149 lines
4.9 KiB
TeX
149 lines
4.9 KiB
TeX
% vim: set ts=2 sw=2 et tw=80:
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\documentclass[12pt,a4paper]{article}
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\usepackage[utf8]{inputenc} \usepackage[margin=2cm]{geometry}
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\usepackage{amstext} \usepackage{amsmath} \usepackage{array}
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\newcommand{\lra}{\Leftrightarrow}
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\title{Howework 3 -- Introduction to Computational Science}
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\author{Claudio Maggioni}
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\begin{document} \maketitle
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\section*{Question 1}
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to be transcribed
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\section*{Question 2}
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\[i = 1 \hspace{1cm} k = 4 \hspace{1cm} \begin{bmatrix}4&2&3&1\\\end{bmatrix}\]
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\[
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l_1 =
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\begin{bmatrix}
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1/8 \\
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1/4 \\
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1/2 \\
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1 \\
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\end{bmatrix}
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\hspace{1cm}
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u_1 = \begin{bmatrix} 32 & 24 & 10 & 11\end{bmatrix}
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\hspace{1cm}
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\]\[
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A_2 = \begin{bmatrix}4 &3 &2& 1\\ 8& 8& 5& 2\\ 16& 12& 10& 5\\ 32& 24& 20 &11 \\\end{bmatrix} - \begin{bmatrix}
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4 & 3 & 5/2 & 11/8 \\
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8 & 6 & 5 & 11/4 \\
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16 & 12 & 10 & 11/2 \\
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32 & 24 & 20 & 11 \\
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\end{bmatrix} =
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\begin{bmatrix}
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0 & 0 & -1/2 & -3/8 \\
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0 & 2 & 0 & -3/4 \\
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0 & 0 & 0 & -1/2 \\
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0 & 0 & 0 & 0 \\
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\end{bmatrix}
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\]
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\[i = 2 \hspace{1cm} k = 2 \hspace{1cm} p = \begin{bmatrix}4&2&3&1\\\end{bmatrix}\]
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\[
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l_2 =\begin{bmatrix}
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0\\1\\0\\0\\
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\end{bmatrix}
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\hspace{1cm}
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u_2 =\begin{bmatrix} 0 & 2 & 0 & -3/4 \end{bmatrix}\]
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\[
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A_3 = \begin{bmatrix}
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0 & 0 & -1/2 & -3/8 \\
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0 & 2 & 0 & -3/4 \\
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0 & 0 & 0 & -1/2 \\
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0 & 0 & 0 & 0 \\
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\end{bmatrix} -
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\begin{bmatrix}
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0 & 0 & 0 & 0 \\
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0 & 2 & 0 & -3/4 \\
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0 & 0 & 0 & 0 \\
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0 & 0 & 0 & 0 \\
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\end{bmatrix}
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=
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\begin{bmatrix}
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0 & 0 & -1/2 & -3/8 \\
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0 & 0 & 0 & 0 \\
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0 & 0 & 0 & -1/2 \\
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0 & 0 & 0 & 0 \\
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\end{bmatrix}
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\]
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\[ i = 3 \hspace{1cm} k = 4 \hspace{1cm} p = \begin{bmatrix}
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4&2&1&3\\
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\end{bmatrix}
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\]
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\[
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l_3 = \begin{bmatrix}
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1 \\0\\0\\0\\
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\end{bmatrix}
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\hspace{1cm}
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u_3 = \begin{bmatrix} 0 & 0& -1/2 & -3/8\\\end{bmatrix}
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\]\[
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A_4 = \begin{bmatrix}
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0 & 0 & -1/2 & -3/8 \\
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0 & 0 & 0 & 0 \\
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0 & 0 & 0 & -1/2 \\
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0 & 0 & 0 & 0 \\
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\end{bmatrix} -
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\begin{bmatrix}
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0 & 0 & -1/2& -3/8\\
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0 & 0 & 0 & 0 \\
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0 & 0 & 0 & 0 \\
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0 & 0 & 0 & 0 \\
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\end{bmatrix} =
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\begin{bmatrix}
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0 & 0 & 0 & 0 \\
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0 & 0 & 0 & 0 \\
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0 & 0 & 0 & -1/2 \\
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0 & 0 & 0 & 0 \\
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\end{bmatrix}
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\]
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\[ i =4 \hspace{1cm} k = 4 \hspace{1cm} p = \begin{bmatrix}
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4&2&1&3\\
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\end{bmatrix}\]
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\[l_4 = \begin{bmatrix} 0 \\0\\1\\0\\\end{bmatrix}
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u_4 = \begin{bmatrix}0&0&0&-1/2\end{bmatrix}
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\]
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\[
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P =
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\begin{bmatrix}
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0 & 0 & 0 & 1 \\
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0 & 1 & 0 & 0 \\
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1 & 0 & 0 & 0 \\
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0 & 0 & 1 & 0 \\
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\end{bmatrix}
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\]
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\[
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L = P *
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\begin{bmatrix}
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1/8 & 0 & 1 & 0 \\
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1/4 & 1 & 0 & 0 \\
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1/2 & 0 & 0 & 1 \\
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1 & 0 & 0 & 0 \\
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\end{bmatrix}
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=
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\begin{bmatrix}
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1 & 0 & 0 & 0 \\
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1/4 & 1 & 0 & 0 \\
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1/8 & 0 & 1 & 0 \\
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1/2 & 0 & 0 & 1 \\
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\end{bmatrix}
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\]
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\[
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U =
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\begin{bmatrix}
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32 & 24 & 20 & 11 \\
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0 & 2 & 0 & -3/4 \\
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0 & 0 & -1/2 & -3/8 \\
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0 & 0 & 0 & -1/2 \\
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\end{bmatrix}
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\]
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\end{document}
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