)</div>
<div class="line"><a name="l00028"></a><span class="lineno"> 28</span>&#160; <span class="keywordflow">return</span> <a class="code" href="../../de/d85/decimal__to__roman__numeral_8cpp.html#a88203bd297e8405160c132faa1187780">fill</a>(<span class="charliteral">&#39;i&#39;</span>, 5 - n) + <span class="stringliteral">&quot;v&quot;</span>;</div>
<div class="line"><a name="l00029"></a><span class="lineno"> 29</span>&#160; <span class="keywordflow">if</span> (n &lt; 9)</div>
<div class="line"><a name="l00030"></a><span class="lineno"> 30</span>&#160; <span class="keywordflow">return</span> <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/string/basic_string.html">std::string</a>(<span class="stringliteral">&quot;v&quot;</span>) + <a class="code" href="../../de/d85/decimal__to__roman__numeral_8cpp.html#a88203bd297e8405160c132faa1187780">fill</a>(<span class="charliteral">&#39;i&#39;</span>, n - 5);</div>
<div class="line"><a name="l00031"></a><span class="lineno"> 31</span>&#160; <span class="keywordflow">if</span> (n &lt; 11)</div>
<div class="line"><a name="l00032"></a><span class="lineno"> 32</span>&#160; <span class="keywordflow">return</span> <a class="code" href="../../de/d85/decimal__to__roman__numeral_8cpp.html#a88203bd297e8405160c132faa1187780">fill</a>(<span class="charliteral">&#39;i&#39;</span>, 10 - n) + <span class="stringliteral">&quot;x&quot;</span>;</div>
<div class="line"><a name="l00033"></a><span class="lineno"> 33</span>&#160; <span class="keywordflow">if</span> (n &lt; 40)</div>
<div class="line"><a name="l00034"></a><span class="lineno"> 34</span>&#160; <span class="keywordflow">return</span> <a class="code" href="../../de/d85/decimal__to__roman__numeral_8cpp.html#a88203bd297e8405160c132faa1187780">fill</a>(<span class="charliteral">&#39;x&#39;</span>, n / 10) + <a class="code" href="../../de/d85/decimal__to__roman__numeral_8cpp.html#a003fb4e1b08279fe4cd50fbbc2782c2d">tolowerRoman</a>(n % 10);</div>
<div class="line"><a name="l00035"></a><span class="lineno"> 35</span>&#160; <span class="keywordflow">if</span> (n &lt; 60)</div>
<div class="line"><a name="l00036"></a><span class="lineno"> 36</span>&#160; <span class="keywordflow">return</span> <a class="code" href="../../de/d85/decimal__to__roman__numeral_8cpp.html#a88203bd297e8405160c132faa1187780">fill</a>(<span class="charliteral">&#39;x&#39;</span>, 5 - n / 10) + <span class="charliteral">&#39;l&#39;</span> + <a class="code" href="../../de/d85/decimal__to__roman__numeral_8cpp.html#a003fb4e1b08279fe4cd50fbbc2782c2d">tolowerRoman</a>(n % 10);</div>
<div class="line"><a name="l00037"></a><span class="lineno"> 37</span>&#160; <span class="keywordflow">if</span> (n &lt; 90)</div>
<div class="line"><a name="l00038"></a><span class="lineno"> 38</span>&#160; <span class="keywordflow">return</span> <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/string/basic_string.html">std::string</a>(<span class="stringliteral">&quot;l&quot;</span>) + <a class="code" href="../../de/d85/decimal__to__roman__numeral_8cpp.html#a88203bd297e8405160c132faa1187780">fill</a>(<span class="charliteral">&#39;x&#39;</span>, n / 10 - 5) + <a class="code" href="../../de/d85/decimal__to__roman__numeral_8cpp.html#a003fb4e1b08279fe4cd50fbbc2782c2d">tolowerRoman</a>(n % 10);</div>
<div class="line"><a name="l00039"></a><span class="lineno"> 39</span>&#160; <span class="keywordflow">if</span> (n &lt; 110)</div>
<div class="line"><a name="l00040"></a><span class="lineno"> 40</span>&#160; <span class="keywordflow">return</span> <a class="code" href="../../de/d85/decimal__to__roman__numeral_8cpp.html#a88203bd297e8405160c132faa1187780">fill</a>(<span class="charliteral">&#39;x&#39;</span>, 10 - n / 10) + <span class="stringliteral">&quot;c&quot;</span> + <a class="code" href="../../de/d85/decimal__to__roman__numeral_8cpp.html#a003fb4e1b08279fe4cd50fbbc2782c2d">tolowerRoman</a>(n % 10);</div>
<div class="line"><a name="l00041"></a><span class="lineno"> 41</span>&#160; <span class="keywordflow">if</span> (n &lt; 400)</div>
<div class="line"><a name="l00042"></a><span class="lineno"> 42</span>&#160; <span class="keywordflow">return</span> <a class="code" href="../../de/d85/decimal__to__roman__numeral_8cpp.html#a88203bd297e8405160c132faa1187780">fill</a>(<span class="charliteral">&#39;c&#39;</span>, n / 100) + <a class="code" href="../../de/d85/decimal__to__roman__numeral_8cpp.html#a003fb4e1b08279fe4cd50fbbc2782c2d">tolowerRoman</a>(n % 100);</div>
<div class="line"><a name="l00043"></a><span class="lineno"> 43</span>&#160; <span class="keywordflow">if</span> (n &lt; 600)</div>
<div class="line"><a name="l00044"></a><span class="lineno"> 44</span>&#160; <span class="keywordflow">return</span> <a class="code" href="../../de/d85/decimal__to__roman__numeral_8cpp.html#a88203bd297e8405160c132faa1187780">fill</a>(<span class="charliteral">&#39;c&#39;</span>, 5 - n / 100) + <span class="charliteral">&#39;d&#39;</span> + <a class="code" href="../../de/d85/decimal__to__roman__numeral_8cpp.html#a003fb4e1b08279fe4cd50fbbc2782c2d">tolowerRoman</a>(n % 100);</div>
<div class="line"><a name="l00045"></a><span class="lineno"> 45</span>&#160; <span class="keywordflow">if</span> (n &lt; 900)</div>
<div class="line"><a name="l00046"></a><span class="lineno"> 46</span>&#160; <span class="keywordflow">return</span> <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/string/basic_string.html">std::string</a>(<span class="stringliteral">&quot;d&quot;</span>) + <a class="code" href="../../de/d85/decimal__to__roman__numeral_8cpp.html#a88203bd297e8405160c132faa1187780">fill</a>(<span class="charliteral">&#39;c&#39;</span>, n / 100 - 5) +</div>
<div class="line"><a name="l00047"></a><span class="lineno"> 47</span>&#160; <a class="code" href="../../de/d85/decimal__to__roman__numeral_8cpp.html#a003fb4e1b08279fe4cd50fbbc2782c2d">tolowerRoman</a>(n % 100);</div>
<div class="line"><a name="l00048"></a><span class="lineno"> 48</span>&#160; <span class="keywordflow">if</span> (n &lt; 1100)</div>
<div class="line"><a name="l00049"></a><span class="lineno"> 49</span>&#160; <span class="keywordflow">return</span> <a class="code" href="../../de/d85/decimal__to__roman__numeral_8cpp.html#a88203bd297e8405160c132faa1187780">fill</a>(<span class="charliteral">&#39;c&#39;</span>, 10 - n / 100) + <span class="stringliteral">&quot;m&quot;</span> + <a class="code" href="../../de/d85/decimal__to__roman__numeral_8cpp.html#a003fb4e1b08279fe4cd50fbbc2782c2d">tolowerRoman</a>(n % 100);</div>
<div class="line"><a name="l00050"></a><span class="lineno"> 50</span>&#160; <span class="keywordflow">if</span> (n &lt; 4000)</div>
<div class="line"><a name="l00051"></a><span class="lineno"> 51</span>&#160; <span class="keywordflow">return</span> <a class="code" href="../../de/d85/decimal__to__roman__numeral_8cpp.html#a88203bd297e8405160c132faa1187780">fill</a>(<span class="charliteral">&#39;m&#39;</span>, n / 1000) + <a class="code" href="../../de/d85/decimal__to__roman__numeral_8cpp.html#a003fb4e1b08279fe4cd50fbbc2782c2d">tolowerRoman</a>(n % 1000);</div>
<div class="line"><a name="l00052"></a><span class="lineno"> 52</span>&#160; <span class="keywordflow">return</span> <span class="stringliteral">&quot;?&quot;</span>;</div>
<div class="line"><a name="l00053"></a><span class="lineno"> 53</span>&#160;}</div>
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Here is the call graph for this function:</div>
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<h2 class="memtitle"><span class="permalink"><a href="#a214743638eff1336f835310049aef979">&#9670;&nbsp;</a></span>toupperRoman()</h2>
<div class="memitem">
<div class="memproto">
<table class="memname">
<tr>
<td class="memname"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/string/basic_string.html">std::string</a> toupperRoman </td>
<td>(</td>
<td class="paramtype">int&#160;</td>
<td class="paramname"><em>n</em></td><td>)</td>
<td></td>
</tr>
</table>
</div><div class="memdoc">
<p>to convert to uppercase Roman Numeral the function works recursively </p>
<div class="fragment"><div class="line"><a name="l00058"></a><span class="lineno"> 58</span>&#160; {</div>
<div class="line"><a name="l00059"></a><span class="lineno"> 59</span>&#160; <span class="keywordflow">if</span> (n &lt; 4)</div>
<div class="line"><a name="l00060"></a><span class="lineno"> 60</span>&#160; <span class="keywordflow">return</span> <a class="code" href="../../de/d85/decimal__to__roman__numeral_8cpp.html#a88203bd297e8405160c132faa1187780">fill</a>(<span class="charliteral">&#39;I&#39;</span>, n);</div>
<div class="line"><a name="l00061"></a><span class="lineno"> 61</span>&#160; <span class="keywordflow">if</span> (n &lt; 6)</div>
<div class="line"><a name="l00062"></a><span class="lineno"> 62</span>&#160; <span class="keywordflow">return</span> <a class="code" href="../../de/d85/decimal__to__roman__numeral_8cpp.html#a88203bd297e8405160c132faa1187780">fill</a>(<span class="charliteral">&#39;I&#39;</span>, 5 - n) + <span class="stringliteral">&quot;V&quot;</span>;</div>
<div class="line"><a name="l00063"></a><span class="lineno"> 63</span>&#160; <span class="keywordflow">if</span> (n &lt; 9)</div>
<div class="line"><a name="l00064"></a><span class="lineno"> 64</span>&#160; <span class="keywordflow">return</span> <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/string/basic_string.html">std::string</a>(<span class="stringliteral">&quot;V&quot;</span>) + <a class="code" href="../../de/d85/decimal__to__roman__numeral_8cpp.html#a88203bd297e8405160c132faa1187780">fill</a>(<span class="charliteral">&#39;I&#39;</span>, n - 5);</div>
<div class="line"><a name="l00065"></a><span class="lineno"> 65</span>&#160; <span class="keywordflow">if</span> (n &lt; 11)</div>
<div class="line"><a name="l00066"></a><span class="lineno"> 66</span>&#160; <span class="keywordflow">return</span> <a class="code" href="../../de/d85/decimal__to__roman__numeral_8cpp.html#a88203bd297e8405160c132faa1187780">fill</a>(<span class="charliteral">&#39;I&#39;</span>, 10 - n) + <span class="stringliteral">&quot;X&quot;</span>;</div>
<div class="line"><a name="l00067"></a><span class="lineno"> 67</span>&#160; <span class="keywordflow">if</span> (n &lt; 40)</div>
<div class="line"><a name="l00068"></a><span class="lineno"> 68</span>&#160; <span class="keywordflow">return</span> <a class="code" href="../../de/d85/decimal__to__roman__numeral_8cpp.html#a88203bd297e8405160c132faa1187780">fill</a>(<span class="charliteral">&#39;X&#39;</span>, n / 10) + <a class="code" href="../../de/d85/decimal__to__roman__numeral_8cpp.html#a214743638eff1336f835310049aef979">toupperRoman</a>(n % 10);</div>
<div class="line"><a name="l00069"></a><span class="lineno"> 69</span>&#160; <span class="keywordflow">if</span> (n &lt; 60)</div>
<div class="line"><a name="l00070"></a><span class="lineno"> 70</span>&#160; <span class="keywordflow">return</span> <a class="code" href="../../de/d85/decimal__to__roman__numeral_8cpp.html#a88203bd297e8405160c132faa1187780">fill</a>(<span class="charliteral">&#39;X&#39;</span>, 5 - n / 10) + <span class="charliteral">&#39;L&#39;</span> + <a class="code" href="../../de/d85/decimal__to__roman__numeral_8cpp.html#a214743638eff1336f835310049aef979">toupperRoman</a>(n % 10);</div>
<div class="line"><a name="l00071"></a><span class="lineno"> 71</span>&#160; <span class="keywordflow">if</span> (n &lt; 90)</div>
<div class="line"><a name="l00072"></a><span class="lineno"> 72</span>&#160; <span class="keywordflow">return</span> <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/string/basic_string.html">std::string</a>(<span class="stringliteral">&quot;L&quot;</span>) + <a class="code" href="../../de/d85/decimal__to__roman__numeral_8cpp.html#a88203bd297e8405160c132faa1187780">fill</a>(<span class="charliteral">&#39;X&#39;</span>, n / 10 - 5) + <a class="code" href="../../de/d85/decimal__to__roman__numeral_8cpp.html#a214743638eff1336f835310049aef979">toupperRoman</a>(n % 10);</div>
<div class="line"><a name="l00073"></a><span class="lineno"> 73</span>&#160; <span class="keywordflow">if</span> (n &lt; 110)</div>
<div class="line"><a name="l00074"></a><span class="lineno"> 74</span>&#160; <span class="keywordflow">return</span> <a class="code" href="../../de/d85/decimal__to__roman__numeral_8cpp.html#a88203bd297e8405160c132faa1187780">fill</a>(<span class="charliteral">&#39;X&#39;</span>, 10 - n / 10) + <span class="stringliteral">&quot;C&quot;</span> + <a class="code" href="../../de/d85/decimal__to__roman__numeral_8cpp.html#a214743638eff1336f835310049aef979">toupperRoman</a>(n % 10);</div>
<div class="line"><a name="l00075"></a><span class="lineno"> 75</span>&#160; <span class="keywordflow">if</span> (n &lt; 400)</div>
<div class="line"><a name="l00076"></a><span class="lineno"> 76</span>&#160; <span class="keywordflow">return</span> <a class="code" href="../../de/d85/decimal__to__roman__numeral_8cpp.html#a88203bd297e8405160c132faa1187780">fill</a>(<span class="charliteral">&#39;C&#39;</span>, n / 100) + <a class="code" href="../../de/d85/decimal__to__roman__numeral_8cpp.html#a214743638eff1336f835310049aef979">toupperRoman</a>(n % 100);</div>
<div class="line"><a name="l00077"></a><span class="lineno"> 77</span>&#160; <span class="keywordflow">if</span> (n &lt; 600)</div>
<div class="line"><a name="l00078"></a><span class="lineno"> 78</span>&#160; <span class="keywordflow">return</span> <a class="code" href="../../de/d85/decimal__to__roman__numeral_8cpp.html#a88203bd297e8405160c132faa1187780">fill</a>(<span class="charliteral">&#39;C&#39;</span>, 5 - n / 100) + <span class="charliteral">&#39;D&#39;</span> + <a class="code" href="../../de/d85/decimal__to__roman__numeral_8cpp.html#a214743638eff1336f835310049aef979">toupperRoman</a>(n % 100);</div>
<div class="line"><a name="l00079"></a><span class="lineno"> 79</span>&#160; <span class="keywordflow">if</span> (n &lt; 900)</div>
<div class="line"><a name="l00080"></a><span class="lineno"> 80</span>&#160; <span class="keywordflow">return</span> <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/string/basic_string.html">std::string</a>(<span class="stringliteral">&quot;D&quot;</span>) + <a class="code" href="../../de/d85/decimal__to__roman__numeral_8cpp.html#a88203bd297e8405160c132faa1187780">fill</a>(<span class="charliteral">&#39;C&#39;</span>, n / 100 - 5) +</div>
<div class="line"><a name="l00081"></a><span class="lineno"> 81</span>&#160; <a class="code" href="../../de/d85/decimal__to__roman__numeral_8cpp.html#a214743638eff1336f835310049aef979">toupperRoman</a>(n % 100);</div>
<div class="line"><a name="l00082"></a><span class="lineno"> 82</span>&#160; <span class="keywordflow">if</span> (n &lt; 1100)</div>
<div class="line"><a name="l00083"></a><span class="lineno"> 83</span>&#160; <span class="keywordflow">return</span> <a class="code" href="../../de/d85/decimal__to__roman__numeral_8cpp.html#a88203bd297e8405160c132faa1187780">fill</a>(<span class="charliteral">&#39;C&#39;</span>, 10 - n / 100) + <span class="stringliteral">&quot;M&quot;</span> + <a class="code" href="../../de/d85/decimal__to__roman__numeral_8cpp.html#a214743638eff1336f835310049aef979">toupperRoman</a>(n % 100);</div>
<div class="line"><a name="l00084"></a><span class="lineno"> 84</span>&#160; <span class="keywordflow">if</span> (n &lt; 4000)</div>
<div class="line"><a name="l00085"></a><span class="lineno"> 85</span>&#160; <span class="keywordflow">return</span> <a class="code" href="../../de/d85/decimal__to__roman__numeral_8cpp.html#a88203bd297e8405160c132faa1187780">fill</a>(<span class="charliteral">&#39;M&#39;</span>, n / 1000) + <a class="code" href="../../de/d85/decimal__to__roman__numeral_8cpp.html#a214743638eff1336f835310049aef979">toupperRoman</a>(n % 1000);</div>
<div class="line"><a name="l00086"></a><span class="lineno"> 86</span>&#160; <span class="keywordflow">return</span> <span class="stringliteral">&quot;?&quot;</span>;</div>
<div class="line"><a name="l00087"></a><span class="lineno"> 87</span>&#160;}</div>
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<div class="ttc" id="abasic_string_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/string/basic_string.html">std::string</a></div><div class="ttdoc">STL class.</div></div>
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<div class="ttc" id="adecimal__to__roman__numeral_8cpp_html_a88203bd297e8405160c132faa1187780"><div class="ttname"><a href="../../de/d85/decimal__to__roman__numeral_8cpp.html#a88203bd297e8405160c132faa1187780">fill</a></div><div class="ttdeci">std::string fill(char c, int n)</div><div class="ttdef"><b>Definition:</b> decimal_to_roman_numeral.cpp:15</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="abasic_istream_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/io/basic_istream.html">std::cin</a></div></div>
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<p>Get list of prime numbers.
<a href="#details">More...</a></p>
<div class="textblock"><code>#include &lt;iostream&gt;</code><br />
<code>#include &lt;vector&gt;</code><br />
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Functions</h2></td></tr>
<tr class="memitem:a3b91f208e8365c95a295cfe3e67df5c3"><td class="memItemLeft" align="right" valign="top"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/vector.html">std::vector</a>&lt; int &gt;&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../de/d9b/prime__numbers_8cpp.html#a3b91f208e8365c95a295cfe3e67df5c3">primes</a> (int max)</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="../../de/d9b/prime__numbers_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>Get list of prime numbers. </p>
<dl class="section see"><dt>See also</dt><dd><a class="el" href="../../d4/d9c/primes__up__to__billion_8cpp.html" title="Compute prime numbers upto 1 billion.">primes_up_to_billion.cpp</a> <a class="el" href="../../d8/ddf/sieve__of__eratosthenes_8cpp.html" title="Get list of prime numbers using Sieve of Eratosthenes Sieve of Eratosthenes is an algorithm to find t...">sieve_of_eratosthenes.cpp</a> </dd></dl>
</div><h2 class="groupheader">Function Documentation</h2>
<a id="ae66f6b31b5ad750f1fe042a706a4e3d4"></a>
<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>
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<div class="line"><a name="l00028"></a><span class="lineno"> 28</span>&#160; <span class="keywordtype">int</span> n;</div>
<div class="line"><a name="l00029"></a><span class="lineno"> 29</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_istream.html">std::cin</a> &gt;&gt; n;</div>
<div class="line"><a name="l00030"></a><span class="lineno"> 30</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/vector.html">std::vector&lt;int&gt;</a> <a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#ad8389ed58fd0ec66df248014775ad1fa">ans</a> = <a class="code" href="../../de/d9b/prime__numbers_8cpp.html#a3b91f208e8365c95a295cfe3e67df5c3">primes</a>(n);</div>
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<div class="line"><a name="l00032"></a><span class="lineno"> 32</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/endl.html">std::endl</a>;</div>
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<h2 class="memtitle"><span class="permalink"><a href="#a3b91f208e8365c95a295cfe3e67df5c3">&#9670;&nbsp;</a></span>primes()</h2>
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<td class="memname"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/vector.html">std::vector</a>&lt;int&gt; primes </td>
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<p>Generate an increasingly large number of primes and store in a list </p>
<div class="fragment"><div class="line"><a name="l00012"></a><span class="lineno"> 12</span>&#160; {</div>
<div class="line"><a name="l00013"></a><span class="lineno"> 13</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/algorithm/max.html">max</a>++;</div>
<div class="line"><a name="l00014"></a><span class="lineno"> 14</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/vector.html">std::vector&lt;int&gt;</a> res;</div>
<div class="line"><a name="l00015"></a><span class="lineno"> 15</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/vector.html">std::vector&lt;bool&gt;</a> numbers(max, <span class="keyword">false</span>);</div>
<div class="line"><a name="l00016"></a><span class="lineno"> 16</span>&#160; <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 2; i &lt; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/algorithm/max.html">max</a>; i++) {</div>
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<div class="line"><a name="l00019"></a><span class="lineno"> 19</span>&#160; res.<a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/vector/push_back.html">push_back</a>(i);</div>
<div class="line"><a name="l00020"></a><span class="lineno"> 20</span>&#160; }</div>
<div class="line"><a name="l00021"></a><span class="lineno"> 21</span>&#160; }</div>
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<div class="line"><a name="l00023"></a><span class="lineno"> 23</span>&#160;}</div>
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<p>A C++ program to demonstrate common Binary Heap Operations.
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<p>C++ Program to find Binary Exponent Iteratively and Recursively.
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<tr class="memitem:aeb48dce0725e63d19147944f41843c73"><td class="memItemLeft" align="right" valign="top">int&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../de/dcf/binary__exponent_8cpp.html#aeb48dce0725e63d19147944f41843c73">binExpo</a> (int <a class="el" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a>, int b)</td></tr>
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<tr class="memitem:a31dbf5f7ceb9c9eec831ef9f7782291f"><td class="memItemLeft" align="right" valign="top">int&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../de/dcf/binary__exponent_8cpp.html#a31dbf5f7ceb9c9eec831ef9f7782291f">binExpo_alt</a> (int <a class="el" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a>, int b)</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="../../de/dcf/binary__exponent_8cpp.html#ae66f6b31b5ad750f1fe042a706a4e3d4">main</a> ()</td></tr>
<tr class="memdesc:ae66f6b31b5ad750f1fe042a706a4e3d4"><td class="mdescLeft">&#160;</td><td class="mdescRight">Main function. <a href="../../de/dcf/binary__exponent_8cpp.html#ae66f6b31b5ad750f1fe042a706a4e3d4">More...</a><br /></td></tr>
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<a name="details" id="details"></a><h2 class="groupheader">Detailed Description</h2>
<div class="textblock"><p>C++ Program to find Binary Exponent Iteratively and Recursively. </p>
<p>Calculate \(a^b\) in \(O(\log(b))\) by converting \(b\) to a binary number. Binary exponentiation is also known as exponentiation by squaring. </p><dl class="section note"><dt>Note</dt><dd>This is a far better approach compared to naive method which provide \(O(b)\) operations.</dd></dl>
<p>Example: 10 in base 2 is 1010. </p><p class="formulaDsp">
\begin{eqnarray*} 2^{10_d} &amp;=&amp; 2^{1010_b} = 2^8 * 2^2\\ 2^1 &amp;=&amp; 2\\ 2^2 &amp;=&amp; (2^1)^2 = 2^2 = 4\\ 2^4 &amp;=&amp; (2^2)^2 = 4^2 = 16\\ 2^8 &amp;=&amp; (2^4)^2 = 16^2 = 256\\ \end{eqnarray*}
</p>
<p> Hence to calculate 2^10 we only need to multiply \(2^8\) and \(2^2\) skipping \(2^1\) and \(2^4\). </p>
</div><h2 class="groupheader">Function Documentation</h2>
<a id="aeb48dce0725e63d19147944f41843c73"></a>
<h2 class="memtitle"><span class="permalink"><a href="#aeb48dce0725e63d19147944f41843c73">&#9670;&nbsp;</a></span>binExpo()</h2>
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<td class="memname">int binExpo </td>
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<td class="paramtype">int&#160;</td>
<td class="paramname"><em>a</em>, </td>
</tr>
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<td class="paramkey"></td>
<td></td>
<td class="paramtype">int&#160;</td>
<td class="paramname"><em>b</em>&#160;</td>
</tr>
<tr>
<td></td>
<td>)</td>
<td></td><td></td>
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<p>Recursive function to calculate exponent in \(O(\log(n))\) using binary exponent. </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">if</span> (b == 0) {</div>
<div class="line"><a name="l00030"></a><span class="lineno"> 30</span>&#160; <span class="keywordflow">return</span> 1;</div>
<div class="line"><a name="l00031"></a><span class="lineno"> 31</span>&#160; }</div>
<div class="line"><a name="l00032"></a><span class="lineno"> 32</span>&#160; <span class="keywordtype">int</span> res = <a class="code" href="../../de/dcf/binary__exponent_8cpp.html#aeb48dce0725e63d19147944f41843c73">binExpo</a>(<a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a>, b / 2);</div>
<div class="line"><a name="l00033"></a><span class="lineno"> 33</span>&#160; <span class="keywordflow">if</span> (b % 2) {</div>
<div class="line"><a name="l00034"></a><span class="lineno"> 34</span>&#160; <span class="keywordflow">return</span> res * res * <a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a>;</div>
<div class="line"><a name="l00035"></a><span class="lineno"> 35</span>&#160; } <span class="keywordflow">else</span> {</div>
<div class="line"><a name="l00036"></a><span class="lineno"> 36</span>&#160; <span class="keywordflow">return</span> res * res;</div>
<div class="line"><a name="l00037"></a><span class="lineno"> 37</span>&#160; }</div>
<div class="line"><a name="l00038"></a><span class="lineno"> 38</span>&#160;}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#a31dbf5f7ceb9c9eec831ef9f7782291f">&#9670;&nbsp;</a></span>binExpo_alt()</h2>
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<td class="memname">int binExpo_alt </td>
<td>(</td>
<td class="paramtype">int&#160;</td>
<td class="paramname"><em>a</em>, </td>
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<td class="paramkey"></td>
<td></td>
<td class="paramtype">int&#160;</td>
<td class="paramname"><em>b</em>&#160;</td>
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<p>Iterative function to calculate exponent in \(O(\log(n))\) using binary exponent. </p>
<div class="fragment"><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="keywordtype">int</span> res = 1;</div>
<div class="line"><a name="l00044"></a><span class="lineno"> 44</span>&#160; <span class="keywordflow">while</span> (b &gt; 0) {</div>
<div class="line"><a name="l00045"></a><span class="lineno"> 45</span>&#160; <span class="keywordflow">if</span> (b % 2) {</div>
<div class="line"><a name="l00046"></a><span class="lineno"> 46</span>&#160; res = res * <a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a>;</div>
<div class="line"><a name="l00047"></a><span class="lineno"> 47</span>&#160; }</div>
<div class="line"><a name="l00048"></a><span class="lineno"> 48</span>&#160; <a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a> = <a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a> * <a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a>;</div>
<div class="line"><a name="l00049"></a><span class="lineno"> 49</span>&#160; b /= 2;</div>
<div class="line"><a name="l00050"></a><span class="lineno"> 50</span>&#160; }</div>
<div class="line"><a name="l00051"></a><span class="lineno"> 51</span>&#160; <span class="keywordflow">return</span> res;</div>
<div class="line"><a name="l00052"></a><span class="lineno"> 52</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>
<p>Give two numbers a, b</p>
<p>int resIterate = binExpo_alt(a, b);</p>
<p>Result of a^b (where '^' denotes exponentiation)</p>
<p><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; resIterate &lt;&lt; <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/endl.html">std::endl</a>;</p>
<div class="fragment"><div class="line"><a name="l00055"></a><span class="lineno"> 55</span>&#160; {</div>
<div class="line"><a name="l00056"></a><span class="lineno"> 56</span>&#160; <span class="keywordtype">int</span> <a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a>, b;<span class="comment"></span></div>
<div class="line"><a name="l00057"></a><span class="lineno"> 57</span>&#160;<span class="comment"> /// Give two numbers a, b</span></div>
<div class="line"><a name="l00058"></a><span class="lineno"> 58</span>&#160;<span class="comment"></span> <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_istream.html">std::cin</a> &gt;&gt; <a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a> &gt;&gt; b;</div>
<div class="line"><a name="l00059"></a><span class="lineno"> 59</span>&#160; <span class="keywordflow">if</span> (<a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a> == 0 &amp;&amp; b == 0) {</div>
<div class="line"><a name="l00060"></a><span class="lineno"> 60</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;Math error&quot;</span> &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="l00061"></a><span class="lineno"> 61</span>&#160; } <span class="keywordflow">else</span> <span class="keywordflow">if</span> (b &lt; 0) {</div>
<div class="line"><a name="l00062"></a><span class="lineno"> 62</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;Exponent must be positive !!&quot;</span> &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="l00063"></a><span class="lineno"> 63</span>&#160; } <span class="keywordflow">else</span> {</div>
<div class="line"><a name="l00064"></a><span class="lineno"> 64</span>&#160; <span class="keywordtype">int</span> resRecurse = <a class="code" href="../../de/dcf/binary__exponent_8cpp.html#aeb48dce0725e63d19147944f41843c73">binExpo</a>(<a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a>, b);<span class="comment"></span></div>
<div class="line"><a name="l00065"></a><span class="lineno"> 65</span>&#160;<span class="comment"> /// int resIterate = binExpo_alt(a, b);</span></div>
<div class="line"><a name="l00066"></a><span class="lineno"> 66</span>&#160;<span class="comment"></span><span class="comment"></span> </div>
<div class="line"><a name="l00067"></a><span class="lineno"> 67</span>&#160;<span class="comment"> /// Result of a^b (where &#39;^&#39; denotes exponentiation)</span></div>
<div class="line"><a name="l00068"></a><span class="lineno"> 68</span>&#160;<span class="comment"></span> <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; resRecurse &lt;&lt; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/endl.html">std::endl</a>;<span class="comment"></span></div>
<div class="line"><a name="l00069"></a><span class="lineno"> 69</span>&#160;<span class="comment"> /// std::cout &lt;&lt; resIterate &lt;&lt; std::endl;</span></div>
<div class="line"><a name="l00070"></a><span class="lineno"> 70</span>&#160;<span class="comment"></span> }</div>
<div class="line"><a name="l00071"></a><span class="lineno"> 71</span>&#160;}</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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<p>Solve the equation \(f(x)=0\) using <a href="https://en.wikipedia.org/wiki/Newton%27s_method">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 />
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Include dependency graph for newton_raphson_method.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="../../de/dd3/newton__raphson__method_8cpp.html#a0a3abbca80bc98e7abcb3ae73abe0f14">MAX_ITERATIONS</a>&#160;&#160;&#160;50000</td></tr>
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<tr class="memitem: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"><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"><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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<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">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">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"></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>
<td></td>
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<p>define \(f(x)\) to find root for </p>
<div class="fragment"><div class="line"><a name="l00026"></a><span class="lineno"> 26</span>&#160; {</div>
<div class="line"><a name="l00027"></a><span class="lineno"> 27</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">// original equation</span></div>
<div class="line"><a name="l00028"></a><span class="lineno"> 28</span>&#160;}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#a3d3f7f41977394680af6ebbed96f3386">&#9670;&nbsp;</a></span>eq_der()</h2>
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<td class="memname">static double eq_der </td>
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<td class="paramname"><em>i</em></td><td>)</td>
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<p>define the derivative function \(f&#39;(x)\) </p>
<div class="fragment"><div class="line"><a name="l00032"></a><span class="lineno"> 32</span>&#160; {</div>
<div class="line"><a name="l00033"></a><span class="lineno"> 33</span>&#160; <span class="keywordflow">return</span> ((3 * <a class="codeRef" 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"><a name="l00034"></a><span class="lineno"> 34</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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<p>Main function </p>
<div class="fragment"><div class="line"><a name="l00037"></a><span class="lineno"> 37</span>&#160; {</div>
<div class="line"><a name="l00038"></a><span class="lineno"> 38</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/random/srand.html">std::srand</a>(<a class="codeRef" 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"><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> z, c = <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/random/rand.html">std::rand</a>() % 100, m, n;</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; <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;\nInitial approximation: &quot;</span> &lt;&lt; c;</div>
<div class="line"><a name="l00044"></a><span class="lineno"> 44</span>&#160; </div>
<div class="line"><a name="l00045"></a><span class="lineno"> 45</span>&#160; <span class="comment">// start iterations</span></div>
<div class="line"><a name="l00046"></a><span class="lineno"> 46</span>&#160; <span class="keywordflow">for</span> (i = 0; i &lt; <a class="code" href="../../de/dd3/newton__raphson__method_8cpp.html#a0a3abbca80bc98e7abcb3ae73abe0f14">MAX_ITERATIONS</a>; i++) {</div>
<div class="line"><a name="l00047"></a><span class="lineno"> 47</span>&#160; m = <a class="code" href="../../de/dd3/newton__raphson__method_8cpp.html#a2003b5b2dcfff0769b957ab5c968b03d">eq</a>(c);</div>
<div class="line"><a name="l00048"></a><span class="lineno"> 48</span>&#160; n = <a class="code" href="../../de/dd3/newton__raphson__method_8cpp.html#a3d3f7f41977394680af6ebbed96f3386">eq_der</a>(c);</div>
<div class="line"><a name="l00049"></a><span class="lineno"> 49</span>&#160; </div>
<div class="line"><a name="l00050"></a><span class="lineno"> 50</span>&#160; z = c - (m / n);</div>
<div class="line"><a name="l00051"></a><span class="lineno"> 51</span>&#160; c = z;</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; <span class="keywordflow">if</span> (std::abs(m) &lt; EPSILON) <span class="comment">// stoping criteria</span></div>
<div class="line"><a name="l00054"></a><span class="lineno"> 54</span>&#160; <span class="keywordflow">break</span>;</div>
<div class="line"><a name="l00055"></a><span class="lineno"> 55</span>&#160; }</div>
<div class="line"><a name="l00056"></a><span class="lineno"> 56</span>&#160; </div>
<div class="line"><a name="l00057"></a><span class="lineno"> 57</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; z &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="l00058"></a><span class="lineno"> 58</span>&#160; <span class="keywordflow">return</span> 0;</div>
<div class="line"><a name="l00059"></a><span class="lineno"> 59</span>&#160;}</div>
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<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>
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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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<p>Computes N^th Fibonacci number given as input argument. Uses custom build arbitrary integers library to perform additions and other operations.
<a href="#details">More...</a></p>
<div class="textblock"><code>#include &lt;cinttypes&gt;</code><br />
<code>#include &lt;ctime&gt;</code><br />
<code>#include &lt;iostream&gt;</code><br />
<code>#include &quot;<a class="el" href="../../d4/d86/large__number_8h_source.html">./large_number.h</a>&quot;</code><br />
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Functions</h2></td></tr>
<tr class="memitem:adf3e906875ab257f7fab3bc1370aa2db"><td class="memItemLeft" align="right" valign="top"><a class="el" href="../../db/d82/classlarge__number.html">large_number</a>&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../de/de4/fibonacci__large_8cpp.html#adf3e906875ab257f7fab3bc1370aa2db">fib</a> (uint64_t n)</td></tr>
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int&#160;</td><td class="memItemRight" valign="bottom"><b>main</b> (int argc, char *argv[])</td></tr>
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<a name="details" id="details"></a><h2 class="groupheader">Detailed Description</h2>
<div class="textblock"><p>Computes N^th Fibonacci number given as input argument. Uses custom build arbitrary integers library to perform additions and other operations. </p>
<p>Took 0.608246 seconds to compute 50,000^th Fibonacci number that contains 10450 digits!</p>
<dl class="section author"><dt>Author</dt><dd><a href="https://github.com/kvedala">Krishna Vedala</a> </dd></dl>
<dl class="section see"><dt>See also</dt><dd><a class="el" href="../../d9/d89/fibonacci_8cpp.html" title="Generate fibonacci sequence.">fibonacci.cpp</a>, <a class="el" href="../../d4/d32/fibonacci__fast_8cpp.html" title="Faster computation of Fibonacci series.">fibonacci_fast.cpp</a>, <a class="el" href="../../de/d47/string__fibonacci_8cpp.html" title="This Programme returns the Nth fibonacci as a string.">string_fibonacci.cpp</a> </dd></dl>
</div><h2 class="groupheader">Function Documentation</h2>
<a id="adf3e906875ab257f7fab3bc1370aa2db"></a>
<h2 class="memtitle"><span class="permalink"><a href="#adf3e906875ab257f7fab3bc1370aa2db">&#9670;&nbsp;</a></span>fib()</h2>
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<td class="memname"><a class="el" href="../../db/d82/classlarge__number.html">large_number</a> fib </td>
<td>(</td>
<td class="paramtype">uint64_t&#160;</td>
<td class="paramname"><em>n</em></td><td>)</td>
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<p>Compute fibonacci numbers using the relation </p><p class="formulaDsp">
\[f(n)=f(n-1)+f(n-2)\]
</p>
<p> and returns the result as a <a class="el" href="../../db/d82/classlarge__number.html">large_number</a> type. </p>
<div class="fragment"><div class="line"><a name="l00024"></a><span class="lineno"> 24</span>&#160; {</div>
<div class="line"><a name="l00025"></a><span class="lineno"> 25</span>&#160; <a class="code" href="../../db/d82/classlarge__number.html">large_number</a> f0(1);</div>
<div class="line"><a name="l00026"></a><span class="lineno"> 26</span>&#160; <a class="code" href="../../db/d82/classlarge__number.html">large_number</a> f1(1);</div>
<div class="line"><a name="l00027"></a><span class="lineno"> 27</span>&#160; </div>
<div class="line"><a name="l00028"></a><span class="lineno"> 28</span>&#160; <span class="keywordflow">do</span> {</div>
<div class="line"><a name="l00029"></a><span class="lineno"> 29</span>&#160; <a class="code" href="../../db/d82/classlarge__number.html">large_number</a> f2 = f1;</div>
<div class="line"><a name="l00030"></a><span class="lineno"> 30</span>&#160; f1 += f0;</div>
<div class="line"><a name="l00031"></a><span class="lineno"> 31</span>&#160; f0 = f2;</div>
<div class="line"><a name="l00032"></a><span class="lineno"> 32</span>&#160; n--;</div>
<div class="line"><a name="l00033"></a><span class="lineno"> 33</span>&#160; } <span class="keywordflow">while</span> (n &gt; 2); <span class="comment">// since we start from 2</span></div>
<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> f1;</div>
<div class="line"><a name="l00036"></a><span class="lineno"> 36</span>&#160;}</div>
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<div class="ttc" id="aclasslarge__number_html"><div class="ttname"><a href="../../db/d82/classlarge__number.html">large_number</a></div><div class="ttdef"><b>Definition:</b> large_number.h:24</div></div>
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