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@@ -100,21 +100,22 @@ $(document).ready(function(){initNavTree('d5/d33/gram__schmidt_8cpp.html','../..
<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 />
<div class="textblock"><code>#include &lt;array&gt;</code><br />
<code>#include &lt;cassert&gt;</code><br />
<code>#include &lt;cmath&gt;</code><br />
<code>#include &lt;array&gt;</code><br />
<code>#include &lt;iostream&gt;</code><br />
<code>#include &quot;math.h&quot;</code><br />
</div><div class="textblock"><div class="dynheader">
Include dependency graph for gram_schmidt.cpp:</div>
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</div><table class="memberdecls">
<tr class="heading"><td colspan="2"><h2 class="groupheader"><a name="namespaces"></a>
Namespaces</h2></td></tr>
<tr class="memitem:d3/d78/namespacelinear__algebra"><td class="memItemLeft" align="right" valign="top"> &#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../d3/d78/namespacelinear__algebra.html">linear_algebra</a></td></tr>
<tr class="memdesc:d3/d78/namespacelinear__algebra"><td class="mdescLeft">&#160;</td><td class="mdescRight">Linear Algebra algorithms. <br /></td></tr>
<tr class="memdesc:d3/d78/namespacelinear__algebra"><td class="mdescLeft">&#160;</td><td class="mdescRight">for io operations <br /></td></tr>
<tr class="separator:"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:d4/d0f/namespacegram__schmidt"><td class="memItemLeft" align="right" valign="top"> &#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../d4/d0f/namespacegram__schmidt.html">gram_schmidt</a></td></tr>
<tr class="memdesc:d4/d0f/namespacegram__schmidt"><td class="mdescLeft">&#160;</td><td class="mdescRight">Functions for <a href="https://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process">Gram Schmidt Orthogonalisation Process</a> <br /></td></tr>
@@ -138,8 +139,6 @@ Functions</h2></td></tr>
</table>
<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_md62"></a>
Algorithm</h3>
@@ -189,15 +188,15 @@ Algorithm</h3>
</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 class="fragment"><div class="line"><a name="l00102"></a><span class="lineno"> 102</span>&#160; {</div>
<div class="line"><a name="l00103"></a><span class="lineno"> 103</span>&#160; <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i &lt; r; ++i) {</div>
<div class="line"><a name="l00104"></a><span class="lineno"> 104</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="l00105"></a><span class="lineno"> 105</span>&#160; <span class="keywordflow">for</span> (<span class="keywordtype">int</span> j = 0; j &lt; c; ++j) {</div>
<div class="line"><a name="l00106"></a><span class="lineno"> 106</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="l00107"></a><span class="lineno"> 107</span>&#160; }</div>
<div class="line"><a name="l00108"></a><span class="lineno"> 108</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="l00109"></a><span class="lineno"> 109</span>&#160; }</div>
<div class="line"><a name="l00110"></a><span class="lineno"> 110</span>&#160;}</div>
</div><!-- fragment -->
</div>
</div>
@@ -241,13 +240,13 @@ Algorithm</h3>
</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>
<div class="fragment"><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; <span class="keywordtype">double</span> <a class="code" href="../../d8/d77/namespacemachine__learning.html#a5305cc18d11943d3dc0d66e70d7e9c44">sum</a> = 0;</div>
<div class="line"><a name="l00062"></a><span class="lineno"> 62</span>&#160; <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i &lt; c; ++i) {</div>
<div class="line"><a name="l00063"></a><span class="lineno"> 63</span>&#160; <a class="code" href="../../d8/d77/namespacemachine__learning.html#a5305cc18d11943d3dc0d66e70d7e9c44">sum</a> += x[i] * y[i];</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; <span class="keywordflow">return</span> <a class="code" href="../../d8/d77/namespacemachine__learning.html#a5305cc18d11943d3dc0d66e70d7e9c44">sum</a>;</div>
<div class="line"><a name="l00066"></a><span class="lineno"> 66</span>&#160;}</div>
</div><!-- fragment --><div class="dynheader">
Here is the call graph for this function:</div>
<div class="dyncontent">
@@ -313,47 +312,58 @@ Here is the call graph for this function:</div>
<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>
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<div class="line"><a name="l00144"></a><span class="lineno"> 144</span>&#160; }</div>
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<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>
<div class="fragment"><div class="line"><a name="l00123"></a><span class="lineno"> 123</span>&#160; {</div>
<div class="line"><a name="l00124"></a><span class="lineno"> 124</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="l00125"></a><span class="lineno"> 125</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; <span class="stringliteral">&quot;Dimension of vector is less than number of vector, hence &quot;</span></div>
<div class="line"><a name="l00126"></a><span class="lineno"> 126</span>&#160; <span class="stringliteral">&quot;\n first &quot;</span></div>
<div class="line"><a name="l00127"></a><span class="lineno"> 127</span>&#160; &lt;&lt; c &lt;&lt; <span class="stringliteral">&quot; vectors are orthogonalised\n&quot;</span>;</div>
<div class="line"><a name="l00128"></a><span class="lineno"> 128</span>&#160; r = c;</div>
<div class="line"><a name="l00129"></a><span class="lineno"> 129</span>&#160; }</div>
<div class="line"><a name="l00130"></a><span class="lineno"> 130</span>&#160; </div>
<div class="line"><a name="l00131"></a><span class="lineno"> 131</span>&#160; <span class="keywordtype">int</span> k = 1;</div>
<div class="line"><a name="l00132"></a><span class="lineno"> 132</span>&#160; </div>
<div class="line"><a name="l00133"></a><span class="lineno"> 133</span>&#160; <span class="keywordflow">while</span> (k &lt;= r) {</div>
<div class="line"><a name="l00134"></a><span class="lineno"> 134</span>&#160; <span class="keywordflow">if</span> (k == 1) {</div>
<div class="line"><a name="l00135"></a><span class="lineno"> 135</span>&#160; <span class="keywordflow">for</span> (<span class="keywordtype">int</span> j = 0; j &lt; c; j++)</div>
<div class="line"><a name="l00136"></a><span class="lineno"> 136</span>&#160; B[0][j] = A[0][j]; <span class="comment">/// First vector is copied as it is.</span></div>
<div class="line"><a name="l00137"></a><span class="lineno"> 137</span>&#160;<span class="comment"></span> }</div>
<div class="line"><a name="l00138"></a><span class="lineno"> 138</span>&#160; </div>
<div class="line"><a name="l00139"></a><span class="lineno"> 139</span>&#160; <span class="keywordflow">else</span> {</div>
<div class="line"><a name="l00140"></a><span class="lineno"> 140</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/array.html">std::array&lt;double, 10&gt;</a></div>
<div class="line"><a name="l00141"></a><span class="lineno"> 141</span>&#160; all_projection{}; <span class="comment">/// array to store projections</span></div>
<div class="line"><a name="l00142"></a><span class="lineno"> 142</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="l00143"></a><span class="lineno"> 143</span>&#160; all_projection[i] = 0; <span class="comment">/// First initialised to zero</span></div>
<div class="line"><a name="l00144"></a><span class="lineno"> 144</span>&#160;<span class="comment"></span> }</div>
<div class="line"><a name="l00145"></a><span class="lineno"> 145</span>&#160; </div>
<div class="line"><a name="l00146"></a><span class="lineno"> 146</span>&#160; <span class="keywordtype">int</span> l = 1;</div>
<div class="line"><a name="l00147"></a><span class="lineno"> 147</span>&#160; <span class="keywordflow">while</span> (l &lt; k) {</div>
<div class="line"><a name="l00148"></a><span class="lineno"> 148</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/array.html">std::array&lt;double, 10&gt;</a></div>
<div class="line"><a name="l00149"></a><span class="lineno"> 149</span>&#160; temp{}; <span class="comment">/// to store previous projected array</span></div>
<div class="line"><a name="l00150"></a><span class="lineno"> 150</span>&#160;<span class="comment"></span> <span class="keywordtype">double</span> factor = NAN; <span class="comment">/// to store the factor by which the</span></div>
<div class="line"><a name="l00151"></a><span class="lineno"> 151</span>&#160;<span class="comment"></span><span class="comment"> /// previous array will change</span></div>
<div class="line"><a name="l00152"></a><span class="lineno"> 152</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="l00153"></a><span class="lineno"> 153</span>&#160; <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i &lt; c; ++i) {</div>
<div class="line"><a name="l00154"></a><span class="lineno"> 154</span>&#160; temp[i] = B[l - 1][i] * factor; <span class="comment">/// projected array created</span></div>
<div class="line"><a name="l00155"></a><span class="lineno"> 155</span>&#160;<span class="comment"></span> }</div>
<div class="line"><a name="l00156"></a><span class="lineno"> 156</span>&#160; <span class="keywordflow">for</span> (<span class="keywordtype">int</span> j = 0; j &lt; c; ++j) {</div>
<div class="line"><a name="l00157"></a><span class="lineno"> 157</span>&#160; all_projection[j] =</div>
<div class="line"><a name="l00158"></a><span class="lineno"> 158</span>&#160; all_projection[j] +</div>
<div class="line"><a name="l00159"></a><span class="lineno"> 159</span>&#160; temp[j]; <span class="comment">/// we take the projection with all the</span></div>
<div class="line"><a name="l00160"></a><span class="lineno"> 160</span>&#160;<span class="comment"></span><span class="comment"> /// previous vector and add them.</span></div>
<div class="line"><a name="l00161"></a><span class="lineno"> 161</span>&#160;<span class="comment"></span> }</div>
<div class="line"><a name="l00162"></a><span class="lineno"> 162</span>&#160; l++;</div>
<div class="line"><a name="l00163"></a><span class="lineno"> 163</span>&#160; }</div>
<div class="line"><a name="l00164"></a><span class="lineno"> 164</span>&#160; <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i &lt; c; ++i) {</div>
<div class="line"><a name="l00165"></a><span class="lineno"> 165</span>&#160; B[k - 1][i] =</div>
<div class="line"><a name="l00166"></a><span class="lineno"> 166</span>&#160; A[k - 1][i] -</div>
<div class="line"><a name="l00167"></a><span class="lineno"> 167</span>&#160; all_projection[i]; <span class="comment">/// subtract total projection vector</span></div>
<div class="line"><a name="l00168"></a><span class="lineno"> 168</span>&#160;<span class="comment"></span><span class="comment"> /// from the input vector</span></div>
<div class="line"><a name="l00169"></a><span class="lineno"> 169</span>&#160;<span class="comment"></span> }</div>
<div class="line"><a name="l00170"></a><span class="lineno"> 170</span>&#160; }</div>
<div class="line"><a name="l00171"></a><span class="lineno"> 171</span>&#160; k++;</div>
<div class="line"><a name="l00172"></a><span class="lineno"> 172</span>&#160; }</div>
<div class="line"><a name="l00173"></a><span class="lineno"> 173</span>&#160; display(r, c, B); <span class="comment">// for displaying orthogoanlised vectors</span></div>
<div class="line"><a name="l00174"></a><span class="lineno"> 174</span>&#160;}</div>
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<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>
<div class="fragment"><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="keywordtype">int</span> r = 0, c = 0;</div>
<div class="line"><a name="l00250"></a><span class="lineno"> 250</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="l00251"></a><span class="lineno"> 251</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="l00252"></a><span class="lineno"> 252</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="l00253"></a><span class="lineno"> 253</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="l00254"></a><span class="lineno"> 254</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="l00255"></a><span class="lineno"> 255</span>&#160; </div>
<div class="line"><a name="l00256"></a><span class="lineno"> 256</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="l00257"></a><span class="lineno"> 257</span>&#160; A{}; <span class="comment">/// a 2-D array for storing all vectors</span></div>
<div class="line"><a name="l00258"></a><span class="lineno"> 258</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="l00259"></a><span class="lineno"> 259</span>&#160; {0}}; <span class="comment">/// a 2-D array for storing orthogonalised vectors</span></div>
<div class="line"><a name="l00260"></a><span class="lineno"> 260</span>&#160;<span class="comment"></span><span class="comment"> /// storing vectors in array A</span></div>
<div class="line"><a name="l00261"></a><span class="lineno"> 261</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="l00262"></a><span class="lineno"> 262</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</div>
<div class="line"><a name="l00263"></a><span class="lineno"> 263</span>&#160; &lt;&lt; <span class="charliteral">&#39;\n&#39;</span>; <span class="comment">/// Input of vectors is taken</span></div>
<div class="line"><a name="l00264"></a><span class="lineno"> 264</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="l00265"></a><span class="lineno"> 265</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="l00266"></a><span class="lineno"> 266</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="l00267"></a><span class="lineno"> 267</span>&#160; }</div>
<div class="line"><a name="l00268"></a><span class="lineno"> 268</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="l00269"></a><span class="lineno"> 269</span>&#160; }</div>
<div class="line"><a name="l00270"></a><span class="lineno"> 270</span>&#160; </div>
<div class="line"><a name="l00271"></a><span class="lineno"> 271</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="l00272"></a><span class="lineno"> 272</span>&#160; </div>
<div class="line"><a name="l00273"></a><span class="lineno"> 273</span>&#160; <span class="keywordtype">double</span> dot = 0;</div>
<div class="line"><a name="l00274"></a><span class="lineno"> 274</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="l00275"></a><span class="lineno"> 275</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="l00276"></a><span class="lineno"> 276</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="l00277"></a><span class="lineno"> 277</span>&#160; dot =</div>
<div class="line"><a name="l00278"></a><span class="lineno"> 278</span>&#160; <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="l00279"></a><span class="lineno"> 279</span>&#160; <span class="keywordflow">if</span> (dot &gt; 0.1) <span class="comment">/// take make the process numerically stable, upper</span></div>
<div class="line"><a name="l00280"></a><span class="lineno"> 280</span>&#160;<span class="comment"></span><span class="comment"> /// bound for the dot product take 0.1</span></div>
<div class="line"><a name="l00281"></a><span class="lineno"> 281</span>&#160;<span class="comment"></span> {</div>
<div class="line"><a name="l00282"></a><span class="lineno"> 282</span>&#160; flag = 0;</div>
<div class="line"><a name="l00283"></a><span class="lineno"> 283</span>&#160; <span class="keywordflow">break</span>;</div>
<div class="line"><a name="l00284"></a><span class="lineno"> 284</span>&#160; }</div>
<div class="line"><a name="l00285"></a><span class="lineno"> 285</span>&#160; }</div>
<div class="line"><a name="l00286"></a><span class="lineno"> 286</span>&#160; }</div>
<div class="line"><a name="l00287"></a><span class="lineno"> 287</span>&#160; <span class="keywordflow">if</span> (flag == 0)</div>
<div class="line"><a name="l00288"></a><span class="lineno"> 288</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;Vectors are linearly dependent\n&quot;</span>;</div>
<div class="line"><a name="l00289"></a><span class="lineno"> 289</span>&#160; <span class="keywordflow">return</span> 0;</div>
<div class="line"><a name="l00290"></a><span class="lineno"> 290</span>&#160;}</div>
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<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>
<div class="fragment"><div class="line"><a name="l00080"></a><span class="lineno"> 80</span>&#160; {</div>
<div class="line"><a name="l00081"></a><span class="lineno"> 81</span>&#160; <span class="keywordtype">double</span> dot =</div>
<div class="line"><a name="l00082"></a><span class="lineno"> 82</span>&#160; <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="l00083"></a><span class="lineno"> 83</span>&#160;<span class="comment"></span> <span class="keywordtype">double</span> anorm =</div>
<div class="line"><a name="l00084"></a><span class="lineno"> 84</span>&#160; <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="l00085"></a><span class="lineno"> 85</span>&#160;<span class="comment"></span> <span class="keywordtype">double</span> factor =</div>
<div class="line"><a name="l00086"></a><span class="lineno"> 86</span>&#160; dot /</div>
<div class="line"><a name="l00087"></a><span class="lineno"> 87</span>&#160; anorm; <span class="comment">/// multiply that factor with every element in a 3rd vector,</span></div>
<div class="line"><a name="l00088"></a><span class="lineno"> 88</span>&#160;<span class="comment"></span><span class="comment"> /// whose initial values are same as the 2nd vector.</span></div>
<div class="line"><a name="l00089"></a><span class="lineno"> 89</span>&#160;<span class="comment"></span> <span class="keywordflow">return</span> factor;</div>
<div class="line"><a name="l00090"></a><span class="lineno"> 90</span>&#160;}</div>
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</table>
</div><div class="memdoc">
<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>
<div class="line"><a name="l00172"></a><span class="lineno"> 172</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;Passed Test Case 1\n &quot;</span>;</div>
<div class="line"><a name="l00173"></a><span class="lineno"> 173</span>&#160; </div>
<div class="line"><a name="l00174"></a><span class="lineno"> 174</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; a2 = {{{3, 1}, {2, 2}}};</div>
<div class="line"><a name="l00175"></a><span class="lineno"> 175</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; b2 = {{0}};</div>
<div class="line"><a name="l00176"></a><span class="lineno"> 176</span>&#160; <span class="keywordtype">double</span> dot2 = 0;</div>
<div class="line"><a name="l00177"></a><span class="lineno"> 177</span>&#160; <a class="code" href="../../d5/d33/gram__schmidt_8cpp.html#aa31ca28f60c880802462335eedc5d91f">linear_algebra::gram_schmidt::gram_schmidt</a>(2, 2, a2, b2);</div>
<div class="line"><a name="l00178"></a><span class="lineno"> 178</span>&#160; flag = 1;</div>
<div class="line"><a name="l00179"></a><span class="lineno"> 179</span>&#160; <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i &lt; 1; ++i)</div>
<div class="line"><a name="l00180"></a><span class="lineno"> 180</span>&#160; <span class="keywordflow">for</span> (<span class="keywordtype">int</span> j = i + 1; j &lt; 2; ++j) {</div>
<div class="line"><a name="l00181"></a><span class="lineno"> 181</span>&#160; dot2 = <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/math/fabs.html">fabs</a>(linear_algebra::gram_schmidt::dot_product(b2[i], b2[j], 2));</div>
<div class="line"><a name="l00182"></a><span class="lineno"> 182</span>&#160; <span class="keywordflow">if</span> (dot2 &gt; 0.1) {</div>
<div class="line"><a name="l00183"></a><span class="lineno"> 183</span>&#160; flag = 0;</div>
<div class="line"><a name="l00184"></a><span class="lineno"> 184</span>&#160; <span class="keywordflow">break</span>;</div>
<div class="line"><a name="l00185"></a><span class="lineno"> 185</span>&#160; }</div>
<div class="line"><a name="l00186"></a><span class="lineno"> 186</span>&#160; }</div>
<div class="line"><a name="l00187"></a><span class="lineno"> 187</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="l00188"></a><span class="lineno"> 188</span>&#160; assert(flag == 1);</div>
<div class="line"><a name="l00189"></a><span class="lineno"> 189</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;Passed Test Case 2\n&quot;</span>;</div>
<div class="line"><a name="l00190"></a><span class="lineno"> 190</span>&#160; </div>
<div class="line"><a name="l00191"></a><span class="lineno"> 191</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; a3 = {{{1, 2, 2}, {-4, 3, 2}}};</div>
<div class="line"><a name="l00192"></a><span class="lineno"> 192</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; b3 = {{0}};</div>
<div class="line"><a name="l00193"></a><span class="lineno"> 193</span>&#160; <span class="keywordtype">double</span> dot3 = 0;</div>
<div class="line"><a name="l00194"></a><span class="lineno"> 194</span>&#160; <a class="code" href="../../d5/d33/gram__schmidt_8cpp.html#aa31ca28f60c880802462335eedc5d91f">linear_algebra::gram_schmidt::gram_schmidt</a>(2, 3, a3, b3);</div>
<div class="line"><a name="l00195"></a><span class="lineno"> 195</span>&#160; flag = 1;</div>
<div class="line"><a name="l00196"></a><span class="lineno"> 196</span>&#160; <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i &lt; 1; ++i)</div>
<div class="line"><a name="l00197"></a><span class="lineno"> 197</span>&#160; <span class="keywordflow">for</span> (<span class="keywordtype">int</span> j = i + 1; j &lt; 2; ++j) {</div>
<div class="line"><a name="l00198"></a><span class="lineno"> 198</span>&#160; dot3 = <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/math/fabs.html">fabs</a>(linear_algebra::gram_schmidt::dot_product(b3[i], b3[j], 3));</div>
<div class="line"><a name="l00199"></a><span class="lineno"> 199</span>&#160; <span class="keywordflow">if</span> (dot3 &gt; 0.1) {</div>
<div class="line"><a name="l00200"></a><span class="lineno"> 200</span>&#160; flag = 0;</div>
<div class="line"><a name="l00201"></a><span class="lineno"> 201</span>&#160; <span class="keywordflow">break</span>;</div>
<div class="line"><a name="l00202"></a><span class="lineno"> 202</span>&#160; }</div>
<div class="line"><a name="l00203"></a><span class="lineno"> 203</span>&#160; }</div>
<div class="line"><a name="l00204"></a><span class="lineno"> 204</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="l00205"></a><span class="lineno"> 205</span>&#160; assert(flag == 1);</div>
<div class="line"><a name="l00206"></a><span class="lineno"> 206</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;Passed Test Case 3\n&quot;</span>;</div>
<div class="line"><a name="l00207"></a><span class="lineno"> 207</span>&#160;}</div>
<div class="fragment"><div class="line"><a name="l00181"></a><span class="lineno"> 181</span>&#160; {</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/container/array.html">std::array&lt;std::array&lt;double, 10&gt;</a>, 20&gt; a1 = {</div>
<div class="line"><a name="l00183"></a><span class="lineno"> 183</span>&#160; {{1, 0, 1, 0}, {1, 1, 1, 1}, {0, 1, 2, 1}}};</div>
<div class="line"><a name="l00184"></a><span class="lineno"> 184</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="l00185"></a><span class="lineno"> 185</span>&#160; <span class="keywordtype">double</span> dot1 = 0;</div>
<div class="line"><a name="l00186"></a><span class="lineno"> 186</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="l00187"></a><span class="lineno"> 187</span>&#160; <span class="keywordtype">int</span> flag = 1;</div>
<div class="line"><a name="l00188"></a><span class="lineno"> 188</span>&#160; <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i &lt; 2; ++i) {</div>
<div class="line"><a name="l00189"></a><span class="lineno"> 189</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="l00190"></a><span class="lineno"> 190</span>&#160; dot1 = <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/math/fabs.html">fabs</a>(</div>
<div class="line"><a name="l00191"></a><span class="lineno"> 191</span>&#160; linear_algebra::gram_schmidt::dot_product(b1[i], b1[j], 4));</div>
<div class="line"><a name="l00192"></a><span class="lineno"> 192</span>&#160; <span class="keywordflow">if</span> (dot1 &gt; 0.1) {</div>
<div class="line"><a name="l00193"></a><span class="lineno"> 193</span>&#160; flag = 0;</div>
<div class="line"><a name="l00194"></a><span class="lineno"> 194</span>&#160; <span class="keywordflow">break</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; }</div>
<div class="line"><a name="l00197"></a><span class="lineno"> 197</span>&#160; }</div>
<div class="line"><a name="l00198"></a><span class="lineno"> 198</span>&#160; <span class="keywordflow">if</span> (flag == 0)</div>
<div class="line"><a name="l00199"></a><span class="lineno"> 199</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;Vectors are linearly dependent\n&quot;</span>;</div>
<div class="line"><a name="l00200"></a><span class="lineno"> 200</span>&#160; assert(flag == 1);</div>
<div class="line"><a name="l00201"></a><span class="lineno"> 201</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;Passed Test Case 1\n &quot;</span>;</div>
<div class="line"><a name="l00202"></a><span class="lineno"> 202</span>&#160; </div>
<div class="line"><a name="l00203"></a><span class="lineno"> 203</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; a2 = {{{3, 1}, {2, 2}}};</div>
<div class="line"><a name="l00204"></a><span class="lineno"> 204</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; b2 = {{0}};</div>
<div class="line"><a name="l00205"></a><span class="lineno"> 205</span>&#160; <span class="keywordtype">double</span> dot2 = 0;</div>
<div class="line"><a name="l00206"></a><span class="lineno"> 206</span>&#160; <a class="code" href="../../d5/d33/gram__schmidt_8cpp.html#aa31ca28f60c880802462335eedc5d91f">linear_algebra::gram_schmidt::gram_schmidt</a>(2, 2, a2, b2);</div>
<div class="line"><a name="l00207"></a><span class="lineno"> 207</span>&#160; flag = 1;</div>
<div class="line"><a name="l00208"></a><span class="lineno"> 208</span>&#160; <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i &lt; 1; ++i) {</div>
<div class="line"><a name="l00209"></a><span class="lineno"> 209</span>&#160; <span class="keywordflow">for</span> (<span class="keywordtype">int</span> j = i + 1; j &lt; 2; ++j) {</div>
<div class="line"><a name="l00210"></a><span class="lineno"> 210</span>&#160; dot2 = <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/math/fabs.html">fabs</a>(</div>
<div class="line"><a name="l00211"></a><span class="lineno"> 211</span>&#160; linear_algebra::gram_schmidt::dot_product(b2[i], b2[j], 2));</div>
<div class="line"><a name="l00212"></a><span class="lineno"> 212</span>&#160; <span class="keywordflow">if</span> (dot2 &gt; 0.1) {</div>
<div class="line"><a name="l00213"></a><span class="lineno"> 213</span>&#160; flag = 0;</div>
<div class="line"><a name="l00214"></a><span class="lineno"> 214</span>&#160; <span class="keywordflow">break</span>;</div>
<div class="line"><a name="l00215"></a><span class="lineno"> 215</span>&#160; }</div>
<div class="line"><a name="l00216"></a><span class="lineno"> 216</span>&#160; }</div>
<div class="line"><a name="l00217"></a><span class="lineno"> 217</span>&#160; }</div>
<div class="line"><a name="l00218"></a><span class="lineno"> 218</span>&#160; <span class="keywordflow">if</span> (flag == 0)</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_ostream.html">std::cout</a> &lt;&lt; <span class="stringliteral">&quot;Vectors are linearly dependent\n&quot;</span>;</div>
<div class="line"><a name="l00220"></a><span class="lineno"> 220</span>&#160; assert(flag == 1);</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/io/basic_ostream.html">std::cout</a> &lt;&lt; <span class="stringliteral">&quot;Passed Test Case 2\n&quot;</span>;</div>
<div class="line"><a name="l00222"></a><span class="lineno"> 222</span>&#160; </div>
<div class="line"><a name="l00223"></a><span class="lineno"> 223</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; a3 = {{{1, 2, 2}, {-4, 3, 2}}};</div>
<div class="line"><a name="l00224"></a><span class="lineno"> 224</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; b3 = {{0}};</div>
<div class="line"><a name="l00225"></a><span class="lineno"> 225</span>&#160; <span class="keywordtype">double</span> dot3 = 0;</div>
<div class="line"><a name="l00226"></a><span class="lineno"> 226</span>&#160; <a class="code" href="../../d5/d33/gram__schmidt_8cpp.html#aa31ca28f60c880802462335eedc5d91f">linear_algebra::gram_schmidt::gram_schmidt</a>(2, 3, a3, b3);</div>
<div class="line"><a name="l00227"></a><span class="lineno"> 227</span>&#160; flag = 1;</div>
<div class="line"><a name="l00228"></a><span class="lineno"> 228</span>&#160; <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i &lt; 1; ++i) {</div>
<div class="line"><a name="l00229"></a><span class="lineno"> 229</span>&#160; <span class="keywordflow">for</span> (<span class="keywordtype">int</span> j = i + 1; j &lt; 2; ++j) {</div>
<div class="line"><a name="l00230"></a><span class="lineno"> 230</span>&#160; dot3 = <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/numeric/math/fabs.html">fabs</a>(</div>
<div class="line"><a name="l00231"></a><span class="lineno"> 231</span>&#160; linear_algebra::gram_schmidt::dot_product(b3[i], b3[j], 3));</div>
<div class="line"><a name="l00232"></a><span class="lineno"> 232</span>&#160; <span class="keywordflow">if</span> (dot3 &gt; 0.1) {</div>
<div class="line"><a name="l00233"></a><span class="lineno"> 233</span>&#160; flag = 0;</div>
<div class="line"><a name="l00234"></a><span class="lineno"> 234</span>&#160; <span class="keywordflow">break</span>;</div>
<div class="line"><a name="l00235"></a><span class="lineno"> 235</span>&#160; }</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; }</div>
<div class="line"><a name="l00238"></a><span class="lineno"> 238</span>&#160; <span class="keywordflow">if</span> (flag == 0)</div>
<div class="line"><a name="l00239"></a><span class="lineno"> 239</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;Vectors are linearly dependent\n&quot;</span>;</div>
<div class="line"><a name="l00240"></a><span class="lineno"> 240</span>&#160; assert(flag == 1);</div>
<div class="line"><a name="l00241"></a><span class="lineno"> 241</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;Passed Test Case 3\n&quot;</span>;</div>
<div class="line"><a name="l00242"></a><span class="lineno"> 242</span>&#160;}</div>
</div><!-- fragment -->
</div>
</div>
</div><!-- contents -->
</div><!-- doc-content -->
<div class="ttc" id="agram__schmidt_8cpp_html_aa8dca7b867074164d5f45b0f3851269d"><div class="ttname"><a href="../../d5/d33/gram__schmidt_8cpp.html#aa8dca7b867074164d5f45b0f3851269d">test</a></div><div class="ttdeci">static void test()</div><div class="ttdef"><b>Definition:</b> gram_schmidt.cpp:155</div></div>
<div class="ttc" id="agram__schmidt_8cpp_html_aa8dca7b867074164d5f45b0f3851269d"><div class="ttname"><a href="../../d5/d33/gram__schmidt_8cpp.html#aa8dca7b867074164d5f45b0f3851269d">test</a></div><div class="ttdeci">static void test()</div><div class="ttdef"><b>Definition:</b> gram_schmidt.cpp:181</div></div>
<div class="ttc" id="afabs_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/numeric/math/fabs.html">std::fabs</a></div><div class="ttdeci">T fabs(T... args)</div></div>
<div class="ttc" id="agram__schmidt_8cpp_html_aed4d308f46ca1d91e348ca260ec1d2fb"><div class="ttname"><a href="../../d5/d33/gram__schmidt_8cpp.html#aed4d308f46ca1d91e348ca260ec1d2fb">linear_algebra::gram_schmidt::dot_product</a></div><div class="ttdeci">double dot_product(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:53</div></div>
<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>
<div class="ttc" id="agram__schmidt_8cpp_html_aed4d308f46ca1d91e348ca260ec1d2fb"><div class="ttname"><a href="../../d5/d33/gram__schmidt_8cpp.html#aed4d308f46ca1d91e348ca260ec1d2fb">linear_algebra::gram_schmidt::dot_product</a></div><div class="ttdeci">double dot_product(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:59</div></div>
<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:79</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>
<div class="ttc" id="aarray_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/container/array.html">std::array</a></div><div class="ttdoc">STL class.</div></div>
<div class="ttc" id="agram__schmidt_8cpp_html_aa31ca28f60c880802462335eedc5d91f"><div class="ttname"><a href="../../d5/d33/gram__schmidt_8cpp.html#aa31ca28f60c880802462335eedc5d91f">linear_algebra::gram_schmidt::gram_schmidt</a></div><div class="ttdeci">void gram_schmidt(int r, const int &amp;c, const std::array&lt; std::array&lt; double, 10 &gt;, 20 &gt; &amp;A, std::array&lt; std::array&lt; double, 10 &gt;, 20 &gt; B)</div><div class="ttdef"><b>Definition:</b> gram_schmidt.cpp:107</div></div>
<div class="ttc" id="agram__schmidt_8cpp_html_aa31ca28f60c880802462335eedc5d91f"><div class="ttname"><a href="../../d5/d33/gram__schmidt_8cpp.html#aa31ca28f60c880802462335eedc5d91f">linear_algebra::gram_schmidt::gram_schmidt</a></div><div class="ttdeci">void gram_schmidt(int r, const int &amp;c, const std::array&lt; std::array&lt; double, 10 &gt;, 20 &gt; &amp;A, std::array&lt; std::array&lt; double, 10 &gt;, 20 &gt; B)</div><div class="ttdef"><b>Definition:</b> gram_schmidt.cpp:121</div></div>
<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>
<div class="ttc" id="anamespacemachine__learning_html_a5305cc18d11943d3dc0d66e70d7e9c44"><div class="ttname"><a href="../../d8/d77/namespacemachine__learning.html#a5305cc18d11943d3dc0d66e70d7e9c44">machine_learning::sum</a></div><div class="ttdeci">T sum(const std::vector&lt; std::valarray&lt; T &gt;&gt; &amp;A)</div><div class="ttdef"><b>Definition:</b> vector_ops.hpp:232</div></div>
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