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Krzysztof kuhy Rudnicki
ffa5dad02a
The 250-line cap applies to prose too, and the reports were the worst
offenders -- ENUME's projectA.tex alone was 1,876 lines. Each is now a
preamble plus one \input{} per section, which is how a long LaTeX document is
normally organised; not one byte of prose was rewritten.
"The PDF is unchanged" is checked by a script rather than by eye, because
hashing the PDF cannot work: pdflatex stamps a /CreationDate and an /ID into
every build, so two runs of the SAME source already differ. tex_fingerprint.sh
compares what a reader sees instead -- page count plus a hash of the extracted
text -- and it is stable across rebuilds of untouched sources, which is the
property that makes it usable as a gate.
It earned its keep immediately. Both ECOTE reports quote an example LaTeX
document inside an lstlisting block, so \end{document} appears in the middle
of the file; splitting on the FIRST one stranded everything after it. Same
page count, different text, caught automatically. EOPSY lab3 exposed a second
bug: a \section{} title that runs onto the next line.
Result: 19 .tex over the cap -> 1. The survivor,
ELAC/.../03-building-nfa-...tex, is 418 lines of consecutive tikz figures
with no subsection to split on and needs a hand-placed boundary.
Verified: 29 splits across two passes (section, then subsection for the
pieces still over), every one reporting `unchanged` from
scripts/tex_fingerprint.sh --check.
Co-Authored-By: Claude Opus 5 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_0172dj34FDpao6wsix8ZuCzE
36 lines
1.6 KiB
TeX
36 lines
1.6 KiB
TeX
\section{Theoretical introduction}
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\subsection{Eigenvalues}
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Eigenvalues of matrix \textbf{A} are defined as a pair of number $\lambda$ and vector \textbf{v} such that:
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\[ \mathbf{A}\mathbf{v} = \lambda\mathbf{v} \]
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This can be rewritten as:
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\[ (\mathbf{A} - \lambda\mathbf{I})\mathbf{v} = 0 \]
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And further we get characteristic equation:
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\[ det(\mathbf{A} - \lambda\mathbf{I}) = 0 \]
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\newpage
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\subsection{QR method for finding eigenvalues}
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First we start by transforming matrix \textbf{A} into tridiagonal form. It increases the effectiveness of calculations.
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Single step of QR method consists of:
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\[ \textbf{A}^{(k)} = \textbf{Q}^{(k)}\textbf{R}^{(k)} \]
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\[ \textbf{A}^{(k+1)} = \textbf{R}^{(k)}\textbf{Q}^{(k)} \]
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$\textbf{Q}^{(k)}$ being orthogonal, therefore:
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\[ \textbf{R}^{(k)} = (\textbf{Q}^{(k)}))^{-1}\textbf{A}^{(k)} = \textbf{Q}^{(k)T}\textbf{A}^{(k)}\textbf{Q}^{(k)} \]
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therefore:
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\[ \textbf{A}^{(k+1)} = \textbf{Q}^{(k)T}\textbf{A}^{(k)}\textbf{Q}^{(k)} \]
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If we have symmetric matrix \textbf{A} it converges to the diagonal matrix $diag(\lambda_i)$
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Since this method can be slowly convergent we use shifts. Single iteration of QR method with shifts looks like this:
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\begin{equation}
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\begin{split}
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\textbf{A}^{(k)} - p_k\textbf{I} &= \textbf{Q}^{(k)}\textbf{R}^{(k)} \\
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\textbf{A}^{(k+1)} &= \textbf{R}^{(k)}\textbf{Q}^{(k)} + p_k\textbf{I} \\
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&= \textbf{Q}^{(k)T}(\textbf{A}^{(k)} - p_k\textbf{I})\textbf{Q}^{(k)} + p_k\textbf{I} \\
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&=\textbf{Q}^{(k)T}\textbf{A}^{(k)}\textbf{Q}^{(k)} \\
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\end{split}
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\end{equation}
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Wher $p_k$ should be chosen as a best estimate of $\lambda_{i+1}$
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\newpage
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