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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
152 lines
4.5 KiB
TeX
152 lines
4.5 KiB
TeX
\section{a) False position method}
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\subsection{Problem}
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We have to find zeros of the function
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\[ f(x) = -2.1 + 0.3x - xe^{-x} \]
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In the interval $[-5; 10]$ using false position method.
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\subsection{Theoretical Introduction}
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\emph{False position} method also called \emph{regula falsi} in fancier circles is similar to the bisection method, the difference is that the interval we use $[a_n, b_n]$ is divided into two subintervals. We have:
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\begin{itemize}
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\item $\alpha$ - The root
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\item $a_n$ - 'left' interval
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\item $b_n$ - 'right' interval
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\item $f(a_n)$ - Value at left interval
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\item $f(b_n)$ - Value at right interval
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\end{itemize}
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We get:
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\[ \frac{f(b_n) - f(a_n)}{b_n - a_n} = \frac{f(b_n) - 0}{b_n - c_n} \]
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From which we get:
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\[ c_n = b_n - \frac{f(b_n)(b_n - a_n)}{f(b_n) - f(a_n)} = \frac{a_nf(b_n) - b_n f(a_n)}{f(b_n)-f(a_n)} \]
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Then we choose next interval as in the bisection method so: we calculate products of function values at $a_n$ and $b_n$ and that subinterval is selected for the next iteration of \emph{false position} method. This subinterval corresponds to the negative product value.
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\subsubsection{Properties of \emph{false position method}}
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This method is always convergent, simillary to bisection method, since it will always choose and shorten the interval which contains the root. If the function is continous and differentiable the method is linearly convergent. That being said the convergence may become sluggish. It can happen if for example one of the endpoints of the intervals will remain the same and the iterating will not shorten the interval to 0. One of the examples of functions that lead to that are barrier functions used in constrained optimization methods.
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\paragraph{Improvement to the method}
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In order to improve the formula and avoid aforementioned situation we can take smaller value of the function for the value that does not change.
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For right end:
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\[ c_n = \frac{a_n\frac{f(b_n)}{2} - b_nf(a_n)}{\frac{f(b_n)}{2} - f(a_n)} \]
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And for left end:
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\[ c_n = \frac{a_nf(b_n) - b_n\frac{f(a_n)}{2}}{f(b_n)- \frac{f(a_n)}{2}} \]
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This is called \emph{modified regula falsi} or \emph{Illinois algorithm}.
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It is superlinearly convergent, globally convergent and length of intervals we get in each iterations converges to zero.
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\subsection{Results}
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\begin{center}
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\includegraphics[scale=0.25]{task1falsepositionoverall.eps}
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\end{center}
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\begin{center}
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\includegraphics[scale=0.25]{task1falsepositionzommedleft.eps}
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\end{center}
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\begin{center}
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\includegraphics[scale=0.25]{task1falsepositionzommedright.eps}
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\end{center}
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\begin{center}
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\begin{tabular}{| c c c |}
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\hline
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Iteration & root & f(root) \\
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\hline
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1 & -0.17673 &-1.9421 \\
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\hline
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2 & -0.32943 &-1.7409 \\
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\hline
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3 & -0.39578 &-1.6308 \\
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\hline
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4 &-0.61812 &-1.1386 \\
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\hline
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5 &-0.76152 &-0.69765 \\
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\hline
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6 &-0.84516 &-0.38573 \\
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\hline
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7 &-0.89015 &-0.19911 \\
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\hline
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8 &-0.91304 &-0.098733 \\
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\hline
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9 & -0.92431 &-0.047922 \\
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\hline
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10 & -0.92976 & -0.02301 \\
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\hline
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11 &-0.93237 & -0.01099 \\
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\hline
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12 &-0.93362 &-0.005236 \\
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\hline
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13 &-0.93421 &-0.0024915 \\
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\hline
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14 &-0.9345 &-0.0011848 \\
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\hline
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15 &-0.93463 &-0.0005633 \\
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\hline
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16 &-0.93469 &-0.00026777 \\
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\hline
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17 &-0.93473 &-0.00012728 \\
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\hline
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18 &-0.93474 &-6.0499e-05 \\
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\hline
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19 &-0.93475 &-2.8756e-05 \\
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\hline
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20 &-0.93475 & -1.3668e-05 \\
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\hline
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21 &-0.93475 & -6.4964e-06 \\
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\hline
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22 &-0.93475 & -3.0878e-06 \\
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\hline
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23 &-0.93475 & -1.4676e-06 \\
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\hline
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24 &-0.93475 & -6.9758e-07 \\
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\hline
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25 &-0.93475 & -3.3157e-07 \\
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\hline
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26 &-0.93475 & -1.5759e-07 \\
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\hline
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27 &-0.93475 & -7.4906e-08 \\
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\hline
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28 &-0.93475 & -3.5603e-08 \\
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\hline
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29 &-0.93475 & -1.6922e-08 \\
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\hline
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30 &-0.93475 & -8.0433e-09 \\
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\hline
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31 &-0.93475 & -3.823e-09 \\
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\hline
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32 &-0.93475 & -1.8171e-09 \\
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\hline
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33 &-0.93475 & -8.6368e-10 \\
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\hline
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34 &-0.93475 & -4.1051e-10 \\
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\hline
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35 &-0.93475 & -1.9512e-10 \\
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\hline
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36 &-0.93475 & -9.2741e-11 \\
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\hline
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\end{tabular}
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\end{center}
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\begin{center}
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\begin{tabular}{| c c c |}
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\hline
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Iteration & root & f(root) \\
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\hline
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1 & 7.0323 & 0.0034663 \\
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\hline
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2 & 7.0212 & 9.0344e-05 \\
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\hline
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3 & 7.0211 & 4.6349e-05 \\
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\hline
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4 & 7.0208 & -4.3921e-05 \\
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\hline
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5 & 7.0209 & 4.8939e-11 \\
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\hline
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\hline
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\end{tabular}
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\end{center}
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