\section{Theoretical introduction}
\subsection{Eigenvalues}
Eigenvalues of matrix \textbf{A} are defined as a pair of number $\lambda$ and vector \textbf{v} such that:
\[ \mathbf{A}\mathbf{v} = \lambda\mathbf{v} \]
This can be rewritten as:
\[ (\mathbf{A} - \lambda\mathbf{I})\mathbf{v} = 0 \]
And further we get characteristic equation:
\[ det(\mathbf{A} - \lambda\mathbf{I}) = 0 \]

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\subsection{QR method for finding eigenvalues}
First we start by transforming matrix \textbf{A} into tridiagonal form. It increases the effectiveness of calculations.

Single step of QR method consists of:
\[ \textbf{A}^{(k)} = \textbf{Q}^{(k)}\textbf{R}^{(k)} \]
\[ \textbf{A}^{(k+1)} = \textbf{R}^{(k)}\textbf{Q}^{(k)} \]
$\textbf{Q}^{(k)}$ being orthogonal, therefore:
\[ \textbf{R}^{(k)} = (\textbf{Q}^{(k)}))^{-1}\textbf{A}^{(k)} = \textbf{Q}^{(k)T}\textbf{A}^{(k)}\textbf{Q}^{(k)} \]
therefore:
\[ \textbf{A}^{(k+1)} = \textbf{Q}^{(k)T}\textbf{A}^{(k)}\textbf{Q}^{(k)} \]
If we have symmetric matrix \textbf{A} it converges to the diagonal matrix $diag(\lambda_i)$

Since this method can be slowly convergent we use shifts. Single iteration of QR method with shifts looks like this:
\begin{equation}
\begin{split}
\textbf{A}^{(k)} - p_k\textbf{I} &= \textbf{Q}^{(k)}\textbf{R}^{(k)} \\
\textbf{A}^{(k+1)} &= \textbf{R}^{(k)}\textbf{Q}^{(k)} + p_k\textbf{I} \\
&= \textbf{Q}^{(k)T}(\textbf{A}^{(k)} - p_k\textbf{I})\textbf{Q}^{(k)} + p_k\textbf{I} \\
&=\textbf{Q}^{(k)T}\textbf{A}^{(k)}\textbf{Q}^{(k)} \\
\end{split}
\end{equation}

Wher $p_k$ should be chosen as a best estimate of $\lambda_{i+1}$

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