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\documentclass[12pt,a4paper]{article}

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\usepackage{amstext} \usepackage{amsmath} \usepackage{array}
\newcommand{\lra}{\Leftrightarrow}

\title{Howework 3 -- Introduction to Computational Science}

\author{Claudio Maggioni}

\begin{document} \maketitle 
\section*{Question 1}
to be transcribed
\section*{Question 2}

\[i = 1 \hspace{1cm} k = 4 \hspace{1cm} \begin{bmatrix}4&2&3&1\\\end{bmatrix}\]
\[
l_1 = 
\begin{bmatrix}
       1/8 \\
       1/4 \\
       1/2 \\
       1 \\
\end{bmatrix}
\hspace{1cm}
u_1 = \begin{bmatrix} 32 & 24 & 10 & 11\end{bmatrix}
\hspace{1cm}
\]\[
A_2 = \begin{bmatrix}4 &3 &2& 1\\ 8& 8& 5& 2\\ 16& 12& 10& 5\\ 32& 24& 20 &11 \\\end{bmatrix} - \begin{bmatrix}
       4         &     3 &             5/2 &          11/8  \\   
       8          &    6  &            5    &         11/4    \\ 
      16           &  12   &          10     &        11/2     \\
      32            & 24    &         20      &       11       \\
\end{bmatrix} =
\begin{bmatrix}
       0     &         0  &           -1/2    &       -3/8 \\    
       0      &        2   &           0       &      -3/4   \\  
       0       &       0    &          0        &     -1/2     \\
       0        &      0      &        0         &     0       \\
\end{bmatrix}
\]

\[i = 2 \hspace{1cm} k = 2 \hspace{1cm} p = \begin{bmatrix}4&2&3&1\\\end{bmatrix}\]
\[
l_2 =\begin{bmatrix}
0\\1\\0\\0\\
\end{bmatrix}
\hspace{1cm}
u_2 =\begin{bmatrix} 0  &            2   &           0     &        -3/4 \end{bmatrix}\]
\[    
A_3 =  \begin{bmatrix}
       0     &         0  &           -1/2    &       -3/8 \\    
       0      &        2   &           0       &      -3/4   \\  
       0       &       0    &          0        &     -1/2     \\
       0        &      0      &        0         &     0       \\
\end{bmatrix} - 
\begin{bmatrix}
       0       &       0  &            0        &      0       \\
       0        &      2   &           0         &    -3/4     \\
       0         &     0    &          0          &    0       \\
       0          &    0     &         0           &   0       \\
\end{bmatrix}
 =
\begin{bmatrix}
       0       &       0      &       -1/2     &      -3/8  \\   
       0        &      0       &       0        &      0      \\ 
       0         &     0        &      0         &    -1/2     \\
       0          &    0         &     0          &    0       \\
\end{bmatrix}
\]

\[ i = 3 \hspace{1cm} k = 4 \hspace{1cm} p = \begin{bmatrix}
4&2&1&3\\
\end{bmatrix}
\]
\[
l_3 = \begin{bmatrix}
1 \\0\\0\\0\\
\end{bmatrix}
\hspace{1cm}
u_3 = \begin{bmatrix} 0 & 0& -1/2 & -3/8\\\end{bmatrix}     
\]\[
A_4 = \begin{bmatrix}
       0       &       0      &       -1/2     &      -3/8  \\   
       0        &      0       &       0        &      0      \\ 
       0         &     0        &      0         &    -1/2     \\
       0          &    0         &     0          &    0       \\
\end{bmatrix} -
\begin{bmatrix}
       0       &       0 &            -1/2&           -3/8\\     
       0        &      0  &            0   &           0    \\   
       0         &     0   &           0    &          0      \\ 
       0          &    0    &          0     &         0       \\
\end{bmatrix} =
\begin{bmatrix}
       0       &       0  &            0 &             0 \\      
       0        &      0   &           0  &            0   \\    
       0         &     0    &          0   &          -1/2   \\  
       0          &    0     &         0    &          0       \\
       \end{bmatrix}
       \]

\[ i =4 \hspace{1cm} k = 4 \hspace{1cm} p = \begin{bmatrix}
4&2&1&3\\
\end{bmatrix}\]
\[l_4 = \begin{bmatrix} 0 \\0\\1\\0\\\end{bmatrix}
u_4 = \begin{bmatrix}0&0&0&-1/2\end{bmatrix}
\]

\[
P =
\begin{bmatrix}
       0       &       0  &            0    &          1   \\   
       0        &      1   &           0      &        0    \\  
       1         &     0    &          0     &         0     \\  
       0          &    0     &         1       &       0 \\
\end{bmatrix}
\]
\[
L = P *       
\begin{bmatrix}
 1/8       &     0         &     1       &       0       \\
       1/4  &          1    &          0  &            0   \\    
       1/2   &         0     &         0   &           1     \\  
       1       &       0      &        0    &          0       \\
\end{bmatrix}
=
\begin{bmatrix}
       1       &       0    &          0     &         0   \\    
       1/4      &      1     &         0      &        0     \\  
       1/8       &     0      &        1       &       0       \\
       1/2        &    0       &       0        &      1       \\
       \end{bmatrix}
       \]
       
\[
U =
\begin{bmatrix}
      32      &      24     &        20    &         11      \\ 
       0        &      2     &         0    &         -3/4     \\
       0         &     0      &       -1/2   &        -3/8     \\
       0          &    0       &       0      &       -1/2     \\
\end{bmatrix}
\]


\end{document}