Update README

Janitaws/finish text document

README.md

@@ -4,41 +4,48 @@
 
 - [Project](https://anderkve.github.io/FYS3150/book/projects/project2.html)
 
+## Requirements
+
+To compile the C++ programs, you need to have these packages installed:
+
+- Armadillo
+- Openmp
+
+To run the Python scripts, you need to have these packages:
+
+- numpy
+- matplotlib
+- pandas
+- seaborn
+
 ## Compile and run C++ programs
 
-Compiling and linking is done using Make, make sure you are in the root folder. To create executable files, run shell commands 
+Compiling and linking is done using Make, make sure you are in the root directory. 
 
-```shell
-cd src/
-make
-```
-To run `script-name` 
-
-```shell
-./main
-```
-
-There are two ways of compiling the code.
-
-The first way is to go into the src directory and compile
+There are two alternative ways to compile the code. The first alternative is to change 
+directory to `src/` and compile
 
 ```shell
     cd src && make
 ```
 
-or you could use make in the project directory
+The second alternative does not involve changing directory to `src/`, use 
+make in the projects root directory
 
 ```shell
     make code
 ```
 
-and it will compile everything with you needing to move into **src**.
-
-You can run any of the compiled programs by going into the src directory
-and then doing
+You can run any of the compiled programs by changing directory to `src/`, then
 
 ```shell
-    ./program-name
+    ./test_suite
+    ./main
+```
+
+Remove object files and executables from `src/`
+```shell
+    make clean
 ```
 
 
@@ -64,7 +71,7 @@ If you want to generate the documentation you can do
     make docs
 ```
 
-in the project root, and the documentation will be made into the **docs** 
+in the project root, and the documentation will be made into the `docs/` 
 directory.
 
 ## Generate project document
@@ -75,4 +82,4 @@ If you want to recompile the Pdf file, you can do
     make latex
 ```
 
-and a Pdf file will be produced inside the **latex** directory.
+and a Pdf file will be produced inside the `latex/` directory.

doc/html/main_8cpp.html

@@ -79,7 +79,10 @@ $(function() {
 <a href="#details">More...</a></p>
 <div class="textblock"><code>#include &lt;cassert&gt;</code><br />
 <code>#include &lt;cmath&gt;</code><br />
+<code>#include &lt;ctime&gt;</code><br />
 <code>#include &lt;iostream&gt;</code><br />
+<code>#include &lt;omp.h&gt;</code><br />
+<code>#include &lt;ostream&gt;</code><br />
 <code>#include &quot;<a class="el" href="utils_8hpp_source.html">utils.hpp</a>&quot;</code><br />
 <code>#include &quot;<a class="el" href="matrix_8hpp_source.html">matrix.hpp</a>&quot;</code><br />
 <code>#include &quot;<a class="el" href="jacobi_8hpp_source.html">jacobi.hpp</a>&quot;</code><br />
@@ -101,7 +104,7 @@ int&#160;</td><td class="memItemRight" valign="bottom"><b>main</b> ()</td></tr>
 </table>
 <a name="details" id="details"></a><h2 class="groupheader">Detailed Description</h2>
 <div class="textblock"><p>Main program for Project 2. </p>
-<p>The program performs the Jacobi rotation method. The size of the matrix, and number of transformations performed are written to file. Eigenvector correstonding to the 3 smallest eigenvalues for matrices of size 6x6 and 100x100 are written to file.</p>
+<p>The program performs the Jacobi rotation method. The size of the matrix, and number of transformations performed are written to file. Eigenvector corresponding to the 3 smallest eigenvalues for matrices of size 6x6 and 100x100 are written to file.</p>
 <dl class="section author"><dt>Author</dt><dd>Cory Alexander Balaton (coryab) </dd>
 <dd>
 Janita Ovidie Sandtrøen Willumsen (janitaws) </dd></dl>

doc/html/matrix_8hpp.html

@@ -82,6 +82,7 @@ $(function() {
 <p>Function prototypes for creating tridiagonal matrices.  
 <a href="#details">More...</a></p>
 <div class="textblock"><code>#include &lt;armadillo&gt;</code><br />
+<code>#include &lt;omp.h&gt;</code><br />
 </div>
 <p><a href="matrix_8hpp_source.html">Go to the source code of this file.</a></p>
 <table class="memberdecls">

doc/html/matrix_8hpp_source.html

@@ -81,22 +81,23 @@ $(function() {
 <div class="line"><a id="l00011" name="l00011"></a><span class="lineno">   11</span><span class="preprocessor">#define __MATRIX__</span></div>
 <div class="line"><a id="l00012" name="l00012"></a><span class="lineno">   12</span> </div>
 <div class="line"><a id="l00013" name="l00013"></a><span class="lineno">   13</span><span class="preprocessor">#include &lt;armadillo&gt;</span></div>
-<div class="line"><a id="l00014" name="l00014"></a><span class="lineno">   14</span> </div>
-<div class="line"><a id="l00028" name="l00028"></a><span class="lineno">   28</span>arma::mat <a class="code hl_function" href="matrix_8hpp.html#aa524feaf9f44790470df4cb99bb714af">create_tridiagonal</a>(</div>
-<div class="line"><a id="l00029" name="l00029"></a><span class="lineno">   29</span>    <span class="keyword">const</span> arma::vec&amp; a, </div>
-<div class="line"><a id="l00030" name="l00030"></a><span class="lineno">   30</span>    <span class="keyword">const</span> arma::vec&amp; d, </div>
-<div class="line"><a id="l00031" name="l00031"></a><span class="lineno">   31</span>    <span class="keyword">const</span> arma::vec&amp; e);</div>
-<div class="line"><a id="l00032" name="l00032"></a><span class="lineno">   32</span> </div>
-<div class="line"><a id="l00047" name="l00047"></a><span class="lineno">   47</span>arma::mat <a class="code hl_function" href="matrix_8hpp.html#aa524feaf9f44790470df4cb99bb714af">create_tridiagonal</a>(<span class="keywordtype">int</span> n, <span class="keywordtype">double</span> a, <span class="keywordtype">double</span> d, <span class="keywordtype">double</span> e);</div>
-<div class="line"><a id="l00048" name="l00048"></a><span class="lineno">   48</span> </div>
-<div class="line"><a id="l00060" name="l00060"></a><span class="lineno">   60</span>arma::mat <a class="code hl_function" href="matrix_8hpp.html#a564688c96a8e4282e995a0663545d627">create_symmetric_tridiagonal</a>(<span class="keywordtype">int</span> n, <span class="keywordtype">double</span> a, <span class="keywordtype">double</span> d);</div>
-<div class="line"><a id="l00061" name="l00061"></a><span class="lineno">   61</span> </div>
-<div class="line"><a id="l00075" name="l00075"></a><span class="lineno">   75</span><span class="keywordtype">double</span> <a class="code hl_function" href="matrix_8hpp.html#af42e501cbf71bec9c77ececc22723df9">max_offdiag_symmetric</a>(arma::mat&amp; A, <span class="keywordtype">int</span>&amp; k, <span class="keywordtype">int</span>&amp; l);</div>
-<div class="line"><a id="l00076" name="l00076"></a><span class="lineno">   76</span> </div>
-<div class="line"><a id="l00077" name="l00077"></a><span class="lineno">   77</span><span class="preprocessor">#endif</span></div>
-<div class="ttc" id="amatrix_8hpp_html_a564688c96a8e4282e995a0663545d627"><div class="ttname"><a href="matrix_8hpp.html#a564688c96a8e4282e995a0663545d627">create_symmetric_tridiagonal</a></div><div class="ttdeci">arma::mat create_symmetric_tridiagonal(int n, double a, double d)</div><div class="ttdoc">Create a symmetric tridiagonal matrix.</div><div class="ttdef"><b>Definition:</b> matrix.cpp:44</div></div>
+<div class="line"><a id="l00014" name="l00014"></a><span class="lineno">   14</span><span class="preprocessor">#include &lt;omp.h&gt;</span></div>
+<div class="line"><a id="l00015" name="l00015"></a><span class="lineno">   15</span> </div>
+<div class="line"><a id="l00029" name="l00029"></a><span class="lineno">   29</span>arma::mat <a class="code hl_function" href="matrix_8hpp.html#aa524feaf9f44790470df4cb99bb714af">create_tridiagonal</a>(</div>
+<div class="line"><a id="l00030" name="l00030"></a><span class="lineno">   30</span>    <span class="keyword">const</span> arma::vec&amp; a, </div>
+<div class="line"><a id="l00031" name="l00031"></a><span class="lineno">   31</span>    <span class="keyword">const</span> arma::vec&amp; d, </div>
+<div class="line"><a id="l00032" name="l00032"></a><span class="lineno">   32</span>    <span class="keyword">const</span> arma::vec&amp; e);</div>
+<div class="line"><a id="l00033" name="l00033"></a><span class="lineno">   33</span> </div>
+<div class="line"><a id="l00048" name="l00048"></a><span class="lineno">   48</span>arma::mat <a class="code hl_function" href="matrix_8hpp.html#aa524feaf9f44790470df4cb99bb714af">create_tridiagonal</a>(<span class="keywordtype">int</span> n, <span class="keywordtype">double</span> a, <span class="keywordtype">double</span> d, <span class="keywordtype">double</span> e);</div>
+<div class="line"><a id="l00049" name="l00049"></a><span class="lineno">   49</span> </div>
+<div class="line"><a id="l00061" name="l00061"></a><span class="lineno">   61</span>arma::mat <a class="code hl_function" href="matrix_8hpp.html#a564688c96a8e4282e995a0663545d627">create_symmetric_tridiagonal</a>(<span class="keywordtype">int</span> n, <span class="keywordtype">double</span> a, <span class="keywordtype">double</span> d);</div>
+<div class="line"><a id="l00062" name="l00062"></a><span class="lineno">   62</span> </div>
+<div class="line"><a id="l00076" name="l00076"></a><span class="lineno">   76</span><span class="keywordtype">double</span> <a class="code hl_function" href="matrix_8hpp.html#af42e501cbf71bec9c77ececc22723df9">max_offdiag_symmetric</a>(arma::mat&amp; A, <span class="keywordtype">int</span>&amp; k, <span class="keywordtype">int</span>&amp; l);</div>
+<div class="line"><a id="l00077" name="l00077"></a><span class="lineno">   77</span> </div>
+<div class="line"><a id="l00078" name="l00078"></a><span class="lineno">   78</span><span class="preprocessor">#endif</span></div>
+<div class="ttc" id="amatrix_8hpp_html_a564688c96a8e4282e995a0663545d627"><div class="ttname"><a href="matrix_8hpp.html#a564688c96a8e4282e995a0663545d627">create_symmetric_tridiagonal</a></div><div class="ttdeci">arma::mat create_symmetric_tridiagonal(int n, double a, double d)</div><div class="ttdoc">Create a symmetric tridiagonal matrix.</div><div class="ttdef"><b>Definition:</b> matrix.cpp:45</div></div>
 <div class="ttc" id="amatrix_8hpp_html_aa524feaf9f44790470df4cb99bb714af"><div class="ttname"><a href="matrix_8hpp.html#aa524feaf9f44790470df4cb99bb714af">create_tridiagonal</a></div><div class="ttdeci">arma::mat create_tridiagonal(const arma::vec &amp;a, const arma::vec &amp;d, const arma::vec &amp;e)</div><div class="ttdoc">Create a tridiagonal matrix.</div><div class="ttdef"><b>Definition:</b> matrix.cpp:12</div></div>
-<div class="ttc" id="amatrix_8hpp_html_af42e501cbf71bec9c77ececc22723df9"><div class="ttname"><a href="matrix_8hpp.html#af42e501cbf71bec9c77ececc22723df9">max_offdiag_symmetric</a></div><div class="ttdeci">double max_offdiag_symmetric(arma::mat &amp;A, int &amp;k, int &amp;l)</div><div class="ttdoc">Find the off-diagonal element with the largest absolute value.</div><div class="ttdef"><b>Definition:</b> matrix.cpp:49</div></div>
+<div class="ttc" id="amatrix_8hpp_html_af42e501cbf71bec9c77ececc22723df9"><div class="ttname"><a href="matrix_8hpp.html#af42e501cbf71bec9c77ececc22723df9">max_offdiag_symmetric</a></div><div class="ttdeci">double max_offdiag_symmetric(arma::mat &amp;A, int &amp;k, int &amp;l)</div><div class="ttdoc">Find the off-diagonal element with the largest absolute value.</div><div class="ttdef"><b>Definition:</b> matrix.cpp:50</div></div>
 </div><!-- fragment --></div><!-- contents -->
 <!-- start footer part -->
 <hr class="footer"/><address class="footer"><small>

include/matrix.hpp

@@ -11,6 +11,7 @@
 #define __MATRIX__
 
 #include <armadillo>
+#include <omp.h>
 
 /** @brief Create a tridiagonal matrix.
  *  

latex/output/eigenvector_10.csv

@@ -0,0 +1,13 @@
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latex/output/eigenvector_100.csv

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+8.8118811881e-01,5.1313579361e-02,9.5560678437e-02,-1.2664793139e-01,5.1313583715e-02,9.5560677562e-02,-1.2664793125e-01
+8.9108910891e-01,4.7213741091e-02,8.8953928103e-02,-1.2038155214e-01,4.7213743269e-02,8.8953927731e-02,-1.2038155232e-01
+9.0099009901e-01,4.3068225595e-02,8.2003031564e-02,-1.1306769867e-01,4.3068226575e-02,8.2003032435e-02,-1.1306769690e-01
+9.1089108911e-01,3.8881045999e-02,7.4734883659e-02,-1.0477000196e-01,3.8881044154e-02,7.4734883337e-02,-1.0477000506e-01
+9.2079207921e-01,3.4656245355e-02,6.7177597583e-02,-9.5560676903e-02,3.4656246833e-02,6.7177599493e-02,-9.5560677562e-02
+9.3069306931e-01,3.0397926577e-02,5.9360420731e-02,-8.5519850964e-02,3.0397921832e-02,5.9360418568e-02,-8.5519847548e-02
+9.4059405941e-01,2.6110186824e-02,5.1313584318e-02,-7.4734880388e-02,2.6110188805e-02,5.1313583715e-02,-7.4734883337e-02
+9.5049504950e-01,2.1797196485e-02,4.3068226072e-02,-6.3299628759e-02,2.1797195856e-02,4.3068226575e-02,-6.3299628182e-02
+9.6039603960e-01,1.7463116664e-02,3.4656247119e-02,-5.1313584252e-02,1.7463115529e-02,3.4656246833e-02,-5.1313583715e-02
+9.7029702970e-01,1.3112143651e-02,2.6110184365e-02,-3.8881045879e-02,1.3112140764e-02,2.6110188805e-02,-3.8881044154e-02
+9.8019801980e-01,8.7484791764e-03,1.7463117510e-02,-2.6110190428e-02,8.7484808507e-03,1.7463115529e-02,-2.6110188805e-02
+9.9009900990e-01,4.3763588619e-03,8.7484790494e-03,-1.3112138188e-02,4.3763573469e-03,8.7484808507e-03,-1.3112140764e-02
+1.0000000000e+00,0.0000000000e+00,0.0000000000e+00,0.0000000000e+00,0.0000000000e+00,0.0000000000e+00,0.0000000000e+00

latex/output/eigenvector_6.csv

@@ -1,9 +0,0 @@
-x,Vector 1,Vector 2,Vector 3
-0.0000000000e+00,0.0000000000e+00,0.0000000000e+00,0.0000000000e+00
-1.4285714286e-01,2.3192061397e-01,-4.1790650593e-01,-5.2112088916e-01
-2.8571428571e-01,4.1790650598e-01,-5.2112088916e-01,-2.3192061388e-01
-4.2857142857e-01,5.2112088920e-01,-2.3192061385e-01,4.1790650595e-01
-5.7142857143e-01,5.2112088915e-01,2.3192061400e-01,4.1790650592e-01
-7.1428571429e-01,4.1790650588e-01,5.2112088921e-01,-2.3192061394e-01
-8.5714285714e-01,2.3192061389e-01,4.1790650591e-01,-5.2112088921e-01
-1.0000000000e+00,0.0000000000e+00,0.0000000000e+00,0.0000000000e+00

latex/output/transform_dense.csv

@@ -1,97 +1,97 @@
 N,T
 5,32
-6,53
-7,73
-8,98
-9,130
-10,157
-11,201
-12,254
-13,293
-14,349
-15,394
-16,458
-17,518
-18,587
-19,649
-20,740
-21,827
-22,911
-23,999
-24,1073
-25,1187
-26,1299
-27,1410
-28,1502
-29,1604
-30,1725
-31,1856
-32,1979
-33,2089
-34,2227
-35,2383
-36,2536
-37,2644
-38,2810
-39,3000
-40,3138
-41,3299
-42,3445
-43,3638
-44,3828
-45,4015
-46,4143
-47,4348
-48,4611
-49,4773
-50,4994
-51,5163
-52,5363
-53,5586
-54,5784
-55,6075
-56,6275
-57,6550
-58,6711
+6,51
+7,68
+8,103
+9,127
+10,160
+11,199
+12,248
+13,281
+14,346
+15,406
+16,459
+17,530
+18,593
+19,659
+20,742
+21,839
+22,892
+23,982
+24,1087
+25,1168
+26,1264
+27,1385
+28,1463
+29,1615
+30,1715
+31,1850
+32,1971
+33,2106
+34,2211
+35,2384
+36,2501
+37,2674
+38,2836
+39,3007
+40,3118
+41,3262
+42,3476
+43,3616
+44,3836
+45,3987
+46,4219
+47,4360
+48,4593
+49,4747
+50,4957
+51,5185
+52,5344
+53,5602
+54,5790
+55,6069
+56,6271
+57,6473
+58,6764
 59,6957
-60,7225
-61,7483
-62,7782
-63,7952
-64,8265
-65,8531
-66,8764
-67,9060
-68,9331
-69,9686
-70,9913
-71,10098
-72,10417
-73,10795
-74,11108
-75,11342
-76,11703
-77,12104
-78,12356
-79,12623
-80,13066
-81,13513
-82,13706
-83,13998
-84,14333
-85,14694
-86,15160
-87,15470
-88,15845
-89,16211
-90,16551
-91,17047
-92,17237
-93,17669
-94,18055
-95,18614
-96,18838
-97,19253
-98,19868
-99,20092
-100,20517
+60,7235
+61,7514
+62,7675
+63,7985
+64,8242
+65,8534
+66,8718
+67,8997
+68,9330
+69,9602
+70,9930
+71,10181
+72,10451
+73,10812
+74,11076
+75,11364
+76,11719
+77,12054
+78,12368
+79,12654
+80,13032
+81,13392
+82,13736
+83,14043
+84,14408
+85,14682
+86,15134
+87,15503
+88,15850
+89,16246
+90,16565
+91,16993
+92,17336
+93,17636
+94,18073
+95,18461
+96,18968
+97,19335
+98,19709
+99,20207
+100,20593

latex/output/transform_tridiag.csv

@@ -6,7 +6,7 @@ N,T
 9,135
 10,177
 11,222
-12,238
+12,264
 13,305
 14,335
 15,417

latex/problems/problem-1.tex

@@ -12,5 +12,5 @@ Scaling will result in a dimensionless variable $\hat{x} = \frac{1}{L}$.
 \end{align*}
 Now we insert the expression into the original equation
 \begin{align*}
-    \frac{d u(\hat{x})}{d\hat{x}^{2}} &= - \frac{F L^{2}}{\gamma} u(\hat{x}) \\
+    \frac{d u(\hat{x})}{d\hat{x}^{2}} &= - \frac{F L^{2}}{\gamma} u(\hat{x}). \\
 \end{align*}

latex/problems/problem-3.tex

@@ -2,7 +2,7 @@
 
 \subsection*{a)}
 
-The function for found the largest off-diagonal can be found in 
+The function to find the largest off-diagonal can be found in 
 \textbf{matrix.hpp} and \textbf{matrix.cpp}.
 
 \subsection*{b)}

latex/problems/problem-5.tex

@@ -1 +1,20 @@
 \section*{Problem 5}
+
+\subsection*{a)}
+We used the Jacobi's rotation method to solve $\boldsymbol{A} \vec{v} = \lambda \vec{v}$, for $\boldsymbol{A}_{(N \cross N)}$ with $N \in [5, 100]$, 
+and increased the matrix size by $3$ rows and columns for every new matrix generated. The number of similarity transformations performed for a tridiagonal matrix 
+of is presented in Figure \ref{fig:transform}. We chose to run the program using dense matrices of same size as the tridiagonal matrices, to compare the scaling data.
+What we see is that the number of similarity transformations necessary to solve the system is proportional to the matrix size. 
+\begin{figure}[h]
+    \centering 
+    \includegraphics[width=0.8\textwidth]{images/transform.pdf}
+    \caption{Similarity transformations performed as a function of matrix size (N), data is presented in a logarithmic scale.}
+    \label{fig:transform}
+\end{figure}
+
+\subsection*{b)}
+For both the tridiagonal and dense matrices we are checking off-diagonal elements above the main diagonal, since these are symmetric matrices.
+The max value is found at index $(k,l)$ and for every rotation of the matrix, we update the remaining elements along row $k$ and $l$. This can lead to an increased 
+value of off-diagonal elements, that previously were close to zero, and extra rotations has to be performed due to these elements. Which suggest that the 
+number of similarity transformations perfomed on a matrix does not depend on its initial number of non-zero elements, making the Jacobi's rotation algorithm as 
+computationally expensive for both dense and tridiagonal matrices of size $N \cross N$.

latex/problems/problem-6.tex

@@ -1 +1,21 @@
 \section*{Problem 6}
+
+\subsection*{a)}
+The plot in Figure \ref{fig:eigenvector_10} is showing the discretization of $\hat{x}$ with $n=10$. 
+The eigenvectors and corresponding analytical eigenvectors have a complete overlap suggesting the implementation of the algorithm is correct. 
+We have included the boundary points for each vector to show a complete solution.
+\begin{figure}[h]
+    \centering
+    \includegraphics[width=0.8\textwidth]{images/eigenvector_10.pdf}
+    \caption{The plot is showing the elements of eigenvector $\vec{v}_{1}, \vec{v}_{2}, \vec{v}_{3}$, corresponding to the three lowest eigenvalues of matrix $\boldsymbol{A} (10 \cross 10)$, against the position $\hat{x}$. The analytical eigenvectors $\vec{v}^{(1)}, \vec{v}^{(2)}, \vec{v}^{(3)}$ are also included in the plot.}
+    \label{fig:eigenvector_10}
+\end{figure}
+
+\subsection*{b)}
+For the discretization with $n=100$ the solution is visually close to a continuous curve, with a complete overlap of the analytical eigenvectors, presented in Figure \ref{fig:eigenvector_100}.
+\begin{figure}
+    \centering
+    \includegraphics[width=0.8\textwidth]{images/eigenvector_100.pdf}
+    \caption{The plot is showing the elements of eigenvector $\vec{v}_{1}, \vec{v}_{2}, \vec{v}_{3}$, corresponding to the three lowest eigenvalues of matrix $\boldsymbol{A} (100 \cross 100)$, against the position $\hat{x}$. The analytical eigenvectors $\vec{v}^{(1)}, \vec{v}^{(2)}, \vec{v}^{(3)}$ are also included in the plot.}
+    \label{fig:eigenvector_100}
+\end{figure}

src/Makefile

@@ -6,6 +6,7 @@ LIBOBJS=$(LIBSRCS:.cpp=.o)
 INCLUDE=../include
 
 CFLAGS=-larmadillo -llapack -std=c++11
+OPENMP=-fopenmp
 
 # Add a debug flag when compiling (For the DEBUG macro in utils.hpp)
 DEBUG ?= 0
@@ -22,14 +23,14 @@ all: test_suite main
 
 # Rules for executables
 main: main.o $(LIBOBJS)
-	$(CC) $^ -o $@ $(CFLAGS) $(DBGFLAG) -I$(INCLUDE)
+	$(CC) $^ -o $@ $(CFLAGS) $(DBGFLAG) -I$(INCLUDE) $(OPENMP)
 
 test_suite: test_suite.o $(LIBOBJS)
-	$(CC) $^ -o $@ $(CFLAGS) $(DBGFLAG) -I$(INCLUDE)
+	$(CC) $^ -o $@ $(CFLAGS) $(DBGFLAG) -I$(INCLUDE) $(OPENMP)
 
 # Rule for object files
 %.o: %.cpp
-	$(CC) -c $^ -o $@ $(DBGFLAG) -I$(INCLUDE)
+	$(CC) -c $^ -o $@ $(DBGFLAG) -I$(INCLUDE) $(OPENMP)
 
 # Make documentation
 doxygen:

src/main.cpp

@@ -4,8 +4,8 @@
  *  The program performs the Jacobi rotation method. 
  *  The size of the matrix, and number of transformations
  *  performed are written to file. 
- *  Eigenvector correstonding to the 3 smallest eigenvalues 
- *  for matrices of size 6x6 and 100x100 are written to file.
+ *  Eigenvector corresponding to the 3 smallest eigenvalues 
+ *  for matrices of size 10x10 and 100x100 are written to file.
  *
  *  @author Cory Alexander Balaton (coryab)
  *  @author Janita Ovidie Sandtrøen Willumsen (janitaws)
@@ -13,7 +13,10 @@
  */
 #include <cassert>
 #include <cmath>
+#include <ctime>
 #include <iostream>
+#include <omp.h>
+#include <ostream>
 
 #include "utils.hpp"
 #include "matrix.hpp"
@@ -28,6 +31,7 @@ void write_transformation_dense(int N)
     ofile << "N,T" << std::endl;
 
     // Increase size of matrix, start at 5 to avoid logic_error of N=4
+    #pragma omp parallel for ordered schedule(static, 1)
     for (int i = 5; i <= N; i++) {
         arma::mat A = arma::mat(i, i).randn();
         A = arma::symmatu(A);
@@ -40,8 +44,11 @@ void write_transformation_dense(int N)
         jacobi_eigensolver(A, 10e-14, eigval, eigvec, 100000, iters, converged);
 
         // Write size, and number of iterations to file
+        #pragma omp ordered
+        {
             ofile << i << "," << iters << std::endl;
         }
+    }
     ofile.close();
 }
 
@@ -58,6 +65,7 @@ void write_transformation_tridiag(int N)
     ofile << "N,T" << std::endl;
 
     // Increase size of matrix, start at 5 to avoid logic_error of N=4
+    #pragma omp parallel for ordered schedule(static, 1)
     for (int i = 5; i <= N; i++) {
         h = 1. / (double) (i+1);
         a = -1. / (h*h), d = 2. / (h*h);
@@ -72,8 +80,11 @@ void write_transformation_tridiag(int N)
         jacobi_eigensolver(A, 10e-14, eigval, eigvec, 100000, iters, converged);
 
         // Write size, and number of iterations to file
+        #pragma omp ordered
+        {
             ofile << i << "," << iters << std::endl;
         }
+    }
     ofile.close();
 }
 
@@ -87,6 +98,7 @@ void write_eigenvec(int N)
 
     // Create tridiagonal matrix
     arma::mat A = create_symmetric_tridiagonal(N, a, d);
+    arma::mat analytic = arma::mat(N, N);
     arma::vec eigval;
     arma::mat eigvec;
 
@@ -96,13 +108,36 @@ void write_eigenvec(int N)
     // Solve using Jacobi rotation method
     jacobi_eigensolver(A, 10e-14, eigval, eigvec, 100000, iters, converged);
 
+    // Build analytic eigenvectors 
+    arma::vec v, analytic_vec = arma::vec(N);
+    for (int i=0; i < N; i++) {
+        v = eigvec.col(i);
+
+        for (int j=0; j < N; j++) {
+            analytic_vec(j) = std::sin(((j+1.)*(i+1.)*M_PI) / (N+1.));
+        }
+
+        analytic_vec = arma::normalise(analytic_vec);
+
+        // Flip the sign of the analytic vector if they are different
+        if (analytic_vec(0)*v(0) < 0.) {
+            analytic_vec *= -1;
+        }
+        analytic.col(i) = analytic_vec;
+    }
+
     std::ofstream ofile;
     // Create file based on matrix size, and write header line to file 
     ofile.open("../latex/output/eigenvector_" + std::to_string(N) + ".csv");
-    ofile << "x,Vector 1,Vector 2,Vector 3" << std::endl;
+    ofile << "x," 
+        << "Vector 1,Vector 2,Vector 3," 
+        << "Analytic 1,Analytic 2,Analytic 3" << std::endl;
 
     // Add boundary value for x=0 
     ofile << scientific_format(0., 16) << "," 
+            << scientific_format(0., 16) << "," 
+            << scientific_format(0., 16) << "," 
+            << scientific_format(0., 16) << "," 
             << scientific_format(0., 16) << "," 
             << scientific_format(0., 16) << ","
             << scientific_format(0., 16) << std::endl;
@@ -113,10 +148,16 @@ void write_eigenvec(int N)
         ofile << scientific_format(x, 16)<< "," 
             << scientific_format(eigvec(i,0), 16) << "," 
             << scientific_format(eigvec(i,1), 16) << "," 
-            << scientific_format(eigvec(i,2), 16) << std::endl;
+            << scientific_format(eigvec(i,2), 16) << ","
+            << scientific_format(analytic(i,0), 16) << ","
+            << scientific_format(analytic(i,1), 16) << ","
+            << scientific_format(analytic(i,2), 16) << std::endl;
     }
     // Add boundary value for x=1
     ofile << scientific_format(1., 16) << "," 
+            << scientific_format(0., 16) << "," 
+            << scientific_format(0., 16) << "," 
+            << scientific_format(0., 16) << ","
             << scientific_format(0., 16) << ","
             << scientific_format(0., 16) << ","
             << scientific_format(0., 16) << std::endl;
@@ -128,7 +169,7 @@ int main()
 {
     write_transformation_tridiag(100);
     write_transformation_dense(100);
-    write_eigenvec(6);
+    write_eigenvec(10);
     write_eigenvec(100);
     return 0;
 }

src/matrix.cpp

@@ -20,6 +20,7 @@ arma::mat create_tridiagonal(
     A(0, 0) = d(0);
     A(0, 1) = e(0);
 
+    #pragma omp parallel for if(n > 50)
     for (int i = 1; i < n-1; i++) {
         A(i, i-1) = a(i-1);
         A(i, i) = d(i);

src/plot.py

@@ -24,29 +24,28 @@ def plot_transformations(save: bool=False) -> None:
         fig.savefig("../latex/images/transform.pdf")
 
 
-
 def plot_eigenvectors(N: int, save: bool=False) -> None:
     # Load data based on matrix size
     path = f"../latex/output/eigenvector_{N}.csv"
     eigvec = pd.read_csv(path, header=0)
 
     fig, ax = plt.subplots()
-    ax.plot(eigvec['x'], eigvec['Vector 1'], label='Vector 1')
-    ax.plot(eigvec['x'], eigvec['Vector 2'], label='Vector 2')
-    ax.plot(eigvec['x'], eigvec['Vector 3'], label='Vector 3')
+    ax.plot(eigvec['x'], eigvec['Vector 1'], label=r'$\vec{v}_{1}$')
+    ax.plot(eigvec['x'], eigvec['Vector 2'], label=r'$\vec{v}_{2}$')
+    ax.plot(eigvec['x'], eigvec['Vector 3'], label=r'$\vec{v}_{3}$')
+    ax.plot(eigvec['x'], eigvec['Analytic 1'], '--', label=r'$\vec{v}^{(1)}$')
+    ax.plot(eigvec['x'], eigvec['Analytic 2'], '--', label=r'$\vec{v}^{(2)}$')
+    ax.plot(eigvec['x'], eigvec['Analytic 3'], '--', label=r'$\vec{v}^{(3)}$')
     ax.set_xlabel(r'Element $\hat{x}_{i}$')
-    ax.set_ylabel(r'Value of element $v_{i}$')
-    ax.legend()
+    ax.set_ylabel(r'Element $v_{i}$')
+    ax.legend(loc='upper left')
 
     # Save to file
     if save is True:
         fig.savefig(f"../latex/images/eigenvector_{N}.pdf")
 
      
-
-
 if __name__ == '__main__':
     plot_transformations(True)
-    plot_eigenvectors(6, True)
+    plot_eigenvectors(10, True)
     plot_eigenvectors(100, True)
-    # plt.show()