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<p><a href="https://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process">Gram Schmidt Orthogonalisation Process</a>
<a href="#details">More...</a></p>
<div class="textblock"><code>#include &lt;iostream&gt;</code><br />
<code>#include &lt;cassert&gt;</code><br />
<code>#include &lt;cmath&gt;</code><br />
<code>#include &lt;array&gt;</code><br />
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Include dependency graph for gram_schmidt.cpp:</div>
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<tr class="memitem:aed4d308f46ca1d91e348ca260ec1d2fb"><td class="memItemLeft" align="right" valign="top">double&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../d5/d33/gram__schmidt_8cpp.html#aed4d308f46ca1d91e348ca260ec1d2fb">linear_algebra::gram_schmidt::dot_product</a> (const <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/array.html">std::array</a>&lt; double, 10 &gt; &amp;x, const <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/array.html">std::array</a>&lt; double, 10 &gt; &amp;y, const int &amp;c)</td></tr>
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<tr class="memitem:a46e459aff5eafffb5ad9ceb129b4d274"><td class="memItemLeft" align="right" valign="top">double&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../d5/d33/gram__schmidt_8cpp.html#a46e459aff5eafffb5ad9ceb129b4d274">linear_algebra::gram_schmidt::projection</a> (const <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/array.html">std::array</a>&lt; double, 10 &gt; &amp;x, const <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/array.html">std::array</a>&lt; double, 10 &gt; &amp;y, const int &amp;c)</td></tr>
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<tr class="memitem:ab6b70f9680752e9fe2ac4e2cce158c40"><td class="memItemLeft" align="right" valign="top">void&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../d5/d33/gram__schmidt_8cpp.html#ab6b70f9680752e9fe2ac4e2cce158c40">linear_algebra::gram_schmidt::display</a> (const int &amp;r, const int &amp;c, const <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/array.html">std::array</a>&lt; <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/array.html">std::array</a>&lt; double, 10 &gt;, 20 &gt; &amp;B)</td></tr>
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<tr class="memitem:aa31ca28f60c880802462335eedc5d91f"><td class="memItemLeft" align="right" valign="top">void&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../d5/d33/gram__schmidt_8cpp.html#aa31ca28f60c880802462335eedc5d91f">linear_algebra::gram_schmidt::gram_schmidt</a> (int r, const int &amp;c, const <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/array.html">std::array</a>&lt; <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/array.html">std::array</a>&lt; double, 10 &gt;, 20 &gt; &amp;A, <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/array.html">std::array</a>&lt; <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/array.html">std::array</a>&lt; double, 10 &gt;, 20 &gt; B)</td></tr>
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<tr class="memitem:aa8dca7b867074164d5f45b0f3851269d"><td class="memItemLeft" align="right" valign="top">static void&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../d5/d33/gram__schmidt_8cpp.html#aa8dca7b867074164d5f45b0f3851269d">test</a> ()</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="../../d5/d33/gram__schmidt_8cpp.html#ae66f6b31b5ad750f1fe042a706a4e3d4">main</a> ()</td></tr>
<tr class="memdesc:ae66f6b31b5ad750f1fe042a706a4e3d4"><td class="mdescLeft">&#160;</td><td class="mdescRight">Main Function. <a href="../../d5/d33/gram__schmidt_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><a href="https://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process">Gram Schmidt Orthogonalisation Process</a> </p>
<p><br />
</p>
<p>Takes the input of Linearly Independent Vectors, returns vectors orthogonal to each other.</p>
<h3><a class="anchor" id="autotoc_md61"></a>
Algorithm</h3>
<p>Take the first vector of given LI vectors as first vector of Orthogonal vectors. Take projection of second input vector on the first vector of Orthogonal vector and subtract it from the 2nd LI vector. Take projection of third vector on the second vector of Othogonal vectors and subtract it from the 3rd LI vector. Keep repeating the above process until all the vectors in the given input array are exhausted.</p>
<p>For Example: In R2, Input LI Vectors={(3,1),(2,2)} then Orthogonal Vectors= {(3, 1),(-0.4, 1.2)}</p>
<p>Have defined maximum dimension of vectors to be 10 and number of vectors taken is 20. Please do not give linearly dependent vectors</p>
<dl class="section author"><dt>Author</dt><dd><a href="https://github.com/Akanksha-Gupta920">Akanksha Gupta</a> </dd></dl>
</div><h2 class="groupheader">Function Documentation</h2>
<a id="ab6b70f9680752e9fe2ac4e2cce158c40"></a>
<h2 class="memtitle"><span class="permalink"><a href="#ab6b70f9680752e9fe2ac4e2cce158c40">&#9670;&nbsp;</a></span>display()</h2>
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<td class="memname">void linear_algebra::gram_schmidt::display </td>
<td>(</td>
<td class="paramtype">const int &amp;&#160;</td>
<td class="paramname"><em>r</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">const int &amp;&#160;</td>
<td class="paramname"><em>c</em>, </td>
</tr>
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<td class="paramkey"></td>
<td></td>
<td class="paramtype">const <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/array.html">std::array</a>&lt; <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/array.html">std::array</a>&lt; double, 10 &gt;, 20 &gt; &amp;&#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 print the orthogonalised vector</p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">r</td><td>number of vectors </td></tr>
<tr><td class="paramname">c</td><td>dimenaion of vectors </td></tr>
<tr><td class="paramname">B</td><td>stores orthogonalised vectors</td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>void </dd></dl>
<div class="fragment"><div class="line"><a name="l00088"></a><span class="lineno"> 88</span>&#160; {</div>
<div class="line"><a name="l00089"></a><span class="lineno"> 89</span>&#160; <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i &lt; r; ++i) {</div>
<div class="line"><a name="l00090"></a><span class="lineno"> 90</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;Vector &quot;</span> &lt;&lt; i + 1 &lt;&lt; <span class="stringliteral">&quot;: &quot;</span>;</div>
<div class="line"><a name="l00091"></a><span class="lineno"> 91</span>&#160; <span class="keywordflow">for</span> (<span class="keywordtype">int</span> j = 0; j &lt; c; ++j) {</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; B[i][j] &lt;&lt; <span class="stringliteral">&quot; &quot;</span>;</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; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; <span class="charliteral">&#39;\n&#39;</span>;</div>
<div class="line"><a name="l00095"></a><span class="lineno"> 95</span>&#160; }</div>
<div class="line"><a name="l00096"></a><span class="lineno"> 96</span>&#160;}</div>
</div><!-- fragment -->
</div>
</div>
<a id="aed4d308f46ca1d91e348ca260ec1d2fb"></a>
<h2 class="memtitle"><span class="permalink"><a href="#aed4d308f46ca1d91e348ca260ec1d2fb">&#9670;&nbsp;</a></span>dot_product()</h2>
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<td class="memname">double linear_algebra::gram_schmidt::dot_product </td>
<td>(</td>
<td class="paramtype">const <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/array.html">std::array</a>&lt; double, 10 &gt; &amp;&#160;</td>
<td class="paramname"><em>x</em>, </td>
</tr>
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<td class="paramkey"></td>
<td></td>
<td class="paramtype">const <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/array.html">std::array</a>&lt; double, 10 &gt; &amp;&#160;</td>
<td class="paramname"><em>y</em>, </td>
</tr>
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<td class="paramtype">const int &amp;&#160;</td>
<td class="paramname"><em>c</em>&#160;</td>
</tr>
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<td></td>
<td>)</td>
<td></td><td></td>
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</div><div class="memdoc">
<p>Dot product function. Takes 2 vectors along with their dimension as input and returns the dot product. </p><dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">x</td><td>vector 1 </td></tr>
<tr><td class="paramname">y</td><td>vector 2 </td></tr>
<tr><td class="paramname">c</td><td>dimension of the vectors</td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>sum </dd></dl>
<div class="fragment"><div class="line"><a name="l00053"></a><span class="lineno"> 53</span>&#160; {</div>
<div class="line"><a name="l00054"></a><span class="lineno"> 54</span>&#160; <span class="keywordtype">double</span> <a class="code" href="../../d8/d77/namespacemachine__learning.html#a5305cc18d11943d3dc0d66e70d7e9c44">sum</a> = 0;</div>
<div class="line"><a name="l00055"></a><span class="lineno"> 55</span>&#160; <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i &lt; c; ++i) {</div>
<div class="line"><a name="l00056"></a><span class="lineno"> 56</span>&#160; <a class="code" href="../../d8/d77/namespacemachine__learning.html#a5305cc18d11943d3dc0d66e70d7e9c44">sum</a> += x[i] * y[i];</div>
<div class="line"><a name="l00057"></a><span class="lineno"> 57</span>&#160; }</div>
<div class="line"><a name="l00058"></a><span class="lineno"> 58</span>&#160; <span class="keywordflow">return</span> <a class="code" href="../../d8/d77/namespacemachine__learning.html#a5305cc18d11943d3dc0d66e70d7e9c44">sum</a>;</div>
<div class="line"><a name="l00059"></a><span class="lineno"> 59</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="#aa31ca28f60c880802462335eedc5d91f">&#9670;&nbsp;</a></span>gram_schmidt()</h2>
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<td class="memname">void linear_algebra::gram_schmidt::gram_schmidt </td>
<td>(</td>
<td class="paramtype">int&#160;</td>
<td class="paramname"><em>r</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">const int &amp;&#160;</td>
<td class="paramname"><em>c</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/container/array.html">std::array</a>&lt; <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/array.html">std::array</a>&lt; double, 10 &gt;, 20 &gt; &amp;&#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/container/array.html">std::array</a>&lt; <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/array.html">std::array</a>&lt; double, 10 &gt;, 20 &gt;&#160;</td>
<td class="paramname"><em>B</em>&#160;</td>
</tr>
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<p>Function for the process of Gram Schimdt Process </p><dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">r</td><td>number of vectors </td></tr>
<tr><td class="paramname">c</td><td>dimension of vectors </td></tr>
<tr><td class="paramname">A</td><td>stores input of given LI vectors </td></tr>
<tr><td class="paramname">B</td><td>stores orthogonalised vectors</td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>void </dd></dl>
<p>we check whether appropriate dimensions are given or not.</p>
<p>First vector is copied as it is.</p>
<p>array to store projections</p>
<p>First initialised to zero</p>
<p>to store previous projected array</p>
<p>to store the factor by which the previous array will change</p>
<p>projected array created</p>
<p>we take the projection with all the previous vector and add them.</p>
<p>subtract total projection vector from the input vector</p>
<div class="fragment"><div class="line"><a name="l00108"></a><span class="lineno"> 108</span>&#160; {</div>
<div class="line"><a name="l00109"></a><span class="lineno"> 109</span>&#160; <span class="keywordflow">if</span> (c &lt; r) { <span class="comment">/// we check whether appropriate dimensions are given or not.</span></div>
<div class="line"><a name="l00110"></a><span class="lineno"> 110</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></div>
<div class="line"><a name="l00111"></a><span class="lineno"> 111</span>&#160; &lt;&lt; <span class="stringliteral">&quot;Dimension of vector is less than number of vector, hence \n first &quot;</span></div>
<div class="line"><a name="l00112"></a><span class="lineno"> 112</span>&#160; &lt;&lt; c &lt;&lt; <span class="stringliteral">&quot; vectors are orthogonalised\n&quot;</span>;</div>
<div class="line"><a name="l00113"></a><span class="lineno"> 113</span>&#160; r = c;</div>
<div class="line"><a name="l00114"></a><span class="lineno"> 114</span>&#160; }</div>
<div class="line"><a name="l00115"></a><span class="lineno"> 115</span>&#160; </div>
<div class="line"><a name="l00116"></a><span class="lineno"> 116</span>&#160; <span class="keywordtype">int</span> k = 1;</div>
<div class="line"><a name="l00117"></a><span class="lineno"> 117</span>&#160; </div>
<div class="line"><a name="l00118"></a><span class="lineno"> 118</span>&#160; <span class="keywordflow">while</span> (k &lt;= r) {</div>
<div class="line"><a name="l00119"></a><span class="lineno"> 119</span>&#160; <span class="keywordflow">if</span> (k == 1) {</div>
<div class="line"><a name="l00120"></a><span class="lineno"> 120</span>&#160; <span class="keywordflow">for</span> (<span class="keywordtype">int</span> j = 0; j &lt; c; j++) B[0][j] = A[0][j]; <span class="comment">///First vector is copied as it is.</span></div>
<div class="line"><a name="l00121"></a><span class="lineno"> 121</span>&#160;<span class="comment"></span> }</div>
<div class="line"><a name="l00122"></a><span class="lineno"> 122</span>&#160; </div>
<div class="line"><a name="l00123"></a><span class="lineno"> 123</span>&#160; <span class="keywordflow">else</span> {</div>
<div class="line"><a name="l00124"></a><span class="lineno"> 124</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/array.html">std::array&lt;double, 10&gt;</a> all_projection{}; <span class="comment">///array to store projections</span></div>
<div class="line"><a name="l00125"></a><span class="lineno"> 125</span>&#160;<span class="comment"></span> <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i &lt; c; ++i) {</div>
<div class="line"><a name="l00126"></a><span class="lineno"> 126</span>&#160; all_projection[i] = 0; <span class="comment">///First initialised to zero</span></div>
<div class="line"><a name="l00127"></a><span class="lineno"> 127</span>&#160;<span class="comment"></span> }</div>
<div class="line"><a name="l00128"></a><span class="lineno"> 128</span>&#160; </div>
<div class="line"><a name="l00129"></a><span class="lineno"> 129</span>&#160; <span class="keywordtype">int</span> l = 1;</div>
<div class="line"><a name="l00130"></a><span class="lineno"> 130</span>&#160; <span class="keywordflow">while</span> (l &lt; k) {</div>
<div class="line"><a name="l00131"></a><span class="lineno"> 131</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/array.html">std::array&lt;double, 10&gt;</a> temp{}; <span class="comment">///to store previous projected array</span></div>
<div class="line"><a name="l00132"></a><span class="lineno"> 132</span>&#160;<span class="comment"></span> <span class="keywordtype">double</span> factor; <span class="comment">///to store the factor by which the previous array will change </span></div>
<div class="line"><a name="l00133"></a><span class="lineno"> 133</span>&#160;<span class="comment"></span> factor = <a class="code" href="../../d5/d33/gram__schmidt_8cpp.html#a46e459aff5eafffb5ad9ceb129b4d274">projection</a>(A[k - 1], B[l - 1], c);</div>
<div class="line"><a name="l00134"></a><span class="lineno"> 134</span>&#160; <span class="keywordflow">for</span>(<span class="keywordtype">int</span> i = 0; i &lt; c; ++i)</div>
<div class="line"><a name="l00135"></a><span class="lineno"> 135</span>&#160; temp[i] = B[l - 1][i] * factor; <span class="comment">///projected array created</span></div>
<div class="line"><a name="l00136"></a><span class="lineno"> 136</span>&#160;<span class="comment"></span> <span class="keywordflow">for</span> (<span class="keywordtype">int</span> j = 0; j &lt; c; ++j) {</div>
<div class="line"><a name="l00137"></a><span class="lineno"> 137</span>&#160; all_projection[j] = all_projection[j] + temp[j]; <span class="comment">///we take the projection with all the previous vector and add them.</span></div>
<div class="line"><a name="l00138"></a><span class="lineno"> 138</span>&#160;<span class="comment"></span> }</div>
<div class="line"><a name="l00139"></a><span class="lineno"> 139</span>&#160; l++;</div>
<div class="line"><a name="l00140"></a><span class="lineno"> 140</span>&#160; }</div>
<div class="line"><a name="l00141"></a><span class="lineno"> 141</span>&#160; <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i &lt; c; ++i) {</div>
<div class="line"><a name="l00142"></a><span class="lineno"> 142</span>&#160; B[k - 1][i] = A[k - 1][i] - all_projection[i]; <span class="comment">///subtract total projection vector from the input vector</span></div>
<div class="line"><a name="l00143"></a><span class="lineno"> 143</span>&#160;<span class="comment"></span> }</div>
<div class="line"><a name="l00144"></a><span class="lineno"> 144</span>&#160; }</div>
<div class="line"><a name="l00145"></a><span class="lineno"> 145</span>&#160; k++;</div>
<div class="line"><a name="l00146"></a><span class="lineno"> 146</span>&#160; }</div>
<div class="line"><a name="l00147"></a><span class="lineno"> 147</span>&#160; display(r, c, B); <span class="comment">//for displaying orthogoanlised vectors</span></div>
<div class="line"><a name="l00148"></a><span class="lineno"> 148</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>
<dl class="section return"><dt>Returns</dt><dd>0 on exit </dd></dl>
<p>a 2-D array for storing all vectors</p>
<p>a 2-D array for storing orthogonalised vectors</p>
<p>storing vectors in array A</p>
<p>Input of vectors is taken</p>
<p>To check whether vectors are orthogonal or not</p>
<p>take make the process numerically stable, upper bound for the dot product take 0.1</p>
<div class="fragment"><div class="line"><a name="l00213"></a><span class="lineno"> 213</span>&#160; {</div>
<div class="line"><a name="l00214"></a><span class="lineno"> 214</span>&#160; <span class="keywordtype">int</span> r=0, c=0;</div>
<div class="line"><a name="l00215"></a><span class="lineno"> 215</span>&#160; <a class="code" href="../../d5/d33/gram__schmidt_8cpp.html#aa8dca7b867074164d5f45b0f3851269d">test</a>(); <span class="comment">// perform self tests</span></div>
<div class="line"><a name="l00216"></a><span class="lineno"> 216</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;Enter the dimension of your vectors\n&quot;</span>;</div>
<div class="line"><a name="l00217"></a><span class="lineno"> 217</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_istream.html">std::cin</a> &gt;&gt; c;</div>
<div class="line"><a name="l00218"></a><span class="lineno"> 218</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;Enter the number of vectors you will enter\n&quot;</span>;</div>
<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_istream.html">std::cin</a> &gt;&gt; r;</div>
<div class="line"><a name="l00220"></a><span class="lineno"> 220</span>&#160; </div>
<div class="line"><a name="l00221"></a><span class="lineno"> 221</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/array.html">std::array&lt;std::array&lt;double, 10&gt;</a>, 20&gt;</div>
<div class="line"><a name="l00222"></a><span class="lineno"> 222</span>&#160; A{}; <span class="comment">///a 2-D array for storing all vectors</span></div>
<div class="line"><a name="l00223"></a><span class="lineno"> 223</span>&#160;<span class="comment"></span> <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/array.html">std::array&lt;std::array&lt;double, 10&gt;</a>, 20&gt; B = {</div>
<div class="line"><a name="l00224"></a><span class="lineno"> 224</span>&#160; {0}}; <span class="comment">/// a 2-D array for storing orthogonalised vectors</span></div>
<div class="line"><a name="l00225"></a><span class="lineno"> 225</span>&#160;<span class="comment"></span><span class="comment"> /// storing vectors in array A</span></div>
<div class="line"><a name="l00226"></a><span class="lineno"> 226</span>&#160;<span class="comment"></span> <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i &lt; r; ++i) {</div>
<div class="line"><a name="l00227"></a><span class="lineno"> 227</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;Enter vector &quot;</span> &lt;&lt; i + 1 &lt;&lt;<span class="charliteral">&#39;\n&#39;</span>; <span class="comment">///Input of vectors is taken</span></div>
<div class="line"><a name="l00228"></a><span class="lineno"> 228</span>&#160;<span class="comment"></span> <span class="keywordflow">for</span> (<span class="keywordtype">int</span> j = 0; j &lt; c; ++j) {</div>
<div class="line"><a name="l00229"></a><span class="lineno"> 229</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;Value &quot;</span> &lt;&lt; j + 1 &lt;&lt; <span class="stringliteral">&quot;th of vector: &quot;</span>;</div>
<div class="line"><a name="l00230"></a><span class="lineno"> 230</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_istream.html">std::cin</a> &gt;&gt; A[i][j];</div>
<div class="line"><a name="l00231"></a><span class="lineno"> 231</span>&#160; }</div>
<div class="line"><a name="l00232"></a><span class="lineno"> 232</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="charliteral">&#39;\n&#39;</span>;</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; </div>
<div class="line"><a name="l00235"></a><span class="lineno"> 235</span>&#160; <a class="code" href="../../d5/d33/gram__schmidt_8cpp.html#aa31ca28f60c880802462335eedc5d91f">linear_algebra::gram_schmidt::gram_schmidt</a>(r, c, A, B);</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; <span class="keywordtype">double</span> dot = 0;</div>
<div class="line"><a name="l00238"></a><span class="lineno"> 238</span>&#160; <span class="keywordtype">int</span> flag = 1; <span class="comment">///To check whether vectors are orthogonal or not</span></div>
<div class="line"><a name="l00239"></a><span class="lineno"> 239</span>&#160;<span class="comment"></span> <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i &lt; r - 1; ++i) {</div>
<div class="line"><a name="l00240"></a><span class="lineno"> 240</span>&#160; <span class="keywordflow">for</span> (<span class="keywordtype">int</span> j = i + 1; j &lt; r; ++j) {</div>
<div class="line"><a name="l00241"></a><span class="lineno"> 241</span>&#160; dot = <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/math/fabs.html">fabs</a>(linear_algebra::gram_schmidt::dot_product(B[i], B[j], c));</div>
<div class="line"><a name="l00242"></a><span class="lineno"> 242</span>&#160; <span class="keywordflow">if</span> (dot &gt; 0.1) <span class="comment">/// take make the process numerically stable, upper bound for the dot product take 0.1</span></div>
<div class="line"><a name="l00243"></a><span class="lineno"> 243</span>&#160;<span class="comment"></span> {</div>
<div class="line"><a name="l00244"></a><span class="lineno"> 244</span>&#160; flag = 0;</div>
<div class="line"><a name="l00245"></a><span class="lineno"> 245</span>&#160; <span class="keywordflow">break</span>;</div>
<div class="line"><a name="l00246"></a><span class="lineno"> 246</span>&#160; }</div>
<div class="line"><a name="l00247"></a><span class="lineno"> 247</span>&#160; }</div>
<div class="line"><a name="l00248"></a><span class="lineno"> 248</span>&#160; }</div>
<div class="line"><a name="l00249"></a><span class="lineno"> 249</span>&#160; <span class="keywordflow">if</span> (flag == 0) <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;Vectors are linearly dependent\n&quot;</span>; </div>
<div class="line"><a name="l00250"></a><span class="lineno"> 250</span>&#160; <span class="keywordflow">return</span> 0;</div>
<div class="line"><a name="l00251"></a><span class="lineno"> 251</span>&#160;}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#a46e459aff5eafffb5ad9ceb129b4d274">&#9670;&nbsp;</a></span>projection()</h2>
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<td class="memname">double linear_algebra::gram_schmidt::projection </td>
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<td class="paramtype">const <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/array.html">std::array</a>&lt; double, 10 &gt; &amp;&#160;</td>
<td class="paramname"><em>x</em>, </td>
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<td class="paramtype">const <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/array.html">std::array</a>&lt; double, 10 &gt; &amp;&#160;</td>
<td class="paramname"><em>y</em>, </td>
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<td class="paramtype">const int &amp;&#160;</td>
<td class="paramname"><em>c</em>&#160;</td>
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<p>Projection Function Takes input of 2 vectors along with their dimension and evaluates their projection in temp</p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">x</td><td>Vector 1 </td></tr>
<tr><td class="paramname">y</td><td>Vector 2 </td></tr>
<tr><td class="paramname">c</td><td>dimension of each vector</td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>factor </dd></dl>
<p>The dot product of two vectors is taken</p>
<p>The norm of the second vector is taken.</p>
<p>multiply that factor with every element in a 3rd vector, whose initial values are same as the 2nd vector.</p>
<div class="fragment"><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; <span class="keywordtype">double</span> dot = <a class="code" href="../../d5/d33/gram__schmidt_8cpp.html#aed4d308f46ca1d91e348ca260ec1d2fb">dot_product</a>(x, y, c); <span class="comment">///The dot product of two vectors is taken</span></div>
<div class="line"><a name="l00074"></a><span class="lineno"> 74</span>&#160;<span class="comment"></span> <span class="keywordtype">double</span> anorm = <a class="code" href="../../d5/d33/gram__schmidt_8cpp.html#aed4d308f46ca1d91e348ca260ec1d2fb">dot_product</a>(y, y, c); <span class="comment">///The norm of the second vector is taken.</span></div>
<div class="line"><a name="l00075"></a><span class="lineno"> 75</span>&#160;<span class="comment"></span> <span class="keywordtype">double</span> factor = dot / anorm; <span class="comment">///multiply that factor with every element in a 3rd vector, whose initial values are same as the 2nd vector.</span></div>
<div class="line"><a name="l00076"></a><span class="lineno"> 76</span>&#160;<span class="comment"></span> <span class="keywordflow">return</span> factor;</div>
<div class="line"><a name="l00077"></a><span class="lineno"> 77</span>&#160;}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#aa8dca7b867074164d5f45b0f3851269d">&#9670;&nbsp;</a></span>test()</h2>
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<td class="memname">static void test </td>
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<p>Test Function. Process has been tested for 3 Sample Inputs </p><dl class="section return"><dt>Returns</dt><dd>void </dd></dl>
<div class="fragment"><div class="line"><a name="l00155"></a><span class="lineno"> 155</span>&#160; {</div>
<div class="line"><a name="l00156"></a><span class="lineno"> 156</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/array.html">std::array&lt;std::array&lt;double, 10&gt;</a>, 20&gt; a1 = {</div>
<div class="line"><a name="l00157"></a><span class="lineno"> 157</span>&#160; {{1, 0, 1, 0}, {1, 1, 1, 1}, {0, 1, 2, 1}}};</div>
<div class="line"><a name="l00158"></a><span class="lineno"> 158</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/array.html">std::array&lt;std::array&lt;double, 10&gt;</a>, 20&gt; b1 = {{0}};</div>
<div class="line"><a name="l00159"></a><span class="lineno"> 159</span>&#160; <span class="keywordtype">double</span> dot1 = 0;</div>
<div class="line"><a name="l00160"></a><span class="lineno"> 160</span>&#160; <a class="code" href="../../d5/d33/gram__schmidt_8cpp.html#aa31ca28f60c880802462335eedc5d91f">linear_algebra::gram_schmidt::gram_schmidt</a>(3, 4, a1, b1);</div>
<div class="line"><a name="l00161"></a><span class="lineno"> 161</span>&#160; <span class="keywordtype">int</span> flag = 1;</div>
<div class="line"><a name="l00162"></a><span class="lineno"> 162</span>&#160; <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i &lt; 2; ++i)</div>
<div class="line"><a name="l00163"></a><span class="lineno"> 163</span>&#160; <span class="keywordflow">for</span> (<span class="keywordtype">int</span> j = i + 1; j &lt; 3; ++j) {</div>
<div class="line"><a name="l00164"></a><span class="lineno"> 164</span>&#160; dot1 = <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/math/fabs.html">fabs</a>(linear_algebra::gram_schmidt::dot_product(b1[i], b1[j], 4));</div>
<div class="line"><a name="l00165"></a><span class="lineno"> 165</span>&#160; <span class="keywordflow">if</span> (dot1 &gt; 0.1) {</div>
<div class="line"><a name="l00166"></a><span class="lineno"> 166</span>&#160; flag = 0;</div>
<div class="line"><a name="l00167"></a><span class="lineno"> 167</span>&#160; <span class="keywordflow">break</span>;</div>
<div class="line"><a name="l00168"></a><span class="lineno"> 168</span>&#160; }</div>
<div class="line"><a name="l00169"></a><span class="lineno"> 169</span>&#160; }</div>
<div class="line"><a name="l00170"></a><span class="lineno"> 170</span>&#160; <span class="keywordflow">if</span> (flag == 0) <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;Vectors are linearly dependent\n&quot;</span>;</div>
<div class="line"><a name="l00171"></a><span class="lineno"> 171</span>&#160; assert(flag == 1);</div>
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<div class="ttc" id="agram__schmidt_8cpp_html_a46e459aff5eafffb5ad9ceb129b4d274"><div class="ttname"><a href="../../d5/d33/gram__schmidt_8cpp.html#a46e459aff5eafffb5ad9ceb129b4d274">linear_algebra::gram_schmidt::projection</a></div><div class="ttdeci">double projection(const std::array&lt; double, 10 &gt; &amp;x, const std::array&lt; double, 10 &gt; &amp;y, const int &amp;c)</div><div class="ttdef"><b>Definition:</b> gram_schmidt.cpp:71</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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<li><a href="https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/probability/binomial_dist.cpp">Binomial Dist</a></li>
<li><a href="https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/probability/poisson_dist.cpp">Poisson Dist</a></li>
</ul>
<h1><a class="anchor" id="autotoc_md44"></a>
<h1><a class="anchor" id="autotoc_md45"></a>
Range Queries</h1>
<ul>
<li><a href="https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/range_queries/fenwick_tree.cpp">Fenwick Tree</a></li>
@@ -353,7 +358,7 @@ Range Queries</h1>
<li><a href="https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/range_queries/segtree.cpp">Segtree</a></li>
<li><a href="https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/range_queries/sparse_table.cpp">Sparse Table</a></li>
</ul>
<h1><a class="anchor" id="autotoc_md45"></a>
<h1><a class="anchor" id="autotoc_md46"></a>
Search</h1>
<ul>
<li><a href="https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/search/binary_search.cpp">Binary Search</a></li>
@@ -368,7 +373,7 @@ Search</h1>
<li><a href="https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/search/ternary_search.cpp">Ternary Search</a></li>
<li><a href="https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/search/text_search.cpp">Text Search</a></li>
</ul>
<h1><a class="anchor" id="autotoc_md46"></a>
<h1><a class="anchor" id="autotoc_md47"></a>
Sorting</h1>
<ul>
<li><a href="https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/sorting/bead_sort.cpp">Bead Sort</a></li>
@@ -404,7 +409,7 @@ Sorting</h1>
<li><a href="https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/sorting/tim_sort.cpp">Tim Sort</a></li>
<li><a href="https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/sorting/wiggle_sort.cpp">Wiggle Sort</a></li>
</ul>
<h1><a class="anchor" id="autotoc_md47"></a>
<h1><a class="anchor" id="autotoc_md48"></a>
Strings</h1>
<ul>
<li><a href="https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/strings/brute_force_string_searching.cpp">Brute Force String Searching</a></li>

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d5/ddb/bogo__sort_8cpp.html
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@@ -136,7 +136,7 @@ Functions</h2></td></tr>
<a name="details" id="details"></a><h2 class="groupheader">Detailed Description</h2>
<div class="textblock"><p>Implementation of <a href="https://en.wikipedia.org/wiki/Bogosort">Bogosort algorithm</a> </p>
<p>In computer science, bogosort (also known as permutation sort, stupid sort, slowsort, shotgun sort, random sort, monkey sort, bobosort or shuffle sort) is a highly inefficient sorting algorithm based on the generate and test paradigm. Two versions of this algorithm exist: a deterministic version that enumerates all permutations until it hits a sorted one, and a randomized version that randomly permutes its input.Randomized version is implemented here.</p>
<h3><a class="anchor" id="autotoc_md65"></a>
<h3><a class="anchor" id="autotoc_md67"></a>
Algorithm</h3>
<p>Shuffle the array untill array is sorted.</p>
<dl class="section author"><dt>Author</dt><dd><a href="https://github.com/imdeep2905">Deep Raval</a> </dd></dl>