% vim: set ts=2 sw=2 et tw=80: \documentclass[12pt,a4paper]{article} \usepackage[utf8]{inputenc} \usepackage[margin=2cm]{geometry} \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} \[i=1 \hspace{1cm} l_1 = \begin{bmatrix}1\\1\\5\\-3\\\end{bmatrix} \hspace{1cm} u_1 = \begin{bmatrix} 2 & 1 & 1 & -2 \\\end{bmatrix}\] \[A_2 = \begin{bmatrix} 2 & 1 & 1 &-2 \\ 2 & 2 & -2 & -1 \\ 10 & 4 & 23 & -8 \\ -6 & -2 & 4 & 6 \\\end{bmatrix} - \begin{bmatrix} 2 & 1 & 1 & -2 \\ 2 &1& 1 &-2 \\ 10 & 5 & 5 & -10 \\ -6 & -3 & -3 & 6 \\\end{bmatrix} = \begin{bmatrix}\\& 1 & -3 & 1 \\ & -1 & 18 & 2 \\ & 1 & 7 & 0 \end{bmatrix}\] \[i = 2 \hspace{1cm} l_2 = \begin{bmatrix}\\1 \\-1 \\1\\\end{bmatrix} \hspace{1cm} u_2 = \begin{bmatrix} & 1& -3& 1\\\end{bmatrix}\] \[A_3 = \begin{bmatrix}\\& 1 & -3 & 1 \\ & -1 & 18 & 2 \\ & 1 & 7 & 0 \end{bmatrix} - \begin{bmatrix}\\&1 & -3 & 1 \\ & -1 & 3 & -1 \\ & 1 & -3 & 1 \\\end{bmatrix} = \begin{bmatrix} \\ \\ &&15&3\\&&10&-1\\\end{bmatrix}\] \[i = 3 \hspace{1cm} l_3 = \begin{bmatrix}\\\\1\\2/3\\\end{bmatrix} \hspace{1cm} u_3 = \begin{bmatrix}&& 15 & 3 \\ \end{bmatrix}\] \[A_4 = \begin{bmatrix} \\ \\ &&15&3\\&&10&-1\\\end{bmatrix} - \begin{bmatrix}\\\\&&15 & 3 \\&&10 & 2 \\\end{bmatrix} = \begin{bmatrix}\\\\\\&&&-3\\\end{bmatrix}\] \[i = 4 \hspace{1cm} l_4 = \begin{bmatrix}\\\\\\1\\\end{bmatrix} \hspace{1cm} u_4 = \begin{bmatrix}&&& -3 \\ \end{bmatrix}\] \[L = \begin{bmatrix}1\\1&1\\5&-1&1\\-3 &1 &2/3&1\\\end{bmatrix} \hspace{1cm} U = \begin{bmatrix}2 &1 &1&-2\\&1 &-3&1\\&&15&3\\&&&-3\\\end{bmatrix}\] \[Ly = B \Rightarrow \begin{bmatrix}1\\1&1\\5&-1&1\\-3 &1 &2/3&1\\\end{bmatrix} \begin{bmatrix}y_1\\y_2\\y_3\\y_4\\\end{bmatrix} = \begin{bmatrix}-1\\-3\\36\\18\\\end{bmatrix} \] \[y_1 = -1\] \[y_2 = -3 - (-1) \cdot 1 = -2\] \[y_3 = 36 - 1 \cdot 2 - (-1) \cdot 5 = 39\] \[y_4 = 18 - \frac{2}{3} \cdot 39 - (-2) - (-3) \cdot (-1) = -9\] \[Ux = Y \Rightarrow \begin{bmatrix}2 &1 &1&-2\\&1 &-3&1\\&&15&3\\&&&-3\\\end{bmatrix} \begin{bmatrix}x_1\\x_2\\x_3\\x_4\\\end{bmatrix} = \begin{bmatrix}-1\\-2\\39\\-9\\\end{bmatrix} \] \[x_4 = 3\] \[x_3 = \frac{39 - 3 - 3}{15} = 2\] \[x_2 = \frac{-2 -3 - (-3) \cdot 2}{1} = 1\] \[x_1 = \frac{ -1 - (-2) \cdot 3 - 1 \cdot 2 - 1 \cdot 1}{2} = 1\] \[x = \begin{bmatrix}1\\1\\2\\3\\\end{bmatrix}\] \section*{Question 2} \[i = 1 \hspace{1cm} k = 4 \hspace{1cm} \begin{bmatrix}4&2&3&1\\\end{bmatrix}\] \[ l_1 = \begin{bmatrix} 1/8 \\ 1/4 \\ 1/2 \\ 1 \\ \end{bmatrix} \hspace{1cm} u_1 = \begin{bmatrix} 32 & 24 & 10 & 11\end{bmatrix} \hspace{1cm} \]\[ A_2 = \begin{bmatrix}4 &3 &2& 1\\ 8& 8& 5& 2\\ 16& 12& 10& 5\\ 32& 24& 20 &11 \\\end{bmatrix} - \begin{bmatrix} 4 & 3 & 5/2 & 11/8 \\ 8 & 6 & 5 & 11/4 \\ 16 & 12 & 10 & 11/2 \\ 32 & 24 & 20 & 11 \\ \end{bmatrix} = \begin{bmatrix} 0 & 0 & -1/2 & -3/8 \\ 0 & 2 & 0 & -3/4 \\ 0 & 0 & 0 & -1/2 \\ 0 & 0 & 0 & 0 \\ \end{bmatrix} \] \[i = 2 \hspace{1cm} k = 2 \hspace{1cm} p = \begin{bmatrix}4&2&3&1\\\end{bmatrix}\] \[ l_2 =\begin{bmatrix} 0\\1\\0\\0\\ \end{bmatrix} \hspace{1cm} u_2 =\begin{bmatrix} 0 & 2 & 0 & -3/4 \end{bmatrix}\] \[ A_3 = \begin{bmatrix} 0 & 0 & -1/2 & -3/8 \\ 0 & 2 & 0 & -3/4 \\ 0 & 0 & 0 & -1/2 \\ 0 & 0 & 0 & 0 \\ \end{bmatrix} - \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 2 & 0 & -3/4 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{bmatrix} = \begin{bmatrix} 0 & 0 & -1/2 & -3/8 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1/2 \\ 0 & 0 & 0 & 0 \\ \end{bmatrix} \] \[ i = 3 \hspace{1cm} k = 4 \hspace{1cm} p = \begin{bmatrix} 4&2&1&3\\ \end{bmatrix} \] \[ l_3 = \begin{bmatrix} 1 \\0\\0\\0\\ \end{bmatrix} \hspace{1cm} u_3 = \begin{bmatrix} 0 & 0& -1/2 & -3/8\\\end{bmatrix} \]\[ A_4 = \begin{bmatrix} 0 & 0 & -1/2 & -3/8 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1/2 \\ 0 & 0 & 0 & 0 \\ \end{bmatrix} - \begin{bmatrix} 0 & 0 & -1/2& -3/8\\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{bmatrix} = \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1/2 \\ 0 & 0 & 0 & 0 \\ \end{bmatrix} \] \[ i =4 \hspace{1cm} k = 4 \hspace{1cm} p = \begin{bmatrix} 4&2&1&3\\ \end{bmatrix}\] \[l_4 = \begin{bmatrix} 0 \\0\\1\\0\\\end{bmatrix} u_4 = \begin{bmatrix}0&0&0&-1/2\end{bmatrix} \] \[ P = \begin{bmatrix} 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ \end{bmatrix} \] \[ L = P * \begin{bmatrix} 1/8 & 0 & 1 & 0 \\ 1/4 & 1 & 0 & 0 \\ 1/2 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 1/4 & 1 & 0 & 0 \\ 1/8 & 0 & 1 & 0 \\ 1/2 & 0 & 0 & 1 \\ \end{bmatrix} \] \[ U = \begin{bmatrix} 32 & 24 & 20 & 11 \\ 0 & 2 & 0 & -3/4 \\ 0 & 0 & -1/2 & -3/8 \\ 0 & 0 & 0 & -1/2 \\ \end{bmatrix} \] \section*{Question 4} A = 1 4 8 3 4 20 40 28 8 40 89 71 3 28 71 114 ans = 1 4 8 3 S = 1 4 8 3 4 16 32 12 8 32 64 24 3 12 24 9 A = 0 0 0 0 0 4 8 16 0 8 25 47 0 16 47 105 ans = 0 2 4 8 S = 0 0 0 0 0 4 8 16 0 8 16 32 0 16 32 64 A = 0 0 0 0 0 0 0 0 0 0 9 15 0 0 15 41 ans = 0 0 3 5 S = 0 0 0 0 0 0 0 0 0 0 9 15 0 0 15 25 A = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 16 ans = 0 0 0 4 \end{document}