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Commit 2ac6bbf1 authored by Krueger Jasmin's avatar Krueger Jasmin
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continue work on pseudocode for the validation of Hensel

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Latex/pseudocode.tex
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......@@ -513,13 +513,6 @@ Comme $\left|\delta_g\right| < r_g$, $\left|\delta_h\right| < r_h$, $\left|g-\ti
\left(\delta_g + \epsilon\delta_g +t(h-\tilde{h})\delta_g+t\left(g-\tilde{g}\right)\delta_h+t\delta_g\delta_h\right)\%\tilde{g}\\
\left(s\left(h-\tilde{h}\right)\delta_g+\left(1+\epsilon\right)\delta_h + s\left(g-\tilde{g}\right)\delta_h+s\delta_g\delta_h\right)\%\tilde{h}
\end{matrix}\right)\right|\\
&\overset{\delta_g\%\tilde{g}=\delta_g}{\overset{\delta_h\%\tilde{h}= \delta_h}{=}} \left|-\left(\begin{matrix}
\delta_g\\
\delta_h
\end{matrix}\right) + \left(\begin{matrix}
\left(\delta_g + \epsilon\delta_g +t(h-\tilde{h})\delta_g+t\left(g-\tilde{g}\right)\delta_h+t\delta_g\delta_h\right)\%\tilde{g}\\
\left(s\left(h-\tilde{h}\right)\delta_g+\left(1+\epsilon\right)\delta_h + s\left(g-\tilde{g}\right)\delta_h+s\delta_g\delta_h\right)\%\tilde{h}
\end{matrix}\right)\right|\\
&\overset{\delta_g\%\tilde{g}= \delta_g}{\overset{\delta_h\%\tilde{h}=\delta_h}{=}} \left|-\left(\begin{matrix}
\delta_g\\
\delta_h
......@@ -531,7 +524,7 @@ Comme $\left|\delta_g\right| < r_g$, $\left|\delta_h\right| < r_h$, $\left|g-\ti
\left(\epsilon\delta_g +t(h-\tilde{h})\delta_g+t\left(g-\tilde{g}\right)\delta_h+t\delta_g\delta_h\right)\%\tilde{g}\\
\left(s\left(h-\tilde{h}\right)\delta_g+\epsilon\delta_h + s\left(g-\tilde{g}\right)\delta_h+s\delta_g\delta_h\right)\%\tilde{h}
\end{matrix}\right)\right|\\
&= \left|\left(\begin{matrix}
\Bigg[&= \left|\left(\begin{matrix}
\left(\epsilon\delta_g +t(h-\tilde{h})\delta_g+t\left(g-\tilde{g}\right)\delta_h+t\delta_g\delta_h\right) \\
\left(s\left(h-\tilde{h}\right)\delta_g+\epsilon\delta_h + s\left(g-\tilde{g}\right)\delta_h+s\delta_g\delta_h\right)
\end{matrix}\right.\right.\\
......@@ -546,7 +539,24 @@ Comme $\left|\delta_g\right| < r_g$, $\left|\delta_h\right| < r_h$, $\left|g-\ti
& \qquad \qquad \qquad \qquad \left.\left. \begin{matrix}
-\tilde{g}\cdot\overline{\overline{\left(\left(\epsilon + t(h-\tilde{h})+t\delta_h\right)\delta_g + t(g-\tilde{g})\delta_h\right)}\cdot\dot{g}}\\
-\tilde{h}\cdot\overline{\overline{\left(\left(\epsilon +s(g-\tilde{g}) +s\delta_g\right)\delta_h + s(h-\tilde{h})\delta_g\right)}\cdot \dot{h}}
\end{matrix}\right)\right|
\end{matrix}\right)\right|\\
&\leq \left|\left(\begin{matrix}
\left(\epsilon\cdot[\pm r_g] +t\cdot[\pm r_h][\pm r_g]+t[\pm r_g][\pm r_h]+t[\pm r_g][\pm r_h]\right) \\
\left(s[\pm r_h][\pm r_g]+\epsilon[\pm r_h] + s[\pm r_g][\pm r_h]+s[\pm r_g][\pm r_h]\right)
\end{matrix}\right.\right.\\
& \qquad \left.\left.\begin{matrix}
- \tilde{g}\cdot\overline{\overline{\left(\epsilon[\pm r_g] +t[\pm r_h][\pm r_g]+t[\pm r_g][\pm r_h]+t[\pm r_g][\pm r_h]\right)}\cdot\dot{g}}\\
-\tilde{h}\cdot \overline{\overline{\left(s[\pm r_h][\pm r_g]+\epsilon[\pm r_h] + s[\pm r_g][\pm r_h]+s[\pm r_g][\pm r_h]\right)}\cdot\dot{h}}
\end{matrix}\right)\right|\Bigg]\\
&\leq \left|\left(\begin{matrix}
\left(\epsilon[\pm r_g] +t[\pm r_h][\pm r_g]+t[\pm r_g][\pm r_h]+t[\pm r_g][\pm r_h]\right)\%\tilde{g}\\
\left(s[\pm r_h][\pm r_g]+\epsilon[\pm r_h] + s[\pm r_g][\pm r_h]+s[\pm r_g][\pm r_h]\right)\%\tilde{h}
\end{matrix}\right)\right|\\
&= \left|\left(\begin{matrix}
\left(\epsilon[\pm r_g] +3t[\pm r_h][\pm r_g]\right)\%\tilde{g}\\
\left(3s[\pm r_h][\pm r_g]+\epsilon[\pm r_h] \right)\%\tilde{h}
\end{matrix}\right)\right|\\
&=:\mu (r_g,r_h)
% & \leq \left|\left(\begin{matrix}
% r_g\\
% r_h
......@@ -586,116 +596,119 @@ Comme $\left|\delta_g\right| < r_g$, $\left|\delta_h\right| < r_h$, $\left|g-\ti
% \end{matrix}\right)\right|\\
\end{align*}
ou alors, en notant $\dot{g} := rev(\tilde{g})^{-1}$ et $rev(a):= rev_{\deg a}(a)$:
\begin{align*}
\left|N_P(g,h)-N_P(g',h')\right| &= \left|N_P(g,h)-N_P(g+\delta_g,h+\delta_h)\right|\\
&= \left|N_P(g,h) - \left(N_P(g,h) + \left(\begin{matrix}
\delta_g\\
\delta_h
\end{matrix}\right) - \left(\begin{matrix}
t\left(h\delta_g+g\delta_h+\delta_g\delta_h\right)\%\tilde{g}\\
s\left(h\delta_g+g\delta_h+\delta_g\delta_h\right)\%\tilde{h}
\end{matrix}\right)\right)\right|\\
&= \left|-\left(\begin{matrix}
\delta_g\\
\delta_h
\end{matrix}\right) + \left(\begin{matrix}
t\left(h\delta_g+g\delta_h+\delta_g\delta_h\right)\%\tilde{g}\\
s\left(h\delta_g+g\delta_h+\delta_g\delta_h\right)\%\tilde{h}
\end{matrix}\right)\right|\\
&= \left|-\left(\begin{matrix}
\delta_g\\
\delta_h
\end{matrix}\right) + \left(\begin{matrix}
\left(\delta_g + \epsilon\delta_g +t(h-\tilde{h})\delta_g+t\left(g-\tilde{g}\right)\delta_h+t\delta_g\delta_h\right)\%\tilde{g}\\
\left(s\left(h-\tilde{h}\right)\delta_g+\left(1+\epsilon\right)\delta_h + s\left(g-\tilde{g}\right)\delta_h+s\delta_g\delta_h\right)\%\tilde{h}
\end{matrix}\right)\right|\\
% ou alors, en notant $\dot{g} := rev(\tilde{g})^{-1}$ et $rev(a):= rev_{\deg a}(a)$:
% \begin{align*}
% \left|N_P(g,h)-N_P(g',h')\right| &= \left|N_P(g,h)-N_P(g+\delta_g,h+\delta_h)\right|\\
% &= \left|N_P(g,h) - \left(N_P(g,h) + \left(\begin{matrix}
% \delta_g\\
% \delta_h
% \end{matrix}\right) - \left(\begin{matrix}
% t\left(h\delta_g+g\delta_h+\delta_g\delta_h\right)\%\tilde{g}\\
% s\left(h\delta_g+g\delta_h+\delta_g\delta_h\right)\%\tilde{h}
% \end{matrix}\right)\right)\right|\\
% &= \left|-\left(\begin{matrix}
% \delta_g\\
% \delta_h
% \end{matrix}\right) + \left(\begin{matrix}
% t\left(h\delta_g+g\delta_h+\delta_g\delta_h\right)\%\tilde{g}\\
% s\left(h\delta_g+g\delta_h+\delta_g\delta_h\right)\%\tilde{h}
% \end{matrix}\right)\right|\\
% &= \left|-\left(\begin{matrix}
% \delta_g\\
% \delta_h
% \end{matrix}\right) + \left(\begin{matrix}
% \left(\delta_g + \epsilon\delta_g +t(h-\tilde{h})\delta_g+t\left(g-\tilde{g}\right)\delta_h+t\delta_g\delta_h\right)\%\tilde{g}\\
% \left(s\left(h-\tilde{h}\right)\delta_g+\left(1+\epsilon\right)\delta_h + s\left(g-\tilde{g}\right)\delta_h+s\delta_g\delta_h\right)\%\tilde{h}
% \end{matrix}\right)\right|\\
% \left(
\Bigg[\left(\begin{matrix}
M_{1\_}\\
M_{2\_}
\end{matrix}\right)
&:=
\left(\begin{matrix}
\left(\delta_g + \epsilon\delta_g +t(h-\tilde{h})\delta_g+t\left(g-\tilde{g}\right)\delta_h+t\delta_g\delta_h\right)\\
\left(s\left(h-\tilde{h}\right)\delta_g+\left(1+\epsilon\right)\delta_h + s\left(g-\tilde{g}\right)\delta_h+s\delta_g\delta_h\right)
\end{matrix}\right)\Bigg]\\
&\approx \left|-\left(\begin{matrix}
\delta_g\\
\delta_h
\end{matrix}\right) +
\left(\begin{matrix}
M_{1\_} - \tilde{g}\cdot rev\left(rev\left(M_{1\_}\right)\cdot\dot{g}\right)\\
M_{2\_} - \tilde{h}\cdot rev\left(rev\left(M_{2\_}\right)\cdot\dot{h}\right)
\end{matrix}\right)\right|\\
\Bigg[M &= \left(\begin{matrix}
1+\epsilon+t(h-\tilde{h}) & t(g-\tilde{g})\\
s(h-\tilde{h}) & 1+\epsilon+s(g-\tilde{g})
\end{matrix}\right)
\left(\begin{matrix}
\delta_g\\
\delta_h
\end{matrix}\right)
+
\left(\begin{matrix}
t\delta_g\delta_h\\
s\delta_g\delta_h
\end{matrix}\right)\\
& =: Z \left(\begin{matrix}
\delta_g\\
\delta_h
\end{matrix}\right)
+
\left(\begin{matrix}
t\delta_g\delta_h\\
s\delta_g\delta_h
\end{matrix}\right)\\
\overline{a} &:= rev(a)\Bigg]\\
&= \left|-\left(\begin{matrix}
\delta_g\\
\delta_h
\end{matrix}\right) + M -
\left(\begin{matrix}
\tilde{g} & \\
& \tilde{h}
\end{matrix}\right)
\overline{
\left(\begin{matrix}
\dot{g} & \\
& \dot{h}
\end{matrix}\right)\overline{M}}\right|\\
&= \left|-\left(\begin{matrix}
\delta_g\\
\delta_h
\end{matrix}\right) + Z \left(\begin{matrix}
\delta_g\\
\delta_h
\end{matrix}\right)
+
\left(\begin{matrix}
t\delta_g\delta_h\\
s\delta_g\delta_h
\end{matrix}\right) \right.\\
& \qquad \qquad \qquad \qquad \qquad \left.-
\left(\begin{matrix}
\tilde{g} & \\
& \tilde{h}
\end{matrix}\right)
\overline{
\left(\begin{matrix}
\dot{g} & \\
& \dot{h}
\end{matrix}\right)
\overline{\left(Z \left(\begin{matrix}
\delta_g\\
\delta_h
\end{matrix}\right)
+
\left(\begin{matrix}
t\delta_g\delta_h\\
s\delta_g\delta_h
\end{matrix}\right)\right)}}\right|
\end{align*}
% \Bigg[\left(\begin{matrix}
% M_{1\_}\\
% M_{2\_}
% \end{matrix}\right)
% &:=
% \left(\begin{matrix}
% \left(\delta_g + \epsilon\delta_g +t(h-\tilde{h})\delta_g+t\left(g-\tilde{g}\right)\delta_h+t\delta_g\delta_h\right)\\
% \left(s\left(h-\tilde{h}\right)\delta_g+\left(1+\epsilon\right)\delta_h + s\left(g-\tilde{g}\right)\delta_h+s\delta_g\delta_h\right)
% \end{matrix}\right)\Bigg]\\
% &\approx \left|-\left(\begin{matrix}
% \delta_g\\
% \delta_h
% \end{matrix}\right) +
% \left(\begin{matrix}
% M_{1\_} - \tilde{g}\cdot rev\left(rev\left(M_{1\_}\right)\cdot\dot{g}\right)\\
% M_{2\_} - \tilde{h}\cdot rev\left(rev\left(M_{2\_}\right)\cdot\dot{h}\right)
% \end{matrix}\right)\right|\\
% \Bigg[M &= \left(\begin{matrix}
% 1+\epsilon+t(h-\tilde{h}) & t(g-\tilde{g})\\
% s(h-\tilde{h}) & 1+\epsilon+s(g-\tilde{g})
% \end{matrix}\right)
% \left(\begin{matrix}
% \delta_g\\
% \delta_h
% \end{matrix}\right)
% +
% \left(\begin{matrix}
% t\delta_g\delta_h\\
% s\delta_g\delta_h
% \end{matrix}\right)\\
% & =: Z \left(\begin{matrix}
% \delta_g\\
% \delta_h
% \end{matrix}\right)
% +
% \left(\begin{matrix}
% t\delta_g\delta_h\\
% s\delta_g\delta_h
% \end{matrix}\right)\\
% \overline{a} &:= rev(a)\Bigg]\\
% &= \left|-\left(\begin{matrix}
% \delta_g\\
% \delta_h
% \end{matrix}\right) + M -
% \left(\begin{matrix}
% \tilde{g} & \\
% & \tilde{h}
% \end{matrix}\right)
% \overline{
% \left(\begin{matrix}
% \dot{g} & \\
% & \dot{h}
% \end{matrix}\right)\overline{M}}\right|\\
% &= \left|-\left(\begin{matrix}
% \delta_g\\
% \delta_h
% \end{matrix}\right) + Z \left(\begin{matrix}
% \delta_g\\
% \delta_h
% \end{matrix}\right)
% +
% \left(\begin{matrix}
% t\delta_g\delta_h\\
% s\delta_g\delta_h
% \end{matrix}\right) \right.\\
% & \qquad \qquad \qquad \qquad \qquad \left.-
% \left(\begin{matrix}
% \tilde{g} & \\
% & \tilde{h}
% \end{matrix}\right)
% \overline{
% \left(\begin{matrix}
% \dot{g} & \\
% & \dot{h}
% \end{matrix}\right)
% \overline{\left(Z \left(\begin{matrix}
% \delta_g\\
% \delta_h
% \end{matrix}\right)
% +
% \left(\begin{matrix}
% t\delta_g\delta_h\\
% s\delta_g\delta_h
% \end{matrix}\right)\right)}}\right|
% \end{align*}
%\printbibliography
......
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