21 lines
1.5 KiB
TeX
21 lines
1.5 KiB
TeX
\section*{Problem 6}
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\subsection*{a)}
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The plot in Figure \ref{fig:eigenvector_6} is showing the discretization of $\hat{x}$ with $n=6$.
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The eigenvectors and corresponding analytical eigenvectors have a complete overlap suggesting the implementation of the algorithm is correct.
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We have included the boundary points for each vector to show a complete solution.
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\begin{figure}[h]
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\centering
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\includegraphics[width=0.8\textwidth]{images/eigenvector_6.pdf}
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\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.}
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\label{fig:eigenvector_6}
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\end{figure}
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\subsection*{b)}
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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}.
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\begin{figure}
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\centering
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\includegraphics[width=0.8\textwidth]{images/eigenvector_100.pdf}
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\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.}
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\label{fig:eigenvector_100}
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\end{figure} |