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wip midterm

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Claudio Maggioni 2020-04-01 23:25:27 +02:00
parent 78df22d27e
commit a79701a817
2 changed files with 76 additions and 8 deletions
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midterm/midterm.tex
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@ -37,7 +37,8 @@ $$1|101010101010|010_F = -(1+2^{-1} +2^{-3} +2^{-5} +2^{-7} +2^{-9} + 2^{-11}) *
$$1|0000 0000 0000|001_F = 2^{-3}_{10} = 0.125_{10}$$
\subsection*{Point d)}
$$1|1111 1111 1111|111_F = (1 + 2^{-1}+ 2^{-2} + 2^{-3} +2^{-4} + 2^{-5} +2^{-6} + 2^{-7} + 2^{-8} + 2^{-9}+ 2^{-10}+ 2^{-11}+ 2^{-12}) * {2^3}_{10} = $$
$$1|1111 1111 1111|111_F = $$
$$=(1 + 2^{-1}+ 2^{-2} + 2^{-3} +2^{-4} + 2^{-5} +2^{-6} + 2^{-7} + 2^{-8} + 2^{-9}+ 2^{-10}+ 2^{-11}+ 2^{-12}) * {2^3}_{10} = $$
$$ = 15.998046875_{10}$$
\subsection*{Point e)}
@ -169,7 +170,7 @@ u_4 =
\end{bmatrix}
\]
\[
\l_5 =
l_5 =
\begin{bmatrix}
0 \\ 0 \\ 0 \\ 0 \\ 1 \\
\end{bmatrix},
@ -196,7 +197,7 @@ u_5 =
\]
\subsection*{Point b)}
\[
\l_1 =
l_1 =
\begin{bmatrix}
1 \\ 2 \\ 3 \\ 4 \\ 5 \\
\end{bmatrix},
@ -214,7 +215,7 @@ e^{T}_1 =
\end{bmatrix}
\]
\[
\l_2 =
l_2 =
\begin{bmatrix}
0 \\ 1 \\ 2 \\ 3 \\ 4 \\
\end{bmatrix},
@ -232,7 +233,7 @@ e^{T}_2 =
\end{bmatrix}
\]
\[
\l_3 =
l_3 =
\begin{bmatrix}
0 \\ 0 \\ 1 \\ 2 \\ 3 \\
\end{bmatrix},
@ -250,7 +251,7 @@ e^{T}_3 =
\end{bmatrix}
\]
\[
\l_4 =
l_4 =
\begin{bmatrix}
0 \\ 0 \\ 0 \\ 1 \\ 2 \\
\end{bmatrix},
@ -267,5 +268,72 @@ e^{T}_4 =
0 & 0 & 0 & -2 & 1 \\
\end{bmatrix}
\]
\subsection{Point c)}
\subsection*{Point c)}
$$Ly = b$$
\[
L =
\begin{bmatrix}
1 & 0 & 0 & 0 & 0 \\
2 & 1 & 0 & 0 & 0 \\
3 & 2 & 1 & 0 & 0 \\
4 & 3 & 2 & 1 & 0 \\
5 & 4 & 3 & 2 & 1 \\
\end{bmatrix},
b =
\begin{bmatrix}
1 \\ 2 \\ 3 \\ 4 \\ 5 \\
\end{bmatrix}
\]
\[
L =
\begin{bmatrix}
1 & 0 & 0 & 0 & 0 \\
2 & 1 & 0 & 0 & 0 \\
3 & 2 & 1 & 0 & 0 \\
4 & 3 & 2 & 1 & 0 \\
5 & 4 & 3 & 2 & 1 \\
\end{bmatrix},
b =
\begin{bmatrix}
1 \\ 2 \\ 3 \\ 4 \\ 5 \\
\end{bmatrix}
\]
$$y_1 = \frac{1}{1} = 1$$
$$y_2 = \frac{2 - 2 \cdot 1}{1} = 0$$
$$y_3 = \frac{3 - 3 \cdot 1 - 2 \cdot 0}{1} = 0$$
$$y_4 = \frac{4 - 4 \cdot 1 - 3 \cdot 0 - 2 \cdot 0}{1} = 0$$
$$y_5 = \frac{5 - 5 \cdot 1 - 4 \cdot 0 - 3 \cdot 0 - 2 \cdot 0}{1} = 0$$
\[
y =
\begin{bmatrix}
1 \\ 0 \\ 0 \\ 0 \\ 0 \\
\end{bmatrix}
\]
\subsection*{Point d)}
$$Ux = y$$
\[
U =
\begin{bmatrix}
1 & 1 & 1 & 1 & 1 \\
0 & 2 & 2 & 2 & 2 \\
0 & 0 & 3 & 3 & 3 \\
0 & 0 & 0 & 4 & 4 \\
0 & 0 & 0 & 0 & 5 \\
\end{bmatrix}
y =
\begin{bmatrix}
1 \\ 0 \\ 0 \\ 0 \\ 0 \\
\end{bmatrix}
\]
$$x_5 = \frac{0}{5} = 0$$
$$x_4 = \frac{0 - 4 \cdot 0}{4} = 0$$
$$x_3 = \frac{0 - 3 \cdot 0 - 3 \cdot 0}{3} = 0$$
$$x_2 = \frac{0 - 2 \cdot 0 - 2 \cdot 0 - 2 \cdot 0}{2} = 0$$
$$x_1 = \frac{0 - 1 \cdot 0 - 1 \cdot 0 - 1 \cdot 0 - 1 \cdot }{1} = 0$$
\[
x =
\begin{bmatrix}
0 \\ 0 \\ 0 \\ 0 \\ 0 \\
\end{bmatrix}
\]
\end{document}