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<div class="headertitle"><div class="title">linear_recurrence_matrix.cpp</div></div>
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</div><!--header-->
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<div class="contents">
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<div class="fragment"><div class="line"><a id="l00001" name="l00001"></a><span class="lineno"> 1</span> </div>
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<div class="line"><a id="l00020" name="l00020"></a><span class="lineno"> 20</span><span class="preprocessor">#include <cassert></span> </div>
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<div class="line"><a id="l00021" name="l00021"></a><span class="lineno"> 21</span><span class="preprocessor">#include <cstdint></span></div>
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<div class="line"><a id="l00022" name="l00022"></a><span class="lineno"> 22</span><span class="preprocessor">#include <iostream></span> </div>
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<div class="line"><a id="l00023" name="l00023"></a><span class="lineno"> 23</span><span class="preprocessor">#include <vector></span> </div>
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<div class="line"><a id="l00024" name="l00024"></a><span class="lineno"> 24</span> </div>
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<div class="line"><a id="l00029" name="l00029"></a><span class="lineno"> 29</span><span class="keyword">namespace </span><a class="code hl_namespace" href="../../dd/d47/namespacemath.html">math</a> {</div>
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<div class="line"><a id="l00036" name="l00036"></a><span class="lineno"> 36</span><span class="keyword">namespace </span><a class="code hl_namespace" href="../../d9/dd1/namespacelinear__recurrence__matrix.html">linear_recurrence_matrix</a> {</div>
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<div class="line"><a id="l00050" name="l00050"></a><span class="lineno"> 50</span><span class="keyword">template</span> <<span class="keyword">typename</span> T = <span class="keywordtype">int</span>64_t></div>
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<div class="line"><a id="l00051" name="l00051"></a><span class="lineno"> 51</span>std::vector<std::vector<T>> matrix_multiplication(</div>
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<div class="line"><a id="l00052" name="l00052"></a><span class="lineno"> 52</span> <span class="keyword">const</span> std::vector<std::vector<T>>& _mat_a,</div>
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<div class="line"><a id="l00053" name="l00053"></a><span class="lineno"> 53</span> <span class="keyword">const</span> std::vector<std::vector<T>>& _mat_b, <span class="keyword">const</span> int64_t mod = 1000000007) {</div>
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<div class="line"><a id="l00054" name="l00054"></a><span class="lineno"> 54</span> <span class="comment">// assert that columns in `_mat_a` and rows in `_mat_b` are equal</span></div>
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<div class="line"><a id="l00055" name="l00055"></a><span class="lineno"> 55</span> assert(_mat_a[0].size() == _mat_b.size());</div>
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<div class="line"><a id="l00056" name="l00056"></a><span class="lineno"> 56</span> std::vector<std::vector<T>> _mat_c(_mat_a.size(),</div>
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<div class="line"><a id="l00057" name="l00057"></a><span class="lineno"> 57</span> std::vector<T>(_mat_b[0].size(), 0));</div>
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<div class="line"><a id="l00061" name="l00061"></a><span class="lineno"> 61</span> <span class="keywordflow">for</span> (uint32_t i = 0; i < _mat_a.size(); ++i) {</div>
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<div class="line"><a id="l00062" name="l00062"></a><span class="lineno"> 62</span> <span class="keywordflow">for</span> (uint32_t j = 0; j < _mat_b[0].size(); ++j) {</div>
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<div class="line"><a id="l00063" name="l00063"></a><span class="lineno"> 63</span> <span class="keywordflow">for</span> (uint32_t k = 0; <a class="code hl_function" href="../../d4/d18/composite__simpson__rule_8cpp.html#a1b74d828b33760094906797042b89442">k</a> < _mat_b.size(); ++<a class="code hl_function" href="../../d4/d18/composite__simpson__rule_8cpp.html#a1b74d828b33760094906797042b89442">k</a>) {</div>
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<div class="line"><a id="l00064" name="l00064"></a><span class="lineno"> 64</span> _mat_c[i][j] =</div>
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<div class="line"><a id="l00065" name="l00065"></a><span class="lineno"> 65</span> (_mat_c[i][j] % mod +</div>
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<div class="line"><a id="l00066" name="l00066"></a><span class="lineno"> 66</span> (_mat_a[i][<a class="code hl_function" href="../../d4/d18/composite__simpson__rule_8cpp.html#a1b74d828b33760094906797042b89442">k</a>] % mod * _mat_b[<a class="code hl_function" href="../../d4/d18/composite__simpson__rule_8cpp.html#a1b74d828b33760094906797042b89442">k</a>][j] % mod) % mod) %</div>
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<div class="line"><a id="l00067" name="l00067"></a><span class="lineno"> 67</span> mod;</div>
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<div class="line"><a id="l00068" name="l00068"></a><span class="lineno"> 68</span> }</div>
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<div class="line"><a id="l00069" name="l00069"></a><span class="lineno"> 69</span> }</div>
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<div class="line"><a id="l00070" name="l00070"></a><span class="lineno"> 70</span> }</div>
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<div class="line"><a id="l00071" name="l00071"></a><span class="lineno"> 71</span> <span class="keywordflow">return</span> _mat_c;</div>
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<div class="line"><a id="l00072" name="l00072"></a><span class="lineno"> 72</span>}</div>
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<div class="line"><a id="l00081" name="l00081"></a><span class="lineno"> 81</span><span class="keyword">template</span> <<span class="keyword">typename</span> T = <span class="keywordtype">int</span>64_t></div>
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<div class="line"><a id="l00082" name="l00082"></a><span class="lineno"> 82</span><span class="keywordtype">bool</span> is_zero_matrix(<span class="keyword">const</span> std::vector<std::vector<T>>& _mat) {</div>
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<div class="line"><a id="l00083" name="l00083"></a><span class="lineno"> 83</span> <span class="keywordflow">for</span> (uint32_t i = 0; i < _mat.size(); ++i) {</div>
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<div class="line"><a id="l00084" name="l00084"></a><span class="lineno"> 84</span> <span class="keywordflow">for</span> (uint32_t j = 0; j < _mat[i].size(); ++j) {</div>
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<div class="line"><a id="l00085" name="l00085"></a><span class="lineno"> 85</span> <span class="keywordflow">if</span> (_mat[i][j] != 0) {</div>
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<div class="line"><a id="l00086" name="l00086"></a><span class="lineno"> 86</span> <span class="keywordflow">return</span> <span class="keyword">false</span>;</div>
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<div class="line"><a id="l00087" name="l00087"></a><span class="lineno"> 87</span> }</div>
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<div class="line"><a id="l00088" name="l00088"></a><span class="lineno"> 88</span> }</div>
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<div class="line"><a id="l00089" name="l00089"></a><span class="lineno"> 89</span> }</div>
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<div class="line"><a id="l00090" name="l00090"></a><span class="lineno"> 90</span> <span class="keywordflow">return</span> <span class="keyword">true</span>;</div>
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<div class="line"><a id="l00091" name="l00091"></a><span class="lineno"> 91</span>}</div>
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<div class="line"><a id="l00092" name="l00092"></a><span class="lineno"> 92</span> </div>
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<div class="line"><a id="l00103" name="l00103"></a><span class="lineno"> 103</span><span class="keyword">template</span> <<span class="keyword">typename</span> T = <span class="keywordtype">int</span>64_t></div>
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<div class="line"><a id="l00104" name="l00104"></a><span class="lineno"> 104</span>std::vector<std::vector<T>> matrix_exponentiation(</div>
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<div class="line"><a id="l00105" name="l00105"></a><span class="lineno"> 105</span> std::vector<std::vector<T>> _mat, uint64_t <a class="code hl_function" href="../../dd/d47/namespacemath.html#afcd07701d73ed65cd616bcba02737f3d">power</a>,</div>
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<div class="line"><a id="l00106" name="l00106"></a><span class="lineno"> 106</span> <span class="keyword">const</span> int64_t mod = 1000000007) {</div>
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<div class="line"><a id="l00112" name="l00112"></a><span class="lineno"> 112</span> <span class="keywordflow">if</span> (is_zero_matrix(_mat)) {</div>
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<div class="line"><a id="l00113" name="l00113"></a><span class="lineno"> 113</span> <span class="keywordflow">return</span> _mat;</div>
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<div class="line"><a id="l00114" name="l00114"></a><span class="lineno"> 114</span> }</div>
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<div class="line"><a id="l00115" name="l00115"></a><span class="lineno"> 115</span> </div>
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<div class="line"><a id="l00116" name="l00116"></a><span class="lineno"> 116</span> std::vector<std::vector<T>> _mat_answer(_mat.size(),</div>
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<div class="line"><a id="l00117" name="l00117"></a><span class="lineno"> 117</span> std::vector<T>(_mat.size(), 0));</div>
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<div class="line"><a id="l00118" name="l00118"></a><span class="lineno"> 118</span> </div>
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<div class="line"><a id="l00119" name="l00119"></a><span class="lineno"> 119</span> <span class="keywordflow">for</span> (uint32_t i = 0; i < _mat.size(); ++i) {</div>
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<div class="line"><a id="l00120" name="l00120"></a><span class="lineno"> 120</span> _mat_answer[i][i] = 1;</div>
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<div class="line"><a id="l00121" name="l00121"></a><span class="lineno"> 121</span> }</div>
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<div class="line"><a id="l00122" name="l00122"></a><span class="lineno"> 122</span> <span class="comment">// exponentiation algorithm here.</span></div>
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<div class="line"><a id="l00123" name="l00123"></a><span class="lineno"> 123</span> <span class="keywordflow">while</span> (<a class="code hl_function" href="../../dd/d47/namespacemath.html#afcd07701d73ed65cd616bcba02737f3d">power</a> > 0) {</div>
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<div class="line"><a id="l00124" name="l00124"></a><span class="lineno"> 124</span> <span class="keywordflow">if</span> (<a class="code hl_function" href="../../dd/d47/namespacemath.html#afcd07701d73ed65cd616bcba02737f3d">power</a> & 1) {</div>
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<div class="line"><a id="l00125" name="l00125"></a><span class="lineno"> 125</span> _mat_answer = matrix_multiplication(_mat_answer, _mat, mod);</div>
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<div class="line"><a id="l00126" name="l00126"></a><span class="lineno"> 126</span> }</div>
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<div class="line"><a id="l00127" name="l00127"></a><span class="lineno"> 127</span> <a class="code hl_function" href="../../dd/d47/namespacemath.html#afcd07701d73ed65cd616bcba02737f3d">power</a> >>= 1;</div>
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<div class="line"><a id="l00128" name="l00128"></a><span class="lineno"> 128</span> _mat = matrix_multiplication(_mat, _mat, mod);</div>
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<div class="line"><a id="l00129" name="l00129"></a><span class="lineno"> 129</span> }</div>
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<div class="line"><a id="l00130" name="l00130"></a><span class="lineno"> 130</span> </div>
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<div class="line"><a id="l00131" name="l00131"></a><span class="lineno"> 131</span> <span class="keywordflow">return</span> _mat_answer;</div>
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<div class="line"><a id="l00132" name="l00132"></a><span class="lineno"> 132</span>}</div>
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<div class="line"><a id="l00133" name="l00133"></a><span class="lineno"> 133</span> </div>
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<div class="line"><a id="l00154" name="l00154"></a><span class="lineno"> 154</span><span class="keyword">template</span> <<span class="keyword">typename</span> T = <span class="keywordtype">int</span>64_t></div>
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<div class="line"><a id="l00155" name="l00155"></a><span class="lineno"> 155</span>T get_nth_term_of_recurrence_series(</div>
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<div class="line"><a id="l00156" name="l00156"></a><span class="lineno"> 156</span> <span class="keyword">const</span> std::vector<std::vector<T>>& _mat,</div>
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<div class="line"><a id="l00157" name="l00157"></a><span class="lineno"> 157</span> <span class="keyword">const</span> std::vector<std::vector<T>>& _base_cases, uint64_t nth_term,</div>
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<div class="line"><a id="l00158" name="l00158"></a><span class="lineno"> 158</span> <span class="keywordtype">bool</span> constant_or_sum_included = <span class="keyword">false</span>) {</div>
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<div class="line"><a id="l00159" name="l00159"></a><span class="lineno"> 159</span> assert(_mat.size() == _base_cases.back().size());</div>
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<div class="line"><a id="l00160" name="l00160"></a><span class="lineno"> 160</span> </div>
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<div class="line"><a id="l00165" name="l00165"></a><span class="lineno"> 165</span> <span class="keywordflow">if</span> (nth_term < _base_cases.back().size() - constant_or_sum_included) {</div>
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<div class="line"><a id="l00166" name="l00166"></a><span class="lineno"> 166</span> <span class="keywordflow">return</span> _base_cases.back()[nth_term - constant_or_sum_included];</div>
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<div class="line"><a id="l00167" name="l00167"></a><span class="lineno"> 167</span> } <span class="keywordflow">else</span> {</div>
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<div class="line"><a id="l00172" name="l00172"></a><span class="lineno"> 172</span> std::vector<std::vector<T>> _res_matrix =</div>
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<div class="line"><a id="l00173" name="l00173"></a><span class="lineno"> 173</span> matrix_exponentiation(_mat, nth_term - _base_cases.back().size() +</div>
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<div class="line"><a id="l00174" name="l00174"></a><span class="lineno"> 174</span> 1 + constant_or_sum_included);</div>
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<div class="line"><a id="l00175" name="l00175"></a><span class="lineno"> 175</span> </div>
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<div class="line"><a id="l00180" name="l00180"></a><span class="lineno"> 180</span> std::vector<std::vector<T>> _res =</div>
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<div class="line"><a id="l00181" name="l00181"></a><span class="lineno"> 181</span> matrix_multiplication(_base_cases, _res_matrix);</div>
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<div class="line"><a id="l00182" name="l00182"></a><span class="lineno"> 182</span> </div>
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<div class="line"><a id="l00183" name="l00183"></a><span class="lineno"> 183</span> <span class="keywordflow">return</span> _res.back().back();</div>
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<div class="line"><a id="l00184" name="l00184"></a><span class="lineno"> 184</span> }</div>
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<div class="line"><a id="l00185" name="l00185"></a><span class="lineno"> 185</span>}</div>
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<div class="line"><a id="l00186" name="l00186"></a><span class="lineno"> 186</span>} <span class="comment">// namespace linear_recurrence_matrix</span></div>
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<div class="line"><a id="l00187" name="l00187"></a><span class="lineno"> 187</span>} <span class="comment">// namespace math</span></div>
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<div class="line"><a id="l00188" name="l00188"></a><span class="lineno"> 188</span> </div>
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<div class="line"><a id="l00193" name="l00193"></a><span class="lineno"> 193</span><span class="keyword">static</span> <span class="keywordtype">void</span> <a class="code hl_function" href="../../d6/d2c/caesar__cipher_8cpp.html#ae1a3968e7947464bee7714f6d43b7002">test</a>() {</div>
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<div class="line"><a id="l00194" name="l00194"></a><span class="lineno"> 194</span> <span class="comment">/*</span></div>
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<div class="line"><a id="l00195" name="l00195"></a><span class="lineno"> 195</span><span class="comment"> * Example 1: [Fibonacci</span></div>
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<div class="line"><a id="l00196" name="l00196"></a><span class="lineno"> 196</span><span class="comment"> * series](https://en.wikipedia.org/wiki/Fibonacci_number);</span></div>
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<div class="line"><a id="l00197" name="l00197"></a><span class="lineno"> 197</span><span class="comment"> *</span></div>
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<div class="line"><a id="l00198" name="l00198"></a><span class="lineno"> 198</span><span class="comment"> * [fn-2 fn-1] [0 1] == [fn-1 (fn-2 + fn-1)] => [fn-1 fn]</span></div>
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<div class="line"><a id="l00199" name="l00199"></a><span class="lineno"> 199</span><span class="comment"> * [1 1]</span></div>
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<div class="line"><a id="l00200" name="l00200"></a><span class="lineno"> 200</span><span class="comment"> *</span></div>
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<div class="line"><a id="l00201" name="l00201"></a><span class="lineno"> 201</span><span class="comment"> * Let A = [fn-2 fn-1], and B = [0 1]</span></div>
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<div class="line"><a id="l00202" name="l00202"></a><span class="lineno"> 202</span><span class="comment"> * [1 1],</span></div>
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<div class="line"><a id="l00203" name="l00203"></a><span class="lineno"> 203</span><span class="comment"> *</span></div>
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<div class="line"><a id="l00204" name="l00204"></a><span class="lineno"> 204</span><span class="comment"> * Since, A.B....(n-1 times) = [fn-1 fn]</span></div>
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<div class="line"><a id="l00205" name="l00205"></a><span class="lineno"> 205</span><span class="comment"> * we can multiply B with itself n-1 times to obtain the required value</span></div>
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<div class="line"><a id="l00206" name="l00206"></a><span class="lineno"> 206</span><span class="comment"> */</span></div>
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<div class="line"><a id="l00207" name="l00207"></a><span class="lineno"> 207</span> std::vector<std::vector<int64_t>> fibonacci_matrix = {{0, 1}, {1, 1}},</div>
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<div class="line"><a id="l00208" name="l00208"></a><span class="lineno"> 208</span> fib_base_case = {{0, 1}};</div>
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<div class="line"><a id="l00209" name="l00209"></a><span class="lineno"> 209</span> </div>
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<div class="line"><a id="l00210" name="l00210"></a><span class="lineno"> 210</span> assert(math::linear_recurrence_matrix::get_nth_term_of_recurrence_series(</div>
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<div class="line"><a id="l00211" name="l00211"></a><span class="lineno"> 211</span> fibonacci_matrix, fib_base_case, 11) == 89LL);</div>
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<div class="line"><a id="l00212" name="l00212"></a><span class="lineno"> 212</span> assert(math::linear_recurrence_matrix::get_nth_term_of_recurrence_series(</div>
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<div class="line"><a id="l00213" name="l00213"></a><span class="lineno"> 213</span> fibonacci_matrix, fib_base_case, 39) == 63245986LL);</div>
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<div class="line"><a id="l00214" name="l00214"></a><span class="lineno"> 214</span> <span class="comment">/*</span></div>
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<div class="line"><a id="l00215" name="l00215"></a><span class="lineno"> 215</span><span class="comment"> * Example 2: [Tribonacci series](https://oeis.org/A000073)</span></div>
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<div class="line"><a id="l00216" name="l00216"></a><span class="lineno"> 216</span><span class="comment"> * [0 0 1]</span></div>
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<div class="line"><a id="l00217" name="l00217"></a><span class="lineno"> 217</span><span class="comment"> * [fn-3 fn-2 fn-1] [1 0 1] = [(fn-2) (fn-1) (fn-3 + fn-2 + fn-1)]</span></div>
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<div class="line"><a id="l00218" name="l00218"></a><span class="lineno"> 218</span><span class="comment"> * [0 1 1]</span></div>
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<div class="line"><a id="l00219" name="l00219"></a><span class="lineno"> 219</span><span class="comment"> * => [fn-2 fn-1 fn]</span></div>
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<div class="line"><a id="l00220" name="l00220"></a><span class="lineno"> 220</span><span class="comment"> *</span></div>
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<div class="line"><a id="l00221" name="l00221"></a><span class="lineno"> 221</span><span class="comment"> * [0 0 1]</span></div>
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<div class="line"><a id="l00222" name="l00222"></a><span class="lineno"> 222</span><span class="comment"> * Let A = [fn-3 fn-2 fn-1], and B = [1 0 1]</span></div>
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<div class="line"><a id="l00223" name="l00223"></a><span class="lineno"> 223</span><span class="comment"> * [0 1 1]</span></div>
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<div class="line"><a id="l00224" name="l00224"></a><span class="lineno"> 224</span><span class="comment"> *</span></div>
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<div class="line"><a id="l00225" name="l00225"></a><span class="lineno"> 225</span><span class="comment"> * Since, A.B....(n-2 times) = [fn-2 fn-1 fn]</span></div>
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<div class="line"><a id="l00226" name="l00226"></a><span class="lineno"> 226</span><span class="comment"> * we will have multiply B with itself n-2 times to obtain the required</span></div>
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<div class="line"><a id="l00227" name="l00227"></a><span class="lineno"> 227</span><span class="comment"> * value ()</span></div>
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<div class="line"><a id="l00228" name="l00228"></a><span class="lineno"> 228</span><span class="comment"> */</span></div>
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<div class="line"><a id="l00229" name="l00229"></a><span class="lineno"> 229</span> </div>
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<div class="line"><a id="l00230" name="l00230"></a><span class="lineno"> 230</span> std::vector<std::vector<int64_t>> tribonacci = {{0, 0, 1},</div>
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<div class="line"><a id="l00231" name="l00231"></a><span class="lineno"> 231</span> {1, 0, 1},</div>
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<div class="line"><a id="l00232" name="l00232"></a><span class="lineno"> 232</span> {0, 1, 1}},</div>
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<div class="line"><a id="l00233" name="l00233"></a><span class="lineno"> 233</span> trib_base_case = {</div>
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<div class="line"><a id="l00234" name="l00234"></a><span class="lineno"> 234</span> {0, 0, 1}}; <span class="comment">// f0 = 0, f1 = 0, f2 = 1</span></div>
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<div class="line"><a id="l00235" name="l00235"></a><span class="lineno"> 235</span> </div>
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<div class="line"><a id="l00236" name="l00236"></a><span class="lineno"> 236</span> assert(math::linear_recurrence_matrix::get_nth_term_of_recurrence_series(</div>
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<div class="line"><a id="l00237" name="l00237"></a><span class="lineno"> 237</span> tribonacci, trib_base_case, 11) == 149LL);</div>
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<div class="line"><a id="l00238" name="l00238"></a><span class="lineno"> 238</span> assert(math::linear_recurrence_matrix::get_nth_term_of_recurrence_series(</div>
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<div class="line"><a id="l00239" name="l00239"></a><span class="lineno"> 239</span> tribonacci, trib_base_case, 36) == 615693474LL);</div>
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<div class="line"><a id="l00240" name="l00240"></a><span class="lineno"> 240</span> </div>
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<div class="line"><a id="l00241" name="l00241"></a><span class="lineno"> 241</span> <span class="comment">/*</span></div>
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<div class="line"><a id="l00242" name="l00242"></a><span class="lineno"> 242</span><span class="comment"> * Example 3: [Pell numbers](https://oeis.org/A000129)</span></div>
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<div class="line"><a id="l00243" name="l00243"></a><span class="lineno"> 243</span><span class="comment"> * `f(n) = 2* f(n-1) + f(n-2); f(0) = f(1) = 2`</span></div>
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<div class="line"><a id="l00244" name="l00244"></a><span class="lineno"> 244</span><span class="comment"> *</span></div>
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<div class="line"><a id="l00245" name="l00245"></a><span class="lineno"> 245</span><span class="comment"> * [fn-2 fn-1] [0 1] = [(fn-1) fn-2 + 2*fn-1)]</span></div>
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<div class="line"><a id="l00246" name="l00246"></a><span class="lineno"> 246</span><span class="comment"> * [1 2]</span></div>
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<div class="line"><a id="l00247" name="l00247"></a><span class="lineno"> 247</span><span class="comment"> * => [fn-1 fn]</span></div>
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<div class="line"><a id="l00248" name="l00248"></a><span class="lineno"> 248</span><span class="comment"> *</span></div>
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<div class="line"><a id="l00249" name="l00249"></a><span class="lineno"> 249</span><span class="comment"> * Let A = [fn-2 fn-1], and B = [0 1]</span></div>
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<div class="line"><a id="l00250" name="l00250"></a><span class="lineno"> 250</span><span class="comment"> * [1 2]</span></div>
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<div class="line"><a id="l00251" name="l00251"></a><span class="lineno"> 251</span><span class="comment"> */</span></div>
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<div class="line"><a id="l00252" name="l00252"></a><span class="lineno"> 252</span> </div>
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<div class="line"><a id="l00253" name="l00253"></a><span class="lineno"> 253</span> std::vector<std::vector<int64_t>> pell_recurrence = {{0, 1}, {1, 2}},</div>
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<div class="line"><a id="l00254" name="l00254"></a><span class="lineno"> 254</span> pell_base_case = {</div>
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<div class="line"><a id="l00255" name="l00255"></a><span class="lineno"> 255</span> {2, 2}}; <span class="comment">// `f0 = 2, f1 = 2`</span></div>
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<div class="line"><a id="l00256" name="l00256"></a><span class="lineno"> 256</span> </div>
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<div class="line"><a id="l00257" name="l00257"></a><span class="lineno"> 257</span> assert(math::linear_recurrence_matrix::get_nth_term_of_recurrence_series(</div>
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<div class="line"><a id="l00258" name="l00258"></a><span class="lineno"> 258</span> pell_recurrence, pell_base_case, 15) == 551614LL);</div>
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<div class="line"><a id="l00259" name="l00259"></a><span class="lineno"> 259</span> assert(math::linear_recurrence_matrix::get_nth_term_of_recurrence_series(</div>
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<div class="line"><a id="l00260" name="l00260"></a><span class="lineno"> 260</span> pell_recurrence, pell_base_case, 23) == 636562078LL);</div>
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<div class="line"><a id="l00261" name="l00261"></a><span class="lineno"> 261</span> </div>
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<div class="line"><a id="l00262" name="l00262"></a><span class="lineno"> 262</span> <span class="comment">/*</span></div>
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<div class="line"><a id="l00263" name="l00263"></a><span class="lineno"> 263</span><span class="comment"> * Example 4: Custom recurrence relation:</span></div>
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<div class="line"><a id="l00264" name="l00264"></a><span class="lineno"> 264</span><span class="comment"> * Now the recurrence is of the form `a*f(n-1) + b*(fn-2) + ... + c`</span></div>
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<div class="line"><a id="l00265" name="l00265"></a><span class="lineno"> 265</span><span class="comment"> * where `c` is the constant</span></div>
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<div class="line"><a id="l00266" name="l00266"></a><span class="lineno"> 266</span><span class="comment"> * `f(n) = 2* f(n-1) + f(n-2) + 7; f(0) = f(1) = 2, c = 7`</span></div>
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<div class="line"><a id="l00267" name="l00267"></a><span class="lineno"> 267</span><span class="comment"> *</span></div>
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<div class="line"><a id="l00268" name="l00268"></a><span class="lineno"> 268</span><span class="comment"> * [1 0 1]</span></div>
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<div class="line"><a id="l00269" name="l00269"></a><span class="lineno"> 269</span><span class="comment"> * [7, fn-2, fn-1] [0 0 1]</span></div>
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<div class="line"><a id="l00270" name="l00270"></a><span class="lineno"> 270</span><span class="comment"> * [0 1 2]</span></div>
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<div class="line"><a id="l00271" name="l00271"></a><span class="lineno"> 271</span><span class="comment"> * = [7, (fn-1), fn-2 + 2*fn-1) + 7]</span></div>
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<div class="line"><a id="l00272" name="l00272"></a><span class="lineno"> 272</span><span class="comment"> *</span></div>
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<div class="line"><a id="l00273" name="l00273"></a><span class="lineno"> 273</span><span class="comment"> * => [7, fn-1, fn]</span></div>
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<div class="line"><a id="l00274" name="l00274"></a><span class="lineno"> 274</span><span class="comment"> * :: Series will be 2, 2, 13, 35, 90, 222, 541, 1311, 3170, 7658, 18493,</span></div>
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<div class="line"><a id="l00275" name="l00275"></a><span class="lineno"> 275</span><span class="comment"> * 44651, 107802, 260262, 628333, 1516935, 362210, 8841362, 21344941,</span></div>
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<div class="line"><a id="l00276" name="l00276"></a><span class="lineno"> 276</span><span class="comment"> * 51531251</span></div>
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<div class="line"><a id="l00277" name="l00277"></a><span class="lineno"> 277</span><span class="comment"> *</span></div>
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<div class="line"><a id="l00278" name="l00278"></a><span class="lineno"> 278</span><span class="comment"> * Let A = [7, fn-2, fn-1], and B = [1 0 1]</span></div>
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<div class="line"><a id="l00279" name="l00279"></a><span class="lineno"> 279</span><span class="comment"> * [0 0 1]</span></div>
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<div class="line"><a id="l00280" name="l00280"></a><span class="lineno"> 280</span><span class="comment"> * [0 1 2]</span></div>
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<div class="line"><a id="l00281" name="l00281"></a><span class="lineno"> 281</span><span class="comment"> */</span></div>
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<div class="line"><a id="l00282" name="l00282"></a><span class="lineno"> 282</span> </div>
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<div class="line"><a id="l00283" name="l00283"></a><span class="lineno"> 283</span> std::vector<std::vector<int64_t>></div>
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<div class="line"><a id="l00284" name="l00284"></a><span class="lineno"> 284</span> custom_recurrence = {{1, 0, 1}, {0, 0, 1}, {0, 1, 2}},</div>
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<div class="line"><a id="l00285" name="l00285"></a><span class="lineno"> 285</span> custom_base_case = {{7, 2, 2}}; <span class="comment">// `c = 7, f0 = 2, f1 = 2`</span></div>
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<div class="line"><a id="l00286" name="l00286"></a><span class="lineno"> 286</span> </div>
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<div class="line"><a id="l00287" name="l00287"></a><span class="lineno"> 287</span> assert(math::linear_recurrence_matrix::get_nth_term_of_recurrence_series(</div>
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<div class="line"><a id="l00288" name="l00288"></a><span class="lineno"> 288</span> custom_recurrence, custom_base_case, 10, 1) == 18493LL);</div>
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<div class="line"><a id="l00289" name="l00289"></a><span class="lineno"> 289</span> assert(math::linear_recurrence_matrix::get_nth_term_of_recurrence_series(</div>
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<div class="line"><a id="l00290" name="l00290"></a><span class="lineno"> 290</span> custom_recurrence, custom_base_case, 19, 1) == 51531251LL);</div>
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<div class="line"><a id="l00291" name="l00291"></a><span class="lineno"> 291</span> </div>
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<div class="line"><a id="l00292" name="l00292"></a><span class="lineno"> 292</span> <span class="comment">/*</span></div>
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<div class="line"><a id="l00293" name="l00293"></a><span class="lineno"> 293</span><span class="comment"> * Example 5: Sum fibonacci sequence</span></div>
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<div class="line"><a id="l00294" name="l00294"></a><span class="lineno"> 294</span><span class="comment"> * The following matrix evaluates the sum of first n fibonacci terms in</span></div>
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<div class="line"><a id="l00295" name="l00295"></a><span class="lineno"> 295</span><span class="comment"> * O(27. log2(n)) time.</span></div>
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<div class="line"><a id="l00296" name="l00296"></a><span class="lineno"> 296</span><span class="comment"> * `f(n) = f(n-1) + f(n-2); f(0) = 0, f(1) = 1`</span></div>
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<div class="line"><a id="l00297" name="l00297"></a><span class="lineno"> 297</span><span class="comment"> *</span></div>
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<div class="line"><a id="l00298" name="l00298"></a><span class="lineno"> 298</span><span class="comment"> * [1 0 0]</span></div>
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<div class="line"><a id="l00299" name="l00299"></a><span class="lineno"> 299</span><span class="comment"> * [s(f, n-1), fn-2, fn-1] [1 0 1]</span></div>
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<div class="line"><a id="l00300" name="l00300"></a><span class="lineno"> 300</span><span class="comment"> * [1 1 1]</span></div>
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<div class="line"><a id="l00301" name="l00301"></a><span class="lineno"> 301</span><span class="comment"> * => [(s(f, n-1)+f(n-2)+f(n-1)), (fn-1), f(n-2)+f(n-1)]</span></div>
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<div class="line"><a id="l00302" name="l00302"></a><span class="lineno"> 302</span><span class="comment"> *</span></div>
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<div class="line"><a id="l00303" name="l00303"></a><span class="lineno"> 303</span><span class="comment"> * => [s(f, n-1)+f(n), fn-1, fn]</span></div>
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<div class="line"><a id="l00304" name="l00304"></a><span class="lineno"> 304</span><span class="comment"> *</span></div>
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<div class="line"><a id="l00305" name="l00305"></a><span class="lineno"> 305</span><span class="comment"> * => [s(f, n), fn-1, fn]</span></div>
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<div class="line"><a id="l00306" name="l00306"></a><span class="lineno"> 306</span><span class="comment"> *</span></div>
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<div class="line"><a id="l00307" name="l00307"></a><span class="lineno"> 307</span><span class="comment"> * Sum of first 20 fibonacci series:</span></div>
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<div class="line"><a id="l00308" name="l00308"></a><span class="lineno"> 308</span><span class="comment"> * 0, 1, 2, 4, 7, 12, 20, 33, 54, 88, 143, 232, 376, 609, 986, 1596, 2583,</span></div>
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<div class="line"><a id="l00309" name="l00309"></a><span class="lineno"> 309</span><span class="comment"> * 4180, 6764</span></div>
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<div class="line"><a id="l00310" name="l00310"></a><span class="lineno"> 310</span><span class="comment"> * f0 f1 s(f,1)</span></div>
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<div class="line"><a id="l00311" name="l00311"></a><span class="lineno"> 311</span><span class="comment"> * Let A = [0 1 1], and B = [0 1 1]</span></div>
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<div class="line"><a id="l00312" name="l00312"></a><span class="lineno"> 312</span><span class="comment"> * [1 1 1]</span></div>
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<div class="line"><a id="l00313" name="l00313"></a><span class="lineno"> 313</span><span class="comment"> * [0 0 1]</span></div>
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<div class="line"><a id="l00314" name="l00314"></a><span class="lineno"> 314</span><span class="comment"> */</span></div>
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<div class="line"><a id="l00315" name="l00315"></a><span class="lineno"> 315</span> </div>
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<div class="line"><a id="l00316" name="l00316"></a><span class="lineno"> 316</span> std::vector<std::vector<int64_t>> sum_fibo_recurrence = {{0, 1, 1},</div>
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<div class="line"><a id="l00317" name="l00317"></a><span class="lineno"> 317</span> {1, 1, 1},</div>
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<div class="line"><a id="l00318" name="l00318"></a><span class="lineno"> 318</span> {0, 0, 1}},</div>
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<div class="line"><a id="l00319" name="l00319"></a><span class="lineno"> 319</span> sum_fibo_base_case = {</div>
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<div class="line"><a id="l00320" name="l00320"></a><span class="lineno"> 320</span> {0, 1, 1}}; <span class="comment">// `f0 = 0, f1 = 1`</span></div>
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<div class="line"><a id="l00321" name="l00321"></a><span class="lineno"> 321</span> </div>
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<div class="line"><a id="l00322" name="l00322"></a><span class="lineno"> 322</span> assert(math::linear_recurrence_matrix::get_nth_term_of_recurrence_series(</div>
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<div class="line"><a id="l00323" name="l00323"></a><span class="lineno"> 323</span> sum_fibo_recurrence, sum_fibo_base_case, 13, 1) == 609LL);</div>
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<div class="line"><a id="l00324" name="l00324"></a><span class="lineno"> 324</span> assert(math::linear_recurrence_matrix::get_nth_term_of_recurrence_series(</div>
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<div class="line"><a id="l00325" name="l00325"></a><span class="lineno"> 325</span> sum_fibo_recurrence, sum_fibo_base_case, 16, 1) == 2583LL);</div>
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<div class="line"><a id="l00326" name="l00326"></a><span class="lineno"> 326</span> <span class="comment">/*</span></div>
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<div class="line"><a id="l00327" name="l00327"></a><span class="lineno"> 327</span><span class="comment"> * Example 6: [Tribonacci sum series](https://oeis.org/A000073)</span></div>
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<div class="line"><a id="l00328" name="l00328"></a><span class="lineno"> 328</span><span class="comment"> * [0 0 1 1]</span></div>
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<div class="line"><a id="l00329" name="l00329"></a><span class="lineno"> 329</span><span class="comment"> * [fn-3 fn-2 fn-1 s(f, n-1)] [1 0 1 1]</span></div>
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<div class="line"><a id="l00330" name="l00330"></a><span class="lineno"> 330</span><span class="comment"> * [0 1 1 1]</span></div>
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<div class="line"><a id="l00331" name="l00331"></a><span class="lineno"> 331</span><span class="comment"> * [0 0 0 1]</span></div>
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<div class="line"><a id="l00332" name="l00332"></a><span class="lineno"> 332</span><span class="comment"> *</span></div>
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<div class="line"><a id="l00333" name="l00333"></a><span class="lineno"> 333</span><span class="comment"> * = [fn-2, fn-1, fn-3 + fn-2 + fn-1, (fn-3 + fn-2 + fn-1 + s(f, n-1))]</span></div>
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<div class="line"><a id="l00334" name="l00334"></a><span class="lineno"> 334</span><span class="comment"> *</span></div>
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<div class="line"><a id="l00335" name="l00335"></a><span class="lineno"> 335</span><span class="comment"> * => [fn-2, fn-1, fn, fn + s(f, n-1)]</span></div>
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<div class="line"><a id="l00336" name="l00336"></a><span class="lineno"> 336</span><span class="comment"> *</span></div>
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<div class="line"><a id="l00337" name="l00337"></a><span class="lineno"> 337</span><span class="comment"> * => [fn-2, fn-1, fn, s(f, n)]</span></div>
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<div class="line"><a id="l00338" name="l00338"></a><span class="lineno"> 338</span><span class="comment"> *</span></div>
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<div class="line"><a id="l00339" name="l00339"></a><span class="lineno"> 339</span><span class="comment"> * Sum of the series is: 0, 0, 1, 2, 4, 8, 15, 28, 52, 96, 177, 326, 600,</span></div>
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<div class="line"><a id="l00340" name="l00340"></a><span class="lineno"> 340</span><span class="comment"> * 1104, 2031, 3736, 6872, 12640, 23249, 42762</span></div>
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<div class="line"><a id="l00341" name="l00341"></a><span class="lineno"> 341</span><span class="comment"> *</span></div>
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<div class="line"><a id="l00342" name="l00342"></a><span class="lineno"> 342</span><span class="comment"> * Let A = [fn-3 fn-2 fn-1 s(f, n-1)], and</span></div>
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<div class="line"><a id="l00343" name="l00343"></a><span class="lineno"> 343</span><span class="comment"> * [0 0 1 1]</span></div>
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<div class="line"><a id="l00344" name="l00344"></a><span class="lineno"> 344</span><span class="comment"> * B = [1 0 1 1]</span></div>
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<div class="line"><a id="l00345" name="l00345"></a><span class="lineno"> 345</span><span class="comment"> * [0 1 1 1]</span></div>
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<div class="line"><a id="l00346" name="l00346"></a><span class="lineno"> 346</span><span class="comment"> * [0 0 0 1]</span></div>
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<div class="line"><a id="l00347" name="l00347"></a><span class="lineno"> 347</span><span class="comment"> *</span></div>
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<div class="line"><a id="l00348" name="l00348"></a><span class="lineno"> 348</span><span class="comment"> * Since, A.B....(n-2 times) = [fn-2 fn-1 fn]</span></div>
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<div class="line"><a id="l00349" name="l00349"></a><span class="lineno"> 349</span><span class="comment"> * we will have multiply B with itself n-2 times to obtain the required</span></div>
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<div class="line"><a id="l00350" name="l00350"></a><span class="lineno"> 350</span><span class="comment"> * value</span></div>
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<div class="line"><a id="l00351" name="l00351"></a><span class="lineno"> 351</span><span class="comment"> */</span></div>
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<div class="line"><a id="l00352" name="l00352"></a><span class="lineno"> 352</span> </div>
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<div class="line"><a id="l00353" name="l00353"></a><span class="lineno"> 353</span> std::vector<std::vector<int64_t>> tribonacci_sum = {{0, 0, 1, 1},</div>
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<div class="line"><a id="l00354" name="l00354"></a><span class="lineno"> 354</span> {1, 0, 1, 1},</div>
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<div class="line"><a id="l00355" name="l00355"></a><span class="lineno"> 355</span> {0, 1, 1, 1},</div>
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<div class="line"><a id="l00356" name="l00356"></a><span class="lineno"> 356</span> {0, 0, 0, 1}},</div>
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<div class="line"><a id="l00357" name="l00357"></a><span class="lineno"> 357</span> trib_sum_base_case = {{0, 0, 1, 1}};</div>
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<div class="line"><a id="l00358" name="l00358"></a><span class="lineno"> 358</span> <span class="comment">// `f0 = 0, f1 = 0, f2 = 1, s = 1`</span></div>
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<div class="line"><a id="l00359" name="l00359"></a><span class="lineno"> 359</span> </div>
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<div class="line"><a id="l00360" name="l00360"></a><span class="lineno"> 360</span> assert(math::linear_recurrence_matrix::get_nth_term_of_recurrence_series(</div>
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<div class="line"><a id="l00361" name="l00361"></a><span class="lineno"> 361</span> tribonacci_sum, trib_sum_base_case, 18, 1) == 23249LL);</div>
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<div class="line"><a id="l00362" name="l00362"></a><span class="lineno"> 362</span> assert(math::linear_recurrence_matrix::get_nth_term_of_recurrence_series(</div>
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<div class="line"><a id="l00363" name="l00363"></a><span class="lineno"> 363</span> tribonacci_sum, trib_sum_base_case, 19, 1) == 42762LL);</div>
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<div class="line"><a id="l00364" name="l00364"></a><span class="lineno"> 364</span>}</div>
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<div class="line"><a id="l00365" name="l00365"></a><span class="lineno"> 365</span> </div>
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<div class="line"><a id="l00370" name="l00370"></a><span class="lineno"> 370</span><span class="keywordtype">int</span> <a class="code hl_function" href="../../dd/d1e/generate__parentheses_8cpp.html#gae66f6b31b5ad750f1fe042a706a4e3d4">main</a>() {</div>
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<div class="line"><a id="l00371" name="l00371"></a><span class="lineno"> 371</span> <a class="code hl_function" href="../../d6/d2c/caesar__cipher_8cpp.html#ae1a3968e7947464bee7714f6d43b7002">test</a>(); <span class="comment">// run self-test implementations</span></div>
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<div class="line"><a id="l00372" name="l00372"></a><span class="lineno"> 372</span> <span class="keywordflow">return</span> 0;</div>
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<div class="line"><a id="l00373" name="l00373"></a><span class="lineno"> 373</span>}</div>
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<div class="ttc" id="acaesar__cipher_8cpp_html_ae1a3968e7947464bee7714f6d43b7002"><div class="ttname"><a href="../../d6/d2c/caesar__cipher_8cpp.html#ae1a3968e7947464bee7714f6d43b7002">test</a></div><div class="ttdeci">void test()</div><div class="ttdef"><b>Definition</b> <a href="../../d6/d2c/caesar__cipher_8cpp_source.html#l00100">caesar_cipher.cpp:100</a></div></div>
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<div class="ttc" id="acomposite__simpson__rule_8cpp_html_a1b74d828b33760094906797042b89442"><div class="ttname"><a href="../../d4/d18/composite__simpson__rule_8cpp.html#a1b74d828b33760094906797042b89442">numerical_methods::simpson_method::k</a></div><div class="ttdeci">double k(double x)</div><div class="ttdoc">Another test function.</div><div class="ttdef"><b>Definition</b> <a href="../../d4/d18/composite__simpson__rule_8cpp_source.html#l00117">composite_simpson_rule.cpp:117</a></div></div>
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<div class="ttc" id="agenerate__parentheses_8cpp_html_gae66f6b31b5ad750f1fe042a706a4e3d4"><div class="ttname"><a href="../../dd/d1e/generate__parentheses_8cpp.html#gae66f6b31b5ad750f1fe042a706a4e3d4">main</a></div><div class="ttdeci">int main()</div><div class="ttdoc">Main function.</div><div class="ttdef"><b>Definition</b> <a href="../../dd/d1e/generate__parentheses_8cpp_source.html#l00110">generate_parentheses.cpp:110</a></div></div>
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<div class="ttc" id="anamespacelinear__recurrence__matrix_html"><div class="ttname"><a href="../../d9/dd1/namespacelinear__recurrence__matrix.html">linear_recurrence_matrix</a></div><div class="ttdoc">Functions for Linear Recurrence Matrix implementation.</div></div>
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<div class="ttc" id="anamespacemath_html"><div class="ttname"><a href="../../dd/d47/namespacemath.html">math</a></div><div class="ttdoc">for assert</div></div>
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<div class="ttc" id="anamespacemath_html_afcd07701d73ed65cd616bcba02737f3d"><div class="ttname"><a href="../../dd/d47/namespacemath.html#afcd07701d73ed65cd616bcba02737f3d">math::power</a></div><div class="ttdeci">uint64_t power(uint64_t a, uint64_t b, uint64_t c)</div><div class="ttdoc">This function calculates a raised to exponent b under modulo c using modular exponentiation.</div><div class="ttdef"><b>Definition</b> <a href="../../d0/d6d/modular__exponentiation_8cpp_source.html#l00035">modular_exponentiation.cpp:35</a></div></div>
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