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<div class="headertitle"><div class="title">ode_semi_implicit_euler.cpp</div></div>
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<div class="contents">
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<a href="../../d3/d06/ode__semi__implicit__euler_8cpp.html">Go to the documentation of this file.</a><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="l00037" name="l00037"></a><span class="lineno"> 37</span> </div>
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<div class="line"><a id="l00038" name="l00038"></a><span class="lineno"> 38</span><span class="preprocessor">#include <cmath></span></div>
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<div class="line"><a id="l00039" name="l00039"></a><span class="lineno"> 39</span><span class="preprocessor">#include <ctime></span></div>
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<div class="line"><a id="l00040" name="l00040"></a><span class="lineno"> 40</span><span class="preprocessor">#include <fstream></span></div>
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<div class="line"><a id="l00041" name="l00041"></a><span class="lineno"> 41</span><span class="preprocessor">#include <iostream></span></div>
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<div class="line"><a id="l00042" name="l00042"></a><span class="lineno"> 42</span><span class="preprocessor">#include <valarray></span></div>
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<div class="line"><a id="l00043" name="l00043"></a><span class="lineno"> 43</span></div>
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<div class="foldopen" id="foldopen00053" data-start="{" data-end="}">
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<div class="line"><a id="l00053" name="l00053"></a><span class="lineno"><a class="line" href="../../d3/d06/ode__semi__implicit__euler_8cpp.html#abaeae8f62a018d197f0187a1c80a90fe"> 53</a></span><span class="keywordtype">void</span> <a class="code hl_function" href="../../d3/d06/ode__semi__implicit__euler_8cpp.html#abaeae8f62a018d197f0187a1c80a90fe">problem</a>(<span class="keyword">const</span> <span class="keywordtype">double</span> &x, std::valarray<double> *y,</div>
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<div class="line"><a id="l00054" name="l00054"></a><span class="lineno"> 54</span> std::valarray<double> *dy) {</div>
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<div class="line"><a id="l00055" name="l00055"></a><span class="lineno"> 55</span> <span class="keyword">const</span> <span class="keywordtype">double</span> omega = 1.F; <span class="comment">// some const for the problem</span></div>
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<div class="line"><a id="l00056" name="l00056"></a><span class="lineno"> 56</span> dy[0][0] = y[0][1]; <span class="comment">// x dot</span></div>
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<div class="line"><a id="l00057" name="l00057"></a><span class="lineno"> 57</span> dy[0][1] = -omega * omega * y[0][0]; <span class="comment">// y dot</span></div>
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<div class="line"><a id="l00058" name="l00058"></a><span class="lineno"> 58</span>}</div>
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</div>
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<div class="line"><a id="l00059" name="l00059"></a><span class="lineno"> 59</span></div>
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<div class="foldopen" id="foldopen00066" data-start="{" data-end="}">
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<div class="line"><a id="l00066" name="l00066"></a><span class="lineno"><a class="line" href="../../d3/d06/ode__semi__implicit__euler_8cpp.html#af3adf7b092a87868917ee5fb4255192b"> 66</a></span><span class="keywordtype">void</span> <a class="code hl_function" href="../../d3/d06/ode__semi__implicit__euler_8cpp.html#af3adf7b092a87868917ee5fb4255192b">exact_solution</a>(<span class="keyword">const</span> <span class="keywordtype">double</span> &x, std::valarray<double> *y) {</div>
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<div class="line"><a id="l00067" name="l00067"></a><span class="lineno"> 67</span> y[0][0] = std::cos(x);</div>
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<div class="line"><a id="l00068" name="l00068"></a><span class="lineno"> 68</span> y[0][1] = -std::sin(x);</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>
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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="foldopen" id="foldopen00082" data-start="{" data-end="}">
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<div class="line"><a id="l00082" name="l00082"></a><span class="lineno"><a class="line" href="../../d6/d60/group__ode.html#ga827bf009831ddc477c5fa8891d5cb35f"> 82</a></span><span class="keywordtype">void</span> <a class="code hl_function" href="../../d6/d60/group__ode.html#ga827bf009831ddc477c5fa8891d5cb35f">semi_implicit_euler_step</a>(<span class="keyword">const</span> <span class="keywordtype">double</span> dx, <span class="keyword">const</span> <span class="keywordtype">double</span> &x,</div>
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<div class="line"><a id="l00083" name="l00083"></a><span class="lineno"> 83</span> std::valarray<double> *y,</div>
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<div class="line"><a id="l00084" name="l00084"></a><span class="lineno"> 84</span> std::valarray<double> *dy) {</div>
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<div class="line"><a id="l00085" name="l00085"></a><span class="lineno"> 85</span> <a class="code hl_function" href="../../d3/d06/ode__semi__implicit__euler_8cpp.html#abaeae8f62a018d197f0187a1c80a90fe">problem</a>(x, y, dy); <span class="comment">// update dy once</span></div>
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<div class="line"><a id="l00086" name="l00086"></a><span class="lineno"> 86</span> y[0][0] += dx * dy[0][0]; <span class="comment">// update y0</span></div>
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<div class="line"><a id="l00087" name="l00087"></a><span class="lineno"> 87</span> <a class="code hl_function" href="../../d3/d06/ode__semi__implicit__euler_8cpp.html#abaeae8f62a018d197f0187a1c80a90fe">problem</a>(x, y, dy); <span class="comment">// update dy once more</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> dy[0][0] = 0.f; <span class="comment">// ignore y0</span></div>
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<div class="line"><a id="l00090" name="l00090"></a><span class="lineno"> 90</span> y[0] += dy[0] * dx; <span class="comment">// update remaining using new dy</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>
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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="foldopen" id="foldopen00103" data-start="{" data-end="}">
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<div class="line"><a id="l00103" name="l00103"></a><span class="lineno"><a class="line" href="../../d6/d60/group__ode.html#ga3874d294ab00fe9ce8731c5b7991a089"> 103</a></span><span class="keywordtype">double</span> <a class="code hl_function" href="../../d6/d60/group__ode.html#ga3874d294ab00fe9ce8731c5b7991a089">semi_implicit_euler</a>(<span class="keywordtype">double</span> dx, <span class="keywordtype">double</span> x0, <span class="keywordtype">double</span> x_max,</div>
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<div class="line"><a id="l00104" name="l00104"></a><span class="lineno"> 104</span> std::valarray<double> *y,</div>
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<div class="line"><a id="l00105" name="l00105"></a><span class="lineno"> 105</span> <span class="keywordtype">bool</span> save_to_file = <span class="keyword">false</span>) {</div>
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<div class="line"><a id="l00106" name="l00106"></a><span class="lineno"> 106</span> std::valarray<double> dy = y[0];</div>
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<div class="line"><a id="l00107" name="l00107"></a><span class="lineno"> 107</span> </div>
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<div class="line"><a id="l00108" name="l00108"></a><span class="lineno"> 108</span> std::ofstream fp;</div>
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<div class="line"><a id="l00109" name="l00109"></a><span class="lineno"> 109</span> <span class="keywordflow">if</span> (save_to_file) {</div>
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<div class="line"><a id="l00110" name="l00110"></a><span class="lineno"> 110</span> fp.open(<span class="stringliteral">"semi_implicit_euler.csv"</span>, std::ofstream::out);</div>
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<div class="line"><a id="l00111" name="l00111"></a><span class="lineno"> 111</span> <span class="keywordflow">if</span> (!fp.is_open()) {</div>
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<div class="line"><a id="l00112" name="l00112"></a><span class="lineno"> 112</span> std::perror(<span class="stringliteral">"Error! "</span>);</div>
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<div class="line"><a id="l00113" name="l00113"></a><span class="lineno"> 113</span> }</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::size_t L = y->size();</div>
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<div class="line"><a id="l00117" name="l00117"></a><span class="lineno"> 117</span> </div>
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<div class="line"><a id="l00118" name="l00118"></a><span class="lineno"> 118</span> <span class="comment">/* start integration */</span></div>
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<div class="line"><a id="l00119" name="l00119"></a><span class="lineno"> 119</span> std::clock_t t1 = std::clock();</div>
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<div class="line"><a id="l00120" name="l00120"></a><span class="lineno"> 120</span> <span class="keywordtype">double</span> x = x0;</div>
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<div class="line"><a id="l00121" name="l00121"></a><span class="lineno"> 121</span> <span class="keywordflow">do</span> { <span class="comment">// iterate for each step of independent variable</span></div>
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<div class="line"><a id="l00122" name="l00122"></a><span class="lineno"> 122</span> <span class="keywordflow">if</span> (save_to_file && fp.is_open()) {</div>
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<div class="line"><a id="l00123" name="l00123"></a><span class="lineno"> 123</span> <span class="comment">// write to file</span></div>
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<div class="line"><a id="l00124" name="l00124"></a><span class="lineno"> 124</span> fp << x << <span class="stringliteral">","</span>;</div>
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<div class="line"><a id="l00125" name="l00125"></a><span class="lineno"> 125</span> <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i < L - 1; i++) {</div>
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<div class="line"><a id="l00126" name="l00126"></a><span class="lineno"> 126</span> fp << y[0][i] << <span class="stringliteral">","</span>;</div>
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<div class="line"><a id="l00127" name="l00127"></a><span class="lineno"> 127</span> }</div>
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<div class="line"><a id="l00128" name="l00128"></a><span class="lineno"> 128</span> fp << y[0][L - 1] << <span class="stringliteral">"\n"</span>;</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> <a class="code hl_function" href="../../d6/d60/group__ode.html#ga827bf009831ddc477c5fa8891d5cb35f">semi_implicit_euler_step</a>(dx, x, y, &dy); <span class="comment">// perform integration</span></div>
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<div class="line"><a id="l00132" name="l00132"></a><span class="lineno"> 132</span> x += dx; <span class="comment">// update step</span></div>
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<div class="line"><a id="l00133" name="l00133"></a><span class="lineno"> 133</span> } <span class="keywordflow">while</span> (x <= x_max); <span class="comment">// till upper limit of independent variable</span></div>
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<div class="line"><a id="l00134" name="l00134"></a><span class="lineno"> 134</span> <span class="comment">/* end of integration */</span></div>
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<div class="line"><a id="l00135" name="l00135"></a><span class="lineno"> 135</span> std::clock_t t2 = std::clock();</div>
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<div class="line"><a id="l00136" name="l00136"></a><span class="lineno"> 136</span> </div>
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<div class="line"><a id="l00137" name="l00137"></a><span class="lineno"> 137</span> <span class="keywordflow">if</span> (fp.is_open())</div>
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<div class="line"><a id="l00138" name="l00138"></a><span class="lineno"> 138</span> fp.close();</div>
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<div class="line"><a id="l00139" name="l00139"></a><span class="lineno"> 139</span> </div>
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<div class="line"><a id="l00140" name="l00140"></a><span class="lineno"> 140</span> <span class="keywordflow">return</span> <span class="keyword">static_cast<</span><span class="keywordtype">double</span><span class="keyword">></span>(t2 - t1) / CLOCKS_PER_SEC;</div>
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<div class="line"><a id="l00141" name="l00141"></a><span class="lineno"> 141</span>}</div>
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</div>
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<div class="line"><a id="l00142" name="l00142"></a><span class="lineno"> 142</span></div>
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<div class="line"><a id="l00144" name="l00144"></a><span class="lineno"> 144</span></div>
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<div class="foldopen" id="foldopen00153" data-start="{" data-end="}">
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<div class="line"><a id="l00153" name="l00153"></a><span class="lineno"><a class="line" href="../../d3/d06/ode__semi__implicit__euler_8cpp.html#aa13517b8e5de1b75592052db7f7e237f"> 153</a></span><span class="keywordtype">void</span> <a class="code hl_function" href="../../d3/d06/ode__semi__implicit__euler_8cpp.html#aa13517b8e5de1b75592052db7f7e237f">save_exact_solution</a>(<span class="keyword">const</span> <span class="keywordtype">double</span> &X0, <span class="keyword">const</span> <span class="keywordtype">double</span> &X_MAX,</div>
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<div class="line"><a id="l00154" name="l00154"></a><span class="lineno"> 154</span> <span class="keyword">const</span> <span class="keywordtype">double</span> &step_size,</div>
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<div class="line"><a id="l00155" name="l00155"></a><span class="lineno"> 155</span> <span class="keyword">const</span> std::valarray<double> &Y0) {</div>
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<div class="line"><a id="l00156" name="l00156"></a><span class="lineno"> 156</span> <span class="keywordtype">double</span> x = X0;</div>
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<div class="line"><a id="l00157" name="l00157"></a><span class="lineno"> 157</span> std::valarray<double> y = Y0;</div>
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<div class="line"><a id="l00158" name="l00158"></a><span class="lineno"> 158</span> </div>
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<div class="line"><a id="l00159" name="l00159"></a><span class="lineno"> 159</span> std::ofstream fp(<span class="stringliteral">"exact.csv"</span>, std::ostream::out);</div>
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<div class="line"><a id="l00160" name="l00160"></a><span class="lineno"> 160</span> <span class="keywordflow">if</span> (!fp.is_open()) {</div>
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<div class="line"><a id="l00161" name="l00161"></a><span class="lineno"> 161</span> std::perror(<span class="stringliteral">"Error! "</span>);</div>
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<div class="line"><a id="l00162" name="l00162"></a><span class="lineno"> 162</span> <span class="keywordflow">return</span>;</div>
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<div class="line"><a id="l00163" name="l00163"></a><span class="lineno"> 163</span> }</div>
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<div class="line"><a id="l00164" name="l00164"></a><span class="lineno"> 164</span> std::cout << <span class="stringliteral">"Finding exact solution\n"</span>;</div>
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<div class="line"><a id="l00165" name="l00165"></a><span class="lineno"> 165</span> </div>
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<div class="line"><a id="l00166" name="l00166"></a><span class="lineno"> 166</span> std::clock_t t1 = std::clock();</div>
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<div class="line"><a id="l00167" name="l00167"></a><span class="lineno"> 167</span> <span class="keywordflow">do</span> {</div>
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<div class="line"><a id="l00168" name="l00168"></a><span class="lineno"> 168</span> fp << x << <span class="stringliteral">","</span>;</div>
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<div class="line"><a id="l00169" name="l00169"></a><span class="lineno"> 169</span> <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i < y.size() - 1; i++) {</div>
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<div class="line"><a id="l00170" name="l00170"></a><span class="lineno"> 170</span> fp << y[i] << <span class="stringliteral">","</span>;</div>
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<div class="line"><a id="l00171" name="l00171"></a><span class="lineno"> 171</span> }</div>
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<div class="line"><a id="l00172" name="l00172"></a><span class="lineno"> 172</span> fp << y[y.size() - 1] << <span class="stringliteral">"\n"</span>;</div>
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<div class="line"><a id="l00173" name="l00173"></a><span class="lineno"> 173</span> </div>
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<div class="line"><a id="l00174" name="l00174"></a><span class="lineno"> 174</span> <a class="code hl_function" href="../../d3/d06/ode__semi__implicit__euler_8cpp.html#af3adf7b092a87868917ee5fb4255192b">exact_solution</a>(x, &y);</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="l00176" name="l00176"></a><span class="lineno"> 176</span> x += step_size;</div>
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<div class="line"><a id="l00177" name="l00177"></a><span class="lineno"> 177</span> } <span class="keywordflow">while</span> (x <= X_MAX);</div>
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<div class="line"><a id="l00178" name="l00178"></a><span class="lineno"> 178</span> </div>
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<div class="line"><a id="l00179" name="l00179"></a><span class="lineno"> 179</span> std::clock_t t2 = std::clock();</div>
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<div class="line"><a id="l00180" name="l00180"></a><span class="lineno"> 180</span> <span class="keywordtype">double</span> total_time = <span class="keyword">static_cast<</span><span class="keywordtype">double</span><span class="keyword">></span>(t2 - t1) / CLOCKS_PER_SEC;</div>
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<div class="line"><a id="l00181" name="l00181"></a><span class="lineno"> 181</span> std::cout << <span class="stringliteral">"\tTime = "</span> << total_time << <span class="stringliteral">" ms\n"</span>;</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> fp.close();</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>
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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="foldopen" id="foldopen00189" data-start="{" data-end="}">
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<div class="line"><a id="l00189" name="l00189"></a><span class="lineno"><a class="line" href="../../d3/d06/ode__semi__implicit__euler_8cpp.html#a0ddf1224851353fc92bfbff6f499fa97"> 189</a></span><span class="keywordtype">int</span> <a class="code hl_function" href="../../dd/d1e/generate__parentheses_8cpp.html#gae66f6b31b5ad750f1fe042a706a4e3d4">main</a>(<span class="keywordtype">int</span> argc, <span class="keywordtype">char</span> *argv[]) {</div>
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<div class="line"><a id="l00190" name="l00190"></a><span class="lineno"> 190</span> <span class="keywordtype">double</span> X0 = 0.f; <span class="comment">/* initial value of x0 */</span></div>
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<div class="line"><a id="l00191" name="l00191"></a><span class="lineno"> 191</span> <span class="keywordtype">double</span> X_MAX = 10.F; <span class="comment">/* upper limit of integration */</span></div>
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<div class="line"><a id="l00192" name="l00192"></a><span class="lineno"> 192</span> std::valarray<double> Y0 = {1.f, 0.f}; <span class="comment">/* initial value Y = y(x = x_0) */</span></div>
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<div class="line"><a id="l00193" name="l00193"></a><span class="lineno"> 193</span> <span class="keywordtype">double</span> step_size;</div>
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<div class="line"><a id="l00194" name="l00194"></a><span class="lineno"> 194</span> </div>
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<div class="line"><a id="l00195" name="l00195"></a><span class="lineno"> 195</span> <span class="keywordflow">if</span> (argc == 1) {</div>
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<div class="line"><a id="l00196" name="l00196"></a><span class="lineno"> 196</span> std::cout << <span class="stringliteral">"\nEnter the step size: "</span>;</div>
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<div class="line"><a id="l00197" name="l00197"></a><span class="lineno"> 197</span> std::cin >> step_size;</div>
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<div class="line"><a id="l00198" name="l00198"></a><span class="lineno"> 198</span> } <span class="keywordflow">else</span> {</div>
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<div class="line"><a id="l00199" name="l00199"></a><span class="lineno"> 199</span> <span class="comment">// use commandline argument as independent variable step size</span></div>
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<div class="line"><a id="l00200" name="l00200"></a><span class="lineno"> 200</span> step_size = std::atof(argv[1]);</div>
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<div class="line"><a id="l00201" name="l00201"></a><span class="lineno"> 201</span> }</div>
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<div class="line"><a id="l00202" name="l00202"></a><span class="lineno"> 202</span> </div>
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<div class="line"><a id="l00203" name="l00203"></a><span class="lineno"> 203</span> <span class="comment">// get approximate solution</span></div>
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<div class="line"><a id="l00204" name="l00204"></a><span class="lineno"> 204</span> <span class="keywordtype">double</span> total_time = <a class="code hl_function" href="../../d6/d60/group__ode.html#ga3874d294ab00fe9ce8731c5b7991a089">semi_implicit_euler</a>(step_size, X0, X_MAX, &Y0, <span class="keyword">true</span>);</div>
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<div class="line"><a id="l00205" name="l00205"></a><span class="lineno"> 205</span> std::cout << <span class="stringliteral">"\tTime = "</span> << total_time << <span class="stringliteral">" ms\n"</span>;</div>
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<div class="line"><a id="l00206" name="l00206"></a><span class="lineno"> 206</span> </div>
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<div class="line"><a id="l00207" name="l00207"></a><span class="lineno"> 207</span> <span class="comment">/* compute exact solution for comparion */</span></div>
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<div class="line"><a id="l00208" name="l00208"></a><span class="lineno"> 208</span> <a class="code hl_function" href="../../d3/d06/ode__semi__implicit__euler_8cpp.html#aa13517b8e5de1b75592052db7f7e237f">save_exact_solution</a>(X0, X_MAX, step_size, Y0);</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> <span class="keywordflow">return</span> 0;</div>
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<div class="line"><a id="l00211" name="l00211"></a><span class="lineno"> 211</span>}</div>
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</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="agroup__ode_html_ga3874d294ab00fe9ce8731c5b7991a089"><div class="ttname"><a href="../../d6/d60/group__ode.html#ga3874d294ab00fe9ce8731c5b7991a089">semi_implicit_euler</a></div><div class="ttdeci">double semi_implicit_euler(double dx, double x0, double x_max, std::valarray< double > *y, bool save_to_file=false)</div><div class="ttdoc">Compute approximation using the semi-implicit-Euler method in the given limits.</div><div class="ttdef"><b>Definition</b> <a href="#l00103">ode_semi_implicit_euler.cpp:103</a></div></div>
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<div class="ttc" id="agroup__ode_html_ga827bf009831ddc477c5fa8891d5cb35f"><div class="ttname"><a href="../../d6/d60/group__ode.html#ga827bf009831ddc477c5fa8891d5cb35f">semi_implicit_euler_step</a></div><div class="ttdeci">void semi_implicit_euler_step(const double dx, const double &x, std::valarray< double > *y, std::valarray< double > *dy)</div><div class="ttdoc">Compute next step approximation using the semi-implicit-Euler method.</div><div class="ttdef"><b>Definition</b> <a href="#l00082">ode_semi_implicit_euler.cpp:82</a></div></div>
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<div class="ttc" id="aode__semi__implicit__euler_8cpp_html_aa13517b8e5de1b75592052db7f7e237f"><div class="ttname"><a href="../../d3/d06/ode__semi__implicit__euler_8cpp.html#aa13517b8e5de1b75592052db7f7e237f">save_exact_solution</a></div><div class="ttdeci">void save_exact_solution(const double &X0, const double &X_MAX, const double &step_size, const std::valarray< double > &Y0)</div><div class="ttdef"><b>Definition</b> <a href="#l00153">ode_semi_implicit_euler.cpp:153</a></div></div>
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<div class="ttc" id="aode__semi__implicit__euler_8cpp_html_abaeae8f62a018d197f0187a1c80a90fe"><div class="ttname"><a href="../../d3/d06/ode__semi__implicit__euler_8cpp.html#abaeae8f62a018d197f0187a1c80a90fe">problem</a></div><div class="ttdeci">void problem(const double &x, std::valarray< double > *y, std::valarray< double > *dy)</div><div class="ttdoc">Problem statement for a system with first-order differential equations. Updates the system differenti...</div><div class="ttdef"><b>Definition</b> <a href="#l00053">ode_semi_implicit_euler.cpp:53</a></div></div>
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<div class="ttc" id="aode__semi__implicit__euler_8cpp_html_af3adf7b092a87868917ee5fb4255192b"><div class="ttname"><a href="../../d3/d06/ode__semi__implicit__euler_8cpp.html#af3adf7b092a87868917ee5fb4255192b">exact_solution</a></div><div class="ttdeci">void exact_solution(const double &x, std::valarray< double > *y)</div><div class="ttdoc">Exact solution of the problem. Used for solution comparison.</div><div class="ttdef"><b>Definition</b> <a href="#l00066">ode_semi_implicit_euler.cpp:66</a></div></div>
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