Finish problem 5 and 6 with plot

latex/problems/problem-5.tex

@@ -1 +1,20 @@
-\section*{Problem 5}
+\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}
+\section*{Problem 6}
+
+\subsection*{a)}
+The plot in Figure \ref{fig:eigenvector_6} is showing the discretization of $\hat{x}$ with $n=6$. 
+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_6.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 $\boldsymbol{A} (6 \cross 6)$, 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_6}
+\end{figure}
+
+\subsection*{b)}
+For the discretization with $n=100$ the solution is visually close to a continous 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}