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\contentsline {section}{Introduction}{3}%
\contentsline {section}{Solving a system of linear equations}{3}%
\contentsline {subsection}{Gauss elimination algorithm}{3}%
\contentsline {subsection}{LU decomposition}{3}%
\contentsline {subsection}{Diagonalization: introduction}{4}%
\contentsline {section}{Problem 1}{4}%
\contentsline {subsection}{(1) LU decomposition implementation}{4}%
\contentsline {subsubsection}{Partial pivoting}{4}%
\contentsline {subsubsection}{(1.1) Solving a linear system}{5}%
\contentsline {subsubsection}{(1.2) Comparing with \texttt {matlab lu} built-in function}{5}%
\contentsline {subsubsection}{(1.3) Decomposition of a matrix}{5}%
\contentsline {subsection}{(2) Puzzle board game}{6}%
\contentsline {subsubsection}{(2.1) Results of \texttt {test\_puzzle.m} script}{6}%
\contentsline {subsection}{(3) Helicopter power formula: dimensional analysis}{7}%
\contentsline {subsubsection}{(3.1) Approaching the problem}{7}%
\contentsline {subsubsection}{(3.2) + (3.3), Adding a constraint}{8}%
\contentsline {section}{The eigenvalue problem and diagonalization}{8}%
\contentsline {subsection}{Power method}{8}%
\contentsline {subsection}{Jacobi method}{9}%
\contentsline {section}{Problem 2}{9}%
\contentsline {subsection}{(1) Power iteration methods implementation}{9}%
\contentsline {subsection}{(2) Eigenmodes of a vibrating string}{9}%
\contentsline {subsubsection}{(2.1) Formalization of the problem}{9}%
\contentsline {subsubsection}{(2.2) Implementation}{10}%
\contentsline {subsubsection}{(2.3) + (2.4), Find the first eigenvalues}{10}%
\contentsline {subsection}{(3) Jacobi method implementation}{11}%
\contentsline {subsubsection}{(3.1) + (3.2) Comparing classic and cyclic jacobi methods}{12}%
\contentsline {subsubsection}{(3.3) Applying the axis aligned rotation matrix $J$}{12}%
\contentsline {subsection}{(4) Landau levels in a square-lattice model}{13}%
\contentsline {subsubsection}{(4.1) Formalization of the problem}{13}%
\contentsline {subsubsection}{(4.2) Boundary conditions: finite-size and periodic models}{15}%
\contentsline {subsubsection}{(4.3) Energy spectra, $B = 0$}{15}%
\contentsline {subsubsection}{(4.4) Landau levels, periodic $B > 0$}{17}%
\contentsline {subsubsection}{(4.5) Hall edge states, finite-size $B > 0$}{18}%
\contentsline {section}{Conclusion}{18}%
\contentsline {section}{Appendix: matlab codes}{19}%
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