mirror of
https://github.com/kuhyx/WUT_Computer_Science.git
synced 2026-11-10 19:07:11 +01:00
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
53 lines
2.0 KiB
TeX
53 lines
2.0 KiB
TeX
\section{Discussion of results}
|
|
\hyperlink{function2_plot}{Code for generating plots}\\
|
|
As we can see error for 'B' system of equations increased after our residual correction. But that's just one matrix and vector pair, after testing more pairs we get following graphs (Tested on systems of equations from a) and b) respectively, with maximum size of matrices and vectors equal to 100):
|
|
\begin{center}
|
|
\includegraphics[scale=0.5]{errorsA.eps}
|
|
\end{center}
|
|
|
|
\begin{center}
|
|
\includegraphics[scale=0.5]{errorsB.eps}
|
|
\end{center}
|
|
As we can see our residual correction method \textbf{does} work for Matrix B and Vector B and decreases the error drastically.
|
|
|
|
\subsection{Errors in b)}
|
|
Matrix B suffers from relatively huge errors. Where do they come from?
|
|
Matrix B is a variant of Hilbert matrix.
|
|
Equation for generating elements of this matrix is as follows:
|
|
\[ b_{ij} = \frac{5}{8(i + j + 1)} \]
|
|
Equation for generating elements of the Hilbert matrix is as follows:
|
|
\[ b_{ij} = \frac{1}{(i + j - 1)} \]
|
|
The only thing that changes are the constants, we have 5 instead of 1 in the numerator and 8 in front of the summation in the denominator and +1 instead of -1. For sufficiently huge 'i' and 'j' those differences hardly matter.
|
|
Hilbert matrix is badly conditioned. It's condtion number for \textit{n} size hilbert matrix is equal to:
|
|
\[ \mathcal{O}(\frac{e^{3.5255n}}{sqrt(n)}) \]
|
|
Let's calculate condition number of matrix B in matlab:
|
|
\begin{simplechar}
|
|
\begin{lstlisting}
|
|
cond(matrixB(10))
|
|
ans = 2.428027097978043e+14
|
|
\end{lstlisting}
|
|
\end{simplechar}
|
|
|
|
And compare it with condition number of matrix A:
|
|
\begin{simplechar}
|
|
\begin{lstlisting}
|
|
cond(matrixA(10))
|
|
ans = 4.550344127923193
|
|
\end{lstlisting}
|
|
\end{simplechar}
|
|
As we can see the difference is huge, conditon number of matrix B for 10 elements is $10^{14}$ times bigger than for matrix A. This explains big errors for matrix B.
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\addtocontents{toc}{\protect\newpage}
|
|
\chapter{Problem 3 - Solving a system of n linear equations - iterative algorithm}
|
|
|